ADVANCES IN IMAGING AND ELECTRON PHYSICS VOLUME 135
EDITOR-IN-CHIEF
PETER W. HAWKES CEMES-CNRS Toulouse, France
ASSOCIATE EDITOR
BENJAMIN KAZAN Palo Alto, California
HONORARY ASSOCIATE EDITOR
TOM MULVEY
Advances in
Imaging and Electron Physics Edited by
PETER W. HAWKES CEMES-CNRS Toulouse, France
VOLUME 135
Elsevier Academic Press 525 B Street, Suite 1900, San Diego, California 92101-4495, USA 84 Theobald’s Road, London WC1X 8RR, UK
This book is printed on acid-free paper. Copyright ß 2005, Elsevier Inc. All Rights Reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopy, recording, or any information storage and retrieval system, without permission in writing from the Publisher. The appearance of the code at the bottom of the first page of a chapter in this book indicates the Publisher’s consent that copies of the chapter may be made for personal or internal use of specific clients. This consent is given on the condition, however, that the copier pay the stated per copy fee through the Copyright Clearance Center, Inc. (www.copyright.com), for copying beyond that permitted by Sections 107 or 108 of the U.S. Copyright Law. This consent does not extend to other kinds of copying, such as copying for general distribution, for advertising or promotional purposes, for creating new collective works, or for resale. Copy fees for pre-2005 chapters are as shown on the title pages. If no fee code appears on the title page, the copy fee is the same as for current chapters. 1076-5670/2005 $35.00 Permissions may be sought directly from Elsevier’s Science & Technology Rights Department in Oxford, UK: phone: (þ44) 1865 843830, fax: (þ44) 1865 853333, E-mail:
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1
CONTENTS
Contributors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Future Contributions . . . . . . . . . . . . . . . . . . . . . . . .
vii ix xi
Optics, Mechanics, and Hamilton–Jacobi Skeletons Sylvain Bouix and Kaleem Siddiqi I. II. III. IV. V. VI. VII.
Introduction . . . . . . . . . . . . . . . . . . . . . Properties of Skeletons. . . . . . . . . . . . . . . . Skeletonization Techniques . . . . . . . . . . . . . Optics, Mechanics, and Hamilton–Jacobi Skeletons Homotopy-Preserving Medial Sets . . . . . . . . . Examples . . . . . . . . . . . . . . . . . . . . . . . Conclusion . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . .
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1 2 13 15 26 32 36 36
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41 57 68 84 95 96
Dynamic Force Microscopy and Spectroscopy Hendrik HO¨lscher and AndrE´ Schirmeisen I. II. III. IV. V.
Introduction to Dynamic Force Microscopy . . Dynamic Force Microscopy in Air and Liquids Noncontact AFM in Vacuum . . . . . . . . . . Dynamic Force Spectroscopy . . . . . . . . . . Conclusion . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . .
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Generalized Almost-Cyclostationary Signals Luciano Izzo and Antonio Napolitano I. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 II. Higher-Order Characterization . . . . . . . . . . . . . . . . . 114 III. Linear Time-Variant Transformations of GACS Signals . . . . 143 v
vi
CONTENTS
IV. Sampling of GACS Signals . . . . . . . . . . . . . . . . . . . V. Time-Frequency Representations of GACS Signals . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . .
190 201 220
Virtual Optical Experiments Robert Thalhammer I. II. III. IV. V. VI. VII. VIII. IX.
Introduction . . . . . . . . . . . . . . . . . . . . . Modulation of the Refractive Index . . . . . . . . Measurement Techniques . . . . . . . . . . . . . . Modeling Optical Probing Techniques . . . . . . . Virtual Experiments and the Optimization Strategy Free Carrier Absorption Measurements . . . . . . Internal Laser Deflection Measurements . . . . . . Interferometric Techniques . . . . . . . . . . . . . Conclusion and Outlook . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . .
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226 229 236 251 274 276 291 308 324 326
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
335
CONTRIBUTORS
Numbers in parentheses indicate the pages on which the authors' contributions begin.
Sylvain Bouix (1), Department of Psychiatry, Boston VA Healthcare System, Harvard Medical School, and Surgical Planning Laboratory, MRI Division, Brigham and Women’s Hospital, Harvard Medical School, Boston, Massachusetts 02115 Hendrik HO¨lscher (41), Center for NanoTechnology (CeNTech) and Institute of Physics, University of Mu¨nster, 48149 Mu¨nster, Germany Luciano Izzo (103), Dipartimento di Ingegneria Elettronica e delle Telecomunicazioni, Universita` di Napoli Federico II, 80125 Napoli, Italy Antonio Napolitano (103), Dipartimento di Ingegneria Elettronica e delle Telecomunicazioni, Universita` di Napoli Federico II, 80125 Napoli, Italy AndrE´ Schirmeisen (41), Center for NanoTechnology (CeNTech) and Institute of Physics, University of Mu¨nster, 48149 Mu¨nster, Germany Kaleem Siddiqi (1), Center for Intelligent Machines & School of Computer Science, McGill University, Montreal, Quebec H3A 2A7, Canada Robert Thalhammer (225), Infineon Technologies, 81730 Munich, Germany
vii
PREFACE This new volume of Advances in Imaging and Electron Physics contains four chapters from the areas of imaging science, atomic force microscopy, signal theory, and semiconductor laser physics. It opens with a study by S. Bouix and K. Siddiqi on the representation of two-dimensional and threedimensional forms by means of skeletons and of the algorithms used to compute them. This leads the authors to a description of new, more eYcient algorithms, which they illustrate with several examples. A fascinating aspect of this work is the role played by classical mechanics and geometrical optics: readers with these backgrounds will find themselves immediately at home. All types of microscopy are welcome in the pages of these Advances and the second contribution, by H. Ho¨lscher and A. Schirmeisen, presents dynamic force microscopy and spectroscopy at length, with discussion of applications in air and in liquid (tapping-mode AFM) and a separate section on noncontact AFM in vacuum. The authors then turn to spectroscopy, which is likewise explored in careful detail. This clear presentation, which enables the reader to understand the instrument and related techniques and familiarizes him with current preoccupations, will, I am sure, be very welcome. The class of almost cyclostationary signals occupies an important place in signal theory and much recent progress in the understanding and manipulation of such signals has been achieved by the contributors of the third review, L. Izzo and A. Napolitano. This is in fact a short monograph on the subject, from which the reader can comprehend the nature of the topic and the various ways in which it can be analyzed. They cover characterization, linear time-variant transformations, sampling, and time–frequency representation of generalized almost cyclostationary signals. The final chapter, by R. Thalhammer, is in a diVerent area again. ‘Virtual optical experiments’ presents the model introduced by the author and G. Wachutka for simulating the internal laser probing techniques that are employed to explore the interior of semiconductors. Although such techniques had proved very useful, interpretation of the results was hampered by eVects caused by the measurement process itself and by properties of the probing beam. The virtual experiments of R. Thalhammer and G. Wachutka form a basis for a full theoretical study of the situation and furnish an estimate of the reliability of any measurements. This long
ix
x
PREFACE
contribution gives a clear account of all this work and will surely be the standard reference for some time to come. I am most grateful to all the contributors for the eVorts they have made to make their work accessible to a wide range of readers and conclude with a list of material planned for future volumes. Peter Hawkes
FUTURE CONTRIBUTIONS
G. Abbate New developments in liquid-crystal-based photonic devices S. Ando Gradient operators and edge and corner detection A. Asif Applications of noncausal Gauss-Markov random processes in multidimensional image processing C. Beeli Structure and microscopy of quasicrystals M. Bianchini, F. Scarselli, and L. Sarti Recursive neural networks and object detection in images G. Borgefors Distance transforms A. Bottino Retrieval of shape from silhouette A. Buchau Boundary element or integral equation methods for static and time-dependent problems B. Buchberger Gro¨bner bases J. Caulfield Optics and information sciences C. Cervellera and M. Muselli The discrepancy-based approach to neural network learning T. Cremer Neutron microscopy H. Delingette Surface reconstruction based on simplex meshes A. R. Faruqi Direct detection devices for electron microscopy xi
xii
FUTURE CONTRIBUTIONS
R. G. Forbes Liquid metal ion sources J. Y.-l. Forrest Grey systems and grey information E. Fo¨ rster and F. N. Chukhovsky X-ray optics A. Fox The critical-voltage eVect P. Geuens and D. van Dyck (vol. 136) The S-state model for electron channelling in high-resolution electron microscopy G. Gilboa, N. Sochen, and Y. Y. Zeevi (vol. 136) Real and complex PDE-based schemes for image sharpening and enhancement L. Godo and V. Torra Aggregation operators A. Go¨ lzha¨ user Recent advances in electron holography with point sources H. Harmuth and B. MeVert (vol. 137) Dogma of the continuum and the calculus of finite diVerences in quantum physics K. Hayashi X-ray holography M. I. Herrera The development of electron microscopy in Spain D. Hitz Recent progress on HF ECR ion sources D. P. Huijsmans and N. Sebe Ranking metrics and evaluation measures K. Ishizuka Contrast transfer and crystal images K. Jensen Field-emission source mechanisms
FUTURE CONTRIBUTIONS
L. Kipp Photon sieves G. Ko¨ gel Positron microscopy T. Kohashi Spin-polarized scanning electron microscopy W. Krakow Sideband imaging R. Leitgeb Fourier domain and time domain optical coherence tomography B. Lencova´ Modern developments in electron optical calculations R. Lenz (vol. 138) Aspects of colour image processing W. Lodwick Interval analysis and fuzzy possibility theory R. Lukac Weighted directional filters and colour imaging L. Macaire, N. Vandenbroucke, and J.-G. Postaire Color spaces and segmentation M. Matsuya Calculation of aberration coeYcients using Lie algebra S. McVitie Microscopy of magnetic specimens L. Mugnier, A. Blanc, and J. Idier Phase diversity K. Nagayama (vol. 138) Electron phase microscopy S. A. Nepijko, N. N. Sedov, and G. Scho¨ nhense (vol. 136) Measurement of electric fields on the object surface in emission electron microscopy M. A. O’Keefe Electron image simulation
xiii
xiv
FUTURE CONTRIBUTIONS
J. OrloV and X. Liu (vol. 138) Optics of a gas field-ionization source D. Oulton and H. Owens Colorimetric imaging N. Papamarkos and A. Kesidis The inverse Hough transform K. S. Pedersen, A. Lee, and M. Nielsen The scale-space properties of natural images E. Rau Energy analysers for electron microscopes H. Rauch The wave-particle dualism E. Recami Superluminal solutions to wave equations ˇ eha´ cˇ ek, Z. Hradil, J. Perˇina, S. Pascazio, P. Facchi, and M. Zawisky J. R Neutron imaging and sensing of physical fields G. Ritter Lattice-based artifical neural networks J.-F. Rivest Complex morphology G. Schmahl X-ray microscopy G. Scho¨ nhense, C. M. Schneider, and S. A. Nepijko Time-resolved photoemission electron microscopy F. Shih General sweep mathematical morphology R. Shimizu, T. Ikuta, and Y. Takai Defocus image modulation processing in real time S. Shirai CRT gun design methods N. Silvis-Cividjian and C. W. Hagen Electron-beam-induced deposition
FUTURE CONTRIBUTIONS
T. Soma Focus-deflection systems and their applications Q. F. Sugon Geometrical optics in terms of CliVord algebra W. Szmaja Recent developments in the imaging of magnetic domains I. Talmon Study of complex fluids by transmission electron microscopy I. J. Taneja (vol. 138) Divergence measures and their applications M. E. Testorf and M. Fiddy Imaging from scattered electromagnetic fields, investigations into an unsolved problem M. Tonouchi Terahertz radiation imaging N. M. Towghi Ip norm optimal filters Y. Uchikawa Electron gun optics K. Vaeth and G. Rajeswaran Organic light-emitting arrays J. Valde´ s (vol. 138) Units and measures, the future of the SI D. Walsh (vol. 138) The importance-sampling Hough transform G. G. Walter Recent studies on prolate spheroidal wave functions C. D. Wright and E. W. Hill Magnetic force microscopy B. Yazici Stochastic deconvolution over groups M. Yeadon Instrumentation for surface studies
xv
ADVANCES IN IMAGING AND ELECTRON PHYSICS, VOL. 135
Optics, Mechanics, and Hamilton–Jacobi Skeletons SYLVAIN BOUIX AND KALEEM SIDDIQI†
Department of Psychiatry, Boston VA Healthcare System, Harvard Medical School, and Surgical Planning Laboratory, MRI Division, Brigham and Women’s Hospital, Harvard Medical School, Boston, Massachusetts 02115 † Center for Intelligent Machines & School of Computer Science, McGill University, Montreal, Quebec H3A 2A7, Canada
I. Introduction . . . . . . . . . . . . . . II. Properties of Skeletons . . . . . . . . . . A. Definitions . . . . . . . . . . . . . . B. Global Structure of the Skeleton . . . . . . 1. Homotopy Type . . . . . . . . . . . C. Local Structure of the Skeleton . . . . . . 1. Classification of 3D Skeletal Points . . . . 2. Classification of 2D Skeletal Points . . . . III. Skeletonization Techniques . . . . . . . . . IV. Optics, Mechanics, and Hamilton–Jacobi Skeletons A. Medial Sets and the Eikonal Equation . . . . B. Hamiltonian Derivation of the Eikonal Equation 1. Variational Principles . . . . . . . . . 2. Fermat’s Principle . . . . . . . . . . C. Average Outward Flux . . . . . . . . . V. Homotopy-Preserving Medial Sets . . . . . . A. 2D Simple Points . . . . . . . . . . . B. 3D Simple Points . . . . . . . . . . . C. Average Outward Flux–Ordered Thinning . . D. The Algorithm and Its Complexity . . . . . E. Labeling the Medial Set . . . . . . . . . VI. Examples . . . . . . . . . . . . . . . A. 2D Medial Sets . . . . . . . . . . . . B. 3D Medial Sets . . . . . . . . . . . . VII. Conclusion. . . . . . . . . . . . . . . References . . . . . . . . . . . . . . .
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I. INTRODUCTION Motivated largely by biological considerations, Blum (1967, 1973) introduced the notion of a skeleton for representing two-dimensional (2D) and three-dimensional (3D) forms. His essential idea was to consider a ball as a geometric primitive placed within the volume occupied by the object and to ISSN 1076-5670/05 DOI: 10.1016/S1076-5670(04)35001-9
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Copyright 2005, Elsevier Inc. All rights reserved.
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BOUIX AND SIDDIQI
dilate it until it could no longer be grown without penetrating the object’s surface. The locus of all such maximal inscribed balls along with their radii comprised the skeleton. Constructions related to the skeleton, such as the cut locus, had been considered in the literature prior to Blum’s work. However, Blum’s eVorts revitalized interest in such descriptions and led to subsequent studies on the properties of symmetry sets and central sets by mathematicians (Mather, 1983; Millman, 1980; Yomdin, 1981). Blum’s intuition was driven by the insight that such representations could oVer significant advantages for the analysis and manipulation of 2D and 3D forms. To enumerate a few: (1) As interior representations they could be used to describe both geometric and mechanical operations applicable on the object’s interior, such as bending, widening, and elongation; (2) they could provide positional, orientational, and metric (size) information in any locality of the interior of an object; (3) their branching topology could be used to describe the underlying ‘‘part’’ structure of the object, and (4) they could be used to generate object-relative coordinate systems for object interiors and their neighborhoods. Over the past few decades skeletons have become popular tools for object modeling in a variety of fields, including computer vision, computer-aided design, graphics, and medical image analysis. A large number of algorithms have been developed to compute skeletal representations, typically tailored to specific applications in these domains. Nonetheless, the computation of skeletons in a way that is accurate, stable in the presence of slight perturbations to the object, and numerically eYcient remains a challenge. This article details our eVorts toward the development of such algorithms, which are motivated by a formulation that arises from considerations of a variational principle in classical mechanics and geometric optics. The article is organized as follows. We begin with a survey of properties of skeletons in Section II. We then review the classes of skeletonization algorithms that have been developed in the literature in Section III. We develop a formal connection between the skeletonization process and variational principles in classical mechanics and geometric optics in Section IV. This development leads to new algorithms for computing 2D and 3D skeletons, which are described in Section V. We present several numerical examples in Section VI and conclude with a discussion in Section VII. II. PROPERTIES
OF
SKELETONS
In this section we review several of the known properties of the skeleton, with a focus on the case of objects whose boundaries are fixed. There is related work on the type of transitions that a skeleton can undergo when a
OPTICS, MECHANICS, AND HAMILTON–JACOBI SKELETONS
3
one-parameter family of deformations is applied to the boundary (Giblin and Kimia, 2002, 2003), but this will not be covered here. We begin by reviewing the technical definitions of the skeleton that are most commonly used in the literature. We then discuss the global properties of the skeleton, such as its invariance to certain transformations, its reversibility and finally its homotopy equivalence with the original object. We then detail the work of Giblin and Kimia (2003, 2004) on the local form of the 2D and 3D skeleton. A. Definitions Most of the following definitions arise in the mathematical morphology literature (e.g., see Serra, 1982). We use a number of concepts from topology in this section to better understand the properties of the skeleton. We thus begin by reviewing some important concepts of topology from Armstrong (1997). Definition 2.1 (Open Ball). Let x 2 Rn and let r > 0. The n-dimensional open ball Br of radius r at x, is the collection of points of distance less than r from x in Euclidean n-space. Explicitly, the open ball with center x and radius r is defined by Br ðxÞ ¼ fy : k y x k < rg; where k.k is Euclidean distance. Definition 2.2 (Bounded Set). A set X Rn is bounded if there exists a real number r > 0 such that X is contained in some ball Br of radius r. Definition 2.3 (Interior Point). A point x 2 Rn is an interior point of X
Rn, if there exists an > 0 such that B(x) X. The set of all interior points ˚. of X is called the interior of X and is denoted X Definition 2.4 (Open Set). A set X is open if every point x 2 X is an interior point. Definition 2.5 (Limit Point). A point x 2 Rn is a limit point of X Rn, if for any > 0, B(x) \ X 6¼ ;. Note that x may or may not be an element of X. Definition 2.6 (Closure). The closure X of a set X is the union of X with its limit points. If X ¼ X, then we say that X is a closed set. Definition 2.7 (Compact Set). A set X is said to be compact if it is closed and bounded. We can now formally define the class of objects under study:
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BOUIX AND SIDDIQI
Definition 2.8 (Object). An object X is a non-empty, bounded, open set of Rn. The skeleton can also be defined for larger classes of sets, including unbounded sets, but these are typically not meaningful in the context of most applications in computer vision, image analysis, or computer-aided design. There are several definitions of medial loci in the literature, some of which are equivalent. We focus on the most common two, both of which have been popularized by Blum (1973). The first one, which we term the skeleton, is based on the notion of a maximal inscribed ball (Figure 1). Definition 2.9 (Maximal Inscribed Ball). An open ball B ¼ Br(x) included in an object X is maximal if there is no other open ball B0 ¼ Br0 (x0 ) included in X entirely containing B. Formally, B is a maximal inscribed ball if 8B0 ;
B B0 X ) B0 ¼ B:
Definition 2.10 (Skeleton). The skeleton of an object X, Sk(X ), is the locus of the centers of all maximal inscribed balls of X: SkðX Þ ¼ fx 2 X; 9r 0; Br ðxÞ is a maximal ball of Xg Definition 2.11 (Skeleton Transform). The skeleton transform of an object X, ST(X), is the skeleton Sk(X) together with the radius function, defining for each point in Sk(X) the radius of the maximal inscribed balls. The second definition, which we refer to as the medial set, corresponds to Blum’s idea of a grass fire (Blum, 1973). Assume that the object is an isotropic homogeneous flammable material in a nonflammable surrounding space and that its boundary is set on fire. The fire will propagate inward until
FIGURE 1. The subtle diVerences between skeletons, medial sets, and central sets: x1, x2, x3 2 = Me(X), x1 2 Sk(x) but x2, x3 2 = Sk(X), and finally, x1, x2, x3 2 Sk(X).
OPTICS, MECHANICS, AND HAMILTON–JACOBI SKELETONS
5
two or more flame fronts collide. The location of all such collisions, is the locus of the medial set (see Figure 1). Interpreting the medial set this way leads to the following definition. Definition 2.12 (Medial Set). The medial set (axis in 2D, surface in 3D) of X, Me(X), is the set of points of X simultaneously reached by grass fires initiated from at least two diVerent points of @X. This is equivalent to the set of points of X for which there are at least two closest points of @X, in the sense of Euclidean distance. These definitions lead to very similar objects; in fact, the only diVerence between the medial set and the skeleton is that some limit points of Me are included in Sk but not in Me. At such points, it is possible to have a maximal inscribed ball that has only one contact point with the boundary and thus belongs to the skeleton but not the medial set. In 3D, this maximal ball is called a sphere of curvature. We have the following relationship between skeletons and medial sets (Matheron, 1988; Serra, 1982): MeðXÞ SkðXÞ;
ð1Þ
MeðXÞ ¼ SkðXÞ:
ð2Þ
Here E denotes the topological closure of a set E (i.e., the union of E and its limit points). The set Sk(X), which we call the central set after Yomdin (1981) is also closely related to the skeleton and medial set. It has certain topological properties with respect to homotopy equivalence between the skeleton and the object as we will see in Section II.C.1. Figure 1 illustrates some of the subtle diVerences between central sets, medial sets, and skeletons in 2D. In the remainder of this article, the precise object being referred to will depend on the context—whether or not the inclusion of all limit points is necessary for the analysis. B. Global Structure of the Skeleton We now review some of the global properties of the skeleton, as studied in Serra (1982). Property 2.1 (Invariance under Translation and Rotation). Sk(X) is invariant under translation and rotation. Given a translation or a rotation g SkðgðXÞÞ ¼ gðSkðXÞÞ: This is a crucial property as most applications of the skeleton to object description require invariance to rigid body transformations.
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BOUIX AND SIDDIQI
Property 2.2 (Reversibility). Given Sk(X) and the radius rx associated with the maximal inscribed ball at each point of Sk(X), it is possible to recover the object. X is the union of the balls centered at Sk(X) of radius rx: X¼
[ Bðx; rx Þ
x2SkðxÞ
In other words, the set Sk(X) together with its associated radius function is equivalent to the set X. The skeleton and its radius information is thus a complete representation of the object. Property 2.3 (Thickness).
The interior of Sk(X) is empty.
The skeleton is of dimension strictly less than that of the object. It consists of curves and points in 2D and of surfaces, curves, and points in 3D. Thus it provides a compact representation of the object. All the above properties also hold for the central set and the medial set. Whereas a technical concern might arise when considering the reconstruction of an object from Me, in fact the missing limit points do not play a role since the objects are defined to be open sets of Rn. 1. Homotopy Type It is important that the ‘‘essential structure’’ of the object be preserved by its skeleton. For example in 2D, the skeleton and the object should have the same number of connected components and holes. In 3D, the skeleton and the object should have the same number of connected components, holes, and cavities. This idea of ‘‘topological equivalence’’ can be formalized by the notion of homotopy equivalence. Definition 2.13 (Homeomorphism). A homeomorphism f of topological spaces is a continuous, bijective map such that f 1 is also continuous. We also say that two spaces are homeomorphic if such a map exists. Definition 2.14 (Homotopic Maps). Two continuous maps f0 : X 7! Y and f1 : X 7! Y are homotopic if there is a continuous map F : X [0, 1] 7! Y such that F(x, 0) ¼ f0(x) and F(x, 1) ¼ f1(x). F is called a homotopy between f0 and f1. Definition 2.15 (Homotopy Equivalent). Two sets X and Y are homotopy equivalent (or of the same homotopy type) if there exists two continuous maps f : X 7! Y and g : Y 7! X such that the composition f s g is homotopic to IdY, the identity map of Y, and the composition g s f is homotopic to IdX, the identity map of X.
OPTICS, MECHANICS, AND HAMILTON–JACOBI SKELETONS
7
Homeomorphism ignores the space in which surfaces are embedded; for example, mirror images are homeomorphic, as are M€ obius strips with an even number of half-twists, and M€ obius strips with an odd number of half-twists. However, sets of diVerent dimensions cannot be homeomorphic as the mapping from the higher-dimensional set to the lower-dimensional one cannot be inverted. As an example, a 3D solid sphere and a 2D disk are not homeomorphic. Hence, one can already conclude that the skeleton Sk(X) is not homeomorphic to the object X, as they are of diVerent dimensions. Homotopy, on the other hand, compares sets regardless of dimension. The homotopy type represents the ‘‘essential structure’’ of a set, that is, what is preserved when the set is compressed or dilated but without any cutting or gluing. For example, a solid cube, a solid square, and a point, which are not homeomorphic, are homotopy equivalent. One can easily be convinced that a solid torus and an annulus are homotopy equivalent, and so too are a hollow cube and a hollow sphere. Intuitively, it would seem natural that the skeleton and the object are homotopy equivalent; however, proving this is a rather diYcult task. Matheron (1988) was one of the first to study the topological properties of skeletons. He showed that Sk(X) is connected if X is connected and bounded, but did not prove homotopy equivalence between X and Me, Sk or Sk(X). Wolter (1992) analyzed the property of the central set Sk(X) and found homotopy equivalence if X is bounded by a piecewise C2 boundary in the plane or a C2 boundary in higher dimensions. This result was already known for objects with C1 boundaries (Yomdin, 1981). Recently, Lieutier (2002) claimed to have a proof of homotopy equivalence between Me(X) and any open bounded subset of Rn. Proving homotopy equivalence in Rn turns out to be diYcult. However, it is possible to ensure homotopy equivalence between a digital object defined on a rectangular (2D) or cubic (3D) lattice and its associated digital medial set, as seen in Section V. C. Local Structure of the Skeleton We now consider the spatial arrangement of the skeleton in a local neighborhood of each skeletal point x. We will see that only certain configurations are generically possible. For example, in 2D, every point on the skeleton can be classified as either an interior curve point, an endpoint of a curve, or a branch point connecting three curves (Giblin and Kimia, 2003). Our analysis will focus primarily on the local structure of the 3D central set, Me(X), but we also review the 2D results at the end of this section.
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BOUIX AND SIDDIQI
1. Classification of 3D Skeletal Points The local structure and classification of the 3D central set has recently been reviewed by Giblin and Kimia (2004). This classification is based on singularity theory and more specifically on the notion of contact (Arnold, 1991). In all of the local structure analysis, the object under study is Me(X) and the boundary of the object @X is assumed to be a single, infinitely diVerentiable, closed surface. Let a sphere be in contact at a point x with a boundary element. We are interested in classifying the generic types of contact between a sphere and a surface. Our definition of genericity is based on the notion of degrees of freedom. Definition 2.16 (Degree of Freedom). The number of degrees of freedom in a problem, distribution, and so on, is the number of parameters which may be independently varied. For example a sphere has 4 degrees of freedom, the 3D position (xc, yc, zc) of its center and its radius r. Similarly, a point on a 3D surface S has 2 degrees of freedom given by (u, v), the parametrization of the surface S(u, v). Definition 2.17 (Generic Contact). A contact is generic if the number of conditions to obtain it is less than or equal to the number of degrees of freedom of the contact. A sphere has 4 degrees of freedom; a point on a surface has 2. Thus, we call a generic contact between a sphere and one surface point, any contact that is defined by at most 6 conditions. For example, if a surface and a sphere have one second-order contact (i.e., they touch, share tangent planes, and curvatures at a point), 4 degrees of freedom are used. One condition is to have the sphere pass through the point, two are to have their tangent planes coincide, and the last one is to have the curvatures coincide. A more complex example is a sphere touching and sharing tangent planes (first-order contact) with n surface points. Here the number of degrees of freedom is 4 þ 2n and the number of conditions is 3n. Therefore, generically, a sphere can have at most four first-order contact points with a surface in R3. Any contact with more conditions than degrees of freedom is nongeneric and can be deformed into a generic one by a small perturbation of the boundary element. Following Giblin and Kimia (2004), we now enumerate the diVerent generic types of contact between a sphere and a surface at a point x and the local form of their intersection in the neighborhood of x: A1: The tangent planes of the sphere and the boundary element coincide at the contact point, but the sphere is not a sphere of curvature (1=r 6¼ k1) and (1=r 6¼ k2) (Figure 2a). Generically,
OPTICS, MECHANICS, AND HAMILTON–JACOBI SKELETONS
9
FIGURE 2. Cross sections of examples of the diVerent types of contacts between a sphere of radius 1 (light gray) and a smooth surface patch (dark gray). The curvature k of the corresponding line of curvature, parametrized by t, is given for each figure. See text for details.
FIGURE 3. The generic intersections between a sphere and a smooth surface patch. See text for details.
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BOUIX AND SIDDIQI
A2:
A3:
A4:
D 4:
the intersection between the sphere and the surface patch in the neighborhood of the contact point is either a point (Figure 3a) or a cross (Figure 3b). The sphere is one of the spheres of curvatures (1=r ¼ k1 and 1=r 6¼ k2) or (1=r 6¼ k1 and 1=r ¼ k2), but 1=r is not an extremum of curvature along the corresponding line of curvature (Figure 2b). The generic intersection of the sphere and the surface patch is a cusp (Figure 3d). The sphere is a sphere of curvature at a ridge point. ((1=r ¼ k1 and 1=r 6¼ k2) or (1=r 6¼ k1 and 1=r ¼ k2)) and 1=r is an extremum of curvature along the corresponding line of curvature [i.e., the first derivative of the curvature in this direction is zero (Figure 2c)]. The intersection is a point if the larger curvature is a maximum or the smaller curvature is a minimum (Figure 3a) and is otherwise a cross (Figure 3c). The sphere is a sphere of curvature at a turning point. The curvature has an inflexion point [i.e., its first and second derivatives in the direction of the line of curvature are equal to zero (Figure 2d)]. The intersection is a cusp (Figure 3e). The sphere is a sphere of curvature at an umbilic point (1=r ¼ k1 ¼ k2). The intersection is a line (Figure 3f ).
The 3D central set is the locus of the centers of all maximal inscribed spheres with at least two contact points on the boundary of the object, along
FIGURE 4. A medial manifold is shown in the center of the sphere and the two surface patches to which it corresponds. Each point Q on the medial manifold is associated with two distinct points P1, P2 on the object’s surface to which it is closest in the sense of Euclidean distance.
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11
with its limit points. Here the object X is assumed to be a non-empty, bounded, open set of R3 whose boundary @X is a smooth (infinitely diVerentiable) closed surface. Intuitively, each point Q on the medial manifold is associated with at least two distinct points P1, P2 on the object’s surface to which it is closest in the sense of Euclidean distance (Figure 4). Giblin and Kimia (2004) classify the type of points on the 3D skeleton using the nature of contact between the maximal inscribed spheres within X and its boundary @X. In particular, the number of contact points and their order determine the type of point (surface, curve, rim, point) that the center of the sphere represents on the 3D skeleton. For a center of a sphere to be on Me, the sphere is required to be maximal and at least bitangent. To be on Me, the sphere is only required to be maximal, but a higher-order contact is expected. A maximal inscribed sphere is by definition completely included in the object. Thus, its intersection with the boundary in the neighborhood of each contact point includes only that contact point. Therefore, the types A2, A4, and D4 cannot be 3D skeletal points (Figure 3). Let Akn represent a 3D skeletal point whose maximal sphere has k contact points of the type An. According to Giblin and Kimia (2004), the type of points that can generically be seen on the 3D skeleton are the following: A12: The sphere has A1 contact with two distinct points. The skeleton is locally a smooth piece of surface, or medial manifold, whose tangent plane bisects the chord linking the two surface points. A3: The sphere has A3 contact with the boundary. This is the limiting case of A21 points as they approach the boundary of the medial manifold. The medial surface is locally the border or rim of a medial manifold. We call A3 points rim points. A13: The sphere has A1 contact with three distinct points on the boundary. The skeleton is locally the intersection curve of three medial manifolds. A1A3: The sphere has A1 contact at one point and A3 contact at another distinct point. The skeleton is locally the intersection point between an A3 curve and an A31 curve. 4 A1 : The sphere has A1 contact at four distinct points. The skeleton is locally the intersection point of four A31 curves. These types of 3D skeletal points are illustrated in Figure 5. In order to better understand how these types arise, Figure 6 depicts a rectangular parallelepiped with its associated 3D skeleton, computed using the algorithm developed in Section V. The formal classification of 3D skeletal points and their local geometry leads to the following description: A 3D skeleton is generically organized
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BOUIX AND SIDDIQI
FIGURE 5. The generic local structures of the 3D skeleton (Giblin and Kimia, 2004). A21 points form a smooth medial manifold, A3 points correspond to the rim of a medial manifold, A31 points represent the intersection curve of three medial manifolds, an A41 point is the intersection point of four A31 curves, and an A1A3 point is the intersection point between an A3 curve and an A31 curve.
FIGURE 6. A rectangular parallelepiped is shown on the left and its corresponding labeled skeleton is shown on the right. Points on the skeleton are labeled using the classification of Giblin and Kimia (2004).
OPTICS, MECHANICS, AND HAMILTON–JACOBI SKELETONS
13
into manifolds (A21 ) bounded by one type of curve (A3) on their free end and attached to two other manifolds at another type of curve (A31 ). An A3 curve can only end at A1A3 points where it must meet an A31 curve. An A31 curve can end at an A1A3 point or intersect three other A31 curves at an A41 point. One can imagine a simple representation where each node is a medial manifold (A21 connected component along with its A3 boundary) connected to its neighboring manifolds with an edge (A31 curves and A41 point). The latter description seems more intuitive but does not capture entirely the structure of the medial surface. The formal classification suggests, as Giblin and Kimia (2004) pointed out, a hypergraph medial representation of shape. The nodes are A41 or A1 A3 points, connected by A31 or A3 edges, themselves connected by A21 hyper edges. 2. Classification of 2D Skeletal Points Points on the 2D medial set can be classified in a similar manner as shown by Giblin and Kimia (2003). The classification in 2D relies on the following two types of generic contacts between a circle and the bounding curve of the object: A1: The tangent to the circle and the boundary element coincide at the contact point, but is not a circle of curvature. A3: The circle is a circle of curvature. A 2D skeletal point is generically either the end of a branch (A3), an interior point of a branch (A21 ), or the junction of three branches (A31 ). III. SKELETONIZATION TECHNIQUES A large number of techniques for computing skeletons of discrete objects have been developed in the computer vision and pattern analysis literature. This section presents a brief overview of these methods, which can be broadly organized into four classes. First, the Blum (1967) definition of the medial set in terms of a grass fire has served as a model for skeletonization. Imagine the inside of the object to be a field of grass, and let the background be a nonflammable material. If the boundary of the object is lit, it can be shown that distinct points on it will collide precisely at positions on the medial set. The skeleton can thus be obtained by simulating the evolution of the boundary and detecting such points of collision (Kimmel et al., 1995; Leymarie and Levine, 1992; Xia, 1989). For example, Tek and Kimia (1999) have proposed an approach for calculating symmetry maps, which is based on the combination of a
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BOUIX AND SIDDIQI
wavefront propagation technique with an exact (analytic) distance function. Such methods give good results and preserve many properties of the skeleton in 2D. However, a generalization to 3D proves to be diYcult. Second, methods based on thinning also attempt to realize Blum’s (1967) grass fire formulation, but they do so by peeling away layers from an object while retaining special points (Arcelli and Sanniti di Baja, 1985; Borgefors et al., 1999; Lee and Kashyap, 1994; Manzanera et al., 1999). At each step, a lattice point is removed if its removal does not modify the topology of the object (i.e., it is a simple point) and if it is not an extremity of the current construction of the skeleton (i.e., it is a nonterminal point). The peeling process is repeated until no more points are removable. These methods typically rely only on local computations, since points that are simple and nonterminal can be identified by examining their immediate neighborhood. Whereas the notion of a simple point for objects sampled on a 2D or 3D rectangular lattice is well defined (Malandain et al., 1993), the characterization of a terminal point is often based on heuristics. Hence, thinning methods are quite sensitive to rigid body transformations applied to the original object and can fail to localize medial points accurately (Borgefors et al., 1999). As a consequence, from such skeletons one can typically reconstruct only a coarse approximation to the object (Bertrand, 1995; Lee and Kashyap, 1994; Manzanera et al., 1999). Third, it has been shown that under appropriate smoothness conditions the vertices of the Voronoi graph of a set of boundary points converges to the skeleton as the boundary sampling rate becomes infinitely dense (Schmitt, 1989). This property has been exploited to develop skeletonization algorithms in 2D (Brandt and Algazi, 1992; Ogniewicz, 1993), as well as extensions to 3D (Amenta et al., 2001; Attali and Montanvert, 1997; Goldak et al., 1991; N€ af et al., 1996; Sheehy et al., 1996; Sherbrooke et al., 1996). Voronoi methods oVer a number of advantages: they are computationally eYcient, are accurate, reversible, invariant under rigid body transformations, and can provide homotopy equivalent skeletons. Unfortunately, they are extremely sensitive to even slight perturbations of the boundary and a pruning algorithm is almost always necessary to remove unwanted peripheral branches (or sheets in 3D). Fourth, some methods exploit the fact that the locus of skeletal points coincides with the singularities of a Euclidean distance function to the boundary. These approaches attempt to detect local maxima of the distance function, or the corresponding discontinuities in its derivatives (Arcelli and di Baja, 1992; Gomes and Faugeras, 1999; Leymarie and Levine, 1992). Unfortunately, when working with discrete objects the detection of local maxima by simple thresholding is not suYcient to guarantee the properties of thinness and homotopy equivalence. Potential solutions are developed in
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15
Malandain and Fernandez-Vidal (1998), where points identified on the medial locus are combined using a topological reconstruction process, and in Pudney (1998), where lattice points are removed in order of their distance from the boundary while anchoring end points and centers of maximal inscribed balls. In the following section we show how Blum’s grass fire flow can be related to a variational principle that arises in considerations of geometric optics. Thus, techniques that have been developed in optics and classical mechanics can be applied to the analysis of medial loci. This analysis leads to the development of a new algorithm for the numerical computation of 2D and 3D medial sets, which will be discussed in detail in Section V.
IV. OPTICS, MECHANICS,
AND
HAMILTON–JACOBI SKELETONS
We begin by developing the connection between the grass fire flow and the formation of the skeleton. Let W(t) be the moving boundary (front) of the object (assumed to be a closed curve in 2D or a closed surface in 3D), ^ the unit inward normal to W and t a parameter to denote the family of N evolved fronts. Let W0 be the initial boundary of the object. The motion of the front is given by @W ^ ¼ f N; @t
ð3Þ
where the function f ¼ f(x) is the speed of the front and is related to the refractive index n of the medium. If n is a function solely of position x, the medium is said to be isotropic. If it depends on both position and orientation, the medium is anisotropic. Furthermore, if n does not vary with position, the medium is said to be homogeneous. In the case of Blum’s grass fire flow, f is the scalar constant 1=n and thus the medium is homogeneous and isotropic. Unfortunately, given an arbitrary boundary W0 no direct analytical method exists to detect the shocks of this equation. A. Medial Sets and the Eikonal Equation A numerical approach to simulating flows of the type in Eq. 3 while handling topological changes is to use level set methods developed by Osher and Sethian (1988) and Sethian (1996). Below we focus the discussion on the case of a monotonically advancing front where f ¼ f(x) has fixed sign for all points x in the domain of W. Let f(x) be a graph of the solution, obtained by superimposing all the evolved fronts in time (see Figure 7 for a 2D example).
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FIGURE 7. A geometric view of a monotonically advancing front [Eq. (3)]. f(x, y) is a graph of the ‘‘solution’’ surface, the level sets of which are the evolved curves. The motion ^ represented here is @W @t ¼ N, where W0 is a circle in the x-y plane. Adapted from Sethian (1996).
In other words, f(x) is the time at which the front crosses a point x in the medium. We thus have fðWðtÞÞ ¼ t
ð4Þ
Taking the total derivative of f with respect to time we get d fðWðtÞÞ ¼ 1 dt
ð5Þ
@f @W ¼1 @x @t
ð6Þ
@W ¼ 1: @t
ð7Þ
rf Now substitute for
@W @t
using Eq. (3), ^ ¼1 rf f N 1 k rf k2 ¼ 2 f
ð8Þ
Eq. (8) is a form of the well-known eikonal equation, a key concept of geometrical optics (Luneburg, 1964; Stavroudis, 1972). It is also at the core of a multitude of other scientific problems. A number of algorithms have been recently developed to numerically solve this equation, including Sethian’s fast marching method (Sethian, 1996), which systematically constructs f using only upwind values; Rouy and Tourin’s (1992) viscosity
OPTICS, MECHANICS, AND HAMILTON–JACOBI SKELETONS
17
solutions approach; and Sussman et al.’s (1994) level set method for incompressible two-phase flows. However, none of these methods address the issue of shock detection explicitly, and more work has to be done to track shocks. In the next section, we consider an alternate framework for solving the eikonal equation, which is based on the canonical equations of Hamilton. The technique is widely used in classical mechanics and rests on the use of a Legendre transformation (see Arnold, 1989; Shankar, 1994), which takes a system of d second-order diVerential equations to a mathematically equivalent system of 2d first-order diVerential equations. B. Hamiltonian Derivation of the Eikonal Equation 1. Variational Principles We begin by reviewing Lagrangian and Hamiltonian methods for solving minimization problems (Arnold, 1989; Shankar, 1994). Formally, let Z t1 _ tÞ dt; c¼ Lðq; q; ð9Þ t0
be a functional over the space of curves {(q, t) : q(t) ¼ q, t0 t t1}. _ t) is called the Lagrangian of the problem. We are interested in L(q, q, finding the curve g ¼ {(q, t) : q(t) ¼ q, t0 t t1}, such that c(g) is an extremum. The procedure is well known and leads to the Euler–Lagrange theorem. Theorem 4.1 (Euler–Lagrange Equation). The curve g is an extremal of Rt _ t) dt on the space of curves joining (q0, t0) the functional cðgÞ ¼ t01 L(q, q, and (q1, t1), if and only if the following Euler–Lagrange equations d @L @L ¼0 dt @ q_ @q
ð10Þ
are satisfied along the curve g. This is a system of d second-order equations whose solutions depend on the 2d boundary conditions q(t0) ¼ q0 and q(t1) ¼ q1. This system of d second-order equations can be transformed in a system of 2d first-order equations, which is usually easier to solve. The transformation makes use of the theory of Hamiltonian canonical variables. The key to the method is to exchange the roles of q_ by the canonical variable p ¼ @L @ q_ , commonly referred to as the momentum, and replace the _ t) with the function H (q, p, t), called the Hamiltonian, Lagrangian L(q, q,
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such that the velocities now become the derived quantities q_ ¼
@H @p
This can be done by applying the following Legendre transformation: _ tÞ H ðq; p; tÞ ¼ p q_ Lðq; q;
ð11Þ
_ are written as functions of p’s. It is a simple exercise to where the q’s verify that the above expression for the velocities q_ then holds. This transformation is possible if the Lagrangian L is nondegenerate (i.e., the determinant of its Hessian is non-zero). One can also take partial derivatives of the Hamiltonian with respect to the q’s and verify that @H @L ¼ @q @q Using Eq. (10), equations:
@L @q
can be replaced with p_ to give Hamilton’s canonical p_ ¼
@H ; @q
q_ ¼
@H @p
ð12Þ
Thus, in the Hamiltonian formalism one starts with the initial positions and momenta (q(t0), p(t0)) and integrates Eq. (12) to obtain the phase space (q(t), p(t)) of the system. Using Equations (9) and (11), it is straightforward to see that Z t1 cðgÞ ¼ p q_ H dt ð13Þ t0
These results are summarized in the following theorem: Theorem 4.2 (Hamilton’s Equations). Let p ¼ @L @ q_ and substitute in Eq. (10). The system of d second-order Euler–Lagrange equations p_ @L @q ¼ 0, is equivalent to the system of 2d first-order equations (Hamilton’s equations) p_ ¼
@H ; @q
q_ ¼
@H ; @p
_ t) is the Legendre transform of the Lagrangian where H (q, p, t) ¼ p q_ (q, q, _ viewed as a function of q. A comparison of the Lagrangian and Hamiltonian formalisms is presented in Table 1. In the case that the extremals emanating from the point (q0, t0) do not intersect elsewhere, but instead form a so-called central field of extremals,
19
OPTICS, MECHANICS, AND HAMILTON–JACOBI SKELETONS TABLE 1 A COMPARISON OF THE LAGRANGIAN AND HAMILTONIAN FORMALISMS The Lagrangian formalism _ The state of the system is described by (q, q). The state may be represented by a point moving with a velocity in a d-dimensional configuration space. The d coordinates evolve according to d second-order equations. For a given L several trajectories may pass through a given point in the configuration space.
The Hamiltonian formalism The state of the system is described by (q, p). The state may be represented by a point in a 2d-dimensional phase space. The 2n coordinates and momenta obey 2d first-order equations. For a given H only one trajectory passes through a given point in the phase space.
From Shankar, 1994.
one can define the action function f as the solution functional of our variational problem fðq; tÞ ¼ minðcðgÞÞ Rg _ tÞ dt ¼ g Lðq; q; R ¼ g p q_ H dt
ð14Þ
where g is an extremal curve. It can be shown that p ¼ @f @q and that the action function satisfies the Hamilton–Jacobi equation (Arnold, 1989). @f @f þ H q; ; t ¼ 0: ð15Þ @t @q We now have all the necessary tools to formalize the connection between geometric optics and the monotonically advancing front by Eq. (3). 2. Fermat’s Principle Like most laws of classical physics, the equations of geometric optics can be derived from a variational principle (Stavroudis, 1972). In this context, the variational principle is called Fermat’s principle, which states that a ray always chooses a trajectory that minimizes the optical path length.1 Consider a (possibly inhomogeneous) isotropic medium with n(x) its refractive index. The following calculations can then be carried out in arbitrary dimensions, although for simplicity we focus on the 2D case. The (2D) trajectory g(t) ¼ 1
More precisely, the path must be a local extremum and in rare cases may in fact be a maximum (Luneburg, 1964).
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(x(t), y(t)) connecting two points g(t0) ¼ q0 and g(t1) ¼ q1 in the medium minimizes the following integral Rq cðgÞ ¼ q 1 n ds 0 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R t1 dx2 dy2 þ cðgÞ ¼ t0 nðx; yÞ dt ð16Þ dt dt R t1 dx dy cðgÞ ¼ t0 Lðx; y; ; ; tÞ dt; dt dt where ds is the line element along the ray. In order to proceed with the derivation, we must ensure that the Lagrangian L is nondegenerate. Unfortunately, the determinant of the Hessian of L is equal to zero because the variational problem is independent of the parameter t. Fortunately, the analysis can be done by choosing the projected coordinate y as the new variable of integration. sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dx2 ds ¼ dx2 þ dy2 ¼ 1 þ dy: ð17Þ dy Eq. (16) then becomes
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dx2 dy cðgÞ ¼ y0 nðx; yÞ 1 þ dy Ry ¼ y01 Lðx; x0 Þ dy; R y1
ð18Þ
where x0 ¼ dx dy L is sometimes referred to as the Fermat Lagrangian. We can apply the Euler–Lagrange Eq. (10) to the Fermat Lagrangian [Eq. (18)] d nx0 @n pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ x0 2 ¼ 0: dy 1 þ x0 2 @x
ð19Þ
This second-order partial diVerential equation is called the ray equation and is related to the eikonal equation. In order to develop this connection we turn to the theory of Hamiltonian canonical variables. First, we write the Hamiltonian using Equations (11) and (18)
with
H ðx; px Þ ¼ px x0 Lðx; x0 Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 ¼ p x x n 1 þ x0 2 ; px ¼
@L nx0 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 @x 1 þ x0 2
ð20Þ
ð21Þ
OPTICS, MECHANICS, AND HAMILTON–JACOBI SKELETONS
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We may solve for x0 in terms of px, x0 ¼
@H px ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; 2 @px n p2x
ð22Þ
which leads to the Fermat Hamiltonian pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi H ðx; px Þ ¼ n2 p2x
ð23Þ
Now, let us assume a central field of extremals, and define fðx; yÞ ¼ ming ðcðgÞÞ. Substituting Eq. (23) into Eq. (15) yields sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi @f @f2 ¼ n2 ð24Þ @y @x Squaring both sides, one obtains k rf k2 ¼ n2 ;
ð25Þ 1 nðqÞ.
which is the eikonal equation [see Eq. (8)] with speed fðqÞ ¼ Observe that the speed of the front f(q) is inversely proportional to the refractive index of the medium n(q). We summarize the results in the following theorem. Theorem 4.3 (Geometrical Optics in Isotropic Media). According to Fermat’s principle, the trajectory of a ray in an inhomogeneous isotropic medium minimizes the following integral sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z t1 Z y1 dx2 _ dt ¼ cðgÞ ¼ dy nðqÞ k qk nðx; yÞ 1 þ dy t0 y0 Its corresponding d second-order Euler–Lagrange equations lead to the ray equation d nx0 @n pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ x0 2 ¼ 0: 0 dy 1 þ x 2 @x Transforming the problem into Hamiltonian form, we obtain a system of 2d first-order equations dpx @H ¼ @x dy dx @H ¼ ; dy @px
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where the Hamiltonian is given by H ðx; px Þ ¼ n2 p2x . Assuming a central field of extremals, a graph of the solution surface
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BOUIX AND SIDDIQI
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dx2 dy fðx; yÞ ¼ minðcðgÞÞ ¼ nðx; yÞ 1 þ g dy g Z
exists and is called the action function. Furthermore, @f @f ¼ H x; @y @x ffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 @f ¼ n2 @x which leads to the eikonal equation krfk2 ¼ n2 Hence, there is an explicit connection between medial sets, a monotonically front from an object’s boundary, Fermat’s principle, and the eikonal equation. If one sets n ¼ 1=f ¼ 1, one gets the grass fire flow [Eq. (3)] @W ^ ¼ N; @t whose shocks correspond to one of the Blum (1973) definitions of the medial set. Thus the medial set is the locus of positions where two or more front ^ is not defined). If we turn back to the variational meet (i.e., where N formulation, we obtain Rt _ dt cðgÞ ¼ t01 nðqÞ k qk ð26Þ ¼
R y1 y0
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dx2 dy; 1þ dy
ð27Þ
which is nothing other than the Euclidean distance kq(t1) q(t0)k between q(t1) and q(t0), with the extremal curve g being a straight line. The associated action function is the Euclidean distance function fðqÞ ¼ min kq x k; x2W0
ð28Þ
where W0 ¼ W(t0) is the initial boundary of the object and the inward normals to the evolving front W are given by ^ ¼ rf N These constructs are illustrated for the outline of a panther shape in Figure 8 (top and middle). Technically f as an action function [Eq. (14)] is
OPTICS, MECHANICS, AND HAMILTON–JACOBI SKELETONS
23
FIGURE 8. The Euclidean distance function f to the boundary of a panther shape (top) with brightness proportional to increasing distance, its gradient vector field rf (center), and the associated average outward flux (bottom). Whereas the smooth regime of the vector field gives zero flux (medium gray), strong singularities give large negative values (dark gray) in the interior of the object. Adapted from Dimitrov et al. (2000).
not defined at medial points where the assumption of a central field of extrema is broken. Fortunately, f in its form in Eq. (28) is defined and is continuous over Rn. Thus we can conclude that the medial set corresponds to locations where the Euclidean distance function f is singular and hence rf is multivalued.
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BOUIX AND SIDDIQI
This result is not novel (e.g., Matheron (1988) noticed that f is diVerentiable and that krfk ¼ 1 for all points q not lying on the medial set). However, by interpreting the grass fire flow as an optical phenomenon following Fermat’s principle, we provide a view motivated by considerations in classical mechanics. This view also leads to a very useful algorithmic framework for the computation of the medial set as shown in the following sections. The key to this method is to study the vector field defined by rf for all points in the interior of the object. We shall exploit the fact that the limiting behavior of the average outward flux of this vector field through a shrinking circular neighborhood can be used to distinguish medial points from nonmedial ones. C. Average Outward Flux We approach the discrimination of medial points, which coincide with the shocks of the grass fire flow, from nonmedial ones, by computing the average outward flux of the vector field rf about a point x. Let C ¼ @R be the bounding curve of R parametrized by s. The outward flux of a vector field through @R is defined as the curve integral. Z ^ ds OF ¼ rf N C
^ is the unit outward normal at each point of C, and ds is the arcHere N length element. Using the divergence theorem one can relate the outward flux to the divergence of the vector field rf RR R ^ ð29Þ R divðrfÞ dA C rf N ds The average outward flux is defined as the outward flux through the boundary of a region R, normalized by the length of the boundary R rf ds ð30Þ AOF ¼ C lengthðCÞ It is a standard fact that the outward flux, or equivalently the integral of the divergence of rf, measures the degree to which the flow generated by rf is area preserving (or volume preserving in 3D) for the region over which it is computed. To elaborate, the outward flux (and hence also the average outward flux) is negative if the area enclosed by the region @R is shrinking under the action of the Hamiltonian flow, positive if it is growing, and zero otherwise. This quantity is clearly strongly dependent on the shape of the region R. When considering a region R that contains a medial point, unfortunately the standard form of the divergence theorem does not
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apply since the vector field rf is multivalued. Instead, the limiting behavior of the average outward flux as the region R shrinks to a point can be considered. If R is a convex region shrunk by a constant factor in every direction, Damon (2005) has shown that for nonmedial points, this limit is zero and that for medial points there is a constant cR > 0, depending on the shape of the region R, such that the average outward flux approaches a ^ 0 >, where N ^ 0 is now strictly negative number bounded above by cR < rf N a one-sided normal to the medial set. The outward flux on the other hand tends to zero at every point, medial or not. The proof of these results relies on an alternate form of the divergence theorem that can be applied in regions intersecting the medial set,2 and which is developed in some detail in (Damon, 2005; Dimitrov, 2003; Dimitrov et al., 2003).
Thus, the limiting behavior of the average outward flux for shrinking circular regions provides an eVective way of detecting skeletal points, where the vector field rf is multivalued. Nonmedial points give values that are numerically close to zero and medial points give large negative values. Algorithm 1 describes a discrete implementation of the average outward flux and Figure 8 illustrates its computation on the silhouette of a panther shape, where values close to zero are shown in medium gray. All computations are carried out on a rectangular lattice, although the bounding surface is shown in interpolated form. Strictly speaking, the average outward flux is desired only in the limit as the region shrinks to a point. However, the average outward flux over a very small neighborhood (a circle in 2D or a sphere in 3D) provides a suYcient approximation to the limiting values. A threshold on the average outward flux yields a close approximation to the 2
Such a form is necessary because the divergence of rf is not defined at medial points, where rf is multivalued.
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medial set, as used in Siddiqi et al. (1999). However, in general it is impossible to guarantee that the result obtained by simple thresholding is homotopic to the original shape. A high threshold may yield a connected set but cannot guarantee that it is thin. A low threshold can yield a thin set but it may be disconnected. The solution, as shown in the subsequent section, is to introduce additional constraints to ensure that the resulting medial set is homotopic to the shape. The essential idea is to incorporate a homotopypreserving thinning process, where the removal of points is guided by the average outward flux values. This leads to a robust and eYcient algorithm for computing 2D and 3D medial sets. V. HOMOTOPY-PRESERVING MEDIAL SETS Our goal is to combine the average outward flux computation with a digital thinning process, where points are removed without altering the object’s topology. In digital topology, a point is said to be simple if its removal does not change the topology of the object. In 2D, we shall consider rectangular lattices, where a point is a unit square with eight neighbors, as shown in Figure 9 (left). Hence, a 2D digital point is simple if its removal does not disconnect the object or create a hole. In 3D, we shall consider cubic lattices, where a point is a unit cube with 6 faces, 12 edges, and 8 vertices. Hence, a 3D digital point is simple if its removal does not disconnect the object, create a hole, or create a cavity (Kong and Rosenfeld, 1989). We should note that the 2D version of the algorithm was first developed in Dimitrov et al. (2000), and we review it here for completeness. More recent results clarifying properties of the limiting behavior of the average outward flux for shrinking circular regions are presented in Damon (2005), Dimitrov (2003), and Dimitrov et al. (2003).
FIGURE 9. Left: A 3 3 neighborhood of a candidate point for removal P. Right: An example neighborhood graph for which P is simple. There is no edge between neighbors 6 and 8 (see text).
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A. 2D Simple Points Consider the 3 3 neighborhood of a 2D digital point x contained within an object and select those neighbors that are also contained within the object. Now construct a neighborhood graph by placing edges between all pairs of neighbors (not including x) that are 4-adjacent or 8-adjacent to one another. If any of the 3-tuples {2, 3, 4}, {4, 5, 6}, {6, 7, 8}, or {8, 1, 2}, are nodes of the graph, remove the corresponding diagonal edges {2, 4}, {4, 6}, {6, 8}, or {8, 2}, respectively. This ensures that there are no degenerate cycles in the neighborhood graph (cycles of length 3). Now, observe that if the removal of x disconnects the object, or introduces a hole, the neighborhood graph will not be connected, or will have a cycle, respectively. Conversely, a connected graph that has no cycles is a tree. Hence, we have a criterion to decide whether or not x is simple: Proposition 5.1. A 2D digital point x is simple if and only if its 3 3 neighborhood graph, with cycles of length 3 removed, is a tree. A straightforward way of determining whether or not a connected graph is a tree is to check that its Euler characteristic jVj jEj (the number of vertices minus the number of edges) is identical to 1. This check only has to be performed locally, in the 3 3 neighborhood of a point P. Figure 9 (right) shows an example neighborhood graph for which P is simple and hence can be removed. B. 3D Simple Points In 3D a digital point can have three types of neighbors. Two points are 6-neighbors if they share a face; two points are 18-neighbors if they share a face or an edge; and two points are 26-neighbors if they share a face, an edge, or a vertex. This induces three n-connectivities, where n 2 {6, 18, 26}, as well as three n- neighborhoods for x, Nn(x). An n-neighborhood without its central point is defined as Nn ¼ Nn(x)\{x}. An object A is n-adjacent to an object B, if there exist two points x 2 A and y 2 B such that x is an n-neighbor of y. A n-path from x1 to xk is a sequence of points x1, x2, . . . , xk, such that for all xi, 1 < i k, xi1 is n-adjacent to xi. An object represented by a set of points O is n-connected, if for every pair of points (xi, xj) 2 O O, there is a n-path from xi to xj. Based on these definitions, Malandain et al. (1993) provide a topological classification of a point x in a cubic lattice by computing two numbers: C: the number of 26-connected components 26-adjacent to x in O \ N26
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C : the number of 6-connected components 6-adjacent to x in O \ N18. An important result with respect to our goal of thinning is the following: Theorem 5.1 (Malandain et al. 1993).
x is simple if C(x) ¼ 1 and C (x) ¼ 1.
We can now determine whether or not the removal of a point will alter the topology of a digital object. When preserving homotopy is the only concern, simple points can be removed sequentially until no more simple points are left. The resulting set will be thin and homotopic to the object. However, without a further criterion the relationship to the medial set will be uncertain since the locus of surviving points depends entirely on the order in which the simple points are removed. In the current context, we have derived a natural criterion for ordering the thinning, based on the average outward flux of the gradient vector field of the Euclidean distance function. C. Average Outward Flux–Ordered Thinning Recall from Section IV.C that the average outward flux of the gradient vector field of the Euclidean distance function can be used to distinguish nonmedial points from medial ones. This quantity tends to zero for the former, but approaches a negative number below a constant times < rf, N0 > for the latter, where N0 is the one-sided normal to the medial axis or surface. Hence, the average outward flux provides a natural measure of the ‘‘strength’’ of a medial point for numerical computations. The essential idea is to order the thinning such that the weakest points are removed first and to stop the process when all surviving points are not simple, or have a total average outward flux below some chosen (negative) value, or both. This will accurately localize the medial set and also ensure homotopy with the original object. Unfortunately the result is not guaranteed to be a thin set (i.e., one without an interior). One way of satisfying this last constraint is to define an appropriate notion of an end point. Such a point would correspond to the end point of a curve (in 2D or 3D), or a point on the rim of a surface, in 3D. The thinning process would proceed as before, but the threshold criterion for removal would be applied only to end points. Hence, all surviving points that were not end points would not be simple and the result would be a thin set. In 2D, an end point will be viewed as any point that could be the end of a 4-connected or 8-connected digital curve. It is straightforward to see that such a point may be characterized as follows: Proposition 5.2 (2D end point). A 2D point x could be an end point of a 1-pixel–thick digital curve if, in a 3 3 neighborhood, it has a single neighbor, or it has two neighbors, both of which are 4-adjacent to one another.
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In 3D, the characterization of an end point is more diYcult. An end point is either the end of a 26-connected curve, or a corner or point on the rim of a 26-connected surface. Proposition 5.3 (3D end point). In R3, if there exists a plane that passes through a point x such that the intersection of the plane with the object includes an open curve that ends at x, then x is an end point of a 3D curve, or is on the rim or corner of a 3D surface. This criterion can be discretized easily to 26-connected digital objects by examining 9 digital planes in the 26-neighborhood of x as in (Pudney, 1998). D. The Algorithm and Its Complexity The essential idea behind the flux-ordered thinning process is to remove simple points sequentially, ordered by their average outward flux, until a threshold is reached. Subsequently, simple points are removed if they are not end points. The procedure converges when all remaining points are either not simple or are end points. The thinning process can be made very eYcient
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by observing that a point that does not have at least one background point as an immediate neighbor cannot be removed, since this would create a hole or a cavity. Therefore, the only potentially removable points are on the border of the object. This suggests the implementation of the thinning process using a heap data structure. A full description of the procedure is given in Algorithm 2. The approach is computationally very eYcient. With n the total number of digital points within the original volume and k the number of points within the object, the worst-case complexity can be shown to be O(n) þ O(k log(k)) (Siddiqi et al., 2002).
E. Labeling the Medial Set The classification of digital points on the 2D medial set is quite straightforward. Consider a circular path through the digital neighbors of such a point P (see Figure 9) and let n be the number of times this path intersects the medial set. The three generic possibilities are: (1) n ¼ 1, in which case P is an endpoint. (2) n ¼ 2, in which case P is an interior (curve) point. (3) n ¼ 3, in which case P is a branch point. The labeling of the 3D medial set is more subtle and relies on the classification of Malandain et al. (1993). Specifically, the numbers C and C , described in Section V.B, can be used to classify curve points, surface points, border points, and junction points (Table 2). However, junction points can be misclassified as surface points when certain special configurations of voxels occur, and these cases have to be TABLE 2 THE TOPOLOGICAL CLASSIFICATION OF MALANDAIN ET AL. (1993) C
C
Type
0 Any 1 1 1 2 2 >2 >2
Any 0 1 2 >2 1 >2 1 2
Interior point Isolated point Border (simple) point Curve point Curves junction Surface point Surface-curve(s) junction Surfaces junction Surfaces-curves junction
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FIGURE 10. The 3D medial set of a cylinder is labeled as border points, surface points, curve points, and junction points.
dealt with using a new definition for simple surfaces (Malandain et al., 1993). Let x be a surface point (C ¼ 2 and C ¼ 1). Let Ax and Bx be the two connected components of O \ N18 6-adjacent to x. Two surface points x and y are in an equivalence relation if there exists a 26-path x0, x1, . . . , xi, . . . , xn with x0 ¼ x and xn ¼ y such that for i 2 [0, . . . , n 1], (Axi \ Axiþ1 6¼ ; and Bxi \ Bxiþ1 6¼ ;) or (Axi \ Bxiþ1 6¼ ; and Bxi \ Cxiþ1 6¼ ;). A simple surface is then defined as any equivalence class of this equivalence relation. We use this definition in our framework to find all the misclassified junctions. If the 26-neighborhood of a previously classified surface point x is not a simple surface, then x is a junction point. Figure 10 illustrates the labeling of the 3D medial set of a cylinder as two simple sheets connected by a 3D digital curve through two junction points. The same definition can be used to extract the individual simple surfaces comprising the medial set of a 3D object. The idea is to find an unmarked surface point on a medial surface and use it as a ‘‘source’’ to build its associated simple surface using a depth first search strategy. The next simple surface is built from the next unmarked surface point and so on, until all surface points are marked.
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VI. EXAMPLES We now illustrate these algorithms for computing medial sets with several 2D and 3D examples.
FIGURE 11. The topology-preserving medial axis (top) is extracted from the average outward flux map of Figure 8. The reconstruction as the envelope of the maximal inscribed disks of the medial axis is overlaid in gray on the original shape (bottom). Adapted from Dimitrov et al. (2000).
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A. 2D Medial Sets We assume that the input is a 2D binary array where the foreground and background are identified by distinct values. The implementation then uses an exact (signed) distance function to a piecewise circular arc interpolation of the boundary, which allows for subpixel computations (details are presented in Dimitrov et al. (2000, 2003)). Following this, Algorithms 1 and 2 are used with the same average outward flux threshold for each example. Figure 11 (middle) shows the subpixel medial axis for the panther silhouette. The accuracy of the representation is illustrated in Figure 11 (bottom), where the shape is reconstructed as the envelope of the maximal inscribed discs associated with each medial axis point. Figure 12 depicts subpixel medial axes for a number of other shapes.
FIGURE 12. Subpixel medial sets for a range of 2D shapes, obtained by average outward flux–ordered thinning. We thank Pavel Dimitrov for providing his implementation of the algorithm presented in Dimitrov et al. (2003).
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B. 3D Medial Sets Next we illustrate the algorithm with both synthetic data and graphical models of 3D objects. In both cases we assume that the input is a 3D binary array. We then use the D-Euclidean distance function (Borgefors, 1984), which provides a good approximation to the true distance function and apply Algorithms 1 and 2. Once again, the only free parameter is the choice of the average outward flux threshold below which the removal of end points is blocked. For these examples, the value was selected so that approximately 25% of the points within the volume had a lower average outward flux. Figure 13 depicts the 3D medial sets of a rectangular parallelopiped and a cylinder, as well as the reconstructions of the original objects by superimposing the maximal inscribed spheres at the locus of all 3D medial points. As expected, the medial set of the parallelopiped is comprised of planes bisecting the adjacent faces and the medial set of the cylinder consists of two cone-like structures that intersect and share a central sheet. Figure 14 depicts the medial sets of several graphical models of varying complexity.
FIGURE 13. First column: The original 3D objects. Second column: The corresponding average outward flux–based medial sets. Third column: The objects reconstructed from the medial sets in the second column.
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FIGURE 14. Medial sets for a range of 3D shapes, obtained by average outward flux– ordered thinning. The gray scale map on the skeletons represents the radius function, with values increasing from red to blue. (Voxelized objects courtesy Juan Zhang and Peter Savadjiev)
These graphical models were originally described in the Virtual Reality Modeling Language (VRML) format. Each model was then rescaled to fit in a 128 128 128 cubic lattice and was then voxelized using a level set–based implementation of a surface extraction method on the cloud of 3D (discrete) surface points (Zhao et al., 2001; Savadjiev et al., 2003). The gray scale map on the skeleton of each binary volume shown in Figure 14 represents the value of the radius function at each medial point, as indicated by the associated gray scale bars.
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VII. CONCLUSION In summary, we have shown in this article that considerations from classical mechanics and geometric optics lead to fresh insights into the computation of medial sets. It is our hope that the algorithms detailed here will find wide use in a number of domains. In fact, both the 2D and 3D numerical implementations are relatively straightforward and are being used by our colleagues for a variety of applications in computer vision and image analysis. We are presently preparing a detailed account of some of the recent work on medial representations in the literature, with a broad coverage of the mathematics, the algorithms, and the applications. This account will appear in the form of a book and will include contributions by several authors. We point the interested reader to Siddiqi and Pizer (2005). ACKNOWLEDGMENTS We are grateful to James Damon, Pavel Dimitrov, Carlos Phillips, Allen Tannenbaum, and Steve Zucker for collaborations toward the development of the skeletonization techniques reviewed in this article. We thank Peter Savadjiev and Juan Zhang for their help with the numerical examples. This work was supported by the Canadian Foundation for Innovation, the Natural Sciences and Engineering Research Council of Canada, and FQRNT Quebec.
REFERENCES Amenta, N., Choi, S., and Kolluri, R. (2001). The power crust, unions of balls, and the medial axis transform. Computational Geometry: Theory and Applications 19(2), 127–153. Arcelli, C., and di Baja, G. S. (1992). Ridge points in Euclidean distance maps. Pattern Recognition Letters 13(4), 237–243. Arcelli, C., and Sanniti di Baja, G. (1985). A width-independent fast thinning algorithm. IEEE Transactions on Pattern Analysis and Machine Intelligence 7(4), 463–474. Armstrong, M. A. (1997). Basic Topology. New York: Springer-Verlag. Arnold, V. I. (1989). Mathematical Methods of Classical Mechanics. New York: SpringerVerlag. Arnold, V. I. (1991). The Theory of Singularities and Its Applications. Cambridge, MA: Cambridge University Press. Attali, D., and Montanvert, A. (1997). Computing and simplifying 2D and 3D continuous skeletons. Computer Vision and Image Understanding 67(3), 261–273.
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ADVANCES IN IMAGING AND ELECTRON PHYSICS, VOL. 135
Dynamic Force Microscopy and Spectroscopy ¨ LSCHER AND ANDRE´ SCHIRMEISEN HENDRIK HO Center for NanoTechnology (CeNTech) and Institute of Physics, University of Mu¨nster, 48149 Mu¨nster, Germany
I. Introduction to Dynamic Force Microscopy . . . . . . A. Basic Principle of Atomic Force Microscopy . . . . . B. Tip-Sample Forces in Atomic Force Microscopy. . . . C. The Stability Advantage of the Dynamic Mode . . . . D. The Harmonic Oscillator as a Model for Dynamic AFM . II. Dynamic Force Microscopy in Air and Liquids . . . . . A. Applications in Air and Liquid . . . . . . . . . . B. Analysis of Tapping-Mode AFM . . . . . . . . . C. Q Control . . . . . . . . . . . . . . . . . III. Noncontact AFM in Vacuum . . . . . . . . . . . A. Frequency-Modulation Detection Scheme . . . . . . B. Origin of the Frequency Shift . . . . . . . . . . C. Experimental Applications . . . . . . . . . . . D. Calculation of the Frequency Shift . . . . . . . . E. Simulation of NC-AFM Results . . . . . . . . . IV. Dynamic Force Spectroscopy . . . . . . . . . . . A. Determining Forces from Frequencies . . . . . . . B. Analysis of the Tip-Sample Interaction Forces . . . . C. Measurement of Energy Dissipation . . . . . . . . V. Conclusion . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . .
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DYNAMIC FORCE MICROSCOPY
The direct measurement of the force interaction between two distinct molecules has been the motivation for scientists for many years. The fundamental forces responsible for the solid state of matter can be directly investigated by the force detection between defined single molecules and atoms. Nonetheless, until very recently, these forces could not be quantitatively measured by atomic force microscopy for a single atomic bond. How can, be reliably forces, which may be as small as one billionth of one Newton measured? How can one single pair of atoms be identified as the source of the force interaction?
ISSN 1076-5670/05 DOI: 10.1016/S1076-5670(04)35002-0
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Copyright 2005, Elsevier Inc. All rights reserved.
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These were the thoughts of Binnig et al. (1986) nearly 20 years ago as they built the first atomic force microscope (AFM). Their idea was to use the technical concepts of a scanning tunneling microscope (STM) to measure interatomic forces. This microscopy technique was also invented by Binnig et al. (1982) only 5 years before and was honored by a Nobel prize for its inventors in 1986. Today, atomic force microscopy is a standard tool in nanotechnology, enabling the characterization of sample surfaces down to the atomic scale. This section provides an introduction to the basic principle of atomic force microscopy and explain its two main imaging modes. A. Basic Principle of Atomic Force Microscopy What is now the basic principle of an atomic force microscope that allows the measurement of forces at the atomic scale? The answer to this question is surprisingly simple: It is the same mechanical principle that is used in the kitchen to measure the weight of one pound of flour (i.e., its gravitational force) by a scale. A spring with a defined elasticity is elongated or compressed by an arbitrary force (e.g., your pound of flour). The compression Dz of the spring (with spring constant cz) is a direct measure of the force F exerted, which in the regime of elastic deformation obeys Hooke’s law: F ¼ cz Dz:
ð1Þ
The only diVerence from the kitchen scale is the sensitivity of the measurement. The ‘‘spring’’ is a bendable cantilever with a stiVness of 0.01 N/m to 10 N/m. Since interatomic forces are in the range of some nN, the cantilever will be deflected by 0.01 nm to 100 nm. Consequently, the precise detection of the cantilever bending is the key feature of an atomic force microscope. If a suYciently sharp tip is directly attached to the cantilever, the interacting forces between this tip and the sample through the bending of the cantilever can be measured. In 1986 Binnig, Quate, and Gerber presented exactly this concept for their first AFM (Binnig et al., 1986). They measured the deflection of a cantilever with sub-a˚ ngstro¨ m precision by a scanning tunneling microscope and used an aluminum foil as the spring. The tip was a piece of diamond glued to this homemade cantilever (Figure 1). With this setup they imaged sample surfaces down to the nanometer scale. During the past years the experimental setup has been modified and atomic force microscopy became an everyday tool in nanotechnology. Cantilevers are now produced by standard microfabrication techniques,
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FIGURE 1. The basic concept of the first atomic force microscope (left) built in 1985 by Binnig, Quate, and Gerber. A sharp diamond tip glued to an gold foil scanned the surface, while the bending of the cantilever was detected with an STM. The ultimate goal was to measure the force between the front atom of the tip and a specific sample atom (right). (Reproduced from Binnig et al., 1986.)
FIGURE 2. (a) Scanning electron micrograph of a rectangular silicon cantilever of 105 mm length and 14.5 mm width. (b) Scanning electron micrograph of the end of a V-shaped silicon cantilever. (Images courtesy U. D. Schwarz, Yale University.)
mostly from silicon and silicon nitride as rectangular or V-shaped cantilevers (Figure 2). Typical dimensions are as follows: lengths of 100–300 mm, widths of 10–30 mm, and thicknesses of 0.3–5 mm. Spring constants and resonance frequencies of cantilevers depend on the mode of operation. For contact AFM measurements they are about 0.01 to 1 N/m and 5 to 100 kHz, respectively. In a typical force microscope, cantilever deflections in the range ˚ to a few micrometers are measured. This corresponds to a force from 0.1 A sensitivity ranging from 1013 N to 105 N. The detection methods to measure cantilever bending have also been improved. Most-commercial AFMs use the so-called laser beam deflection scheme shown in Figure 3a. The bending and torsion of cantilevers can be detected by a laser beam reflected at their backside. The reflected laser spot is detected with a sectioned photodiode. The diVerent parts are read out separately. A four-quadrant diode usually is used to detect the normal as
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¨ LSCHER AND SCHIRMEISEN HO
FIGURE 3. (a) Principle of an atomic force microscope working with the laser beam deflection method. Bending and torsion of the cantilever are measured simultaneously by measuring the lateral and vertical deflection of a laser beam while the sample is scanned in the x-y plane. The laser beam deflection is determined using a four-quadrant photo diode: If A, B, C, and D are proportional to the intensity of the incident light of the corresponding quadrant, the signal (A þ B)(C þ D) is a measure for the bending and (A þ C)(B þ D) a measure for the torsion of the cantilever. (b) Bending and torsion of rectangular cantilevers. The torsion of the cantilever (middle) is solely due to lateral forces acting in x direction, whereas both forces acting normal to the surface (Fz) as well as acting in plane in y direction (Fy) caused a bending of the cantilever (bottom).
well as the torsional movements of the cantilever for lateral friction measurements. With the cantilever at equilibrium the spot is adjusted such that the upper and the lower sections show the same intensity. If the cantilever bends up or down, the spot moves, and the diVerence signal between upper and lower section is a measure of the bending. In order to enhance sensitivity, several research groups have adapted an interferometer system to measure the cantilever deflection. A thorough comparison of diVerent detection methods is given by Marti (1999). A feedback system, which controls the vertical z-position of the tip on the sample surface, keeps the deflection of the cantilever (and thus the force between tip and sample) constant. Moving the tip relative to the sample in the x-y-plane of the surface by means of a piezoelectric drive, the actual z-position of the tip is recorded as a function of the lateral x-y position with (ideally) sub-a˚ ngstro¨ m precision. The obtained data represent a map of equal forces, which is analyzed and visualized by computer processing. This is a ‘‘static’’ measurement; hence it is called static mode. Another term is contact mode, because tip and sample are in direct mechanical contact.
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This type of measurement is historically the oldest one. This widespread technique can be used to obtain nanometer resolution images on a wide variety of surfaces. Furthermore, it has the advantage that not only the deflection, but also the torsion of the cantilever, can be measured. As shown by Mate et al. (1987), this lateral force can be directly correlated to the friction between tip and sample. Thus atomic force microscopy can be extended to friction force microscopy (FFM). Figure 4 presents typical applications of an atomic force microscope driven in contact mode. The images represent a measurement of a L-a-dipalmitoylphophatidychline (DPPC) film adsorbed on a mica substrate. The lateral force was simultaneously recorded with the topography and shows a contrast between the DPPC film and the substrate. This eVect can be attributed to the diVerent frictional forces on DPPC and mica. As an alternative to the contact mode the cantilever can be excited to vibrate near its resonant frequency close to the sample surface. Under the influence of tip-sample forces the resonant frequency (and consequently also amplitude and phase) of the cantilever will change and serve as the measurement parameters. This is called the dynamic mode. If the tip is approached towards the surface, the oscillation parameters amplitude and phase are influenced by the tip-surface interaction, and can therefore be used as feedback channels. For example, a certain set-point for the amplitude is given, and the feed-back loop will adjust the tip-sample distance such that the amplitude remains constant. The controller parameter is recorded as a function of the lateral position of the tip with respect to the sample and the scanned image essentially represents the surface topography. The technical realization of dynamic mode AFMs is based on the same key components as a static AFM setup. Again, typically the method of laser deflection sensing is used as shown in Figure 5. The cantilever is mounted on a piezo-element, which allows the cantilever beam to oscillate. The reflected laser beam is analyzed for oscillation amplitude and phase diVerence between excitation and vibration. Depending on the mode of operation, a feedback mechanism will adjust the oscillation parameters and/or tip-sample distance during the scanning. The setup can be operated in air, UHV, and even in fluids. This allows measuring a wide range of surface properties. An image obtained with this experimental setup is shown in Figure 4c and d. To compare the results with the dynamic mode, the sample is again DPPC adsorbed on a mica substrate. While in contact mode the frictional forces are measured simultaneously with the topography, in dynamic mode the phase between excitation and oscillation is acquired as an additional channel. The phase image gives information about the diVerent material properties of DPPC and the mica substrate. The link
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¨ LSCHER AND SCHIRMEISEN HO
FIGURE 4. (a) Atomic force microscopic image obtained in contact mode of a monomolecular DPPC (L-a-dipalmitoyl-phophatidycholine) film adsorbed on mica. The image is color coded (i.e., dark areas represent the mica substrate and light areas the DPPC film). (b) The simultaneously recorded friction image shows lower friction on the film (dark areas) as on the substrate (light areas). (c) A dynamic force microscopic image of the same sample measured at another position with a dierent cantilever reveals a comparable topography of the adsorbed film. (d) The phase contrast is directly related to the topography (i.e., the phase is dierent between substrate and DPPC film). All images are 5 mm 5 mm in size. The sample was prepared by M. Hirtz using the technique described in Gleiche et al. (2000). (AFM images by J.-E. Schmutz and D. Ebeling, University of Mu¨ nster.)
between phase signal and a specific material property, however, is not straightforward as it will be discussed in II. DFMs can be driven in several operational modes. As already mentioned the amplitude or the phase can serve as the feedback parameter to control the cantilever sample distance. Expressions such as noncontact mode, intermittent-contact mode, and tapping mode are found in the literature for these modes. Furthermore, the cantilever can be driven in the so-called
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FIGURE 5. Setup of a DFM operated in the AM or tapping mode. A laser beam is deflected by the back side of the cantilever, and its deflection is detected by a split photo diode. The cantilever vibration is caused by an external frequency generator driving an excitation piezo. A lock-in amplifier is used to compare the cantilever driving with its oscillation. The amplitude signal is held constant by a feedback loop controlling the cantilever sample distance.
self-excitation mode, which can be divided further into the constantexcitation and constant-amplitude mode. All these modes are systematically analyzed and categorized in the next sections of this review article. Nonetheless, we first discuss the most important tip-sample forces relevant in atomic force microscopy. B. Tip-Sample Forces in Atomic Force Microscopy Many sample properties related to tip-sample forces can be detected with an AFM. The obtained contrast depends on the operational mode and the actual tip-sample interactions. Before detailing the operational modes of atomic force microscopy, the most important tip-sample interactions are discussed. Figure 6 shows the schematic shape of the interaction force the tip senses during an approach toward the sample surface. Upon approach of the tip
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¨ LSCHER AND SCHIRMEISEN HO
FIGURE 6. Schematic shape of the tip-sample force during an approach of the tip toward the sample surface. First the tip senses attractive, adhesive forces. After the minimum the force becomes repulsive and tip and sample are elastically deformed.
toward the sample, the negative attractive forces, representing, for example, van der Waals or electrostatic interaction forces, increase until a minimum is reached. This turnaround point is due to the onset of repulsive forces, caused by Pauli repulsion, which start to dominate on further approach. With a further reduction of the tip-sample distance the tip is pushed into the sample surface and both are elastically deformed. In general, the tip-sample interaction is a sum of diVerent forces. They can be roughly divided into attractive and repulsive components. The most important forces are summarized in the following: Van der Waals forces are caused by fluctuating induced electric dipoles in atoms and molecules. The distance dependence of this force for two distinct molecules follows 1/z7. For simplicity solid bodies are often assumed to consist of many independent noninteracting molecules, and the van der Waals forces of these bodies are obtained by simple summation. For example, for a sphere over a flat surface the van der Waals force is given by FvdW ðzÞ ¼
AH R ; 6z2
ð2Þ
where R is the radius of the sphere and AH is the Hamaker constant, which is typically in the range of 0.1 aJ (Israelachvili, 1992). This geometry is often used to approximate the van der Waals forces between tip and sample. Due to the 1/z2 dependency van der Waals forces are long-range forces compared to other forces relevant in atomic force microscopy. Capillary forces are important under ambient conditions. Water molecules condense at the sample surface (and also on the tip) and cause the occurrence of an adsorption layer. Consequently, the AFM tip penetrates through this layer when approaching the sample surface. At the tip-sample contact, a water meniscus is formed, which causes a very strong attractive
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force. For soft samples such forces often lead to unacceptable deformations of the surface. However, this eVect is circumvented by measuring directly in liquids. Alternatively, capillary forces can be avoided by performing the experiments in a glove box with dry gases or in vacuum. Pauli or ionic repulsion forces are the most important forces in conventional contact mode AFM. The Pauli exclusion principle forbids that the charge clouds of two electrons showing the same quantum numbers can have some significant overlap; first, the energy of one of the electrons has to be increased. This yields a repulsive force. Additionally, overlap of the charge clouds of electrons can cause an insuYcient screening of the nuclear charge, leading to ionic repulsion of Coulombic nature. The Pauli and the ionic repulsion are nearly hardwall potentials. Thus, for tip and sample in intimate contact, most of the (repulsive) interaction is carried by the atoms directly at the interface. The Pauli repulsion is of purely quantum mechanical origin, and semi-empirical potentials are mostly used to allow an easy and fast calculation. A well-known model is the Lennard-Jones potential, which combines shortrange repulsive interactions with longrange attractive van der Waals interactions: r 6 r0 12 0 VLJ ðzÞ ¼ E0 2 ; ð3Þ z z where E0 is the bonding energy and r0 the equilibrium distance. In this case the repulsion is described by an inverse power law with n ¼ 12. The term with n ¼ 6 describes the attractive van der Waals potential between two atoms/molecules. Another approach was given by Morse considering the repulsion by an exponential increase with a decay length in the sub-a˚ ngstro¨ m region. If only an approximate accuracy is needed, even hard-wall potentials can be applied. Elastic forces and deformations can occur if the tip exerts a certain finite force on the sample. Since this deformation aVects the eVective contact area the knowledge on the acting elastic forces and the corresponding deformation mechanics of the contact is an important issue in atomic force microscopy. The indentation of a sphere into a flat surface was already analyzed in 1881 by H. Hertz (see, e.g., Johnson, 1985). However, this model does not include adhesion forces, which have to be considered at the nanometer scale. Two extreme cases were analyzed by Johnson et al. (1971) and Derjaguin et al. (1975). The model of Johnson, Kendall, and Roberts (JKR model; 1971) considers only the adhesion forces inside the contact area, whereas the model of Derjaguin, Muller, and Toporov (DMT model) includes only the adhesion outside the contact area by an oVset in the Hertz approach. Various models to analyze the contact mechanics in the
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¨ LSCHER AND SCHIRMEISEN HO
intermediate regime were suggested by other authors (see, e.g., Schwarz, 2003, for a recent overview). Friction forces are counteracting the movement of the tip during the scan process. These forces dissipate the kinetic energy of the moving tip-sample contact into the surface or tip material. This can be due to permanent changes in the surface itself, by scratching or indenting, or by the excitation of lattice vibration (i.e., phonons) in the material. Chemical binding forces arise from the overlap of molecular orbitals due to specific bonding states between the tip and the surface molecules. These forces are extremely short-ranged and can be exploited to achieve atomic resolution imaging of surfaces. Since these forces are also specific to the chemical identity of the molecules, it might be possible someday to identify the chemical character of the surface atoms with AFM scans. Magnetic and electrostatic forces are of long-range character and might be attractive or repulsive. They are usually measured when the tip is not in contact with the surface (i.e., ‘‘noncontact’’ mode). For magnetic forces, magnetic materials have to be used for tip or tip coating. Well-defined electrical potentials between tip and sample are necessary for the measurement of electrostatic forces. More detailed information on intermolecular and surface forces relevant to atomic force microscopy measurements can be found in the monographs of Israelachvili (1992) and Sarid (1994). Figure 7 summarizes the most important forces. In principle every type of force can be measured with an AFM. Nonetheless, the actual sensitivity to a specific force might depend the actual mode of operation. Therefore, we analyze the most important diVerences between static and dynamic mode measurements in the next section.
FIGURE 7. Summary of the forces relevant in atomic force microscopy. (Image courtesy of Udo Schwarz, Yale University.)
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C. The Stability Advantage of the Dynamic Mode To obtain a basic understanding of the diVerences between dynamic and static mode, it is very instructive to analyze the underlying physics by a simplified model shown in Figure 8. Whereas the cantilever is represented by a spring with spring constant cz, the force interaction between tip and surface is modeled by a second spring. The validity of this assumption is limited to oscillation amplitudes considerably smaller than the interaction range of the tip-sample forces. In this case the spring constant of the tipsample spring is given by the force gradient, at the average tip-sample distance. The new eVective spring constant of the cantilever is then given by ceff ðzÞ ¼ cz
@Fts ðzÞ : @z
ð4Þ
From the simple harmonic oscillator (neglecting any damping eVects) we find that the resonance frequency f of the system is shifted by Df from the free resonant frequency f0 due to the force interaction ts ðzÞ cz @F@z ceff ¼ ; f ¼ ð f0 þ Df Þ ¼ 4p2 m 4p2 m
2
2
ð5Þ
FIGURE 8. Simplified model to illustrate the influence of the tip-sample forces on the resonance frequency of the cantilever. The tip is attached to a cantilever with spring constant cz, and the force interaction is modeled by a spring with a stiness equal to the force gradient cts ðzÞ ¼ ð@Fts Þ=ð@zÞ. Note that the force interaction spring is not constant, but depends on the tip-sample distance z.
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where m* represents the eVective mass of the cantilever. In the approximation that Df is much smaller than f0 we can write Df 1 @Fts ðzÞ : ¼ f0 2cz @z
ð6Þ
Therefore, we find that the frequency shift of the cantilever resonance is proportional to the force gradient of the tip-sample interaction. Although the above consideration is based on a very simplified model, it shows qualitatively that in dynamic force microscopy the oscillation frequency depends on the force gradient, while static force microscopy measures the force itself. In principle, the force curve can be calculated from the force gradient and vice versa (neglecting a constant oVset). Therefore, it seems that the two methods are equivalent and the choice depends on whether the beam deflection or the frequency shift can be measured with better precision at the cost of technical eVort. However, one important issue for the operation of the AFM has been neglected so far: the mechanical stability of the measurement. In static AFM the tip is slowly approached toward the surface and the attractive forces between tip and sample have to be counteracted by the restoring force of the cantilever. This fails, however, if the force gradient of the tip-sample forces is larger than the spring constant of the cantilever. Mathematically speaking, an instability occurs if cz <
@Fts ðzÞ : @z
ð7Þ
In this case the attractive forces can no longer be sustained by the cantilever and the tip ‘‘jumps’’ toward the sample surface (Burnham et al., 1989). This eVect strongly influences AFM measurements done in the static mode. This is clearly visible by looking at a typical force-vs-distance curve shown in Figure 9. Here the force is seen acting on the tip recorded during an approach and retraction movement of the cantilever. Upon approach of the cantilever toward the sample, the attractive forces acting on the tip bend the cantilever toward the sample surface. At a specific point close to the sample surface, these forces can be no longer sustained by the cantilever spring and the tip ‘‘jumps’’ toward the sample surface. Now tip and sample are in direct mechanical contact. A further approach toward the sample surface pushes the tip into the sample. Since the spring constant of the cantilever is much softer than the elasticity of the sample, the bending of the cantilever increases almost linearly. If the cantilever is now retracted from the surface the tip stays in contact with the sample, because it is strongly attracted by the sample due to adhesive forces and the force Fadh is necessary to disconnect the tip from
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FIGURE 9. A typical force-vs-distance curve obtained in static mode. The cantilever is approached toward the sample surface. Due to strong attractive forces, it ‘‘jumps’’ (snap-in) toward the sample surface at a specific position. During retraction the tip is strongly attracted by the surface and the snap-out point is considerably behind the snap-in point. This results in a hysteresis between approach and retraction.
the surface. The snap-out point is always at a larger distance from the surface than the snap-in, which results in a hysteresis between approach and retraction of the cantilever. This phenomenon of mechanical instability is often referred to as the jump-to-contact. Due to these jumps large parts of the tip-sample interaction cannot be measured. The forces very close to the sample surface, where the interaction between the foremost tip atom and the atoms in the surface is most prominent cannot be measured. Second, the jump-to-contact often causes the tip to change the very last tip or surface atoms. A smooth, careful approach needed to measure the full force curve does not seem feasible. The goal to measure the chemical interaction forces of two single molecules may become impossible. How can this dilemma be resolved? The simplest solution of the jump-tocontact problem is to choose a suYciently stiV spring, so that the instability described by Eq. (7) does not occur. This seems simple at first glance but is a challenging task in practice. Chemical bonding forces extend over a distance range of about 0.1 nm. Typical binding energies of a couple of eV will lead to adhesion forces of the order of some nN. Force gradients will therefore reach values of some 10–30 N/m. A spring for stable force measurements must be as stiV as some 100 N/m, to ensure that no instability occurs (a safety factor
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¨ LSCHER AND SCHIRMEISEN HO
of 10 seems to be a minimum requirement, since usually one cannot be sure a priori, that only one atom will dominate the interaction). In order to measure the nN interaction force, a static cantilever deflection of less than 0.01 nm has to be detected. This is possible with special AFMs in UHV as shown by Jarvis et al. (1996) and Cross et al. (1998). Nevertheless, this is too challenging for standard beam deflection AFM setups commonly used at room temperature in ambient conditions. On the other hand, a trick can be used to enhance the counteracting force of the cantilever: if we oscillate the cantilever with large amplitude, it is virtually stiVer at the lower turning point of the oscillation close to the sample surface (see Figure 5). The oscillation amplitude can be chosen such that the restoring force of the cantilever is at all points stronger than the adhesion force. Mathematically speaking, the measurement is stable as long as the cantilever spring force Fc ¼ cz A is larger than the attractive tip-sample force Fts (Giessibl, 1997). Still, the equilibrium point of the oscillation is far away from the point of closest contact of the tip and surface atoms. Now, the total force curve can be probed by varying the equilibrium point of the oscillation (i.e., by adjusting the z-piezo). In practical applications amplitudes of 10–100 nm are used with cantilever spring constants of about 10 N/m. This means, that the oscillation amplitude is much larger than the force interaction range. The above simplification for Eq. (6), that the force gradient remains constant within one oscillation cycle, does not hold anymore and measurement stability is gained at the cost of a simple quantitative analysis of the experiments. Nonetheless, dynamic AFM has first been used to obtain ‘‘true’’ atomic resolution images of clean surfaces in vacuum by Giessibl (1995) and Sugawara et al. (1995). D. The Harmonic Oscillator as a Model for Dynamic AFM An oscillating cantilever has three degrees of freedom: the amplitude, the frequency, and the phase diVerence between excitation and oscillation. Let us consider the damped driven harmonic oscillator. The cantilever is mounted on a piezoelectric element that is oscillating with amplitude ad at the driving frequency fd zd ðtÞ ¼ ad cosð2pfd tÞ:
ð8Þ
We assume that the cantilever spring obeys Hooke’s law. Second, we introduce a friction force that is proportional to the speed of the cantilever motion, whereas a denotes the damping coeYcient. With Newton’s first law we find for the oscillating system the following equation of motion for the position of the cantilever tip
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55
m€zðtÞ ¼ a_z ðtÞ cz zðtÞ þ cz zd ðtÞ:
ð9Þ pffiffiffiffiffiffiffiffiffiffiffiffi The eigenfrequency of the cantilever is given f0 ¼ 1=2p cz =m . We further define the dimensionless quality factor Q ¼ 2pf0 m =a, anti-proportional to the damping coeYcient a. The quality factor describes the decay of the oscillation amplitude after switching oV the external excitation (ad 0). After some basic math this results in the following diVerential equation: €zðtÞ þ
2pf0 z_ ðtÞ þ 4p2 f02 zðtÞ ¼ 4p2 f02 ad cosð2pfd tÞ: Q
ð10Þ
The solution is a linear combination of two terms (Albrecht et al., 1991). Starting from rest, and switching on the piezo-excitation at t ¼ 0, the amplitude will increase from zero to the final magnitude and reach a steady state, where amplitude, phase, and frequency of the oscillation stay constant over time. The steady-state solution zðt 0Þ follows the external excitation with amplitude A and phase diVerence f zðt 0Þ ¼ A cosð2pfd t þ fÞ
ð11Þ
The transient solution follows ztrans ðtÞ ¼ Atrans expðpft=QÞ cosð2pfd t þ ftrans Þ:
ð12Þ
We emphasize the important fact that the exponential term causes the amplitude to change exponentially with time constant t ¼ pf =Q. In vacuum conditions, only the internal dissipation due to bending of the cantilever is present, and Q reaches values of 10,000–30,000 at typical resonant frequencies of 100–500 kHz. This results in a relatively long transient regime of t ¼ 10–30 ms, which limits the possible operational time for the dynamic mode. This eVect was analyzed in detail by Albrecht et al. (1991). They demonstrated that changes in the measured amplitude, which reflect a change in the tip-sample force, will have a time lag of some ms, which is very slow considering one wants to scan a 256 256 point image within a few minutes. In air, however, viscous damping due to air friction dominates and Q goes down to 200–400, resulting in a time constant well below the millisecond level. This response time is fast enough to use the amplitude as a measurement parameter. If we evaluate the steady-state solution in the diVerential equation, we find the following well-known solution for amplitude and phase of the oscillation as a function of the excitation frequency ad f02 ffi Að fd Þ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð fd2 f02 Þ2 þ ð f0 fd =QÞ2
ð13Þ
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¨ LSCHER AND SCHIRMEISEN HO
FIGURE 10. Amplitude (a) and phase (b) versus the driving frequency for the damped harmonic oscillator. The solid line represents the resonance curves without any tip-sample interaction. The dashed line represents the shifted curves due to an attractive tipsample interaction. Since the cantilever is oscillated with a fixed driving frequency this results into a change of amplitude and phase. This eVect is indicated by arrows if the cantilever is oscillated at its eigenfrequency.
tanfð fd Þ ¼
fd f0 : Qð fd2 f02 Þ
ð14Þ
Amplitude and phase diagrams are depicted in Figure 10. As can be seen from Eq. (13), the amplitude will reach its maximum near the eigenfrequency f0. The exact maximum, however, is at a frequency diVerent from f0, if Q has a finite value. The damping causes the resonant frequency to shift from f0 to f0 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 : ð15Þ f0 ¼ f0 1 2Q2 This frequency shift is negligible for Q factors of 100 and above, which is the case for most applications in vacuum or air. However, for measurements in liquids, Q can be as small as 1 and f0 diVers significantly from f0. As will be discussed later, it is also possible to modify the Q factor by using a special excitation method. The Q factor controls also the width of the amplitude and the steepness of the phase curve. Both parameters are strongly related to the sensitivity of dynamic AFM. It can be shown that the minimum detectable force gradients are proportional to the inverse Q factor (Albrecht et al., 1991; Martin et al., 1987) @Fts 1 / pffiffiffiffi : ð16Þ @z min Q
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Consequently, high Q factors increase the sensitivity, but on the other hand, the acquisition time is prohibitively long due to the transient time t. In the case that the excitation frequency is equal to the resonant frequency of the undamped cantilever fd ¼ f0 , we find the useful relation A0 ¼ Qad
for
fd ¼ f0
ð17Þ
for the maximal resonance amplitude A0. Since f0 f0 in most practical cases, we find that Eq. (17) holds true for exciting the cantilever at its resonance. Using similar arguments as for the amplitude curve, it can be shown that the phase becomes approximately 90 degrees in the resonance case. We also see that in order to reach vibration amplitudes of some 10 nm, the excitation ˚ for cantilever oscillations in air and amplitude has to be smaller than 1 A smaller than 1 pm for typical cantilevers operated in vacuum. These simple examples show also that very small external vibrations (i.e., noise in the laboratory) might result in much larger oscillations of the cantilever (i.e., much larger noise in the experiment). So far we have not considered an additional force term, describing the interaction between the probing tip and the sample. For very small oscillation amplitudes this can be done with the simple spring model introduced in Section I.C. As shown by Eq. (4) the additional tip-sample force can be modeled as a shift of the eVective spring constant of the cantilever. The dashed lines in Figure 10 show that this causes a shift of the resonance curves, resulting in a change of amplitude and phase. Consequently, these two parameters can be used to detect the tip-sample interactions. However, for typical DFM experiments the oscillation amplitude is as large as 10–100 nm and the cantilever tip will experience the whole range of tip-sample interaction during one single oscillation cycle, rather than one constant ‘‘spring.’’ Consequently, more advanced mathematical approaches must be used to analyze amplitude, phase, and frequency. II. DYNAMIC FORCE MICROSCOPY
IN
AIR
AND
LIQUIDS
The amplitude modulation (AM) or tapping mode is the most widespread operational mode of dynamic force microscopy. It is the commonly used technique to measure the topography of samples in ambient conditions or liquids. Due to the oscillation of the cantilever strong repulsive forces acting on the sample can be prevented in most cases. This makes the technique especially interesting for the imaging of soft biological samples. Furthermore, in contact mode it has been observed that weakly immobilzed structures, as single macromolecules, are pushed away during the scanning
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process. To overcome this disadvantage the tapping mode was developed (Hansma et al., 1994; Putman et al., 1994; Zhong et al., 1993). Still, as discussed in this section, a hysteresis can be also observed in the tapping mode due to unstable oscillation. Sometimes these problems can be avoided by an active control of the Q factor as shown in the last subsection. A. Applications in Air and Liquid In the amplitude modulation or tapping mode the cantilever is excited externally with a fixed frequency close to its resonance. Oscillation amplitude and phase during approach of tip and sample serve as the experimental observation channels. A diagram of a typical tapping-mode AFM setup was already shown in Figure 5. The oscillation amplitude and the phase, detected with the photo diode, are analyzed with a lock-in amplifier. The amplitude is compared to the set-point, and the diVerence or error signal is used to adjust the z-piezo (i.e., the probe-sample distance). The external modulation unit supplies the signal for the excitation piezo, and at the same time the oscillation signal serves as the reference for the lock-in amplifier. During one oscillation cycle with amplitudes of 10–100 nm the tipsample interaction will range over a wide distribution of forces, including attractive as well as repulsive forces. We will therefore measure a convolution of the force distance curve with the oscillation trajectory. This complicates the interpretation of AM-AFM measurements appreciably. Nonetheless, as shown by the following applications the tapping mode is very useful to measure surface topography and other parameters down to the nanometer scale. The tapping mode has been successfully employed (and is mostly used) in ambient conditions and in liquids. In addition to the topographic information, the phase signal can be acquired during the scan, yielding the so-called phase image. It has been shown that the phase signal is closely related to the energy dissipated in the tip-sample contact (Cleveland et al., 1998). Highresolution imaging has been extensively performed in the area of material science. Due to its technical relevance the investigation of polymers has been the focus of many studies (e.g., a recent review about AFM imaging on polymers in Magonov, 2004). In Figure 11 the topography of a diblock copolymer (BC0.26-3A0.53F8H10) at diVerent magnifications is shown (Sivaniah et al., 2001). On the large scan (a) the large-scale structure of the microphase-separated PS cylinders (within a PI matrix) lying parallel to the substrate can be seen. In the high-resolution image (b) a surface substructure of regular domes can be seen, which were found to be related to the cooling process during the polymer preparation.
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FIGURE 11. Tapping mode images of BC0.26-3A0.53F8H10 at (a) low resolution and (b) high resolution. The height scale is 10 nm. (Reproduced from Sivaniah et al., 2001.)
Imaging in liquids opens up the avenue for the investigation of biological samples in their natural environment. For example, Mo¨ ller et al. (1999) have obtained high-resolution images of the topography of hexagonally packed intermediate (HPI) layer of Deinococcus radiodurans with tappingmode AFM. Another impressive example for the imaging of DNA in liquid is shown in Figure 12. Jiao et al. (2001) measured the time evolution of a DNA strand interacting with a molecule as shown by a sequence of images acquired in liquid over a time range of several minutes. B. Analysis of Tapping-Mode AFM During an approach of the oscillating cantilever toward the sample surface the resonant frequency of the cantilever changes due to the tipsample force, as already discussed with the simplified spring model introduced in Section I.D. If the cantilever was excited exactly at its eigenfrequency before the approach, it will be excited oV-resonance as the tipsample forces start to act. This in return changes amplitude and phase (see Eq. [13] and [14]) serving as the measurement signals. Consequently, a diVerent amplitude causes a change of the encountered eVective force. We can see already from this simple gedanken-experiment that the interpretation of amplitude and phase curves is highly complex. The analysis needs advanced analytical and numerical methods to determine the actual amplitude and phase relationship for a specific tip-sample force. The inverse problem, however, is even more complicated. In fact, there is no quantitative theory for AM-AFM available, which allows the experimentalist
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FIGURE 12. Dynamic p53-DNA interactions observed by time-lapse tapping mode AFM imaging in solution. Both p53 protein and DNA were weakly adsorbed to a mica surface by balancing the buer conditions. (a) A p53 protein molecule (arrow) was bound to a DNA fragment. The protein (b) dissociated from and then (c) reassociated with the DNA fragment. (d) A downward movement of the DNA with respect the protein occurred, constituting a ‘‘sliding’’ event whereby the protein changes its position on the DNA. Image size: 620 nm. Color scale (height) range: 4 nm. Time units: minutes, seconds. (Image courtesy of T. Sch€ aVer, University of Mu¨ nster.)
to unambiguously convert the experimental data curves back to a force distance relationship. In order to analyze the tapping mode in more detail, we discuss the features of amplitude and phase versus distance curves. In Figure 13 corresponding experimental curves are presented. The measurements were performed in air with a Si-cantilever approaching a Si-wafer, with a cantilever resonant frequency of 299.95 kHz (Anczykowski, 1999b). Clearly, the amplitude and phase curves exhibit specific features like instabilities and hysteresis. Amplitude and phase change rather abruptly at certain points dapp and dret when the cantilever sample distance is decreased and increased. Besides, the amplitude or phase-distance curves do not resemble the tipsample force in a simple direct manner. Additionally, we find a hysteresis between approach and retraction (i.e., dapp < dret ). In order to gain some qualitative insight into the complex relationship between the tip-sample forces and oscillation parameters, we resort to
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FIGURE 13. Amplitude and phase diagrams with excitation frequency (a) below (b) exactly at and (c) above the resonant frequency for tapping-mode AFM from experiments with a Si-cantilever on a Si-wafer in air. (Adapted from Anczykowki (1999b) with permission from the author.)
numerical simulations. Anczykowski et al. (1996a,b; 1998) analyzed the oscillation trajectory of the cantilever under the influence of a given force model. Van der Waals interactions were considered as the only eVective, attractive forces. Mechanical relaxations of tip and sample surface were treated in the limits of continuum theory with the numerical MYD/BHW approach (Hughes and White, 1979; Muller et al., 1980). The cantilever trajectory was determined by solving the equation of motion including the tip-sample forces. The results of the simulation for the amplitude and phase of the tip oscillation as a function of piezo-position are presented in Figure 14. It must be kept in mind that the z position of the probe is not equivalent with the real tip-sample distance at equilibrium position, since the cantilever might bend statically due to the interaction forces. The behavior of the cantilever can be divided into three diVerent driving regimes of the driving frequency. Therefore, we distinguish three cases, where the beam is oscillated below its resonant frequency f0, exactly at f0 and above f0. As an example, let us start to discuss the discontinuous features in the AFM spectroscopy curves of the first case, where the excitation frequency is smaller than f0. Consider the oscillation amplitude as a function of excitation frequency in Figure 10 in conjunction with a typical force curve as depicted in Figure 9. Upon approach of probe and sample, attractive forces will lower the eVective resonant frequency of the oscillator. Therefore, the excitation
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FIGURE 14. Amplitude and phase diagrams with excitation frequency (a) below (b) exactly at and (c) above the resonant frequency for tapping-mode AFM from the numerical simulations. Additionally, the bottom diagrams show the interaction forces at the point of closest tip-sample distance (i.e., the lower turnaround point of the oscillation). (Adapted from Anczykowki (1999b) with permission from the author.)
frequency will now be closer to the resonant frequency, causing the vibration amplitude to increase. This in return reduces the tip-sample distance, which again gives rise to a stronger attractive force. The system becomes unstable until the point dapp is reached, where repulsive forces stop the self-enhancing instability. This can be clearly observed in Figure 14. Consequently, large parts of the force distance curve cannot be detected due to this instability. In the second case, where the excitation equals the free resonant frequency, only a small discontinuity is observed on reduction of the piezo position. Here a shift of the resonant frequency toward smaller values, induced by the attractive force interaction, will reduce the oscillation amplitude. The distance between tip and sample is therefore reduced as well, and the self-amplifying eVect with the sudden instability does not occur, as long as repulsive forces are not encountered. However, at closer tip-sample distances, repulsive forces will cause the resonant frequency to shift again
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toward higher values, increasing the amplitude with decreasing tip-sample distance. Therefore a self-enhancing instability will also occur in this case, but here at the crossover from purely attractive forces to the regime where repulsive forces occur. Correspondingly, a small kink in the amplitude curve can be observed in Figure 14. An even more clear indication of this eVect is manifested by the sudden change in the phase signal at dapp. In the last case, with fd > f0 , the eVect of amplitude reduction due to the resonance frequency shift is even larger. Again, we find no instability in the amplitude signal during approach in the attractive force regime. But as soon as the repulsive force regime is reached, the instability occurs due to the induced positive frequency shift. Consequently, a large jump in the phase curve from values smaller than 90 degrees toward values larger than 90 degrees is observed. The small change in the amplitude curve is not resolved in the simulated curves in Figure 14; however, it can be clearly seen in the experimental curves in Figure 13. Qualitatively, all prominent features of the simulated curves can also be found in the experimental data sets. Hence, the above model seems to capture the important factors necessary for an appropriate description of the experimental situation. But what is the reason for this unexpected behavior? Turning to the numerical simulations again, with access to all physical parameters, allows understand the underlying processes. The lower part of Figure 14 also shows the interaction force between the tip and the sample at the point of closest approach (i.e., the sample-sided turnaround point of the oscillation). Exactly at the points of the discontinuities, the total interaction force changes from the net-attractive regime to the attractive-repulsive regime also termed the intermittent contact regime. As soon as a minimum distance is reached, the tip starts to also experience repulsive forces, which completely changes the oscillation behavior. In other words, the dynamic system switches between two oscillatory states. Directly related to this fact is the second phenomenon, the hysteresis eVect. There are separate curves for the approach of the probe toward the surface and the retraction. This seems to be somewhat counterintuitive, since the tip is constantly approaching toward and retracting from the surface, and the average values of amplitude and phase should be independent of the direction of the average tip-sample distance movement. A hysteresis between approach and retraction within one oscillation, due to dissipative processes, should directly influence amplitude and phase. However, no dissipation models were included in the simulation. In this case, the hysteresis in Figure 14 is due to the fact that the oscillation jumps into diVerent modes, the systems exhibits bistability (see, e.g., Gleyzes et al., 1991). This eVect is often observed in oscillators under the influence of nonlinear forces (see, e.g., Landau and Lifshift, 1990).
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FIGURE 15. Amplitude versus driving frequency with (solid line) and without (dashed line) tip-sample forces. Due to the interaction forces the amplitude curve is multivalued and has unstable branches where the cantilever jumps into more stable oscillations states.
Figure 15 shows an interpretation of these eVects. The change of the amplitude curve due to tip-sample forces (solid line) is compared with the undistorted case (dashed line). In contrast to the linear, harmonic oscillator the amplitude curve is strongly deformed by the additional tip-sample interaction and has more than one value for each driving frequency. Therefore, some branches of the curve are unstable and the cantilever oscillation jumps into another oscillation state. The arrow, for example, marks such an instability occurring during the approach toward the sample surface. A more detailed discussion about the instabilities in tapping mode can be found in Ku¨ hle et al. (1997, 1998), Wang (1998), Garcı´a et al. (1999), Aime´ et al. (1999), and Lee et al. (2002). In conclusion although a qualitative interpretation of the interaction forces is possible, the AM- or tappingmode AFM is not suitable to gain direct quantitative knowledge of tip-sample force interactions. However, it is a very useful tool for imaging nanometer-sized structures in a wide variety of setups, in air or even in liquid. We find that there exist two distinct modes for the externally excited oscillation: the net-attractive and the intermittent contact mode, describing which kind of forces govern the tip-sample interaction. The phase can be used as an indicator, in which mode the system is running. In particular, if the free resonant frequency of the cantilever is higher than the excitation frequency, the system cannot stay in the net-attractive regime, due to a self-enhancing instability. Since in many applications involving soft and delicate biological samples strong repulsive forces should be avoided, the tappingmode AFM should be operated at frequencies equal or above the
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free resonant frequency (San Paulo & Garcı´a, 2000). Even then statistical changes of tip-sample forces during the scan might induce a sudden jump into the intermittent contact mode, and the above explained hysteresis will tend to keep the system in this mode. It is therefore of high importance to tune the oscillation parameters in such a way, that the AFM stays in the netattractive regime (Kru¨ ger et al., 1997). A concept, which achieves this task, is the Q-control system, which is discussed in detail in the next subsection. C. Q Control We have already discussed the virtues of a high Q value for high-sensitivity measurements: the minimum detectable force gradient was inversely proportional to the square root of Q (see the discussion in Section I.D). In vacuum Q mainly represents the internal dissipation of the cantilever during oscillation, an internal damping factor. Little damping is obtained by using highquality cantilevers, which are etched from defect-free, single-crystal silicon wafers. Under ambient or liquid conditions, the quality factor is dominated by dissipative interactions between the cantilever and the surrounding medium, and Q values can be as low as 100 for air or even 1 in liquid. Still, is it somehow possible to compensate for the damping eVect by exciting the cantilever in a sophisticated way? The shape of the resonance curves can be influenced toward higher or lower Q values by an additional amplitude feedback loop. This so-called Q-control method, where the amplitude feedback is mediated directly by the excitation piezo was invented by Anczykowski et al. (1998). Their approach has the advantage that no additional mechanical setups are necessary. Any conventional AFM can be upgraded with an additional electronics box to control the Q factor. A DFM with Q control has an additional feedback circuit consisting of a time (‘‘phase’’) shifter and a gain amplifier. The signal of the displacement sensor is fed into an amplifier and is subsequently used to excite the piezo driving the cantilever in addition to the external function generator. The time delay between the cantilever displacement and the excitation is adjusted by the time (or phase) shifter. This feedback is comparable to the technique used for measurement in vacuum (see Section III). It can be mathematically described by an additional force FQ ¼ gcz zðt t0 Þ
ð18Þ
in the equation of motion. The basic idea of the feedback loop now is to compensate the damping force of the cantilever FDamp ¼ ð2pf0 m=QÞ_z ðtÞ. Assuming a sinusoidal cantilever oscillation and an appropriate time shift corresponding to 90 degrees
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ðt0 ¼ 0:25=f0 Þ or 270 degrees ðt0 ¼ 0:75=f0 Þ, it can be shown that the eVective Q value can be adjusted to Qeff ¼
1 : 1=Q g
ð19Þ
The -sign depends on the actual phase shift ( ¼ 90 degrees and þ ¼ 270 degrees). This equation shows that the damping of the oscillator can be enhanced or weakened by the choice of the phase and the gain factor, respectively. The feedback loop therefore allows varying of the eVective quality factor QeV of the complete dynamic system. Hence, the method is termed Q control. Figure 16 shows experimental data on the eVect of Q control on the amplitude and phase as a function of the external excitation frequency. While Q ¼ 353 without Q control, the quality factor of the system operated in air can be increased or decreased to 1191 and 119, respectively. Q control oVers the advantage of increasing the parameter space of stable AFM operation in tapping-mode AFM. Consider the resonance curve of Figure 10a. When approaching the tip toward the surface there are two competing mechanisms: bringing the tip closer to the sample results in increasing attractive forces. Conversely, for the case fd > f0 , the resonant frequency of the cantilever is shifted toward smaller values due to the
FIGURE 16. Amplitude versus driving frequency with and without Q control. The natural Q factor of the system is Q ¼ 353. It can be increased to 1191 or decreased to 119. (Experiment: D. Ebeling, University of Mu¨ nster.)
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attractive forces, which causes the amplitude to become smaller preventing a tip-sample contact. This is the desirable regime, where stable operation of the AFM is possible in the net-attractive regime. However, as explained previously, below a certain tip-sample separation dapp, the system switches suddenly into the intermittent contact mode, where surface modifications are likely due to the onset of strong repulsive forces. The steeper the amplitude curve, the larger the regime of stable, net-attractive AFM operation. Figure 10a shows that the steepness of the amplitude curve is governed by the quality factor Q. A high Q therefore facilitates stable operation of the DFM. An example is shown in Figure 17. A surface scan of an ultrathin organic film is acquired in tapping-mode under ambient conditions. First, the inner square is scanned without the Q enhancement, and then a wider surface area was scanned with applied Q control. The highquality factor provides a larger parameter space for operating the AFM in the net-attractive regime, allowing good resolution of the delicate organic surface structure. Without the Q control the surface structures are deformed or even destroyed due to the strong repulsive tip-sample interactions (Chi et al., 2000; Gao et al., 2001; Zou et al., 2002). This also allowed imaging of DNA structures without predominantly depressing the soft material during imaging. It was then
FIGURE 17. Tapping mode image of a Langmuir–Blodgett film (ethyl-2,3-dihydroxyoctadecanoate) on mica. The sample was first scanned in tapping mode (inner square). The subsequent imaging of a larger surface area with Q control demonstrates that the distortion of the polymer can be prevented. (Image courtesy of B. Anczykowski, nanoAnalytics GmbH; and L. Chi, University of Mu¨ nster.)
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possible to observe a DNA diameter close to the theoretical value with the Q control feedback (Pignataro et al., 2002). As previously stated, Q Control can be used to enhance the damping in the oscillating system. This would decrease the sensitivity of the system, but on the other hand the response time of the amplitude change is decreased as well. For tapping mode applications, where highspeed scanning is the goal, Q control reduces the relaxation time (Sulchek et al., 2002). The Q control works also in liquids where this method was used to enhance the resolution (Humphris et al., 2000; Tamayo et al., 2001). Furthermore, it has been suggested to use Q control to increase the decay time of ringdown experiments (Ho¨ lscher, 2002b). In conclusion, we have shown that by applying an additional feedback circuit to the dynamic AFM system it is possible to influence the quality factor Q of the cantilever oscillation, expanding the versatility of the DFM toward high-resolution, high-speed, and low-force scanning. III. NONCONTACT AFM
IN
VACUUM
Obtaining highresolution images with an AFM requires preparation of clean sample surfaces without unwanted adsorbates. Therefore, these experiments are usually performed in ultrahigh vacuum (UHV). As a consequence most dynamic force microscope experiments done in UHV use the so-called frequency modulation (FM) detection scheme introduced by Albrecht et al. (1991). In this mode the cantilever is self-oscillated, in contrast to the AM- or tapping-mode discussed in the previous section. This FM technique enables the imaging of singlepoint defects on clean sample surfaces in vacuum, and its resolution is comparable with the scanning tunneling microscope (STM) without the restriction to conducting surfaces. During the last decade the term noncontact atomic force microscopy (NC-AFM) has been established because it is commonly believed that a repulsive, destructive contact between tip and sample is prevented by this technique. This section presents the basic principles of the experimental setup the origin and calculation of the detected frequency shift, applications, and simulation the experimental results. A. Frequency-Modulation Detection Scheme As mentioned in Section I.D, the external excitation of the cantilevers with high Q factors in vacuum limits the aquisition time (bandwidth) of a DFM. The time constant t for the amplitude to adjust to a diVerent tip-sample
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force scales with 1/Q. In vacuum applications, Q is in the range of 10,000–30,000, which means that t is in the range of some 10 ms. This is too long to scan an image with high resolution. This problem is avoided by the FM-detection scheme introduced by Albrecht et al. (1991). The basic setup of a DFM using this driving mechanism is schematically shown in Figure 18. The movement of the cantilever is measured with a displacement sensor. This signal is then fed into an amplifier possessing an automatic gain control (AGC) and is subsequently used to excite the piezo oscillating the cantilever. The time delay between the excitation signal and cantilever deflection is adjusted by a time (‘‘phase’’) shifter to a value t0 ¼ 1=ð4f0 Þ corresponding to 90 degrees, since this ensures an oscillation near resonance. Two diVerent modes have been established: the constant amplitude mode (Albrecht et al., 1991), where the oscillation amplitude A is kept at a constant value by the AGC, and the constant excitation mode
FIGURE 18. The schematic setup of a dynamic force microscope using the frequency modulation technique. This experimental setup is often used in UHV. A significant feature is the positive feedback of the self-driven cantilever. The detector signal is amplified and phase shifted before it is used to drive the piezo. The measured quantity is the frequency shift due to the tip-sample interaction, which serves as the control signal for the cantilever-sample distance.
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(Ueyama et al., 1998), where the excitation amplitude is kept constant. The following focuses on the original constant amplitude mode. The key feature of the described setup is the positive feedback loop that oscillates the cantilever, always at its resonance frequency f (the cantilever serves as the frequency-determining element). This is in contrast to an external driving of the cantilever by a frequency generator, where the driving frequency fd is not necessarily the resonant frequency of the cantilever. If the cantilever oscillates near the sample surface, the tip-sample interaction alters its resonant frequency, which is then diVerent from the eigenfrequency f0 of the free cantilever. The actual value of the resonant frequency depends on the nearest tip-sample distance and the oscillation amplitude. The measured quantity is the frequency shift Df, which is defined as the diVerence between both frequencies ðDf :¼ f f0 Þ. The detection method received its name from the frequency demodulator (FM detector). The cantilever driving, however, is independent of this part of the setup. Other setups use a phase-locked-loop (PLL) to detect the frequency and to oscillate the cantilever exactly with the frequency measured by the PLL (Loppacher et al., 1998). For imaging the frequency shift Df is used to control the cantilever sample distance. Thus the frequency shift is constant and the acquired data represent planes of constant Df, which can be related to the surface topography in many cases (see Section III.C). The recording of the frequency shift as a function of the tip-sample distance or the oscillation amplitude can be used to determine the tip-sample force with high resolution (see Section IV). B. Origin of the Frequency Shift Before presenting experimental results obtained in vacuum with this set-up, we analyze the origin of the frequency shift. A good insight into the cantilever dynamics is given by looking at the tip potential displayed in Figure 19. If the cantilever is far away from the sample surface, the tip moves in a symmetric parabolic potential (dotted line), and its oscillation is harmonic. In such a case, the tip motion is sinusoidal and the resonance frequency is given by the eigenfrequency f0 of the cantilever. However, if the cantilever approaches the sample surface, the potential—which determines the tip oscillation—is modified to an eVective potential VeV (solid line) given by the sum of the parabolic potential and the tip- sample interaction potential Vts (dashed line). This eVective potential diVers from the original parabolic potential and shows an asymmetric shape. As a result of this modification of the tip potential the oscillation becomes anharmonic, and the resonance frequency of the cantilever depends on the
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FIGURE 19. The frequency shift in dynamic force microscopy is caused by the tip-sample interaction potential (dashed line), which alters the harmonic cantilever potential (dotted line). Therefore, the tip moves in an anharmonic and asymmetric eective potential (solid line). zmin is the minimum position of the eVective potential.
oscillation amplitude A. Since the eVective potential experienced by the tip changes also with the nearest distance D, the frequency shift is a functional of both parameters ð) Df :¼ Df ðD; AÞÞ. Figure 20 shows some experimental frequency shift versus distance curves for diVerent oscillation amplitudes. The experiments were performed with an AFM designed for operation in UHV and at low temperatures (Allers et al., 1998). (The specific features of the experimental setup are described in more detail in Section III.C.) The rectangular-shaped silicon cantilever used for this experiment had a spring constant of 38 N/m and an eigenfrequency of 171 kHz. The sample was cooled down to T ¼ 80 K by liquid nitrogen cooling. Tip and sample were grounded during the recording of the data set. The obtained experimental frequency shift vs. distance curves show a behavior expected from the simple model explained above. All curves show a similar overall shape, but diVer in magnitude in dependence of the oscillation amplitude and the nearest tip-sample distance. During the approach of the cantilever toward the sample surface, the frequency shift decreases and reaches a minimum. With a further reduction of the nearest tip-sample distance, the frequency shift increases again and becomes positive. For smaller oscillation amplitudes, the minimum of the Df (z)-curves is deeper and the slope after the minimum is steeper than for larger amplitudes (i.e., the overall eVect is larger for smaller amplitudes). This can be explained by the simple potential model as well: decreasing the amplitude A for a fixed nearest distance D moves the minimum of the
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FIGURE 20. Experimental frequency shift versus distance curves acquired with a silicon tip ˚ ) in UHV at low temperature. The and a graphite sample for diVerent amplitudes (54–180 A curves are shifted along the x-axes to make them comparable. The zero line and the contact point z0 are defined by the force laws described in Section IV.
eVective potential closer to the sample surface. Therefore, the relative perturbation of the harmonic cantilever potential increases, which increases also the absolute value of the frequency shift. C. Experimental Applications The excitement about the NC-AFM technique in UHV was driven by the first results of Giessibl (1995), who achieved imaging of the true atomic structure of the Si(111)–77-surface with this technique. Moreover, Sugawara et al. (1995) observed the motion of single atomic defects on InP (Indiumarsenid) with true atomic resolution. However, imaging on conducting or semi-conducting surfaces is also possible with the STM, and these first NC-AFM images provided little new information on surface properties. The true potential of NC-AFM lies in the imaging of nonconducting surface with atomic precision, which was first demonstrated by Bammerlin et al. (1996) on NaCl. A long-standing question about the surface reconstruction of the
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FIGURE 21. Imaging of a NiO(001) sample surface with a noncontact AFM. (a) Surface ˚ . (b) A dopant atom step and an atomic defect. The lateral distance between two atoms is 4.17 A ˚ higher as the other atoms. (Images courtesy of is imaged as a light protrusion about 0.1 A W. Allers and S. Langkat, University of Hamburg.)
technological relevant material aluminium oxide could be answered by Barth and Reichling (2001), who imaged the atomic structure of the high temperature phase of a-Al2O3(0001). The high-resolution capabilities of noncontact atomic force microscopy are nicely demonstrated by the images shown in Figure 21. Allers et al. (2001) resolved atomic steps and defects resolved with atomic resolution on nickel oxide. Today such a resolution is routinely obtained by diVerent research groups (for an overview see, Garcı´a and Pere´ z, 2002; Giessibl, 2003; Morita et al., 2002). Using the NC-AFM technique the highest resolution is obtained by using stable microscopes in UHV at low temperatures. To show the technical eVort required for such measurements, we present the experimental setup of an ATF designed and built by Allers et al. (1998) at the University of Hamburg. Figure 22 shows a schematic drawing of the complete system. The AFM is built into a UHV system that comprises three vacuum chambers: one for cantilever and sample preparation, which also serves as a transfer chamber; one for analysis purposes; and a main chamber in the middle, with a UHVcompatible bath cryostat attached underneath. The middle chamber houses the AFM head and a specially designed vertical transfer mechanism, based on a double chain, which allows lowering the microscope into the cryostat. Advantages of this mechanism are a reduced need for room height in comparison to other transfer mechanisms, low thermal conductivity to decrease the heat load on the cryostat, and mechanical decoupling from the rest of the system, which enhances the vibration isolation of the AFM. Further damping of the system was achieved by mounting the complete UHV and LT equipment on a table carried by pneumatic damping legs, which in turn stand on a separate foundation to decouple it from building
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FIGURE 22. Front view of a low-temperature UHV-AFM system. The AFM head has a height of about 100 mm and a diameter of 40 mm while the complete systems together with the electronics fills a complete laboratory. (Image courtesy of W. Allers and A. Schwarz, University of Hamburg.)
vibrations. The cryostat and dewar are separated from the rest of the UHV system by a bellow. In addition, the dewar is surrounded by sand for isolation from acoustic vibrations. The microscope allows in situ tip and sample exchange at room temperature in the main chamber. It is cooled in the bath cryostat by either liquid nitrogen or liquid helium, where it reaches temperatures down to 10 K. At the same time, the sample may be heated separately to any temperature up to 400 K. The bath cryostat has significant advantages for high-resolution imaging: cooling the complete
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microscope eliminates most of the thermal drift problems, keeps the creep eVects of the piezo actuators at a minimum, and reduces the thermally induced vibrations of the cantilever. Finally, using a bath cryostat instead of a flow cryostat avoids most of the noise input caused by the flowing and boiling cryogenic liquid. The cantilever bending is detected with an all-fiber interferometer. Such a design allows that all electronic components remain outside the vacuum and at room temperature. For dynamic mode measurements, the cantilever is oscillated by a thin piezo attached underneath the cantilever stage. A resonance loop with constant amplitude is realized by driving this piezo with the preamp signal, regulated by an automatic gain– controlled amplifier. A tunable signal delay allows selecting the optimum phase shift within the loop. The frequency is converted into a voltage by a quadrature FM detector. Using such a setup it is possible to image even thin xenon films deposited on highly oriented pyrolithic graphite (HOPG). This demonstrates the ability to map fragile, true van der Waals crystals (Allers et al., 1999). Within a xenon crystal, the atoms are, as in all noble gas crystals, entirely bound by van der Waals forces, which is why their electronic configuration is nearly identical to the free atoms. This represents a major diVerence to other materials and no chemical bonds are expected to play a significant role in the AFM image formation. The HOPG substrate is expected to have only negligible influence on the electronic configuration of the adsorbed xenon atoms. For the measurements on the xenon thin film, the freshly cleaved and precooled HOPG substrate was exposed to a 1.3 105 mbar atmosphere of the noble gas for 3 minutes at temperatures well below 60 K. The thickness of the resulting xenon film is not exactly known; force microscopy images acquired on large scales, however, did not show any uncovered areas of the substrate. Figure 23 shows an atomic-scale image of the xenon film. The observed maxima are arranged in a structure with sixfold symmetry, as expected for atoms at the (111) surface of a material crystallizing in the fcc structure. The ˚ 10%, distance between individual maxima was determined to be 4.5 A which is in good agreement with the nearest-neighbor distance in a xenon ˚ ). Therefore, the positions of the maxima can be single crystal (4.35 A identified as the positions of the xenon atoms at the surface. The apparent atomic corrugation was about 25 pm. Therefore, it is straightforward to identify the maxima as the positions of the xenon atoms at the sample surface. Such a direct correlation between the maxima of the image and the surface structure, however, might fail for more complex surfaces as shown in the following by the example of graphite. Using the same microscope Allers et al. (1999) also imaged the graphite substrate, which is a layered material. The carbon atoms are covalently
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¨ LSCHER AND SCHIRMEISEN HO
FIGURE 23. Three-dimensional representation of the NC-AFM image of a xenon thin film deposited on a graphite substrate. The experiment was done at 22 K with a cantilever possessing an eigenfrequency of 160 kHz and a spring constant of 40 N/m. The presented data ˚ . The set was taken at a constant frequency shift of 92 Hz with an oscillation amplitude of 94 A corrugation height is about 25 pm. (Image courtesy of W. Allers, University of Hamburg.)
bonded and arranged in a honeycomb structure within the (0001) plane. Individual layers of (0001) planes stick together only by weak van der Waals forces and show an ABA . . . stacking. For scanning probe methods it is an interesting feature that three distinctive sites exist on the surface. There are carbon atoms with (a-type) and without (b-type) a neighbor in the next graphite layer. The hollow site (H-site) is in the center of the hexagon. Neither contact-mode AFM nor STM images show a hexagonal atomic structure. For both methods, the contrast is well understood and exhibits ˚ . Therefore, the protrusions with a sixfold symmetry and a distance of 2.46 A questions arises how the graphite (0001) surface is imaged by NC-AFM? The answer to this question is the experimental image in Figure 24. It was obtained with the same microscope, but diVerent experimental para˚ , T ¼ 22 K. meters: f0 ¼ 160 kHz, Df ¼ 63 Hz, cz ¼ 35 N/m, A ¼ 88 A The NC-AFM image exhibits a trigonal structure of maxima and minima ˚ . This structure is diVerent from the expected with a distance of 2.46 A hexagonal atomic structure of graphite (0001) shown in the inset in Figure 24. A detailed analysis showed that the images consist of two diVerent types of maxima and minima. The overall contrast is dominated by a main maximum and an absulute minimum. The corrugation amplitude between these is 12 pm. Nonetheless, a closer look at the line section reveals that
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FIGURE 24. An experimental NC-AFM images of graphite (0001) obtained in UHV at low ˚ 10 A ˚ ). The corresponding line section along the [1100]-direction temperature (image size: 10 A (solid line) exhibits two dierent types of minima and maxima (see text). The simulation (dashed line) shows the same features as the experiment and enables the identification of the positions of the carbon atoms. Interestingly, the H-sites are imaged as the higher maxima, whereas the positions of the carbon atoms appear as two diVerent minima.
there is also an intermediate maximum and minimum. The lateral distance ˚, between the main maximum and the intermediate maximum is about 1.42 A which corresponds to the nearest-neighbor distance between the carbon atoms. Thus, with the assumption that protrusions represent atomic lattice sites, the main maxima may be classified as the a-type and the lower maxima as the b-type atom (or vice versa). However, the distance between the lower ˚ . Therefore, all specific minima and the intermediate minima is also 1.42 A features of the experimental image—minima and maxima—would fit to the lattice positions of the carbon atoms. Consequently, it is not clear from the start which features of the image correspond to the actual positions of the carbon atoms. An assignment of the observed features is only possible by a comparison of theory and experiment. This has been done by a simulation method explained in Section III.E. The outcome enables the direct comparison of the NC-AFM image with the atomic structure of the graphite (0001) surface, allowing an identification of the actual positions of the carbon atoms in the NC-AFM images. As indicated in Figure 24 by the line section (dashed line), the H-sites appear as maxima, whereas the a- and b-type atoms are imaged as two diVerent minima. Consequently, the real atomic structure of HOPG appears as the inversion of the topography in noncontact atomic force microscopy. This contrast inversion is strongly related to the very small nearestneighbor distance of the carbon atoms. Due to this characteristic feature of the surface structure of the graphine layers, the attractive force is largest above the H-sites. At these positions the tip apex interacts with six nearest
¨ LSCHER AND SCHIRMEISEN HO
78
carbon atoms. If it is placed directly above a carbon atom, it interacts only with one plus three carbon atoms. This argument is supported by the images of a single-walled carbon nanotube obtained by Ashino et al. (2004). Carbon atoms in the nanotube have the same atomic structure as in the graphene layers in HOPG, but the sheets are rolled up to cylindrical tubes. The overall contrast on the nanotube is the same as on graphite and it can be concluded that the hollow sites appear as maxima, if the tip stays in the attractive regime of the interaction forces. Hembacher et al. (2003) found a diVerent contrast measuring in the repulsive regime of the tip-sample forces using a tuning fork as sensor (Giessibl, 1998). They detected the tunneling current together with the frequency shift and observed a triangular structure in the STM image and hexagonal rings of carbon atoms in the AFM image. D. Calculation of the Frequency Shift As described in Section III.A, it is a specific feature of the FM technique that the cantilever is ‘‘self-driven’’ by a positive feedback loop. Due to this experimental setup, the corresponding equation of motion is diVerent from the case of an externally driven cantilever discussed in Section II. The external driving must be replaced by another term to describe the self-driving mechanism correctly. Therefore, the equation of motion is given by m€zðtÞ þ
2pf0 m z_ ðtÞ þ cz zðtÞ ¼ Fts ½zðtÞ; z_ ðtÞ þ gcz zðt t0 Þ : |fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl} Q
ð20Þ
driving
where z :¼ zðtÞ represents the position of the tip at the time t; cz, m, and Q are the spring constant, the eVective mass, and the quality factor of the cantilever, respectively. Fts ¼ ð@Vts Þ=ð@zÞ is the tip-sample interaction force. The term on the right describes the active feedback of the system by the amplification of the displacement signal by the gain factor g measured at the retarded time t t0 . The frequency shift can be calculated from the above equation of motion with the ansatz zðtÞ ¼ A cosð2pftÞ describing the stationary solutions of Eq. (20). This approach is satisfied by the following assumptions, which are typically fulfilled in NC-AFM experiments: 1. The tip-sample interaction force is a function of the actual tip position z(t) and its velocity z_ ðtÞð) Fts :¼ Fts ½zðtÞ; z_ ðtÞÞ. The exact form of this function, however, depends on many diVerent parameters, such as the material properties of tip and sample, the shape of the tip, the bias voltage, etc.
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79
2. The tip oscillations are nearly sinusoidal, since the nonlinear interaction force is typically much smaller than the retracting cantilever force. This condition has to be fulfilled to avoid jump-to-contacts of the tip toward the sample surface, i.e., jFts j cz A for all z(t) (see Section I.C). 3. The microscope is driven in the often used constant amplitude mode and the amplitude A is held constant through the variation of the gain factor g by the AGC ()A ¼ const). Introducing this ansatz into Eq. (20), the obtained equation is then simplified in the following way. After a multiplication with cos(2pft), the equation is integrated over one period of oscillation t ¼ ½0; 1=f . The same procedure is repeated after a multiplication with sin(2pft).1 The result is a set of two coupled trigonometric equations (Ho¨ lscher et al., 2001b): f2 f2 2f g cosð2pft0 Þ ¼ 0 2 Acz f0 1 f 2f g sinð2pft0 Þ ¼ Q f0 Acz
1=f Z
Fts ½zðtÞ; z_ ðtÞ cosð2pftÞ dt;
ð21Þ
0
1=f Z
Fts ½zðtÞ; z_ ðtÞ sinð2pftÞ dt
ð22Þ
0
These equations can be solved numerically, if one is interested in the exact dependency of the tip-sample interaction force Fts and the time delay t0 on the oscillation frequency f and the gain factor g. Fortunately, a detailed analysis shows that the results of a DFM experiment are mainly determined by the tip-sample force and only very slightly by the time delay, if t0 is set to an optimal value by the experimentalist before an approach of the tip toward the sample surface. These values of the time delay are specific resonance values corresponding to 90 degrees (i.e., t0 ¼ 1=4f0 ) and can be easily found by a minimization of the gain factor as a function of the time delay. Therefore, it can be assumed that cosð2pft0 Þ 0 and sinð2pft0 Þ 1 and the two coupled Eqs. (21) and (22) can be decoupled f2 Df ffi 0 Acz
1=f Z0
Fts ½zðtÞ; z_ ðtÞ cosð2pf0 tÞ dt; 0
1 2f0 jgj ffi þ Q Acz
1
ð23Þ
1=f Z0
Fts ½zðtÞ; z_ ðtÞ sinð2pf0 tÞ dt 0
This treatment of the equation is equivalent to a Fourier transformation.
ð24Þ
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¨ LSCHER AND SCHIRMEISEN HO
Since no assumption was made about the specific force law describing the tip-sample interaction Fts, these equations are valid for every type of interaction as long as the resulting cantilever oscillations are nearly sinusoidal. Assuming reasonable tip-sample forces it can be shown that the frequency shift is determined by the average of the tip-sample force between forward and backward movement, whereas the gain factor is directly related to dissipative processes such as hysteresis or viscous damping. This result is related to the odd and even weighting functions (cos and sin) in both integrals of Eqs. (23) and (24) (Du¨ rig, 2000b). For the further analysis of the NC-AFM mode, handy formulas to calculate the frequency shift for given force laws are very useful. As mentioned in Section I.B, some tip-sample forces can be described by inverse-power laws of the type Fn ðzÞ ¼
Cn : zn
ð25Þ
By substituting this force law into Eq. (23), comparatively simple analytic solutions of the integral can be found for small n f0 C 1 d ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p Df ðn ¼ 1Þ ¼ 1 ð26Þ cz A 2 d 2 A2 Df ðn ¼ 2Þ ¼
f0 C 2 1 cz ðd 2 A2 Þ3=2
ð27Þ
Df ðn ¼ 3Þ ¼
3 f0 C3 d 2 cz ðd 2 A2 Þ5=2
ð28Þ
Df ðn ¼ 4Þ ¼
1 f0 C4 A2 þ 4d 2 2 cz ðd 2 A2 Þ7=2
ð29Þ
Df ðn ¼ 5Þ ¼
5 f0 C5 dð3A2 þ 4d 2 Þ 8 cz ðd 2 A2 Þ9=2
ð30Þ
Df ðn ¼ 6Þ ¼
3 f0 C6 A4 þ 12A2 d 2 þ 8d 4 : 8 cz ðd 2 A2 Þ11=2
ð31Þ
These equations might be especially valuable for the measurement of longrange forces (van der Waals, magnetic or electrostatic), which are quite often described by a force law following Eq. (25). An application is shown in Figure 26 for the long-range van der Waals force.
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81
The integral in Eq. (23) can be further simplified with the assumption that the decay length of the tip-sample interaction is much smaller than the oscillation amplitude ðD AÞ. This large amplitude limit was first analyzed by Giessibl (1997). This condition is fulfilled for typical NC-AFM experiments, where the amplitudes are larger than 5 nm. In this case, the ‘‘right’’ part (zmin z D þ 2A) of the eVective potential is nearly parabolic, and only its ‘‘left’’ part (D z zmin) is deformed (see Figure 19). First, we transform the integral Eq. (23) with u ¼ cosð2pf0 tÞ to f0 Df ¼ pcz A
Z1 1
u Fts ½D þ Að1 þ uÞ pffiffiffiffiffiffiffiffiffiffiffiffiffi du 1 u2
ð32Þ
Because of the assumed large amplitude the weighting termpin integral ffiffiffipthis pffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffi Eq. (32) can be expanded at u ¼ 1 to u= 1 u2 1=ð 2 1 xÞ. Applying now the transformation z ¼ D þ A(1 þ u) and extending the upper limit of the integral to infinity, the frequency shift can be simplified to (Du¨rig, 1999) 1 f0 Df ¼ pffiffiffi 2p cz A3=2
Z1 D
Fts ðzÞ pffiffiffiffiffiffiffiffiffiffiffiffi dz: zD
ð33Þ
It is interesting to note that the integral in this equation is independent of the amplitude. It depends solely on the tip-sample force and the tip-sample distance D. The experimental parameters (cz, f0, and A) appear only as prefactors. Consequently, it is possible to define the normalized frequency shift (Giessibl, 1997) gðzÞ :¼
cz A3=2 Df ðzÞ f0
ð34Þ
This is a very useful quantity to compare experiments obtained with diVerent amplitudes and cantilevers. Additionally, it is easier to link theoretical calculations with experiments (see Section III.E). The validity of Eq. (34) is nicely demonstrated by the application of this equation to the frequency shift curves already presented in Figure 20. As shown in Figure 25 all curves obtained for diVerent amplitudes result into one universal g curve, which depends only on the actual tip-sample distance D. Furthermore, Eq. (33) helps to calculate the frequency shift for inverse force laws. If we introduce the force law [Eq. (25)] into this equation, the frequency shift can be calculated from (Giessibl, 1997) f0 Cn Df pffiffiffi I1 ðnÞ; 2pcz A2=3 Dn1=2
ð35Þ
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82
FIGURE 25. Application of Eq. (34) to the frequency shift curves shown in Figure 20. The ˚. normalized frequency shift g is nearly identical for all amplitudes ranging from 54 to 180 A
TABLE 1 Samples of Often Used Values of the Integral I1(n) n
1
2
3
4
5
6
7
8
13
I1(n)
p
1 2p
3 8p
5 16 p
0.273p
0.246p
0.226p
0.209p
0.161p
where I1(n) is given by Z1 I1 ðnÞ :¼ 1
8 p < 1 1 3 . . . ð2n 3Þ dy ¼ n :p ð1 þ y2 Þ 2n1 ðn 1Þ!
n¼1 n > 1:
ð36Þ
Table 1 contains some values of the integral function I1(n). It can be shown that the accuracy of this approximation increases not only with the oscillation amplitudes A, but also with larger values of n. Figure 26 demonstrates the high precision of formula Eq. (35) (solid lines) by a comparison with exact numerical results (symbols) for the Lennard–Jones force.
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83
FIGURE 26. Comparison between the exact numerical results for the frequency shift (symbols) and some approximation formulas. (a) An application for the long-range van der Waals force Eq. (2) with the parameters R ¼ 12 nm and AH ¼ 0.1 aJ. The deviation between the exact numerical result (symbol) and Eq. (27) (solid line) is negligible. The approximation Eq. (35) is shown by a dashed line and yields good agreement only if the nearest tip-sample distance is small compared to the amplitude. The calculation using the force gradient method Eq. (6) (dashed-dotted lines) is nearly useless. (b) The formula for the large amplitude limit Eq. (35) works best for short-range force as shown for three dierent oscillation amplitudes using the shortrange Lennard–Jones force Eq. (3) and the approximation formula Eq. (35) (solid lines). The exact results (symbols) are well reproduced.
E. Simulation of NC-AFM Results For the simulation of complete NC-AFM images it is very practical to use reduced units and to compute the normalized frequency shift g given by 1 g ¼ pffiffiffi 2p
Z1 D
Fts ðzÞ pffiffiffiffiffiffiffiffiffiffiffiffi dz: zD
ð37Þ
For the simulation one can choose then a certain feedback parameter gconst and determine the corresponding nearest distance D at diVerent scan positions (x, y), i.e., solves numerically the equation g(D, x, y) ¼ gconst. This procedure models the behavior of the feedback in the experimental setup and results in maps of constant frequency shifts similar to the experiment. In some cases the sample surface can be suYciently described by the sum of pair-wise Lennard–Jones potentials (see Section I.B). 12 6 ! N N X X r0 r0 Vts ¼ VLJ ðri Þ ¼ E0 2 ; ð38Þ r ri i i¼1 i¼1
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¨ LSCHER AND SCHIRMEISEN HO
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where ri :¼ ðx xi Þ2 þ ðy yi Þ2 þ ðz zi Þ2 represents the distance between the point-like tip apex and the ith surface atom. Consequently, the tip-sample interaction force Fts is given by 13 7 ! N N X @VLJ @ri 12E0 X r0 r0 z ¼ : ð39Þ Fts ¼ ri @ri @z r0 i¼1 ri ri i¼1 With this tip-sample force the corresponding normalized frequency shift can easily be calculated with Eq. (37) for the large-amplitude limit. The equilibrium distance r0 and the bonding energy are the only input parameters needed for the simulation (besides the atomic positions). Due to the comparably simple form of the tip-sample force Eq. (39), complete NC-AFM images can be calculated in this way with acceptable computing time (Giessibl and Bielefeldt, 2000; Ho¨ lscher et al., 2000b, 2001a). An example of the described approach is shown in Figure 24. The experimental line section (solid line) is well reproduced by the stimulation (dashed line) and reveals the contrast inversion on HOPG. However, this simulation method is limited to nonreactive surfaces where the tip-sample forces can be approximated by empirical models. In many cases, however, the tip-sample forces must be calculated by more advanced approaches that use sophisticated models for the calculation of the tipsample interactions. Using ab initio methods the tip-sample forces for diVerent tips were calculated on various sample surfaces such as Si(111), InP(110), GaAs(110), and CaF2. A detailed discussion can be found in Pere´ z et al. (1997, 1998); Ke et al. (1999, 2001); To´ bik et al. (1999, 2001); Bennewitz et al. (2000); Foster et al. (2002); and Sasaki et al. (2002). IV. DYNAMIC FORCE SPECTROSCOPY Since its invention in 1986 the AFM has been widely used to study tipsample interactions for various material combinations. Unfortunately, such investigations were often strongly hindered close to the sample surface by a ‘‘jump to contact’’ (see Section I.C). This problem, however, is avoided using large oscillation amplitudes in the dynamic mode. In Section III.B, we showed how the frequency shift can be calculated for a given tip-sample interaction law. Recalling the motivation for atomic force microscopy in Section I.A, the inverse problem is even more interesting: How can the tip-sample interaction be determined from frequency shift data? The answer to this question is given is this section and it leads to the dynamic force spectroscopy (DFS) technique, which is a direct extension of the noncontact AFM technique.
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85
A. Determining Forces from Frequencies The basic idea of DFS can be understood by recalling the reason for the shift of the resonance frequency in the dynamic mode (see Section II.B). From the introductory courses of theoretical physics it is known that the period of oscillation T of a particle in a potential VeV can be calculated by the integral Dþ2A pffiffiffiffiffiffiffi Z dz pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; TðEÞ ¼ 2m E Veff ðzÞ
ð40Þ
D
where E is the energy of the system. In our case, the energy is directly related to the amplitude A. Unfortunately, it is not possible to solve the inverted problem (i.e., the eVective potential VeV (z) cannot be determined from the function T(E) without further assumptions). The reason is that the inverted function z(VeV) is two-valued: each value of VeV corresponds to two diVerent values of z (see, Landau and Lifshitz, 1990). In the special case of an oscillating cantilever, this problem can be solved by dividing the potential VeV (z) into two parts: ‘‘left’’ and ‘‘right’’ of the minimum zmin (see Figure 19). Since the tip-sample interaction potential Vts(z) decreases rapidly with the tip-sample distance, it is usually very small on the ‘‘right’’ side. Therefore, it can be assumed that this part is parabolic 1 Veff ðzÞ ¼ cz ðz zmin Þ2 for z zmin ; ð41Þ 2 where zmin is the position of the minimum of the eVective potential. As a result of this assumption, the integration of Eq. (40) from zmin to D þ 2A leads to T0 pffiffiffiffiffiffiffi þ 2m TðEÞ ¼ 2
zZmin
D
dz pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; E Veff ðzÞ
ð42Þ
where T0 ¼ 1/f0 is the period of oscillation of the free cantilever. Now the function VeV (z) is reversible for all z zmin, and the integral in Eq. (42) can be inverted. Following the calculations of Landau and Lifshitz (1990), we obtain for the ‘‘left’’ amplitude 1 jzmin DðVeff Þj ¼ pffiffiffiffiffiffiffi p 2m
V Zeff
0
T0 2 TðEÞ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dE: p Veff E
ð43Þ
Since the frequency shift Df :¼ f0 f is the measured quantity in dynamic force microscopy, we transform Eq. (43) to
86
¨ LSCHER AND SCHIRMEISEN HO
ZA0 jzmin DðA0 Þj ¼ 0
0A f0 Df ðA0 Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dA0 ; f0 þ Df ðA0 Þ A20 A0 2
ð44Þ
where A0 represents the amplitude in the parabolic (‘‘right’’) part of the eVective potential. It is straightforward to apply this formula to a frequency shift versus amplitude curve, and to determine the corresponding tip-sample interaction potential from the diVerence between parabolic and eVective potential 1 Vts ðDÞ ¼ cz ðA20 jzmin DðA0 Þj2 Þ 2
ð45Þ
If this method is applied to experimental frequency shift versus amplitude curves, it has to be taken into account that the amplitude A0 is not identical with the experimental amplitude, but this eVect can be considered by a simple search algorithm that optimizes A0 (Ho¨lscher et al., 1999). The detailed analysis shows that the method is unique. Moreover, the systematic error made by assumption Eq. (41) can be neglected for realistic tip-sample potentials if the cantilever-sample distance is large enough. An application of this method to experimental data taken in UHV is shown in Figure 27 for a silicon tip and a graphite sample. The cantilever used for this study had a spring constant of 38 N/m and an eigenfrequency of 177 kHz. The Df (A)-curves displayed in Figure 27a were measured for four diVerent cantilever-sample distances at room temperature. All curves show the same typical overall behavior, but diVer significantly in quantity, depending on the cantilever-sample distance. Despite these diVerences, the reconstruction of the tip-sample potential using these experimental data sets leads to identical results (see Figure 27b). The corresponding tip-sample forces are shown in Figure 27c. The fact that tip-sample potentials and forces can be determined with this accuracy without a jump-to-contact demonstrates the advantage of dynamic force spectroscopy in comparison to force versus distance curves discussed in Section I.C. The complete attractive part of the force curve can be measured without a discontinuity in this way. Nonetheless, the above approach presented to dynamic force spectroscopy is diVerent from the commonly used technique to measure tip-sample forces. For most experimental setups it is more straightforward to measure the frequency shift with a fixed amplitude and as a function of the actual cantilever sample distance. Such curves were already presented in Figure 28 (see Section III.B). The calculation of the tip-sample interactions from such data sets is also possible as first demonstrated by Du¨ rig (1999). His method is based on the inversion of the integral in Eq. (33). It can be transformed to
DYNAMIC FORCE MICROSCOPY AND SPECTROSCOPY
87
FIGURE 27. (a) Four measured Df(Aexp)-curves recorded in ultrahigh vacuum with a silicon tip and a graphite sample for various cantilever-sample distances. The curves are individually shifted along the x-axes to fit all into the same graph. The subtracted oVset is ˚ , 162 A ˚ , 130 A ˚ , and 108 A ˚ , respectively. (b) The tip-sample potential calculated from the 216 A experimental data shown in (a). The zero point of the x-axis is arbitrarily chosen. (c) The corresponding tip-sample forces.
pffiffiffi cz A3=2 @ Z Df ðzÞ pffiffiffiffiffiffiffiffiffiffiffiffi dz; Fts ðDÞ ¼ 2 f0 @D zD 1
ð46Þ
D
which allows a direct calculation of the tip-sample interaction force from the frequency shift versus distance curves. An application of this formula to the experimental frequency shift versus distance curves already presented in Section III.B is shown in Figure 28. The resultant force curves are nearly identical although obtained with diVerent oscillation amplitudes.
88
¨ LSCHER AND SCHIRMEISEN HO
FIGURE 28. The tip-sample-force calculated with the experimental data shown in Figure 20 using the formula Eq. (46) is shown by symbols. The force Fts Eq. (47) is plotted by a dasheddotted line. The best fit using the force law Fc is displayed by a solid line. The border between ‘‘contact’’ and ‘‘noncontact’’ force is marked by the position z0.
Since Eq. (33) was derived with the condition that the resonance amplitude is considerably larger than the decay length of the tip-sample interaction, the same restriction applies for Eq. (46). This limitation is comparable to assumption Eq. (41) used for the derivation of the first method. However, using more advanced algorithms, it is also possible to determine forces from dynamic force spectroscopy experiments without the restriction to use large amplitudes. The numerical approach of Gotsmann et al. (1999, 2001) works in every regime. Du¨rig (2000a), Giessibl (2001), and Sader and Jarvis (2004) introduced methods that are not restricted to the large amplitude regime. The resolution of dynamic force spectroscopy can be driven down to the atomic scale. Lantz et al. (2001) measured frequency shift versus distance curves at diVerent lattice sites of the Si(111)–(77) surface (see Figure 29). In this way they were able to distinguish diVerences in the bonding forces between inequivalent adatoms of the 7 7 surface reconstruction of silicon. The concept of dynamic force spectroscopy can be also extended to 3D-force spectroscopy by mapping the complete force field above the sample surface (Ho¨ lscher et al., 2002a). Figure 30a shows a schematic of the measurement principle. Frequency shift versus distance curves are recorded on a
DYNAMIC FORCE MICROSCOPY AND SPECTROSCOPY
89
FIGURE 29. Atomic resolution image of the Si(111)–(77) surface. Frequency shift versus distance curves (a) were measured at the positions marked in (b). (Image reproduced from Lantz et al., 2001.)
FIGURE 30. (a) Principle of 3D-force spectroscopy. The cantilever oscillates near the sample surface and measure the frequency shift in a x-y-z-box. The 3D surface shows the ˚ 10 A ˚ ) obtained immediately before the recording topography of the sample (image size: 10 A of the spectroscopy field. (b) The reconstructed force field of NiO (001) shows atomic resolution. The data are taken along the line shown in (a).
matrix of points perpendicular to the sample surface. Using Eq. (46) the complete 3D force field between tip and sample can be recovered with atomic resolution. Figure 30b shows a cut through the force field as a 2D map.
90
¨ LSCHER AND SCHIRMEISEN HO
If the NC-AFM is capable of measuring forces between single atoms with sub-nN precision, why should it not be possible to also exert forces with this technique? In fact, the new and exciting field of nanomanipulation would be driven to a whole new dimension, if defined forces can be reliably applied to single atoms or molecules. In this respect, Loppacher et al. (2003) achieved to push on diVerent parts of an isolated Cu-TBBP molecule, which is known to possess four rotatable legs. They measured the force-distance curves while one of the legs was pushed by the AFM tip and turned by 90 degrees, thus being able to measure the energy which was dissipated during ‘‘switching’’ this molecule between diVerent conformational states. The manipulation of single silicon atoms with the NC-AFM was demonstrated by Oyabu et al. (2003), who removed single atoms from a Si(111)–7 7 surface with the AFM tip and could subsequently deposit atoms from the tip on the surface again. The possibility to simultaneously exert and measure forces during single atom or molecule manipulation is an exciting new application of highresolution NC-AFM experiments. B. Analysis of the Tip-Sample Interaction Forces Since the above presented dynamic force spectroscopy methods can be used to measure the tip-sample interactions with high resolution, this approach opens a direct way to compare experiments with theoretical models and predictions. As an example we analyze the force curves already presented in Figure 28. Giessibl (1997) suggested describing the force between the tip and the sample by a combination of a long-range (van der Waals) and a short-range (Lennard–Jones) term (see Section I.B). The long-range part describes the van der Waals interaction of the tip, modeled as a sphere with a specific radius, with the surface. The short-range Lennard–Jones term is a superposition of the attractive van der Waals interaction of the last tip apex atom with the surface and the Coulomb repulsion. For a tip with the radius R, this assumption results in the tip-sample force: r 7 AH R 12E0 r0 13 0 Fts ðzÞ ¼ 2 þ ð47Þ 6z r0 z z Since this approach does not explicitly consider elastic contact forces between tip and sample, we call this the ‘‘noncontact’’ force law in the following. A fit of this equation to the experimental tip-sample force curve is shown in Figure 28 by a solid line; the obtained parameters are AHR ¼ 2.4 1027 Jm, ˚ , and E0 ¼ 3 eV (Ho¨ lscher et al., 2000a). The regime right from r0 ¼ 3.4 A the minimum fits well to the experimental data, but the deep and wide
DYNAMIC FORCE MICROSCOPY AND SPECTROSCOPY
91
minimum of the experimental curves cannot be described accurately with the noncontact force. This is caused by the steep increase of the Lennard–Jones force in the repulsive regime () Fts / 1/r13 for z < r0). The specific choice of the short-range force does not matter; the obtained agreement is not significantly better with other choices. The elastic contact behavior can be described with the assumption that the overall shape of tip and sample changes only slightly until point contact is reached and that, after the formation of this point contact, the tip-sample forces are given by the well-known Hertz theory (see, e.g., Johnson, 1985; Landau and Lifshitz, 1991). This theory considers only the repulsive elastic force between tip and sample but no additional deformation due to attractive forces (see, e.g., Schwarz (2003) for a discussion about this topic). The easiest way is to consider the attractive forces by a simple oVset (Hertz-plusoVset model). This approach coincides with the DMT model (Derjaguin et al., 1975). It results in a force law of the type Fc ¼ G0 ðz0 zÞ3=2 þ Fad
for
z z0
ð48Þ
The first term in this equation describes the elastic behavior of a Hertzian contact, where z0 is the point of contact, and G0 is a constant that depends on the elasticity of tip and sample, and on the shape of the tip (Landau and Lifshitz, 1991). The oVset Fad is the adhesion force between tip and sample surface. Since the experimental tip-sample force shows a reasonable agreement with the noncontact force until its minimum, we define the contact ˚ , and therefore Fad point by this minimum, that is, z0 :¼ min{Fts(z)} ¼ 3.7 A :¼ Fts (z0) ¼ 6.7 nN. With this choice the connection between the noncontact and contact force and their corresponding force gradients is continuous. A fit of Eq. (48) to the experimental data is shown in Figure 28 by a solid line (G0 ¼ 5.8 105 nN/m3/2). The experimental force curves agree quite well with the contact force law for distances D < z0. Therefore, the overall behavior of the experimentally obtained force curves can be suYciently described by a combination of long-range (van der Waals), short-range (Lennard–Jones), and contact (Hertz/DMT) forces. The result demonstrates that not only noncontact, but also elastic contact forces can be quantitatively measured by dynamic force spectroscopy. C. Measurement of Energy Dissipation Dynamic AFM methods have proven their great potential for imaging surface structures at the nanoscale, and we have also discussed methods that allow assessment of forces between distinct single molecules. However, there is another physical mechanism, which can be analyzed with the dynamic
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mode: the energy dissipation due to nonconservative tip-sample forces. In constant-amplitude FM mode, where the quantitative interpretation of the image contrast is possible, there is an intuitive distinction between conservative and dissipative tip-sample interactions. While we have shown the correlation between forces and frequency shifts of the oscillating system, we have neglected one experimental input channel. The excitation amplitude, which is necessary to keep the oscillation amplitude constant, is a direct indication of the energy dissipated during one oscillation cycle. Du¨ rig (2000b) has shown that in self-excitation mode (with an excitation-oscillation phase diVerence of 90 degrees), conservative and dissipative interactions can be strictly separated. In part this energy is dissipated in the cantilever itself; another part is due to external viscous forces in the surrounding medium. But more interestingly, some energy is dissipated at the tip-sample interaction. This is the focus of the following paragraph. In contrast to conservative forces acting at the tip-sample junction, which at least in vacuum can be understood in terms of van der Waals, electrostatic, and chemical interactions, the dissipative processes are comparatively poorly understood. Stowe et al. (1999) have shown that if a voltage potential is applied between tip and sample, charges are induced in the sample surface, which will follow the tip motion (in their case, the oscillation was parallel to the surface). Due to the finite resistance of the sample material, energy will be dissipated during the charge movement. This eVect has been exploited to image the doping level of semiconductors. Energy dissipation has also been observed while imaging magnetic materials. Liu and Gru¨ tter (1998) found that energy dissipation due to magnetic interactions was enhanced at the boundaries of magnetic domains, which was attributed to domain wall oscillations. But also in the absence of external electromagnetic fields energy dissipation was observed in close proximity of tip and sample within one nanometer. Clearly, mechanical surface relaxations must give rise to energy losses. One could model the AFM tip as a small hammer, hitting the surface at high frequency, possibly resulting in phonon excitations. From a continuum mechanics point of view, we assume that the mechanical relaxation of the surface is not only governed by elastic responses. Viscoelastic eVects of soft surfaces will also render a significant contribution to energy dissipation. The whole area of phase imaging in tapping mode is concerned with those eVects (Anczykowski et al., 1999a; Cleveland et al., 1998; Garcia et al., 1998; Tamayo and Garcia, 1997). In the atomistic view the last tip atom can be envisaged to change position while yielding to the tip-sample force field. A strictly reversible change of position would not result in a loss of energy. Still, it has been pointed out by Sasaki and Tsukada (2000) that a change in atom position would result in a change in the force interaction itself. Therefore it is possible that the tip atom
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changes position at diVerent tip-surface distances during approach and retraction, eVectively causing an atomic-scale hysteresis to develop. In fact, HoVmann et al. (2001) have measured short-range energy dissipation for a tungsten tip on silicon in UHV. A theoretical model, explaining this eVect on the basis of a two–energy state system was developed. However, a clear understanding of the underlying physical mechanism is still lacking. Nonetheless, the dissipation channel has been used to image surfaces with atomic resolution (Guggisberg et al., 2000; Lu¨ thi et al., 1997). Instead of feedbacking the distance on the frequency shift, the excitation amplitude in FM-mode can be used as the control signal. The Si(111)–(7 7) reconstruction was successfully imaged in this mode (see also Figure 31). The step edges of monoatomic NaCl islands on singlecrystalline copper have also rendered atomic resolution contrast in the dissipation channel (Bennewitz et al., 2000). The dissipation processes discussed so far are mostly in the configuration where the tip is oscillated perpendicular to the surface. Friction is usually referred to as the energy loss due to lateral movement of solid bodies in contact. It is interesting to note in this context that Israelachvili (1992) has pointed out a quantitative relationship between lateral and vertical (with respect to the surface) dissipation. He states that the hysteresis in vertical force-distance curves should equal the energy loss in lateral friction. An experimental confirmation of this conjecture at the molecular level is still missing. Physical interpretation of energy dissipation processes at the atomic scale seems to be diYcult at this point. Nonetheless, we can find a quantitative relation between the energy loss per oscillation cycle DE and the experimental parameters in dynamic AFM (Cleveland et al., 1998):
FIGURE 31. An NC-AFM image of the Si(111)–(7 7) surface. The atomic structure can be seen simultaneously in the topography (a), the tunneling current (b), and the dissipation channel (c). Defects are marked by circles and can be seen in all three channels. (Image reproduced from Guggisberg et al. (2000)).
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A 2 fd DE ¼ pcz Aad sinðfÞ Q f0
ð49Þ
Interestingly, this equation is obtained without strong assumptions except that the motion of the oscillating cantilever is nearly sinusoidal. Then Eq. (49) follows directly from the conservation of energy principle, and it is applicable to most dynamic AFM modes. For example, in FM-mode AFM the oscillation frequency f changes due to tip-sample interaction while at the same time the oscillation amplitude A is kept constant by adjusting the drive amplitude ad. By measuring these quantities Eq. (49) can be applied to determine the average power dissipated by the tip-sample interaction. In spectroscopy applications usually ad (z) is not measured directly, but the gain factor g(z) (see Section III.D) proportional to ad (z) is recorded. With the help of Eq. (17) we can write: ad ðzÞ ¼
A0 gðzÞ ; Qg0
ð50Þ
where A0 and g0 are the amplitude and gain factor at large tip-sample distances where the tip-sample interactions are negligible. Now let us consider the tapping-mode or AM-AFM. In this case the cantilever is driven at a fixed frequency and with constant drive amplitude, while the oscillation amplitude and phase shift may change when the probing tip interacts with the sample surface. Assuming that the driving frequency fd is chosen to be close to f0, Eq. (49) can be further simplified by using Eq. (17) for the free oscillation amplitude A0. This yields cz DE ¼ p ðAA0 sinðfÞ A2 Þ ð51Þ Q This equation implies that if the oscillation amplitude A is kept constant by the feedback loop during scanning, as is commonly done in tapping mode, simultaneously acquired phase data can be interpreted in terms of energy dissipation (Anczykowski et al., 1999a; Cleveland et al., 1998; Garcı´a et al., 1999; Tamayo and Garcia, 1998). When analyzing such phase images (Chen et al., 1998; Magonov et al., 1997; Pickering et al., 1998) one has to be careful, because the phase may also change due to the transition from netattractive (f > 90 degrees) to intermittent contact (f < 90 degrees) interaction between the tip and the sample. If phase measurements are performed close to the instability point where the oscillation switches from the netattractive to the intermittent-contact regime, a large contrast in the phase channel is observed. However, this contrast is not due to dissipative processes. Only a variation of the phase signal within one regime will give information about the tip-sample dissipative processes.
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Although the energy dissipation in NC-AFM seems to be an interesting channel of additional information about surface properties, the underlying physical mechanisms are far from being understood. Further experiments and theoretical simulations are necessary to shed light on this important issue. V. CONCLUSION In summary, we have presented an overview of the basic principles of atomic force microscopy driven in the static as well as the dynamic mode. The main advantage of the dynamic mode is the possibility to control the tip-sample distance without unwanted jump to contacts. The application of the tapping mode in air and liquids is an everyday tool in nanotechnology enabling the imaging down to the nanometer scale. The ultimate ‘‘true’’ atomic resolution, however, is limited to vacuum conditions using the frequency-modulation technique. In contrast to the tapping mode, this oscillation technique allows the quantitative interpretation of the tipsample interactions. The detailed analyses reveal that the frequency shift is given by the tip-sample force, whereas the gain factor is directly related to the energy dissipation. A very promising application of the dynamic mode is its extension to dynamic force spectroscopy. Its basic idea to measure forces through frequency shifts may seem puzzling, but the derived formulas give a direct access to the tip-sample interactions. The high-resolution capability of this approach is proven by the obtained 3D force fields with atomic resolution. The question of the precise characterization of the exact tip geometry remains unanswered. The chemical identity of the final atoms at the tip end influence the image contrast, but up to now they are not identified in standard AFM applications. Using predefined tips with single-walled nanotubes is one way out of this dilemma. Alternatively, the AFM can be combined with a field ion microscope (FIM), which allows to image the tips with atomic resolution. ACKNOWLEDGMENTS We would like to thank all colleagues who contributed to this work with their images and experimental results: W. Allers, B. Anczykowski, L. Chi, D. Ebeling, M. Hirtz, S. Langkat, J.-E. Schmutz, T. Sch€aVer, and U. D. Schwarz. Furthermore, we acknowledge the continuous support of H. Fuchs
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at the University of Mu¨ nster. The work was funded in part by the Deutsche Forschungsgemeinschaft (DFG) (Grant No. HO 2237/2-1) and the German Federal Ministry of Education and Research (BMBF) (Grant No. 03N8704).
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ADVANCES IN IMAGING AND ELECTRON PHYSICS, VOL. 135
Generalized Almost-Cyclostationary Signals* LUCIANO IZZO AND ANTONIO NAPOLITANO Dipartimento di Ingegneria Elettronica e delle Telecomunicazioni Universita` di Napoli Federico II, 80125 Napoli, Italy
I. Introduction . . . . . . . . . . . . . . . . . . . . A. Generalized Almost-Cyclostationary Signals . . . . . . . . B. Nonstochastic Approach for Signal Analysis . . . . . . . . C. Review on Higher-Order Cyclostationarity . . . . . . . . D. Outline . . . . . . . . . . . . . . . . . . . . . II. Higher-Order Characterization . . . . . . . . . . . . . . A. Introduction . . . . . . . . . . . . . . . . . . . B. Strict Sense Characterization . . . . . . . . . . . . . C. Generalized Cyclic Moments . . . . . . . . . . . . . 1. Temporal Parameters . . . . . . . . . . . . . . . 2. Spectral Parameters . . . . . . . . . . . . . . . D. Generalized Cyclic Cumulants . . . . . . . . . . . . . 1. Temporal Parameters . . . . . . . . . . . . . . . 2. Spectral Parameters . . . . . . . . . . . . . . . E. Estimation of the Generalized Cyclic Statistics . . . . . . . F. Examples of GACS Signals . . . . . . . . . . . . . . 1. Chirp Signal . . . . . . . . . . . . . . . . . . 2. Nonuniformly Sampled Signal . . . . . . . . . . . . G. Summary . . . . . . . . . . . . . . . . . . . . III. Linear Time-Variant Transformations of GACS Signals . . . . . A. Introduction . . . . . . . . . . . . . . . . . . . B. FOT Deterministic and Random Linear Systems . . . . . . 1. FOT Deterministic and Random Systems . . . . . . . . 2. FOT Deterministic Linear Systems . . . . . . . . . . 3. Impulse-Response Function Decomposition for FOT Random LTV Systems . . . . . . . . . . . . . . . . . . C. Higher-Order System Characterization in the Time Domain . . D. Higher-Order System Characterization in the Frequency Domain E. Ergodicity of the Output Signal of a LTV System . . . . . .
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* Reprinted with permission from: L. Izzo and A. Napolitano, ‘‘The higher-order theory of generalized almost-cyclostationary time-series,’’ IEEE Trans. Signal Processing, Vol. 46, pp. 2975–2989, Nov. 1998. L. Izzo and A. Napolitano, ‘‘Linear time-variant transformations of generalized almost-cyclostationary signals, Part I: Theory and method,’’ IEEE Trans. signal Processing, Vol. 50, pp. 2947–2961, Dec. 2002. L. Izzo and A. Napolitano, ‘‘Linear time-variant transformations of generalized almost-cyclostationary signals, Part II: Developments and applications,’’ IEEE Trans. Signal Processing, Vol. 50, pp. 2962–2975, Dec. 2002. L. Izzo and A. Napolitano, ‘‘Sampling of generalized almost-cyclostationary signals,’’ IEEE Trans. Signal Processing, Vol. 51, pp. 1546–1556, June 2003.
ISSN 1076-5670/05 DOI: 10.1016/S1076-5670(04)35003-2
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Copyright 2004, IEEE All rights reserved.
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F. Countability of the Set of the Output Cycle Frequencies. . . . . . 1. Analysis of LTV Systems. . . . . . . . . . . . . . . . 2. The Special Case of FOT Deterministic LTV Systems . . . . . G. LAPTV Filtering . . . . . . . . . . . . . . . . . . . H. Product Modulation . . . . . . . . . . . . . . . . . . I. Multipath Doppler Channels . . . . . . . . . . . . . . . J. Summary . . . . . . . . . . . . . . . . . . . . . . IV. Sampling of GACS Signals . . . . . . . . . . . . . . . . . A. Introduction . . . . . . . . . . . . . . . . . . . . . B. Discrete-Time ACS Signals . . . . . . . . . . . . . . . . C. Sampling of GACS Signals . . . . . . . . . . . . . . . . D. Conjecturing the Nonstationarity Type of the Continuous-Time Signal E. Summary . . . . . . . . . . . . . . . . . . . . . . V. Time-Frequency Representations of GACS Signals. . . . . . . . . A. Introduction . . . . . . . . . . . . . . . . . . . . . B. Second-Order GACS Signals . . . . . . . . . . . . . . . C. Time-Frequency Representations of GACS Signals . . . . . . . D. Signal Feature Extraction . . . . . . . . . . . . . . . . Appendices . . . . . . . . . . . . . . . . . . . . . . Appendix A . . . . . . . . . . . . . . . . . . . . . . Appendix B . . . . . . . . . . . . . . . . . . . . . . Appendix C . . . . . . . . . . . . . . . . . . . . . . Appendix D . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . .
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I. INTRODUCTION This section introduces the class of second-order generalized almost-cyclostationary signals in the classical stochastic approach. Then, the alternative characterization in the nonstochastic (or fraction-of-time probability) framework is considered. Moreover, within such a framework, a review on the higher-order cyclostationarity is provided. Finally, the outline of the topics treated in Section II through V is presented. A. Generalized Almost-Cyclostationary Signals The theory of second- and higher-order almost-cyclostationary (ACS) signals has been developed and applied to several signal-processing and communication problems, such as weak-signal detection, parameter estimation, system identification, blind-adaptive spatial filtering, and so faith (see Dandawate´ and Giannakis, 1994, 1995; Dehay and Hurd, 1994; Gardner, 1988a,b, 1993, 1994; Gardner and Spooner, 1994; Gladyshev, 1963; Hurd, 1991; Izzo and Napolitano, 1996c, 1997a; Napolitano, 1995; Spooner and Gardner, 1994, and references therein).
GENERALIZED ALMOST-CYCLOSTATIONARY SIGNALS
105
A finite-power continuous-time complex-valued stochastic process x(t) is said second-order almost-cyclostationary in the wide sense or, equivalently, almost-periodically correlated, if its autocorrelation function E{x(t þ t)x*(t)}, with E{} denoting statistical expectation, is an almost-periodic function (Besicovitch, 1932; Bohr, 1933; Corduneanu, 1989) of t with frequencies not depending on t. That is, the autocorrelation function is the limit of an uniformly convergent sequence of trigonometric polynomials in t: X an ðtÞe j2pan t ; Efxðt þ tÞx ðtÞg ¼ ð1:1Þ R xx n2I
where I is a countable set, the frequencies an (not depending on t) are referred an ðtÞ, called cyclic autocorrelato as cycle frequencies, and the coeYcients R xx tion functions, are given by Z T=2 an ðtÞ ≜ lim 1 Efxðt þ tÞx ðtÞge j2pan t dt R xx ð1:2Þ T!1 T T=2
hEfxðt þ tÞx ðtÞge j2pan t it : Let us define the function of the two variables (a, t) xx ða; tÞ ≜ hEfxðt þ tÞx ðtÞge j2pat i ; R t
ð1:3Þ
which is called two-variable cyclic autocorrelation function or, if it does not generate ambiguity, cyclic autocorrelation function. Its magnitude and phase are the amplitude and phase, respectively, of the finite-strength additive complex sinewave component at frequency a contained in the autocorrelation function with lag t. According to Eqs. (1.1) and (1.2), the two-variable cyclic autocorrelation function Eq. (1.3) of ACS processes is nonzero only in correspondence of a countable set of values of a. That is, 8 a n ðtÞ; a ¼ an n 2 I
ð1:5Þ
Hurd (1991) shows that the ACS processes are characterized by the following conditions: 1. The set A ≜ [ At t2R
is countable.
ð1:6Þ
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2. The autocorrelation function E{x(t þ t)x*(t)} is uniformly continuous in t and t. xx ð0; tÞ ≜ hEfxðt þ tÞ 3. The time-averaged autocorrelation function R x ðtÞgit is continuous for t ¼ 0 (and, hence, for every t). 4. The process is mean-square continuous, that is sup Efjxðt þ tÞ xðtÞj2 g ! 0 t2R
as t ! 0
ð1:7Þ
In Figure 1, the support in the (a, t) plane of the two-variable cyclic xx ða; tÞ of an ACS signal is reported. According autocorrelation function R to Condition 1, such a support is constituted by the lines a ¼ an, n 2 I, that is, lines parallel to the t axis in correspondence of the cycle frequencies. If the autocorrelation function is an almost-periodic function of t whose (generalized) Fourier series expansion has both coeYcients and frequencies depending on the lag parameter t, then the process is said to be generalized almost-cyclostationary (GACS): X ðnÞ ðtÞe j2pan ðtÞt : Efxðt þ tÞx ðtÞg ¼ ð1:8Þ R xx n2I
In Eq. (1.8), the frequencies an(t) are referred to as lag-dependent cycle frequencies and the coeYcients, referred to as generalized cyclic autocorrelation functions, are given by
xx ða; tÞ of FIGURE 1. Support in the (a, t) plane of the cyclic autocorrelation function R an ACS signal.
GENERALIZED ALMOST-CYCLOSTATIONARY SIGNALS
ðnÞ ðtÞ ≜ Efxðt þ tÞx ðtÞge j2pan ðtÞt : R xx t
107 ð1:9Þ
It can be shown that for GACS signals the two-variable cyclic autocorrelation function can be expressed in terms of the generalized cyclic autocorrelation functions by the following relationship X ðnÞ xx ða; tÞ ¼ ðtÞda a ðtÞ R ð1:10Þ R xx n n2I
where da is the Kronecker delta, that is, da ¼ 1 for a ¼ 0 and da ¼ 0 for a 6¼ 0. Moreover, it results that At ¼ [ fa 2 R : a ¼ an ðtÞg: n2I
ð1:11Þ
xx ða; tÞ is Thus, for GACS signals the support in the (a, t) plane of R constituted by a countable set of curves described by the equations a ¼ an ðtÞ; n 2 I. In Figure 2, the support in the (a, t) plane of the cyclic autocorrelation xx ða; tÞ of a GACS signal is reported. function R From the previous considerations, it follows that the ACS signals are the sub-class of GACS signals such that the lag-dependent cycle frequencies an(t) are constant with respect to t and are equal to the cycle frequencies. Moreover, for GACS signals that are not ACS the set A defined in Eq. (1.6) is uncountable.
xx ða; tÞ of a FIGURE 2. Support in the (a, t) plane of the cyclic autocorrelation function R GACS signal.
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B. Nonstochastic Approach for Signal Analysis The signal analysis approach used in the following sections is nonstochastic, and is also referred to as the fraction-of-time (FOT) probability framework (Gardner, 1988a, 1994). It is an alternative approach where signals are modeled as single functions of time (time series) rather than sample paths of stochastic processes. Such an approach is more appropriate when the ensemble of realizations does not exist and is introduced just to create an (abstract) mathematical model, the stochastic process, for the only time series at hand. Thus, the FOT probability framework turns out to be particularly useful when the stochastic process is not ergodic or it is subject to a processing destroying ergodicity properties (see Section III). In the FOT probability approach, statistical parameters are defined through infinite-time averages of a single time series (and of products of time- and frequency-shifted versions of this time series) rather than ensemble averages of a stochastic process. Moreover, starting from this single time series, a (possibly time-varying) probability distribution function can be constructed, which leads to the expectation operation and all the familiar probabilistic concepts and parameters, such as stationarity, cyclostationarity, nonstationarity, independence, mean, variance, moments, cumulants, and so on. Estimators of the FOT probabilistic parameters are obtained by considering finite-time averages of the same quantities involved in the infinitetime averages. Therefore, assuming the previously-mentioned parameters exist, that is, the infinite-time averages exist, their asymptotic estimators converge by definition to the true values, which are exactly the infinite-time averages without the necessity of requiring ergodicity properties as in the stochastic process framework. Thus, in the FOT probability framework, the kind of convergence of the estimators to be considered as the data-record length approaches infinity is the convergence of the function sequence of the finite-time averages (indexed by the data-record length). Therefore, unlike the stochastic process framework where convergence must be intended, for example, in the ‘‘stochastic meansquare sense’’ (Dandawate´ and Gianakis, 1995; Dehay and Hurd, 1994; Hurd, 1991; Lacoume et al., 1997) or ‘‘almost sure sense’’ (Hurd and Les´kow, 1992; Li and Cheng, 1997) or ‘‘in distribution,’’ the convergence in the FOT probability framework must be considered ‘‘pointwise,’’ in the ‘‘temporal mean-square sense’’ (Weiner, 1930) (see also Appendix A), or in the ‘‘sense of generalized functions (distributions)’’ (PfaVelhuber, 1975). The functional analysis approach for signal processing based on infinitetime averages was first introduced by Norbert Wiener (1930) in the classical paper with reference to time-invariant statistics of ordinary functions of time; subsequently, it was developed in Bass (1971), Brown (1958),
109
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Furstenberg (1960), Hofstetter (1964), Kac and Steinhaus (1938), Kac (1959), Les´kow and Napolitano (2002), and Urbanik (1958) and then extended to the case of distributions (generalized functions) in PfaVelhuber (1975). The term fraction-of-time probability was first introduced in Hofstetter (1964). Moreover, in Wold (1948) an isometric isomorphism (Wold isomorphism) between a stationary stochastic process and the Hilbert space generated by a single sample path was singled out and a rigorous link between the FOT probability and the stochastic process frameworks in the stationary case was established. The case of time-variant FOT statistics of ACS time series was widely treated by William A. Gardner in (1988a), and then in Gardner (1991, 1994) with reference to the second-order statistics and in Gardner and Brown (1991), Gardner and Spooner (1994) and Spooner and Gardner (1994) for the higher-order statistics. Furthermore, the Wold isomorphism was extended to cyclostationary sequences in Hurd and Koski (2004). Moreover, in Izzo and Napolitano (1996a, 1998b, 2002b,c) the class of the GACS time series was introduced and characterized in the FOT probability framework. In the time-variant nonstochastic framework for ACS and GACS time series, the almost-periodic functions (Besicovitch, 1932; Bohr, 1933; Corduneanu, 1989) play the same role played by the deterministic signals in the stochastic-process framework and the expectation operator is the almostperiodic component extraction operator, that is, the operator that extracts all the finite-strength additive sine wave components of its argument. Therefore, in the case of processes exhibiting suitable ergodicity properties, all the results presented here can be interpreted in the stochastic process framework by substituting the almost-periodic component extraction operator with the statistical expectation operator. Specifically, for any finite-power time series x(t), let us consider the decomposition xðtÞ ≜ xp ðtÞ þ xr ðtÞ where xp ðtÞ ≜
X
x e j2pt
ð1:12Þ
ð1:13Þ
is an almost-periodic function and xr(t) a residual term not containing any finite-strength additive sinewave component, that is, hxr ðtÞe j2pat it 0; 8a 2 R:
ð1:14Þ
The almost-periodic component extraction operator E{a}{} is defined by Efag fxðtÞg ≜ xp ðtÞ:
ð1:15Þ
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Let x(t) be real valued and let us denote by G the set of frequencies (for any x) of the almost-periodic component of the function of t U(x x(t)), where ( 1; x 0 UðxÞ ≜ ð1:16Þ 0; x < 0 is the unit step function. It can be shown that the function of x fGg
FxðtÞ ðxÞ ≜ Efag fUðx xðtÞÞg
ð1:17Þ
for any t is a valid cumulative distribution function except for the rightcontinuity property. That is, it has values in [0, 1], is nondecreasing, fGg fGg FxðtÞ ð 1Þ ¼ 0, and FxðtÞ ðþ1Þ ¼ 1. Moreover, its derivative with respect fGg to x, say fxðtÞ ðxÞ, is a valid probability density function, and it results that Z fGg Efag fxðtÞg ¼ x fxðtÞ ðxÞ dx ð1:18Þ R
which is a result formally analogous to that in the classical stochastic process framework. For an almost-periodic signal x(t), it results x(t) xp(t) and, hence, Efag fxðtÞg ¼ xðtÞ:
ð1:19Þ
That is, the almost-periodic functions are the deterministic signals in the FOT probability framework. All the other signals are the random signals (see also Section II.B). Note that in this work ‘‘random’’ is not synonymous with ‘‘stochastic.’’ In fact, the adjective stochastic is adopted, as usual, when an ensemble of realizations or sample paths exist, whereas the adjective random is referred to a single function of time, namely, a signal or a system impulse-response function. Therefore, to avoid ambiguities, when necessary, deterministic and random signals or systems in the FOT probability sense will be referred to as FOT deterministic and FOT random, respectively. Analogously, a second-order characterization for the real-valued time series x(t) can be obtained by using the almost-periodic component extraction operator as the expectation operator. Specifically, the function of x1 and x2 fG g
t FxðtþtÞxðtÞ ðx1 ; x2 Þ ≜ Efag fUðx1 xðt þ tÞÞUðx2 xðtÞÞg
ð1:20Þ
is a valid second-order joint cumulative distribution function for any fixed t and t, except for the right-continuity property with respect to x1 and x2, where Gt is the set of frequencies (for any (x1, x2)) of the almost-periodic component of the function of t Uðx1 xðt þ tÞÞUðx2 xðtÞÞ. Moreover, the
GENERALIZED ALMOST-CYCLOSTATIONARY SIGNALS
111
fG g
t second-order derivative with respect to x1 and x2 of FxðtþtÞxðtÞ ðx1 ; x2 Þ, say fGt g fxðtþtÞxðtÞ ðx1 ; x2 Þ, is a valid second-order joint probability density function. Furthermore, it results that the autocorrelation function Efag fxðt þ tÞxðtÞg can be expressed as
E
fag
fxðt þ tÞxðtÞg ¼ ¼ ¼
R
fG g
2
R X
t x1 x2 fxðtþtÞxðtÞ ðx1 ; x2 Þ dx1 dx2
a2At X n2I
Rxx ða; tÞej2pat
ð1:21Þ
j2pan ðtÞt RðnÞ xx ðtÞe
where At is a countable set defined analogously to Eq. (1.5), Rxx ða; tÞ ≜ hxðt þ tÞxðtÞe j2pat it
ð1:22Þ
is the (nonstochastic) (two-variable) cyclic autocorrelation function, and j2pan ðtÞt ð1:23Þ RðnÞ xx ðtÞ ≜ xðt þ tÞxðtÞe t is the (nonstochastic) generalized cyclic autocorrelation function. If for some t the set At contains not only the element a ¼ 0, then the time series is GACS (or, in particular, ACS) in the wide sense. From these time series sine waves can be generated by using quadratic or higher-order nonlinear time-invariant transformations. In the special case of ACS time series, it has been shown that algorithms based on the theory of generated sine waves turn out to be asymptotically unaVected by both noise and interference and then are highly tolerant to noise and interference in practice. Finally, note that the almost-periodic component of a time series (or a lag product time series) can be extracted by exploiting the synchronized averaging identity (Gardner, 1988a, 1990) or the algorithms proposed by Sethares and Staley (1999). C. Review on Higher-Order Cyclostationarity The theory of GACS signals presented in the following sections extends the theory of higher-order cyclostationarity that was introduced in the FOT probability framework by Gardner and Spooner (1994) and Spooner and Gardner (1994) and subsequently developed in Flagiello et al. (2000), Izzo and Napolitano (1996c, 1997a, 1998a), Napolitano (1995), and Napolitano and Spooner (2000, 2001). The following brief review introduces the necessary terminology. A continuous-time complex-valued time series x(t) is said to exhibit Nth-order wide-sense cyclostationarity with cycle frequency a 6¼ 0, for a given conjugation configuration, if the Nth-order cyclic temporal moment function (CTMF)
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Rax ðtÞ≜ RaxðÞ1 ;...;xðÞN ðtÞ * + N Y ≜ xðÞn ðt þ tn Þe j2pat n¼1
ð1:24Þ
t
exists and is not zero for some delay vector t (Gardner and Spooner, 1994). In (1.24), t ≜ ½t1 ; . . . ; tN ⊤ and x ≜ ½xðÞ1 ; . . . ; xðÞN ⊤ are column vectors, and (*)n represents the nth optional complex conjugation. The N-fold Fourier transform S ax ð f Þ of the CTMF, which is called the Nth-order cyclic spectral moment function (CSMF), can be written as S ax ð f Þ ¼ Sxa ð f 0 Þdð f ⊤ 1 aÞ
ð1:25Þ
⊤
where d() is Dirac delta function, 1 ≜ ½1; . . . ; 1 , and the prime sign denotes the operator that transforms v ¼ ½v1 ; . . . ; vN ⊤ into v0 ¼ ½v1 ; . . . ; vN 1 ⊤ . The function Sxa ð f 0 Þ, referred to as the Nth-order reduced-dimension CSMF (RD-CSMF), can be expressed as the (N 1)-fold Fourier transform of the Nth-order reduced-dimension CTMF defined as Rax ðt0 Þ ≜ Rax ðtÞjtN ¼0 :
ð1:26Þ
For N ¼ 2 and conjugation configuration [xx*], the RD-CTMF is coincident with the cyclic autocorrelation function Raxx ðt1 Þ, whereas, for conjugation configuration [xx] it is coincident with the conjugate cyclic autocorrelation function Raxx ðt1 Þ that can be nonzero only if the signal is noncircular (Picinbono, 1994). Gardner and Spooner (1994) showed that the RD-CSMF Sxb ( f 0 Þ can contain impulsive terms if the vector f with PN 1 fN ¼ b n¼1 fn lies on the b-submanifold, that is, if there exists at least one partition {m1, , mp} of {1, , N} with p > 1 such that each sum P ami ¼ n2mi fn is an jmijth-order cycle frequency of x(t), where jmij is the number of elements in mi. A well-behaved frequency-domain function that characterizes a signal’s higher-order cyclostationarity can be obtained starting from the Nth-order cyclic temporal cumulant function (CTCF), that is, the coeYcient
Cbx ðtÞ ≜ cum xðÞn ðt þ tn Þ; n ¼ 1; . . . ; N e j2pbt t ð1:27Þ of the Fourier-series expansion of the Nth-order temporal cumulant function (Gardner and Spooner (1994). Its N-fold Fourier transform is the Nthorder cyclic spectral cumulant function P bx ð f Þ, which can be written as P bx ð f Þ ¼ Pbx ð f 0 Þdð f ⊤ 1 bÞ, where the Nth-order cyclic polyspectrum (CP) Pbx ð f 0 Þ is the (N 1)-fold Fourier transform of the reduced-dimension CTCF (RD-CTCF) Cxb ðt0 Þ obtained by setting tN ¼ 0 into Eq. (1.27). The CP turns out to be a well-behaved function (i.e., it does not contain impulsive terms) under the mild assumption that the time series x(t) and
GENERALIZED ALMOST-CYCLOSTATIONARY SIGNALS
113
x(t þ t) are asymptotically (j t j! 1) independent. Moreover, except on a b-submanifold, it is coincident with the RD-CSMF Sxb ð f 0 Þ. Second- and higher-order cycle frequencies are obtained as linear combinations of parameters such as the baud rate and the carrier frequency. Therefore, cyclostationarity-based signal-processing algorithms are potentially signal selective, the selectivity being obtained by a suitable choice of the working cycle frequency, provided that the useful and disturbance signals exhibit at least one diVerent cycle frequency. D. Outline Section II introduces the class of the GACS time series (Izzo and Napolitano,1998b, 2002b). Time series belonging to this class are characterized by multivariate statistical functions that are almost-periodic functions of time whose Fourier series expansions can exhibit coeYcients and frequencies depending on the lag shifts of the time series. Moreover, the union over all the lag shifts of the lag-dependent frequency sets is not necessarily countable. ACS time series turn out to be the subclass of GACS time series for which the frequencies do not depend on the lag shifts and the union of the previously-mentioned sets is countable. The higher-order characterization of GACS time series in the strict and wide sense is provided in the nonstochastic (or FOT probability) framework. Generalized cyclic moment and cumulant functions (in both time and frequency domains) are introduced and relationships among them are stated. Section III addresses the problem of the linear time-variant (LTV) filtering of GACS signals in the FOT probability framework (Izzo and Napolitano, 2002b,c). The adopted approach is particularly useful as an alternative to the classical stochastic one, when stochastic systems transform ergodic input signals into nonergodic output signals, as it happens with several channel models encountered in the practice. Systems are classified as deterministic or random in the FOT probability framework. Moreover, the new concept of expectation in the FOT probability sense of the impulse-response function of a system is introduced. For the LTV systems, the higher-order system characterization in the time domain is provided in terms of the system temporal moment function, which is the kernel of the operator that transforms the finite-strength additive sine wave components contained in the input lag product into the finite-strength additive sine wave components contained in the output lag product. The higher-order characterization in the frequency domain is also provided and input/output relationships are derived in terms of temporal and spectral moment and cumulant functions. The countability of the set of the output cycle frequencies is studied with
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reference to LTV systems for both ACS and GACS not containing any ACS component input signals. Thus, the linear almost-periodically time-variant filtering and the product modulation are considered in detail. Moreover, several Doppler channel models are analyzed. In all these examples, it is shown that the FOT probability approach allows characterization of the system and its output in terms of statistical functions that can be measured by a single time series. Furthermore, the usefulness of considering the linear filtering problem within the class of the GACS signals is clarified and several pitfalls arising from continuing to adopt for the observed time series the ACS model when the increasing of the data-record length makes the GACS model more appropriate are pointed out. Section IV addresses, the problem of sampling a continuous-time GACS signal (Izzo and Napolitano, 2003). It is shown that the discrete-time signal constituted by the samples of a GACS signal is a discrete-time ACS signal. Thus, discrete-time ACS signals can arise not only from the sampling of continuous-time ACS signals, but also from the sampling of a wider class of nonstationary signals, that is, the continuous-time GACS signals. Relationships between generalized cyclic statistics of a continuous-time GACS signal and cyclic statistics of the discrete-time ACS signal constituted by its samples are derived. The problem of aliasing in the domain of the cycle frequencies is considered and a condition ensuring that the cyclic temporal moment function of the discrete-time signal can be obtained by sampling that of the continuous-time signal is determined. Finally, it is shown that, starting from the sampled signal, the GACS or ACS nature of the continuous-time signal can be conjectured, provided that the analysis parameters such as the sampling period, padding factor, and data-record length are properly chosen. Section V expresses time-frequency representations for GACS signals in terms of generalized cyclic statistics (Izzo and Napolitano, 1997b). The Wigner–Ville distribution and the ambiguity function are examined in detail and the special case of ACS signals is considered. Moreover, the problem of signal feature extraction based on a single-record estimation is addressed.
II. HIGHER-ORDER CHARACTERIZATION A. Introduction This section deals with time series belonging to a class wider than that of ACS time-series. Specifically, it deals with continuous-time time series whose multivariate statistical functions are almost-periodic functions of time
GENERALIZED ALMOST-CYCLOSTATIONARY SIGNALS
115
exhibiting Fourier series expansions with coeYcients and frequencies that can depend on the lag shifts of the time series. Moreover, the union over all the lag shifts of the lag-dependent frequency sets is not necessarily countable. Time series belonging to the class under consideration are called therein the generalized almost-cyclostationary time series. The class of GACS time series includes, as a subclass, the ACS time series that are obtained when the cycle frequencies do not depend on the lag shifts. Examples of GACS time series not belonging to the subclass of ACS time series arise from some linear time-variant transformations of ACS time series, such as channels introducing a time-variant delay (Izzo and Napolitano, 1995b, 1996b, 1998b, 2002b,c). Chirp signals and several angle-modulated and time-warped communication signals are further examples. The signal analysis framework utilized is that of the FOT probability, where statistical parameters are defined through infinite-time averages of a single time series rather than ensemble averages of a stochastic process (see Section I.B). In Section II.B, GACS time series are introduced and characterized in the strict sense. Moreover, some results for ACS time series derived in Gardner and Brown (1991) are extended to the GACS case. Section II.C.1. shows that for GACS time series the cyclic temporal moment functions (Gardner and Spooner, 1994) are not necessarily continuous functions of the lag vector even if the temporal moment function is continuous. In particular, it is shown that those GACS time series not belonging to the class of ACS time series exhibit a time-averaged autocorrelation function discontinuous in the origin. Moreover, it is pointed out that the spectral characterization in terms of cyclic spectral moment functions (Gardner and Spooner, 1994) can result inadequate. Therefore, generalized cyclic moments in both time and frequency domains are introduced to characterize GACS time series in the wide sense. Furthermore, it is shown that, under mild conditions, the Nth-order temporal moment function can be expressed as a sum of complex sinusoids whose amplitudes and frequencies are called Nth-order generalized cyclic temporal moment functions and (moment) lag-dependent cycle frequencies, respectively. Section II.D introduces a similar representation for the Nthorder temporal cumulant function, which can be expressed in terms of generalized cyclic temporal cumulant functions and (cumulant) lag-dependent cycle frequencies. Then, starting from such a representation, the characterization in the frequency domain in terms of generalized cyclic spectral cumulant functions is presented. Furthermore, relationships among the introduced functions are derived. Section II.E briefly addresses the problem of estimation of the introduced generalized cyclic statistics. Finally, Section II.F considers two examples of GACS time series. It is worth noting that the adopted definition of GACS time series is in agreement with that of almost-periodically correlated signal given in Dehay
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IZZO AND NAPOLITANO
and Hurd (1994) and Hurd (1991), with reference to second-order statistics. Note that, however, in Dehay and Hurd (1994), and Hurd (1991), the entire theory is practically limited to GACS signals with second-order (reduced dimension) cyclic temporal moment functions that are continuous functions of the lag parameter, that is, second-order ACS signals. B. Strict Sense Characterization This section is aimed at characterizing in the strict sense time series in the FOT probability framework. Such a characterization generalizes that proposed by Gardner and Brown (1991) to the case in which the set of frequencies of the joint probability density function depends on the lag shifts of the time series. Let xðtÞ ≜ xr ðtÞ þ jxi ðtÞ; t 2 R, be a continuous-time complex-valued finite-power time series. If the set Gt,j of all frequencies of the finite-strength additive sine wave components contained in the function of t U x ð1t þ t; jÞ ≜
N Y
Uðxrn xr ðt þ tn ÞÞ Uðxin xi ðt þ tn ÞÞ
ð2:1Þ
n¼1
is countable for each of the column vectors t ≜ ½t1 ; . . . ; tN T 2 RN and j ≜ j r þ jj i ≜ ½xr1 þ jxi1 ; . . . ; xrN þ jxiN T 2 CN , and, moreover, also the set Gt ≜ [j2CN Gt;j is countable for each t 2 RN , then the time series is said to be Nth-order generalized almost cyclostationary in the strict sense (Izzo and Napolitano, 1998b). Following the proof given in Gardner and Brown (1991) with reference to almost-cyclostationary time series, it can be shown that the function fG g
t Fxðtþt ðjÞ ≜ EfGt g fU x ð1t þ t; jÞg 1 ÞxðtþtN Þ X ¼ Fxg ðt; jÞej2pgt
ð2:2Þ
g2Gt
is a valid almost-periodically time-varying joint cumulative distribution function for each fixed value of t, except for the right continuity property with respect to each xrn and xin variable. In Eq. (2.2), F gx ðt; jÞ ≜ hU x ð1s þ t; jÞe j2pgs is
ð2:3Þ
fg is the almost-periodic component extraction operator, that is, the and E operator that extracts all the finite-strength additive sine wave components (with frequencies ranging in the set Gt) present in its argument. The operator EfGt g fg in the following will be denoted by Efag f:g if not indicating the set Gt does not generate ambiguity. It plays, in the FOT fGt g
GENERALIZED ALMOST-CYCLOSTATIONARY SIGNALS
117
probability framework, the same role played by the expectation operator in the stochastic process framework. Furthermore, for a stochastic process X(t) with almost-periodic joint probability density function and cycloergodicity properties, the expected value of any function of X(t), say g(X(t)), is coincident with the almost-periodic component Efag fgðxðtÞÞg for almost all sample-paths x(t) of X (t) (Gardner, 1988a, 1994; Gardner and Brown, 1991). Thus, from Eqs. (2.1) and (2.2) it follows that in the FOT probability framework the joint cumulative distribution function is introduced analogously as in the stochastic process framework by considering the expected value of the indicator of the event ft 2 R : xr ðt þ tn Þ xrn; xi ðt þ tn Þ xin ; n ¼ 1; . . . ; Ng. From Eq. (2.2) it follows that the function Eq. (2.1) can be decomposed into the sum of its almost-periodic component (the deterministic component) and a residual term ‘U ðt; t; jÞ not containing any finite-strength additive sine wave component: fG g
t U x ð1t þ t; jÞ ¼ Fxðtþt ðjÞ þ ‘U ðt; t; jÞ 1 Þ...xðtþtN Þ
with 1 lim T!1 T
Z
T=2 T=2
‘U ðt; t; jÞe j2pgt dt 0
8g 2 R:
ð2:4Þ
ð2:5Þ
In the special case where the set G ≜ [ Gt t2RN
ð2:6Þ
is countable, the time series x(t) is said to be Nth-order almost cyclostationary in the strict sense, the sum in Eq. (2.2) can be extended to the set G, and fGg
Fxðtþt1 Þ...xðtþtN Þ ðjÞ ≜ EfGg fU x ð1t þ t; jÞg
ð2:7Þ
is a valid cumulative distribution function except for the right continuity property with respect to each xrn and xin variable (Gardner and Brown, 1991). The 2Nth-order derivative (in the sense of generalized functions) of the cumulative distribution function fG g
@ 2N fGt g Fxðtþt ðjÞ 1 Þ...xðtþtN Þ @xr1 @x . . . @x @x rN iN (i1 ) N Y ð2:8Þ fGt g dðxrn xr ðt þ tn ÞÞdðxin xi ðt þ tn ÞÞ ¼E n¼1 P g j2pgt ¼ g2Gt f x ðt; jÞe
t fxðtþt ðjÞ ≜ 1 Þ...xðtþtN Þ
turns out to be a valid almost-periodically time-varying joint probability density function for each fixed value of t. In Eq. (2.8)
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IZZO AND NAPOLITANO
* f gx ðt; jÞ ≜
N Y
+ dðxrn xr ðv þ tn ÞÞ dðxin xi ðv þ tn ÞÞ e
j2pgv
n¼1
ð2:9Þ v
is the 2Nth-order derivative of the corresponding Fourier coeYcient [Eq. (2.3)] of the joint cumulative distribution function [Eq. (2.2)]. Let us note that if the time series x(t) is almost periodic, then the function Eq. (2.1), which is a memoryless nonlinear transformation of x(t), is in turn almost periodic and hence it is coincident with its almost-periodic component Eq. (2.2) [‘U ðt; t; jÞ 0 in Eq. (2.4)]. Therefore, the probability density function Eq. (2.8) can be expressed as N Y fGt g fxðtþt ðjÞ ¼ dðxrn xr ðt þ tn ÞÞ dðxin xi ðt þ tn ÞÞ ð2:10Þ 1 Þ...xðtþtN Þ n¼1
that is, almost-periodic time series are deterministic in the FOT probability framework. All the signals that are not almost-periodic functions (and, hence, also the GACS and ACS signals) are the random signals in the FOT probability framework. Signals not containing any almost-periodic component (that is, any finite-strength additive sine wave component) are the FOT zero-mean signals and are said FOT purely random signals. The following lag-shift invariance property for the set Gt holds: ð2:11Þ Gtþ1D Gt : In fact, for each D 2 R it results that fG
g
tþ1D Fxðtþt ðjÞ 1 þDÞ...xðtþtN þDÞ X ≜ U x ð1v þ t þ 1D; jÞ e j2pgv v e j2pgt
g2Gtþ1D
¼
X
g2Gtþ1D [Gt
¼
U x ð1v þ t þ 1D; jÞ e j2pgv v e j2pgt
ð2:12Þ
X U x ð1ðv þ DÞ þ t; jÞe j2pgðvþDÞ vþD e j2pgðtþDÞ
g2Gt
fG g
t ¼ FxðtþDþt ðjÞ: 1 Þ...xðtþDþtN Þ
fG g
t Moreover, from Eq. (2.12) it follows that Fxðtþt ðjÞ is a function 1 Þ...xðtþtN Þ of 1t þ t. Consequently, from Eq. (2.4) it follows that also ‘U (t, t; j) is a function of 1t þ t and, hence, in the following will be denoted by ‘U (1t þ t; j). Finally, the time series x(t) and x(t þ t) are said asymptotically (jtj ! 1) independent in the FOT probability sense if
fG
g
fG g
fG g
½t;0 0 FxðtþtÞxðtÞ ðjÞ ! FxðtþtÞ ðxr1 ; xi1 ÞFxðtÞ0 ðxr2 ; xi2 Þ
ð2:13Þ
GENERALIZED ALMOST-CYCLOSTATIONARY SIGNALS
119
as jtj ! 1 (see Brown, 1987; Gardner, 1988a, and Gardner and Brown, 1991, for the case of ACS signals). C. Generalized Cyclic Moments In this section, generalized cyclic moments in both time and frequency domains are introduced to characterize GACS time series in the wide sense (Izzo and Napolitano, 1996a, 1998b, 2002b). 1. Temporal Parameters In the FOT probability framework, a continuous-time possibly complexvalued time series x(t) is said to exhibit Nth-order wide-sense cyclostationarity with cycle frequency a 6¼ 0, for a given conjugation configuration, if the Nth-order cyclic temporal moment function * + N Y a ðÞn j2pat Rx ðtÞ ≜ x ðt þ tn Þe ð2:14Þ n¼1
t
is not zero for some t (Gardner and Spooner, 1994). In Eq. (2.14), the convergence of the infinite averaging with respect to t is assumed in the temporal mean-square sense (see Appendix A). The more general convergence in the sense of distributions (generalized functions) is discussed in PfaVelhuber (1975), with reference to stationary time series and the results can be extended with minor changes to time series exhibiting cyclostationarity. Note that, for N ¼ 2, (*)1 absent, (*)2 present, and t2 ¼ 0 the CTMF reduces to the cyclic autocorrelation function Eq. (1.22). If the set At ≜ fa 2 R : Rax ðtÞ 6¼ 0g
ð2:15Þ
is countable for each t, then the time series is said to be Nth-order generalized almost-cyclostationary in the wide sense (for the considered conjugation configuration) and the almost-periodic function ( ) N Y fAt g ðÞn Rx ðt; tÞ ≜ E x ðt þ tn Þ ð2:16Þ X n¼1 ¼ Rax ðtÞej2pat a2At
which is called the temporal moment function, is a valid moment function, that is, it can be expressed as Z N Y fGt g ðxrn þ jxin ÞðÞn fxðtþt ðjÞdj r dj i : ð2:17Þ Rx ðt; tÞ ¼ 1 Þx...ðtþtN Þ R2N n¼1
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IZZO AND NAPOLITANO
In fact, accounting for Eq. (2.9), one has Z Y N ðxrn þ jxin ÞðÞn fxg ðt; jÞdj r dj i R2N
n¼1
¼
Z Y N R2N
*
ðxrn þ jxin ÞðÞn
n¼1 N
Y d xrn xr ðv þ tn Þ d xin xi ðv þ tn Þ e j2pgv
R
* ¼
n¼1
R
2N
N
Y ðxrn þ jxin ÞðÞn d xrn xr ðv þ tn Þ
+ dj r dj i v
n¼1
E :d xin xi ðv þ tn Þ dj r dj i e j2pgv v * + N
ðÞn Y ¼ e j2pgv xr ðv þ tn Þ þ jxi ðv þ tn Þ n¼1
v
¼ Rgx ðtÞ
ð2:18Þ whose derivation assumes the order of integration and time averaging can be interchanged and exploits the sampling property of Dirac’s delta function. From Eqs. (2.8) and (2.18), Eq. (2.17) easily follows. Note that Eq. (2.17) generalizes to GACS time series the theorem of almost-periodic component extraction, which was stated in Gardner and Brown (1991) with reference to ACS time series. From the previous discussion it follows that the Nth-order lag product can be expressed as a sum of its almost-periodic component (i.e., its deterministic component in the FOT probability framework) and a residual term not containing any finite-strength additive sinewave component, that is, Lx ð1t þ tÞ ≜
N Y xðÞn ðt þ tn Þ n¼1
ð2:19Þ
¼ Rx ðt; tÞ þ ‘x ðt; tÞ with h‘x ðt; tÞe j2pat it 0;
8a 2 R:
ð2:20Þ
The almost-periodic component of a time series (or a lag product time series) can be extracted by exploiting the synchronized averaging identity (Gardner, 1988a, 1990) or the algorithms proposed by Sethares and Staley (1999).
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GENERALIZED ALMOST-CYCLOSTATIONARY SIGNALS
Let us note that from Eq. (2.18) it follows immediately that, for each t, At Gt :
ð2:21Þ
Therefore, the countability of Gt ensures that of At. In the special case where the set A ≜ [ At
ð2:22Þ
t2RN
is countable, the time series x(t) is said to be Nth-order almost cyclostationary in the wide sense. Note that this set is countable for N ¼ 2 if the function Rx ðt; tÞjt2 ¼0 is uniformly continuous in t and t1 (Dehay and Hurd, 1994). A useful characterization can be introduced for the GACS time series observing that, accounting for the countability of At for each t, the support in the (a, t) space of the CTMF is constituted by a set of N-dimensional manifolds (Choquet-Bruhat and DeWitt-Morette, 1982) defined by the implicit equations Fz0 ða; tÞ ¼ 0;
z0 2 W 0 ;
ð2:23Þ
where W 0 is a countable set. That is, it results that supp fRax ðtÞg ≜ clfða; tÞ 2 At RN : Rax ðtÞ 6¼ 0g ¼ cl 0 [ fða; tÞ 2 R RN : Fz0 ða; tÞ ¼ 0; Rax ðtÞ 6¼ 0g z 2W 0
¼ cl [ fða; tÞ 2 R Dz : a ¼ az ðtÞg z2W
ð2:24Þ where cl denotes closure, W is a countable set and, in the last equality, each manifold described by an implicit equation Fz0 ða; tÞ ¼ 0 has been decomposed into a countable set of manifolds each described by the explicit equation a ¼ az(t), where each function az(t), called Nth-order (moment) lag-dependent cycle frequency, is defined in Dz RN and is not necessarily a continuous function of t. In Eq. (2.24), Dz1 \ Dz2 ¼ ; for z1 6¼ z2. Therefore, set At can be written as At ¼ [ fa 2 R : a ¼ az ðtÞg z2W
ð2:25Þ
where the Nth-order (moment) lag-dependent cycle frequencies az(t) are such that for each ða; tÞ 2 At RN there exists at most one z 2 W such that a ¼ az(t). Starting from Eq. (2.25), the temporal moment function Eq. (2.16) can be expressed in terms of a Fourier series expansion where the sum ranges over a set not depending on t [as in Eq. (2.16)], whereas the frequencies depend on t:
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Rx ðt; tÞ ¼
X
Rx;z ðtÞe j2paz ðtÞt
ð2:26Þ
z2W
where the functions Rx,z(t), which are called the generalized CTMFs (GCTMFs), are given by 8* + N > < Y ðÞn x ðt þ tn Þe j2paz ðtÞt ; 8t 2 Dz ð2:27Þ Rx;z ðtÞ ≜ n¼1 > t : 0; elsewhere and are not necessarily continuous functions of t. In representation Eq. (2.25) we have that 8t it results that az1(t) 6¼ az2(t) for z1 6¼ z2. However, if more functions az1(t), . . . , azk(t) are defined in K (not necessarily coincident) neighborhoods, say I1, . . . , IK, of the same point t0, only one of them is defined in t 0 and, moreover, it results that lim
Dt!0 t0 þDt2Ik
azk ðt0 þ DtÞ ¼ a0 ;
k ¼ 1; . . . ; K
ð2:28Þ
then it is convenient to consider in Eq. (2.25) az1 ðt0 Þ ¼ . . . ¼ azK ðt 0 Þ ¼ a0 and to put Rx;zk ðt 0 Þ ≜
lim
Dt!0 t 0 þDt2Ik
Rx;zk ðt 0 þ DtÞ;
k ¼ 1; . . . ; K:
ð2:29Þ
The set of points t 0 such that Eq. (2.28) holds for some zk 2 W will be denoted by Dx and is assumed to be countable. With convention Eq. (2.29), it can be shown that CTMFs and GCTMFs are related by the following relationship: X Rax ðtÞ ¼ Rx;z ðtÞda az ðtÞ : ð2:30Þ z2W
Rax ðt1 Þej2pat
Thus, the function represents the sum of all the finite-strength sinewaves with frequency a contained in Eq. (2.26) when t ¼ t1. Moreover, 8t 2 Dz Dx Rx;z ðtÞ ¼ Rax ðtÞja¼az ðtÞ :
ð2:31Þ
For GACS time series that are not ACS, even if the set At and the temporal moment function Rx(t, t) are continuous functions of the lag vector t, the CTMFs are not necessarily continuous functions of t. Specifically, according to Eq. (2.30), they can result to be constituted by sums of Kronecker’s delta functions depending on t. It is well known that for every finite-power time series x(t) the conventional time-averaged autocorrelation function R0xx ðtÞ is continuous in t ¼ 0
123
GENERALIZED ALMOST-CYCLOSTATIONARY SIGNALS
if and only if the cross-correlation function R0xy ðtÞ is continuous for any t 2 R and for any finite-power time series y(t) (Lee, 1967; pp. 74–78). Therefore, if one defines the time series 2 3 6Y 7 6 N ðÞ 7 n ðt þ t Þ7 e j2pat ya ðtÞ ≜ 6 x n 7 6 4 n¼1 5
ð2:32Þ
n6¼k
then the Nth-order CTMF of x(t) can be written as Rax ðtÞ ¼ R0xðÞk ya ðtk Þ:
ð2:33Þ
Consequently, the time-averaged autocorrelation function R0xx ðtÞ is continuous in t ¼ 0 if and only if the CTMFs are continuous in t for all Nth-order cycle frequencies a and for all orders N. Therefore, all GACS time series that are not ACS exhibit time-averaged autocorrelation functions discontinuous in t ¼ 0. In particular, accounting for the property R0xx ðtÞ ¼ R0xx ð tÞ, it follows that R0xx ðtÞ contains the additive term x2 dt , where x2 is the time-averaged power of x(t). Such a property should not be confused with that examined in Hurd (1974), where the discontinuity of the time-varying autocorrelation function is considered. Note that the class of time series considered here is coincident with that in Izzo and Napolitano (2002b,c; 2003) and is wider than that analyzed in Izzo and Napolitano (1998b), where the continuity with respect to t for both Rx ðt; tÞ and At was assumed and, consequently, lag-dependent cycle frequencies az(t) and GCTMFs Rx;z ðtÞ continuous with respect to t were obtained. Accounting for Eqs. (2.11) and (2.21), the following lag-shift invariance property holds for any D 2 R: Atþ1D At :
ð2:34Þ
Consequently, for all real numbers D, from Eq. (2.25) it follows that az ðt þ 1DÞ ¼ az ðtÞ:
ð2:35Þ
Thus, it can be easily shown that Rx;z ðt þ 1DÞ ¼ Rx;z ðtÞe j2paz ðtÞD : Therefore, accounting for Eqs. (2.26) and (2.36), it results that X Rx;z ðt þ 1tÞ Rx ðt; tÞ ¼ z2W
ð2:36Þ
ð2:37Þ
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IZZO AND NAPOLITANO
In other words, the temporal moment function Rx ðt; tÞ of a GACS time series is a function of 1t þ t and hence in the following, with a little abuse of notation, it will be denoted by Rx ð1t þ tÞ: Rx ðt; tÞ Rx ð1t þ tÞ:
ð2:38Þ
Consequently, from Eq. (2.19), we will also have that ‘x ðt; tÞ ‘x ð1t þ tÞ:
ð2:39Þ
Note that Eqs. (2.35) and (2.36) suggest that the dimensions of the lagdependent cycle frequencies and of the GCTMFs can be reduced without information loss. In fact, if one defines the Nth-order reduced-dimension (moment) lag-dependent cycle frequencies a z ðt0 Þ ≜ az ðtÞjtN ¼0
ð2:40Þ
and the Nth-order reduced-dimension GCTMFs (RD-GCTMFs) Rx;z ðt0 Þ ≜ Rx;z ðtÞ ≜ jtN ¼0 ;
ð2:41Þ
it can be easily shown that the Nth-order lag-dependent cycle frequencies and the Nth-order GCTMFs can be expressed as az ðtÞ ¼ az ðt 1tN Þ ¼ a z ðu0 Þju0 ¼t0 1tN
ð2:42Þ
Rx;z ðtÞ ¼ Rx;z ðu0 Þej2pa z ðu ÞtN ju0 ¼t0 1tN ;
ð2:43Þ
and 0
respectively. For N ¼ 2, (*)1 absent, and (*)2 present, the RD-GCTMF in Eq. (2.41) is called the generalized cyclic autocorrelation function [see Eq. (1.23)]. Moreover, Eq. (2.36) generalizes to GACS time series the result derived in Gardner and Spooner (1994) for the case of ACS time series (see Eq. 46 in Gardner and Spooner, 1994). By substituting Eqs. (2.42) and (2.43) into Eq. (2.26), the following expression for the temporal moment function is obtained X 0 Rx ð1t þ tÞ ¼ Rx;z ðt0 1tN Þej2pa z ðt 1tN ÞðtþtN Þ : ð2:44Þ z2W
Let us note that for the special case of ACS time series the functions az(t) are independent of t and then there exists a one-to-one correspondence between the elements z belonging to W and the cycle frequencies a belonging to the countable set A. Moreover, for each a and z such that az(t) ¼ a, it results that Rx;z ðtÞ ¼ Rax ðtÞ;
ð2:45Þ
GENERALIZED ALMOST-CYCLOSTATIONARY SIGNALS
125
that is, for ACS time series, the generalized cyclic temporal moment functions are coincident with the cyclic temporal moment functions. Furthermore, for ACS signals, Eq. (2.44) specializes to X Rx ð1t þ tÞ ¼ Rax ðt0 1tN Þej2pa:ðtþtN Þ : ð2:46Þ a2A
Finally, we say that the GACS signal x(t) contains an ACS component if there exists at least one lag-dependent cycle frequency az(t) that assumes a constant value within a set with nonzero Lebesgue measure in RN . 2. Spectral Parameters The almost-cyclostationary time series can be characterized in the frequency domain by the Nth-order reduced-dimension CSMF Sxa ð f 0 Þ [see Eq. (1.25)], which is the (N 1)-fold Fourier transform of the reduced-dimension CTMF. Such a spectral characterization, however, is not appropriate for those GACS time series that do not belong to the class of ACS time series, that is, when the set A is not countable. In fact, in such a case, the expression of the reduced-dimension CTMFs can contain Kronecker delta functions depending on t0 [see Eq. (2.30) with tN ¼ 0]. Consequently, the reduceddimension CSMFs can be infinitesimal. In this section, a useful characterization in the spectral domain is provided for those GACS time series that are not necessarily ACS. In the following, all the Fourier transforms are assumed to exist at least in the sense of distributions (generalized functions) (Zemanian, 1987). The N-fold Fourier transform of the Nth-order GCTMF of a time series x(t) Z ⊤ S x;z ð f Þ ≜ Rx;z ðtÞe j2pf t dt ð2:47Þ RN
is called the Nth-order generalized CSMF (GCSMF). Moreover, accounting for Eq. (2.43), it can be expressed as Z
0⊤ 0 S x;z ð f Þ ¼ Rx;z ðt0 Þd a z ðt0 Þ f ⊤ 1 e j2pf t dt0 : ð2:48Þ RN 1
Let us now consider the Nth-order spectral moment function defined by ( ) N
Y ðÞn fL0 g XT t; ð Þn fn S x ð f Þ ≜ lim E ð2:49Þ T!1
n¼1
where ( )n denotes an optional minus sign to be considered only when the optional conjugation (*)n is present,
126
IZZO AND NAPOLITANO
Z XT ðt; f Þ ≜
tþT=2
t T=2
xðuÞe j2pfu du
ð2:50Þ
and L0 is the set of possible Nth-order cycle frequencies of the time series XT(t, f) when T ! 1. In Appendix B, it is shown that L 0 contains only the element a ¼ 0 and, moreover, the spectral moment function can be expressed in terms of GCSMFs by the relationship X Sxð f Þ ¼ S x;z ð f Þ: ð2:51Þ z2W
Furthermore, accounting for Eqs. (2.47) and (2.26) with t ¼ 0, from Eq. (2.51) it follows that Z ⊤ Sxð f Þ ¼ Rx ðtÞe j2pf t dt: ð2:52Þ RN
Note that, as follows immediately from Eq. (2.45), for the ACS time series the GCSMFs are coincident with the CSMFs. Moreover, in the following, it is shown that the GCSMFs of GACS time series not containing any ACS component are not impulsive, unlike those of the ACS time series. For an ACS time series x(t), there is a one-to-one correspondence between the elements z 2 Wx and the cycle frequencies a 2 Ax and, moreover, the GCSMFs are coincident with the cyclic spectral moment functions, which, can be expressed as in Gardner and Spooner (1994) [see Eq. (1.25)]. S x;z ðlÞ S ax ðlÞ ¼ Sxa ðl0 Þdðl⊤ 1 aÞ 2 3 p X X Y bm 4 5 Pxmii ðl0mi Þdðbmi l⊤ ¼ mi 1Þ P
ð2:53Þ
b⊤ 1¼a i¼1
where P is the set of distinct partitions of {1, . . . , N}, each constituted by the subsets {mi : i ¼ 1, . . . , p}, jmij is the number of elements in mi, and xmi is the jmij-dimensional vector whose components are those of x having indices in mi. bm In Eq. (2.53), Pxmii ðl0mi Þ is the jmijth-order cyclic polyspectrum at cycle frequency bmi of {x(*)n(t), n 2 mi} and b ≜ ½bm1 ; . . . ; bmp ⊤ . Thus, the GCSMFs of ACS time series are impulsive. By considering the N-fold Fourier transform of both sides of Eq. (2.36) with D ¼ t, one has S x;z ð f Þe j2pf where
⊤
1t
¼ S x;z ð f Þ ! Az ðt; f Þ f
ð2:54Þ
GENERALIZED ALMOST-CYCLOSTATIONARY SIGNALS
Az ðt; f Þ ≜
R RN
e j2paz ðtÞt e j2pf
⊤
t
dt
127 ð2:55Þ
z ðt; f 0 Þdð f ⊤ 1Þ: ¼A
¯ z(t, f 0 ) is In Eq. (2.55), !f denotes N-fold convolution with respect to f, A 0 the (N 1)-fold Fourier transform of e j2pa z ðt Þt with respect to t 0 and the last equality has been obtained accounting for Eq. (2.42). Finally, by substituting Eq. (2.55) into Eq. (2.54), one obtains that S x;z ð f Þe j2pf
⊤
1t
z ðt; f 0 Þdð f ⊤ 1Þ ¼ S x;z ð f Þ !½A f
ð2:56Þ
from which it follows that the GCSMF S x;z ð f Þ can be impulsive if and only if both the functions convolved in the right-hand-side are impulsive, that is, z ðt; f 0 Þ is impulsive in f 0 , which occurs only if if and only if the function A 0 a z ðt Þ is constant in a set of values of t0 with nonzero Lebesgue measure in RN 1 , i.e., if x(t) contains an ACS component (see Section II.C.1.). D. Generalized Cyclic Cumulants In this section, generalized cyclic cumulants in both time and frequency domains are introduced to characterize GACS time series in the wide sense (Izzo and Napolitano, 1996a, 1998b, 2002b). 1. Temporal Parameters The definition of the Nth-order temporal cumulant function of a continuoustime complex-valued time series x(t) given in [Spooner and Gardner (1994)] for ACS time series, can be extended to GACS time series:
Cx ð1t þ tÞ ≜ cum xðÞn ðt þ tn Þ; n ¼ 1; . . . ; N @N ≜ ð jÞN @o( 1 . . . @o ð2:57Þ "N # ) N X f Gt g ðÞn loge E exp j on x ðt þ tn Þ : v¼0
n¼1
It can be expressed as Cx ð1t þ tÞ ¼
X P
" ð 1Þ
p 1
ð p 1Þ!
p Y i¼1
# Rxmi ð1t þ tmi Þ
ð2:58Þ
where v ≜ ½o1 ; . . . ; oN ⊤ , P is the set of distinct partitions of {1, . . . , N}, each constituted by the subsets {mi : i ¼ 1, . . . , p}, jmij is the number of elements in mi, xmi is the jmij-dimensional vector whose components are those of x having indices in mi, and, according to definition Eq. (2.16),
128
IZZO AND NAPOLITANO
( Rxmi ð1t þ tmi Þ ¼ E
Y
fGtmi g
) x
ðÞn
n2mi
ðt þ tn Þ :
ð2:59Þ
Moreover, according to that shown in Gardner and Spooner (1994) for ACS time series, " # p X Y Rx ð1t þ tÞ ¼ Cxmi ð1t þ tmi Þ ð2:60Þ P
i¼1
where
Cxmi ð1t þ tmi Þ ≜ cum xðÞn ðt þ tn Þ; n 2 mi :
ð2:61Þ
The function Cx ð1t þ tÞ is a valid cumulant function, that is, it satisfies all the properties of the ordinary cumulant function in the stochastic process framework. For example, if the time series x(t) and x(t þ t) are asymptotically (jtj ! 1) independent in the FOT probability sense (see Section II.B.), then, for tn, n 6¼ k fixed, it results limjtk j!1 Cx ð1t þ tÞ ¼ 0. Since, for GACS time series, Gt is a countable set, from Eq. (2.57) it follows that the temporal cumulant function is almost periodic in t and can be expressed as X Cx ð1t þ tÞ ¼ Cbx ðtÞej2pbt ð2:62Þ b2Bt
where Cbx ðtÞ ≜ hCx ð1t þ tÞe j2pbt it is the Nth-order cyclic temporal cumulant function and
Bt ≜ b 2 R : Cbx ðtÞ 6¼ 0
ð2:63Þ
ð2:64Þ
N
is a countable set 8t 2 R . The support in the (b,t) space of the CTCF is constituted by a set of N-dimensional manifolds that can be described by the explicit equations b ¼ bx ðtÞ; x 2 WC
ð2:65Þ
where WC is a countable set and the functions bx(t), which are called the Nth-order (cumulant) lag-dependent cycle frequencies, are not necessarily continuous functions of t. Thus, for each t, Bt ¼ [ fb 2 R : b ¼ bx ðtÞg x2WC
and the almost-periodic function of time Cx ð1t þ tÞ can be written as
ð2:66Þ
GENERALIZED ALMOST-CYCLOSTATIONARY SIGNALS
Cx ð1t þ tÞ ¼
X
Cx;x ðtÞej2pbx ðtÞt
129 ð2:67Þ
x2WC
where the functions Cx;x ðtÞ, which are called the Nth-order generalized cyclic temporal cumulant functions (GCTCFs), are given by Cx ð1t þ tÞe j2pbx ðtÞt t ; 8t 2 Dx Cx;x ðtÞ ≜ ð2:68Þ 0; elsewhere where Dx is the domain of bx(t). In representation (2.66) we have that 8t it results that bx1(t) 6¼ bx2(t) for x1 6¼ x2. However, if more functions bx1(t), . . . , bxK(t) are defined in K (not necessarily coincident) neighborhoods, say I1, . . . , IK, of the same point t 0, only one of them is defined in t 0 and, moreover, it results that lim
Dt!0 t0 þDt2Ik
bxk ðt0 þ DtÞ ¼ b0 ; k ¼ 1; . . . ; K;
ð2:69Þ
then it is convenient to consider in Eq. (2.66) bx1(t0) ¼ . . . ¼ bxK (t0) ¼ b0 and to put Cx;xk ðt0 Þ ≜
lim
Dt!0 t0 þDt2Ik
Cx;xk ðt 0 þ DtÞ; k ¼ 1; . . . ; K:
ð2:70Þ
The set of points t0 such that Eq. (2.69) holds for some xk 2 WC is denoted by D0x and assumed to be countable. With convention Eq. (2.70), it can be shown that CTCFs and GCTCFs are related by the following relationship: X Cbx ðtÞ ¼ Cx;x ðtÞdb bx ðtÞ : ð2:71Þ x2Wc
Moreover, 8t 2 Dx D0x , Cx;x ðtÞ ¼ Cbx ðtÞjb¼bx ðtÞ :
ð2:72Þ
For any real number D, accounting for Eq. (2.11), from Eq. (2.57) and Eq. (2.62) it follows that Bt þ 1D Bt
ð2:73Þ
bx ðt þ 1DÞ ¼ bx ðtÞ
ð2:74Þ
and hence from which it results that
130
IZZO AND NAPOLITANO
Cx;x ðt þ 1DÞ ¼ Cx;x ðtÞej2pbx ðtÞD :
ð2:75Þ
Moreover, if one introduces the Nth-order reduced-dimension GCTCF (RD-GCTCF) Cx;x ðt 0 Þ ≜ Cx;x ðtÞjtN ¼0 one has
ð2:76Þ
Cx;x ðtÞ ¼ Cx;x ðu0 Þej2pb x ðu ÞtN ju0 ¼t0 1tN
ð2:77Þ
ðt0 Þ ≜ b ðtÞj b x x tN ¼0
ð2:78Þ
where
0
are the reduced-dimension cumulant lag-dependent cycle frequencies. Moreover, it can be easily shown that ðu0 Þj 0 0 bx ðtÞ ¼ bx ðt 1tN Þ ¼ b x u ¼t 1tN and, by substituting Eq. (2.77) and Eq. (2.79) into Eq. (2.67) X 0 Cx ð1t þ tÞ ¼ Cx;x ðt0 1tN Þej2pb x ðt 1tN ÞðtþtN Þ :
ð2:79Þ
ð2:80Þ
x2Wc
The Nth-order GCTCF can be expressed in terms of GCTMFs of order less than or equal to N by substituting Eq. (2.58) into Eq. (2.68) and accounting for Eq. (2.30): " # p X XY p 1 Cx;x ðtÞ ¼ ð 1Þ ðp 1Þ! Rxmi ;zi ðtmi Þdbx ðtÞ az ðtÞ⊤ 1 ð2:81Þ z2W i¼1
P
where zi 2 Wmi ði ¼ 1; . . . ; pÞ; z ≜ ½z1 ; . . . ; zp ⊤ ; W ≜ ½Wm1 ; . . . ; Wmp ⊤ ; az ðtÞ ≜ ½az1 ðtm1 Þ; . . . ; azp ðt mp Þ⊤ , and, according to Eq. (2.27), 8* + > < Y ðÞn j2pazi ðtmi Þt x ðt þ tn Þe ; t mi 2 D z i ð2:82Þ Rxmi ;zi ðtmi Þ ¼ n2mi > t : 0; elsewhere where Dzi is set in which azi(tmi) is defined. Furthermore, the Nth-order GCTMF can be expressed in terms of GCTCFs of order less than or equal to N, by substituting Eq. (2.60) into Eq. (2.27), in which the lag product has been replaced by Rx ð1t þ tÞ: " # p X XY Cxmi ;xi ðt mi Þdaz ðtÞ bj ðtÞ⊤ 1 Rx;z ðtÞ ¼ ð2:83Þ P
j2W c i¼1
where WC ≜ [WCm1 , . . . ,WCmp ]⊤ and Eq. (2.71) has been accounted for.
GENERALIZED ALMOST-CYCLOSTATIONARY SIGNALS
131
In Gardner and Spooner (1994), it is shown that for ACS time series such that x(t) and x(t þ t) are asymptotically (jtj ! 1) independent, the reduceddimension cyclic temporal cumulant function Cxb ðt 0 Þ is summable whereas, in general, the reduced-dimension cyclic temporal moment function Rax ðt0 Þ is not. In the following, a similar result will be found for GACS time series. From Eq. (2.83) with tN ¼ 0, accounting for Eq. (2.77) and assuming that each partition is ordered so that mp always contains N as its last element, it results that " Y X X p 1 0 0 Rx;z ðt Þ ¼ Cx;z ðt Þ þ P p6¼1
j2W C i¼1
ðt 0 1tr Þtr Þ Cxmi ;xi ðt 0mi 1tri Þ exp ð j2pb xi mi i i :Cxm ;xp ðt0m Þd 0 Pp 1 0 p
p
a z ðt Þ
i¼1
ð2:84Þ
bxi ðtmi Þ b xp ðtmp Þ
where, for each partition, tri is the last element in mi. Therefore, the RDGCTMF is the sum of two contributions: the RD-GCTCF, which, under mild conditions, converges to zero as kt0 k ! 1 and the remaining term, which is not convergent as kt0 k ! 1. In fact, under the assumption that the time series x(t) and x(t þ tn) (n ¼ 1,. . ., N 1) are asymptotically (jtnj ! 1) independent in the FOT probability sense, accounting for the well-known independence property of cumulants, it results that Cx,z(t0 ) ! 0 as kt 0 k ! 1. Moreover, for each fixed partition, when kt 0 k ! 1 with the constraints tri ! 1, tm0 i ¼ 1tri (i ¼ 1, . . . p 1) and tm0 p finite, one has lim
p 1 Y
kt0 k!1 i¼1 t0mi ¼1tri ; t 0mp finite
Cxmi xi ; ðt0mi 1tri Þ
ðt0 1tr Þtr Cx x ðt0 Þ :exp j2pb x i mi i i mp; i mp
¼
ð2:85Þ
p 1
Y ð0Þtr ; Cxmi ;ji ð0ÞCmp ;jp ðt0mp Þ lim exp j2pb xi i tri !1
i¼1
which is not convergent. Consequently, for each t it results that Rx;z ðt0 Þ 0
) C ðt0 Þ 0 (/ x;z
ð2:86Þ
that is, fbx ðtÞgx2Wc faz ðtÞgz2W
ð2:87Þ
132
IZZO AND NAPOLITANO
and hence Bt At :
ð2:88Þ
In other words, the set of the moment lag-dependent cycle frequencies includes, as a subset, the one of the cumulant lag-dependent cycle frequencies. Finally, let us note that if two time series x1(t) and x2(t) are statistically independent in the FOT probability sense (see Section II.B.), then, reasoning as in the stochastic process framework, for the time series y(t) ¼ x1(t) þ x2(t) it can be shown that Cy ð1t þ tÞ ¼ Cx1 ð1t þ tÞ þ Cx2 ð1t þ tÞ
ð2:89Þ
Cy;x ðtÞ ¼ Cx1 ;x ðtÞ þ Cx2 ;x ðtÞ:
ð2:90Þ
and hence
2. Spectral Parameters In this section, all the Fourier transforms are assumed to exist at least in the sense of distributions (generalized functions) (Zemanian, 1987). The N-fold Fourier transform of the Nth-order GCTCF of a time series x(t) Z ⊤ P x;x ð f Þ ≜ Cx;x ðtÞe j2pf t dt ð2:91Þ RN
is called the Nth-order generalized cyclic spectral cumulant function (GCSCF) which, accounting for Eq. (2.77), can be expressed as Z
0 ðt0 Þ f ⊤ 1 e j2pf ⊤ dt0 : Cx;x ðt 0 Þd b ð2:92Þ P x;x ð f Þ ¼ x RN 1
Moreover, the (N 1)-fold Fourier transform of Cx,x(t0 ) is called the Nthorder generalized cyclic polyspectrum, which, for ACS time series, is coincident with the cyclic polyspectrum. Note that, under the assumption of asymptotic independence for the time series x(t), if there exists an e > 0 such that jCx,x(t 0 )j ¼ o(kt0 k Nþ1 e) as kt 0 k ! 1, then the RD-GCTCF is absolutely integrable and, therefore, Fourier transformable in the ordinary sense. Let us now consider the Nth-order spectral cumulant function n o ðÞ P x ð f Þ ≜ lim cum XT n ðt; ð Þn fn Þ; n ¼ 1; . . . ; N : ð2:93Þ T!1
133
GENERALIZED ALMOST-CYCLOSTATIONARY SIGNALS
It can be written as
n o ðÞ P x ðf Þ ¼ lim cum0 XT n ðt; ð Þn fn Þ; n ¼ 1; . . . ; N T!1 " * +# p X Y ðÞ Y p 1 n ¼ lim ð 1Þ ðp 1Þ! XT ðt; ð Þn fn Þ T!1
¼
X P
"
i¼1
P
ð 1Þp 1 ðp 1Þ!
n2mi
p Y S xmi ð f mi Þ
#
t
ð2:94Þ
i¼1
where in the first equality cum0 denotes the cumulant evaluated by substituting E{Gt}{} with hit in definition Eq. (2.57). By considering the N-dimensional Fourier transform of the right-hand side of Eq. (2.58) with t ¼ 0 and comparing the result with Eq. (2.94), one obtains that Z ⊤ P x ðf Þ ¼ Cx ðtÞe j2pf t dt: ð2:95Þ RN
Moreover, taking into account the N-dimensional Fourier transform of the right-hand side of Eq. (2.67) with t ¼ 0, the Nth-order spectral cumulant function can be expressed in terms of Nth-order GCSCFs by the relationship X Pxð f Þ ¼ P x;x ð f Þ: ð2:96Þ x2Wc
Finally, let us note that for the ACS time series the GCSCFs are coincident with the cyclic spectral cumulant functions. E. Estimation of the Generalized Cyclic Statistics Appropriate estimators for the cyclic statistics of ACS time series have been proposed in Gardner (1988a), Spooner and Gardner (1994) within the FOT probability framework and in Dandawate´ and Giannakis (1994, 1995), Dehay and Hurd (1994), Hurd (1991), and Hurd and Les´kow (1992) in the classical stochastic process framework. In this section, the problem of estimating the higher-order (generalized) cyclic statistics of those GACS time series not belonging to the class of ACS time series is briefly considered in the FOT probability framework (see also Appendix A). Results on the estimation problem in the stochastic process framework are presented in Napolitano (2004). Assuming that t 2 Dz and t 2 = Dz0 for any z0 6¼ z (see Eq. (2.27)) and the lag-dependent cycle frequency az (t) is known, the function
134
IZZO AND NAPOLITANO
RxT ;z ðt0 ; tÞ ≜
1 T
Z Y N R n¼1
rect
u t 0 xðÞn ðu þ tn Þe j2paz ðtÞu du T
ð2:97Þ
where rect(t) ¼ 1 for jtj 1/2 and rect(t) ¼ 0 otherwise and T is the observation time, turns out to be an estimate of the GCTMF, as follows immediately observing that lim RxT ;z ðt0 ; tÞ ¼ Rx;z ðtÞ
T!1
ð2:98Þ
where the convergence is in the temporal mean-square sense. In Appendix A, this kind of convergence is discussed and the definition of both bias and variance of estimators in the nonstochastic approach is given. For t 2 Dx and t 2 = Dx0 for any x0 6¼ x [see Eq. (2.68)], the function " # p Y X X CxT ;j ðt0 ; tÞ ≜ ð 1Þp 1 ðp 1Þ! RxTmi ;zi ðt0 ; t mi Þdbx ðtÞ az ðtÞ⊤ 1 z2W i¼1
P
ð2:99Þ converges to the theoretical GCTCF as T ! 1: lim CxT ;x ðt0 ; tÞ ¼ Cx;x ðtÞ:
T!1
ð2:100Þ
When the set of the lag-dependent cycle frequencies {az(t)}z2W is unknown, accounting for Eq. (2.24), it can be estimated starting from the support supp fRaxT ðt0 ; tÞg ≜ clfða; tÞ 2 R RN : RaxT ðt0 ; tÞ 6¼ 0g
ð2:101Þ
where RaxT ðt0 ; tÞ ≜
1 T
Z Y N R n¼1
rect
u t 0 xðÞn ðu þ tn Þe j2pau du T
ð2:102Þ
is the estimate of the Nth-order CTMF given in Spooner and Gardner (1994). In fact, in the limit for T ! 1, the support Eq. (2.101) has zero measure in RNþ1 and is constituted by a countable set of manifolds described by the implicit equations a ¼ az (t), z 2 W. Moreover, estimates az,T(t) of the functions az(t) are, for each fixed value of t, the estimated frequencies of the almost-periodic component contained in the lag product, which can be obtained, for example, by exploting the algorithm proposed in Dehay and Hurd (1996). Note that estimated lag-dependent cycle frequencies can be
GENERALIZED ALMOST-CYCLOSTATIONARY SIGNALS
135
substituted to the true values in the estimator Eq. (2.97) of the GCTMF, provided that the maximum absolute error in the estimate is much smaller than the cycle frequency resolution 1/T, that is, max jaz;T ðtÞ az ðtÞj % t2R
N
1 : T
ð2:103Þ
In Napolitano (2004) it is shown that, in the stochastic process framework, for N ¼ 2 and t2 ¼ 0 the function Eq. (2.102) turns out to be a meansquare consistent estimator of the (conjugate) cyclic autocorrelation function as a function of (a, t1), provided that some mixing conditions expressed in terms of summability of the second and fourth-order cumulants of x(t) are satisfied. Such conditions generally hold for stochastic processes with finite or approximately finite memory. As regards the parameters in the frequency domain, the function " # Z N ⊤ 1 t0 þZ=2 Y ðÞn S xT ;z ðt0 ; f Þz ≜ XT ðt; ð Þn fn Þ ! Az ðt; f Þ ej2pf 1t dt ð2:104Þ Z t0 Z=2 n¼1 f where Az(t, f ) is defined in Eq. (2.55), is an estimator for the GCSMF, since it can be shown that lim lim S xT ;z ðt0 ; f Þz ¼ S x;z ð f Þ:
T!1 Z!1
ð2:105Þ
F. Examples of GACS Signals In this section, two examples of GACS signals are presented (Izzo and Napolitano, 1998b). The first one is the chirp signal, which is widely considered in various application fields (e.g., physics, sonar, radar, communications). The second one, is a nonuniformly sampled signal. Further examples are given in Section III. 1. Chirp Signal Let us consider the chirp signal xðtÞ ≜ expð jpct2 Þ
ð2:106Þ
where the nonzero real parameter c is the chirp rate. The cyclic temporal moment function at the cycle frequency a can be obtained by substituting Eq. (2.106) into Eq. (2.14):
136
IZZO AND NAPOLITANO
Rax ðtÞ ¼ exp jpc1ð Þ⊤ tð2Þ da c1ð Þ⊤ t d1ð Þ⊤ 1
ð2:107Þ
where 1ð Þ ≜ ½ð Þ1 1; . . . ,ð ÞN 1⊤ and tð2Þ ≜ ½t21 ; . . . ; t2N ⊤ : Equation (2.107) shows that the CTMF is nonzero only when the number of conjugated and unconjugated entries of the vector x is the same and, moreover, only on the hyperplane in the (a, t) space defined by a ¼ c1ð Þ⊤ t:
ð2:108Þ
Therefore, since the set of t’s such that Eq. (2.108) holds has zero Lebesgue measure in RN, the CSMFs are infinitesimal (see Section II.C.2). A spectral characterization for the chirp signal can be obtained by utilizing the introduced generalized cyclic spectral moment functions. In fact, by comparing Eq. (2.107) with Eq. (2.30), it results that the set W contains just one element and, moreover,
and
az ðtÞ ¼ c1ð Þ⊤ t
ð2:109Þ
Rx;z ðtÞ ¼ exp jpc1ð Þ⊤ tð2Þ d1ð Þ⊤ 1
ð2:110Þ
which is nonzero only when N is even and the number of conjugated and unconjugated entries of the vector x is the same. Therefore, the GCSMF is given by p
1 S x;z ð f Þ ¼ N=2 exp j 1ð Þ⊤ f ð2Þ d1ð Þ⊤ 1 ð2:111Þ c jcj where f ð2Þ ≜ ½ f12 ; . . .; fN2 ⊤ : Figure 3a shows the real part and Fig. 3b the support in the (a, t) plane of the second-order RD-CTMF Rax ðtÞ with (*)1 absent and (*)2 present (i.e., the cyclic autocorrelation function), for a chirp signal with c ¼ 0:002=Ts2 , where Ts is the sampling period, as estimated by 512 samples (for a discussion on the aliasing issue, see Section IV.C). The linear behavior of the reduced-dimension lag-dependent cycle frequency a z ðtÞ ¼ ct is quite evident. The function obtained by setting a ¼ ct is just the real part of the second-order RD-GCTMF Rx,z(t), that is, the generalized cyclic autocorrelation function. Note that in Section IV.D it is explained that the GACS nature of a continuous-time signal can only be conjectured starting from the discrete-time signal constituted by its samples. As regards the GCTCFs, accounting for Eq. (2.110) and Eq. (2.81), it results that
GENERALIZED ALMOST-CYCLOSTATIONARY SIGNALS
137
FIGURE 3. (a) Graph of real part and (b) support in the (a, t) plane of the cyclic autocorrelation function Raxx ðtÞ for a chirp signal with chirp rate c ¼ 0:002=Ts2 .
Cbx ðtÞ ¼
X P
" ð 1Þp 1 ðp 1Þ!
p XY z2W i¼1
ð Þ⊤ ð2Þ P p exp jpc1mi tmi d1ð Þ⊤ 1 db c ð Þ⊤ 1 t mi m m i¼1 mi
i i ð Þ⊤ ð2Þ ¼ exp jpc1 t db c1ð Þ⊤ t " # p X Y p 1 : ð 1Þ ðp 1Þ! d1ð Þ⊤ 1 P
i¼1
mi
mi
ð2:112Þ
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where 1mi is the jmij-dimensional vector [1, . . . , 1]⊤. Therefore, the set WC contains just one element and, moreover, bx ðtÞ ¼ c1ð Þ⊤ t and
Cx;x ðtÞ ¼ exp jpc1ð Þ⊤ tð2Þ
X
"
P
ð2:113Þ
p Y ð Þp 1 ðp 1Þ! d1ð Þ⊤ 1 i¼1
mi
mi
# :
ð2:114Þ
Let us note that for the chirp signal the RD-GCTCFs do not converge to zero as kt0 k ! 1. In fact, the time series x(t) and x(t þ tn) are not asymptotically independent: xðt þ tn Þ ¼ xðtÞdðt; tn Þ
ð2:115Þ
t2n Þ
where d(t, tn) is the sinewave exp½ jpcð2ttn þ which, for any tn, is a deterministic signal in the FOT probability sense. Finally, it is noteworthy that the chirp signal, as each signal which is random in the FOT probability framework and deterministic in the stochastic process framework, is not ergodic for the cumulants (and hence for the GCTCFs and GCSCFs), since all stochastic cumulants (N 2) are identically zero. 2. Nonuniformly Sampled Signal Uniformly sampled signals can be modeled as the product of a continuoustime signal by a train of impulses with constant period. In such a case, when the continuous-time signal is strictly bandlimited and ACS, its higher-order cyclostationarity features can be easily determined provided that the sampling rate is suYciently high (Napolitano, 1995; Izzo and Napolitano, 1996c). A more realistic model considers nonuniformly spaced impulse trains, that is, with a ‘‘period’’ not constant with time. Let us consider the nonuniformly sampled signal xðtÞ ≜ wðtÞsðtÞ;
ð2:116Þ
where sðtÞ ≜
þ1 X
dðt kTp ðtÞÞ
ð2:117Þ
k¼ 1
is an impulse train whose ‘‘period’’ Tp(t) is a slowly varying function of t and w(t) is a finite-power time series exhibiting Nth-order wide-sense stationarity (WSS) (i.e., such that Raw ðtÞ ≢ 0 only for a ¼ 0).
GENERALIZED ALMOST-CYCLOSTATIONARY SIGNALS
139
According to definition Eq. (2.14), by using the Poisson’s sum formula þ1 þ1 X X 1 dðt kTp ðuÞÞ ¼ e j2pmt=Tp ðuÞ ð2:118Þ T ðuÞ p m¼ 1 k¼ 1 specialized for u ¼ t, it results that * " !#+ N N Y X X a Rs ðtÞ ¼ fp ðt þ tn Þ exp j2p ð Þn mn fp ðt þ tn Þ : ðt þ tn Þ at n¼1
m2ZN
n¼1
ð2:119Þ where fp(t) ≜ 1/Tp(t) and m ≜ [m1, . . . , mN]⊤. In general, from this equation the analytical expression of the lag-dependent cycle frequencies cannot be easily derived. However, the lag-dependent cycle frequencies can be estimated following the approach presented in Section II.E. Moreover, according to the WSS exhibited by w(t), the lag-dependent cycle frequencies of x(t) are the same as those of s(t) and, furthermore, the GCTMFs of x(t) are given by [see Eq. (3.136)] Rx;z ðtÞ ¼ R0w ðtÞRs;z ðtÞ:
ð2:120Þ
Note that, when the maximum variation of the lag-dependent cycle frequencies is not greater than the cycle frequency resolution 1/T (with T the datarecord length), then the sampling ‘‘period’’ can be considered practically constant and hence the sampled signal can be modeled as ACS rather than GACS. Figure 4a shows the magnitude and Figure 4b the support in the (a, t) plane of the cyclic autocorrelation function Raxx ðtÞ of a signal x(t) obtained by uniformly sampling a signal w(t) exhibiting WSS and with power spectral 0 ð f Þ ¼ ð1 þ f 2 =B2 Þ 4 with B ¼ 0.0025/T . The period of the density Sww s impulse train has been fixed at Tp0 ¼ 4Ts and the estimate has been obtained on the basis of 128 samples. The almost-cyclostationary behavior of x(t) is evident since the support of Raxx ðtÞ is contained in lines parallel to the t axis. Figure 5a shows the magnitude and Figure 5b the support of Raxx ðtÞ with reference to a nonuniformly spaced impulse train with Tp(t) ¼ Tp0/(1 þ 0.5 cos(2p f0t)), where f0 ¼ 0.0005/Ts, and 128 samples have been processed. In such a case, the dependence of the cycle frequencies on t cannot be appreciated. However, when the sample size is increased to 16384, the generalized cyclostationary nature of the sampled signal x(t) becomes evident (see Figure 6). Finally, it is worthwhile to underline that the continuous-time nonuniformly sampled signal x(t) is GACS. On the contrary, in Section IV.C it is shown that the discrete-time signal constituted by the samples of a continuous-time GACS signal is always ACS.
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FIGURE 4. (a) Graph of magnitude and (b) support in the (a, t) plane of the cyclic autocorrelation function Raxx ðtÞ for a signal obtained by sampling a colored signal exhibiting WSS by an ideal impulse train with period Tp0 ¼ 4Ts . A sample size of 128 samples has been used.
G. Summary In this section, the higher-order characterization of the generalized almostcyclostationary time series has been addressed in the nonstochastic framework. For such a class of time series, multivariate statistical functions are almost-periodic functions of time whose Fourier series expansions can exhibit coeYcients and frequencies depending on the lag shifts of the time
GENERALIZED ALMOST-CYCLOSTATIONARY SIGNALS
141
FIGURE 5. (a) Graph of magnitude and (b) support in the (a, t) plane of the cyclic autocorrelation function Raxx ðtÞ for a signal obtained by sampling a colored signal exhibiting WSS by an impulse train with ‘‘period’’ Tp ðtÞ ¼ Tp0 =ð1 þ 0:5 cosð2pf0 tÞÞ, where Tp0 ¼ 4Ts and f0 ¼ 0:0005=Ts . A sample size of 128 samples has been used.
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IZZO AND NAPOLITANO
FIGURE 6. (a) Graph of magnitude and (b) support in the (a, t) plane of the cyclic autocorrelation function Raxx ðtÞ for a signal obtained by sampling a colored signal exhibiting WSS by an impulse train with ‘‘period’’ Tp ðtÞ ¼ Tp0 =ð1 þ 0:5 cosð2pf0 tÞÞ, where Tp0 ¼ 4Ts and f0 ¼ 0:0005=Ts . A sample size of 16384 samples has been utilized.
GENERALIZED ALMOST-CYCLOSTATIONARY SIGNALS
143
series. Moreover, the union over all the lag shifts of the lag-dependent frequency sets is not necessarily countable and the time-averaged autocorrelation function can be discontinuous in the origin. Almost-cyclostationary time series are the subclass of GACS time series for which the frequencies do not depend on the lag shifts and the union of the previously-mentioned sets is countable. The GACS time series have been characterized in the strict sense and some known results for the subclass of ACS time series have been extended to the GACS case. Generalized cyclic moments in both time and frequency domains have been defined and their properties have been discussed. Moreover, generalized cyclic temporal and spectral cumulants have also been introduced. For ACS time series, the generalized cyclic statistics have been shown to reduce to the corresponding (nongeneralized) cyclic statistics.
III. LINEAR TIME-VARIANT TRANSFORMATIONS
OF
GACS SIGNALS
A. Introduction In several problems of interest in signal processing and communications, stochastic processes are processed by linear time-variant systems. Depending on the nature of the system, it can be modeled as deterministic or stochastic. Deterministic systems operate only on the time variable of the input stochastic process and treat as a constant the variable belonging to the outcome space (Papoulis, 1991, paragraph 10–2). Thus, two identical sample paths of the input give rise to two identical sample paths of the output. Hence, deterministic systems transform deterministic input signals into deterministic output signals. On the contrary, stochastic systems operate on both time and variable belonging to the outcome space. Thus, in general, they transform deterministic signals into stochastic processes. Many communication channels can be physically modeled as stochastic LTV systems. In Middleton (1967a), a statistical theory of reverberation and related first-order scattered fields is developed by describing the scattering mechanism by a LTV filter response; in Middleton (1967b), the second-order statistics of the reverberation (nonstationary) processes are determined in detail. In Bello (1963), a statistical characterization of stochastic LTV channels is carried out in terms of correlation functions of system functions defined in both time and frequency domains. In Tsao (1984), the problem of LTV filtering is addressed by stochastic diVerential equations.
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In all these classical papers, the stochastic LTV systems are described by an ensemble of impulse-response functions. Thus, the system functions are defined through ensemble averages of quantities related to the impulseresponse function, to its mono- or bi-dimensional Fourier transform, or products of time-and/or frequency-shifted versions of them. For such systems, even if the input stochastic process possesses amenable ergodicity properties, the output process, in general, is not ergodic (Gardner, 1990), and, therefore, neither the output process statistical functions nor the system functions can be estimated by a single sample path of the input and output stochastic processes. In this section, the problem of LTV filtering is addressed in the FOT probability framework. Thus, such a problem is treated in a new perspective with respect to the classical one of the stochastic process framework. This new approach is motivated by the necessity of characterizing in a useful way the output signal of randomly fluctuating linear channels which, in the stochastic process framework, in general, provide nonergodic outputs. On the contrary, in the FOT probability approach adopted here, measurements based on a finite data-record length asymptotically approach, as the datarecord length approaches infinity, the theoretical statistical functions, provided that the system impulse-response function and the input signal are suYciently regular to assure the considered infinite-time averages exist in the temporal mean-square sense. The (linear and nonlinear) systems are classified as deterministic or random in the FOT probability framework. Deterministic systems are those that map deterministic (i.e., constant, periodic, or almost-periodic) inputs into deterministic outputs. They include the linear almost-periodically timevariant (LAPTV) systems, as well as the systems that perform a time scale changing. Random systems are all transformations that cannot be classified as deterministic. The random linear systems include chirp modulators, modulators whose carrier is a pseudo-noise sequence, channels introducing timevarying delays, and systems that perform a time windowing. Note that throughout this section ‘‘random’’ is not synonymous with ‘‘stochastic.’’ In fact, the adjective stochastic is adopted, as usual, when an ensemble of realizations or sample paths exist, whereas the adjective random is referred to a single function of time, namely a signal or a system impulse-response function. Therefore, to avoid ambiguities, when necessary, deterministic and random systems in the FOT probability sense will be referred to as FOT deterministic and FOT random systems, respectively. The LTV systems are decomposed into the parallel connection of their FOT deterministic and purely random components and the concept of expectation (in the FOT sense) of the impulse-response function is introduced. The
GENERALIZED ALMOST-CYCLOSTATIONARY SIGNALS
145
characterization in terms of higher-order statistics of the LTV systems is performed by analyzing the way in which periodicities present in the input Nth-order lag product are transformed into periodicities contained in the output Nth-order lag product. Such a characterization turns out to be particularly useful for GACS signals, since their Nth-order lag product can be decomposed into an almost-periodic component, which is called the Nthorder temporal moment function, and a residual term (see Section II.C.1). The choice of operating in the GACS context is useful in practice since ACS signals processed by LTV systems can generate GACS signals that are not necessarily ACS. Moreover, communication signals with parameters, such as the baud rate or the carrier frequency, slowly varying with time can be modeled as GACS and can be thought as obtained by linear (not periodically) time-variant transformations of ACS signals. It is shown that the Nth-order lag product of the system impulse-response function can be decomposed into two terms. The former, which will be referred to as the system temporal moment function, is by definition the kernel of the linear (with respect to the lag product) operator that transforms the almost-periodic component of the input lag product (i.e., the input temporal moment function), into the almost-periodic component of the output lag product. The latter is the kernel of the operator that transforms any almost-periodic component of the input lag product into a component of the output lag product not containing any almost-periodic component. The proposed decomposition of the Nth-order lag product of the system impulse-response function allows one to easily describe the behavior of the LTV systems in terms of input/output relations involving generalized cyclic statistics. Therefore, input/output relations in terms of generalized cyclic temporal and spectral moment functions are derived with reference to both FOT deterministic and FOT random linear systems. Moreover, for FOT deterministic systems, the input/output relations in terms of moment functions and cumulant functions are shown to be the same. The lack of ergodicity of the output signal of a stochastic LTV system in the stochastic process framework is discussed with reference to moment and cumulant functions. Then, the countability of the set of the output cycle frequencies is studied. Such a problem turns out to be relevant in the FOT probability approach since, in this framework, the almost-periodic component extraction operator is the analogous of the stochastic expectation operator in the classical stochastic process framework. Thus, the output-signal FOT moments and cumulants exist only if the output lag product contains finitestrength additive sinewave components, that is, a countable set of output cycle frequencies exists. If such a set is empty or more than countable, we have identically zero or divergent, respectively, generalized cyclic statistics.
146
IZZO AND NAPOLITANO
The countability of the set of the output cycle frequencies is studied with reference to the general case of LTV systems for both ACS and GACS not containing any ACS component input signals. Moreover, the special case of FOT deterministic LTV systems is analyzed in more detail. Thus, a characterization of the output signal is provided. Furthermore, several special cases of LTV systems are analyzed in detail. Specifically, LAPTV systems, systems performing product modulation, and several kinds of Doppler channels are considered. These results allow to properly characterize the output of several classes of LTV channels of interest in communications in terms of statistical functions that can be estimated by a single data record. Such a characterization is important as a first step to properly identify or equalize a channel. In the considered examples, it is shown that in practical situations diVerent system or signal models should be adopted depending on the data-record length. For example, for large data-record length possible time variations of timing parameters of the signals must be taken into account, making the GACS model more appropriate than the ACS model. Moreover, it is shown that GACS signals not containing any ACS component, filtered by a LAPTV system, give rise to signals with identically zero (generalized) cyclic statistics. Then, filtered versions of such signals exhibit estimated (generalized) cyclic statistics that are asymptotically zero when the data-record length approaches infinity. Multipath Doppler channels are also analyzed. They generate at the output several replicas of the input signal, each characterized by a diVerent complex amplitude, delay, time scaling factor, and frequency shift. Such channels are encountered very often in radar and mobile communication problems. In radar applications, they account for the presence of multiple slow fluctuating point scatterers. In mobile communications, such models are appropriate when the mobile station receives the signal transmitted by the base station through multiple trajectories, each characterized by a diVerent complex attenuation, length, and radial speed of the mobile station with respect to the base station. It is shown that multipath Doppler channels can be modelled as LAPTV systems or, more generally, as FOT deterministic LTV systems, depending on the length of the data-record adopted to observe the input and output signals and the bandwidth of the input signal. B. FOT Deterministic and Random Linear Systems In this section, systems are classified as deterministic or random in the FOT probability sense. Then, a characterization is provided for both deterministic and random linear systems (Izzo and Napolitano, 1999, 2002b).
GENERALIZED ALMOST-CYCLOSTATIONARY SIGNALS
147
1. FOT Deterministic and Random Systems In the FOT probability framework, a deterministic system is defined as a possibly complex (and not necessarily linear) system that for every deterministic (i.e., almost periodic) input time series delivers a deterministic output time series (Izzo and Napolitano, 1995a). Systems that are not FOT deterministic will be referred to as FOT random systems. Therefore, for the input time series xðtÞ ¼ e j2plt
ð3:1Þ
a FOT deterministic system delivers the output almost-periodic time series X yðtÞ ¼ G0 e j2pt ð3:2Þ 2El
where, for each fixed l, El ≜ f1 ðlÞ; . . . ; n ðlÞ; . . .g is the countable set of the output frequencies (depending on the input frequency l) and the G0 are complex coeYcients. The set of points ð; lÞ 2 El R such that G0 6¼ 0 is constituted by a set of not necessarily continuous curves defined by the implicit equations Fs0 ð; lÞ ¼ 0;
s0 2 O0
ð3:3Þ
where O0 is a countable set. That is, it results that clfð; lÞ 2 El R : G0 6¼ 0g 0 ¼ cl [ 0 fð; lÞ 2 R R : Fs0 ð; lÞ ¼ 0; G 6¼ 0g s0 2O
ð3:4Þ
¼ cl [ fð; lÞ 2 R D’s ðlÞ : ¼ ’s ðlÞg s2O
where O is a countable set and, in the last equality, each curve described by the implicit equation Fs0 ð; lÞ ¼ 0 has been decomposed into a countable set of curves, each described by the explicit equation ¼ ’s ðlÞ, where ’s() can always be chosen among the monotonic real functions and such that ’s ðlÞ 6¼ ’s0 ðlÞ for s 6¼ s0 , and D’s is the domain of ’s(). Therefore, defined the functions 0 G j¼’s ðlÞ ; l 2 D’s Gs ðlÞ ≜ ð3:5Þ 0; elsewhere the output almost-periodic time series Eq. (3.2) can be written as X Gs ðlÞe j2p’s ðlÞt : yðtÞ ¼
ð3:6Þ
s2O
Note that, for a given system, the functions ’s() and Gs () are not univocally determined, since, in general, for each curve described by an implicit
148
IZZO AND NAPOLITANO
equation, several decompositions into curves described by explicit equations are possible. Moreover, if more functions ’s1 ðlÞ; . . . ; ’sK ðlÞ are defined in K (not necessarily coincident) neighborhoods of the same point l0, all have the same limit, say ’0, for l ! l0, and only one is defined in l0, then it is convenient to assume all the functions defined in l0 with ’s1 ðl0 Þ ¼ ¼ ’sK ðl0 Þ ¼ ’0 and, consequently, to define Gsi ðl0 Þ ≜ lim Gsi ðlÞ; l!l0
i ¼ 1; . . . ; K
ð3:7Þ
where, for each i, the limit is made with l ranging in the neighborhood of l0 where the function ’si(l) is defined. Note that a mild regularity assumption on the system is that a small change in the input frequency l gives rise to small changes in the output frequencies n ðlÞ 2 El . Thus, the n(l) are continuous functions (not necessarily invertible) of the input frequency l, that is, the set El is continuous with respect to l. In such a case, the functions Fs0 (, l) and, hence, the functions ’s(l) are continuous in their domain. Finally, let us observe that the FOT deterministic systems are those called ‘‘stationary’’ by Claasen and Mecklenbr€auker in Claasen and Mecklenbr€ auker (1982). 2. FOT Deterministic Linear Systems Let us now consider a linear time-variant system. Its input/output relationship is given by Z yðtÞ ¼ hðt; uÞxðuÞ du ð3:8Þ R
where h(t, u) is the system impulse-response function. By Fourier transforming both sides of Eq. (3.8), one obtains the input/output relationship in the frequency domain Z Yð f Þ ≜ yðtÞe j2pft dt R Z ð3:9Þ ¼ Hð f ; lÞX ðlÞ dl R
where the transmission function H( f, l) [10] (also referred to as Zadeh’s bifrequency function (Bello, 1963)) is the double Fourier transform of the impulse-response function: Z Hð f ; lÞ ≜ hðt; uÞe j2pð ft luÞ dt du: ð3:10Þ R2
GENERALIZED ALMOST-CYCLOSTATIONARY SIGNALS
149
In Eqs. (3.9) and (3.10), and in the following, the Fourier transforms areassumed to exist in the sense of distributions (generalized functions) (Zemanian, 1987). Since the transmission function Eq. (3.10) represents the Fourier transform of the output y(t) corresponding to the input Eq. (3.1), for FOT deterministic linear systems the transmission function can be obtained by Fourier transforming the right-hand side of Eq. (3.6): X Hð f ; lÞ ¼ Gs ðlÞdð f ’s ðlÞÞ ð3:11aÞ s2O
¼
X
Hs ð f Þdðl cs ð f ÞÞ
ð3:11bÞ
s2O
where the functions cs(), referred to as the frequency mapping functions, are the inverse functions of ’s(), and the functions Gs() and Hs(), accounting for the fact that dð f ’s ðlÞÞ ¼ jc_ s ð f Þjdðl c_ s ð f ÞÞ and dðl cs ð f ÞÞ ¼ j’_ s ðlÞjdð f ’s ð f ÞÞ [Zemanian, 1987; Section 1.7], are linked by the relationships Hs ð f Þ ¼ jc_ s ð f ÞjGs ðcs ð f ÞÞ
ð3:12Þ
and ð3:13Þ Gs ðlÞ ¼ j’_ s ðlÞjHs ð’s ðlÞÞ _ with cs ðÞ and ’_ s ðÞ denoting the derivative of cs() and ’s(), respectively. Moreover, accounting for Eq. (3.10), from Eqs. (3.11a) and (3.11b) it follows that the impulse-response function of FOT deterministic linear systems can be expressed as XZ Gs ðlÞej2p’s ðlÞt e j2plu dl ð3:14aÞ hðt; uÞ ¼ R
s2O
¼
XZ s2O
R
Hs ð f Þe j2pcs ð f Þu e j2pft df :
ð3:14bÞ
By substituting Eqs. (3.11a) and (3.11b) into Eq. (3.9), one obtains the input/output relationship for FOT deterministic linear systems in the frequency domain: XZ Yð f Þ ¼ Gs ðlÞdð f ’s ðlÞÞX ðlÞ dl ð3:15aÞ s2O
¼
X s2O
R
Hs ð f ÞX ðcs ð f ÞÞ
ð3:15bÞ
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IZZO AND NAPOLITANO
from which it follows that yðtÞ ¼
XZ s2O
¼
X
R
Gs ðlÞX ðlÞej2p’s ðlÞt dl
hs ðtÞ ! xcs ðtÞ
s2O
where ! denotes convolution and Z X ðcs ð f ÞÞej2pft df : xcs ðtÞ ≜ R
ð3:16aÞ
ð3:16bÞ
ð3:17Þ
In other words, the output of FOT deterministic LTV systems is constituted by frequency compressed or stretched and filtered versions of the input. It can be shown that the parallel and cascade concatenation of FOT deterministic LTV systems is still a FOT deterministic LTV system. The class of FOT deterministic LTV systems includes that of the LAPTV systems which, in turn, includes, as special cases, both linear periodically time-variant and linear time-invariant (LTI) systems. For LAPTV systems, the frequency mapping functions cs ( f ) are linear with unitary slope, that is, cs ð f Þ ¼ f s;
s2O
and then the impulse-response function can be expressed as X hðt; uÞ ¼ hs ðt uÞej2psu :
ð3:18Þ
ð3:19Þ
s2O
Moreover, the systems performing time-scale changing are FOT deterministic. In such a case, the impulse-response function is given by hðt; uÞ ¼ dðu stÞ
ð3:20Þ
where s 6¼ 0 is the scale factor, the set O contains just one element, cs ð f Þ ¼
f s
ð3:21Þ
1 : jsj
ð3:22Þ
and Hs ð f Þ ¼
Furthermore, decimators and interpolators are FOT deterministic discretetime systems (Izzo and Napolitano, 1998a). The subclass of FOT deterministic LTV systems obtained by considering O containing only one element was studied, in the stochastic process framework, in Franaszek (1967) and Franaszek and Liu (1967) with reference to
GENERALIZED ALMOST-CYCLOSTATIONARY SIGNALS
151
the continuous-time case and in Liu and Franaszek (1969) with reference to the discrete-time case. The most important property of these systems, as evidenced in Franaszek (1967), Franaszek and Liu (1967), and Liu and Franaszek (1969) is that they preserve in the output the wide-sense stationarity of the input random process. In Claasen and Mecklenbr€auker (1982) the class of LTV systems called here FOT deterministic are analyzed considering a concept of stationarity with reference to a single time series. Linear time-variant systems that cannot be modeled as FOT deterministic systems include chirp modulators, modulators whose carrier is a pseudonoise sequence, channels introducing a time-varying delay, and systems performing time windowing. In fact, all these systems do not deliver an almost-periodic function when they are excited by a sinewave. Finally, note that, on the basis of the introduced terminology, a deterministic system in the stochastic process framework can be classified either as FOT deterministic or FOT random in the FOT probability sense, depending on the behavior of its impulse-response function. Moreover, a stochastic system can exhibit an impulse-response function whose sample paths are either FOT deterministic or FOT random. 3. Impulse-Response Function Decomposition for FOT Random LTV Systems In the FOT probability framework, a GACS time series x(t) can be decomposed into its deterministic (i.e., almost periodic) and purely random components: xðtÞ≜ Efag fxðtÞg þ xr ðtÞ ¼ xp ðtÞ þ xr ðtÞ
ð3:23Þ
where xp(t) is the almost-periodic component of x(t) and xr(t) is the purely random component such that hxr ðtÞe j2pat it 0;
8a 2 R:
ð3:24Þ
The decomposition [Eq. (3.23)] in the FOT probability framework is analogous to the decomposition, in the stochastic process framework, of a stochastic process into its statistical average and (zero mean) residual term. Moreover, in the stochastic process framework, a similar decomposition into deterministic and purely random components can be considered for the impulse-response function of a LTV system (Bello, 1963), since the statistical expectation operator can be applied in the same way to stochastic processes and stochastic system impulse-response functions. In fact, in such
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IZZO AND NAPOLITANO
a framework, the randomness of both processes and systems is due to the dependence on the variable belonging to the outcome space, which is not linked to the time variables. In the FOT probability framework, instead, the almost-periodic extraction operator is not able to provide the deterministic component also for the systems, since in this framework the randomness is a consequence of the time behavior of the functions. As an example, let us consider the impulse-response function hðt þ t; tÞ ¼ dðð1 sÞt stÞ of a system performing a time-scale changing [see Eq. (3.20)]. Even though the system is FOT deterministic, by applying the almost-periodic component extraction operator to hðt þ t; tÞ to obtain its deterministic component, one would obtain the incongruous result that the deterministic component of hðt þ t; tÞ should be identically zero, unless the system were LTI (s ¼ 1). In the FOT probability framework, a useful and congruous decomposition of a LTV system into the parallel concatenation of two subsystems, that will be referred to as its FOT deterministic and FOT purely random components, can be obtained by writing the impulse-response function in the following way: hðt; uÞ ≜ hD ðt; uÞ þ hR ðt; uÞ:
ð3:25Þ
In Eq. (3.25), hD(t, u) denotes the impulse-response function of the subsystem that for any almost-periodic input delivers an almost-periodic output. Its analytic expression is therefore given by Eqs. (3.14a) or (3.14b) and it will be referred to as the impulse-response function of the FOT deterministic component of the LTV system. The function hR(t, u) is the impulse-response function of the subsystem that for any almost-periodic input delivers an output signal not containing any finite-strength additive sinusoidal component. Such a system will be referred to as the FOT purely random component of the LTV system. Thus, for FOT deterministic LTV systems, it results that hðt; uÞ hD ðt; uÞ:
ð3:26Þ
hðt; uÞ hR ðt; uÞ
ð3:27Þ
Systems for which will be referred to as FOT purely random systems. A summary of the meanings of FOT deterministic and FOT random for both signals and systems is reported in Table 1. By substituting Eqs. (3.23) and (3.25) into the input/output relationship Eq. (3.8), one obtains that the almost-periodic component of the output y(t) can be expressed as
GENERALIZED ALMOST-CYCLOSTATIONARY SIGNALS
153
TABLE 1 Summary of the Meanings of FOT Deterministic and FOT Random for Both Signals and Systems
Signals Systems
FOT deterministic
FOT random
FOT purely random
Almost-periodic (AP) functions Transform AP functions into AP functions
Not FOT deterministic Not FOT deterministic
Not containing any finite-strength additive sine wave Transform AP functions into signals not containing any finite-strength additive sine wave
Z Efag fyðtÞg ¼
hD ðt; uÞ xp ðuÞ du Z hR ðt; uÞ xp ðuÞ du þ Efag ZR fag þE hD ðt; uÞ xr ðuÞ du ZR fag þE hR ðt; uÞ xr ðuÞ du :
R
ð3:28Þ
R
The second term in the right-hand side of Eq. (3.28) is identically zero since, by definition, hR(t, u) transforms almost-periodic inputs into signals not containing finite-strength additive sinusoidal components. Moreover, under the mild assumption that the deterministic component hD(t, u) is suYciently regular so that the functions Gs(l) do not contain impulsive terms in l, from (3.14a) it follows that ( ) Z XZ fag fag j2p’s ðlÞt hD ðt; uÞ xr ðuÞ du ¼ E Gs ðlÞ Xr ðlÞe dl ¼ 0 E R
s2O
R
ð3:29Þ where the absence of impulsive terms in the Fourier transform Xr( f ) of xr(t) has been accounted for [see Eq. (3.24)]. Furthermore, let us observe that the fourth term in the right-hand side of Eq. (3.28) could give nonzero contribution if and only if there exists statistical dependence, in the FOT probability sense, between the input signal and the system. This occurrence, however, means that functional dependence should exist between the functions hRð; Þ and xr ðÞ, which is in contradiction with the linearity assumption for the system. Then, unlike the stochastic process framework, in the FOT probability framework linear systems cannot be statistically dependent on
154
IZZO AND NAPOLITANO
the input signal and, hence, also the fourth term in the right-hand side of Eq. (3.28) is zero, which leads to Z Efag fyðtÞg ¼ hD ðt; uÞ xp ðuÞ du: ð3:30Þ R
Note that, since xp ðtÞ ¼ E fxðtÞg and Efag fyðtÞg are the expectations (in the FOT sense) of the input and output signals, respectively, by analogy with the stochastic process framework, hD(t, u) can be interpreted as the expectation (in the FOT sense) of the impulse-response function h(t, u). It is worthwhile to note that the expectation of h(t, u) cannot be obtained, in general, by extracting the almost-periodic component of h(t, u) as it happens in the case of the signals. In the case where a functional dependence between x(t) and h(t, u) exists, then Eq. (3.30) does not hold. For example, if xðtÞ xr ðtÞ 2 R (e.g., any zero-mean real GACS signal) and hðt; uÞ hRðt; uÞ ¼ xr ðt þ tÞ dðu tÞ, it results that Z hR ðt; uÞ xr ðuÞ du Efag fyðtÞg ¼ Efag fag
R
¼ E fxr ðt þ tÞxr ðtÞg X ¼ Rxr ;z ðtÞej2paz ð½t;0Þt fag
ð3:31Þ
z
where the functions Rxr ;z ðtÞ are the generalized cyclic autocorrelation functions of xr(t). It is worthwhile to note that, in the stochastic process framework, when a stochastic process passes through a LTV system with random impulseresponse function (i.e., which is in turn a stochastic process), a result analogous to Eq. (3.30) can be obtained, provided that the almost-periodic component extraction operator is substituted with the statistical expectation (i.e., the ensemble average) operator and the FOT deterministic component of the impulse-response function is substituted with the ensemble average of the stochastic impulse-response function and, moreover, the input signal and the system are statistically independent. The time-varying ensemble average of the output stochastic process, however, can be estimated by a single sample path only if appropriate ergodicity conditions involving both the input stochastic process and the stochastic LTV system are verified. The result of Eq. (3.30), instead, allows one to determine the time-varying FOT expectation of y(t), i.e., the finite-strength additive sine wave components of y(t), starting from a single available time series, independently of the possible existence of underlying (ergodic or not) stochastic processes.
GENERALIZED ALMOST-CYCLOSTATIONARY SIGNALS
155
As an example of application of the previous results, let us evaluate the almost-periodic component of the pulse-amplitude–modulated (PAM) signal xPAM ðtÞ ¼ xd ðtÞ ! qðtÞ
ð3:32Þ
where q(t) is a finite-energy pulse and xd(t) is the ideal sampled signal xd ðtÞ ¼
þ1 X
xðkÞ dðt kTs Þ:
ð3:33Þ
k¼ 1
The signal xPAM(t) can be interpreted as a LTI (and, hence, deterministic) transformation of the random signal xd(t) statistically independent of the LTI system. Therefore, according to Eq. (3.30), it results that Efag fxPAM ðtÞg ¼ Efag fxd ðtÞg ! qðtÞ þ1 X Ef~ag fxðkÞg qðt kTs Þ ¼
ð3:34Þ
k¼ 1 f~ag
where E fg denotes the discrete-time almost-periodic component extraction operator, and the second equality is proved in Appendix C. Note that Eq. (3.34) is a result formally analogous to that obtained in the stochastic process framework (Franks, 1969; Section 8.3). C. Higher-Order System Characterization in the Time Domain In this section, the system temporal moment function is introduced to provide the higher-order characterization of LTV systems in the time domain. Moreover, input/output relationships for LTV systems excited by GACS time series are derived in terms of temporal moment and cumulant functions (Izzo and Napolitano, 1999, 2002b). The Nth-order lag product of the output y(t) of a LTV system, accounting for Eqs. (2.19) and (3.8), is given by Ly ð1t þ tÞ ≜
N Y yðÞn ðt þ tn Þ n¼1
Z ¼
N Y hðÞn ðt þ tn ; t þ sn ÞLx ð1t þ sÞ ds
ð3:35Þ
RN n¼1
where s ≜ ½s1 ; . . . ; sN ⊤ . Therefore, the hidden periodicities of the input signal that are regenerated by the input Nth-order lag product are transformed into periodicities of the output Nth-order lag product by a linear (with respect to the lag product) operator whose kernel is the Nth-order lag product of the impulse-response function.
156
IZZO AND NAPOLITANO
Let us consider the following decomposition of the Nth-order lag product of the impulse-response function: N Y
hðÞn ðt þ tn ; un Þ ≜ Rh ð1t þ t; uÞ þ ‘h ð1t þ t; uÞ
ð3:36Þ
n¼1
where u ≜ ½u1 ; . . . ; uN T . In Eq. (3.36), the function Rh (1t þ t, u), which will be referred to as the system temporal moment function, is by definition the kernel of the linear (with respect to the lag product) operator that transforms the almost-periodic component of the input lag product, that is, the input temporal moment function, into an almost-periodic component of the output lag product (which, at this point, cannot yet be recognized to be the whole almost-periodic component, that is the output temporal moment function). Moreover, the function ‘h (1t þ t, u) is the kernel of the operator that transforms any almost-periodic component of the input lag product into a FOT purely random component (i.e., not containing any additive finite-strength sinusoidal component) of the output lag product. In order to obtain a powerful expression of the system temporal moment function, let us consider the input lag product T
Lx ð1t þ sÞ Rx ð1t þ sÞ ¼ ej2pl
ð1tþsÞ
ð3:37Þ
T
where l ≜ ½l1 ; . . . ; lN . The almost-periodic component of the corresponding output lag product Ly;l ð1t þ tÞ can be written as X Efag fLy;l ð1t þ tÞg ¼ G0 ðtÞe j2pt ð3:38Þ 2El;t
where, for each fixed (l, t), El,t is a countable set and G0 ðtÞ are complex functions. Moreover, note that either El,t or G0 ðtÞ depend on the choice of the conjugation configuration of the factors of the output lag product. Since the set El,t is countable, the set of points ð; lÞ 2 El;t RN such 0 that G ðtÞ 6¼ 0 is constituted by a countable set of N-dimensional manifolds (Choquet-Bruhat and DeWitte-Morette, 1982) that can be described by explicit equations. Therefore, it results that cl fð; lÞ 2 El;t RN : G0 ðtÞ 6¼ 0g ¼ cl [ fð; lÞ 2 R Dy ðtÞ : ¼ ’y ðl; tÞg
ð3:39Þ
y2Y
where Y is a countable set, ’y ð; Þ are real functions (which depend on the choice of the conjugation configuration of the factors of the lag products) such that ’y ðl; tÞ 6¼ ’y0 ðl; tÞ for y 6¼ y0 , and Dy(t) is the domain of ’y(, t). Thus, defined the functions
GENERALIZED ALMOST-CYCLOSTATIONARY SIGNALS
Gy ðl; tÞ ≜
G0 ðtÞj¼’y ðl;tÞ ; l 2 Dy ðtÞ 0; elsewhere
157 ð3:40Þ
the almost-periodic component of the output lag product can be written as X Efag fLy;l ð1t þ tÞg ¼ Gy ðl; tÞe j2p’y ðl;tÞt : ð3:41Þ y2Y
Furthermore, since the input lag product [Eq. (3.37)] does not contain the residual term ‘x(1t þ s), from the definition of system temporal moment function it results that Z ⊤ Efag fLy; l ð1t þ tÞg ¼ Rh ð1t þ t; uÞe j2pl u du ð3:42Þ RN
from which, accounting for Eq. (3.41), it follows that the system temporal moment function can be expressed as XZ ⊤ Rh ð1t þ t; uÞ ¼ Gy ðl; tÞe j2p’y ðl;tÞt e j2pl u dl: ð3:43Þ y2Y
RN
Note that, for a given system, the functions ’y(, ) and Gy(, ) are not univocally determined since, in general, for each manifold several representations by explicit equations are possible. Furthermore, if there exist vectors l0 such that more functions ’yi ðl; tÞ i ¼ 1; . . . ; K have the same limit, say ’0(t), as l ! l0 , then it is convenient to assume all the functions defined in l0 with ’yi ðl0 ; tÞ ¼ ’0 ðtÞ and consequently to define Gyi ðl0 ; tÞ ¼ lim Gyi ðl; tÞ; l!l0
i ¼ 1; . . . ; K
ð3:44Þ
where, for each i, the limit is made with l ranging in the neighborhood of l0 where ’yi(l, t) is defined. If the functions ’y(l, t) are constant with respect to l for any y 2 Y, then the function Rh ð1t þ t; uÞ is almost periodic in t. Finally, if for N ¼ 2 Rh ð1t þ t; uÞ is periodic in t, then it is the FOT counterpart of the system intercorrelation function introduced in Duverdier et al. (1999) for the class of (stochastic) cyclostationary systems. It can be shown that for all real numbers D the following lag-shift invariance property holds El;tþ1D El;t
ð3:45Þ
G0 ðt þ 1DÞ ¼ G0 ðtÞej2pD ;
ð3:46Þ
and, moreover,
158
IZZO AND NAPOLITANO
from which it follows that ’y ðl; t þ 1DÞ ¼ ’y ðl; tÞ
ð3:47Þ
Gy ðl; t þ 1DÞ ¼ Gy ðl; tÞej2p’y ðl;tÞD
ð3:48Þ
and
where, in the derivation of Eq. (3.48), Eq. (3.40) has been accounted for. Since in the FOT probability framework the almost-periodic component extraction operator is the expectation operator, for N ¼ 2 the function Efag fLy; l ð1t þ tÞg is the correlation between the output signals corresponding to the inputs ej2pl1 t and ej2pl2 t [see Eqs. (3.37) and (3.41)]. Moreover, from Eqs. (3.45)–(3.48) it follows that the correlation function Efag fLy; l ð1t þ tÞg depends on t1 t2. A similar property, however, does not hold in general for the dependence on l1 and l2. In the special case where, for all y 2 Y, Gy ðl1 ; l2 ; t1 ; t2 Þ ¼ Gy ðDl; DtÞ
ð3:49Þ
’y ðl1 ; l2 ; t1 ; t2 Þ ¼ ’y ðDl; DtÞ
ð3:50Þ
where Dl ≜ l1 l2 and Dt ≜ t1 t2 , the function defined in Eq. (3.41) is the counterpart in the FOT probability framework of the spaced-frequency spaced-time correlation function defined in the stochastic process framework (Proakis, 1995). Moreover, its Fourier transform with respect to the variable Dt evaluated for Dl ¼ 0 is the FOT counterpart of the Doppler power spectrum of the channel defined in the stochastic process framework. Furthermore, the double Fourier transform with respect to Dl and Dt of the right-hand-side of Eq. (3.41) is the FOT counterpart of the scattering function of the channel (Proakis, 1995). Let us observe that for N ¼ 1 the system temporal moment function is coincident with the FOT deterministic component of the impulse-response function, that is, Rh ðt; uÞ hD ðt; uÞ:
ð3:51Þ
Moreover, the set Y is coincident with O, the functions Gy(l1, t1) and ’y (l1, t1) do not depend on t1 and are coincident with the functions Gs(l1) and ’s ðl1 Þðs ¼ yÞ, respectively, and the functions ’s() can be chosen among the monotonic functions. The output temporal moment function of a LTV system, by substituting Eqs. (2.19) and (3.36) into Eq. (3.35), can be written as
GENERALIZED ALMOST-CYCLOSTATIONARY SIGNALS
159
Z Efag fLy ð1t þ tÞg ¼
Rh ð1t þ t; 1t þ sÞRx ð1t þ sÞ ds Z ‘h ð1t þ t; 1t þ sÞRx ð1t þ sÞ ds þ Efag RN Z þ Efag Rh ð1t þ t; 1t þ sÞ ‘x ð1t þ sÞ ds RN Z þ Efag ‘h ð1t þ t; 1t þ sÞ‘x ð1t þ sÞ ds : RN
ð3:52Þ
RN
The second term in the right-hand side of Eq. (3.52) is identically zero since ‘h ð1t þ t; 1t þ sÞ is by definition the kernel of the operator that transforms the almost-periodic component of the input lag product into a FOT purely random component of the output lag product. Moreover, under the mild assumption that the system temporal moment function is suYciently regular so that the functions Gy(l, t) do not contain impulsive terms in l, accounting for Eq. (3.43), one has Z fag Rh ð1t þ t; 1t þ sÞ‘x ð1t þ sÞds E RN ( ) XZ ð3:53Þ fag j2p’y ðl;tÞt ¼E Gy ðl; tÞLx ðlÞe dl y2Y
RN
¼0 where Lx(l) is the Fourier transform of the function ‘x(u) and, hence, does not contain impulsive terms in l [in fact, Eq. (2.20) with ‘x (t, t) ‘x (1t þ t) holds for any t]. Furthermore, also the fourth term in the right-hand side of Eq. (3.52) is zero since it could give nonzero contribution only in the case of functional dependence between the functions ‘h(,) and ‘x(), that is, only in the case of functional dependence between the functions hR(,) and xr(), which is in contrast with the linearity assumption (see Section III.B.3). Therefore, the temporal moment function of the output y(t) can be written as Efag fLy ð1t þ tÞg Ry ð1t þ tÞ Z ¼ Rh ð1t þ t; 1t þ sÞ Rx ð1t þ sÞds:
ð3:54Þ
RN
Equation (3.54) is the input/output relationship in terms of temporal moment functions for LTV systems and is formally analogous to that obtained in the stochastic process framework. Its advantage with respect to the corresponding formula in the stochastic process framework is that it is the asymptotic result of a time average measure without the necessity of
160
IZZO AND NAPOLITANO
ergodicity assumptions. Note that, since Rx (1t þ t) and Ry (1t þ t) are the expectations (in the FOT sense) of the input and output lag products, respectively, by analogy with the stochastic process framework, Rh (1t þ t, 1t þ s) can be interpreted as the expectation (in the FOT sense) of the impulse-response function lag product and this justifies its name ‘‘system temporal moment function.’’ Furthermore, Rh (1t þ t, 1t þ s) cannot be obtained, in general, by extracting the almost-periodic component of the impulse-response function lag product as it happens in the case of signals. Let us observe that, in general, for N > 1, the Nth-order lag product of the FOT deterministic component hD (t, u) of the impulse-response function is not coincident with the system temporal moment function Rh (1t þ t, u). However, in the special case of FOT deterministic LTV systems, accounting for Eqs. (3.14a) and (3.26), one has N Y hðÞn ðt þ tn ; un Þ n¼1
¼
X Z
s2ON
N Y
RN n¼1
ð Þ
j2pws n GðÞ sn ðð Þn ln Þe
ðlð Þ Þ⊤ ð1tþtÞ j2pl⊤ u
e
ð3:55Þ dl
where ws ðlð Þ Þ ≜ ½ð Þ1 ’s1 ðð Þ1 l1 Þ; ð ÞN jsN ðð ÞN lN Þ⊤ : Therefore, by comparing Eq. (3.43) with Eq. (3.55), we get ð Þ
Rh ð1t þ t; uÞ ¼
N Y hðÞn ðt þ tn ; un Þ n¼1 N Y n¼1
ðÞ hD n ðt
ð3:56Þ þ t n ; un Þ
which is a result formally analogous to that obtained in the stochastic process framework. Moreover, in such a case there exists a one-to-one correspondence between the elements y 2 Y and the vectors s ≜ [s1, , sN]⊤ 2 ON, ð Þ ⊤ ’y ðl; tÞ ¼ wð Þ Þ 1 ðindependent of tÞ s ðl
ð3:57Þ
and Gy ðl; tÞ ¼
N Y ð Þ j2pws ðlð Þ Þ⊤ t n GðÞ : sn ðð Þn ln Þe
ð3:58Þ
n¼1
Finally, in Appendix C it is shown that for FOT deterministic LTV systems the input/output relationship in terms of temporal cumulant functions is the same as that given in Eq. (3.54) in terms of temporal moment functions, that is,
161
GENERALIZED ALMOST-CYCLOSTATIONARY SIGNALS
Z Cy ð1t þ tÞ ¼
RN
Rh ð1t þ t; 1t þ sÞ Cx ð1t þ sÞds:
ð3:59Þ
Such a result is analogous to that for deterministic LTV systems in the stochastic process framework. Moreover, Eq. (3.59) generalizes the result derived in Napolitano (1995) for LAPTV systems excited by ACS signals. D. Higher-Order System Characterization in the Frequency Domain In this section, random and deterministic LTV systems in the FOT probability sense are characterized in the frequency domain. Moreover, input/ output relationships in terms of spectral moment and cumulant functions are derived for LTV systems excited by GACS time series (Izzo and Napolitano, 2002b). By assuming that all Fourier transforms considered in the following exist at least in the sense of distributions (Zemanian, 1987), one has that the N-fold Fourier transform of both sides of Eq. (3.54) gives the input/output relationship in terms of spectral moment functions: Z Sy ð f Þ ¼ S h ð f ; lÞS x ðlÞ dl ð3:60Þ RN
where the function
Z
S h ð f ; lÞ ≜
R
2N
Rh ðt; uÞe j2pð f
⊤
t l⊤ uÞ
dtdu
ð3:61Þ
referred to as the system spectral moment function, accounting for Eq. (3.43), can be expressed as X Gy ðl; f Þ ð3:62Þ S h ð f ; lÞ ¼ y2Y
with
Z Gy ðl; f Þ ≜ ¼
R
Z
N
Gy ðl; tÞe j2pf
RN 1
⊤
t
dt ð3:63Þ
Gy ðl; ½t 0 ; 0Þ
dð’y ðl; ½t0 ; 0Þ f ⊤ 1Þe j2pf 0
0⊤ 0 t
dt0
where in the derivation of the last equality, Eq. (3.48) has been accounted for. In the special case of FOT deterministic LTV systems, by substituting Eqs. (3.57) and (3.58) into Eq. (3.63), one obtains that
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IZZO AND NAPOLITANO
Gy ðl; f Þ ¼
N
Y ðÞ Gsn n ðð Þn ln Þd fn ð Þn ’sn ðð Þn ln Þ
ð3:64aÞ
N
Y n ðð Þ f Þd l ð Þ c ðð Þ f Þ HsðÞ n n n n sn n n n
ð3:64bÞ
n¼1
¼
n¼1
where, in the derivation of Eq. (3.64b), Eq. (3.12) has been accounted for. Moreover, from Eq. (3.62), accounting for Eqs. (3.11a) and (3.11b), it follows that N Y S h ð f ; lÞ ¼ H ðÞn ðð Þn fn ; ð Þn ln Þ: ð3:65Þ n¼1
Finally, let us note that since for FOT deterministic LTV systems the input/output relationship in terms of temporal cumulant functions is the same as that in terms of temporal moment functions (see Appendix C), accounting for Eq. (3.60), one obtains that the input/output relationship in terms of spectral cumulant functions can be expressed as Z Pyð f Þ ¼ S h ð f ; lÞP x ðlÞdl: ð3:66Þ RN
E. Ergodicity of the Output Signal of a LTV System In this section, the lack of ergodicity of the output signal of a stochastic LTV system in the stochastic process framework is discussed. Such a discussion suggests one of the most important motivations to consider the LTV filtering in the FOT probability framework (Izzo and Napolitano, 2002b). A stochastic process is said to be ergodic (Gardner, 1994) in the stochastic TMF [TCF] if it is equal to the TMF [TCF] of almost every sample path. The stochastic TMF is defined as the statistical expectation of the lag product, whereas the stochastic TCF can be expressed by Eq. (2.58) in which, however, the involved moment functions are stochastic. For stochastic systems in the stochastic process framework, in general the input/output relations in terms of stochastic TMFs and TCFs can be diVerent from those in the FOT probability framework, that is, those in terms of their possible asymptotic estimators (i.e., TMFs and TCFs, respectively). In other words, in general such systems transform ergodic input stochastic processes into nonergodic output stochastic processes. In the stochastic process framework, stochastic systems, whose sample paths of the impulse-response function are impulse-response functions of deterministic systems in the FOT probability sense, destroy the input
163
GENERALIZED ALMOST-CYCLOSTATIONARY SIGNALS
ergodicity properties. In fact, let us consider a stochastic LTV system characterized by the impulse-response function H (t, u) and excited by a stochastic process X(t) statistically independent of the system. The Nth-order stochastic temporal moment and cumulant functions of the output process Y(t) can be written as ( ) N Y ðÞ R Y ð1t þ tÞ ≜ E Y n ðt þ tn Þ Z ¼
RN
n¼1(
E
N Y n¼1
) HðÞn ðt þ tn ; t þ sn Þ
( E
N Y
) X ðÞn ðt þ sn Þ ds
n¼1
ð3:67Þ Y ð1t þ tÞ ≜ cumfY ðÞn ðt þ tn Þ; n ¼ 1; . . . ; Ng C ( ) Z X" p Y Y p 1 ðÞn ¼ ð 1Þ ðp 1Þ! E H ðt þ tn ; t þ sn Þ RN P n2mi i¼1 3 q XY X ð1t þ sm ;n Þ5ds C mi ;vj i j Pmi j¼1
ð3:68Þ where E{} denotes statistical expectation, the overbar has been adopted to denote stochastic statistical functions, Pmi is the set of distinct partitions of the elements of mi, each constituted by the subsets {nj : j ¼ 1, . . ., q}, jnjj is the number of elements in nj, and Xmi,nj is the jnjj-dimensional vector whose components are those (possibly conjugate) of Xmi having indices in nj. From Eqs. (3.67) and (3.68), it follows that the input/output relations in terms of stochastic TMFs and TCFs are diVerent. However, since almost every sample path of the considered stochastic system is a deterministic system in the FOT probability sense, then the input/output relations in terms of TMFs and TCFs (which are candidate asymptotic estimators of their stochastic counterparts) are the same (see Appendix C). Therefore, such a system transforms in a diVerent way the stochastic TMFs and TCFs and their candidate asymptotic estimators and, hence, it transforms ergodic stochastic processes into nonergodic stochastic processes. Let us note that, with reference to the moments, a stochastic system with impulse-response function with FOT deterministic sample-paths transforms an ACS ergodic input stochastic process X(t) into a nonergodic output process Y(t). In fact, for any ACS ergodic input it results that X ð1t þ tÞ ¼ Rx ð1t þ t) for almost all sample-paths x(t) (Dandawate´ and R Giannakis¸, 1995; Gardner, 1994), but for the systems under consideration
164
IZZO AND NAPOLITANO
( E
N Y
) H
ðÞn
ðt þ tn ; t þ sn Þ
6¼
n¼1
N Y
hðÞn ðt þ tn ; t þ sn Þ
ð3:69Þ
n¼1
for almost all sample paths h(t, u). Hence, the kernel of the integral that transforms Rx ð1t þ tÞ into Ry ð1t þ t) [see Eq. (3.54) with Eq. (3.56) sub X ð1t þ t) into R Y ð1t þ tÞ stituted into] is diVerent from that transforming R Y ð1t þ tÞ 6¼ Ry ð1t þ t). [see Eq. (3.67)] and, consequently, R Examples of stochastic systems with the previously-mentioned behavior in the stochastic process framework are the stochastic LTV systems with impulse-response functions whose sample paths are deterministic in the FOT probability sense and whose randomness is due to the presence of random parameters. A notable case is the widely adopted model for the noncoherent channel, that is, the stochastic system with impulse-response function H (t, u) ¼ R(t) ejY(t) d(u t), where, for a fixed t, R(t) is a Rayleigh random variable, Y(t) is a uniformly distributed random variable, and the dependence on t in R(t) and Y(t) can be neglected in the observation interval. Further examples are carrier modulators introducing random phase shifts and the Doppler channel that transforms a sine wave with a fixed frequency into one whose frequency is a random variable (Papoulis, 1991, paragraph 10–3). Stochastic linear systems destroying ergodicity whose randomness is not due to a random parameter in the impulse-response function are, for example, the stochastic bandlimited LTI systems, that is, those systems whose impulseresponse function sample paths are almost all bandlimited functions. A further drawback of considering the random filtering problem in the stochastic process framework is evident if one considers the input signal modeled as a deterministic signal x(t) (with possibly unknown, but nonrandom, parameters). In such a case, the input/output relationships for random channels Eqs. (3.67) and (3.68) become ( ) Z N N Y Y ðÞ Y ð1t þ tÞ ¼ R E H n ðt þ tn ; t þ sn Þ xðÞn ðt þ sn Þ ds ð3:70Þ RN
C Y ð1t þ tÞ ¼
Z
X
RN
Y
P
x
ðÞj
n¼1
n¼1
" ð 1Þ
p 1
ðp 1Þ!
#
p Y i¼1
( E
Y n2mi
) ðÞn
H
ðt þ tn ; t þ sn Þ
ð3:71Þ
ðt þ sj Þ ds:
j2mi
Consequently, in the stochastic process framework, for the same input signal x(t), one obtains diVerent output statistical functions depending on the choice of modeling such a signal as a sample path of a stochastic process [Eqs. (3.67) and (3.68)] or as a deterministic (with possible unknown
GENERALIZED ALMOST-CYCLOSTATIONARY SIGNALS
165
parameters) signal Eqs. (3.70) and (3.71)]. Such a choice, of course, influences the possible match between the output signal statistical functions and their asymptotic estimators. It is worthwhile to note that a similar drawback is not present in the FOT approach adopted here. F. Countability of the Set of the Output Cycle Frequencies In this section, the countability of the set of the output cycle frequencies of a LTV system excited by a GACS signal is discussed (Izzo and Napolitano, 1995a,b, 2002c). Such a problem turns out to be relevant in the FOT probability approach since, in such a framework, the almost-periodic component extraction operator is the analogous of the stochastic expectation operator in the classical stochastic process framework (see Section II.B). Let us consider an LTV system excited by a time series x(t). By substituting the expression in Eq. (3.43) of the system temporal moment function into the input/output relationship [Eq. (3.54)] and accounting for Eqs. (2.26) and (2.36), one obtains the following expression for the Nth-order temporal moment function of the output time series y(t) XXZ Ry ð1t þ tÞ ¼ Gy ðl; tÞej2p’y ðl;tÞt S x;z ðlÞdl ð3:72Þ RN
y2Y z2Wx
from which, by taking the coeYcient of the periodic component at frequency a, one obtains the output CTMF: Ray ðtÞ ≜ Ry ð1t þ tÞe j2pat t XXZ ð3:73Þ ¼ Gy ðl; tÞ d’y ðl;tÞ a S x;z ðlÞdl: y2Y z2Wx
RN
Thus, the set of the Nth-order output cycle frequencies for each value of t is given by Ay;t ≜ fa 2 R : Ray ðtÞ 6¼ 0g Z ¼ [ [ a2R: Gy ðl; tÞ d’y ðl;tÞ a S x;z ðlÞdl 6¼ 0 : y2Y z2Wx
R
ð3:74Þ
N
An alternative expression for the output CTMF can be obtained by substituting Eq. (3.43) into Eq. (3.54), taking the sine wave component at frequency a, and accounting for Eqs. (2.48) and (2.51): Z XXZ 0 a Ry ðtÞ ¼ Gy ð½l0 ; a z ðu0 Þ l ⊤ 1; tÞRx;z ðu0 Þ N 1 N 1 R ð3:75Þ y2Y z2Wx R 0⊤ 0 0 j2pl u 0 0 d’y ð½l0 ;a z ðu0 Þ l ⊤ 1;tÞ a du dl : e
166
IZZO AND NAPOLITANO
In the following, the general case of LTV systems and the special case of FOT deterministic LTV systems will be considered in order to discuss the countability of the cycle frequency set of the output lag-product waveforms. To this end, two subclasses of GACS input signals will be considered. The former is the class of the GACS signals not containing any ACS component, that is, such that none of the lag-dependent (moment) cycle frequencies az(t) is constant in a set of values of t with nonzero Lebesgue measure in RN (see Section II.C.1). The latter is the subclass of the ACS signals, that is, such that all the lag-dependent cycle frequencies are constant with t. 1. Analysis of LTV Systems a. GACS Input Not Containing ACS Components. Let us assume that the input signal x(t) is GACS not containing ACS components and that the LTV system is suYciently regular so that the functions Gy(l, t) do not contain impulses in l. Such an assumption on the system means that we are not considering ideal resonator systems, that is, systems such that the output lag product contains finite-strength additive sine waves only in correspondence of particular values of frequencies in the input lag product. Since x(t) does not contain ACS components, S x,z(l) does not contain impulsive terms in l (see Section II.C.2) and, hence, nonempty sets in the right-hand side of Eq. (3.74) can be obtained only if the supports of the integrand functions have nonzero Lebesgue measure in RN. Consequently, the following inclusion relationship holds: Ay;t [ Ayt
ð3:76Þ
y2Y
where Ayt ≜ fa 2 R : measN fl 2 RN : ’y ðl; tÞ ¼ ag 6¼ 0g
ð3:77Þ
N
with measN{} denoting the Lebesgue measure in R . For fixed values of y and t, ð3:78Þ ’y ðl; tÞ ¼ a is the equation of an N-dimensional manifold in the (N þ 1)-dimensional space (a, l). Therefore, for each set L RN such that measN {l 2 L : ’y(l, t) ¼ a} > 0, the corresponding set of values of a satisfying Eq. (3.78) must be at most countable. In other words, each set Ayt is at most countable and hence, accounting for Eq. (3.76), the set Ay,t is at most countable, that is, the output time series y(t) is GACS (or possibly a zero-power signal if, for N ¼ 2, (*)1 absent, and (*)2 present, Ay,t is an empty set for all t 2 R2). b. ACS input. In the case where the input signal x(t) is ACS, there exists a one-to-one correspondence between the elements z 2 Wx and the cycle
GENERALIZED ALMOST-CYCLOSTATIONARY SIGNALS
167
frequencies g 2 Ax and, moreover, the GCSMF S x;z ðlÞ can be expressed by Eq. (2.53), which substituted into Eq. (3.73), leads to XXX X Z Ray ðtÞ ¼ Gy ðvm ðl; bÞ; tÞ d’y ðvm ðl;bÞ;tÞ a N p y2Y g2Ax P b⊤ 1¼g R p Y bm Pxmii ðl0mi Þdl0m1 . . . dl0mp i¼1
ð3:79Þ
where vm ðl; bÞ ≜ ½l0m1; bm1 l0m1 ⊤ 1; . . . ; l0mp; bmp l0mp ⊤ 1 with m ≜ ½m1 ; . . .; mp ⊤ and b ≜ ½bm1; . . .; bmp ⊤ . Moreover, under the mild assumption that the time series x(t) and x(t þ t) are asymptotically (jtj ! 1) independent in the FOT probability sense (see Section II.B), the cyclic polyspectra bm Pxmii ðl0mi Þ are not impulsive (Gardner and Spooner, 1994) and, hence, the following inclusion relationship holds: Ay;t [
[
[
[
y2Y g2Ax m2P b2Bm1 Bmp
Aðy;g;m;bÞ t
ð3:80Þ
where Bmi denotes the set of jmijth-order cumulant cycle frequencies bmi of {x(*)n(t), n 2 mi}, and n n ðy;g;m;bÞ At ≜ a 2 R : measN p ½l0m1 ; ; l0mp 2 RN p : o o ð3:81Þ ’y ðvm ðl; bÞ; tÞ ¼ a; b⊤ 1 ¼ g > 0 : Thus, reasoning as in the case of GACS input, it follows that each ðy;g;m;bÞ is empty or countable and, hence, the set Ay,t is in turn empty set At or countable, that is, the output time series y(t) is GACS (or possibly zero power). As an example, let us consider the Doppler channel existing between a transmitter and a receiver with nonzero relative acceleration, that is, a purely random LTV system [see Eq. (3.27)] characterized by the impulseresponse function hðt; uÞ ¼ dðu t þ DðtÞÞ
ð3:82Þ
where DðtÞ ≜ d0 þ d1 t þ d2 t2 ;
d2 6¼ 0:
ð3:83Þ
The almost-periodic component of the output lag product when the input ⊤ lag product is e j2pl (1tþs) is given by [see Eq. (3.41)] n o ð Þ⊤ ⊤ ð Þ Efag fLy; l ð1t þ tÞg ¼ e j2pl ð1tþtÞ Efag e j2pl D ð1tþtÞ ð3:84Þ
168
IZZO AND NAPOLITANO
where Dð Þ ð1t þ tÞ ≜ ½ð Þ1 Dðt þ t1 Þ; . . . ; ð ÞN Dðt þ tN Þ⊤ . Moreover, in the special case of D(t) given by Eq. (3.83), it results that n o ð Þ ⊤ ð Þ ⊤ ð Þ ð Þð2Þ Efag e j2pl D ð1tþtÞ ¼ e j2pl ½d0 1 þd1 t þd2 t ð3:85Þ ð Þ ⊤ ð Þ e j2pl ½d1 1 þ2d2 t t dl⊤ 1ð Þ d2 where 1ð Þ ≜ ½ð Þ1 1; . . . ; ð ÞN 1⊤ ; tð Þ ≜ ½ð Þ1 t1 ; . . . ; ð ÞN tN ⊤ ; tð Þð2Þ ≜ ½ð Þ1 t21 ; . . . ; ð ÞN t2N ⊤ , and the fact that Z þ1 cosðat þ bt2 Þ dt; b 6¼ 0 0
is finite, and, hence
D E 2 e j2pðatþbt Þ ¼ da db
ð3:86Þ
t
has been accounted for. From Eqs. (3.84) and (3.85) and accounting for Eq. (3.41), it follows that for the LTV system with impulse-response function Eq. (3.82), Eq. (3.83), the set Y contains only one element and, for d2 6¼ 0, Gy ðl; tÞ ¼ e j2pl
ð Þ⊤
½ð1 d1 Þt d2 tð2Þ
’y ðl; tÞ ¼ 2d2 lð Þ⊤ t;
dlð Þ⊤ 1
ð3:87Þ
for lð Þ⊤ 1 ¼ 0:
ð3:88Þ
Thus, accounting for Eq. (3.43), it results that Z ð Þ ⊤ Rh ð1t þ t; uÞ ¼ e j2pl ½u F ðt;tÞ dlð Þ⊤ 1 dl RN
ð3:89Þ
where
with
Fð Þ ðt; tÞ ≜ ½ð Þ1 Fðt; t1 Þ; . . . ; ð ÞN Fðt; tN Þ⊤ ¼ ð1 d1 Þtð Þ d2 tð Þð2Þ 2d2 tð Þ t
ð3:90Þ
Fðt; tÞ ≜ ð1 d1 Þt d2 t2 2d2 tt:
ð3:91Þ
Note that, even if F( )(t, t) is not function of 1t þ t; ej2pl F ðt;tÞ dlð Þ⊤ 1 is. By substituting the expression of the system temporal moment function Eq. (3.89) into the input/output relationship Eq. (3.54), one obtains that Z ⊤ ð Þ Ry ð1t þ tÞ ¼ ej2pl F ðt;tÞ dlð Þ⊤ 1 S x ðlÞ dl: ð3:92Þ ⊤
ð Þ
RN
The presence of the Kronecker delta in the integrand function implies that the output TMF can contain finite-strength additive sine wave components
169
GENERALIZED ALMOST-CYCLOSTATIONARY SIGNALS
only if the input spectral moment function S x ðlÞ contains impulsive terms, that is, if the input signal contains an ACS component (see Section II.C.2). In particular, if the input signal is ACS, it results that X X S x ðlÞdlð Þ⊤ 1 ≜ S ax ðlÞdlð Þ⊤ 1 ¼ Sxa ðl0 Þdða l⊤ 1Þdlð Þ⊤ 1 ð3:93Þ a2Ax
a2Ax
Sxa ðl0 Þ
where is the reduced-dimension cyclic spectral moment function (Gardner and Spooner, 1994), which can be impulsive and is given by Eq. (2.53). If all the optional conjugations are absent, from Eq. (3.93) and accounting for the identity dða l⊤ 1Þdl⊤ 1 ¼ dðl⊤ 1Þda
ð3:94Þ
S x ðlÞdl⊤ 1 ¼ Sx0 ðl0 Þdðl⊤ 1Þ:
ð3:95Þ
it follows that Thus, by substituting Eq. (3.95) into Eq. (3.92) and accounting for Eq. (2.53), one obtains that Ry ð1t þ tÞ ¼
R
8 > > > > <
RN 1
ej2pl
0⊤
½F0 ðt;t 0 Þ 1Fðt;tN Þ
9 > 3> > > p 1 = X X bmp Y bmi 0 0 0 ⊤ 4 P0x ðl Þ þ Pxmp ðlmp Þ Pxmi ðlmi Þdðbmi lmi 1Þ5 dl0 > > > > i¼1 b⊤ 1¼0 > > P > > : ; p6¼1 2
ð3:96Þ where, in the derivation, the sampling property of Dirac’s delta function has been accounted for and it is assumed, without loss of generality, that each partition is ordered such that mp always contains N as its last element. Thus, by indicating with ri the last element of each subset mi (i ¼ 1, . . ., N 1) and performing the inverse Fourier transforms in Eq. (3.96), one has 0 0 Ry ð1t þ tÞ ¼ Cx0 ðF 2 ðt; t Þ 1Fðt; tN ÞÞ X X bmp 4 þ Cxmp ðF0mp ðt; t 0mp Þ 1Fðt; tN ÞÞ P p6¼1
b⊤ 1¼0
3
p 1 Y bm e j2pbmi ½Fðt;tri Þ Fðt;tN Þ Cxmii ðF0mi ðt; t 0mi Þ 1Fðt; tri ÞÞ5 i¼1
ð3:97Þ
170
IZZO AND NAPOLITANO
where Cxb ðt0 Þ is the reduced-dimension cyclic temporal cumulant function, which is the inverse Fourier transform of the cyclic polyspectrum Pbx ð f 0 Þ (Gardner and Spooner, 1994); (see also Section I.C). Finally, by substituting the expression of F(t, t) given by Eq. (3.91) into Eq. (3.97) with tN ¼ 0 and taking the coeYcient of the finite-strength additive sinewave component at frequency a, one obtains the expression for the Nth-order reduced-dimension CTMF Ray ðt 0 Þ. For example, in the case of a zero mean signal x(t), that is, a signal not containing any finite-strength additive sine wave component and, moreover, such that x(t) and x(t þ t) are asymptotically (jtj ! 1) independent, for N ¼ 4, one has X 0 Rayyyy ðt1 ; t2 ; t3 Þ ¼ Cxxxx ð0; 0; 0Þdt1 dt2 dt3 da þ Rbxx2 ð0ÞRbxx1 ð0Þ ⊤
b 1¼0 h j2pb1 ½ð1 d1 Þt2 d2 t22 daþb1 2d2 t2 dt3 dt1 t2 e 2
ð3:98Þ
þ dt2 dt1 t3 ej2pb1 ½ð1 d1 Þt3 d2 t3 daþb1 2d2 t3 i 2
þ dt1 dt2 t3 ej2pb1 ½ð1 d1 Þt3 d2 t3 daþb1 2d2 t3
where, in the derivation of Eq. (3.98), the fact that the equality in Eq. (3.97) is in the temporal mean-square sense and the fact that, for an asymptotically independent time series x(t) it results that, for any order N, lim Cxb ðt0 tÞ ¼ 0;
jtj!1
8t0 6¼ 0
ð3:99Þ
have been accounted for. Note that, by comparing Eq. (3.98) with Eq. (2.30), it follows that the GCTMFs are discontinuous functions of t1, t2, and t3. Furthermore, for N ¼ 2 and t2 ¼ 0 one has Fðt; t1 Þ Fðt; t2 Þjt2 ¼0 ¼ ð1 d1 Þt1 d2 t21 2d2 t1 t
ð3:100Þ
from which we have 0 ðFðt; t Þ Fðt; 0ÞÞ Ryy ð1t þ ½t1 ; 0Þ ¼ Cxx 1 0 ðð1 d Þt d t2 2d t tÞ ¼ Cxx 1 1 2 1 2 1 0 ð0Þd d Rayy ðt1 Þ ¼ Cxx a t1
ð3:101Þ ð3:102Þ
which are coincident with the autocorrelation function and the cyclic autocorrelation function, respectively, in the case of x(t) real signal. In order to corroborate the eVectiveness of the previously presented theoretical results, a simulation experiment has been realized. In this experiment (and in those presented in the following), time has been discretized with sampling increment Ts ¼ T/M, where T is the data-record length and M is the number of samples (for a discussion on the aliasing issue, see Section IV.C). The parameters of the Doppler channel have been assumed to be d0 ¼ 0Ts,
GENERALIZED ALMOST-CYCLOSTATIONARY SIGNALS
171
d1 ¼ 4.5, and d2 ¼ 1.52676 10 3/Ts. Moreover, it has been selected, as input signal to the Doppler channel, the double side-band (DSB) signal xðtÞ ¼ sðtÞ cosð2pf0 tÞ
ð3:103Þ
where f0 ¼ 0.06/Ts and s(t) is a colored fourth-order wide-sense stationary signal that has been obtained by passing fourth-order wide-sense stationary white noise through the LTI filter H0( f ) ¼ (1 þ jf/B0) 8, with B0 ¼ 0.00035/Ts. It is well known that x(t) is wide-sense cyclostationary with fourth-order cyclic temporal moment functions nonzero only in correspondence of the cycle frequencies a ¼ )2f0, a ¼ )4f0 (and a ¼ 0) (Napolitano, 1995), Thus, the support in the (a, t1) plane of a slice with t2 ¼ t1 and t3 ¼ 0 of the reduced-dimension CTMF of x(t) is confined on the five lines with equations a ¼ 0, a ¼ )2f0, and a ¼ )4f0. Figure 7 shows (a) the magnitude and (b) the support in the (a, t1) plane of the slice for t2 ¼ t1, t3 ¼ 0 of the fourth-order reduced-dimension (i.e., t4 ¼ 0) CTMF of the input DSB signal x(t) estimated by M ¼ 211 samples. The estimated magnitude and support of the slice of the reduced-dimension CTMF of the output signal y(t) are reported in Figures 8a and b, respectively. The generalized cyclostationary nature of y(t) is evident. Note that, in Section IV.D it is explained that the GACS nature of a continuous-time signal can only be conjectured starting from the discrete-time signal constituted by its samples. 2. The Special Case of FOT Deterministic LTV Systems Let us now assume that the LTV system is FOT deterministic and suYciently regular so that the functions Gy(l, t) do not contain impulses in l. By substituting into Eq. (3.72) the expressions from Eqs. (3.57) and (3.58) of ’y(l, t) and Gy (l, t) for FOT deterministic LTV systems, one obtains N
X Z Y n ’ Ry ð1t þ tÞ ¼ l ð Þ ’_ sn ð Þn ln HsðÞ sn n n n N ð3:104Þ s2ON R n¼1 ð Þ
S x ðlÞej2pws
ðlð Þ Þ⊤ ð1tþtÞ
dl
where Eqs. (2.51) and Eqs. (3.13) have been accounted for. Moreover, by using the change of variables ( )n ln ¼ csn( fn), with csn() being the inverse function of ’sn() (see Section III.B.2), Eq. (3.104) becomes X Bx;s ð1t þ tÞ ð3:105Þ Ry ð1t þ tÞ ¼ s2On
where
172
IZZO AND NAPOLITANO
FIGURE 7. Graph of the (a) magnitude and (b) support in the (a, t1) plane of the slice for t2 ¼ t1 and t3 ¼ 0 of the fourth-order reduced-dimension CTMF Raxxxx ðt1 ; t2 ; t3 Þ of the input DSB signal x(t).
GENERALIZED ALMOST-CYCLOSTATIONARY SIGNALS
173
FIGURE 8. Graph of the (a) magnitude and (b) support in the (a, t1) plane of the slice for t2 ¼ t1 and t3 and t3 ¼ 0 of the fourth-order reduced-dimension CTMF Rayyyy ðt1 ; t2 ; t3 Þ of the output y(t) to the purely random Doppler channel defined in Eqs. (3.82) and (3.83) and excited by the DSB signal x(t).
174
IZZO AND NAPOLITANO
Z Bx;s ð1t þ tÞ ≜
N
Y ⊤ ðÞ ð Þ f ej2pf ð1tþtÞ df Hsn n ð Þn fn S x c ð Þ s
RN n¼1
ð3:106Þ ð Þ with c ð Þ Þ ≜ ½ð Þ1 cs1 ðð Þ1 f 1 Þ; . . . ; ð ÞN csN ðð ÞN fN Þ⊤ . s ðf
a. GACS Input not Containing ACS Components. If hs ðtÞ 2 L2 ðRÞ 8s 2 O, then also Hs ð f Þ 2 L2 ðRÞ 8s 2 O. In addition, if x(t) is GACS not containing any ACS component so that the SMF S x ðlÞ is not impulsive in l (see ðÞn Section II.C.2) and, moreover, is measurable and bounded, then PN n¼1 Hsn N N ð Þ ð Þ ðð Þn fn ÞS x ðc s ð f ÞÞ 2 L2 ðR Þ and, hence, Bx;s ð1t þ tÞ 2 L2 ðR Þ (as a function of t). Consequently, if Bx;s ð1t þ tÞ is regular for ktk ! 1, then it is infinitesimal as ktk ! 1 and, hence, as j t j! 1. Therefore, Bx;s ð1t þ tÞ, as function of t, is a function with zero power so that the product Bx;s1 ð1t þ t1 ÞBx;s2 ð1t þ t2 Þ does not contain any finite-strength additive sinewave component. Thus, the TMF Ry ð1t þ tÞ is zero in the temporal mean-square sense. The same result is found if Hs ð f Þ 2 L1 ðRÞ and S x ðlÞ is measurable and bounded. Therefore, in general, FOT deterministic systems excited by GACS signals not containing any ACS component deliver signals with zero power. Such a result could not hold if some hs ðtÞ 2 = L2 ðRÞ ore some 1 Hs ð f Þ 2 = L ðRÞ. In Section III.I the case in which hs(t) is impulsive is treated in detail [see also the example of Eq. (3.113)]. b. ACS Input. In the case where the signal x(t) is ACS, by substituting the expression of the GCSMF [Eq. (2.53)] valid for ACS signals into Eq. (3.73) specialized to the case of FOT deterministic systems [i.e., with Eqs. (3.57) and (3.58) substituted into] and accounting for the sampling property of Dirac’s delta function, one obtains that X XX X Z a Ry ðtÞ ¼ N g2A t s2O0
P b⊤ 1¼g
RN p
p Y 0⊤ ri @GðÞ sri ðð Þri ðbmi lmi 1ÞÞ i¼1 ej2pKm;s ðl;tÞ d
p Y i¼1
Km;s ðl; tÞ ≜
p X i¼1
n2mi fri g
1 A n GðÞ sn ðð Þn ln Þ
ð3:107Þ
Km;s ðl;1Þ a
bm 0 0 Pxmii ðlmi Þdlm1
with
Y
0
. . . dlmp
0 0 ð Þ ⊤ 0 ½ð Þri ’sr ð Þri ðbmi lm⊤i 1Þ tri þ ws0 ðlð Þ mi Þ tmi ð3:108Þ i
mi
ð Þ
where ri denotes the last element in mi and wsmi ðlð Þ mi Þ is the jmij-dimensional
GENERALIZED ALMOST-CYCLOSTATIONARY SIGNALS
175
vector whose elements are ð Þn ’sn ðð Þn ln Þ with n 2 mi. Moreover, under the mild assumption that the time series x(t) and x(t þ t) are asymptotically bm ðjtj ! 1Þ independent (see Section II.B.), the cyclic polyspectra Pxmii ðl0mi Þ are not impulsive (Gardner and Spooner, 1994) and, hence, the following inclusion relationship holds: Ay;t [
[
[
[
s2ON g2Ax m2P b2Bm1 ... Bmp
where ðs;g;m;bÞ
At
Aðs;g;m;bÞ t
n n ≜ a 2 R : measN p ½l0m1 ; . . . ; l0mp 2 RN p: o o Km;s ðl; 1Þ ¼ a; b⊤ 1 ¼ g > 0
ð3:109Þ
ð3:110Þ
which is independent of t and in the following will be denoted by A(s,g,m,b). In Eq. (3.110), the set corresponding to the partition with p ¼ N can be not empty only if x(t) contains finite-strength additive sine wave components. In fact, for p ¼ N it results that ( ) N
X ⊤ ðs;g;m;bÞ A ≜ a2R: ð Þn ’sn ð Þn bn ¼ a; b 1 ¼ g ð3:111Þ n¼1
⊤
where m ¼ ½1; . . . ; N and b ¼ ½b1 ; . . . ; bN ⊤ with bn frequencies of the additive almost-periodic component in x(t). From the countability of the bn’s, the countability of the set A(s,g,m,b) follows immediately. With regard to the sets corresponding to partitions with p < N, Eq. (3.110) can be written as ( ( Aðs;g;m;bÞ ¼
a 2 R : measjmi j 1 l0mi 2 Rjmi j 1 :
0 ð Þ ð Þri ’sr ð Þri ðbmi l0mi ⊤ 1Þ þ ws0m ðlð Þ Þ⊤ 1 ¼ ki ; m i i i ) ) p X ki ¼ a b⊤1 ¼ g > 0; i ¼ 1; . . . ; p;
ð3:112Þ
i¼1
from which it follows that a suYcient condition to ensure that some set A(s,g,m,b) is nonempty is that some function ’s () contains a linear part with unit slope, that is, that the system contains a LAPTV component. In the special case where s ¼ s01, it is not necessary that the linear part has unit slope; that is, it is suYcient that the system contains a time-scale changing component. Finally, note that since the sets A(s,g,m,b) do not depend on t, if the output y(t) has finite power, then it is ACS. As an example, let us consider the Doppler channel existing between a transmitter and a receiver with constant relative speed, that is, an FOT
176
IZZO AND NAPOLITANO
deterministic LTV system characterized by the impulse-response function hðt; uÞ ¼ dðu t þ ðd0 þ d1 tÞÞ:
ð3:113Þ
Such a system introduces a linearly time-varying delay D(t) ¼ d0 þ d1t, that is, it performs a time-scale changing [see Eqs. (3.20) through (3.22)] and introduces a constant delay in the output signal: yðtÞ ¼ xðð1 d1 Þt d0 Þ ¼ xðð1 d1 ÞtÞ ! dðt d0 =ð1 d1 ÞÞ:
ð3:114Þ
By assuming, for the sake of simplicity, d0 ¼ 0 and by substituting Eqs. (3.20) through (3.22) into Eqs. (3.105) and (3.106), it results that Ry ð1t þ tÞ ¼ Rx ðsð1t þ tÞÞ
ð3:115Þ
Ray ðtÞ ¼ Ra=s x ðstÞ
ð3:116Þ
where s ≜ 1 d1 . Therefore, if the input signal is the DSB signal defined in Eq. (3.103), then the output signal y(t) is in turn DSB with cycle frequencies )2f0s and )4f0s. Figure 9 shows (a) the magnitude and (b) the support in the (a, t1) plane of the slice for t2 ¼ t1 and t3 ¼ 0 of the reduced-dimension fourth-order CTMF of the output DSB signal y(t) estimated by M ¼ 211 samples. The input DSB signal has been assumed to be the same as that considered in the experiment described in Section III.F.1. and the Doppler channel parameters have been fixed at d0 ¼ 0Ts and d1 ¼ 1:7. The simulation results are in accordance with the theoretical results. Finally, it is worthwhile to note the diVerent behavior of the two Doppler channels Eqs. (3.82) and (3.83) and (3.113)] in the presence of an ACS input signal. Specifically, the Doppler channel [Eqs. (3.82) and (3.83)] is a FOT purely random LTV system and, then, the output signal is a GACS signal. On the contrary, the Doppler channel Eq. (3.113) is a FOT deterministic LTV system and, hence, the output signal is an ACS signal. G. LAPTV Filtering This section considers the LAPTV filtering of GACS signals (Izzo and Napolitano, 2000a, 2002c). Let us consider an LAPTV system, that is, an FOT deterministic LTV system with impulse-response function X hðt; uÞ ¼ hs ðt uÞe j2psu ð3:117Þ s2O
where O is the countable set of the frequency shifts introduced by the system.
GENERALIZED ALMOST-CYCLOSTATIONARY SIGNALS
177
FIGURE 9. Graph of the (a) magnitude and (b) support in the (a, t1) plane of the slice for t2 ¼ t1 and t3 ¼ 0 of the fourth-order reduced-dimension CTMF Rayyyy ðt1 ; t2 ; t3 Þ of the output y(t) to the FOT deterministic Doppler channel defined in Eq. (3.113) and excited by the DSB signal x(t).
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Accounting for Eqs. (3.12) and (3.18), it results that ’s ðlÞ ¼ l þ s;
s2O
ð3:118Þ
and ð3:119Þ Gs ðlÞ ¼ Hs ðl þ sÞ; s 2 O where the functions Hs() are Fourier transforms of the functions hs(). Then, by specializing Eqs. (3.57) and (3.58), one has ’y ðl; tÞ ¼ l⊤ 1 þ sð Þ⊤ 1
ð3:120Þ
and Gy ðl; tÞ ¼
N Y * + j2pðlþsð Þ Þ⊤ t n ð Þ l þ s HsðÞ n e n n n
ð3:121Þ
n¼1
where s( ) ≜ [( )1 s1, . . ., ( )N sN]⊤. The case of LAPTV systems excited by ACS signals is widely treated in Napolitano (1995). The general result, valid for any FOT deterministic LTV system, that the output TMF is zero in the temporal mean-square sense when the input is a GACS signal not containing any ACS component (see Section III.F.2) can be analyzed in more detail in the case of LAPTV systems. Let us denote by Dx,y the union of the two sets Dx and Dy, where Dx [Dy] is the set of the points t 2 RN such that some of the diVerent input [output] lag-dependent cycle frequencies assume the same value (see Section II.C.1). Thus, for all t not belonging to Dx,y, it results that the GCTMF of the output time series y(t) can be written as Ry; ðtÞ ≜ hLy ð1t þ tÞe j2pb ðtÞt it X ¼ hBx;s ð1t þ tÞe j2pb ðtÞt it N s2O # *" + N Y X X j2psð Þ⊤ ð1tþtÞ ðÞn j2pb ðtÞt ¼ Rx;z ð1t þ tÞe ! hsn ðtn Þ e t
s2ON z2Wx
¼
X h
s2O
Rb s x
ð Þ⊤1
ðtÞe
j2psð Þ⊤ t
i
t
N
n¼1
N Y ðÞn ! hsn ðtn Þ n¼1
t
b¼b ðtÞ
ð3:122Þ which, in the case of LTI systems, reduces to Ry; ðtÞ ¼ Rbx ðtÞ ! t
N Y n¼1
hðÞn ðt
n Þ
b¼b ðtÞ
:
ð3:123Þ
GENERALIZED ALMOST-CYCLOSTATIONARY SIGNALS
179
In Eqs. (3.122) and (3.123), !t denotes N-dimensional convolution with respect to t and, in the derivation of the second and fourth equality in Eq. (3.122), Eqs. (3.105) (specialized to LAPTV systems) and (2.30), respectively, have been accounted for. Moreover, for all t 2 Dx;y it results that Ry; ðtÞ ¼
lim
Dt!0 tþDt2RN Dx;y
Ry; ðt þ DtÞ:
ð3:124Þ
Taking into account Eq. (2.30) and the formal relationship dt t0 ! dðtÞ ¼ dt t0
ð3:125Þ
Equations (3.122) through (3.124) reveal that the output GCTMFs can be not identically zero only if the input time series contains ACS components (in which case the output time series is ACS) unless some function hsn() (or h ()) contains impulsive terms, as it occurs in the case of systems introducing constant time delays or frequency shifts. Therefore, every LAPTV (or, in particular, LTI) filtering of a GACS signal not containing ACS components delivers a signal with all the GCTMFs identically zero, that is, with zero power. Indeed, GACS signals not containing ACS components exhibit timeaveraged autocorrelation function Rxx ðt1 ;0Þ containing the additive term x2 dt1 (see Section II.C.1), so that the power x2 is uniformly spread over an infinite bandwidth. Then, the output of every filter Hs( f ) with finite bandwidth is a signal with zero power. To corroborate such a result by an example, let us consider the output y(t) of a LTI filter with transfer function Hð f Þ ¼ ð1 þ jð f fh Þ=Bh Þ 4 þ ð1 þ jð f þ fh Þ=Bh Þ 4 , where fh ¼ 0:015=Ts and Bh ¼ 0:005=Ts : The input GACS signal x(t) is that obtained at the output of the Doppler channel considered in Section III.F.1. The estimate of the magnitude of the cyclic a autocorrelation function Ryy* (t1) of y(t) is reported in Figure 10 and as a function of the lag t1 and the cycle frequency a, for a data-record length of 211 and 214 points, respectively. The approach of the estimates to the identically zero value as the data-record length increases is evident. Finally, let us observe that in several estimation problems signals can be modeled as ACS or GACS depending on the data-record length. This fact puts some limitations in the performances obtainable with some signal processing algorithms adopted in communication applications where ACS signals are processed by LAPTV systems (see, e.g., the cyclic Wiener filtering for interference removal; Gardner, 1994). In fact, if the data-record length is increased too much in order to gain a better immunity against the eVects of noise and interference, it can happen that the ACS model for the input signal is not appropriate anymore but, rather, a GACS model needs to be considered since possible time variations of timing parameters of the signals (not
180
IZZO AND NAPOLITANO
FIGURE 10. Graph of magnitude of the cyclic autocorrelation function Rayy ðt1 Þ of the output y(t) to a LTI system excited by a GACS input signal, estimated by a data-record length of (a) 211 and (b) 214 samples.
evidenced with smaller data-record length) must be taken into account (see, e.g., the example in Section II.F.2). Consequently, increasing the datarecord length too much does not have, for example, the beneficial eVect of improving the reliability of the output-signal cyclic statistic estimates but, rather, gives rise to cyclic statistics (and generalized cyclic statistics) that are asymptotically zero. Therefore, there exists an upper limit to the maximum
GENERALIZED ALMOST-CYCLOSTATIONARY SIGNALS
181
usable data-record length and, consequently, there exists a limit to the minimum acceptable signal-to-noise ratio for cyclostationarity-based algorithms which are, in principle, intrinsically immune to the eVects of noise and interference, provided that the data-record length approaches infinity. The identically zero (generalized) cyclic statistics of the LAPTV filtered GACS signals are consequence of the properties of the single observed time series (e.g., the possible time variation of a timing parameter, such as carrier frequency or baud rate). In contrast to this, statistical functions of a stochastic process can be identically zero as a consequence of the presence, in the stochastic process model, of a random parameter whose eVect is to make the stochastic expectations equal to zero. In such a case, however, in general the stochastic process is not ergodic and, hence, the FOT framework is more appropriate. H. Product Modulation Let us consider the LTV system that performs a product modulation of the input GACS time series x(t), that is, whose input/output relationship is yðtÞ ¼ cðtÞxðtÞ
ð3:126Þ
where c(t) is a GACS time series statistically independent of x(t) in the FOT probability sense and with Nth-order TMF X Rc ð1t þ tÞ ¼ Rc;x ðtÞe j2pgx ðtÞt : ð3:127Þ x2Wc
The system impulse-response function is hðt; uÞ ¼ cðtÞdðt uÞ:
ð3:128Þ
Then, the system is purely random [see Eq. (3.27)] if c(t) does not contain finite-strength additive sine waves. Accounting for Eqs. (3.126) and (3.127), the almost-periodic component ⊤ of the output lag product corresponding to the input lag product e j2pl (1tþs) is given by X ⊤ ⊤ Rc;x ðtÞe j2pl t e j2p½gx ðtÞþl 1t : ð3:129Þ Efag fLy;l ð1t þ tÞg ¼ x2Wc
Then, by comparing Eq. (3.129) with Eq. (3.41), it results that there is a one-to-one correspondence between the elements y of the set Y and the elements x of Wc and, moreover, ’y ðl; tÞ ¼ gx ðtÞ þ l⊤ 1
ð3:130Þ
182
IZZO AND NAPOLITANO ⊤
Gy ðl; tÞ ¼ Rc;x ðtÞej2pl t :
ð3:131Þ
Furthermore, from Eq. (3.43) it follows that the temporal moment function of the product modulation transformation can be written as Rh ð1t þ t; uÞ ¼ Rc ð1t þ tÞdð1t þ t uÞ
ð3:132Þ
where d(u) is N-dimensional Dirac’s delta function. Hence, accounting for Eq. (3.54), the input/output relationship in terms of TMFs is Ry ð1t þ tÞ ¼ Rc ð1t þ tÞRx ð1t þ tÞ
ð3:133Þ
which is formally analogous to that obtained in the stochastic process framework. Note that, Eq. (3.132) confirms the interpretation (see Section III.C) of Rh(1t þ t, u) as the expectation (in the FOT sense) of the lag product of the impulse-response function. In fact, Rc(1t þ t) is just the almost-periodic component (that is, the expectation in the FOT sense) of the lag product of the signal c(t). By substituting Eq. (3.127) into Eq. (3.133) and taking into account that the expression of the Nth-order TMF of x(t) is X Rx ð1t þ tÞ ¼ Rx;z ðtÞe j2paz ðtÞt ; ð3:134Þ z2Wx
one obtains that the potential lag-dependent cycle frequencies of the output GACS time series are b ðtÞ ¼ az ðtÞ þ gx ðtÞ;
ðz; xÞ 2 Wx Wc
ð3:135Þ
Ry; ðtÞ ≜ hRy ð1t þ tÞe j2pb ðtÞt it X X ¼ Rx;z ðtÞRc;x ðtÞdb ðtÞ ½az ðtÞþgx ðtÞ
ð3:136Þ
and, moreover,
z2Wx x2Wc
for all t 2 = Dx [ Dy . Note that, in the special case in which x(t) and c(t) are both ACS time series, Eq. (3.136) reduces to Eq. (29) of Spooner and Gardner (1994). Finally, let us observe that when the time series c(t) is almost periodic, the transformation Eq. (3.126) is LAPTV (and hence FOT deterministic) and therefore, according to Eq. (3.56), it results that Rc ð1t þ tÞ ¼
N Y cðÞn ðt þ tn Þ: n¼1
ð3:137Þ
GENERALIZED ALMOST-CYCLOSTATIONARY SIGNALS
183
Moreover, accounting for Eqs. (3.59) and (3.133), the input/output relationship in terms of temporal cumulant functions is the same as that in terms of temporal moment functions: Cy ð1t þ tÞ ¼ Rc ð1t þ tÞCx ð1t þ tÞ:
ð3:138Þ
I. Multipath Doppler Channels Let us consider the reflection of a transmitted signal xðtÞ ¼ Ref~ xðtÞe j2pfc t g
ð3:139Þ
on a target, embedded in a homogeneous medium, that can be described by the slowly fluctuating point target model (Van Trees, 1971). The received signal is given by yðtÞ ¼ Ref~ yðtÞej2pfc t g ¼ Refb~ xðt DðtÞÞej2pfc ðt DðtÞÞ g
ð3:140Þ
where Re{} denotes real part, b is the complex attenuation, and D(t) is the round-trip time-varying delay introduced by the channel: , 2 DðtÞ DðtÞ ¼ R t : ð3:141Þ c 2 In Eq. (3.141), R(t) is the time-varying distance between the transmitter and the target at time t and c is the medium propagation speed. In the case of constant radial speed w of the target with respect to the transmitter, i.e., if R(t) ¼ R0 þ wt, then the delay D(t) depends linearly on t, say D(t) ¼ d0 þ d1t, where d0 ≜ 2R0/(c þ w) and d1 ≜ 2w/(c þ w). Thus, the ~ðtÞ by the LTV transformation with impulsesignal ~ yðtÞ is obtained from x response function hðt; uÞ ¼ a dðu st þ d0 Þe j2pnt
ð3:142Þ
where s ≜ 1 d1 is the time-scale factor, n ≜ d1 fc is the Doppler shift introduced by the channel, and a ≜ be j2pfd0. In the special case where fc ¼ 0 and a ¼ 1, this channel reduces to that considered in the example in Section III.F.2. Moreover, in the case of multiple slow fluctuating point scatterers, the LTV system is a multipath Doppler channel with impulse-response function hðt; uÞ ¼
K X
ak dðu sk t þ dk Þej2pnk t
k¼1
where K is the number of the channel paths.
ð3:143Þ
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IZZO AND NAPOLITANO
By double Fourier transforming the right-hand side of Eq. (3.143), one obtains the transmission function defined in Eq. (3.10) Hð f ; lÞ ¼
K X ak e j2pldk dð f nk lsk Þ
ð3:144Þ
k¼1
from which, accounting for Eq. (3.11a), it follows that the multipath Doppler channel is an FOT deterministic LTV system with O ¼ {1, . . . , K}, ’k ðlÞ ¼ sk l þ nk ; Gk ðlÞ ¼ ak e j2pldk ;
k ¼ 1; . . . ; K
ð3:145Þ
k ¼ 1; . . . ; K:
ð3:146Þ
Equivalently, accounting for Eq. (3.12), we get 1 ck ð f Þ ¼ ð f nk Þ; k ¼ 1; . . . ; K sk Hk ð f Þ ¼
ak j2pð f nk Þdk =sk e ; jsk j
ð3:147Þ
k ¼ 1; . . . ; K
ð3:148Þ
Therefore, by substituting Eqs. (3.145) and (3.146) into Eqs. (3.57) and (3.58) and then the result into Eq. (3.43), one obtains the following expression for the system temporal moment function of the multipath Doppler channel (Izzo and Napolitano, 2000b, 2002c): N
X Y ð Þ⊤ ðÞ Rh ð1t þ t; uÞ ¼ akn n e j2pnk ð1tþtÞ dðsk + ð1t þ tÞ d k uÞ k2IKN
n¼1
ð3:149Þ ⊤
⊤
⊤
ð Þ
where k ≜ ½k1 ; . . . ; kN , sk ≜ ½sk1 ; . . . ; skN , d k ≜ ½dk1 ; . . . ; dkN , nk ≜ ½ð Þ1 nk1 ; . . . ; ð ÞN nkN ⊤ , IK ≜ {1, . . . , K}, and + denotes the Hadamard matrix product. By substituting Eq. (3.149) into Eq. (3.54), one obtains the TMF of the output signal N
X Y ð Þ⊤ ðÞ R~y ð1t þ tÞ ¼ akn n ej2pnk ð1tþtÞ Rx~ ðsk + ð1t þ tÞ d k Þ ð3:150Þ k2IKN
n¼1
where the equality must be intended in the temporal mean-square sense (see Appendix A). In general, the time-scale factors of diVerent paths are diVerent, that is, sh 6¼ sk for h 6¼ k. Therefore, since the equality in Eq. (3.150) is in the temporal mean-square sense, not all terms in the right-hand-side of Eq. (3.150) give a contribution to the almost-periodic component of the ~(t) Nth-order output lag product. In fact, under the mild assumption that x
GENERALIZED ALMOST-CYCLOSTATIONARY SIGNALS
185
~(t þ t) are asymptotically (jtj ! 1) independent in the FOT probabiland x ity framework (see Section II.B), it results that if sh 6¼ sk for h 6¼ k, then, for t ¼ 1t0, jt0j ! 1, one has Rx~ ðs + ð1t þ tÞÞ !
N Y Efag f~ xðÞn ðsn t þ sn tn Þg
ð3:151Þ
n¼1
~(*)h(sht þ shth) and x ~(*)k(skt þ sktk) turn since for any h 6¼ k the time series x out to be time shifted of infinite quantities one another. Moreover, note that the same factorization of the TMF is obtained for jtj ! 1 since also in this ~(*)h(sht þ sh th) and x ~(*)k(skt þ sk tk) are time shifted of case the time series x infinite quantities one another. If at least one of the sk is diVerent from all the others, it results that even if not all the entries sk are diVerent, at least two of them are not equal, so that in the asymptotic (jtj ! 1) factorization of Rx~ ðs + ð1t þ tÞÞ there is at least one term of the kind E{a}{~ x(*)n (snt þ sntn)}. ~(t) does not contain any finite-strength additive sine wave compoThus, if x nent, then lim Rx~ ðs + ð1t þ tÞÞ ¼ 0:
jtj!1
ð3:152Þ
Consequently, when at least one of the sk is diVerent from all the others, the function of t Rx~ ðs + ð1t þ tÞÞ has zero power and, hence, the product Rx~ ðs1 + ð1t þ t1 ÞÞ Rx~ ðs2 + ð1t þ t2 ÞÞ does not contain finite-strength additive sine wave components. Therefore, by assuming in (3.150) that at least one of the sk is diVerent from all the others, the terms with s 6¼ 1sn (that is, with k 6¼ 1kn) are zero in the temporal mean-square sense, so that Eq. (3.150) becomes R~y ð1t þ tÞ ¼
N
X Y ð Þ⊤ ðÞ ak n e j2pnk 1 ð1tþtÞ Rx~ ðsk ð1t þ tÞ 1dk Þ
k2IK
ð3:153Þ
n¼1
where the equality is in the temporal mean-square sense and the fact that s1kn ¼ 1skn has been accounted for. Finally, from Eq. (3.153) we have Ra~y ðtÞ ¼
N
X Y ð Þ⊤ ðÞ ak n ej2pnk 1 t
k2IK
X
zk 2Wx
n¼1
Rx;zk ðsk t 1dk Þdaz
k
ðsk tÞ ða 1ð Þ⊤ 1nk Þ=sk
N
X Y ð Þ⊤ ðÞ ¼ ak n e j2pnk 1 t k2IK
n¼1
ða 1ð Þ⊤ 1nk Þ=sk Rx~ ðsk t
1dk Þ
ð3:154Þ
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IZZO AND NAPOLITANO
where 1ð Þ ≜ ½ð Þ1 1; ; ð ÞN 1⊤ and, in the derivation of the last equality, Eq. (2.30) has been accounted for. ~(t) is GACS so is ~yðtÞ and, moreover, From Eq. (3.154) it follows that if x the lag-dependent cycle frequencies of ~yðtÞ are given by bz;k ðtÞ ¼ sk az ðtÞ þ 1ð Þ⊤ 1nk ;
z 2 Wx ; k 2 IK :
ð3:155Þ
~ðtÞ is ACS, ~yðtÞ is in turn ACS. Furthermore, in the special case in which x Let us assume that the observation interval for the estimation of the (generalized) cyclic statistics is [ T/2, T/2]. Thus, equation Eq. (3.150) or Eq. (3.153) must be used to describe the output behavior, depending on the value of the product bandwidth data-record length. In fact, for the signal ~ðtÞ with bandwidth W, it results that x ,, -2w ~ðstÞ ¼ x ~ 1 ~ðtÞ x t ’x ð3:156Þ cþw provided that the condition WT % 1 þ
c w
ð3:157Þ
is satisfied (Van Trees, 1971). Therefore, if such a condition holds for each of the K channel paths, then the time-scale factors sk can be considered unitary and, hence, the functions ’k(l) [see Eq. (3.145)] are linear with unit slope, that is, the multipath Doppler channel can be modeled as a LAPTV system. Consequently, all terms in Eq. (3.150) give nonzero contribution in the temporal mean-square sense and the CTMF Ra~y ðtÞ of the output signal is given by Ra~y ðtÞ ¼
N
ð Þ⊤ X Y ð Þ⊤ a n 1 ðÞ akn n e j2pnk t Rx~ k ðt d k Þ:
k2IKN
ð3:158Þ
n¼1
On the contrary, when Eq. (3.157) is not satisfied, Eq. (3.153) holds and, hence, the CTMFs of the output signal are related to those of the input signal by Eq. (3.154). To corroborate the fact that a diVerent behavior of the (generalized) cyclic statistics can be predicted by Eq. (3.154) or Eq. (3.158) depending on the data-record length adopted for the CTMF estimates, the following experiment has been carried out. Let us consider the output of a two-path Doppler channel characterized by a1 ¼ 1, d1 ¼ 0, s1 ¼ 1, n1 ¼ 0, a2 ¼ 1, d2 ¼ 32Ts, s2 ¼ 0.9998, and v2 ¼ 2.49975 10 5 / Ts, where Ts is the sampling period, and the second path is obtained by considering a relative radial speed v2/c ¼ 10 4. The channel is excited by a binary pulse-amplitude modulated signal with bit rate 1/16Ts and full-duty-cycle rectangular pulse. The estimate
GENERALIZED ALMOST-CYCLOSTATIONARY SIGNALS
187
of the magnitude of the cyclic autocorrelation function of y(t) is reported in Figure 11a and b as a function of the lag t1 and the cycle frequency a, for a data-record length of 213 and 218 points, respectively. For T ¼ 213 Ts it results WT % (1 þ c/w2), then the terms with k 6¼ 1kn must be accounted
FIGURE 11. Graph of magnitude of the cyclic autocorrelation function Ryy a ðt1 Þ of the output y(t) to a multipath Doppler channel excited by a pulse-amplitude–modulated input signal, estimated by a data-record length of (a) 213 and (b) 218 samples.
188
IZZO AND NAPOLITANO
for according to Eq. (3.158) and hence, the channel can be modeled as LAPTV (see Figure 11a). On the contrary, according to Eq. (3.154), such terms disappear for T ¼ 218Ts since, in such a case, WT ’ 1.6 (1 þ c/w2) (see Figure 11b). Briefly, both the above analysis and the previous experiment show that if the product bandwidth data-record length is suYciently small, then the multipath Doppler channel can be modeled as an LAPTV system, since the eVects of the time scaling factors can be neglected. On the contrary, if the data-record length is increased (e.g., to obtain a high noise immunity in cyclostationarity-based algorithms) the time scaling factors must be taken into account and the system cannot be modeled as LAPTV but, rather, as FOT deterministic. In Napolitano (2003), for the output of a multipath Doppler channel with ACS input, the second-order spectral characterization is performed in the stochastic process framework. It is shown that, if at least one of the sk is diVerent from the others, then the output can be modeled as a spectrally correlated stochastic process whose Loe`ve bifrequency spectrum has the support in the bifrequency plane constituted by lines with slopes sk1/sk2. Moreover, if the values of the sk’s are unknown, then only the density of the Loe`ve bifrequency spectrum on lines with unit slope (sk1 ¼ sk2) can be consistently estimated. Finally, let us note that if the attenuation introduced by the channel cannot be assumed constant, the appropriate model is the Doppler-spread channel (or time-selective fading channel) for which the output corresponding to the input signal Eq. (3.139) is given by yðtÞ ¼ Ref~ yðtÞej2pfc t g ¼ RefAðt DðtÞ=2Þ~ xðt DðtÞÞej2pfc ðt DðtÞÞ g
ð3:159Þ
(Van Trees, 1971), where A(t) is the time-varying attenuation. Therefore, the ~ðtÞej2pfc t into ~yðtÞe j2pfct has impulse-response LTV channel that transforms x function hðt; uÞ ¼ Aðt DðtÞ=2Þdðu t þ DðtÞÞ:
ð3:160Þ
Consequently, the output statistics can be evaluated considering first the eVects of the time-varying delay (see the examples of Sections III.F.1 and 2) and then the eVects of the product modulation introduced by A(t D(t)/2) (see Section III.H). The model [Eq. (3.159)] was studied in Blanco and Hill (1979) in the stochastic process framework. Moreover, the special case of A(t) and D(t) periodic functions with the same period was considered in Duverdier and Lacaze (1996).
GENERALIZED ALMOST-CYCLOSTATIONARY SIGNALS
189
J. Summary In this section, the problem of linear time-variant filtering of GACS signals has been addressed in the nonstochastic framework. Systems have been classified as deterministic or random in the fraction-of-time probability framework. Moreover, the higher-order characterization of linear time-variant systems has been provided in the time domain by the system temporal moment function, which is the operator that transforms the almost-periodic component of the input lag product into the almost-periodic component of the output lag product. The linear systems have also been characterized in the frequency domain and input/output relationships have been provided in terms of both temporal and spectral moment and cumulant functions. The usefulness of the proposed approach has been demonstrated by the statistical characterization of those systems that in the stochastic approach are modelled as random. In fact, such systems, which include, for example, the fading communications channels, in general transform ergodic input signals into nonergodic output signals. The countability of the cycle frequency set of the output signal of a LTV system has been studied. Then, it has been shown that, unless the output signal is zero power, general LTV systems deliver GACS signals if they are excited either by ACS signals or GACS signals not containing any ACS component. The FOT deterministic LTV systems deliver ACS signals when excited by ACS signals, provided that they contain a LAPTV or a time-scale changing component. Moreover, they deliver an output zero-power signal when excited by GACS signals not containing any ACS component, unless the impulse-response function contains impulsive terms. The special cases of LAPTV filtering, product modulation, and Doppler channel filtering have been analyzed in detail. In the considered examples, it has been shown that in practical situations diVerent system or signal models should be adopted depending on the data-record length. With reference to LAPTV filtering, it has been shown that there exists an upper limit in the maximum usable data-record length and, consequently, there exists a limit to the minimum acceptable signal-to-noise ratio for cyclostationarity-based algorithms when the increasing of the data-record length makes for the input signal the GACS model more appropriate than the ACS one. With regard to the multipath Doppler channels, it has been shown that they can be modeled as LAPTV or FOT deterministic systems depending on the value of the product bandwidth data-record length. If the product bandwidth data-record length is suYciently small, then the multipath Doppler channel should be modeled as an LAPTV system, whereas, if the data-record length is not small (e.g., to obtain a high-noise immunity in cyclostationarity-based algorithms) the system cannot be modeled as LAPTV but, rather, as FOT deterministic. As a consequence, the output
190
IZZO AND NAPOLITANO
estimated (generalized) cyclic statistics obey to two diVerent models, depending on the available data-record length for the estimates. Simulation results for the described applications have been presented.
IV. SAMPLING
OF
GACS SIGNALS
A. Introduction The proper statistical characterization of the nonstationary signal at the output of a time-varying channel is the first step to properly perform detection, equalization, and demodulation. Thus, since in most of signal-processing applications continuous-time signals are subject to sampling operations, the link between continuous- and discrete-time is of great interest. This section addresses the problem of sampling a continuous-time GACS signal. It is shown that the discrete-time signal constituted by the samples of a continuous-time GACS signal is a discrete-time ACS signal, whether the continuous-time signal is ACS or not. Moreover, relationships between higher-order generalized cyclic statistics of a GACS signal and higher-order cyclic statistics of the discrete-time signal of its samples are determined and some results derived in Izzo and Napolitano (1996c) and Napolitano (1995) for the case of ACS signals are extended to the more general case of GACS signals. The problem of aliasing in the domain of the cycle frequencies is considered and a condition ensuring that the cyclic temporal moment function of the discrete-time signal can be obtained by sampling that of the continuous-time signal is determined. Spectral parameters of sampled GACS signals have not been considered here since a continuous-time GACS signal that is not ACS exhibits cyclic spectral moment functions that are infinitesimal (see Section II.C.2). Spectral parameters of sampled ACS signals are considered in Izzo and Napolitano (1996c) and Napolitano (1995). The continuous-time GACS signals, ACS or not, give rise to a sampled discrete-time ACS signal. Thus, the nonstationarity type of the sampled signal does not allow us to determine if the underlying continuous-time GACS signal is ACS or not unless further a priori information is available. However, it is shown how the GACS or ACS nature of the continuous-time signal can be conjectured from the behavior of the cyclic temporal moment function of the sampled ACS signal. Specifically, it is shown that, under some regularity conditions, the analysis parameters (e.g., sampling frequency, data-record length, padding factor) can be chosen such that the lag-dependent cycle frequency variations within a sampling period are
GENERALIZED ALMOST-CYCLOSTATIONARY SIGNALS
191
suYciently small so that a slice of the cyclic temporal moment function of a continuous-time GACS signal over a discrete grid for both lags and cycle frequencies can appear as if its support were piecewise continuous. That is, it can appear as if the lag-dependent cycle frequencies were varying with continuity with respect to the lag parameters. Consequently, if the lagdependent cycle frequencies are piecewise constant, then the underlying continuous-time signal should be conjectured to be ACS, otherwise, it should be conjectured to be GACS but not ACS. B. Discrete-Time ACS Signals In this section, some results on the strict and wide sense characterization in the FOT probability framework of discrete-time ACS signals are provided since they will be used in the sequel. Further results can be found in Izzo and Napolitano (1998a). For a treatment in the stochastic approach, see Giannakis (1998). Let us consider a discrete-time complex-valued finite-power time series ~ m;j of all frequencies of the finitex(k) ≜ xr(k) þ jxi(k), k 2 Z. If the set G strength additive sinewave components contained in the function of k N Y ~ x ð1k þ m; jÞ ≜ U Uðxrn xr ðk þ mn ÞÞUðxin xi ðk þ mn ÞÞ ð4:1Þ n¼1
is countable for each of the column vectors m ≜ ½m1 ; . . . ; mN ⊤ 2 ZN and j ≜ j r þ jj i ≜ [xr1 þ jxi1, . . ., xrN þ jxiN]⊤ 2 CN and, moreover, also the set ~m ≜ [ N G ~ m;j is countable for each m 2 ZN, then the time series is said to G j2C be Nth-order almost-cyclostationary in the strict sense. Note that, whereas in the continuous-time case the set G defined in Eq. (2.6) can be countable or ~ ≜ [m2ZN G ~ m is always not, in the discrete-time case, the corresponding set G countable. Consequently, all the discrete-time signals for which the function Eq. (4.1) contains finite-strength additive sinewave components are ACS in the strict sense. Therefore, unlike in the continuous-time case, in the discretetime case a class of GACS (in the strict sense) signals extending that of the ~ uncountable. ACS ones cannot be introduced by considering G It can be shown that the function F~ xðkþm1 Þ xðkþmN Þ ðjÞ ≜ Ef~ag fU x ð1k þ m; jÞg X ~g ¼ F~ x ðm; jÞe j2p~gk
ð4:2Þ
~m ~g2G
is a valid joint cumulative distribution function for each fixed value of m except for the right-continuity property with respect to each xrn and
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IZZO AND NAPOLITANO
~ In xin variable. Moreover, the sum can be equivalently extended to G. Eq. (4.2), K X 1 ~ x ð1k þ m; jÞe j2p~gk U k!1 2K þ 1 K¼ K
~ ~g ðm; jÞ ≜ lim F x
ð4:3Þ
and Ef~ag fg is the almost-periodic component extraction operator for discrete-time series. Consequently, the 2Nth-order derivative of the joint cumulative distribution function Eq. (4.2) with respect to xr1, xi1, . . . , xrN, xiN turns out to be a valid joint probability density function almost periodic in k: f~xðkþm1 ÞxðkþmN Þ ðjÞ ≜
@ 2N ~ xðkþm Þ xðkþm Þ ðjÞ F 1 N @xr1 @xi1 @xrN dxiN X ~g ~ fx ðm; jÞe j2p~gk ¼
ð4:4Þ
~m ~g2G
~g where each Fourier coeYcient f~x ðm; jÞ is the 2Nth-order derivative of the corresponding Fourier coeYcient [Eq. (4.3)] of the joint cumulative distribution function shown in Eq. (4.2). In the FOT probability framework, a discrete-time finite-power possibly complex-valued time series x(k) is said to exhibit Nth-order wide-sense cyclo-stationarity with cycle frequency ~a 2 = Z, for a given conjugation configuration, if the Nth-order CTMF K Y N X 1 xðÞn ðk þ mn Þe j2p~ak k!1 2K þ 1 k¼ K n¼1
~ ~a ðmÞ ≜ lim R x
ð4:5Þ
exists and is not zero for some column vector m ≜ ½m1 ; ; mN ⊤ 2 ZN . In Eq. (4.5), x ≜ [x(*)1 (k), , x(*)N(k)]⊤, and the convergence of the infinite averaging with respect to k is assumed in the temporal mean-square sense or, more generally, in the sense of distributions (generalized functions). If the set ~ m ≜ f~ A a 2 ½ 1=2; 1=2½:
~ ~a ðmÞ 6¼ 0g R x
ð4:6Þ
is countable for each m, then the Nth-order lag product can be expressed as a sum of its almost-periodic component and a residual term not containing any finite-strength additive sine wave component, that is, ~ x ð1k þ mÞ ≜ L
N Y xðÞn ðk þ mn Þ n¼1
~ x ð1k þ mÞ þ ~‘x ð1k þ mÞ: ¼R In Eq. (4.7), the almost-periodic function
ð4:7Þ
GENERALIZED ALMOST-CYCLOSTATIONARY SIGNALS
( ~ x ð1k þ mÞ ≜ E R
f~ag
X
¼
~m ~a2A
N Y xðÞn ðk þ mn Þ
193
)
n¼1
ð4:8Þ
~ ~a ðmÞe j2p~ak R x
which is called the temporal moment function, is a valid moment function and the residual term is such that K X 1 ~‘x ð1k þ mÞe j2p~ak 0; K!1 2K þ 1 k¼ K
lim
Moreover, it can be shown that ~ x ð1k þ mÞ ¼ R
R R
2N
8~a 2 ½ 1=2; 1=2½:
N Y ðxrn þ jxin ÞðÞn
ð4:9Þ
ð4:10Þ
n¼1
f~xðkþm1 ÞxðkþmN Þ ðjÞdj r dj i : Note that, whereas in the continuous-time case the set A defined by Eq. (2.22) can be countable or not, in the discrete-time case, the corresponding set ~≜ [ A ~m A
ð4:11Þ
m2ZN
is always countable. Consequently, all the discrete-time signals for which the Nth-order lag product contains finite-strength additive sine wave components are ACS in the wide sense and the sum in Eq. (4.8) can be extended ˜. over the set A The Nth-order temporal cumulant function of a discrete-time complexvalued ACS signal x(k) can be expressed as " # p X Y p 1 ~ ~ Cx ð1k þ mÞ ¼ ð 1Þ ðp 1Þ! Rx ð1k þ mm Þ : ð4:12Þ i¼1
P
mi
i
Moreover, it turns out to be an almost-periodic function of k that can be written as X b~ ~ ~ ðmÞe j2pbk C~x ð1k þ mÞ ¼ ð4:13Þ C x ~ B ~ b2
where ~
K X 1 ~ ~x ð1k þ mÞe j2pbk C K!1 2K þ 1 k¼ K
b C~x ðmÞ ≜ lim
ð4:14Þ
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IZZO AND NAPOLITANO
and the sets ~≜ [ B ~m B
ð4:15Þ
~ ~ 2 ½ 1=2; 1=2½: C ~b ðmÞ 6¼ 0 ~m ≜ b B x
ð4:16Þ
m2ZN
are countable. C. Sampling of GACS Signals In this section, the sampling of GACS signals is considered and the link between higher-order generalized cyclic statistics of a continuous-time GACS signal and higher-order cyclic statistics of the discrete-time ACS signal constituted by its samples is established (Izzo and Napolitano, 2001, 2003). Let us consider the discrete-time signal x(k) constituted by the samples of the continuous-time GACS signal xc(t): xðkÞ ≜ xc ðtÞjt¼kTs
k 2 Z;
ð4:17Þ
where Ts denotes the sampling period. Accounting for Eq. (2.2), the sampled version of Eq. (2.4) can be written as ~ x ð1k þ m; jÞ ≜ U x ð1t þ t; jÞj U c t¼kTs ;t¼mTs X ¼ F gxc ðmTs ; jÞe j2pgkTs g2GmTs
ð4:18Þ
þ‘U ð1kTs þ mTs ; jÞ where U xc ð1t þ t; jÞ and Fxgc ðt; jÞ are defined by Eqs. (2.1) and (2.3), respectively, and the fact that ‘U(t, t;j) ‘U(1t þ t;j) was used [see the observation following Eq. (2.12)]. The sampled residual term ‘U(1kTs þ mTs; j) does not contain any finite-strength additive (discrete-time) sine wave component, that is (see Appendix D), K X 1 ‘U ð1kTs þ mTs ; jÞ e j2p~gk 0 K!1 2K þ 1 k¼ K
lim
8~g 2 ½ 1=2; 1=2½: ð4:19Þ
Thus, the almost-periodic function (with respect to k) in Eq. (4.18) is ~ x ð1k þ m; jÞ, that is, coincident with the almost-periodic component of U ~ xðkþm Þxðkþm Þ ðjÞ defined by (4.2). with F 1
N
GENERALIZED ALMOST-CYCLOSTATIONARY SIGNALS
195
The Fourier coeYcients of the joint cumulative distribution function of xc(t) are related to those of the joint cumulative distribution function of the discrete-time signal x(k) by the following relationship (see Appendix D): ~ ~g ðm; jÞ ¼ F x
þ1 X p¼ 1
Fxð~gc þpÞfs ðmTs ; jÞ
ð4:20Þ
where fs ≜ 1/Ts is the sampling frequency. Furthermore, by taking the 2Nthorder derivative with respect to xr1, xi1, . . . , xrN, xiN of both sides of Eq. (4.20), one obtains the relationship between the Fourier coeYcients of the joint probability density function of xc(t) and those of the joint probability density function of the discrete-time signal x(k): ~g f~x ðm; jÞ ¼
þ1 X p¼ 1
fxð~cgþpÞfs ðmTs ; jÞ
ð4:21Þ
where fxgc ðt; jÞ is defined by (2.9). The sampled signal x(k) is ACS in the strict sense whether the continuoustime GACS (in the strict sense) signal xc(t) is ACS or not, that is, independently of the fact that the set G ≜ [t 2 R Gt is countable or not. Such a result follows from the relation (proved in Appendix D) X ~g ~ x ð1k þ m; jÞ ¼ ~ ðm; jÞ ej2p~gk þ ‘U ð1kTs þ mTs ; jÞ U F ð4:22Þ x ~m ~g2G
where ~ m ≜ f~g 2 ½ 1=2; 1=2½: F ~ ~g ðm; jÞ 6¼ 0 for some jg G x
ð4:23Þ
~ m. ~≜[ NG and the sum in Eq. (4.22) can be equivalently made over G m2Z To enlighten the wide-sense cyclostationarity properties of the sampled signal x(k) and their relationship with the generalized cyclostationarity properties of the subsumed GACS signal xc(t), it is appropriate to start by considering the sampled version of Eq. (2.19), that is ~ x ð1k þ mÞ ¼ Lxc ð1t þ tÞj L t¼kTs ;
t¼mTs
~ 0 ð1k þ mÞ þ ~‘0 ð1k þ mÞ ¼R x x
ð4:24Þ
where, accounting for Eq. (2.20), the residual term ~‘0 ð1k þ mÞ ≜ ‘x ð1t þ tÞj c t¼kTs ; x
t¼mTs
ð4:25Þ
does not contain any finite-strength (discrete-time) additive sine wave component (the proof is similar to that provided for Eq. (4.19) in Appendix D). ~ 0 ð1k þ mÞ, accounting for Eqs. (2.16) and (2.26), can be The function R x
196
IZZO AND NAPOLITANO
written as ~ 0 ð1k þ mÞ ≜ Rx ð1t þ tÞj R c t¼kTs ; t¼mTs x ¼
X a2AmTs
¼
X
Raxc ðmTs Þe j2pakTs
Rxc ;z ðmTs Þe j2paz ðmTs ÞkTs :
ð4:26aÞ ð4:26bÞ ð4:26cÞ
z2W
~ 0 ð1k þ mÞ ≜ is an almost-periodic function of k. Furthermore, since Thus, R x 0 ~‘ ð1k þ mÞ ≜ does not contain any finite-strength additive sine wave comx ~ 0 ð1k þ mÞ is coincident with R ~ x ð1k þ mÞ, which is the alponent, then R x ~ x ð1k þ mÞ (see (4.7)). most-periodic component contained in L Let us note that the set AmTs in Eq. (4.26b), accounting for Eqs. (2.15) and (2.25), can be written as AmTs ≜ fa 2 R : Raxc ðmTs Þ 6¼ 0g
[ fa 2 R : a ¼ az ðmTs Þg z2W
~0 g ¼ [ fa 2 R : a ¼ ð~a þ pÞfs ; ~a 2 A m
ð4:27Þ
p2Z
~ 0 is defined by where, A m ~ 0 ≜ f~ a 2 ½ 1=2; 1=2½: ~ a ¼ ða=fs Þ mod 1; a 2 AmTs g: A m
ð4:28Þ
In Eq. (4.28), mod b denotes the modulo b operation with values in [ b/2, b/2[, that is, a Mod b a Mod b 2 ½0; b=2½ a mod b ≜ ð4:29Þ ða Mod bÞ b a Mod b 2 ½b=2; b½ with Mod b being the usual modulo b operation with values in [0, b[. The CTMFs of the sampled signal x(k) are linked to the CTMFs of the continuous-time signal xc(t) by the relationship (see Appendix D) ~ ~a ðmÞ ¼ R x
þ1 X p¼ 1
aþpÞfs Rð~ ðmTs Þ xc
ð4:30Þ
which extends the result derived in Napolitano (1995) for ACS signals to the case of GACS signals. Then, accounting for Eq. (4.27), Eq. (4.26b) can be written as ~ x ð1k þ mÞ ¼ R
þ1 X X ~ 0 p¼ 1 ~a2A m
aþpÞfs Rð~ ðmTs Þej2p~afs kTs xc
ð4:31aÞ
197
GENERALIZED ALMOST-CYCLOSTATIONARY SIGNALS
X
¼
~ ~a ðmÞej2p~ak R x
ð4:31bÞ
~ ~a2A m 0
where, in the second equality, Eq. (4.30) has been accounted for. The sampled signal x(k) is ACS in the wide sense whether the continuoustime GACS (in the wide sense) signal S xc(t) is ACS or not, that is, independently of the fact that the set A ≜ t2R At is countable or not. In fact, from ~0 A ~ m and, comparison of Eq. (4.8) with Eq. (4.31b), it results that A m moreover, the last sum in Eq. (4.31b) can be equivalently extended to the elements of the set ~m ~ ≜ [ A A m2ZN
¼ [
m2ZN
¼ [
m2RN
[ f~ a 2 ½ 1=2; 1=2½: ~a ¼ ða=fs Þ mod 1; Raxc ðmTs Þ 6¼ 0g
a2AmTs
[ f~ a 2 ½ 1=2; 1=2½: ~ a ¼ az ðmTs ÞTs mod 1; Rxc ;z ðmTs Þ 6¼ 0g
z2W
ð4:32Þ ˜ is countwhere Eqs. (4.26b) and (4.26c) have been considered. Note that A ˜ able since it is obtained as the union of the countable sets Am for m ranging in the countable set ZN. Therefore, unlike in the continuous-time case, in the discrete-time case a class of GACS (in the wide sense) signals extending that ˜ uncountable. of the ACS ones cannot be introduced by considering A The CTMFs of a discrete-time ACS signal obtained by sampling a continuous-time GACS signal can be expressed in terms of its GCTMFs by the relationship [obtained by substituting Eq. (2.30) into Eq. (4.30)] X ~ ~a ðmÞ ¼ R Rxc ;z ðmTs Þ x ðz;mÞ2X~a
¼
X
z2W
ð4:33Þ
Rxc ;z ðmTs Þd½az ðmTs ÞTs ~a mod 1
where, accounting for Eq. (4.32), X~a ≜ fðz; mÞ 2 W ZN : az ðmTs ÞTs ¼ ~a mod 1; Rxc ;z ðmTs Þ 6¼ 0g
ð4:34Þ
Let us note that, although the discrete-time TMF is the sampled version of the continuous-time TMF [see Eq. (4.26a)], in general for the CTMFs it results that ~ ~a ðmÞ 6¼ Ra ðtÞj R t¼mTs ;a¼~afs xc x
ð4:35Þ
due to the presence of aliasing in the cycle frequency domain [see Eq. (4.30)]. Moreover, from Eq. (4.30) it follows that there is no aliasing in considering
198
IZZO AND NAPOLITANO
the sampled version of the Nth-order CTMF at a given cycle frequency a if and only if there are no cycle frequencies of xc(t) that diVer from a for an integer multiple of the sampling frequency fs. A necessary and suYcient condition ensuring the absence of aliasing in the whole cycle frequency domain is that the sampling frequency fs is suYciently high so that all the cycle frequencies of the continuous-time signal xc(t) belong to the interval [ fs/2, fs/2]. In fact, in such a case, only the term with p ¼ 0 in the right-hand side of Eq. (4.30) can give nonzero contribution in the base support region and, moreover, it results that ~ ~a ðmÞ ¼ Ra ðtÞj R a 2 ½ 1=2; 1=2½; 8m 2 ZN : x xc t¼mTs ;a¼~afs 8~
ð4:36Þ
Note that, in the special case of continuous-time ACS signals, the necessary and suYcient condition on the cycle frequencies assuring no aliasing for the Nth-order CTMF in the whole cycle frequency domain is verified when the signal is strictly bandlimited with bandwidth less than fs/2N since, in this case, all the Nth-order cycle frequencies are less than fs/2 (Napolitano, 1995). In the more general case of GACS signals, from Eq. (4.33) it follows that a suYcient condition assuring no aliasing in the whole cycle frequency domain is jaz ðmTs ÞTs j
1 8m 2 ZN and 8z 2 W such that Rxc ;z ðmTs Þ 6¼ 0: ð4:37Þ 2
With regard to the GCTCFs of the discrete-time signal x(k) constituted by the samples of the continuous-time signal xc(t), in Appendix D it is shown that þ1 X
~
b C~x ðmÞ ¼
p¼ 1
~
s CðxbþpÞf ðmTs Þ: c
ð4:38Þ
Moreover, by reasoning as for the GCTMFs, one obtains that the CTCFs of the discrete-time ACS signal obtained by sampling a continuous-time GACS signal can be expressed in terms of its GCTCFs by the relationship X ~ b C~x ðmÞ ¼ Cxc ;x ðmTs Þ ðx;mÞ2Yb~
¼
X
x2WC
ð4:39Þ
Cxc ;x ðmTs Þd½bx ðmTs ÞTs b ~ mod 1
where ~ mod 1; Cx ;x ðmTs Þ 6¼ 0g: ð4:40Þ Yb~ ≜ fðx; mÞ 2 WC ZN : bx ðmTs ÞTs ¼ b c
GENERALIZED ALMOST-CYCLOSTATIONARY SIGNALS
199
D. Conjecturing the Nonstationarity Type of the Continuous-Time Signal In the previous section it is shown that both ACS and GACS that are not ACS continuous-time signals, by sampling, give rise to ACS discrete-time signals. Thus, the nonstationarity type of the subsumed continuous-time signal cannot be derived from that of the discrete-time signal of its samples. In this section, it is shown how the possible ACS nature of a continuous-time GACS signal can be conjectured from the behavior of the support of the CTMF of the sampled ACS signal (Izzo and Napolitano, 2002a, 2003). The support in [ 1/2, 1/2 [ ZN of the CTMF (as a function of (~a, m) ) of the discrete-time signal Eq. (4.17), accounting for Eq. (4.33), is given by ~a
~ ðmÞ ¼ [ fð~ supp ½R a; mÞ 2 ½ 1=2; 1=2½ ZN : x z2W
ð4:41Þ
~ a ¼ az ðmTs ÞTs mod 1; Rxc ;z ðmTs Þ 6¼ 0g that is, is contained in the countable set of manifolds described by the equations ~ a ¼ az (mTs)Ts mod 1, z 2 W. Furthermore, by using a M-point DFT based algorithm with a zero-padding factor F for evaluating the CTMF in Eq. (4.41) as a function of ~a and m, the cycle frequency step will be D~ a ¼ 1/MF. Consequently, if the parameters Ts, M, and F are such that the cycle frequency variations within a sampling period are smaller than the (normalized) cycle frequency step, that is, if for all z 2 W the condition jaz ðm1 Ts Þ az ðm2 Ts Þj 2D~a=Ts
ð4:42Þ
N
is satisfied for all m1, m2 2 Z , such that jm1n m2n j 1;
n ¼ 1; . . . ; N;
Rxc ;z ðm1 Ts Þ 6¼ 0; Rxc ;z ðm2 Ts Þ 6¼ 0
ð4:43Þ
then, due to the interpolation made by the graphic software, the support of the slice of the CTMF as a function of (~a, mh) in [ 1/2, 1/2[ Z will appear to be piecewise continuous in correspondence of the continuous tracts of the functions az(t). Thus, in the case of piecewise constant functions az(mTs), according to the considerations following Eq. (2.44), the continuous-time signal should be conjectured to be ACS. On the contrary, when the az(mTs) are not piecewise constant, the continuous-time GACS signal should be conjectured to be not ACS. Let us note that the above-described conjecturing procedure can result to be not adequate when the relationship involving analysis and signal parameters is not appropriate. For example, if jaz ðm1 Ts Þ az ðm2 Ts Þj > 2D~a=Ts
ð4:44Þ
200
IZZO AND NAPOLITANO
for some m1, m2 2 ZN and some z 2 W such that m1n ¼ m2n for n 6¼ h; jm1h m2h j ¼ 1; Rxc ;z ðm1 Ts Þ 6¼ 0; Rxc ;z ðm2 Ts Þ 6¼ 0
ð4:45Þ
then the support of the slice of the CTMF as a function of (~a, mh) will appear discontinuous also in correspondence of the continuous tracts of the functions az(t). Specifically, it will be constituted by small piecewise constant tracts and, hence, the continuous-time signal should be conjectured to be anyway ACS. To illustrate how diVerent relationships among analysis and signal parameters can lead to diVerent conjectures on the nonstationarity type of the continuous-time signal, two simulation experiments have been conducted. In the experiments, time has been discretized with sampling increments Ts ¼ T/M, where T is the data-record length and M ¼ 211 is the number of samples. Moreover, a padding factor F ¼ 2 has been adopted in the DFTs. The subsumed GACS signal xc(t) is that obtained by passing through a channel introducing a time-varying delay the binary-phase shift keyed (BPSK) signal x0 ðtÞ ¼
þ1 X
ak pðt kTp Þ cosð2pf0 tÞ
ð4:46Þ
k¼ 1
where f0 ¼ 0.06/Ts, Tp ¼ 64Ts, ak 2 {)1} are equiprobable symbols, and p(t) is a Tp-width full-duty-cycle rectangular pulse. The linear time-variant system introducing the time-varying delay D(t) has impulse-response function hðt; uÞ ¼ dðu t þ DðtÞÞ
ð4:47Þ
DðtÞ ≜ d0 þ d1 t þ d2 t2 :
ð4:48Þ
where The fourth-order (N ¼ 4) reduced-dimension (t4 ¼ 0) lag-dependent cycle frequencies of the (real-valued) signal xc(t) are [see Eq. (3.98)] 8 > < b1 2d2 t2 dt3 dt1 t2 ð4:49Þ az ðt1 ; t2 ; t3 ; 0Þ ¼ b1 2d2 t3 dt2 dt1 t3 > : b1 2d2 t3 dt1 dt2 t3 with b1 ¼ )2nf0 þ kTp, n 2 {0, 1}, k 2 Z being the second-order cycle frequencies of the BPSK signal x0(t). Thus, the slice for t1 ¼ t2 and t3 ¼ 0 of the fourth-order reduced-dimension CTMF of xc(t) has a support in the (a, t1) plane confined on the lines with equations a ¼ 2d2b1t1.
GENERALIZED ALMOST-CYCLOSTATIONARY SIGNALS
201
In the first experiment, the channel parameters are set to d0 ¼ 0, d1 ¼ 4.5, and d2 ¼ 0.0015/Ts, so that Eqs. (4.42) and (4.43) hold. In Figure 12, the magnitude and the support of the slice for m2 ¼ m1 and m3 ¼ 0 of the reduced-dimension (m4 ¼ 0) CTMF of the sampled signal are represented as functions of ~ a ¼ aTs and m1 ¼ t1/Ts. From this figure, according to the fact that Eqs. (4.42) and (4.43) hold, the non ACS nature of the underlying continuous-time GACS signal xc(t) can be conjectured. In the second experiment, the channel parameters are set to d0 ¼ 7705Ts, d1 ¼ 16.7Ts, and d2 ¼ 0.0076/Ts and, hence, (4.44) and (4.45) are satisfied. In Figure 13 the magnitude and the support of the same slice considered in Figure 12 of the reduced-dimension CTMF of the sampled signal are reported. From such a figure, it is evident that xc(t) should be conjectured to be ACS. E. Summary In this section, the problem of sampling a continuous-time GACS signal has been addressed. It has been shown that the discrete-time signal constituted by the samples of a GACS signal is a discrete-time ACS signal. Thus, discrete-time ACS signals can arise from ACS and nonACS continuous-time GACS signals. Relationships between generalized cyclic statistics of a continuous-time GACS signal and cyclic statistics of the ACS discrete-time signal constituted by its samples have been derived. Probability density functions and temporal moment and cumulant functions have also been considered. The problem of aliasing in the domain of the cycle frequencies has been considered and a condition ensuring that the cyclic temporal moment function of the discrete-time signal can be obtained by sampling that of the continuous-time signal has been determined. Moreover, it has been shown how, starting from the sampled signal, the possible ACS nature of the continuous-time GACS signal can be conjectured, provided that the analysis parameters are chosen so that the lag-dependent cycle frequency variations within a sampling period are suYciently small.
V. TIME-FREQUENCY REPRESENTATIONS
OF
GACS SIGNALS
A. Introduction This section deals with time-frequency representations of second-order GACS signals (Izzo and Napolitano, 1997b). A brief introduction (Section V.B.) on GACS signals is reported here to introduce notation since for
202
IZZO AND NAPOLITANO
FIGURE 12. Graph of the (a) magnitude and (b) support, as functions of ð~a; m1 Þ ~ ~a ðm1 ; ðaTs ; t1 =Ts Þ, of the slice for m2 ¼ m1 and m3 ¼ 0 of the reduced-dimension CTMF R xxxx m2 ; m3 ; 0Þ of the sampled signal in the first experiment.
GENERALIZED ALMOST-CYCLOSTATIONARY SIGNALS
203
FIGURE 13. Graph of the (a) magnitude and (b) support, as functions of ð~a; m1 Þ ~ a~ ðm1 ; ðaTs ; t1 =Ts Þ, of the slice for m2 ¼ m1 and m3 ¼ 0 of the reduced-dimension CTMF R xxxx m2 ; m3 ; 0Þ of the sampled signal in the second experiment.
204
IZZO AND NAPOLITANO
time-frequency representations a symmetric version of the autocorrelation is more appropriate. In Section V.C the Cohen’s general class of time-frequency distributions (Cohen, 1995; Hlawatasch and Boudreaux-Bartels, 1992) of GACS signals is considered and it is shown that any representation belonging to this class is expressed as sum of two terms. The first one involves the generalized cyclic statistics of the signal; the second one is related to the residual term obtained by subtracting to the second-order lag product its almost-periodic component (the time-varying autocorrelation function). The Wigner–Ville distribution is examined in detail. Moreover, the ambiguity function is considered. It is shown that it can be expressed as a sum of impulsive terms related to the generalized cyclic statistics of the signal and a non impulsive component related to the previously-mentioned residual term. Furthermore, the subclass of the ACS signals is examined. In Section V.D, the problem of signal feature extraction is considered. It is shown that the estimation of the cyclic autocorrelation as a function of cycle frequency and lag parameter allows one to determine the lag-dependent cycle frequencies and the generalized cyclic autocorrelation functions annihilating the eVect of the residual term when the collect time increases. Then, an estimate of the time-varying autocorrelation function can be derived. On the contrary, in general, in time-frequency representations the component related to the residual term cannot be separated by the components related to the generalized cyclic statistics. Finally, let us note that the time-frequency distributions and the ambiguity function were originally defined with reference to finite-energy signals. Moreover, finite-power signals can be considered by using Dirac’s delta functions (see, e.g., sine waves and chirp signals in Cohen, 1995). The approach adopted here follows this line since GACS time series have finite power. However, it is note worthy that a diVerent approach is adopted in Gardner (1988a), where for a time-windowed ACS signal, the Wigner– Ville distribution is related to the cyclic periodograms and the ambiguity function is recognized to be equal, but for a scale factor, to the cyclic correlogram. B. Second-Order GACS Signals In the FOT probability framework, a continuous-time complex-valued finitepower time series x(t) is said to exhibit second-order wide-sense cyclostationarity with cycle frequency a 6¼ 0 if the symmetric cyclic autocorrelation function Raxx ðtÞ ≜ hxðt þ t=2Þx ðt t=2Þe j2pat it
ð5:1Þ
GENERALIZED ALMOST-CYCLOSTATIONARY SIGNALS
205
exists and is not zero for some t (Gardner, 1988a). Analogously, the time series is said to exhibit second-order wide-sense conjugate cyclostationarity if the symmetric conjugate cyclic autocorrelation function Raxx ðtÞ ≜ hxðt þ t=2Þxðt t=2Þe j2pat it
ð5:2Þ
exists and is not zero for some t. In the following, we deal with time series exhibiting cyclostationarity. The consideration of time series exhibiting conjugate cyclostationarity will require some obvious minor changes. If the set At ≜ fa 2 R : Raxx ðtÞ 6¼ 0g
ð5:3Þ
is countable for each t, then the time series is said to be secondorder generalized almost-cyclostationary in the wide-sense, and the almostperiodic function Rxx ðt; tÞ ≜ EfAt g fxðt þ t=2Þx ðt t=2Þg X ¼ Raxx ðtÞej2pat
ð5:4Þ
a2At
is referred to as the time-varying symmetric autocorrelation function. Then, the lag product x(t þ t/2)x*(t t/2) can be expressed as the sum of its almost-periodic component and a residual term not containing finitestrength additive sine wave components: xðt þ t=2Þx ðt t=2Þ ≜ Rxx ðt; tÞ þ ‘xx ðt; tÞ
ð5:5Þ
h‘xx ðt; tÞe j2pat it 0; 8a 2 R:
ð5:6Þ
where The support in the (a, t) plane of the symmetric cyclic autocorrelation function can be written as suppfRaxx ðtÞg ≜ clfða; tÞ 2 At R : Raxx ðtÞ 6¼ 0g ¼ cl [ fða; tÞ 2 R Dz : a ¼ a z ðtÞg
ð5:7Þ
z2W
where W is a countable set and the functions a z ðtÞ, z 2 W, are the (reduceddimension) lag-dependent cycle frequencies whose domains are denoted by Dz, z 2 W. By reasoning as in Section II.C.1, it can be shown that the time-varying autocorrelation function in Eq. (5.4) can be expressed as X Rxx ðt; tÞ ¼ Rxx;z ðtÞej2pa z ðtÞt ð5:8Þ z2W
where the functions
206
IZZO AND NAPOLITANO
Rxx;z ðtÞ ≜
hxðt þ t=2Þx ðt t=2Þe j2pa z ðtÞt it ; 0;
8t 2 Dz elsewhere
ð5:9Þ
are the symmetric generalized cyclic autocorrelation functions. The Fourier transform of the generalized cyclic autocorrelation function Z Rxx;z ðtÞe j2pf t dt ð5:10Þ Sxx;z ð f Þ ≜ R
is called the symmetric generalized cyclic spectrum. In the special case of ACS time series, it is coincident with the cyclic spectrum. C. Time-Frequency Representations of GACS Signals All time-frequency distributions for a complex-valued time series x(t) can be obtained from Z Cxx ðt; f Þ ¼ xðu þ t=2Þx ðu t=2Þ fðy; tÞ e j2pyðt uÞ e j2pf t du dt dy R3
ð5:11aÞ Z ¼
R
xðt þ t=2Þx ðt t=2Þ ! Fðt; tÞe j2pf t dt t
where
ð5:11bÞ
Z Fðt; tÞ ≜
R
fðy; tÞe j2pyt dy
ð5:12Þ
and ! denotes convolution with respect to t. The kernel function f(y, t) t determines the distribution and its properties (Cohen, 1995). By substituting Eq. (5.5) into Eq. (5.11b) and accounting for Eqs. (5.8) and (5.10), the expression of the generic time-frequency distribution in terms of generalized cyclic statistics can be obtained: X F Cxx ðt; f Þ ¼ Sxx;z ð f Þ ! AF ð5:13Þ z ðt; f Þ þ Lxx ðt; f Þ f
z2W
where AF z ðt; f Þ ≜ and LF xx ðt; f Þ ≜
Z
ej2pa z ðtÞt ! Fðt; tÞe j2pf t dt
ð5:14Þ
‘xx ðt; tÞ ! Fðt; tÞ e j2pf t dt:
ð5:15Þ
R
Z R
t
t
GENERALIZED ALMOST-CYCLOSTATIONARY SIGNALS
207
In other words, all time-frequency distributions of GACS time series can be expressed as the sum of two contributions. The first one is the sum of all generalized cyclic spectra each spread (in the frequency domain) by the time-varying function AF z ðt; f Þ depending on the corresponding (reduceddimension) lag-dependent cycle frequency a z ðtÞ and the kernel function. The second one is related to the residual term ‘xx* (t, t). Note that by taking N ¼ 2 in (2.55) and F(t, t) ¼ d(t) in Eq. (5.14), it results AF z ð t; f Þ ¼ A z ðt; f Þ. By adopting the kernel function f(y, t) ¼ 1 in (5.11a) and, hence, F(t, t) ¼ d(t) in Eq. (5.11b), one obtains the Wigner–Ville distribution (Cohen, 1995) Z Wxx ðt; f Þ ≜ xðt þ t=2Þx ðt t=2Þ e j2pf t dt ð5:16Þ R
which, accounting for Eqs. (5.13) through (5.15), can be expressed as X Sxx;z ð f Þ ! F fej2pa z ðtÞt g þ F f‘xx ðt; tÞg ð5:17Þ Wxx ðt; f Þ ¼ f t!f
z2W
t!f
(Izzo and Napolitano, 1997b), where F t!f denotes the Fourier transform operator from the t domain to the f domain. In the special case of ACS time series the (reduced dimension) lagdependent cycle frequencies are constant and, hence, Eq. (5.13) reduces to X a Cxx ðt; f Þ ¼ Sxx ð f Þ ! Cða; f Þej2pat þ LF ð5:18Þ xx ðt; f Þ f
a2A
where
Z Cða; f Þ ≜
R2
Fðt; tÞ e j2pðatþf tÞ dt dt
ð5:19Þ
and (2.45) has been accounted for. Moreover, for the Wigner–Ville distribution F(t, t) ¼ d(t), that is, C (a, f) ¼ d(f) and, hence, X a Wxx ðt; f Þ ¼ Sxx ð f Þ ej2pat þ F f‘xx ðt; tÞg ð5:20Þ a2A
t!f
which is just the result derived in Gournay and Nicolas (1995) except for the component depending on ‘xx*(t, t). The absence of such a term in Gournary and Nicolas (1995) stems from the fact that for ACS signals, in the stochastic process framework [adopted in Gournay and Nicolas (1995)], ‘xx*(t, t) is dropped out by the statistical expectation operation. However, let us note that such a residual term is present also in the stochastic process approach when asymptotically mean ACS (AMACS) processes (Boyles and Gardner, 1983) are considered. Moreover, it is worthwhile to underline that the
208
IZZO AND NAPOLITANO
residual term is always present in the single sample-path–based estimate of the Wigner–Ville distribution that, for both ACS and AMACS processes, asymptotically approaches expression Eq. (5.20) for almost all sample paths when the collect time increases, provided that appropriate ergodicity properties are satisfied (Boyles and Gardner, 1983; Gardner, 1994). Multiplerecord estimates of the Wigner–Ville distribution lead to a zero residual term. However, they can be singled out only when the signal is cyclostationary (i.e., all the cycle frequencies are multiple of a fundamental one) and the period of cyclostationarity is known (Ko¨ nig and Bo¨ me, 1994). The ambiguity function Z Axx ðn; tÞ ≜ xðt þ t=2Þx ðt t=2Þ e j2pnt dt ð5:21Þ R
for GACS time series, accounting for Eqs. (5.5) and (5.8), can be expressed in terms of generalized cyclic statistics (Izzo and Napolitano, 1997b): X Rxx;z ðtÞdðn az ðtÞÞ þ F f‘xx ðt; tÞg: ð5:22Þ Axx ðn; tÞ ¼ z2W
t!n
Equation (5.22) shows that the ambiguity function of GACS signals is the sum of some impulsive terms whose supports are curves described by the lagdependent cycle frequencies and whose amplitudes are the generalized cyclic autocorrelation functions and an aperiodic component that, accounting for Eq. (5.6), does not contain impulses. Finally, let us note that, in the special case of ACS time series, Eq. (5.22) specializes to X Axx ðn; tÞ ¼ Raxx ðtÞdðn aÞ þ F f‘xx ðt; tÞg: ð5:23Þ a2A
t!n
D. Signal Feature Extraction In problems of signal feature extraction for GACS time series, in general, no a priori knowledge exists on the possible cyclostationary nature of the signal. Therefore, single sample-path–based estimators of time-frequency distributions and generalized cyclic statistics must be used. The estimators are obtained directly by applying the definitions where, however, integrals and time averages are performed over a finite data record. Then, they asymptotically approach the theoretical values when the observation time increases. Once the lag-dependent cycle frequencies and/or the generalized cyclic statistics have been estimated, the time-varying autocorrelation function
GENERALIZED ALMOST-CYCLOSTATIONARY SIGNALS
209
can be reconstructed, signal parameters can be estimated, and signals can be classified on the basis of their diVerent generalized cyclic statistic characteristics. It is worthwhile to note that, in general, in time-frequency representations of GACS time series, the component related to the residual term ‘xx*(t, t) cannot be separated from the component related to the cyclic statistics [see Eq. (5.13)]. In the special case of ACS time series, however, from Eq. (5.18) it follows that the component related to the cyclic statistics is almost periodic and, hence, algorithms for estimating amplitude and frequencies of almostperiodic signals embedded in noise can be exploited to obtain estimates of the cyclic parameters of interest. The role played by the residual term can be illustrated by an example. Specifically, let us consider the signal Z xðtÞ ¼ hðt; uÞsðuÞdu ð5:24Þ R
where sðtÞ ¼ wðtÞ expð j2pf0 tÞ
ð5:25Þ
hðt; uÞ ¼ dðu t þ DðtÞÞ
ð5:26Þ
and is the impulse-response function of a channel introducing a time-varying delay D(t). Figure 14a shows the magnitude of the cyclic autocorrelation function Raxx ðtÞ, for the signal x(t), as a function of a and t, as estimated by 256 samples. It has been assumed that f0 ¼ 0.04/Ts, where Ts is the sampling period (see Section IV.C for the aliasing issue), and the real signal w(t) is 0 ð f Þ ¼ ð1 þ f 2 =B2 Þ 8 wide-sense stationary with power spectral density Sww with B ¼ 0.015/Ts. Moreover, a time-varying delay D(t) ¼ d1t þ d2t2 with d1 ¼ 0.25 and d2 ¼ 0.02/Ts has been considered. The support of Raxx ðtÞ is constituted by curves described by the reduced-dimension lag-dependent cycle frequencies a z ðtÞ; on each of them, the cyclic autocorrelation function is just equal to the corresponding generalized cyclic autocorrelation function Rxx; zðtÞ. Figure 14b shows the magnitude of the Wigner–Ville distribution Wxx ðt; f Þ for the same signal. The presence of a component related to the residual term is evident. Finally, let us observe that the estimates of the cyclic autocorrelation function and the ambiguity function diVer only for a scaling factor (Gardner, 1988a). Therefore, the above cyclic autocorrelation function– based estimation procedure can also be interpreted in terms of ambiguity function.
210
IZZO AND NAPOLITANO
FIGURE 14. Magnitude of (a) the cyclic autocorrelation function Raxx ðtÞ as a function of a and t and (b) the Wigner–Ville distribution Wxx ðt; f Þ as a function of t and f.
GENERALIZED ALMOST-CYCLOSTATIONARY SIGNALS
211
APPENDICES Appendix A In this appendix, the convergence in the temporal mean-square sense of time averages is briefly discussed. A more comprehensive treatment can be found in Weiner (1930) for stationary time series and in Brown (1987), Gardner (1988a), for almost-cyclostationary time series. The more general convergence in the sense of distributions (generalized functions) of statistical functions defined starting from a single time series is treated in PfaVelhuber (1975). In all the sections, all time averages are assumed to exist and to be convergent in the temporal mean-square sense (t.m.s.s.). That is, given a time series z(t) (possibly obtained as lag product of another time series) and defined Z 1 tþT=2 zb ðtÞT ≜ zðuÞ e j2pbu du ð2:A:1Þ T t T=2 Z 1 þT=2 zb ≜ lim zðuÞ e j2pbu du ð2:A:2Þ T!1 T T=2 it is assumed that lim zb ðtÞT ¼ zb
T!1
that is, lim
T!1
ðt:m:s:s:Þ;
D E jzb ðtÞT zb j2 ¼ 0; t
8b 2 R
ð2:A:3Þ
8b 2 R:
ð2:A:4Þ
It can be shown that if the time series z(t) has finite average-power (i.e., 2 hjz(t)j it < 1), then the set B ≜ {b 2 R : zb 6¼ 0} is countable, the series P 2 jz b2B b j is summable (Brown, 1987), and, accounting for Eq. (2.A.3), it follows that X X lim zb ðtÞT e j2pbt ¼ zb e j2pbt ðt:m:s:s:Þ: ð2:A:5Þ T!1
b2B
b2B
The magnitude and phase of zb are the amplitude and phase of the finitestrength additive complex sine wave with frequency b contained in the time series z(t). Moreover, the right-hand side in Eq. (2.A.5) is just the almostperiodic component contained in the time series z(t). The function zb(t)T is an estimator of zb based on the observation {z(u), u 2 [t T/2, t þ T/2]}. It is worthwhile to underline that, in the FOT probability framework probabilistic functions are defined in terms of the almost-periodic
212
IZZO AND NAPOLITANO
component extraction operation, which plays the same role played by the statistical expectation operation in the stochastic process framework (Gardner, 1988a, 1994) (see also Section II.B). Therefore, biasfzb ðtÞT g ≜ Efag fzb ðtÞT g zb ’ zb ðtÞT t zb n o varfzb ðtÞT g ≜ Efag jzb ðtÞT Efag fzb ðtÞT gj2 D E ’ jzb ðtÞT hzb ðtÞT it j2
ð2:A:6Þ
ð2:A:7Þ
t
where the approximation becomes exact equality in the limit as T ! 1. Thus, unlike the stochastic process framework, where the variance accounts for fluctuations of the estimates over the ensemble of sample paths, in the FOT probability framework the variance accounts for the fluctuations of the estimates in the time parameter t (e.g., the central point of the finite-length time series segment adopted for the estimation). Therefore, the assumption that the estimator asymptotically approaches the true value (the infinite-time average) in the mean-square sense is just equivalent to the statement that the estimator is mean-square consistent in the FOT probability sense. In fact, from Eqs. (2.A.4), (2.A.6), and (2.A.7), it follows that D E jzb ðtÞT zb j2 ’ varfzb ðtÞT g þ jbiasfzb ðtÞT gj2 ð2:A:8Þ t
and this approximation becomes exact as T ! 1. In such a case, estimates obtained by using different time segments asymptotically do not depend on the central point of the segment. Appendix B In this appendix, it is shown that for T ! 1 the set L0 of the Nth-order cycle frequencies of the time series XT (t, f ) [see Eq. (2.50)] contains only the element a ¼ 0. Moreover, the derivation of Eq. (2.51) is presented. Let us consider the function * + N Y ðÞn a j2pat Mx ð f Þ ≜ lim XT ðt; ð Þn f n Þe ð2:B:1Þ T!1
n¼1
t
whose magnitude and phase are amplitude and phase of the finite-strength sine component with frequency a contained in the product QN wave ðÞn X ðt; ð Þ n¼1 T n fn Þ when T ! 1. By substituting Eq. (2.50) into Eq. (2.B.1) and accounting for Eqs. (2.14) and (2.30), one has
GENERALIZED ALMOST-CYCLOSTATIONARY SIGNALS
Max ð f Þ ¼
XZ z2W
R
N
Rx;z ðtÞdaz ðtÞ a f ⊤ 1 e j2pf
⊤
t
dt:
213 ð2:B:2Þ
Moreover, substituting Eqs. (2.42) and (2.43) into Eq. (2.B.2) and accounting for Eq. (2.48), one obtains that X Max ð f Þ ¼ S x;z ð f Þda ð2:B:3Þ z2W
from which it follows immediately that the set L0 contains only the element a ¼ 0. Therefore, the spectral moment function can be expressed as ( ) N Y ðÞn fL0 g S x ð f Þ ≜ lim E XT ðt; ð Þn fn Þ T!1 n¼1 * + N Y ðÞn ð2:B:4Þ ¼ lim X ðt; ð Þ fn T!1
¼
X
n¼1
T
n
t
S x;z ð f Þ
z2W
which is just Eq. (2.51). Appendix C This appendix presents proofs of some results shown in Section III. Proof of Eq. (3.34) With reference to the ideal sampled signal xd(t) defined in Eq. (3.33), we have that for any 0 < E < Ts, it results Z 1 T=2 xd ðtÞ e j2pat dt lim T!1 T T=2 Z þ1 1 T=2 X ¼ lim xðkÞ dðt kTs Þ e j2pat dt T!1 T T=2 k¼ 1 Z KTs þE X þK 1 xðkÞ dðt kTs Þ e j2pat dt ¼ lim K!1 ð2K þ 1ÞTs KT E s k¼ K ð3:A:1Þ Z þ1 þK X 1 1 lim xðkÞ dðt kTs Þ e j2pat dt ¼ Ts K!1 2K þ 1 k¼ K 1 þK X 1 1 lim xðkÞ e j2pakTs ¼ Ts K!1 2K þ 1 k¼ K 1 ~a ¼ x Ts ~a¼aTs where
214
IZZO AND NAPOLITANO K X 1 xðkÞe j2p~ak : K!1 2K þ 1 k¼ K
x~a ≜ lim
ð3:A:2Þ
In the first equality in Eq. (3.A.1), we used the fact that if the limit of f(T) as T ! 1, T 2 Rþ, exists, then it is equal to the limit made on any extracted sequence f~ðKÞ ¼ f ðKTs Þ as K ! 1, K 2 N, where K ¼ ⌊T/Ts⌋, with ⌊⌋ denoting the integer part. Moreover, by denoting with Ef~ag fg the discrete-time almost-periodic component extraction operator, we have þ1 X
Ef~ag fxðkÞg dðt kTs Þ k¼ 1 " # þ1 X X ~a j2p~ak ¼ x e dðt kTs Þ ¼
~ k¼ 1 ~a2A þ1 X X ~a
x
~ ~a2A
¼
X
e j2p~ak Ts d
k¼ 1
x~a e j2p~at=Ts
~ ~a2A
¼
X
x~a e j2p~at=Ts
~ ~a2A
þ1 X
,
t k Ts
-
dðt kTs Þ
ð3:A:3Þ
k¼ 1 þ1 X
1 e j2ppt=Ts Ts p¼ 1
þ1 1 X ~a X x e j2pð~aþpÞt=Ts Ts ~ p¼ 1 ~a2A 1 X aTs j2pat ¼ x e Ts a2A
¼
where ~ ≜ f~ A a 2 ½ 1=2; 1=2½: x~a 6¼ 0g
ð3:A:4Þ
A ≜ fa 2 R : a ¼ ~ a=Ts þ p; p 2 Z; x~a 6¼ 0g
ð3:A:5Þ
and Poisson’s sum formula þ1 X k¼ 1
dðt kTs Þ ¼
þ1 1 X ej2ppt=Ts Ts p¼ 1
ð3:A:6Þ
has been accounted for. Therefore, from Eqs. (3.A.1) and (3.A.3) it follows that the almostperiodic component of the signal xd(t) can be expressed as
GENERALIZED ALMOST-CYCLOSTATIONARY SIGNALS
1 X aTs j2pat x e Ts a2A þ1 X ¼ Ef~ag fxðkÞgdðt kTs Þ
215
Efag fxd ðtÞg ¼
ð3:A:7Þ
k¼ 1
and, hence, the second equality in Eq. (3.34) easily follows. Proof of Eq. (3.59) The Nth-order temporal cumulant function of the output y(t) of a LTV system excited by a GACS time series x(t), accounting for Eqs. (2.58) and (3.54), is given by " # p X Y p 1 Cy ð1t þ tÞ ¼ ð 1Þ ðp 1Þ! Rxmi ð1t þ t mi Þ i¼1 P Z X" p Y ¼ ð 1Þp 1 ðp 1Þ! ð3:A:8Þ RN P i¼1 # Rhmi ð1t þ t mi ; 1t þ smi ÞRxmi ð1t þ smi Þ ds where
( Rxmi ð1t þ smi Þ ≜ E
fag
Y
) x
ðÞk
ðt þ sk Þ
ð3:A:9Þ
k2mi
and Rhmi (1t þ t mi, 1t þ smi) is the jmijth-order temporal moment function of the LTV system which, according to Eq. (3.43), can be written as Rhmi ð1t þ tmi ; 1t þ smi Þ X Z ⊤ ¼ Gymi ðlmi ; t mi Þ e j2p’ymi ðlmi ;tmi Þt e j2plmi ð1tþsmi Þ dsmi : ymi 2Ymi
ð3:A:10Þ
Rjmi j
By assuming that the LTV system is FOT deterministic, there is a oneto-one correspondence between the elements ymi 2 Ymi and the vectors smi 2 Ojmij and, according to Eqs. (3.57) and (3.58), it results that ð Þ ⊤ ’ym ðlmi ; t mi Þ ¼ wð Þ smi ðlmi Þ 1 i
and Gymi ðlmi ; t mi Þ ¼
Y k2mi
ð Þ ð Þ ⊤ k ð Þ l GðÞ ej2pwsmi ðlmi Þ tmi : k sk k
ð3:A:11Þ
ð3:A:12Þ
Therefore, by substituting Eqs. (3.A.11) and (3.A.12) into Eq. (3.A.10), it results that
216
IZZO AND NAPOLITANO
p Y Rhmi ð1t þ tmi ; 1t þ smi Þ i¼1
¼
X Z s2ON
N
Y ð Þ j2pws ðlð Þ Þ⊤ ð1tþtÞ j2pl⊤ ð1tþsÞ n ð Þ l GðÞ e dl sn n n e
ð3:A:13Þ
RN n¼1
¼ Rh ð1t þ t; 1t þ sÞ where the fact that the sets mi are disjoint has been accounted for. Finally, Eq. (3.59) immediately follows by substituting Eq. (3.A.13) into (3.A.8) and taking into account Eq. (2.58). Appendix D In this appendix, proofs of some results presented in Section IV.C, are reported. Proof of Eq. (4.19) For any 0 < E < Ts the argument of the limit in Eq. (4.19) can be written as K X 1 ‘U ð1t þ t; jÞjt¼kTs ;t¼mTs e j2p~gk 2K þ 1 k¼ K Z KTs þE þ1 X 1 ¼ ‘U ð1t þ mT s ; jÞ dðt kTs Þ e j2p~gt=Ts dt ð4:A:1Þ 2K þ 1 KTs E k¼ 1 Z KTs þE þ1 X 1 ‘U ð1t þ mTs ; jÞ e j2pð~g pÞt=Ts dt ¼ ð2K þ 1ÞT s KT E s p¼ 1
where, in the second equality, the Poisson sum formula [Eq. (3.A.6)] has been accounted for. Taking the limit for K ! 1 in Eq. (4.A.1) and accounting for Eq. (2.5), one obtains Eq. (4.19). Proof of Eq. (4.20) By substituting Eq. (4.18) into Eq. (4.3) and accounting for Eq. (4.19), one has ~ ~g ðm; jÞ ¼ F x
X g2GmTs
K X 1 e j2pð~g gTs Þk : K!1 2K þ 1 k¼ K
Fxgc ðmTs ; jÞ lim
ð4:A:2Þ
Finally, by using the limit K þ1 X X 1 e j2p~ak ¼ d~aþp K!1 2K þ 1 p¼ 1 k¼ K
lim
into Eq. (4.A.2), Eq. (4.20) easily follows.
ð4:A:3Þ
GENERALIZED ALMOST-CYCLOSTATIONARY SIGNALS
217
Let us define the set
Proof of Eq. (4.22)
~ 0 ≜ f~g 2 ½ 1=2; 1=2½: ~g ¼ ðg=fs Þ mod 1; g 2 GmT g; G s m
ð4:A:4Þ
where mod b denotes the modulo b operation with values in [ b/2, b/2[ [see Eq. (4.29)]. From Eq. (4.A.4), accounting for Eqs. (2.2) and (4.18), it results that g GmTs ≜ fg jg [2 R : Fxc ðmTs ; jÞ 6¼ 0 for some ~ 0 ; F g ðmT s ; jÞ 6¼ 0 for some jg: ¼ fg 2 R : g ¼ ð~g þ pÞfs ; ~g 2 G m xc p2Z
ð4:A:5Þ Therefore, from Eq. (4.18) it follows that ~ x ð1k þ m; jÞ ¼ U
þ1 X X ~0 ~g2G m
¼
X
p¼ 1
Fxð~gc þpÞfs ðmTs ; jÞej2pð~gþpÞfs kTs þ ‘U ð1kTs þ mTs ; jÞ
~ ~g ðm; jÞej2p~gk þ ‘U ð1kTs þ mTs ; jÞ F x
~ ~g2G m 0
ð4:A:6Þ where, in the second equality, Eq. (4.20) has been used. Thus, by comparing Eq. (4.A.6) with Eq. (4.2) and accounting for Eq. (4.19), it results that the set ~ 0 is coincident with the set G ~ m defined by Eq. (4.23) and, hence, Eq. (4.22) G m is proved. Proof of Eq. (4.30) By taking the Fourier coefficient at frequency a in both sides of Eq. (2.17), one obtains that Z N Y a Rxc ðtÞ ¼ ðxrn þ jxin ÞðÞn fxac ðt; jÞ dj r dj i : ð4:A:7Þ R2N n¼1
Analogously, by taking the Fourier coefficient at frequency ~a in both sides of Eq. (4.10), one has Z N Y ~a ~ ~a ðmÞ ¼ R ðxrn þ jxin ÞðÞn f~x ðm; jÞ dj r dj i x ¼
R2N n¼1 þ 1 Z X
p¼ 1
N Y
R2N n¼1
ð4:A:8Þ ðxrn þ
jxin ÞðÞn fxð~caþpÞfs ðmTs ; jÞ
dj r dj i
where, in the second equality, Eq. (4.21) has been used. Thus, by substituting Eq. (4.A.7) into Eq. (4.A.8), Eq. (4.30) easily follows.
218
IZZO AND NAPOLITANO
Proof of Eq. (4.38) By taking the Fourier coefficient at frequency b in both sides of Eq. (2.58), one has " # p X X Y ami p 1 b ð 1Þ ðp 1Þ! Rxc;mi ðt mi Þ ð4:A:9Þ Cxc ðtÞ ¼ P
a⊤ 1¼b i¼1
where the second summation ranges over all vectors a ≜ ½am1 ; . . . ; amp ⊤ with am ami such that Rxc;mi i ðt mi Þ ≢ 0, and the fact that the sets mi are disjoint has been accounted for. In Eq. (4.A.9), Z 1 T=2 Y ðÞn am Rxc;mi i ðtmi Þ ≜ lim xc ðt þ tn Þe j2pami t dt: ð4:A:10Þ T!1 T T=2 n2m i
~ in both sides of Analogously, by taking the Fourier coefficient at frequency b Eq. (4.12), one obtains " # p X XY ~ b ~ami p 1 ~ ðmÞ ¼ ~ ðmm Þd ⊤ ~ ð 1Þ ðp 1Þ! R ð4:A:11Þ C xm x i ð~ a 1 bÞmod 1 ~ a
P
i
i¼1
~ ≜ ½~am1 ; . . . ; ~amp ⊤ with where the second summation ranges over all vectors a ~ami ~ ðmm Þ ≢ 0. In Eq. (4.A.11), ~ ami 2 ½ 1=2; 1=2½ such that R xm i i K X Y 1 ~ami ~ ðmm Þ ≜ lim xðÞn ðk þ mn Þe j2p~ami k : ð4:A:12Þ R xmi i K!1 2K þ 1 k¼ K n2m i
By substituting Eq. (4.30) into Eq. (4.A.11), it results that 2 X XX X ~ b 4ð 1Þp 1 ðp 1Þ! ... C~x ðmÞ ¼ ~ a
P
q1 2Z
qp 2Z
p Y ð~am þqi Þfs Rxc;mii ðmmi Ts Þdð~a⊤ 1 bÞmod ~ 1
#
ð4:A:13Þ
i¼1
~ and qi, (i ¼ 1, . . . , p) give rise to a sum over all where the summations over a cycle frequencies ~ ami 2 ½ 1=2; 1=2½ such that (~ami þ qi Þfs ¼ ami ði ¼ 1; . . . ; pÞ ~ s a⊤ 1Þ mod fs ¼ 0. That is, and ðbf 2 3 p Y X X ~ b a m p 1 4ð 1Þ ðp 1Þ! Rxc;mi i ðmmi Ts Þ5 C~x ðmÞ ¼ P
¼
XX q2Z
P
2
~ s a⊤ 1¼qfs ;q2Zg i¼1 fa:bf
4ð 1Þp 1 ðp 1Þ!
X
3 p Y ami Rxc;mi ðmmi Ts Þ5
~ s qfs i¼1 a⊤ 1¼bf
ð4:A:14Þ from which, using Eq. (4.A.9), Eq. (4.38) immediately follows.
219
GENERALIZED ALMOST-CYCLOSTATIONARY SIGNALS GLOSSARY OF ACRONYMS AND NOTATIONS hit E{a} {} or EfAt g fg d() d() U() cl vT v0 ! ! n
1 * (*) or (*)n ( ) or ( )n 1( ) n( )
Rax ðtÞ Rax ðt0 Þ Rx ð1t þ tÞ Rx;z ðtÞ
continuous time average with respect to t almost-periodic component extraction operator Dirac delta Kronecker delta unit step function closure (of a set) vector transposition v ¼ ½v1 ; . . . ; vN ⊤ ) v0 ¼ ½v1 ; . . . ; vN 1 ⊤ convolution N-fold convolution with respect to v ≜ ½v1 ; . . . ; vN ⊤ [1, . . . , 1]⊤ complex conjugation optional complex conjugation optional minus sign ½ð Þ1 1; . . . ; ð ÞN 1⊤ ½ð Þ1 v1 ; . . . ; ð ÞN vN ⊤ fraction of time linear time variant linear time invariant linear almost periodically time variant almost cyclostationary generalized almost cyclostationary cyclic temporal moment function reduced-dimension cyclic temporal moment function temporal moment function
P x( f )
generalized cyclic temporal moment function reduced-dimension generalized cyclic temporal moment function (moment) lag-dependent cycle frequency reduced-dimension (moment) lag-dependent cycle frequency cyclic spectral moment function reduced-dimension cyclic spectral moment function spectral moment function temporal cumulant function generalized cyclic temporal cumulant function spectral cumulant function
Rh ð1t þ t; uÞ S h( f, l)
system temporal moment function system spectral moment function
Rx;z ðt 0 Þ az(t) a z ðt 0 Þ S ax ð f Þ Sxa ð f 0 Þ Sx ð f Þ Cx(1t þ t) Cx,x(t)
(1.2) (1.15), (2.2)
(1.16)
FOT LTV LTI LAPTV ACS GACS CTMF RD-CTMF TMF
(1.24), (2.14) (1.26)
GCTMF
(2.16), (2.26), (2.38) (2.27)
RD-GCTMF
(2.41) (2.24), (2.25) (2.40)
CSMF RD-CSMF
(1.25) (1.25)
SMF TCF GCTCF
(2.49), (2.51) (2.58), (2.67) (2.68) (2.93), (2.94), (2.96) (3.43) (3.61)
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ADVANCES IN IMAGING AND ELECTRON PHYSICS, VOL. 135
Virtual Optical Experiments ROBERT THALHAMMER Infineon Technologies, 81730 Munich, Germany
I. Introduction . . . . . . . . . . . . . . . . . II. Modulation of the Refractive Index . . . . . . . . . A. The Plasma-Optical EVect . . . . . . . . . . . 1. The Drude Theory . . . . . . . . . . . . . 2. Absorption Spectra and the Kramers–Kronig Relation. 3. Experimental Results . . . . . . . . . . . . 4. Comparison of the Available Data . . . . . . . B. The Thermo-optical EVect . . . . . . . . . . . 1. Temperature Dependence of the Absorption CoeYcient 2. Temperature Dependence of the Refractive Index . . III. Measurement Techniques . . . . . . . . . . . . . A. Device Characterization Methods . . . . . . . . . B. Internal Laser Probing Techniques . . . . . . . . 1. Free Carrier Absorption Measurements . . . . . . 2. Internal Laser Deflection Measurements. . . . . . 3. Backside Laser Probing . . . . . . . . . . . 4. Fabry–Perot Reflectivity Measurements . . . . . . 5. Fabry–Perot Transmission Measurements . . . . . 6. Mach–Zehnder Interferometry . . . . . . . . . 7. Summary . . . . . . . . . . . . . . . . IV. Modeling Optical Probing Techniques . . . . . . . . A. Simulation Steps—An Overview . . . . . . . . . B. Electrothermal Device Simulation . . . . . . . . . 1. Survey of Electrothermal Device Simulation Models . 2. The Thermodynamic Model . . . . . . . . . . 3. Material Properties . . . . . . . . . . . . . 4. Numerical Methods. . . . . . . . . . . . . C. Calculation of the Refractive Index Modulations. . . . D. Wave Propagation in Inhomogeneous Media . . . . . 1. Algorithms Reported in the Literature . . . . . . 2. The Propagator Matrix . . . . . . . . . . . 3. Boundary Conditions in Propagation Direction . . . 4. Summary of the Algorithm . . . . . . . . . . E. Fourier Optics . . . . . . . . . . . . . . . 1. Image Formation by Aperture Holes . . . . . . . 2. Image Formation by Thin Lenses . . . . . . . . 3. Propagation in Free Space . . . . . . . . . . F. Detector Response . . . . . . . . . . . . . . V. Virtual Experiments and the Optimization Strategy . . . .
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VIII.
IX.
THALHAMMER A. Fundamental Principle . . . . . . . . . . . . . . . B. General Assumptions . . . . . . . . . . . . . . . . Free Carrier Absorption Measurements . . . . . . . . . . . A. Optical Field Distribution of the Probing Beam . . . . . . . B. Longitudinal Averaging/Samples with Cell Structure . . . . . C. The Fabry–Perot EVect . . . . . . . . . . . . . . . D. Spatial Resolution . . . . . . . . . . . . . . . . . E. Optimizing the Optical Setup . . . . . . . . . . . . . 1. Angular Beam Aperture . . . . . . . . . . . . . . 2. Sample Alignment . . . . . . . . . . . . . . . . F. EVects of Surface Recombination . . . . . . . . . . . . 1. Devices with Large Carrier Lifetimes . . . . . . . . . . 2. Devices with Small Carrier Lifetimes . . . . . . . . . . G. Summary . . . . . . . . . . . . . . . . . . . . Internal Laser Deflection Measurements . . . . . . . . . . . A. The Measurement Signal . . . . . . . . . . . . . . . 1. Internal Beam Deflection . . . . . . . . . . . . . . 2. Image Formation by a Thin Lens . . . . . . . . . . . 3. Projection by the Imaging Lens . . . . . . . . . . . . 4. Detector Signal . . . . . . . . . . . . . . . . . 5. Parameters of the Experiment . . . . . . . . . . . . B. Deflection Measurements in Case of Low Power Dissipation. . . C. Simultaneous Free Carrier Absorption Measurements . . . . . D. The Fabry–Perot EVect . . . . . . . . . . . . . . . E. Detector Response . . . . . . . . . . . . . . . . . F. Image Formation Conditions . . . . . . . . . . . . . G. Deflection Measurements of Large Temperature Gradients . . . 1. Test Structures . . . . . . . . . . . . . . . . . 2. EVective Interaction Lengths and the Modified Evaluation Rule H. Summary . . . . . . . . . . . . . . . . . . . . Interferometric Techniques . . . . . . . . . . . . . . . A. Mach–Zehnder Interferometry . . . . . . . . . . . . . B. Backside Laser Probing . . . . . . . . . . . . . . . 1. EVects of the Sample Preparation . . . . . . . . . . . 2. The Measurement Signal . . . . . . . . . . . . . . 3. DiVerential Backside Laser Probing . . . . . . . . . . C. Summary . . . . . . . . . . . . . . . . . . . . Conclusion and Outlook . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . .
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I. INTRODUCTION The continuous progress of physical and technical research in semiconductor physics and microelectronics is supported by accurate and reliable characterization methods. They are indispensable for analyzing the electric, magnetic, mechanic, thermal, optical, and acoustical properties of the samples. Measuring the electrical terminal behavior and external emission
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characteristics (e.g., the optical output power of laser devices) has been state of the art for many years. Recently, however, promising perspectives have been opened as probing the interior of semiconductor samples has become possible. Since the internal distribution of state variables, such as the charge carrier concentrations or the temperature, has thus become accessible by experiments, the scientific understanding of complex internal processes becomes more profound and the validation of theoretical models can be based on a larger diversity of experiments, yielding reliable models of more general validity. In addition, internal probing provides great benefit for the development of semiconductor devices, integrated circuits, and microstructures, as, for example, the heat dissipation has become one of the most crucial problems in microelectronic engineering. Suitable measurement techniques are scanning probe microscopy, electron beam testing, and optical beam testing. An overview of characterization and failure analysis methods is provided in Soden and Anderson (1995) and So¨lkner (1994). Considerable progress has been achieved by the introduction of a novel class of internal laser probing techniques (Boccara et al., 1980; Deboy et al., 1996; Fournier et al., 1986; Goldstein et al., 1993; Heinrich et al., 1986; Jackson et al., 1981), which exploit the electro-optical and the thermooptical eVect. They enable space-resolved and time-resolved measurements of the charge carrier and temperature distributions in the interior of semiconductor samples and devices. However, extracting the measurands from the detector signals has been based on the concepts of geometrical ray tracing (Deboy, 1996) or one-dimensional (1D) models (Seliger et al., 1996a,b), neglecting two major aspects. First, preparing the samples and performing the actual measurement inevitably introduces undesired eVects (e.g., an enhanced surface recombination caused by sawing the dice and polishing the surfaces, or an oblique transition of the probing beam in case the sample is misaligned). Second, fundamental physical constraints arise from the wave propagation of the probing beam (e.g., a limited spatial resolution owing to the finite lateral extension of the beam profile or undesired Fabry–Perot oscillations if the probing beam is multiply reflected oV the polished surfaces). A comprehensive and physically rigorous analysis of the experiments has become possible by the recently proposed model for simulating the measurement process (Thalhammer and Wachutka, 2003a). It covers an electrothermal device simulation of the sample’s operating condition, the calculation of the optical beam propagation through the sample, the lenses, and aperature holes, and finally the simulation of the detector response. The idea of ‘‘virtual experiments’’ (Thalhammer and Wachutka, 2003b) constitutes the key concept for a complete theoretical study of the entire experiment and for
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assessing the accuracy of the corresponding measurement results. Two major advantages are gained by the numerical analysis. First, the theoretical model is able to visualize internal processes and to calculate a variety of physical quantities that cannot be measured. In addition, the spatial and temporal resolution is usually much higher than that of the experimental methods. For instance, the carrier and temperature distribution in unprepared samples is accessible by simulation and can be compared with the extracted measurands of a simulated experiment. It is thus possible to discriminate the electrothermal phenomena of interest against parasitic eVects that are introduced by preparing the sample, performing the actual measurement, and evaluating the detector signals. Second, theoretical studies can replace some experimental tests for optimizing the probing techniques, avoiding the construction of the setups and the processing of the necessary test structures. They help to reduce costs and development time, as virtual experiments reveal quantitative results for the optimum probing conditions and the optimum sample geometry, as well as the sensitivity to each of the parameters. This is especially true as the rapid progress of computer technology produces increasingly powerful machines, which even facilitate the implementation of costly models for complex physical phenomena. This article reviews the fundamental ideas for simulating internal optical probing techniques. It demonstrates representative applications of virtual experiments for assessing the measurement accuracy and the impact of parasitic eVects, as well as for optimizing the probing conditions with respect to a desired purpose, such as a high sensitivity or a large measurement range. Many of these experimental methods have been successfully used for investigating a large variety of samples, such as power devices, microelectromechanical systems (MEMS), and devices that are below 1 mm in size. However, scanning space–resolved profiles works best within devices whose size is on the scale of some tens of microns to a few millimeters. For that reason, most of the examples in this article are chosen from the typical conditions of probing power devices for which many of these laser probing techniques have been invented. The typical features common to all kinds of power device structures are the heavily doped emitters that are only a few microns in depth, while most of the structure consists of a large and weakly doped region. It has to sustain the blocking voltage under reverse biasing conditions. In the forwardconducting state, its resistivity is decreased by several orders of magnitude by injecting charge carriers. Typical doping levels are about 1012 cm3 to 1014 cm3, whereas the injected carrier densities amount to some 1015 cm3 or 1016 cm3. Under these high-injection conditions, electron and hole concentrations are equal.
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Among the power switches, the insulated gate bipolar transistor (IGBT) (see the sample in Figure 3) has become the most popular device for blocking voltages from 600 V to 3.5 kV. The entire chip comprises an array of thousands of single cells. Each of them consists of a pnp bipolar transistor whose base is driven by an n-channel MOSFET. Hence, for a positive gate bias an n-channel is formed in the p-well, through which electrons are drifting into the n-base of the bipolar transistor. As a consequence, holes are injected by the p-emitter at the bottom, thus flooding the n-region with carriers. Although the emitter of the internal pnp transistor is connected to the bottom contact, it has become common practice to call this contact the collector and the top contact the emitter. These terms are taken from transistor operation at the circuit level. Power devices cover a power range from a few watts to some megawatts and must meet specific challenges. In the forwardconducting state, a high carrier injection into the weakly doped bulk region is desirable for the sake of a low forward voltage drop. On the other hand, the turnoV behavior is limited by the stored charge carriers that have to be extracted. The fundamental tradeoV between a high forward conductivity and a fast switching behavior can be improved by ‘‘carrier profile engineering’’—by suitably adjusting the emitters’ eYciencies and by incorporating local recombination centers. However, power dissipation increases up to some 100 kW/cm2 during transient switching, or even more during overload operation. Designing devices that master these excessive operating conditions requires a precise knowledge of the structures’ electrical and thermal properties, and gains significant benefit from probing techniques enabling space- and time-resolved measurements of the internal charge carrier and temperature profiles.
II. MODULATION
OF THE
REFRACTIVE INDEX
This section discusses the underlying physical principle of internal laser probing techniques, that is, the interaction of infrared electromagnetic waves with free carriers (plasma-optical eVect) and lattice vibrations (thermo-optical eVect). The discussion is limited to an overview of the common models and a summary of the most significant experimental results for the dependence of the complex refractive index on carrier concentration and lattice temperature. For further details, readers are referred to the cited publications (see also Deboy, 1996; Seliger, 1998, and the references therein). In this article, the real part of the refractive index at the frequency o of the probing laser will be represented by no since it is common practice in semiconductor physics that the symbol n denotes the concentration of free electrons.
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A. The Plasma-Optical EVect The optical absorption near the band gap of silicon is governed by two major processes, namely band-to-band absorption (interband transitions) and free carrier absorption (intraband transitions) (Sturm and Reaves, 1992). Because the density of states at the band edges is high, the optical excitation of an electron from the valence band to the conduction band depends only very weakly on the carrier concentration (Soref and Bennett, 1987). Therefore, the change of the absorption coeYcient a caused by the injection of carriers originates from an increased free carrier absorption. Together with the associated eVect on the real part of the refractive index no, this will be discussed in the following section. 1. The Drude Theory A classical description of the plasma-optical eVect is known as the Drude model (Ashcroft and Mermin, 1976), which regards the charge carriers as harmonic oscillators with vanishing binding energies. Solving the equation of motion mn;p€rn;p þ
mn;p r_ n;p ¼ qE tcn;p
ð1Þ
in the frequency domain yields the electrical polarization Pn ¼ qnrn and Pp ¼ qprp caused by the displacement of the electrons and the holes, respectively. The refractive index then calculated by expanding the square root ffiffiffiffiffiffiffiffiffiffiffiffi pis of the dielectric constant R ðoÞ ¼ no þ ia=2k0 in terms of 1/o: ! q2 l2 n p n o ¼ nr 2 þ ð2Þ 8p 0 nr c2 mn mp ! q3 l2 n p a¼ 2 þ 2 ; mp mp 4p 0 nr c3 m2 n mn
ð3Þ
where nr, mn, and mp denote the static refractive index and the mobilities of electrons and holes, respectively. Inserting the material parameters of silicon, we obtain for a wavelength l ¼ 1.3 mm: no ¼ 3:5 8:33 1022 cm3 n 5:70 1022 cm3 p
ð4Þ
a ¼ 2:51 1019 cm2 n þ 3:91 1019 cm2 p :
ð5Þ
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However, accurately modeling the plasma-optical eVect must consider all scattering processes that are assisted by phonons or impurities, since free carrier absorption requires the interaction with a third particle for momentum conservation. Although these eVects are implicitly included in the relaxation times tcnp (or the mobility), they are not treated consistently by the Drude model. Besides, parabolic band edges are assumed, which fails for optical excitation to higher states in the same band. 2. Absorption Spectra and the Kramers–Kronig Relation Huang et al. (1990) carried out a quantum mechanical analysis of the absorption coeYcient, which is assumed proportional to the transition rate Z1 W ¼
W ðkÞf ðEk Þ2VNðEk Þ dEk :
ð6Þ
0
Here, W(k) denotes the transition rate for phonon emission or absorption, Ek the electron energy, f(Ek) the distribution function, and N(Ek) the density of states. Huang et al. used the models for acoustic-phonon scattering, optical-phonon scattering, and impurity scattering (Ridley, 1982) and added a description of the nonparabolic band structure. Since the scattering mechanisms are proportional to the free carrier concentration, the absorption coeYcient in the near-infrared spectrum also depends linearly on the electron and hole concentrations. Knowing the frequency-dependent absorption coeYcient, the real part of the refractive index is calculated from the Kramers–Kronig relation (see Huang et al., 1990), which relates the real part no of the refractive index to its imaginary part and thus to the absorption coeYcient a Z1 c aðo0 Þ no ðoÞ ¼ 1 þ P do0 : p o0 2 o2
ð7Þ
0
In this equation the symbol P denotes the Cauchy principal value. For a carrier density of 1016 cm3, which is a typical concentration in the electronhole plasma of forward-biased power devices, Huang et al. reported a concentration dependence proportional to n1.07 and p1.03. A similar strategy is followed by Soref and Bennett (1987). They collected experimental data about the absorption spectra of heavily doped samples (Fan and Becker, 1950; Spitzer and Fan, 1957) and calculated the real part of the refractive index from the Kramers–Kronig relation. Their results are proportional to n1.05 and p0.805 for a wavelength of 1.3 mm and proportional to n1.04 and p0.818 for a wavelength of 1.55 mm, respectively.
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3. Experimental Results The carrier concentration dependence of the absorption coeYcient was measured by several authors from the absorption of uniformly doped samples (Fan and Becker, 1950; Schmid, 1981; Schroeder et al., 1978; Spitzer and Fan, 1957). In this case, the probing beam is mainly attenuated by interacting with the majority carriers, which are thermally generated by the doping atoms. These experiments separately reveal the dependence on both the electron concentration and the hole concentration, that is, @a/@n and @a/@p, respectively. However, probing the carrier distribution in weakly doped samples requires knowledge of the absorption coeYcient under high-injection conditions. The optical absorption of an electron-hole plasma was measured on forward-conducting pin diodes by Horwitz and Swanson (1980). They used a vertically propagating probing beam that penetrates through a grid contact at the top and is reflected at the bottom metallization layer. The detected damping is therefore sensitive to the integrated carrier distribution, which is calculated from the measured diVusion potential across the junctions. As a result, they reported a nonlinear dependence of the absorption coeYcient on the injected carrier density. Laterally irradiating the sample by a monochromatic light source, Schierwater (1975) detected the transmitted intensity at various current densities, at each of which the total stored charge was determined by reverse-recovery measurements (see Section III.A). For a wavelength of 1.3 mm, he obtained the linear dependence Da ¼ 2.80 1018 cm2 Dn on the injected carrier density Dn. Whereas Schierwater’s setup is sensitive to an average concentration of the injected carriers, later experiments used a focused laser beam for a space-resolved scanning of the characteristic carrier concentration profile. Again, calibration was done by measuring the reverserecovery behavior (Hille, 2001) or the diVusion potential (Schlo¨ gl, 2000). A direct measurement of the real part of the refractive index was performed by Yu et al. (1996). They detected the transmittance of a Fabry– Perot resonator while modulating its optical thickness by optically generated charge carriers. However, their results are about one order of magnitude larger than the theoretical predictions based on the Kramers–Kronig relation. The diVerence probably arises from the evaluation strategy, which assumes the decrease of the transmitted light intensity to be merely caused by the modulation of the refractive index but neglects the enhanced free carrier absorption. A diVerent approach is reported by Deboy, 1996; Deboy et al., 1996, who performed simultaneous laser absorption and deflection measurements (Section III.B.2). Comparing the carrier contribution to the deflection signal with the calibrated absorption signal yields the refractive index modulation Dno ¼ 4.58 1021 cm3 Dn.
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4. Comparison of the Available Data Introducing the definitions @no @no @no :¼ þ @C T @n p;T @p n;T
@a @C
T
:¼
@a @n
þ p;T
@a ; @p n;T
ð8Þ
ð9Þ
the isothermal modulations of the complex refractive index under highinjection conditions (n ¼ p) become @no Dno ¼ Dn; ð10Þ @C T Da ¼
@a @C
Dn:
ð11Þ
T
Reported data are compared in Figure 1 and Table 1. For the carrier concentration dependence of the absorption coeYcient, we recognize a satisfying agreement of the data reported by Huang et al., Soref et al., and Horwitz et al., while Schierwater’s and Schlo¨ gl’s experimental results diVer
FIGURE 1. Silicon refractive index (left) and absorption coeYcient (right) for equal carrier concentrations (T ¼ 300 K, l ¼ 1.3 mm).
234
THALHAMMER TABLE 1 CARRIER CONCENTRATION DEPENDENCE OF THE REFRACTIVE INDEX AND THE ABSORPTION COEFFICIENT FOR n ¼ p ¼ 1 1016 cm3, T ¼ 300 K, l ¼ 1.3 mm Reference
ð@no =@CÞT
Drude theory Huang et al. (1990) Soref and Bennett (1987) Schierwater (1975) Horwitz and Swanson (1980) Deboy et al. (1996) Schlo¨ gl (2000)
1.40 1021 cm3 4.36 1022 cm3 3.57 1021 cm3 4.58 1021 cm3
(@a/ @C)T 6.42 5.11 5.92 2.80 4.85
1019 1018 1018 1018 1018
cm2 cm2 cm2 cm2 cm2
8.1 1018 cm2
Most data are published graphically. In this case, the fit functions are adopted from Deboy, 1996.
by a factor of 2. On the other hand, the published data about the carrier concentration dependence of the refractive index vary about one order of magnitude, which is not surprising if the diYculties in measuring this coeYcient are considered. Fortunately, this uncertainty does not constitute a serious problem since, as will be shown later, it is the thermo-optical eVect that predominantly modulates the real part of the refractive index under the typical operating conditions that internal laser probing techniques enable investigating. For all data in this article, the following dependence on the carrier concentration has been assumed (Hille and Thalhammer, 1997): @no ¼ 1:81 1021 cm3 ð12Þ @C T
@a @C
T
¼ 5:11 1018 cm2 :
ð13Þ
B. The Thermo-optical EVect Thermally induced modulations of the complex refractive index originate from three major eVects, namely, the change of the distribution functions of carriers and phonons, the temperature-induced band gap narrowing, and the thermal expansion of the crystal. As temperature rises, the latter mechanism decreases the optical density and, consequently, the refractive index. Since an increase is observed in reality, the two former eVects are obviously decisive.
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1. Temperature Dependence of the Absorption CoeYcient Considering interband transitions with the absorption or the emission of an acoustic or an optical phonon, Macfarlane et al. (1958) developed a model for the temperature dependence of the band-to-band absorption. Their model reproduces accurately the measured absorption coeYcient at elevated temperatures where interband absorption becomes the most significant mechanism at a wavelength of 1.3 mm (Sturm and Reaves, 1990, 1992). A semi-empirical relation accounting for the temperature dependence of free carrier absorption processes was gained from spectral emissivity measurements (Rogne et al., 1996). Temperature-dependent measurements of the absorption coeYcient are reported by Schierwater (1975) and Schlo¨ gl (2000). Either of them carried out his above-mentioned experiment at various temperatures ranging from 150 K to 400 K. From their graphically published data, the temperature dependence of the absorption coeYcient can be extracted and amounts to about (@a/@T )n,p 104 cm1 K1 at a carrier concentration of 1 1016 cm3. However, as temperature rises in the interior of bipolar devices subjected to their typical operating conditions during absorption measurements, the crucial eVect on the absorption coeYcient originates from an enhanced injection of carriers by the emitter regions, which, in turn, increases the free carrier absorption. Hence, among the two contributions daðn; p; TÞ n¼p @a @a dn ¼ ; ð14Þ þ dT @T n;p @C T dT the latter significantly exceeds the former. The temperature dependence (@a/@T )n,p of the absorption coeYcient may therefore be neglected at all. 2. Temperature Dependence of the Refractive Index The temperature coeYcient of the refractive index can be directly measured by various optical techniques. Their basic idea is to detect the thermally induced modulation of the optical sample thickness by ellipsometry (Jellison and Burke, 1986), by interferometry exploiting the Fabry–Perot eVect (Caulley et al., 1994; Hille and Thalhammer, 1997; Icenogle et al., 1976; Magunov, 1992), or by interferometry with respect to a reference beam of constant phase (Seliger et al., 1996b). If the measurements are carried out on electrical devices, self-heating originates the temperature rise which must be determined from the electrical power dissipation (Hille and Thalhammer, 1997), from the saturation currents of a MOSFET (Seliger et al., 1996a), or from the thermal expansion (Deboy, 1996), for example.
236
THALHAMMER TABLE 2 TEMPERATURE DEPENDENCE OF THE REFRACTIVE INDEX FOR A WAVELENGTH OF l ¼ 1.3 mm Reference
ð@no =@TÞn;p
Magunov (1992) Bertolotti et al. (1990) Jellison and Burke Icenogle et al. (1976) Seliger et al. (1996b) Hille and Thalhammer (1997)
2.0 1.8 2.4 1.5 1.6 1.6
104 104 104 104 104 104
K1 K1 K1 K1 K1 K1
Temperature range Room temperature 15 C . . . 35 C 25 C . . . 75 C Room temperature Room temperature 25 C . . . 125 C
Data adapted from Hille and Thalhammer, 1997.
A diVerent method is reported by Bertolotti et al. (1990), whose samples were shaped as prisms and illuminated by a monochromatic light ray. The refractive index of the prism can thus be determined from the minimum refraction angle. A comparison of the various experimental results is presented in Table 2. Some of the diVerences arise as in the experiments the temperature dependence is measured for a specific doping concentration, current density, biasing condition, and so on, but usually not for fixed carrier concentration. Nevertheless, the published data are considered an appropriate approximation for (@no/@T )n, p within acceptable error bounds. In this article, the following temperature coeYcient at 1.3 mm (Hille and Thalhammer, 1997) has been used: @no ¼ 1:60 104 K1 : ð15Þ @T n;p
III. MEASUREMENT TECHNIQUES Electrical measurements of the terminal characteristics—both stationary and transient—have been state of the art in device characterization for many years. Recently, however, various probing methods have been introduced that open promising perspectives for research in semiconductor physics and the development of devices and microstructures, as measuring the internal distribution of physical state variables has become possible. This article is focused on optical characterization methods that exploit the electro-optical and the thermo-optical eVect and enable to extract charge carrier and temperature profiles. Typical representatives are reviewed in this section.
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A. Device Characterization Methods Electrically characterizing the terminal behavior usually constitutes the first approach to the experimental investigation of semiconductor devices. Both static and dynamic measurements at various temperatures are used to extract information about technological parameters and device performance. Capacitance–voltage (C–V ) measurements, for example, are a common technique to determine the doping profile of MOS structures and pin diodes. In addition, pulsed current–voltage (I–V ) measurements reveal a variety of decisive device parameters, such as threshold voltages, current or voltage gain, saturation currents, breakdown voltages, holding currents, or snapback voltages. Considerable eVort has been spent to determine the carrier lifetime and the injected charge which is stored in the bulk of a power device during its forward-conducting state. During turn-oV, the excess carrier concentration is partly reduced by internal recombination and partly extracted by a negative terminal current. Suppressing one of each possibilities is the fundamental idea of two special measurement techniques as follow. First, the stored charge QS can be calculated by integrating the reverse recovery current during a rapid turnoV. This idea was first proposed by HoVmann and Schuster (1964), who applied a sudden blocking pulse to a diode carrying a forward current I. The calculation of QS, however, is based on the assumption that the internal recombination during the switching process is negligible. Satisfying this condition requires an experimental setup with low parasitic wire inductivities, which facilitate a steep current slope during turnoV. An eVective carrier lifetime teV ¼ QS/I can be extracted from these measurements (Schlangenotto and Gerlach, 1969), which have become the most frequently used technique to determine the stored charge (Benda and Spenke, 1967; Berz, 1979, 1980; Kao and Davis, 1970; Kuno, 1964). Readers are referred to these publications for further details about the reverse recovery process and the correlation to the carrier recombination in the interior of the device. Second, the carrier lifetimes are determined by open-circuit–voltage-decay (OCVD) measurements. For that purpose, the forward current flow through a diode is suddenly interrupted, thus suppressing any reverse recovery current. The internal recombination of the excess carrier concentration leads to continuously declining diVusion potentials across the junctions. Therefore, the resulting decrease of the terminal voltage is related to the carrier lifetime. Since the invention of this method for investigating pn-junctions (Gossick, 1953; Lederhandler and Giacoletto, 1955), it has become a common technique for measuring eVective carrier lifetimes in junction devices (Choo and Mazur, 1970; Davies, 1963; Gerlach and Schlangenotto, 1972).
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THALHAMMER
A thorough analysis of the switching process reveals which physical eVects govern each period during turnoV. The corresponding sectors of the current or voltage transients thus provide specific information about, for example, the carrier lifetime under high and weak injection or the minority carrier lifetimes in the heavily doped regions. However, pure measurements of the terminal behavior in principle do not allow scanning the distribution of a physical quantity, as for example, the carrier concentration. This knowledge, however, is of great benefit for physical research (e.g., for studying transport phenomena or carrier capture and emission kinetics), as well as for device development, particularly for adjusting the carrier profile of power devices and thus improving the trade-oV between a low forward voltage drop and a fast turn-oV behavior. The development of experimental techniques for the spatially resolved measurement of the carrier distribution started in 1952. Haynes and Briggs (1952) reported that infrared radiation is emitted by silicon and germanium samples into which carriers are being injected. Although these materials are indirect semiconductors, there is a small fraction of phonon- or excitonassisted radiative recombination which constitutes the physical origin of the detected radiation (Haynes and Westphal, 1956). This eVect can be exploited to determine the distribution of carriers in the interior of forward-biased power devices. For that purpose, a specific sector of the sample is observed through an aperture hole. A photo multiplier is used to amplify the emitted infrared radiation, whose intensity is a measure of the local carrier concentration. The 2D distribution of the carriers is thus obtained by shifting the device along its vertical and horizontal axis (Dannh€auser and Krausse, 1973; Gerlach, 1966; Gerlach et al., 1972; Jo¨ rgens, 1982a; Schierwater, 1975; Schlangenotto et al., 1974). While one of the most remarkable features of recombination-radiation measurements is the excellent spatial resolution of about 6 mm to 10 mm, this technique suVers from several major drawbacks (Bleichner et al., 1990b; Jo¨ rgens, 1982b, 1984). First, it is sensitive to the product np; that is, carriers can only be detected if both types are present simultaneously. Consequently, this method is not applicable for the investigation of the minority carrier distribution. Second, the weak radiation intensity requires highly sensitive preamplifiers and an extensive averaging for a suYcient amount of time, thus raising great demands for stable operating conditions during the experiment. Third, since reabsorption attenuates the radiation that is emitted from the inner regions of the sample, the detected intensity primarily originates in those carriers which recombine close to the device surfaces. Therefore, the measurement results are seriously aVected by surface recombination unless the samples are prepared properly. During the same decades, another probing method for the measurement of carrier distributions has been established, which exploits the dependence
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of the absorption coeYcient on the carrier concentration (Bleichner et al., 1990a; Briggs and Fletcher, 1953; Cooper and Paxman, 1978; Go¨ rtz, 1984; Harrick, 1956; Houston et al., 1976; Schierwater, 1975). While an infrared laser beam traverses the device under test, the latter is subjected to periodic current pulses. Since photon scattering by free carriers causes an enhanced absorption during the duty cycle, the local concentration of the injected carriers can be extracted from the decrease of the transmitted light intensity. A detailed description of this method is given in Section III.B.1. Compared with recombination-radiation measurements, the free carrier absorption technique provides a superior sensitivity and a higher time resolution (Bleichner et al., 1990b; Jo¨ rgens, 1984). It will be demonstrated in Section VI.D that an optimized optical setup provides an improved spatial resolution which compares to recombination-radiation measurements. By the invention of optical probing techniques, the temperature distribution has become accessible to measurements, too. For example, thermal expansion causes surface displacements, which are detected by a laser interferometer (Martin and Ash, 1986). This method has been used for investigating transport phenomena (Suddendorf et al., 1992), testing power devices (Claeys et al., 1994) and opto-electronic devices (Epperlein, 1993), and calibrating another temperature-sensitive measurement technique (Deboy et al., 1996). Another class of characterization methods is based on the temperature dependence of the refractive index. Since the reflection coeYcient is therefore aVected by the surface temperature, the latter can be extracted from the intensity modulation of a laser beam that is reflected oV the chip surface (Claeys et al., 1994; Quintard et al., 1996; Rosencwaig, 1987). Laser reflectance thermometry constitutes a useful technique for investigating thermal and plasma waves, which are excited by a modulated pump laser beam (Rosencwaig, 1987; Suddendorf et al., 1992). Pioneering work has been done by Boccara et al. (1980) and Jackson et al. (1981), who introduced photothermal deflection spectroscopy, also known as the ‘‘mirage eVect.’’ The basic mechanism is the deflection of a probing beam due to thermally induced gradients of the refractive index. As the sample is periodically heated by a pump laser beam, temperature inhomogeneities are built up in the adjacent air around the sample, thus deflecting a second laser beam that is propagating closely above and parallel to the device surface [external mirage eVect (Murphy and Aamodt, 1980; Salazar and Sanchez-Lavega, 1999)]. The same physical principle is the foundation of the internal mirage eVect. For that purpose, the probing beam penetrates the sample and is deflected in the interior of the device (Fournier et al., 1986). Since the refractive index of semiconductors is aVected by both the temperature and the carrier density, this nondestructive and noninvasive
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technique is well suited for investigating electric and thermal transport phenomena and material properties. A similar setup without a pump laser is used for characterizing semiconductor power devices to detect the gradients of carrier concentration and temperature that arise from the electrothermal behavior during typical switching conditions (Deboy, 1996; Deboy et al., 1996). Since parallel surfaces of the investigated devices form a Fabry–Perot resonator cavity whose optical thickness is aVected by modulations of the refractive index, the temperature evolution can be extracted from the oscillations of the reflected or transmitted intensity (Seliger, 1998). As an alternative, the phase shift of the probing beam during a single propagation through the sample is detected by interference with a reference beam (Goldstein et al., 1993, 1994; Heinrich et al., 1986). Fabry–Perot thermometry, free carrier absorption, and laser deflection measurements are representative examples of internal laser probing techniques (see Section III.B). Their excellent spatial and temporal resolution facilitates a more comprehensive analysis of the internal electric and thermal transport phenomena. B. Internal Laser Probing Techniques Optical beam testing (So¨lkner, 1994) is attractive for a variety of reasons. For example, very short light pulses can be generated, the probe environment need not be evacuated, and lenses and laser sources are available at low costs. Among the optical characterization methods, the novel class of internal probing techniques has emerged in recent years, exploiting the dependence of the refractive index on carrier concentration and lattice temperature. These non-destructive and non-invasive techniques thus provide time-resolved information about the distribution of carriers and heat in the interior of semiconductor devices and work in the following manner. During transient switching conditions, variations in temperature and the injection or removal of carriers aVect the complex refractive index in the investigated sample. Detecting the resulting modulations of the absorption, the deflection, or the phase shift of an incident probing beam, information about the local carrier concentration and lattice temperature can be extracted. Space-resolved profiles of these quantities are scanned by shifting the device along its vertical or horizontal axis. Although some of these techniques are also applicable for investigating submicron devices, scanning the carrier concentration or temperature profiles works best within structures that are suYciently large in size. Using internal laser probing techniques therefore promises great benefit especially for the development of power devices. The following section provides an
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overview of the internal laser probing techniques that are already available or concepts that are planned to be realized. 1. Free Carrier Absorption Measurements The carrier distribution of an electron-hole plasma in the intrinsic region of power devices can be determined by free carrier absorption measurements. The optical setup is the same as for laser deflection measurements (see Figure 3), except that the detector comprises a simple pin diode instead of the four-quadrant photo diode. A typical detector signal is shown in Figure 2. The continuous wave (cw) laser beam is focused onto the device being tested which is subjected to periodic current pulses. Since injecting carriers enhances the absorption, a decrease of the transmitted light intensity is detected by the photo diode. To improve the signal-to-noise ratio, the measurement signal is averaged for several hundreds of current pulses. The excess carrier concentration Dn(x0, t) at the current beam position x0 is then extracted from the absorption law @a Dnðx0 ; tÞ ; ð16Þ Ion ðx0 ; tÞ ¼ Ioff ðx0 Þ exp L @C where Ion and IoV denote the transmitted intensities during on-state and oVstate of the device, respectively, and L is the interaction length. To eliminate the unknown incident intensity, which may be subject to a thermal drift during the measurement, the AC component of the photo current is scaled to the DC component. This quantity shall be called the absorption signal (Figure 2). In case of small duty cycles, it is equal to Ion/IoV 1. Repeating
FIGURE 2. Measurement signal for free carrier absorption measurements (pulse duration 100 ms).
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the measurements at diVerent beam positions by shifting the devices along its vertical axis enables a vertical scanning of the carrier concentration profile. The spatial resolution is about 10 mm to 20 mm, while the time resolution is limited by the detector capacity to typically some hundreds of nanoseconds. Commonly used light sources are HeNe lasers with a wavelength of l ¼ 3.39 mm (Bleichner et al., 1990a; Cooper and Paxman, 1978; Go¨ rtz, 1984; Linnros et al., 1993), InGaAs laser diodes (Deboy et al., 1996; Tornblad et al., 1997) (l ¼ 1.3 mm), or an incoherent (amplified spontaneous emission (ASE) light source with a wavelength of 1.55 mm (Hille, 2001; Hille et al., 2000). Special periods during the current pulse can be investigated by pulsing the laser beam or by using a mechanical chopper (Go¨ rtz, 1984). The sample preparation includes polishing the surfaces and depositing an antireflective coating. The latter step is not required if multiple reflections at the parallel surfaces are suppressed by using an incoherent light source or by appropriately aligning the sample to exploit the vanishing reflection coeYcient at the Brewster angle (Go¨ rtz, 1984; Linnros et al., 1993; Tornblad et al., 1997). It should be mentioned that evaluating the detector signal according to Eq. (16) is based on the following assumptions:
The observed intensity decrease originates from an enhanced free carrier absorption. This assumption is invalid, for example, if self-heating modulates the optical thickness no(T )L(T ) of the sample, thus changing the transmittance of the Fabry–Perot cavity, which is formed by the parallel device surfaces.
The lateral extension of the probing beam can be neglected.
The carrier distribution along the beam R L path is constant. Otherwise, the product n(x0)L has to be replaced by 0 nðx0 ; zÞ dz. In this case, Eq. (16) cannot be solved for the unknown carrier distribution analytically.
The optical axis is perpendicular to the device surfaces. If the probing beam traverses the sample at an angle FSi in silicon, the intensity decrease at the beam position x0 is given by 0 1 ZL @a Ion ðx0 Þ ¼ Ioff ðx0 Þ exp@ Dnðx0 þ z tanFSi ; zÞdzA: ð17Þ @C 0
The deflection of the probing beam caused by internal gradients of the refractive index can be neglected. This condition is not satisfied during operating conditions with high-power dissipation. The resulting temperature gradients give rise to a beam displacement which can, for example, easily amount up to 20 mm in the interior of a sample, which is only 800 mm in length (see Figure 36).
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2. Internal Laser Deflection Measurements Inhomogeneous distributions of any state variable aVecting the refractive index cause an associated gradient of the refractive index. Detecting the resulting deflection of a penetrating laser beam constitutes the fundamental principle of the photothermal deflection spectroscopy. This mechanism is also known as the mirage eVect and has been exploited to study electrical and thermal transport phenomena (Fournier et al., 1986). The same principle underlies the internal laser deflection method (Deboy et al., 1996), which has been demonstrated to be a sensitive technique for measuring temperature profiles in the interior of power devices. Except for the detector, the setup (see Figure 3) is similar to the laser absorption method described in the previous section. Under transient switching conditions, both the injection (or removal) of carriers and a local temperature change cause a gradient of the refractive index, deflecting the focused laser beam. A long-distance objective transforms the internal deflection approximately into a parallel shift, which is detected by a four-quadrant photo diode. While the diVerence between the photo currents of two opposite segments is related to the vertical or horizontal deflection, the sum of all photo currents is proportional to the total intensity. Thus, laser deflection and laser absorption measurements can be performed simultaneously. Like the absorption signal, the deflection signal is also scaled to the total transmitted intensity in order to eliminate external disturbances. An example of typical measurement signals is shown in Figure 4. The rapid change of the deflection signal immediately after turn-on and turn-oV (t ¼ 0 and t ¼ 70 ms,
FIGURE 3. Experimental setup for internal laser deflection measurements on an IGBT sample. Adapted from Thalhammer and Wachutka, 2003a. # 2003 OSA.
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FIGURE 4. Deflection signal (solid line) and absorption signal (broken line) for internal laser deflection measurements on a power diode subjected to a current pulse of 70 ms in duration.
respectively) originates from the injection or removal of carriers in the intrinsic region. The thermal contribution to the deflection signal is visible on the time scale of several tens of microseconds. Assuming the paraxial approximation, the scaled deflection signal has been quantitatively related to the refractive index gradient (Deboy, 1996): I1 ðx; tÞ I3 ðx; tÞ I1 ðx; tÞ þ 0 L I2 ðx; tÞ þ I3 ðx; tÞ þ IL4 ðx; tÞ 1 Z ~Q Z @no 1 @no @ ðx; z; tÞ dz þ ðx; z; tÞz dzA: ¼ no F A @x @x
Mðx; tÞ :¼
0
ð18Þ
0
In this equation, I1, I2, I3, and I4 represent the photo currents of the four ~ denote the distance between the device’s rear detector segments. F, A, and Q surface and the imaging lens, the angular aperture of the probing beam, and the slope of the standardized detector response function, respectively. It ~ depends on the laser spot size o on should be mentioned, however, that Q the detector (Hille and Thalhammer, 1997, ch. 5.4.2). Changing the position of the lenses therefore demands a recalibration of the response function. A more general expression that is also valid in case of large beam deflections (Thalhammer, 2005) is derived from Fermat’s principle and Fourier optics (see Section VII.A). Then, the deflection signal M becomes:
2 Q L dno L dno 3 M¼ þO þ LðF Vt d2 Þ ; ð19Þ Vt 2no no dx dx
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where Q, Vt, and d2 denote the absolute detector response function (see Section VII.E), the inverse of the transverse magnification of the imaging lens, and the distance between the imaging lens and the detector, respectively. Solving for the refractive index gradient reveals the desired information on the internal carrier concentration and temperature gradients: L2 2no
M Vt =Q þ LðF Vt d2 Þ
þ OðM 3 Þ ¼
@no ðx; tÞ ¼ @x
@no @n @no @T ðx; tÞ þ ðx; tÞ: @C T @x @T n;p @x
ð20Þ
If the carrier concentration profile is in its stationary state before a significant temperature gradient is built up at the current beam position, the two contributions to the deflection signal can be separated owing to their widely diVering time constants (see Figure 4). This condition does not hold if heat sources are closely located near the beam path. In this case, the carrier contribution must be determined from the absorption signal or an electrothermal device simulation and subtracted from the deflection signal. To prevent disturbances by a thermal drift of the experimental setup, the detector is mounted on a piezo x-y–translation unit, which is controlled by low-pass filters. Thus, the average deflection signals are kept at zero and the detector position is dynamically adjusted in the center of the laser beam. For laser deflection and free carrier absorption measurements, the sample needs to be prepared by dicing to a suitable size and polishing the surfaces. After that, an antireflective coating is deposited, unless the interference of multiply reflected rays in the interior of the device is suppressed by use of an incoherent light source or aligning the sample such that the laser beam propagates at the Brewster angle. While the internal laser deflection technique facilitates an excellent sensitivity of 25 mK /mm for temperature measurements in power devices (Deboy, 1996), it suVers from its limited measurement range as the detector response saturates. During typical transient switching conditions, the maximum detectable temperature gradient with a typical interaction length of a few millimeters is exceeded if the peak temperature rises to only a few degrees Kelvin [cf. (Deboy, 1996, ch. 7.4.2)]. Section VII.G discusses how to overcome this restriction. 3. Backside Laser Probing Like the internal laser deflection method, backside laser probing exploits the dependence of the refractive index on carrier concentration and lattice temperature. The detected quantity, however, is the phase shift D’(t) of a
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vertically propagating probing beam caused by modulations of the optical sample thickness (Thalhammer et al., 1998a): D’ðtÞ ¼ 2
2p l
# ZL " @no @no @no DTðz; tÞ þ Dnðz; tÞ þ Dpðz; tÞ dz: @T n;p @n p;T @p T;n 0
ð21Þ This phase shift is detected by interference with a reference beam that is reflected oV the rear metallization layer. While the laser diode is operating in cw mode (Goldstein et al., 1994; Seliger et al., 1996a, 1998) or in pulsed mode (Goldstein et al., 1994), the device under test is subjected to periodic current pulses. A schematic view of the experimental setup is shown in Figures 5 and 6. Heinrich et al. (1986) proposed an interferometer including a Wollaston prism to split the original laser beam. Their setup has been modified by Goldstein et al. (1993), who exploited Bragg diVraction by the acoustic waves in the interior of an acousto-optic modulator (AOM). The diVraction angles are controlled by the driving frequencies o1 and o2, thus arbitrarily positioning the probing beam and the reference beam is possible. After being reflected and passing the AOM cell again, the two beams interfere on the detector, causing an intensity signal proportional to sin(2Dot þ D’(t)). Two diVerent ways of signal processing can be done. Including a local oscillator with frequency 2Do (Figure 6; Goldstein et al., 1994; Seliger, 1998;
FIGURE 5. Principle of backside laser probing on an IGBT sample. While the probing beam penetrates the active area and is reflected at the top metallization layer, the reference beam is reflected at the collector contact metallization layer. Adapted from Thalhammer and Wachutka, 2003b. # 2003 OSA.
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FIGURE 6. Experimental setup for backside laser probing.
Seliger et al., 1996a) facilitates a highly sensitive detection of either the intensity modulation or the phase modulation of the probing beam. The phase of the local oscillator determines which quantity is actually observed. For the sake of an increased bandwidth, however, the local oscillator is omitted, allowing measurement of the phase modulations during fast transient switching conditions (Fu¨rbo¨ ck et al., 1998; Seliger, 1998). A typical phase shift signal is shown in Figure 7. The two contributions originating from the injection/removal of carriers and the temperature rise during the current pulse can be easily identified owing to their diVering time constants. Although opposite in sign, they are of the same order of magnitude if the little power is dissipated in the interior of the sample. However, for a large temperature rise, which typically occurs as power devices are operating in short-circuit mode, for example, the phase modulation by the carriers can be neglected in view of the thermal contribution. The backside laser probing technique is a valuable method for investigating lateral carrier concentration and temperature inhomogeneities. Its spatial resolution is limited by the laser spot size to approximately 2 mm.
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FIGURE 7. Phase shift signal for operating conditions with lowpower dissipation (pulse duration 50 ms). To illustrate each contribution simulation results are included (broken lines). Adapted from Thalhammer and Wachutka, 2003b. # 2003 OSA.
Although only integral information on the vertical profiles is provided, this technique is useful for characterizing vertical power devices since it is capable of detecting large temperature rises of 100 K or more. As a slight modification of the optical setup, diVerential backside laser probing has been used for investigating IGBTs by Fu¨ rbo¨ ck et al. (1999): directing both beams into the active area of the sample enables a direct measurement of the diVerence between the temperatures at the two beam positions (Goldstein, 1993). The backside laser probing techniques require a special sample preparation: To provide access through the rear side a window of approximately 70 mm 70 mm in size has to be opened in the metallization layer. This is done by photolithographic structured etching with two subsequent etching steps using HNO3 and HF acids. Thus, the metallization can be removed without etching the silicon. The eVect of the contact window on the carrier and temperature distribution is discussed in Section VIII.B.1. As the final preparation step, an antireflective coating is deposited to suppress multiple reflections within the substrate. 4. Fabry–Perot Reflectivity Measurements The reflectivity of a Fabry–Perot resonator is modulated by thermally induced variations of its optical thickness no(T )L(T ). Hence, the temperature evolution within the sample can be detected by monitoring the reflected intensity (Pogany et al., 1998a; Seliger et al., 1997a, 1998). The experimental
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FIGURE 8. Probing the temperature distribution in a lateral SOI IGBT by Fabry–Perot reflectivity measurements. The probing beam is reflected back and forth between the top surface of the wafer and the silicon/oxide interfaces.
setup for investigating lateral SOI power devices (Seliger, 1998) is shown in Figure 8. Denoting the reflection coeYcients at the front surface and at the rear surface by rf and rr, respectively, the reflectivity RFP of a lossy Fabry–Perot resonator of thickness L becomes (Seliger, 1998) RFP ¼
ðrf þ rr eaL Þ2 4rf rr eaL sin2 ð2p l no LÞ
ð1 þ rf rr eaL Þ2 4rf rr eaL sin2 ð2p l no LÞ
:
ð22Þ
As two adjacent maxima of the reflectivity are observed, the optical thickness no(T ) L(T ) has been changed by l/2. The corresponding temperature diVerence DT is therefore equal to l @L @no 1 no þL DT ¼ : ð23Þ 2 @T @T The temperature distribution in lateral SOI IGBTs has been determined by Fabry–Perot reflectivity measurements (Seliger et al., 1997b) by means of a vertically propagating laser beam. On the other hand, scanning temperature profiles in vertical devices requires a laterally impinging probing beam.
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In this case, however, a limited measurement range is to be expected since the amplitude of the intensity oscillations decreases as the internal deflection increases. This problem is inherent to all interferometric techniques that use a laterally propagating beam and is addressed in Section VIII.A. 5. Fabry–Perot Transmission Measurements A related concept is founded on monitoring the transmittance TFP of a Fabry–Perot resonator which is modulated as the temperature changes its optical thickness. TFP ¼
ð1 rf rr Þ2 eaL ð1 þ rf rr eaL Þ2 4rf rr eaL sin2 ð2p l no LÞ
:
ð24Þ
While thermally induced Fabry–Perot oscillations of the transmitted intensity are an undesired eVect during laser absorption measurements, they can be exploited for extracting the temperature rise in the interior of the sample. Choosing a coherent light source and omitting the deposition of an antireflective coating, the same experimental setup as for the absorption measurements can be used (see Section III.B.1). The temperature diVerence between two adjacent maxima is then calculated according to Eq. (23). 6. Mach–Zehnder Interferometry Mach–Zehnder interferometry constitutes another probing concept for scanning the vertical carrier concentration and temperature profiles (Deboy, 1996). The experimental setup is sketched in Figure 9. The modulations of the refractive index give rise to a phase shift of the laterally penetrating probing beam. This phase shift is detected by interference with a reference beam that is guided around the sample. Mach–Zehnder interferometry requires a coherent light source. Multiple reflections between the parallel surfaces must be suppressed by depositing an antireflective coating. However, no measurement results on semiconductor devices based on this method have yet been reported. 7. Summary Internal laser probing techniques exploit the electro-optical and the thermooptical eVect and thus enable time-resolved measurements of the carrier and temperature distribution in the interior of semiconductor samples. Table 3 compares the previously described methods. Although each technique only reveals integral information along the beam path, the sensitivity and the measurement range is expressed in terms of the local carrier concentration and temperature values, assuming typical interaction lengths for the
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FIGURE 9. Experimental setup of the Mach–Zehnder interferometer.
conversion. As far as the probing techniques have not been realized, yet, expected data are presented. IV. MODELING OPTICAL PROBING TECHNIQUES Internal laser probing techniques provide valuable information that crucially enhances our insight into the internal behavior of semiconductor samples and microstructures. Accurately evaluating the measurement signals, however, requires a consistent analysis that exceeds the concepts of geometrical ray tracing or simple 1D models. For that purpose, a physically rigorous model for simulating the entire measurement process has been proposed (Thalhammer and Wachutka 2003a; Thalhammer et al., 1998b). Parasitic eVects that may be introduced as the sample is prepared and inherent phenomena arising from the wave propagation of the probing beam can thus be investigated thoroughly. As will be demonstrated for some representative examples of internal laser probing techniques in later sections, the numerical analysis supports the design of the experiments and improves the interpretation of the measurement results. The following sections provide a detailed discussion of the basic ideas for modeling optical probing techniques. Throughout this sections the coordinate system is chosen in such a way that the laser beam propagates along the positive z-axis.
TABLE 3 COMPARISON OF SOME TYPICAL REPRESENTATIVES OF INTERNAL LASER PROBING TECHNIQUES Free carrier absorption measurements
Laser deflection measurements
Beam propagation Scan direction Detected quantities
Lateral Vertical Carrier concentration
Sensitivity
1014 cm3
Measurement range up to Spatial resolution Calibration Light source Preparation
1018 cm3
Lateral Vertical Gradient of carrier concentration, temperature gradient 1011 cm3/mm 25 mK/mm 1014 cm3/mm 50 mK/mm 10 mm Detector response Incoherent Polishing surfaces
10 mm — Incoherent Polishing surfaces
Backside laser probing
FP reflectivity measurements
FP transmittance measurements
Mach–Zehnder interferometry
Vertical Lateral Carrier concentration, temperature
Vertical Lateral Carrier concentration, temperature
Lateral Vertical Carrier concentration, temperature
Lateral Vertical Carrier concentration, temperature
1014 cm3 5 mK 1019 cm3 >500 K 5 mm — Coherent Etching a window, depositing an antireflective coating
1015 cm3 50 mK 1019 cm3 >500 K 5 mm — Coherent —
1013 cm3 1 mK ?
1013 cm3 1 mK ?
10 mm — Coherent Polishing surfaces
10 mm — Coherent Polishing surfaces, depositing an antireflective coating
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A. Simulation Steps—An Overview Since the optically absorbed power is by several orders of magnitude smaller than the electric power dissipation, the electrothermal processes in the investigated sample are not aVected by the penetrating probing beam. Hence, simulating the measurement process can follow the strategy sketched in Figure 10. The first step is a transient electrothermal device simulation of the operating condition the structure is subjected to during the measurement. The calculation takes into account specific properties of the prepared sample (e.g., a window in the metallization layer [see Section III.B.3] or an enhanced surface recombination rate caused by sawing and polishing the surfaces). This step can be conveniently performed by using one of the commercially available general-purpose simulators, such as DESSISISE (ISE Integrated Systems Engineering AG, 2004), which is based on a consistent electrothermal
FIGURE 10. Strategy for the physically rigorous simulation of a laser probing measurement on a semiconductor structure subjected to a specific operating condition (solid lines). The dashed lines reflect a real experiment. Adapted from Thalhammer and Wachutka (2003a). # 2003 OSA.
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model. Thus, the carrier concentration and temperature profiles n, p, and T are obtained as functions of space and time. From these the space- and time-dependent modulations of the complex refractive index no ¼ no þ ia at the laser frequency o are calculated according to the dependencies summarized in Section IV.C. The third step, calculating beam propagation through the device under test, is one of the most demanding problems. Since the laser field oscillates much faster than any electrical or thermal variable, the optical problem is solved for an appropriately chosen set of points of time t1, t2, t3, . . . at each of which the refractive index distribution is treated stationary. An eYcient algorithm is derived from Maxwell’s equations in the frequency domain (see Section IV.D) As a result, we obtain the field distribution of the reflected and the transmitted waves, Er and Et, respectively, at every point of time under consideration. The image of the emerging wave on the detector ED is formed by an arrangement of lenses, aperture holes, and so on. A suitable model for the beam propagation through this optical elements is gained from Fourier optics (see Section IV.E). Finally, integrating the field distribution over the active areas of the (possibly dynamically translated) detector (see Section III.B.2) reveals the photo currents of each detector segment. From these, the time-dependent measurement signal is constructed. The following sections provide a more detailed discussion of the algorithms used for each simulation step. B. Electrothermal Device Simulation From the physically rigorous point of view, the internal dynamics of semiconductors constitutes a many-particle problem of quantum mechanics. However, among the numerous degrees of freedom only a few macroscopic observables are of particular interest. A rigorous treatment of the many particle problem is therefore neither feasible nor desirable. Hence, the fundamental challenge in semiconductor device modeling is reducing complexity to a physically based set of equations with a limited number of variables. 1. Survey of Electrothermal Device Simulation Models Several semi-classical approaches to device simulation have been proposed that are based on a momentum expansion of Boltzmann’s transport equation. Assuming local equilibrium distribution functions, the first three moments represent particle conservation, momentum conservation, and energy conservation, respectively. From these, Bløtekjær (1970) derived a set of
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equations that is known as the hydrodynamic (HD) model today. Cook and Frey (1982) simplified the HD model to the so-called energy balance (EB) model by neglecting the mean kinetic energy compared to the electron thermal energy. Assuming constant and equal temperatures of the electron gas, the hole gas and the host lattice, we obtain the drift–diVusion (DD) model as a special case of the EB model. From the historical point of view, the DD model was the first approach to semiconductor device simulation and was reported by vanRoosbroeck in 1950. It forms the original basis of many well-known general-purpose device simulators [e.g., PISCES, MEDICI (Synopsys, 2004) ATLAS (Silvaco, 2004)]. Although additional models of lattice heating have been included in some of them afterward, they usually lack of a selfconsistent treatment of carrier and heat flow. A diVerent methodology was published by Wachutka (1990, 1991), who applied the principles of irreversible thermodynamics (Callen, 1985) and linear transport theory (Onsager, 1931a,b) to derive a self-consistent electrothermal formulation, which is known as the thermodynamic (TD) model. Considering the three interacting systems of the electrons, the holes, and the lattice phonons, the associated particle and energy balance equations govern the evolution of the state variables and thus reflect the dynamics of the whole system. This model forms the basis of the general purpose simulator DESSISISE (ISE, 2004) and is summarized in the following sections. 2. The Thermodynamic Model In the thermodynamic model, each of the three subsystems—electrons, holes, and phonons—is represented by state variables, such as n, p, and T. Their contribution to the total energy are, for example, " # @fn dun ¼ Tdsn qfn dn ¼ cn dT þ q T fn dn; ð25Þ @T n;p where Sn, fn, Cn denote the entropy density, the quasi-Fermi level of the electrons, and the specific heat of the electron gas, respectively. In a more general description that also accounts for the dynamic ionization of trap centers, in particular the donor and acceptor levels, additional subsystems with their specific particle densities and temperatures must be regarded (Lades, 2000; Wachutka, 1995). If the ionization energies of the dopants are small compared with the thermal energy, however, a complete ionization of all dopands can be assumed. The dynamic evolution of each extensive parameter X is governed by the associated balance equation
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Y @nðr; tÞ þ div j x ðr; tÞ ¼ ðr; tÞ; @t x
ð26Þ
where x, jx, and Px denote the density of X, the corresponding current density, and the production rate, respectively. Hence, the particle balances for electrons and holes and the energy balance become: q
@nðr; tÞ div J n ðr; tÞ ¼ q½Gðr; tÞ Rðr; tÞ @t
ð27Þ
q
@pðr; tÞ þ div J p ðr; tÞ ¼ q½Gðr; tÞ Rðr; tÞ @t
ð28Þ
@uðr; tÞ þ div J u ðr; tÞ ¼ 0: @t
ð29Þ
Here, n, p, Jn, Jp denote the densities and the electric current densities of electrons and holes, respectively, while u and Ju represent the density and the current density of the total energy. Local gradients of the state variables originate particle and energy currents, which are the subject of Onsager’s (1931a,b) irreversible transport theory. According to his fundamental reciprocity theorem, the bilinear dependence Ps ¼ Sk, j Fk Lkj Fj of the entropy production rate Ps on the driving forces Fk is represented by a symmetric and positive definite matrix Lkj. Consequently, the current densities Jk ¼ Sj Lkj Fj can be expressed in terms of the gradients of the state variables with six independent transport coeYcients: J n ¼ qnmn ð=fn þ Pn =TÞ snp =fp
ð30Þ
J p ¼ qpmp ð=fp þ Pp =TÞ spn =fn
ð31Þ
J u ¼ kt =T þ ðTPn þ fn ÞJ n þ ðTPp þ fp ÞJ p þ snp TPp =fn þspn TPn =fP :
ð32Þ
The independent parameters are represented by the mobilities mn and mp, the coeYcient snp ¼ spn, the thermoelectric powers Pn and Pp, and the thermal conductivity kt. In the general theory, the electron current density may also depend on the gradient of the quasi-Fermi level of the holes, which has been subject of
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many discussions in literature (e.g., Mnatsakanov et al., 1987, 1998). Recently, internal laser probing techniques (see Section III.B.2) have been used for the experimental validation (Schlo¨gl et al., 1998, 2000). However, the corresponding coeYcient snp is small and will therefore be neglected in the following discussions, which is in agreement with the commonly accepted hypothesis. Inserting Eq. (32) and the equations similar to Eq. (25) into Eq. (29) yields the heat flow equation (Wachutka, 1990); ct
@T =ðkt =TÞ ¼ H @t
H ¼ =½ðfn þ Pn TÞJ n þ ðfp þ Pp TÞJ p @fp @p @fn @n q fp T þq fn T @T @t @T @T
ð33Þ
ð34Þ
where the heat generation rate H includes the joule heats of electrons and holes, J 2n =qnmn and J 2p =qpmp , respectively; the generation/recombination heat q(R G) (fp þ TPp fn TPn); the Peltier–Thomson heat JnT =Pn Jp T =Pp; and an additional term caused by transient modulations of the carrier concentrations n and p. In addition to the dynamic Eqs. (27), (28), and (33), state equations have to be satisfied. These are Poisson’s equation =ð=cÞ ¼ qðn p þ NA NDþ Þ;
ð35Þ
which constitutes the conditional equation for the electrostatic potential c, and the Fermi–Dirac statistics for electrons and holes kT n ln fn ðn; p; TÞ ¼ cðn; p; TÞ ð36Þ q nie;n ðTÞ kT p ln fp ðn; p; TÞ ¼ cðn; p; TÞ þ ; q nie;p ðTÞ
ð37Þ
which relate the conjugate variables fn and fp to the thermodynamic variables n, p, and T. Whereas the quasi-Fermi levels fn,p are explicitly given by Eqs. (36) and (37) and can be directly inserted into Eqs. (30) and (31). Poisson’s equation [Eq. (35)] cannot be solved for the potential c analytically. Therefore, c is introduced as an additional variable and we end up with a system of four partial diVerential equations comprising Eqs. (27), (28), (33), and (35), which must be solved numerically.
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3. Material Properties The model sketched in the previous section includes several material coeYcients. These are the dielectric constant , the total heat capacity ct, the eVective intrinsic density ni,eV, the generation/recombination rate R, as well as the transport coeYcients mn, mp, Pn, Pp, and kt. Many of them must cover a variety of physical phenomena and therefore usually exhibit a very complex dependence on the state variables. A detailed overview of common models can be found in Schenk (1998). One of the crucial parameters of the electrothermal model is the total recombination rate R, which complexly depends on the distributions of the charge carriers in phase space. In indirect semiconductors, transitions from the conduction band minimum to the valence band maximum (or vice versa) proceed by the assistance of a third particle for momentum transfer. Radiative recombination can be neglected in view of the recombination via traps in the energy gap (Shockley–Read–Hall recombination), the Auger recombination, and the impact ionization (Chynoweth, 1958; Lackner, 1991; Moll and van Overstraten, 1963; van Overstraten and Man, 1970). The most significant contribution usually originates from the recombination via a trap level in the energy gap. Shockley and Read (1952) and Hall (1952) derived an empirical model by balancing the transition rates between the conduction band, the trap level, and the valence band. The transition probabilities are related to the capture cross-sections and expressed in terms of the so called carrier lifetimes tn and tp. They usually depend on doping concentration, temperature, and electric field (Tyagi and van Overstralten, 1983). As some common dopands for lifetime control (Baliga, 1978; Lisiak and Milnes, 1975), introduce a variety of diVerent trap levels in the energy gap (Azimov et al., 1974; Brotherton et al., 1979), a generalization for two coupled defect levels was derived by Schenk and Krumbein (1995). The motion of carriers as a response to the driving aYnities is limited by scattering events. The most important mechanisms in silicon are phonon scattering and impurity scattering (Brooks, 1951; Caughey and Thomas, 1967; Masetti et al., 1983) electron-hole scattering (Choo, 1972; Conwell and Weisskopf, 1950; Fletcher, 1957), and scattering by interface states (Hartstein et al., 1976; Lombardi et al., 1988; Sah et al., 1972). In large electric fields, a saturation of the carrier mobility is observed (Canali et al., 1975; Caughey and Thomas, 1967). 4. Numerical Methods Solving the coupled system of the partial diVerential Eqs. (27), (28), (33), and (35) involves the following steps: discretization in space and time transforms the partial diVerential equations to a nonlinear system of algebraic
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VIRTUAL OPTICAL EXPERIMENTS
equations. The whole system is either simultaneously solved by the direct Newton algorithm or decoupled and subjected to the iterative Gummel scheme (1964). A suitable space discretization is based on the box discretization method (Bank et al., 1983). Its greatest advantage is the ability to handle arbitrary geometries and nonuniform meshes. The perpendicular bisectors of the edges to the neighboring vertices form the boundary of a box around each vertex. Then, the partial diVerential equations are integrated over the volume of the box, applying the theorem of Gauss to transform the integrated divergence operators. The time discretization scheme follows the TR/BDF2 method (Bank et al., 1985). A trapezoidal diVerentiation formula is applied to determine the step toward an artificial point of time. Based on this interpolation, the real-time step from t ¼ tn to t ¼ tnþ1 is calculated. C. Calculation of the Refractive Index Modulations Any of the models in Section II can be chosen to describe how the complex refractive index is aVected by the carrier concentrations and the temperature. As mentioned previously, however, the published data diVer rather widely, but any of the proposed models depends exactly or approximately linearly on carrier concentration and temperature. For that reason, a first-order Taylor expansion of the complex refractive index can be used to evaluate the resulting eVect on the laser beam (Thalhammer and Wachutka, 2003a): @no @no @no no ðn; p; TÞ ¼ no;0 þ Dn þ Dp þ DT ð38Þ @n p;T @p n;T @T n;p
@a aðn; p; TÞ ¼ a0 þ @n
@a Dn þ @p p;T
@a Dp þ @T n;T
DT:
ð39Þ
n;p
In case of high-injection conditions (n ¼ p), these relations simplify to @no @no Dn þ DT no ðn; p; TÞ ¼ no;0 þ ð40Þ @C T @T n;p
@a aðn; p; TÞ ¼ a0 þ @C
@a Dn þ @T T
DT n;p
ð41Þ
where the definitions (8) and (9) have been used. That means, for modeling the measurement process, it is suYcient to know the expansion coeYcients. The constant terms may be fixed by special parasitic eVects (e.g.,
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THALHAMMER
the Fabry–Perot oscillations of the absorption signal [see Section VI.C]), whose amplitude decreases as a0 increases. But since the probing techniques actually detect the diVerence between oV-state and on-state of the device under test, the desired information provided by the detector signals does not depend on no,0 and a0. The linear expansion coeYcients therefore represent the information that needs to be known. The subsequent discussions are based on the following data: @no @no ¼ 1:81 1021 cm3 ¼ 1:60 104 K1 @C T @T n;p ð42Þ @a @a 18 2 ¼ 5:11 10 cm 104 cm1 K1 : @C T @T n;p Under typical operating conditions of power devices the most important eVects are caused by the carrier concentration dependence of the absorption coeYcient (@a/@C )T and the temperature dependence of the refractive index (@no/@T )n,p. It should be mentioned that in general all the expansion coeYcients depend on the state variables and therefore have to be measured at a representative set of operating points if widely diVering operating conditions are investigated. D. Wave Propagation in Inhomogeneous Media To perform a physically rigorous simulation of the beam propagation through the device under test, we must calculate wave propagation in a medium with an inhomogeneous and in general arbitrary distribution of the complex refractive index. Spatial variations arise for two major reasons. First, diVerent kinds of materials in the structure (e.g., silicon, oxide, or metallization layers) form regions with rather widely diVering refractive indices. Second, additional variations during transient switching of the device are caused by the electro-optical and the thermo-optical eVect. These modulations are small but time-dependent and must be accurately taken into account as they aVect the desired contribution to the detector signals. 1. Algorithms Reported in the Literature A large diversity of numerical methods has been proposed in the literature (see, for example, the summary in Koch, 1989), the most important of which are briefly mentioned here. Since the optical field distribution in the entire sample must be calculated at numerous points of time, the suitability of the algorithms is judged by their computational economy in a large simulation domain.
VIRTUAL OPTICAL EXPERIMENTS
261
For surface integral techniques (e.g., the method of moments [MOM] or the boundary element method [BEM]), a set of basis functions is selected, each of which represents an exact solution of the partial diVerential equation. The fundamental idea is to expand the unknown field distribution in terms of these basis functions, with the unknown variables being the expansion coeYcients. They are determined by the requirement to satisfy the boundary conditions. In particular, they have to minimize the scalar products of the residuals of the governing equations and a set of weighting functions. Common choices for the latter are the basis functions themselves (Galerkin method) or Dirac’s delta distributions at the grid nodes (point matching techniques). A special case is the generalized multipole technique (Hafner, 1990). The electromagnetic vector fields in homogeneous regions are expressed in terms of the electromagnetic multipole functions at diVerent origins, each of them satisfying the Helmholtz equation. The boundary conditions at a finite number of matching points constitute an overdetermined system of equations, which is solved for the expansion coeYcients in the least-square sense. Although the rather small number of unknowns promises an eYcient calculation, this technique can only be used in homogeneous domains and is rather diYcult to set up (Ko¨ rner, 1999). A straightforward and generally applicable technique is gained by finite element or finite diVerence discretization of the wave equation or Maxwell’s equations (Thomas, 1995; Yee, 1966). In order to get an acceptable discretization error, however, the mesh needs to be refined up to 10 nodes per wavelength. Since the latter is about 400 nm in silicon, the typical dimensions of suitable samples for laser probing would require a huge amount of grid nodes. Alternatively, beam propagation methods (Lagasse and Baets, 1987) consider a sequence of parallel planes perpendicular to the optical axis and calculate the field distribution on each plane from the distribution on the respective preceding plane. The relation is derived from Maxwell’s equations (NiederhoV, 1996), or the scalar wave equation (Feit and Fleck, 1978; Roey et al., 1981; Yahel and Last, 1992). It can be analytically solved for the unknown field values, if the paraxial approximation is assumed. Each propagation step therefore just requires a matrix multiplication, thus resulting in a very fast algorithm. However, beam propagation techniques can only be applied if a paraxial wave is propagating in one direction. Although longitudinal reflections can be included by calculating the propagation back and forth, numerical stability is seriously aVected because numerical noise will inevitably excite inherently present parasitic modes whose field distribution is exponentially increasing. Multiple reflections can be calculated in a single computation step by multiplying propagator matrices, which allow for both forward and
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backward traveling waves. To derive such matrices, the wave guide model (Burckhardt, 1966; Yuan and Strojwas, 1991) assumes a longitudinally piecewise constant refractive index and expands the electric and magnetic field distribution in each layer in terms of a Fourier series. The expansion coeYcients are matched so that the fields are continuous across the interfaces, yielding linear relations that define the propagator matrices in the respective layers. Their product relates the field distribution at the emergence plane to that on the entrance plane, which is finally calculated from the radiation boundary conditions in propagation direction. A similar strategy forms the diVerential method (Kirchauer and Selberherr, 1997; Petit, 1980; Yeung, 1988), which merely diVers by how the propagator matrices are derived: a lateral Fourier transformation of Maxwell’s equations leads to a first-order diVerential equation in z direction. Explicit Euler discretization of the latter yields the desired linear relation of the field distributions on two adjacent planes. With these methods, the field distribution on the entrance plane is considered a superposition of forward and backward propagating waves, whereas it is simply equal to the incident wave in the abovementioned beam propagation techniques. An eYcient algorithm for simulating laser probing techniques has been reported by Thalhammer and Wachutka (2003a). It is similar to the wave guide model and the diVerential method. As the major improvement, a diVerent set of variables enables a much coarser discretization without losing accuracy. Furthermore, the propagator matrix is derived by a longitudinal integration of Maxwell’s equations, which is more accurate and stable than the previously-described techniques and can be combined with diVerent lateral discretization methods. The key ideas are outlined in the following sections. 2. The Propagator Matrix First, a linear relation of the field distributions on the entrance plane and the emergence plane is derived that rigorously takes into account an arbitrary superposition of forward and backward propagating waves. a. Basic Assumptions.
The derivation is carried out in two dimensions; that is, all quantities are assumed independent of the y coordinate and derivatives with respect to y vanish.
The field distribution is TE polarized; that is,
EðrÞ ¼ Ey ðrÞey : Consequently, the magnetic field becomes B(r) ¼ Bx(r)ex þ Bz(r)ez.
ð43Þ
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VIRTUAL OPTICAL EXPERIMENTS
c
The dielectric constant R ðr; tÞ ¼ no ðr; tÞ þ i 2o aðr; tÞ
2
varies slowly in view of the oscillations of the electromagnetic field. At any point of time tj, the beam propagation can therefore be calculated for a stationary dielectric constant R(r, tj). Hence, the argument tj is omitted in the following discussion.
The magnetic permeability is equal to m0. Since no polarization dependent eVects have been observed in the measurements, the second constraint does not limit generality. Besides, the two latter assumptions are satisfied naturally. For many problems of practical relevance, the edge lengths of the investigated samples or the characteristic scales along one axis are much larger than the spot size of the probing beam. In this case of approximative translational symmetry, 2D simulations accurately reflect the respective experiment. To cover the general case of 3D structures with similar scales in each direction, the presented algorithm has to be extended to 3D, which is straightforward from the theoretical point of view, but considerably enlarges the computational eVorts for simulating the experiments. b. Computational Variables and their Governing Equations. Based on these assumptions, Maxwell’s equations in the frequency domain become @ Ey ðx; zÞ ¼ ioBx ðx; zÞ @z
ð44Þ
@ o i @2 Bx ðx; zÞ ¼ i 2 R ðx; zÞEy ðx; zÞ Ey ðx; zÞ: @z c o @x2
ð45Þ
The two degrees of freedom are represented by Ey and Bx. The remaining @ component Bz can be calculated from Bz ðx; zÞ ¼ oi @x Ey ðx; zÞ. The key idea for enhancing computational economy consists in replacing the variables Ey and Bx by the electric fields of the forward and backward propagating waves and analytically splitting oV their rapid oscillations. Formally speaking, the following transformation of variables with the global parameter k is used: h i 1 o E F ðx; zÞ :¼ eikz Ey ðx; zÞ Bx ðx; zÞ 2 k ð46Þ i 1 ikz h o B E ðx; zÞ :¼ e Ey ðx; zÞ þ Bx ðx; zÞ : 2 k The electromagnetic fields are reconstructed from the inverse transformation Ey ðx; zÞ ¼ eikz E F ðx; zÞ þ eikz E B ðx; zÞ k ikz F e E ðx; zÞ þ eikz E B ðx; zÞ : Bx ðx; zÞ ¼ o
ð47Þ
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THALHAMMER
If the parameter k is chosen as the propagation constant along the z-axis, the variables EF and EB represent the envelopes of the waves traveling in positive and negative z direction, respectively. EF and EB vary the more slowly the closer k matches the propagation constant in z direction. Although the latter is a priori unknown, the value no oc is a good guess for paraxially propagating waves. A grid refinement of several microns has proven suYcient, which is about two orders of magnitude coarser than the required discretization width for calculations in terms of Ey and Bx. Inserting Eq. (47) into Eqs. (44) and (45) reveals the diVerential equations that EF and EB must satisfy: @ F io2 i @ 2 F ik F E ¼ 2 R E F þ E E @z 2k @x2 2 2c k 2 io i 2ikz @ 2 B ik 2ikz B 2ikz B þ 2 R e E þ e E e E 2k 2 2c k @x2
ð48Þ
@ B io2 i @ 2 B ik B E ¼ 2 R E B E þ E @z 2k @x2 2 2c k io2 i 2ikz @ 2 F ik 2ikz F 2ikz F 2 R e E e E þ e E : 2k 2 2c k @x2
ð49Þ
c. The Propagator Matrix. Performing the numerical solution of Eqs. (48) and (49) on a rectangular tensor grid, diVerent lateral (x-axis) and longitudinal (z-axis) discretization methods can be flexibly combined. The most accurate strategy for the lateral direction is the transformation to kxspace (e.g., Yahel and Last, 1992; Yuan and Strojwas, 1991; Kirchauer and @2 Selberherr, 1997). In this case, the operator @x 2 is represented by the factor k2x , while the product R(x, z) EF,B (x, z) becomes a convolution integral. The discretized algebraic system of equations therefore includes fully occupied matrices, which is the major drawback of this approach. In addition, the expansion in terms of periodic functions involves considerable diYculties if Dirichlet boundary conditions to model reflecting surfaces at x ¼ x1 or x ¼ xNx have to be implemented. An eYcient algorithm becomes possible if finite diVerence discretization is used in the lateral direction (Thalhammer and Wachutka, 2003a). The resulting restriction that the largest grid spacing along the x-axis must not exceed 1/10 of the smallest lateral wavelength lx ¼ 2p/kx is easily satisfied for paraxially propagating waves. The greatest advantage of finite diVerence discretization is that the discretized system of equations can be evaluated @2 very quickly since the operator @x 2 is represented by a band-structured matrix with only the first superdiagonal and subdiagonal lines occupied. Special attention must be paid to the longitudinal discretization to accurately allow for the rapidly oscillating terms e2ikz in Eqs. (48) and (49). For
VIRTUAL OPTICAL EXPERIMENTS
265
that purpose, these equations are integrated analytically over the interval [zn, znþ1], linearly interpolating the computation variables EF,B and the dielectric constant R. Sorting by EF,B(x, zn) and EF,B (x, znþ1) transforms Eqs. (48) and (49) to a matrix equation of the form: 0 1 iDzn 2 2 @ ðxÞ B ðxÞ þ K @ A n n n x x C E F ðx; znþ1 Þ B 4k @ iDzn 2 A E B ðx; znþ1 Þ @ Cn ðxÞ þ Kn @x2 Dn ðxÞ þ 4k x 1 0 ð50Þ ~ n ðxÞ þ iDzn @ 2 ~ n ðxÞ þ K ~ n@2 F A B x C E ðx; zn Þ B 4k x ¼@ : A iDz 2 E B ðx; zn Þ n 2 ~ n ðxÞ þ K ~ n ðxÞ ~ @ @ D C n x 4k x ~ n are functions of o, k, zn, and znþ1, In this equation, the constants Kn and K ~ ~ n ðxÞ, and D ~ n ðxÞ; C ~ n ðxÞ additionally while An (x), Bn(x), Cn(x), Dn(x), An ðxÞ; B depend on R(x, zn) and R(x, znþ1). The second step is the lateral discretization, which is illustrated for the finite diVerence discretization scheme:
2 2f ðxjþ1 Þ 2f ðxj Þ @ f ðxÞ ¼ 2 ðxjþ1 xj1 Þðxjþ1 xj Þ ðxjþ1 xj Þðxj xj1 Þ @x x¼xj ð51Þ 2f ðxj1 Þ : þ ðxjþ1 xj1 Þðxj xj1 Þ Discretizing the diVerential operator @x2 in the matrices of Eq. (50) at x ¼ xj therefore results in three terms at either side of the equation which refer to the positions xj1, xj, and xjþ1: Mn;1 j uðxjþ1 ; znþ1 Þ þ Mn;2 j uðxj ; znþ1 Þ þ Mn;3 j uðxj1 ; znþ1 Þ ¼ Mn;4 j uðxjþ1 ; zn Þ þ Mn;5 j uðxj ; zn Þ þ Mn;6 j uðxj1 ; zn Þ with
uðxj ; zn Þ ¼
E F ðxj ; zn Þ : E B ðxj ; zn Þ
ð52Þ
ð53Þ
k can be easily derived from Eq. (50) by The complex 2 2 matrices Mn;j inserting the discretization rule Eq. (51). Hence, the following relation is obtained:
M2;n Uðznþ1 Þ ¼ M1;n Uðzn Þ
ð54Þ
where the unknown fields at the plane z ¼ zn are summarized in a vector Uðzn Þ ¼ ðE F ðx1 ; zn Þ; E B ðx1 ; zn Þ; . . . ; E F ðxNx ; zn Þ; E B ðxNx ; zn ÞÞT T ¼ ðuT ðx1 ; zn Þ; . . . ; uT ðxNx ; zn ÞÞ :
ð55Þ
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M2,n and M1,n represent band structured matrices with complex 2 2 matrices as their coeYcients and only one occupied superdiagonal and subdiagonal line, for example: 1 0 .. .. .. . . . C B C B Mn;3 j1 Mn;2 j1 Mn;1 j1 C B ð56Þ M2;n ¼ B C: C B Mn;3 j Mn;2 j Mn;1 j A @ .. .. .. . . . 1 M Hence, a matrix Pn :¼ M2;n 1;n can be defined that relates the field distributions on two adjacent planes z ¼ zn and z ¼ znþ1. Multiplying all these matrices finally yields a propagator matrix P, expressing the field distribution at the rear surface z ¼ zNz in terms of the field distribution at the front surface z ¼ z1: NY NY z 1 z 1 1 UðzNz Þ ¼ PUðz1 Þ with P :¼ Pn ¼ M2;n M1;n : ð57Þ n¼1
n¼1
It should be mentioned that P accounts for wave propagation in both directions in a physically rigorous manner since it represents the discretized Maxwell Eqs. (44) and (45). The derivation merely assumes that the fields EF,B and the dielectric constant are accurately represented by a linear interpolation in every interval [zn, znþ1]. 3. Boundary Conditions in Propagation Direction As the field distributions of the incident waves are actually known, the initial value problem of the Eqs. (48) and (49) is transformed to a boundary value problem with one half of the boundary conditions defined at the surfaces z ¼ z1 and z ¼ zNz , respectively. Deriving these so-called radiation boundary conditions (Moore et al., 1988) requires the computational variables to be expressed in terms of forward and backward propagating waves, which are related to the incident waves by the refraction law at the interfaces. a. Boundary Condition in the kx–z-space. Throughout this section, the representation of any quantity in kx–z-space will be denoted by a tilde () above the corresponding symbol. Figure 11 defines of impingh the notation i ~ l;r with wave vector kx ; kl;r ing and emerging waves E at an interface z between two regions of dielectric constants l for z < 0 and r for z > 0. Their amplitudes are related by the refraction law (Hecht and Zajac, 1974). The simplest representation is gained in kx–z-space: l r l r ~ r ðkx ; z ¼ 0Þ þ kz kz E ~ r ðkx ; z ¼ 0Þ; ~ l ðkx ; z ¼ 0Þ ¼ kz kz E E ð58Þ þ l l 2kz 2kz
VIRTUAL OPTICAL EXPERIMENTS
267
FIGURE 11. Refraction of waves at an interface between two regions of dielectric constants l and r. Adapted from Thalhammer and Wachutka (2003a). # 2003 OSA.
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where kl;r l;r k20 k2x : The traveling waves are expressed in terms of the z ¼ electric and magnetic fields and finally in terms of the computation variables ~ F and E ~ B: E ~ F ðkx ; zÞ þ 1 1 k eikz E ~ B ðkx ; zÞ: ð59Þ ~ l;r ðkx ; zÞ ¼ 1 1 k eikz E E l;r 2 2 kl;r k z z F ;B ~ Note that this equation also identifies E as the envelopes of the forward and backward propagating waves if the parameter k matches the propagation constant klz or krz , respectively. ~ lþ is equal to the incident laser field At the entrance plane z ¼ z1, the field E l ~ ~ E i , while E represents the unknown reflected wave. Inserting Eq. (59) into Eq. (58) reveals the boundary condition at the left-hand boundary of the simulation domain 1 k ikz1 ~ F 1 k ikz1 ~ B ~ 1 þ a e E ðkx ; z1 Þ þ 1 a e E ðkx ; z1 Þ E i ðkx Þ ¼ ð60Þ 2 kz 2 kz and the conditional equation of the reflected wave ~ r ðkx Þ ¼ 1 1 k eikz1 E ~ F ðkx ; z1 Þ þ 1 1 þ k eikz1 E ~ B ðkx ; z1 Þ: E 2 kaz 2 kaz
ð61Þ
In these relations, a denotes the dielectric constant in front of the entrance qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi plane (i.e., for z < z1) and kaz is given by kaz ¼ a k20 k2x . Similar conditions apply to the righthand boundary of the simulation ~ r ¼ 0Þ: domain at z ¼ zNz , where no wave impinges from the back ðE
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THALHAMMER
1 k ikzNz ~ F 1 k kzNz ~ B 1 b e 1þ b e E ðkx ; zNz Þ þ E ðkx ; zNz Þ: 0¼ ð62Þ 2 kz 2 kz qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Here, kbz ¼ b k20 k2x and b is the dielectric constant behind the rear surface (i.e., for z > zNz). ~ i ðkx Þ; E ~ r ðkx Þ, and E ~ t ðkx Þ To obtain a compact notation, the components E ~ ~ ~ are summarized in the vectors E i ; E r , and E t , respectively. Nx 2Nx matrices ~ a;b are defined by B h
i 1 k a;b ~ B :¼ 1 a;b eikza;b ; 2 kz h i;2i1 i ~ a;b B :¼ 0 otherwise
h
~ a;b B
i i;2i
1 k :¼ 1 a;b eikza;b for i ¼ 1; . . . ; Nx 2 kz
i; j
ð63Þ where the superscripts a and b refer to the front (z ¼ z1 ¼: za) and the rear (z ¼ zNz ¼: zb) surface of the sample, and the þ or subscripts indicate the propagation along the positive or negative z-axis, respectively. Then, the boundary conditions read ~ 1 Þ and ~ a Uðz E~ i ¼ B þ
b
~ N Þ: ~ Uðz 0¼B z
ð64Þ
Each of these matrix Eq. (64) comprises Nx scalar equations, thus representing the required 2Nx boundary conditions for the 2Nx unknown field com~ [See Eq. (55)]. Similarly, the conditional equations of the ponents in U emerging waves become ~ 1Þ ~ a Uðz E~ r ¼ B
and
~ N Þ: ~ b Uðz E~ t ¼ B z þ
ð65Þ
b. Matrix Representation of the Boundary Conditions in Real Space. To derive the boundary conditions in real space, the refraction law and the relations in the previous section are regarded as operator equations. Since kx qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi @ @2 2 writes as i @x in real space, the operator ka;b is represented by k z a;b 0 þ @x2 . A feasible way to evaluate the square root is the Taylor expansion of 1/kz in terms of kx (Moore et al., 1988; Mur, 1981):
4 1 1 1 @2 @ ¼ 1 þ O : ð66Þ pffiffiffiffiffiffiffi 2 @x2 a;b 4 k @x k 2 a;b 0 kz a;b 0 Inserting this approximation transforms Eqs. (60) to (62) to relations that can be easily discretized in real space. The resulting matrices Ba;b are diagonal or band structured if the series is expanded up to the constant or the quadratic terms, respectively. A second strategy for gaining the matrix representation Ba;b exploits the linearity of the Fourier transformation. Multiplying Eq. (64) by the
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VIRTUAL OPTICAL EXPERIMENTS
transformation matrix Tnm :¼ eikx;m xn results in the boundary conditions E i ¼ Baþ Uðz1 Þ
and
where the desired matrix representation
0 ¼ Bb UðzNz Þ; Ba;b
ð67Þ
in real space is calculated from
~ a;b 1 Ba;b ¼ T B T :
ð68Þ
In order to compare the accuracy of these approaches, the refraction at a silicon–air interface is considered (i.e., the situation illustrated in Figure 11 ~ r ¼ 0). The reflected and transmitted waves are with l ¼ 1, r ¼ 11.7, and E calculated from diVerent formulations of the radiation boundary conditions. As be seen in Figure 12, the results obtained by using the Fourier transformations [see Eq. (68)] excellently agree with the analytical Fresnel formula (Hecht and Zajac, 1974). On the other hand, merely taking into account the constant term in the Taylor expansion [Eq. (66)] introduces a significant error for angles of incidence beyond 5 degrees. Additionally including the quadratic term of the expansion is suYciently accurate for angles of incidence up to about 15 degrees. This approach is preferable against the Fourier transformation method since it results in sparse matrices.
FIGURE 12. Reflection and transmission coeYcient at a silicon–air interface. Simulation results obtained by different implementations of the boundary conditions are compared with Fresnel’s formula.
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THALHAMMER
4. Summary of the Algorithm In summary, beam propagation through the device under test is simulated by performing the following steps. First, the propagator matrix P, which relates the unknowns U ðzNz Þ at the rear surface to the unknowns UðzN1 Þ at the entrance plane, is calculated from Eq. (57). Second, the boundary conditions Eq. (67) are expressed as Baþ Ei ð69Þ Uðz1 Þ ¼ Bb P 0 and solved for the unknown fields at the entrance plane. Finally, the reflected and transmitted waves are extracted: a B Er ð70Þ ¼ Uðz1 Þ: Bbþ P Et In order to obtain the field distribution in the interior of the sample, the field distributions Uðzn Þð1 < n < Nz Þ have to be calculated from Uðz1 Þ by iteratively using Eq. (54). Note that only the right-hand side of Eq. (69) depends on the incident beam. Therefore, the field distribution for several diVerent incident waves can be calculated simultaneously with almost no additional eVort: Baþ E i;1 E i;2 E i;3 ð ðz Þ; U ðz Þ; U ðz Þ; ::: Þ ¼ ::: : ð71Þ U 1 1 2 1 3 1 Bb P 0 0 0 E. Fourier Optics The reflected or transmitted waves are projected onto the detector by focusing lenses and aperture holes. An appropriate model is gained from Fourier optics (Goodman, 1968). Regarding a sequence of planes perpendicular to the optical axis (see Figure 13), the field distribution E(x, y, znþ1) on each plane z ¼ znþ1 is calculated from the field distribution E(x, y, zn) on the respective preceding plane z ¼ zn by applying one of the modulations below. Contrary to the algorithm in the previous section, this model only allows for waves propagating in the positive z direction. 1. Image Formation by Aperture Holes In the limit znþ1 ¼ zn, the image formation by an aperture hole of radius R with the center located at (xc, yc) is described by an amplitude modulation in real space:
VIRTUAL OPTICAL EXPERIMENTS
271
FIGURE 13. For modeling the image formation by optical elements, a sequence of parallel planes z ¼ zn, n ¼ 1, 2, . . . is introduced. The field distribution on plane z ¼ zn þ 1 is calculated from the field distribution on plane z ¼ zn. Adapted from Thalhammer and Wachutka (2003a). # 2003 OSA.
h i Eðx; y; znþ1 Þ ¼ F R2 ðx xc Þ2 ðy yc Þ2 Eðx; y; zn Þ
ð72Þ
where F(x) is the unit step function, which is equal to 1 for x > 0 and equal to 0 for x < 0. 2. Image Formation by Thin Lenses For modeling thick lenses in the optical setup, they can be replaced by an appropriate arrangement on thin lenses. Hence, we can restrict ourselves to considering thin lenses within which the translation of the light ray can be neglected. Hence, only the phase shift ’(x, y) of the wavefront needs to be taken into account. It is proportional to the optical thickness at the position (x, y): ’ðx; yÞ ¼ k0 nL Dðx; yÞ þ k0 ½Dðxc ; yc Þ Dðx; yÞ:
ð73Þ
In this relation, nL and D(x, y) represent the refractive index of the lens and the geometrical thickness at the position (x, y), respectively. Deriving D (x, y) by geometrical considerations, the phase delay on the paraxial approximation can be finally expressed as ’ðx; yÞ ¼ k0 nL D0 þ
k0 2 ðx þ y2 Þ 2f
ð74Þ
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h
i1 (Goodman, 1968), where f ¼ ðnL 1Þ R11 R12 denotes the focal length of the lens. Consequently, the image formation by a thin lens introduces the following phase modulation in real space:
k0 Eðx; y; znþ1 Þ ¼ expðik0 nL D0 Þexp i ðx2 þ y2 Þ Eðx; y; zn Þ: ð75Þ 2f 3. Propagation in Free Space
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi As only waves with a positive propagation constant kz ¼ þ k20 k2x k2y along the optical axis need to be taken into account, the field distribution between the planes z ¼ zn and z ¼ znþ1 can be expressed in terms of the Fourier integral R ~ x ; ky Þ dkx dky : Eðx; y; zÞ ¼ eikx xþiky y eikz ðkx ;ky Þz Eðk ð76Þ At the plane z ¼ zn, this relation constitutes the 2D Fourier expansion of the field distribution E(x, y, zn) in terms of its Fourier transform ~ x ; ky ; zn Þ ¼ eikz ðkx ;ky Þzn Eðk ~ x ; ky Þ. Therefore, propagation from z ¼ zn to Eðk z ¼ znþ1 is modeled by a phase change in kx– ky space: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ~ x ; ky ; zn Þ: ~ x ; ky ; znþ1 Þ ¼ ei k20 k2x k2y ðznþ1 zn Þ Eðk ð77Þ Eðk F. Detector Response The illumination of the detector generates photo currents that are electronically amplified and converted. The switching times of the device under test are assumed to be large enough so that the photo diode capacitance and any delay in the signal processing can be neglected. Since except for the laser deflection technique the detector comprises a simple pin diode, the measurement signal is proportional to the integrated intensity distribution, with the integration boundaries matching the active area of the detector. During laser deflection experiments, the four-quadrant pin diode detector is dynamically translated to keep the DC component of the deflection signal vanishing (see Section III.B.2). Since the duty cycle of the periodic current pulses is very small, this DC component is equal to the deflection signal during the oV-state of the device under test. The intensity distribution along a cutline across the detector is shown in Figure 14, where two opposite segments are represented by the intervals [x1, x2] and [x3, x4]. The associated photo currents I1 and I2 are proportional to the integrated intensities
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FIGURE 14. Intensity distribution on the detector.
Zx2 I1 ðtÞ ¼ const jjE D ðx; tÞjj2 dx
ð78Þ
x1
Zx4 I2 ðtÞ ¼ const
jjE D ðx; tÞjj2 dx:
ð79Þ
x3
While the widths x2 x1 and x4 x3 and the gap x3 x2 between the segments are fixed by the detector geometry, the absolute position x1 is calculated to satisfy the condition I1off ¼ I2off during the oV-state. Then, the deflection signal M is obtained by MðtÞ ¼
I1 ðtÞ I2 ðtÞ : I1 ðtÞ þ I2 ðtÞ
ð80Þ
The simultaneously detected absorption signal, which is actually defined as the AC component of the detected time-dependent intensity distribution scaled to its DC component (see Section III.B.1), is calculated from AðtÞ ¼
I1 ðtÞ þ I2 ðtÞ I1off þ I2off
1:
ð81Þ
Note that these ratios merely depend on the detector geometry, whereas they are not aVected by its (probably unknown) sensitivity and the amplification factors.
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V. VIRTUAL EXPERIMENTS
AND THE
OPTIMIZATION STRATEGY
The previous section presented a physically rigorous model for simulating optical probing techniques. This chapter outlines the basic ideas of ‘‘virtual experiments’’ and defines a systematic strategy of how to investigate the experiment, particularly how to discriminate the electrothermal eVects of interest against the parasitic influences of the sample preparation and the measurement process, and how to optimize the experiments with respect to a desired purpose—for example, a high sensitivity or a large measurement range. A. Fundamental Principle Researchers and device engineers would like to know the carrier and temperature distributions that are present in the interior of unprepared samples or product devices. However, internal laser probing can only be performed on specific samples that are prepared for the measurements (see Section III.B and Table 3). They diVer, for example, with respect to their size or a window in the metallization layer, which is required for backside laser probing. The measurement results therefore suVer from parasitic eVects that are inevitably introduced as the sample is prepared (e.g., an enhanced surface recombination rate caused by sawing and polishing the surfaces). Besides, the actual measurement process introduces some additional errors that originate, for example, from a limited spatial resolution due to the lateral spreading of the beam profile or from an oblique transition of the probing beam if the sample is misaligned. Since the distributions of physical quantities in any structure are accessible by simulation, the concept of ‘‘virtual experiments’’ (Thalhammer and Wachutka, 2003b) has been proposed to tackle the abovementioned problems. The key idea consists of comparing the results of two simulation runs (see Figure 15). On the one hand, an electrothermal device simulation of an unprepared sample yields the carrier concentration nref and temperature distribution Tref that are actually wanted to be known. These data are kept as a fixed reference during the optimization study. On the other hand, virtual experiments are performed; that is, the model outlined in the previous section is used to simulate the measurement process, taking into account the specific properties of the prepared samples. From the calculated detector signals, a carrier concentration nextr and a temperature Textr are extracted according to the evaluation rules that would also be applied for the real measurements [for example, the absorption law Eq. (16) or Eq. (20)]. The diVerence of these profiles allows assessment of the impact of the
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FIGURE 15. Basic strategy for optimizing optical experiments. Adapted from Thalhammer and Wachutka, 2003b. # 2003 OSA.
experiment. Performing these virtual experiments with diVerent setups and sample geometries, the goal of an optimized experiment is achieved if the extracted profiles nextr and Textr match the reference quantities nref and Tref as closely as possible. In addition, the remaining diVerence is a measure of the minimum experimental error. B. General Assumptions The optimization strategy presented above is based on some general issues that are briefly summarized as follows:
Preparing the samples and evaluating the detector signals is considered part of the experiment. That means, purely optimizing the actual measurement process is not desirable. It is rather favored to minimize the overall error introduced by preparing the sample (thus changing its original behavior), performing the actual measurement that suVers from inevitable physical constraints, and evaluating the detector signals that is usually based on some simplifying assumptions.
The above-described concept comprehensively relies on simulations that therefore need to be properly calibrated. Although a possible inaccuracy of the electrothermal simulation will aVect both the calculation of the reference profiles and the first step of the virtual experiment (see Section IV.B), the physical models must accurately reflect the diVerence of the unprepared and the prepared samples’ behavior. The same is true for calculating the refractive index modulations. The associated coeYcients (see Section II) are used for the virtual experiment (see Section IV.C), as well as for evaluating the detector signal (see Eqs. (16), (20), (21), and (23)), but on diVerent spatial scales.
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However, as optical probing techniques provide additional experimental data, they also help to improve the validation and calibration of theoretical models. Thus, more accurate simulations can be performed which, in turn, facilitate the development of more advanced experiments.
Whenever possible, it is preferable that the desired measurands be accurately extracted from the measurement signals via an analytical formula (e.g., Eq. (20)). In case the dependence of the detector signals cannot be solved for the internal physical state variables (e.g., because of a complex nonlocal dependence), inverse modeling may be used. The internal carrier and temperature distribution is represented by a suYciently large number of unknowns, whose initial values are determined by an electrothermal device simulation. Next, the optical module is used for calculating the detector signal. Comparing it with the real measurement signal yields the necessary update of the carrier concentration and temperature distribution. This procedure is iterated until the extracted updates reproduce the current values. In case the measurands are extracted this way, the virtual experiments and the optimization strategy as previously sketched can be applied as well by including the numerical evaluation as an additional part of the experiment.
VI. FREE CARRIER ABSORPTION MEASUREMENTS In previous sections, the concept of virtual experiments for a physically rigorous theoretical study of internal optical probing techniques has been illustrated. Together with analytical models, this concept is now used to perform a comprehensive analysis of the free carrier absorption method. The major goals of these investigations are to enhance the understanding of the actual measurement process; to support the design and optimization of the experiments with respect to a minimum experimental error, a large measurement range, and a minimum sensitivity to parasitic eVects; and to provide quantitative results for the optimum setup and the optimum sample geometry. A. Optical Field Distribution of the Probing Beam The field distribution of a laser beam probing the internal carrier concentration profile in the forward conducting state of a pin power diode is shown in Figure 16. For the sake of clarity, the number of pixels in the figure is reduced. Standing waves caused by multiple reflections at the surfaces are therefore not visible.
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FIGURE 16. Optical field distribution of a Gaussian beam (A ¼ 0.1) incident at a depth x ¼ 60 mm for probing the carrier distribution in a forward-biased pin power diode. Adapted from Thalhammer and Wachutka, 2003b. # 2003 OSA.
For an incident Gaussian wave with an angular aperture of A ¼ 0.1, the beam profile in the interior of the sample is also nearly Gaussian. Locating the focal plane in the center of the sample, the minimum spot diameter is about 12 mm. It increases to about 25 mm at the surfaces of a sample that is 2.5 mm in length. If the probing beam impinges close to the top or bottom surfaces of the device under test, the beam profile is distorted by reflections at the metallization layer, resulting in an undesirable interference pattern with several additional intensity peaks (Figure 17). The simulated field distributions agree excellently with the infrared images taken by the camera included in the optical setup (Figure 18). While the measured profile of a beam penetrating the bulk region is approximately Gaussian (righthand image of Figure 18), a diVraction and interference pattern is observed in case the probing beam propagates close to the metallizations layers (left-hand image of Figure 18). B. Longitudinal Averaging/Samples with Cell Structure Absorption measurements have become a suitable technique for probing the carrier distribution in power semiconductor devices. Most of these samples, however, are built as cell structures with several (often hundreds of )
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FIGURE 17. Optical field distribution of a Gaussian beam (A ¼ 0.1) incident at a depth x ¼ 10 mm for probing the carrier distribution in a forward-biased pin power diode. Adapted from Thalhammer and Wachutka, 2003b. # 2003 OSA.
FIGURE 18. Experimental field distribution at the rear surface for a beam position near the anode boundary (left) and in the center (right) of the sample. The images are taken by an infrared camera during absorption measurements on a pin power diode. Adapted from Thalhammer and Wachutka, 2003b. # 2003 OSA.
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identical devices along the beam path (see Figure 20). In case the carrier distribution is 2D or 3D, as it is in a power transistor or IGBT, for example, the carrier concentration along the beam path is no longer constant as it has been assumed in Eq. (16), but periodic in terms of the cell width D. Expanding the carrier distribution into a Fourier series nðx; y; zÞ ¼ n0 ðx; yÞ þ
1 X
Dnk ðx; yÞ cosðk 2pz=DÞ
ð82Þ
k¼1
transforms the integrated carrier density, which is the decisive quantity aVecting the detector signal, to 1 L
ZL nðx; y; zÞdz ¼ n0 ðx; yÞ þ 0
1 X D sinðk 2pL=DÞ Dnk ðx; yÞ 2pkL k¼1
ð83Þ
¼ n0 ðx; yÞ:
Since the sample length is at least approximately an integer multiple of the cell width, the absorption signal thus reflects the average concentration n0(x, y) at the beam position (x, y). However, this simple consideration neglects the deflection of the probing beam due to refractive index gradients and possible reflections at the metallization layers. The carrier distribution in the top region of an IGBT sample is shown in Figure 20. The space charge regions around the reversed biased p-wells form areas with a higher refractive index, thus acting as focusing lenses. The probing beam is therefore deflected toward the upper boundary of the device, where it traverses regions of lower carrier concentration. As a consequence, the transmitted light intensity may even exceed the transmission during the oV-state. The resulting increase of the absorption signal (Figure 19, depth 10 mm) would be misinterpreted in terms of a negative injected carrier concentration. To obtain reliable results, a minimum distance to the top boundary must be kept, which can be shown to be 20 mm for a sample length of 1 mm. C. The Fabry–Perot EVect Multiple reflections at the polished surfaces of the sample give rise to Fabry– Perot modulations of the transmitted intensity as the optical thickness no L of the sample changes (Hecht and Zajac, 1974). The latter is increased by one wavelength, for example, if the temperature in a device of 2.4 mm in size rises by 1.7 K, which can easily be caused by self-heating during the typical operating conditions of power devices. For instance, a pin diode subjected to
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FIGURE 19. Calculated absorption signal for different beam positions in a sample comprising 25 IGBT cells (pulse duration 70 ms).
FIGURE 20. Carrier distribution around the top p-well of a forward-biased IGBT sample.
a current pulse of 150 A /cm2 dissipates an average power of 110 W/cm3, which increases the temperature by approximately 0.01 K /ms. Therefore, oscillations with a period length of 160 ms are observed on the measurement signal (Figure 21, left). However, the temperature rise for a low-power dissipation may increase the optical sample thickness only by a small fraction of a wavelength. In this case, a linear drift of the signal is observed (Figure 21, right) whose slope is governed by the linear Taylor expansion of the Fabry–Perot transmitivity at
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FIGURE 21. Absorption signal measured on a pin power diode sample. Dissipating high power (current density 150 A/cm2, pulse duration 500 ms) originates Fabry–Perot oscillations of the absorption signal (left). A linear drift is observed in case of a low-power dissipation (right; current density 15 A/cm2, pulse duration 100 ms). Adapted from Thalhammer, 2005. # 2005 AIP.
the initial optical thickness. Hence, the drift may be either positive or negative in sign or may even vanish, if the initial optical thickness has been a half-integer multiple of the wavelength. As typical sample sizes are some thousands of wavelengths and the actual die size is usually not known up to 0.1%, the diVerence of the measured and the simulated signals is not very astonishing. These undesired eVects can be suppressed, for example, by depositing an antireflective coating, polishing the surfaces at a slightly tilted angle of 1 degree to prevent interference of the multiply reflected rays, using an incoherent laser source (Hille, 2001) or aligning the sample appropriately to exploit the vanishing reflection coeYcient at the Brewster angle (Go¨ rtz, 1984). D. Spatial Resolution To estimate how the lateral extension of the probing beam limits the spatial resolution, the carrier concentration is expanded into a Taylor series at the beam position x0 (Hille and Thalhammer, 1997; Thalhammer, 2005) 1 nðxÞ ¼ nðx0 þ DxÞ ¼ nðx0 Þ þ n0 ðx0 ÞDx þ n00 ðx0 ÞDx2 þ ::: 2
ð84Þ
Then, the convolution with a Gaussian beam profile I(x, y, z) of radius w(z) in case of high-injection conditions becomes
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THALHAMMER
w2 ðzÞ 00 w2 ðzÞ n ðx0 Þ þ ::: ¼ nðx0 Þ 1 þ nðxÞIðx; y; zÞ dx dy ¼ nðx0 Þ þ þ ::: 8 8L2D ð85Þ
The integral of this convolution along the beam path governs the totally absorbed power of a Gaussian beam with the z-dependent spot radius sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 zk0 2A 2 wðzÞ ¼ 1þ : ð86Þ k0 A 2no Thus, the carrier concentration extracted from the absorption signal becomes 1 0 L=2
Z L2 2A 1 1 1 C B : nextr ðx0 Þ ¼ nðx0 Þ@1 þ 2 w2 ðzÞ dzA ¼ nðx0 Þ 1 þ 2 2 2 þ 8LD L 2LD k0 A 96L2D n2o L=2
ð87Þ Although this consideration assumes a Gaussian profile and neglects a possible deflection of the probing beam, it reveals some important results. First, the carrier density extracted from the absorption signal is larger than the real concentration. Second, the relative error amounts to 1=2L2D k20 2A þ L2 2A =96L2D n2o , which is only a few percent in the case of typical sample lengths L and diVusion lengths LD. Third, if the ambipolar diVusion length LD is constant in the volume fficovered by the probing beam, pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi the ‘‘apparent’’ diVusion length nextr =n00extr , which is extracted from a measured carrier concentration profile nextr, matches the ‘‘real’’ diVusion ffi pffiffiffiffiffiffiffiffiffi length LD ¼ n=n00 . For a quantitative analysis, the spatial resolution x is defined by the condition that the measured concentration nextr(x0) is equal to the carrier density n(x0 þ x) at the position x0 þ x. Using Eq. (87) and (84), the spatial resolution at the minimum of the carrier distribution is shown to be sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi L2 2A 1 : ð88Þ þ x¼ 2 2 k0 A 48n2o For a typical sample length of 2 mm and an angular aperture of 0.04, the spatial resolution becomes x ¼ 7 mm, which is comparable to that of the recombination radiation measurements (see Section III.A). For further illustration, Figure 22 shows the reference carrier distribution gained by a device simulation, the convolution of the beam profile and this reference carrier distribution, as well as the carrier concentrations extracted
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FIGURE 22. Vertical carrier distribution in a pin diode (sample length 2 mm). The averaged carrier density is calculated by the convolution of the beam profile and the concentration profile. Note that the carrier concentration is plotted on a linear scale. Adapted from Thalhammer, 2005. # 2005 AIP.
from virtual experiments with diVerent angular apertures. To make possible diVerences clearly visible, we follow the common practice for analyzing power devices and plot the carrier densities on a linear axis. Even on this scale, all profiles are in excellent agreement. We can therefore conclude that the lateral extension of the probing beam does not introduce a significant error. E. Optimizing the Optical Setup To analyze the accuracy of an experiment quantitatively, the following quantities are regarded: first, the relative diVerence Erel of the carrier concentration nextr(x) extracted from the detector signal of a virtual experiment and the reference concentration nref (x) obtained by an electrothermal device simulation vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u N 2 u 1 X nextr ðxi Þ 1 : Erel :¼ t ð89Þ N i¼1 nref ðxi Þ
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THALHAMMER
where N denotes the number of sampling points xi. The second optimization criterion arises as a minimum distance to the sample boundaries is required to prevent a corruption of the detector signal caused by diVraction phenomena (see Section VI.A). Quantitatively, the necessary minimum disðxi Þ tance is defined by the position xi, where the relative deviation j nnextr 1j ref ðxi Þ exceeds 10%. 1. Angular Beam Aperture For small angular apertures, the most significant error originates in the large spot diameter. Increasing the angular aperture decreases the minimum spot size but increases the spreading of the probing beam and, consequently, the spot diameter at the device surfaces. Hence, at a given sample length, there is an optimum angular aperture. It increases as a smaller sample size is chosen. The analytical calculation (see Eq. (87)) of the previous section predicts an L2 2 error of 2L2 1k2 2 þ 96L2 An2 , which attains its minimum D 0 A
D o
L Erel;opt ¼ pffiffiffi 4 3no k0 L2D
for an angular aperture of A;opt
sffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi 4 3no ¼ : k0 L
ð90Þ
ð91Þ
Inserting a sample length of L ¼ 800 mm and a diVusion length of 38 mm yields A,opt ¼ 0.085 and Erel,opt ¼ 0.6% at a wavelength of 1.55 mm. The results of virtual experiments are shown in Figure 23. The calculated optimum angular apertures and the relative errors agree quite well with the predictions of the analytical consideration. For example, a sample length of 800 mm requires a minimum distance to the device boundaries of 12 mm. This is a bit larger than the average spot radius Wopt ¼ 8.2 mm, as the probing beam is deflected and its tails are diVracted due to refractive index gradients. However, some important issues should be pointed out. First, this analysis assumes that the carrier distribution is constant along the beam path, in particular, that the surface recombination rate can be neglected. For that reason, the minimum error continuously decreases as the sample length decreases. Second, preparing and handling the sample is only manageable if its length is larger than 0.5 mm. Smaller interaction length can only be realized by using specific test structures (which will be discussed in Section VII.G.1). Third, an acceptable signal-to-noise ratio demands a minimum interaction length of about several 100 mm.
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FIGURE 23. Optimizing the angular aperture of the probing beam. Left, average relative error of the carrier concentration profile extracted from a virtual experiment. Right, minimum distance to the sample boundaries for which to local error is below 10%. Adapted from Thalhammer, 2005. # 2005 AIP.
2. Sample Alignment The previous discussion is based on the assumption that the sample surfaces are perpendicular to the optical axis. However, if the probing beam traverses the sample at an angle FSi, the carrier concentration extracted from the detector signal is governed by the integral 1 nextr ðx0 Þ ¼ L
L=2 Z
nðx0 þ z tanFSi Þ dz:
ð92Þ
L=2
Since FSi is related to the angle of incidence F by the refraction law, Eq. (92) enables an analytical estimation of the relative error, which is plotted as a dashed line in Figure 24. The solid line represents the result of virtual experiments, which also take into account the beam deflection in the interior of the sample. For that reason, the curve is not symmetric with respect to F ¼ 0. It is important to note that a possible misalignment of the device under test results in an error that may easily amount to several 10% and thus
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FIGURE 24. Oblique transition of the probing beam through a pin power diode (sample length 0.8 mm, A ¼ 0.08). Left, carrier concentrations extracted from virtual experiments with different angles of incidence. Right, Relative error as a function of the angle of incidence. The solid line is gained from virtual experiments, while the analytical consideration is based on Eq. (92). Adapted from Thalhammer, 2005. # 2005 AIP.
constitutes a significant source of error. Hence, a precise sample alignment is a crucial precondition of accurate measurement results. As another possible misalignment, the sample might be shifted along the optical axis. It is evident that locating the focal plane of the focusing lens in the center of the device under test is the optimum setup as it enables the smallest average spot diameter. It has been shown that for an interaction length of L ¼ 800 mm and an agular beam aperture of A ¼ 0.08 the theoretical error limit is 0.6% in case of a symmetric field distribution. However, if the focal plane is located at the front surface, the average error increases to 1.2% and the minimum required distance to the boundaries increases from 12 mm to 16 mm. Regarding the absolute values, however, it is obvious that slightly shifting the center of the sample out of the focal plane does not introduce a significant experimental error. F. EVects of Surface Recombination Dicing the samples may change the carrier distribution along the beam path since sawing and polishing the surfaces usually enhances the surface recombination rate. As sample preparation is considered to be part of the experiment (i.e., the decreased carrier concentration at the edges is regarded as a parasitic eVect), the reference carrier distribution is still calculated without surface recombination. In this case, it is no longer desirable to prepare samples with an interaction length as small as possible. We rather expect
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an optimum length that minimizes the overall error caused by both the carrier concentration decrease at the surfaces and the spreading of the probing beam in large devices. In this section, the carrier distribution in the prepared samples is calculated with assuming a surface recombination velocity of 105 cm/s which is suYcient to decrease the carrier densities at the surfaces to their equilibrium values. The following results may therefore be considered the ‘‘worst case,’’ whereas the discussion in the previous section reflects the ‘‘best case’’ of a vanishing surface recombination rate. 1. Devices with Large Carrier Lifetimes In case of large diVusion lengths, the curvature in the vertical direction is small. Therefore, the convolution of the carrier distribution with the lateral beam profile does not introduce a significant error, but the eVects of the surface recombination reach far into the sample. The absorption R L=2 signal is therefore sensitive to the averaged carrier density NðxÞ :¼ L1 L=2 nðx; zÞ dz, which significantly diVers from the reference concentration (Figure 25). Simulating the measurement process for diVerent geometries reveals the optimum arrangement which reproduces the reference profile as closely as possible (Figure 26). For that purpose, the sample has to be 2.5 mm in length, which is about 20 times the diVusion length. The corresponding optimum angular aperture is about 0.02. Thus, the theoretical error limit
FIGURE 25. Carrier distribution in a pin power diode (ambipolar carrier lifetime 8 ms, diffusion length 110 mm). The dots refer to the densities extracted from virtual experiments, including surface recombination effects for three different setups. Adapted from Thalhammer, 2005. # 2005 AIP.
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THALHAMMER
FIGURE 26. Optimizing the probing conditions for samples with a large ambipolar carrier life‐time. Left, average relative error of the carrier concentration profile obtained by virtual experiments. Right, required minimum distance to the boundary for which the local deviation is below 10%. The diVusion length is 110 mm.
for probing devices with large carrier lifetimes is 6% and a minimum distance of 16 mm to the boundaries is required. However, it should be mentioned that the diVusion length cannot be directly extracted from the measured carrier distribution because its curvature diVers from that of the reference profile. An analytical first order correction is gained by expanding the 2D carrier distribution in terms of cosine functions (see Dannh€ auser and Krause, 1973 [Appendix]) Z1 nðx; zÞ ¼ 0
z nðx; gÞ cos g dg L=2
ð93Þ
where each component n(x, g) must satisfy the ambipolar diVusion equation 2 @ 1 2 4 g 2 nðx; gÞ ¼ 2 nðx; gÞ ð94Þ L @x2 LD and the boundary conditions Damb
@nðx; zÞ L ¼ sr nðx; zÞ for z ¼ : @z 2
ð95Þ
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They transform to gj tanðgj Þ ¼
sr L 2Damb
ð96Þ
with an infinite set of discrete solutions gj. Thus, we obtain sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 X x x0 2z 1 1 4 nðx; zÞ ¼ Aj cosh :¼ þ g2j 2 where cos gj 2 L Lj L Lj LD j¼1 ð97Þ and the integrated carrier concentration finally becomes 1 X x x0 sinðgj Þ NðxÞ ¼ Aj cosh : Lj gj j¼1
ð98Þ
Assuming an infinite surface recombination velocity (thus, gj ¼ (2j 1)p/2) and approximating this expansion by the first moment, the desired firstorder correction for the unknown diVusion length LD is obtained as sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 1 p2 ; ð99Þ ¼ LD L21 L2 where L1 is extracted from the curvature of the measured distribution N(x). 2. Devices with Small Carrier Lifetimes In case of a small ambipolar carrier lifetime (i.e., the diVusion length LD is small compared to the die size), the carrier concentration profile steeply decreases at theRsample surfaces. As a consequence, the averaged concentraL=2 tion NðxÞ :¼ L1 L=2 nðx; zÞ dz, matches the carrier concentration without surface recombination. Hence, the most significant error in case of a large sample length is introduced by the limited resolution due to the lateral extension of the probing beam. On the other hand, the decrease of the carrier concentration at the surfaces is no longer negligible in small devices. The optimum sample length is 800 mm, which is again about 20 times the diVusion length. The corresponding optimal angular aperture is about 0.08 (Figure 27), which agrees well with the result of the analytical consideration [Eq. (91)] of the previous section. Also in case of a small diVusion length, the numerical study predicts a lower error limit of 6% and a minimum distance to the metallization layers of about 16 mm. As the averaged concentration N(x) matches the reference profile, the ambipolar diVusion length can be directly extracted from the measured carrier distribution. This results is also evident from Eq. (99), which simplifies to LD L1 for LD L.
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THALHAMMER
FIGURE 27. Carrier distribution in a pin power diode (ambipolar carrier lifetime 0.8 ms, diffusion length 34 mm). The dots mark the concentration values extracted from virtual experiments, including surface recombination eVects for two diVerent setups.
G. Summary The theoretical study of the free carrier absorption measurement technique leads to the following conclusions:
Reflection and diVraction at the top or bottom sample surfaces give rise to unpredictable distortions of the beam profile, thus corrupting the measurement signal. To avoid these eVects, a minimum distance from the device boundaries is required.
Self-heating can easily change the optical thickness of the sample by several wavelengths. To suppress the resulting Fabry–Perot oscillations of the measurement signal, special precautions must be taken (e.g., depositing an antireflective coating or using an incoherent light source).
The space charge regions around the reverse-biased p-wells in IGBT samples act as focusing lenses deflecting the probing beam. It can therefore traverse regions with much lower carrier concentration, resulting in an increase of the absorption signal during the duty cycle. For sample lengths below 1 mm, however, keeping a distance of approximately 20 mm to the top surface enables reliable measurements that are sensitive to the average carrier concentration in the cell structure along the beam path.
The strategy for optimizing the experiment strives for minimizing two criteria: first, the deviation of the carrier concentration extracted by a virtual experiment from the reference concentration obtained by an
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electrothermal simulation, and second, the minimum distance to the device boundaries, which is required to prevent a corruption of the measurement signal.
At any given sample length L, the optimum angular aperture of the probing beam can be calculated from the condition of minimizing the average beam cross-section. The resulting analytical formula reproduces the optimum, which was found numerically by means of virtual experiments with diVerent probing beams.
A surface recombination velocity of 105 cm/s decreases the carrier concentrations at the sample surfaces almost to their equilibrium values. Consequently, an optimum sample length is about 20 diVusion lengths. The corresponding optimum angular aperture can be approximately extracted from the condition of minimizing the average beam cross section. Thus, the measurements can be carried out at an average error of about 6%, if a minimum distance of 16 mm to the top and bottom device surfaces is kept.
The exact position of the focal plane of the probing beam is not critical. Even if it is located at the sample surfaces, instead of in the center, the average error merely increases by a factor of 2.
A possible misalignment of the sample constitutes the most significant source of error. Tilting the sample at an angle of a few degrees results in an error of several 10%. In addition, the necessary minimum distance to the boundaries increases rapidly.
The accuracy of the real measurement results sensitively depends on the precise knowledge of the carrier concentration dependence of the absorption coeYcient. This source of error is not apparent in the numerical studies as they rely on the same value to calculate the refractive index modulations (see Section IV.C) and to evaluate the detector signal. However, since this coeYcient enters as a factor in the evaluation rule, the relative profiles are accurate in any case.
VII. INTERNAL LASER DEFLECTION MEASUREMENTS As another application example of virtual experiments, the laser deflection technique (see Section III.B.2) is discussed in this section. Many of the results for free carrier absorption measurements are expected to apply for deflection measurements as well, for example, the longitudinal averaging in samples comprising a periodic cell structure (Section VI.B) and the lateral convolution of the beam profile and the refractive index distribution (Section VI.D). It is also evident that a crucial source of error originates from a possible misalignment of the sample.
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Internal laser deflection measurements are sensitive to the gradient of the refractive index and are therefore suitable for probing carrier concentration gradients and temperature gradients. Unless the laser beam propagates close to the heat sources, both contributions can usually be separated owing to the widely diVering electric and thermal time constants. Alternatively, the carrier contribution is calculated from the absorption measurements and subtracted from the deflection signal. Laser deflection measurements have been shown to constitute a very sensitive technique for probing small temperature gradients in samples of some millimeters in length (Deboy, 1996). However, some applications (e.g., the characterization of power devices operating under short-circuit conditions) rather require a large measurement range, which is limited by the saturation of the detector response function. The following discussion therefore also pays specific attention to designing the experiments for detecting large temperature gradients. A. The Measurement Signal Extracting the measurands from the detector signal requires a physically consistent equation that relates the deflection signal to the refractive index gradients. An analytical expression also shows which parameters of the experiment aVect the detector signal and therefore may be adjusted to enhance the sensitivity, the measurement range, or any desired property of the optical setup. A derivation strategy that is also valid in case of large temperature gradients is proposed by Thalhammer (2005). The basic idea is to regard the probing beam as a Gaussian beam whose optical axis is deflected in the interior of the sample according to the principles of geometrical optics. The internal deflection is represented by two parameters, namely D~ x and a2, since the field distribution at the rear surface can be considered the intensity distribution of a non-deflected Gaussian beam, which emerges from the focal plane of the focusing lens at an angle a2 and a shift D~ x (Figure 28). Finally, Fourier optics is used to model the projection of this Gaussian beam onto the detector plane. 1. Internal Beam Deflection According to Fermat’s principles, the 2D internal beam path x(z) minimizes the Hamilton integral (Marcuse, 1982): L=2 Z
S¼ L=2
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi no ðx; zÞ 1 þ x0 2 dz
with x0 ¼
dx : dz
ð100Þ
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FIGURE 28. Internal deflection and projection of the probing beam onto the detector plane. Adapted from Thalhammer (2005). # 2005 AIP.
If the refractive index gradient is constant in the volume covered by the probing beam, a complete analytical solution can be derived (Thalhammer, 2005). In case of a vanishing angle of incidence, we obtain z L=2 x x0 ¼ l cosh l ð101Þ l where the definition l :¼
no ðx0 Þ dno =dx
ð102Þ
has been introduced. Thus, the total beam shift Dx0 and the angular deflection a2 become 4 4 L2 L dno L2 L ð103Þ þO þO Dx0 ¼ l ½coshðL=lÞ 1 ¼ ¼ l l 2l dx 2no ðx0 Þ 3 3 dx L L dno L L j þO tanða2 Þ ¼ ¼ sinhðL=lÞ ¼ þ O ¼ : dz z¼L=2 l l l dx no ðx0 Þ ð104Þ
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The equivalent parameters D~ x and at are 6 L L L L L4 L l sinh ¼ 3þlO D~ x ¼ Dx0 tana2 ¼ l cosh 2 l 2 l l 24l ð105Þ sinðat Þ ¼ no sinða2 Þ ¼ no tanh
3 L dno L þO : ¼L l l dx
ð106Þ
2. Image Formation by a Thin Lens To model the projection of the probing onto the detector plane, the image formation by a converging lens of focal length f is considered (see Figure 13; n ¼ 0, d1 :¼ z1 z0, d2 :¼ z3 z2). Using the relations in Eqs. (77) and (75), as well as the necessary Fourier transforms and inverse transforms, the field distribution E0(x, y) on the original plane 0 can be shown to be related to its image E3(x, y) on the plane 3 by the expression (Thalhammer, 2005) E 3 ðxÞ ¼ const e
ik0 ðd1 þd2 Þ
Z1
k0 Vt 2 k0 x exp i exp i ðx þ Vt xÞ2 E 0 ðxÞdx 2f a 1
ð107Þ where 1 Vt :¼ d2 =f 1
2d1 d2 f 1 1 1 and a :¼ ¼ d2 f f d1 d2 2½d1 f ð1 þ Vt Þ ¼ 2½d1 Vt d2 :
ð108Þ
The x-axis is scaled by Vt, which therefore represents the inverse of the transverse magnification. A physical interpretation of the parameter a is gained if the lens is considered to form an image of plane 3. Since that is located at f (1 þ Vt), the parameter a allows for the focusing conditions and is twice the distance between plane 0 and the image of plane 3. 3. Projection by the Imaging Lens Neglecting the curvature of the phase front, the field distribution of a rotated Gaussian beam at the focal plane of the focusing lens (Figure 28) can be written as " # ðx D~ xÞ2 E 0 ðxÞ ¼ const exp 2 ð109Þ exp½ik0 sinðat Þ x: w0 =cos2 ðat Þ
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Equation (107) with d1 ¼ F þ L/2no shows that the resulting image on plane 3 is also a Gaussian profile exp[(x Dx)2/w2]. Its spot radius w and the shift Dx are given by a D~ x aL dno L dno 3 ð110Þ Dx ¼ þO sinðat Þ ¼ 2Vt Vt 2Vt dx no dx w0 w¼ Vt cosðat Þ
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi " # a2 cos4 ðat Þ A a L2 dno 2 L dno 4 1 þO : 1þ 2Vt no dx 2 dx k20 w40 ð111Þ
where we have reused the definitions Vt ¼ f/(d2 f ) and a ¼ 2ðL=2no þ F Vt d2 Þ (see Eq. (108)). In general, the total beam displacement is aVected by the internal shift D~ x and the angular deflection at, but two interesting special cases can be adjusted. First, if the imaging lens projects the focal plane in the sample (plane 0) exactly onto the detector (a ¼ 0), the setup is only sensitive to the internal shift D~ x. This condition, however, is diYcult to realize since a is in the order of some mm and a sin(at) has to be much smaller than D~ x, which is only some microns. Second, if plane 3 is located on the focal plane of the imaging lens (d2 ¼ f ), we obtain Vt ! 1, a ! 1, a/2Vt ! f and the setup is only sensitive to the angular deflection. 4. Detector Signal In case of small deflections, the detector signal is proportional to the beam displacement Dx. However, it has been demonstrated (Hille and Thalhammer, 1997) that the slope Q of the response function depends on the spot size w of the probing beam on the detector. Consequently, any modification of the optical setup, in particular choosing a diVerent beam aperture or shifting the imaging lens, demands a recalibration of the response function. For that purpose, the detector is shifted by the piezo translators during the oV-state of the device under test, recording the corresponding modulation of the deflection signal. This preprocessing step can be included in the program control of the experimental setup. It has been proposed to scale the displacement Dx to the spot size w of the probing beam on the detector (Deboy, 1996; Deboy et al., 1996). However, since the latter is unknown, it is more convenient to consider the detector response with respect to the absolute displacement:
QðwÞ L2 dno L dno 3 M ¼ QðwÞ Dx ¼ þO þ LðF Vt d2 Þ : ð112Þ Vt nw dx 2no dx
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Solving for the refractive index gradient results in the following evaluation rule: dno @no dT @no dn M Vt =QðwÞ þ OðM 3 Þ: ð113Þ þ ¼ L2 ¼ dx @T n;p dx @C T dx 2n þ LðF Vt d2 Þ o
5. Parameters of the Experiment Equation (112) clearly reveals the degrees of freedom that can be used to optimize the experiment with respect to a desired purpose (e.g., a large measurement range). First, the interaction length L. Reducing the sample length obviously decreases the internal deflection at a given gradient of the refractive index, which, in turn, allows to detect higher temperature gradients. On the other hand, the shrinking of the sample size is limited by practical constraints. Designing specific test structures is discussed in Section VII.G. Second, the detector response function. As its saturation is the limiting eVect of the measurement range, a smaller slope of the response function favors the maximum detectable beam displacement, whereas a large slope results in a high sensitivity. Section VII.E addresses how the detector response depends on its geometry. Third, the image formation conditions which are represented by the parameter b :¼ F Vtd2 ¼ F fd2/(d2 f ) ¼ (a L/no)/2. As mentioned above, b can be interpreted as the distance between the rear surface of the sample and the image of the detector plane, which is formed by the imaging lens. Note that the transverse magnification 1/Vt does not contribute an independent parameter since scaling the intensity distribution on the detector has the same eVect as the inverse scaling of the detector itself (except for the bandwidth, which is related to the detector area). Formally speaking, the parameter Vt can be combined with the slope of the response function Q(w). That means, if the parameters of the optical configuration, which are obviously chosen as F, f, d2, are replaced by a diVerent set, namely Vt ¼ f/(d2 f ), F þ d2, and b ¼ F fd2/(d2 f ), only the latter aVects the deflection signal. This is also true in case the image formation is performed by several lenses, as they can be replaced by a single thick lens if all distances are measured to the corresponding principal planes. B. Deflection Measurements in Case of Low Power Dissipation The temperature evolution in power devices has been successfully measured when the samples have been operating with low-power dissipation (Deboy et al., 1996; Hille and Thalhammer, 1997; Thalhammer et al., 1995, 1997).
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Choosing sample lengths of some hundreds of microns to a few millimeters, the angular deflection and the internal shift of the probing beam are suYciently small so that the beam displacement on the detector remains in the linear range of the response function. For virtual experiments, the carrier contributions are calculated from the simulated absorption signals (see Section VI.F) and subtracted from the deflection signals. Thus, the remaining contributions are related to the temperature gradients (Figure 29, left). To gain the absolute temperature profiles, the integration constants are calculated from the total electric power dissipation assuming adiabatic boundary conditions. Figure 29 demonstrates that the extracted temperature gradients agree with the reference gradients up to an error of about 10% to 20%. The deviation predominantly arises as the carrier contributions are only known up this level of accuracy. However, as the integration significantly smoothes the profiles, the error of the absolute temperature profiles becomes almost negligible. Quantitatively, the average relative error defined similar to Eq. (89) amounts to 6.8% for L ¼ 800 mm, A ¼ 0.08 and increases up to 14.5% for L ¼ 2500 mm, A ¼ 0.05. This clearly demonstrates that laser deflection measurements are a suitable method for sensitively and accurately detecting small temperature modulations.
FIGURE 29. Virtual laser deflection experiments for probing the transient evolution of the temperature profile in a pin power diode subjected to a current pulse of 150 A/cm2. Adapted from Thalhammer (2005). # 2005 AIP.
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C. Simultaneous Free Carrier Absorption Measurements Summing up all the photo currents of the four-quadrant detector provides a convenient way to monitor the transmitted light intensity, thus enabling simultaneous absorption measurements without any significant experimental eVort. However, the beam profile on the detector attains its maximum intensity in the center of the four-quadrant diode (i.e., in the inactive area between the segments). Thus, a deflection of the laser beam caused by transient refractive index gradients increases the detected infrared (IR) power, which would be misinterpreted in terms of a lower carrier concentration. Besides, optimizing the setup for absorption measurements favors a larger detector as it more accurately detects the total incident power even in case of a shifted probing beam. On the contrary, it will be shown below that shrinking the detector size enhances the measurement range of the deflection measurements. The impact of diVerent detector types is simulated by virtual experiments (Figure 30). It is evident that the pin diode yields the more accurate carrier concentration profile and suppresses the undesired coupling of the absorption signal and the internal beam deflection. Additionally, the sensitivity to a possible delocation of the focal plane in the device is decreased. Despite the more complicated experimental setup, an additional beam splitter and a separate pin diode detector for simultaneous absorption measurements are therefore preferable.
FIGURE 30. Carrier distribution in a pin power diode extracted from the absorption signal which is recorded during deflection measurements by a four-quadrant photo diode or an additional pin diode (interaction length L ¼ 2.5 mm, A ¼ 0.06). Adapted from Thalhammer (2005). # 2005 AIP.
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D. The Fabry–Perot EVect As a temperature rise modulates the optical sample length, Fabry–Perot oscillations of the absorption signal are observed (see Section VI.C). The standardized deflection signal, however, is not aVected by pure amplitude modulations since any constant factor is canceled Eq. (18). However, in case of planeparallel facets, each of the multiply reflected rays of diVerent order is shifted by a distance that is the larger the more times the ray has been reflected back and forth (Figure 31). The resulting displacement of the total intensity distribution depends on whether individual contributions interfere constructively or destructively (Figure 31). Similar to the free carrier absorption measurements, the thermally induced Fabry–Perot eVect originates a small drift in case of low-power dissipation (Figure 32, left) or complete oscillations if the optical thickness increases by several wavelengths (Figure 32, right). E. Detector Response Denoting the photo currents of two opposite detector segments by I1 and I2, the standardized deflection signal M becomes M¼
I1 I2 I1 =I2 1 : ¼ I1 þ I2 I1 =I2 þ 1
ð114Þ
If one of the photo currents is much greater than the other one, the deflection signal saturates at 1 and any further increase of the beam displacement
FIGURE 31. Origin of the Fabry–Perot effect of the deflection signal. Above, intensity distribution of each of the multiply reflected rays. Below, the total intensity distribution depends on whether individual rays interfere constructively or destructively (for the sake of clarity, each profile is scaled to 1). Adapted from Thalhammer (2005). # 2005 AIP.
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FIGURE 32. Fabry–Perot effect during deflection measurements on a pin power diode in case of low-power dissipation (left, current density 15 A/cm2) and high-power dissipation (right, current density 150 A/cm2). Adapted from Thalhammer (2005). # 2005 AIP.
does not increase the detector signal. Therefore, to gain a large measurement range, the photo currents must be of equal order of magnitude. As they are aVected by the integral of the field distribution over the detector segments, the decisive parameter is the ratio of the beam spot size w and the detector size s. For diVerent detector geometries, the typical shape of the response function is plotted in Figure 33. The gray shaded areas in Figure 34 illustrate the ratio of the photo currents in case the laser spot size w is comparable to the segment size s (detector 2). As the ratio of the photo currents is larger than it would be for a smaller detector (dashed areas, detector 1 in Figure 34), a larger detector enhances the sensitivity, but decreases the measurement range of the optical setup (see Figure 33). Of course, a similar trend can be derived for various widths of the gap g between the detector segments. The smaller the gap, the better the photo currents of opposite segments balance, and, consequently, the larger the measurement range and the smaller the sensitivity. Although the detector geometry aVects the slope of the response function (see Figure 33) and may therefore be chosen to adjust the sensitivity or the measurement range of the setup by up to a factor of 2 (Thalhammer, 2005), some practical constraints must be considered. First, if the displacement is larger than the detector size, the photo currents are induced by the tails of the field distribution. Their intensity decreases exponentially as the beam displacement increases, thus significantly raising the noise level. Second, the response of many available photo detectors is disturbed if the borders of the
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FIGURE 33. Detector response for different detector segment sizes s. Adapted from Thalhammer (2005). # 2005 AIP.
FIGURE 34. Four-quadrant diodes with a small segment size (detector 1) and a large segment size (detector 2). The photocurrents are proportional to the highlighted areas. Adapted from Thalhammer (2005). # 2005 AIP.
active area are significantly illuminated. Therefore, the spot radius and the beam displacement should not exceed the detector size. F. Image Formation Conditions As mentioned in Section VII.A.5, the image formation by the optical setup is characterized by the parameter b ¼ F fd2/(d2 f ), cf. Figure 28. For a fixed sample length L, it can be replaced by a ¼ 2b þ L/no. Increasing a will increase the total displacement Dx as well as the spot radius w on the detector (see Eqs. (110) and (111)). Since the latter eVect is predominant
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FIGURE 35. Deflection signal in case of different image formation conditions. The optical setup is characterized by the parameter b ¼ F fd2/(d2 f ). Adapted from Thalhammer (2005). # 2005 AIP.
and decreases the sensitivity, enhancing the parameter a ¼ 2b þ L/no reduces the deflection signal (Figure 35). Though this eVect might be exploited to adjust the measurement range and the sensitivity by approximately a factor of 2 or 3 (Thalhammer, 2005), the image formation conditions need to be chosen to adjust the beam displacement and the spot size on the detector plane. They should be approximately the detector size for the same reasons mentioned in the previous section. G. Deflection Measurements of Large Temperature Gradients For some applications (e.g., the internal characterization of power devices operating under short circuit conditions), the ability to detect large temperature gradients of up to some K/mm is preferable against a high sensitivity. In this case, however, the beam deflection for die sizes of some 100 microns to a few millimeters becomes too large to be measured by laser deflection experiments as the detector response saturates (Deboy, 1996; Hille and Thalhammer, 1997). As previously discussed, modifying the optical setup (i.e., the detector and the image formation conditions) may enhance the measurement range by about a factor of 3. However, the most eVective method is shrinking the interaction length by preparing suitable test structures, which are illustrated in this section. As a representative situation, we assume a heat source closely located at W the top surface and dissipating an average power density of 2104 cm 2 per
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FIGURE 36. Intensity distribution at the rear surface at different points of time after turnon. The sample is dissipating an average power of 2 104 W/cm2 at its top surface. The probing beam enters at a depth of 40 mm.
square area. This model reflects, for example, the short-circuit operation of IGBTs with a voltage drop of 200 V at a current density of 100 A/cm2. The resulting heat generation rate is almost completely dissipated in the metaloxide-semiconducter (MOS) channels and the space charge region at the top of the device. The crucial problem becomes apparent in Figure 36. Even in a sample of only 800 mm in length, the temperature gradients originate an internal beam displacement of some tens of microns. For instance, a laser beam entering at a depth of 40 mm is shifted by 20 mm after 100 ms. Apart from the diYculties in detecting this deflection, the spatial resolution will seriously suVer from such a large displacement since the measurement signal will be sensitive to the average temperature gradient between x ¼ 20 mm and x ¼ 40 mm. 1. Test Structures To reduce the interaction length while keeping the die size, test structures are regarded with an active area of some 100 mm embedded in a larger silicon substrate (Figure 37). Although the total deflection of the probing beam is smaller, the carrier and temperature distribution in these test structures is diVerent due to the current and heat spreading (Figure 37). Virtual experiments as described in Section V are carried out on test structures with diVerent lengths of the active area. Since we particularly want to investigate the eVects of the current and heat spreading, the reference distribution is calculated from devices with Ls ¼ La, resulting in an
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FIGURE 37. Temperature distribution in a test structure at t ¼ 100 ms after turn-on (sample W length Ls ¼ 800 mm, length of the active area La ¼ 100 mm). The sample is dissipating 2 104 cm in the active area.
FIGURE 38. Temperature gradient extracted from the simulated deflection signal on test structures (sample length Ls ¼ 800 mm, length of active area La, t ¼ 100 ms after turn-on).
approximately 1D temperature distribution. The profiles of the measurand are extracted from the simulated detector signals according to Eq. (113), assuming an interaction length equal to the size of the active area (L ¼ La), and plotted in Figure 38. Two tendencies are clearly visible. First, the saturation of the response function results in a maximum detectable temperature gradient that is the larger, the smaller length of the active area. Second, simply inserting the length La of the active area as the interaction length L significantly overestimates the temperature gradient, particularly in case of a small La. This is especially caused by the first term in the denominator of Eq. (113). The quadratic dependence on L refers to the contribution to the beam
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displacement on the detector, which arises from the internal shift of the probing beam, while the contribution caused by the internal angular deflection depends linearly on L. Hence, the error increases as the absolute value of F Vtd2 is smaller compared to L/2no (i.e., the image of the device rear surface is located closer to the detector plane). Note that La ¼ 800 mm ¼ Ls represents a 1D structure, which therefore does not show 2D heat diVusion. 2. EVective Interaction Lengths and the Modified Evaluation Rule Evaluating the deflection measurements on test structures obviously demands the introduction of an eVective interaction length LeV which reflects the eVects of the current and heat spreading. As the deflection signal includes two contributions with a linear and a quadratic dependence on L Eq. (113), the following definitions are introduced: L=2 Z dTref 1 dT ðx; zÞ dz Leff;lin ðxÞ :¼ dx dx
ð115Þ
L=2
L2eff;qu ðxÞ
L=2 Z dTref 1 dT ðx; zÞz dz: :¼ dx dx
ð116Þ
L=2
These quantities are calculated analytically by integrating the simulated temperature distributions in the interior of the test structures. The diVusion of heat broadens the temperature profile along the beam path, but reduces the maximum temperature rise. These two eVects nearly balance each other in case of the linear eVective interaction length, which therefore almost matches the length La of the active area. On the other hand, the quadratic eVective pffiffiffiffiffiffiffiffiffiffi interaction length can be approximated by the empirical expression Ls La . The evaluation rule therefore needs to be dno @no dT @no dn M Vt =QðoÞ þ OðM 3 Þ; ð117Þ þ ¼ ¼ dx @T n;p dx @C T dx La Ls þ F Vt d2 2no where Vt ¼ f/(d2 f ). Usually, Ls/2no amounts to a few 100 microns while the other terms in the parentheses are in the order of some centimeters. Therefore, the most significant eVect arises from the linear dependence on La, unless the image of the device rear surface is located on the detector plane. In this case, also the quadratic dependence on LaLs is pronounced. As seen in Figure 39, the temperature distributions extracted according to Eq. (117) accurately match the reference temperature profiles. The maximum detectable temperature gradient increases as the active length decreases.
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FIGURE 39. Temperature distribution extracted from the simulated deflection signal on test structures according to Eq. (117) (sample length Ls ¼ 800 mm, length of active area La). When integrating the gradients, the temperature rise at the bottom surface has been set to 0.
La 200 mm turns out to be an optimum geometry, since little additional improvement is gained by a further reducing La. In this case, the measurement range extends up to approximately 0.6 K/mm. Since the deviations in the absolute temperature profiles are even less significant, typical temperature increases up to about 60 K can be detected. Finally, the assumptions the presented formula is based upon should be summarized. First, the total internal displacement and the angular deflection of the probing beam must be small so that the paraxial approximation remains valid. In case of the test structures with a small active area, this restriction even holds during operations with large power dissipation. Second, the gradient of the refractive index is assumed to be constant within the volume covered by the probing beam. Owing to the rather small lateral extension of the probing beam (see Section VI.A), this precondition is satisfied with reasonable accuracy. Third, the proper eVective interaction lengths must be inserted. In particular, La constitutes the length of the active area (i.e., the region within which the electric power is dissipated). However, the length Ls must be modified for very large silicon dies, where the outer regions are not aVected by the heat dissipation in the active area. It is obvious that in this case, the sample length
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Ls has to replaced by the width that is covered by the heat diVusion during the duty cycle. Fourth, this study has been performed with a fixed heat dissipation rate, thus implicitly neglecting the thermal feedback on the electrical device operation. This assumption is invalid, for example, during short-circuit operation of IGBT samples comprising a very small number of cells. Since the MOS saturation currents significantly depend on the channel temperature, the lower peak temperature in the smaller test structures results in a higher short-circuit current which, in turn, increases the heat dissipation rate. Probing test structure with a very small active area will therefore result in a too large temperature gradient extracted from the detector signal. H. Summary The major results of virtual laser deflection experiments can be summarized by the following statements.
A consistent quantitative calculation of the deflection signal is gained by considering the probing beam a Gaussian beam whose optical axis is deflected in the sample according to the principle of geometrical ray tracing. Calculating the projection by the imaging lens from Fourier optics, the field distribution on the detector is shown to also be a Gaussian profile. The deflection signal is aVected by three parameters, namely, the interaction length in the sample, the image formation properties of the imaging lens, and the detector response function.
The temperature distribution in power devices operating with lowpower dissipation can be detected by internal laser deflection measurements with high sensitivity and excellent accuracy.
Summing up the photocurrents of all segments of the four-quadrant detector opens a simple way to calculate the total transmitted intensity. Although laser deflection and free carrier absorption measurements can thus be performed simultaneously, a separate pin diode detector essentially improves the accuracy and the reliability of the absorption measurements.
Thermal modulations of the optical sample thickness cause the Fabry– Perot eVect of the deflection signal. It is observed as a drift or as several oscillations of the detector signal, unless multiple reflections are suppressed by depositing an antireflective coating or using an incoherent light source.
The saturation of the detector response function constitutes the limiting eVect of the measurement range, which can be extended by reducing the detector size. However, an increase by at most a factor of 2 is attainable on account of practical restrictions.
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Changing the image formation conditions of the imaging lens scales the beam shift on the detector. Adjusting the measurement range and the sensitivity this way is limited by practical restrictions, as it also aVects the spot diameter, which should approximately match the detector size.
The most eVective way to increase the measurement range is to reduce the interaction length L. Probing test structures that comprise a small active area embedded in a larger silicon die requires the definition of eVective interaction lengths that reflect the eVects of the additional lateral heat diVusion. Thus, laser deflection measurements can detect temperature gradients up to about 0.6 K/mm in semiconductor devices, which typically correspond to temperature rises of about 60 K if the heat sources are located near the top surface.
VIII. INTERFEROMETRIC TECHNIQUES This section is dedicated to discussing laser probing techniques that are based on interferometry and therefore promise several crucial advantages. First, the periodically oscillating detector signal lacks saturation eVects, thus facilitating a large measurement range limited merely by parasitic eVects. Second, extracting the desired information is based on the fact that the optical path length covered by the probing beam has changed by one wavelength as two subsequent intensity maxima are being observed on the detector. Consequently, there is no need for calibrating the optical setup, such as the magnification of the imaging system or the sensitivity of the detector. Third, as interferometric techniques are sensitive to the optical sample thickness, they do not detect the gradients but directly reveal the absolute value of the refractive index. However, the oscillating signal reflects the temperature evolution and thus requires an integration in time space, but the initial temperature distribution during transient switching is usually known. In particular, it is homogeneous and equal to the ambient temperature for the examples discussed below. As the major drawback, however, the oscillating signal does not allow discriminating whether it originates from an ascending or a descending temperature evolution. Consequently, it can be diYcult to identify the temperature maximum during a transient process. A. Mach–Zehnder Interferometry As a first example, Mach–Zehnder interferometry is discussed (see Section III.B.6). Thermally or electrically induced modulations of the optical sample thickness aVect the phase of the horizontally propagating probing beam.
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The resulting phase shift is detected by interference with a reference beam that is conducted around the sample (see Figure 9). To study the capabilities of this technique, the same situation as in Section VII.G is considered. Close to its top surface, the sample is dissipating a W power of 2104 cm 2 during the period 0 < t < 100 ms. The rising temperature in the die (200 mm wafer thickness, sample length 800 mm) increases its optical thickness. As the phase delay of the probing beam is thus continuously increasing, the superposition of the probing beam and the reference beam alternates between constructive and destructive interference, resulting in oscillations of the detected intensity (Figure 40). The temperature rise DTpp between two peaks can be extracted by DTpp ¼
o L @n @T
l : @L þ no @T
ð118Þ
However, the thermally induced gradients of the refractive index deflect the probing beam, which can shift its position by some tens of microns at the sample’s rear surface (see Figure 36). Consequently, the interference of the probing beam and the reference beam deteriorates as their distance on the detector increases. If the latter becomes too large, the field distribution merely displays diVerent interference fringes with an almost constant integral intensity (Figure 41) and the detector signal approximately assumes an average level (see Figure 40). This saturation eVect is reached the earlier, the larger the perpendicular refractive index gradient along the beam path (i.e., the smaller the distance to the heat source). As the amplitude of the
FIGURE 40. Detector signal of a Mach–Zehnder interferometer for different positions of W the probing beam. The sample is dissipating a heat of 2 104 cm .
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FIGURE 41. Intensity distribution on the detector of a Mach–Zehnder interferometer at different points of time. The magnification of the imaging system has been assumed to be 1. The image of the sample’s top surface is located at x ¼ 0; the initial position of the probing beam is projected onto x ¼ 40 mm.
oscillations decreases below the noise level, the phase evolution can no longer be extracted from the detector signal, thus limiting the measurement range. Figure 42 shows the associated evolution of the phase shift. The latter has been assumed to increase by p at each maximum or minimum of the oscillating detector signal (solid diamonds in Figure 42). It excellently matches the reference curves gained by an electrothermal device simulation and related to the phase shift according to Eq. 118, at least until each of the temperature evolutions attains its respective maximum. Even for beam positions close to the top surface, where the large refractive index gradients strongly deflect the probing beam and, consequently, rapidly decrease the amplitude of the detector signal, the phase evolution can be accurately extracted from the faint oscillations during the duty cycle. It is evident that this is not possible for real experiments if the noise level exceeds the signal amplitude. However, this evaluation method fails in predicting the maximum and the following decrease of the temperature. To demonstrate the accuracy of actual measurement process, the phase shift extraction is modified as follows. At each position of the probing beam, the device simulation result is used to identify that point of time at which the respective temperature
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FIGURE 42. Phase shift evolution extracted from the detector signal of the Mach–Zehnder interferometer. The solid and open symbols are gained from the extrema of the oscillating detector signal (details in the text). Solid lines reflect the reference evolution obtained by an electrothermal device simulation.
evolution is at its maximum. Before (or after) this point of time, the phase shift is increased (or decreased) by p at each extremum of the detector signal. Thus, the resulting phase evolutions (open diamonds in Figure 42) excellently reproduce the reference curves even after the temperature maximum. A similar result is obtained when looking at the temperature profiles that are constructed from the transient phase shift signals (Figure 43). The extracted temperature distributions during the duty cycle are in excellent agreement with the reference profiles gained by an electrothermal device simulation, whereas the above-mentioned correction needs to be applied for points of time after turn-oV. This clearly demonstrates that Mach–Zehnder interferometry constitutes a highly accurate probing technique with a large measurement range but requires additional eVort to identify possible phase shift maxima during transient processes. B. Backside Laser Probing Backside laser probing has been successfully used to investigate a large diversity of semiconductor devices, for example, power MOSFETs (Seliger et al., 1996b), smart power devices (Seliger et al., 1997b), IGBTs (Thalhammer et al., 1998a), memory cells (Seliger, private communication, 1997), ESD protection devices (Fu¨ rbo¨ ck et al., 1998), and MEMS (Pogany et al., 1998b). Contrary to the previously discussed measurement techniques, backside laser probing
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FIGURE 43. Temperature distribution extracted from the phase shift signal of the Mach– Zehnder interferometer. Solid symbols reflect a monotonically increasing phase, whereas the open diamonds are constructed by applying a decreasing phase shift after the respective local temperature maximum.
operates by means of a vertically propagating laser beam. In vertical devices it therefore provides only integral information on the carrier concentration and the temperature profiles. However, since the beam path is approximately parallel to the most significant gradient of the refractive index, the probing beam is not laterally deflected, thus promising a large measurement range. 1. EVects of the Sample Preparation To provide access for the probing beam penetrating the device from the rear side, a window needs to be etched if the bottom of the sample is covered by a metallization layer (see Section III.B.3). This preparation step will probably aVect the internal charge carrier and temperature distribution in vertical devices, such as power devices where the rear metallization is used as one of the device terminals (e.g., as the collector contact of an IGBT). A quantitative analysis has been carried out by simulating multiple IGBT cells (Thalhammer et al., 1998a). The carrier distribution under forward-biased operation is plotted in Figure 44. The figure clearly reveals that current crowding at the collector contact egdes raises the carrier concentration in the window region at the bottom of the device, in particular at the egdes of the window. This phenomenon is typical of potential driven particle currents. It also well known from everyday life as it can be observed, for example, as the water level rises in front of the piers of a bridge in the river.
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FIGURE 44. Carrier distribution in a forward-biased IGBT sample with a window in the collector contact metallization layer.
For a detailed quantitative analysis, horizontal cuts in diVerent depths of the structure are plotted in Figure 45. Compared with an unprepared device, the carrier density at the edges of the window is almost twice as large. The diVerence becomes smaller as the distance to the rear surface increases, and it vanishes at the top surface. The integral of the carrier distribution along a vertical cutline—this is the decisive quantity aVecting the phase shift signal— is therefore about 35% larger than that of an unprepared sample would be. As a consequence of the enhanced current density at the window edges, a local temperature increase is observed (Figure 46). However, the temperature profiles along horizontal cutlines in various depths (Figure 47) clearly reveal that this eVect amounts to only some percent and is therefore negligible. In summary, etching a window in the rear metallization layer originates current crowding at the contact edges. The resulting local increase of the carrier concentration enhances the carrier contribution to the phase shift signal by about 35%, whereas the modulations of the temperature distribution can be neglected. Since the latter is by far the most significant eVect in
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FIGURE 45. Carrier distribution along horizontal cutlines in various depths of an IGBT sample (see Figure 44) prepared for backside laser probing. For symmetry reasons, only the right-hand half of the sample is shown.
samples with a high-power dissipation as, for example, IGBTs operating under short-circuit conditions, the preparation eVects need not be taken into account for evaluating the measurement results in this case. 2. The Measurement Signal The backside laser probing technique operates on a laser beam with an angular aperture of about 0.1 to 0.4. The resulting optical field distribution (Figure 48) attains a minimum spot diameter of about 8 mm at the top surface. The interaction length is given by twice the wafer thickness and is therefore about some hundreds of microns (e.g., 2 200 mm in the sample shown in Figure 48). Consequently, beam spreading increases the spot diameter to only 15 mm at the bottom surface, which promises an excellent spatial resolution. a. Phase shift signal and measurement accuracy. As described in Section III.B.3, the driving frequencies o1 and o2 of the acousto-optic modulator control the positions of the probing beam and the reference beam. Since diVraction within the modulator shifts the frequencies of the laser beams by o1 and o2, respectively, their electric fields on the detector become
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FIGURE 46. Temperature distribution in an IGBT sample with a window in the collector contact metallization layer (short-circuit operation with 500 V collector–emitter voltage, t ¼ 3 ms after turn on).
E pr ðtÞ ¼ E pr;0 exp½iðo þ 2o1 Þt þ iD’ðtÞ; E ref ðtÞ ¼ E ref ;0 exp½iðo þ 2o2 Þt ð119Þ (Seliger, 1998), where o denotes the frequency of the original laser beam. Consequently, the following intensity signal is observed: ID ðtÞ ¼ jE pr;0 j2 þ jE ref ;0 j2 þ 2E pr;0 E ref ;0 cos½2Dot þ D’ðtÞ with Do ¼ o1 o2 :
ð120Þ
To extract the desired phase shift, the intensity signal ID(t) is solved for D’(t), determining the unknown coeYcients jE pr;0 j2 þ jE ref ;0 j2 and 2Epr,0 Eref,0 from the extrema of the oscillations. An alternative strategy, which works well if multiple oscillations of the intensity signal are observed, is based on filtering in frequency space. In case that D’(t) is small compared to 2Dot, the Fourier transform of ID(t) contains three components, namely the constant oVset and the two contributions around 2Do. The latter arises as the original signal is real. Multiplying by a band pass filter around 2Do cancels the constant oVset and the component with the negative
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FIGURE 47. Temperature distribution along horizontal cutlines in various depths of an IGBT sample prepared for backside laser probing (short-circuit operation with 500 V collector– emitter voltage, t ¼ 3 ms). For symmetry reasons, only the right handhalf of the sample is shown.
frequency. Thus, the transformation back to time space yields a complex signal ~I D ðtÞ ¼ const exp½i2Dot þ iD’ðtÞ, the phase of which can be extracted by ~ Im½I D ðtÞ 2Dot þ ’ðtÞ ¼ arctan : ð121Þ Re½~I D ðtÞ Note that an appropriate choice of the filter function is the key for a successful and accurate extraction of the phase evolution. As an example, the intensity signals for Do ¼ 2p 1 MHz are calculated from virtual experiments with diVerent angular apertures and plotted in Figure 49. It is nearly impossible to distinguish the symbols in the figure (i.e., using diVerent angular apertures results in practically the same intensity signal). For further illustration, the extracted phase shift signals are compared with the integrated temperature profile calculated according to Eq. (21) in Figure 50. It is evident that the diVerence amounts to only a few percent for all angular aperatures. The major reasons are that owing to the rather small interaction length of 2 200 mm, the probing beam does not spread out significantly and partial reflections at longitudinal refractive
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FIGURE 48. Optical field distribution in an IGBT sample subjected to backside laser probing. The angular aperture of the laser beam is 0.25. Adapted from Thalhammer and Wachutka, 2003b. # 2003 OSA.
FIGURE 49. Detector signal of virtual backside laser probing measurements on an IGBT sample operating under short-circuit conditions (500 V collector–emitter voltage, t ¼ 3 ms, Do ¼ 2p1 MHz). The solid lines with symbols refer to various angular apertures, namely, A ¼ 0.1 ( ), A ¼ 0.2 (.), A ¼ 0.25 (), and A ¼ 0.4 (/). Since the different probing conditions produce nearly the same intensity signal, the symbols in the figure overlap almost completely. To illustrate the phase shift, a harmonically oszillating intensity signal (i.e., D’(t) 0) is included as the dotted line.
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FIGURE 50. Left, evolution of the phase shift extracted from the detector signal. The solid line represents the integrated temperature profile according to Eq. 21. To illustrate the negligible differences, the right-hand figure shows the difference of the phase shift signals from the integrated temperature profile.
index gradients can be neglected. Hence, the spatial resolution of backside laser probing is excellent and the desired measurands can be accurately extracted from the phase shift signals by using the 1D model of Eq. 21. b. Interpretation of the phase-shift signal. In case low power is dissipated within the sample, two contributions to the phase shift signal [see Eq. (21)] are observed. They arise from transient modulations of the carrier concentrations n and p and the temperature distribution T (see Figure 7). During forward-biased operation, carriers are injected and temperature rises after turn-on. Hence, the two contributions are opposite of sign but of the same order of magnitude (Thalhammer et al., 1998a). Note that probing bipolar devices operating on carrier injection by a transparent emitter at the bottom results in a carrier contribution that is about 35% larger than that of an unprepared sample without the rear metallization window would be. The capabilities of backside laser probing become evident when probing devices operating with large power dissipation. As an example, the shortcircuit operation of an IGBT sample with diVerent collector–emitter voltages is discussed. As the carrier contribution to the phase shift signal can be neglected in view of the thermal contribution, the phase shift signal is a direct measure of the heat DQ(t) stored in the device per area A (Thalhammer et al., 1998a):
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4p @no D’ðtÞ ¼ lct AD @T
ZL ct AD DTðx; tÞ dx ¼ 0
4p @no DQðtÞ: lct AD @T
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ð122Þ
The associated phase shift signals gained from virtual experiments as well as from real measurements are plotted on the left-hand y-scale of Figure 51. The large measurement range that covers a phase shift from a few mrad to several rad is one of the major advantages of the backside laser probing technique, which is therefore able to detect temperature rises from a few mK to some 100 K. Further interesting information is gained if the transient temperature distributions under these operating conditions (Figure 52) are approximated by linear profiles with vanishing temperature rise at the bottom. Thus the integral of Eq. (122) can be expressed in terms of the surface temperature rise
FIGURE 51. IGBT sample operating under short-circuit conditions with different collector– emitter voltages UCE (pulse duration 10 ms). Solid and dashed lines represent the phase shift signals D’(t) when related to the left-hand y-axis. The right-hand y-axis refers to the interpretation in terms of the surface temperature rise DT(t) according to Eq. 123. The result of an electrothermal device simulation is included as the dotted line. Adapted from Thalhammer and Wachutka, 2003b. # 2003 OSA.
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FIGURE 52. Transient evolution of the temperature profiles in an IGBT sample operating under short-circuit conditions with 300 V collector emitte–voltage (pulse duration 10 ms).
DT(z ¼ 0, t) (Thalhammer et al., 1998a; Thalhammer and Wachutka, 2003b): D’ðtÞ ¼
2pL @no DTðx ¼ 0; tÞ: l @T
ð123Þ
Consequently, the phase shift signal can be directly reinterpreted in terms of the surface temperature rise DT. The corresponding scaling is added as an additional y-axis on the right-hand side of Figure 51. To verify the accuracy of this consideration, the actual evolution of the surface temperature rise gained from an electrothermal device simulation is also included in the figure. As can be seen, this interpretation is only valid with an error of about 50% (Figure 51, right), which arises mainly from the approximative representation of the temperature profiles. Nevertheless, it clearly demonstrates that backside laser probing opens a way to detect the hot spots in the investigated structure and to approximately reveal their respective temperature rises. 3. DiVerential Backside Laser Probing As a final application example of virtual experiments, diVerential backside laser probing (Fu¨ rbo¨ ck et al., 1999) is discussed. It has been realized as a slight modification of the optical setup for backside laser probing and operates on both beams positioned in the window region. Thus, lateral temperature inhomogeneities are expected to be directly revealed by the measurement signal.
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FIGURE 53. Absolute phase shift signals near the egdes of the collector contact window and in the center of an IGBT sample operating under short-circuit conditions.
DiVerential backside laser probing measurements on IGBTs operating under short-circuit conditions with a large collector–emitter voltage have been reported (Fu¨ rbo¨ck et al., 1999). The diVerential phase shift signal is governed by two contributions: First, a large rate of heat is dissipated in the channel region and the space charge region at the reverse-biased p-well on top of the sample. Second, the window in the rear metallization layer gives rise to current filaments at the bottom contact edges. Although the heat dissipation rate in this region is comparably small, it essentially contributes to the temperature integral as the filaments extend by more than 100 mm into the sample (Fu¨ rbo¨ck et al., 1999). Consequently, the total absolute phase shift signal near the window edges is slightly larger than that in the center of the sample (Figure 53). Unfortunately, the diVerential phase shift signal is significantly aVected as a consequence of preparing the sample. If the probing beams are placed between two cells and at the edge of the p-well, respectively, a positive peak during the current pulse is observed on the diVerential phase shift signal (Figure 54, left). It is introduced by the probing beam penetrating the channel area and the space charge region at the reverse biased p-well, where a large power is being dissipated during the current pulse. Immediately after turn-oV, however, the heat generation in the current filaments at the window edges is dominating. It is detected by the second probing beam and therefore results in a negative peak of the diVerential phase shift signal. The theoretical predictions of virtual experiments on the shape of the diVerential phase shift signal are excellently confirmed by real measurements (Figure 54, left).
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FIGURE 54. DiVerential phase shift signal of two probing beams positioned between two cells and at the edge of the p-well, respectively (short-circuit operation with 500 V collector– emitter voltage). Left, phase shift signals obtained on a prepared sample. Right, the results of the real and the virtual experiment on prepared samples are corrected as mentioned in the text and compared with the phase shift obtained by an electrothermal device simulation of an unprepared sample.
However, the diVerential phase shift signal of an unprepared device would be completely diVerent (Figure 54, right): As electrothermal device simulation reveals, it shows only one positive peak with two slight maxima reflecting the turn-on and turn-oV losses, respectively. Since thermal diVusion levels out the lateral temperature inhomogeneities arising during the current pulse, the diVerential signal drops to zero immediately after turn-oV. On account of these significant eVects caused by the necessary sample preparation, it is extremely diYcult to infer the temperature distribution in unprepared devices from the results of diVerential phase shift measurements. This may only be achieved by inverse modeling using precisely calibrated simulations. However, to obtain at least an approximative analytical elimination of the preparation eVect, the following correction of the measurement signal has been proposed (Fu¨ rbo¨ ck et al., 1999). Since we know that a fictitious diVerential phase shift signal gained on unprepared samples would vanish after turn-oV, a piecewise linear function is added to the signal measured on the prepared samples. This function is zero before turn-on, increases linearly during the current pulse up to the absolute value of the diVerential signal immediately before turn-oV and decreases exponentially to zero afterward. The thus corrected signals are plotted in the right-hand image of Figure 54 and at least approximately reflect what the diVerential phase shift signal of an unprepared sample would be. The theoretical model predicts that preparing the sample causes the associated measurement signals to become completely diVerent in shape. This prediction is clearly confirmed by the actual shape of the real measurement signal, thus
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emphasizing that virtual experiments constitute a powerful strategy to support the interpretation of the measurement results. C. Summary
Regardless of the image formation and the detector sensitivity, two adjacent interference extrema are observed if the phase of the probing beam is shifted by p. Consequently, the optical setup of interferometric techniques does not need to be calibrated.
The deflection of the probing beam caused by refractive index gradients perpendicular to the beam path limits the measurement range. For Mach– Zehnder interferometry, this eVect is visible on the decreasing amplitude of the oscillating detector signal.
Despite the possibly faint detector signal of the Mach–Zehnder interferometer, the phase shift evolutions can be accurately extracted, even in case of a large deflection of the probing beam. The resulting temperature distributions excellently match the reference profiles gained by device simulation.
Although the diVerence of the phase shift can be accurately determined from two adjacent extrema of the oscillating detector signal, it is diYcult to discriminate whether the phase shift evolution is increasing or decreasing. Additional information is needed to identify possible temperature maxima during a transient process.
Backside laser probing uses a vertically propagating laser beam and thus provides integral information about the carrier and temperature distribution in vertical devices. As the gradients of the refractive index are mainly parallel to the propagation direction of the probing beam, the latter is not deflected laterally. Consequently, backside laser probing oVers a high sensitivity and a very large measurement range.
In vertical devices, the necessary sample preparation (etching a window in the bottom metallization) gives rise to current crowding at the contact edges. The enhanced carrier concentration aVects the carrier contribution to the phase shift signal by about 35%. The corresponding eVect on the temperature distribution can be neglected.
Even in case of a high-angular aperture, the lateral convolution of the laser beam profile and the distributions of charge carrier and temperature does not introduce a detectable error. The phase shift signal can be approximated by the integrated carrier and temperature profiles with an error of only a few percent.
In case of a large power dissipation, the phase shift signal is a measure for the heat stored in the device. Assuming a linear temperature profile allows a rough extraction of the temperature rise at the top surface.
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Positioning both laser beams in the window is the basic idea of diVerential backside laser probing. In this case, however, the sample preparation eVects significantly change the qualitative shape of the diVerential phase shift signal.
IX. CONCLUSION
AND
OUTLOOK
The fundamental ideas of virtual experiments have been reviewed in this article. They have been demonstrated to constitute a powerful methodology for studying the physical principles of the probing techniques and the resulting eVects on sensitiviy, accuracy, and measurement range. The internal laser probing techniques discussed in this article exploit the electrooptical and the thermo-optical eVect. As their common principle, the complex refractive index is modulated by the injection or removal of charge carriers and temperature variations during transient switching of the device under test. Detecting the resulting modulations of the absorption, the deflection, or the phase shift of an incident probing beam allows extraction of information on the space-resolved and time-resolved distributions of charge carriers and temperature in the interior of semiconductor samples. The physical model for simulating the experiments includes an electrothermal device simulation of the sample’s operating condition, the calculation of the refractive index modulations, the simulation of wave propagation through the sample, the lenses, and aperture holes, and finally the simulation of the detector response. One of the major steps is a numerically eYcient algorithm for simulating wave propagation in large computational domains. The key idea for assessing the quality of an experiment consists in comparing the measurands extracted from the simulated detector signals with the data obtained by simulating the operation of the associated unprepared samples. Thus, the desired experimental results can be distinguished from parasitic eVects that have been introduced as the sample has been prepared, the actual measurement process has been performed, and the detector signal has been evaluated. The theoretical analysis reveals some general statements that apply to all of the investigated probing techniques: First, the lateral spreading of the probing beam does not introduce a significant error (i.e., the convolution of the beam profile with the distribution of the complex refractive index is suYciently accurately reflected by regarding the 1D carrier concentration and temperature integrals along the beam path). Choosing the probing conditions according to the theoretically predicted optimum promises measurements of the carrier and temperature distribution with excellent
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accuracy and sensitivity. One of the major sources of error originates from an oblique propagation of the probing beam, which may result from a possible misalignment of the sample. The internal beam deflection caused by refractive index gradients perpendicular to the optical axis limits the measurement range, particularly of those techniques using a laterally propagating probing beam. In case of the mirage eVect, a too large deflection causes the detector response function to saturate, while the detector signal of interferometric techniques disappears when the beams that are supposed to interfere move away from each other. The most eVective way to overcome this restriction is measuring specific test structures comprising a small active area within a silicon die of larger size. It has been demonstrated that the resulting heat-spreading eVects can be approximately corrected by introducing an eVective interaction length that is related to the sample geometry by simple analytical formulas. Because backside laser probing operates by means of a vertically propagating probing beam, the internal beam deflection is usually negligible. Since the oscillating interference signal therefore lacks saturation eVects, backside laser probing is capable of detecting temperature rises of some hundreds of degrees Kelvin. Hence, although this technique provides merely integral information on carrier concentration and temperature, it constitutes one of the most powerful methods for investigating devices subjected to operating conditions with a very large power dissipation. Although the spatial resolution of the currently available setups is excellent for probing microstructures, the trend toward smaller dimensions of microelectronic devices will attract scientists’ interest to the ability of submicron scanning. In this case, virtual experiments can help to deconvolute the interaction of the beam profile and the refractive index distribution by inverse modeling. The internal carrier and temperature distribution is represented by a suYciently large number of unknowns, whose initial values are gained by an electrothermal device simulation or extracted from the detector signals using the available evaluation rules. Simulating the measurement process for this kind of refractive index distribution and comparing the calculated detector signals with actually observed signals yields the necessary update of the carrier concentration and temperature distribution. This procedure is iterated until the extracted updates reproduce the current values. Thus, powerful experiments can be constructed that facilitate the exploration of physical processes on a scale that is well below the spot size of the probing beam. The optical model outlined in Section II does not only provide a powerful strategy for simulating probing techniques, but is also capable of calculating the operating characteristics of optical sensors exploiting the thermo-optical or the electro-optical eVect (Thalhammer et al., 1999). Such types of devices
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have been proposed for monitoring the temperature evolution in microstructures (Cocorullo et al., 1997, 1998; Liu et al., 2000). In principle, since these sensors or probing techniques are actually sensitive to modulations of the refractive index distribution, they can be exploited to detect any physical quantity aVecting the refractive index, such as, for example, mechanical stress, provided the corresponding dependence is suYciently pronounced. Virtual experiments of such techniques can be performed according to the strategy illustrated in Figure 10 if the electrothermal device simulation is extended to a consistent electro-thermo-mechanical simulation. Finally, laser probing is also attractive for samples of diVerent materials (e.g., silicon carbide). As the respective experiments will require specific setups using, for example, a diVerent sample geometry or a smaller wavelength of the probing laser, the development of such techniques will gain significant benefit from virtual experiments. REFERENCES Ashcroft, N. W., and Mermin, N. D. (1976). Solid State Physics. Philadelphia: Holt, Rinehart and Winston. Azimov, S. A., Islamov, L. I., and Sultanov, N. A. (1974). Investigation of the influence of heat treatment on the electrical properties of platinum-doped silicon. Sov. Phys. Semicond. 8, 758. Baliga, B. J. (1978). Recombination level selection criteria for lifetime reduction in integrated circuits. Solid State Electronics 21, 1033. Bank, R. E., Coughran, W. M., Fichtner, W., Grosse, E. H., Rose, D. J., and Smith, R. K. (1985). Transient simulation of silicon devices and circuits. IEEE Trans. Comp. Aided Design 4, 436. Bank, R. E., Rose, D. J., and Fichtner, W. (1983). Numerical methods for semiconductor device simulation. IEEE Trans. Electron Dev. 30, 1031. Benda, H., and Spenke, E. (1967). Reverse recovery processes in silicon power rectifiers. Proc. IEEE. 55, 1331. Bertolotti, M., Bogdanov, V., Ferrari, A., Jascow, A., Nazorova, N., Pikhtin, A., and Schirone, L. (1990). Temperature dependence of the refractive index in semiconductors. J. Opt. Soc. Am. 7, 918. Berz, F. (1979). Step recovery of pin diodes. Solid State Electronics 22, 927. Berz, F. (1980). Ramp recovery in pin diodes. Solid State Electronics 23, 783. Bleichner, H., Nordlander, E., Rosling, M., and Berg, S. (1990a). A time-resolved optical system for spatial characterization of the carrier distribution in a gate-turn-oV thyristor (GTO). IEEE Trans. Instrument. and Measurement 39, 473. Bleichner, H., Rosling, M., Vobecky, J., Lundqvist, M., and Nordlander, E. (1990b). A comparative study of the carrier distributions in dynamically operating GTO’s by means of two optically probed measurement methods, in Proceedings of IEEE International Conference of Power Semiconductor Devices and ICs. Tokyo, Japan: Institute of Electrical Engineers of Japan, p.1095. Bløtekjær, K. (1970). Transport equations for electrons in two-valley semiconductors. IEEE Trans. Electron Dev. 17, 38.
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Boccara, A. C., Fournier, D., and Badoz, J. (1980). Thermo-optical spectroscopy: Detection by the mirage eVect. Appl. Phys. Lett. 36, 130. Briggs, H. B., and Fletcher, R. C. (1953). Absorption of infrared light by free carriers in germanium. Phys. Rev. 91, 1342. Brooks, H. (1951). Scattering by ionized impurities in semiconductors. Phys. Rev. 83, 879. Brotherton, S. D., Bradley, P., and Bicknell, J. (1979). Electrical properties of platinum in silicon. J. Appl. Phys. 50, 3396. Burckhardt, C. B. (1966). DiVraction of a plane wave at a sinusoidally stratified dielectric grating. J. Opt. Soc. Am. 56, 1502. Callen, H. B. (1985). Thermodynamics and an Introduction to Thermostatistics, ed. 2, New York: John Wiley and Sons. Canali, C., Majni, G., Minder, R., and Ottaviani, G. (1975). Electron and hole drift velocity measurements in silicon and their empirical relation to electric field and temperature. IEEE Trans. Electron Dev. 22, 1045. Caughey, D. M., and Thomas, R. E. (1967). Carrier mobilities in silicon empirically related to doping and field. Proc. IEEE Dec., 2192. Caulley, J. A. M., Donnelly, V. M., Vernon, M., and Taha, I. (1994). Temperature dependence of the near-infrared refractive index of silicon, gallium arsenide, and indium phosphide. Phys. Rev. B 49, 7408. Choo, S. C. (1972). Theory of forward-biased diVused-junction pln rectifier. Part I: Exact numerical solutions. IEEE Trans. Electron Dev. 19, 954. Choo, S. C., and Mazur, R. G. (1970). Open circuit voltage decay behavior of junction devices. Solid State Electronics 13, 553. Chynoweth, A. G. (1958). Ionization rates for electrons and holes in silicon. Phys. Rev. 109, 537. Claeys, W., Dilhaire, S., and Quintard, V. (1994). Laser probing of thermal behavior of electronic components and its application in quality and reliability testing. Microelectronic Eng. 24, 411. Cocorullo, G., Corte, F. G. D., Iodice, M., Rendina, I., and Sarro, P. M. (1997). A temperature all- silicon micro-sensor based on the thermo-optic eVect. IEEE Trans. Electron Dev. 44, 766. Cocorullo, G., Corte, F. G. D., Rendina, I., and Sarro, P. M. (1998). Thermo-optic eVect exploitation in silicon microstructures. Sensors and Actuators A 71, 19. Conwell, E., and Weisskopf, V. F. (1950). Theory of impurity scattering in semiconductors. Phys. Rev. 77, 388. Cook, R. K., and Frey, J. (1982). An eYcient technique for two dimensional simulation of velocity overshot eVects in Si and GaAs devices. Compel 1, 65. Cooper, R. W., and Paxman, D. H. (1978). Measurement of charge carrier behavior in pin diodes using a laser technique. Solid State Electronics 21, 865. Dannh€ auser, F., and Krausse, J. (1973). Die r€aumliche Verteilung der Rekombination in legierten Silizium-psn-Gleichrichtern bei Belastung in Durchlaßrichtung. Solid State Electronics 16, 861. Davies, L. W. (1963). The use of pln structures in investigations of transient recombination from high injection levels in semiconductors. Proc. IEEE 51, 1637. Deboy, G. (1996). Charakterisierung von Leistungshalbleitern durch Interne Laserdeflektion. PhD thesis, Munich, Germany: Technische Universit€at Mu¨ nchen. Deboy, G., So¨ lkner, G. E., Wolfgang, E., and Claeys, W. (1996). Absolute measurement of transient carrier concentration and temperature gradients in power semiconductor devices by internal IR-laser deflection. Microelectronic Eng. 31, 299. Epperlein, P.-W. (1993). Micro-temperature measurements on semiconductor laser mirrors by reflectance modulations: A newly developed technique for laser characterization. Jpn. J. Appl. Phys. 32, 5514.
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Index
introduction to, 114–116 spectral parameters for, 125–127, 132–133 in strict sense, 117–119 temporal parameters for, 119–125 Almost-periodic component extraction operator, 109–110, 219 Almost-periodic functions, 193–194 Almost-periodically correlated, 104 AMACSs. See Asymptotically mean almost-cyclostationary signals Ambiguity functions, 204, 208 Amplitude, 54. See also Constant-amplitude mode curves, 57 large, 81 oscillation, 47, 54, 57 phase diagrams, 56 Q control and, 66 Amplitude modulation (AM), 57. See also Tapping mode -AFM, 59, 94 in air/liquids, 58–59, 60 Angular beam apertures, 284, 285 AOM. See Modulators, acousto-optic Asymptotically mean almostcyclostationary signals (AMACSs), 207–208 Atomic force microscopy (AFM), 95. See also Cantilevers; Dynamic force microscopy; Noncontact atomic force microscopy AM-, 59, 94 basic principles of, 42–47 dynamic mode, 45 measurements, 52–53 tapping-mode, 53, 56, 59–60, 61, 62, 64
A Ab initio methods, 84 ABA, 76 Absorption. See also Free carrier absorption measurements optical, 232 spectra, 231 Absorption coeYcient dependence of, 233–234 temperature dependence of, 235 Acquisition times, 68 ACS. See Almost-cyclostationary signals Action functions, 19 AFM. See Atomic force microscopy AGC. See Automatic gain control Algorithm(s). See also Average outward flux complexity of, 29–30 in literature, 260–262 skeleton, 2 summaries, 270 2/topology-preserved thinning, 29–30, 34 Aliasing, 197–198 Almost-cyclostationary signals (ACSs), 219 discrete-time, 191–194, 197 framework for, 109 input, 166–171, 172, 173, 174–176, 177 models, 179 second-order, 104 Almost-cyclostationary signals, higher-order, 104, 111–113 characterization of, 114–143 cyclic statistical estimates of, 133–135 335
336 Atomic force microscopy (AFM), (Cont.) tip-sample forces in, 47–50 in vacuum/noncontact, 68–84 Autocorrelation functions cyclic, 105, 112, 179, 180 generalized cyclic, 106–107, 111 symmetric conjugate cyclic, 205 symmetric cyclic, 204–206 symmetric generalized cyclic, 206 time-averaged, 105 two-variable cyclic, 105–106, 111 Automatic gain control (AGC), 69 Average outward flux (Algorithm 1) computations, 24–26, 34 ordered thinning, 28–29
B Bandwidth, 68 Bass, J., 108 Baud rates, 112–113 BEM. See Boundary element methods Binary-phase shift keyed signal (BPSK), 200 Binnig, G., 42–43 Blum, H., 1–2, 4–5 Boundary element methods (BEM), 261 Bounded Set, 2 BPSK. See Binary-phase shift keyed signal Brewster angle, 242, 245 Brown, W. A., 109 Brown, W. M., 108
C Cantilever(s) bending measurements, 42–44 degrees of freedom with, 54–57 detection methods for, 44 dynamic modes of, 45–47 oscillating, 54–57 rectangular/V-shaped, 43 resonance frequency of, 51–52
INDEX
Si-, 60–61 trajectory, 61–63 Capacitance-voltage measurements (C-V), 237 Carbon atoms, 77–78 Carrier(s). See also Free carriers concentrations, 233–234, 257 distributions, 238–239, 242 frequency, 112–113 lifetimes, 237, 289–290 Cauchy principal value, 231 Central field of extremals, 18–19 Central sets, 5, 9, 10–11 Circuit levels, 229 Closure, 2, 219 Collectors, 229 Compact Set, 2 Complex conjugation, 219 Constant-amplitude mode, 47, 69–70, 79 Constant-excitation mode, 47 Contact mode, 44–45, 49 Continuous-time average, 219 GACS, 114, 190–191, 194, 199–201, 202, 203 Continuum theory, MUD/BHW, 61 Coulomb forces, 90 CSCF. See Cyclic spectral cumulant function CSMF. See Cyclic spectral moment function CTCF. See Cyclic temporal cumulant function CTMF. See Cyclic temporal moment function C-V. See Capacitance-voltage measurements Cyclic autocorrelation functions. See Autocorrelation functions Cyclic spectral cumulant function (CSCF), 112 generalized, 132–133 Cyclic spectral moment function (CSMF), 219 generalized, 125–127
INDEX
Cyclic temporal cumulant function (CTCF), 112 generalized, 129–132 Nth-order, 128–132 Cyclic temporal moment function (CTMF), 111, 119, 219 generalized, 122 Nth-order, 134 of output signals, 186 reduced-dimension generalized, 124 of sampled signals, 196–197, 199–201, 202, 203
D D second-order equations, 17 Damping, 232 DD. See Drift-diVusion model Deflection(s) internal measurement techniques for laser, 243–245 large beam, 244 laser beam, 43, 45–47 signals, 243–245 signals, scaled, 244 spectroscopy photothermal, 239, 245 Deformations, 49 Degree of Freedom, 54–57 Deinococcus radiodurans, 59 Density, of states, 231 Derjaguin, Muller and Toporov model (DMT), 49, 91 DFM. See Dynamic force microscopy DFS. See Dynamic force spectroscopy L-a-dipalmitoylphophatidychline (DPPC), 45, 46 Dirac delta, 219, 261 Dirac statistics, 257 Discretization finite diVerence, 264
337 longitudinal, 264–265 in space, 258–259 in time, 258–259 Distance. See also Euclidean distance curves, 71, 72 curves v. phase, 60–61 Distribution(s) carrier, 238–239, 242 field, 254 functions, 231, 234 temperature, 249 time frequency, 201–209, 210 Wigner-Ville, 207–209, 210 DMT. See Derjaguin, Muller and Toporov model Doppler channels, 164, 175 multipath/LTV, 183–188 spread, 188 Doppler power spectrums, 158 DPPC. See L-adipalmitoylphophatidychline (DPPC) Drift-diVusion model (DD), 255 Drude theory, 230–231 Dynamic force microscopy (DFM), 95. See also Atomic force microscopy; Cantilevers in air, 57–59 analysis of, 53, 56, 59–60, 61, 62, 64 basic setup of, 69–70 feedback loop and, 65–66 introduction to, 41–42 invention of, 84 in liquids, 57–59 operational modes of, 46–47, 57 Q control and, 56, 65–68 Dynamic force spectroscopy (DFS), 84 forces/frequencies in, 85–90 Dynamic mode advantages of, 51–54 AFM, 45
338
E Eikonal equation Fermat’s principle and, 19–24 Hamiltonian derivation of, 17–24 medial sets and, 15–17 variational principles of, 17–19 Electron beam testing, 227 Electrons, 255–257 Electrothermal device simulation material properties of, 258 models, 254–255 numerical methods in, 258–259 optical probing and, 254–259 TD and, 255–257 Euclidean distance, 11, 14. See also Distance D-, 34 function, 22–24 Euler-Lagrange Equation (Theorem 4.1), 17–18 Fermat’s principle and, 20, 21 Excitation mode constant-, 47, 69–70 oscillation and, 54 self, 47
F Fabry-Perot cavity, 242 Fabry-Perot eVect, 235 free carrier absorption measurements and, 279–281 internal laser deflection measurements and, 299, 300 Fabry-Perot oscillations, 260 Fabry-Perot reflectivity measurements, 248–250 Fabry-Perot resonators, 232, 240, 248–249 Fabry-Perot thermometry, 240 Fabry-Perot transmission measurements, 250 Fermat’s principle, 244 Eikonal equation and, 19–24
INDEX
Euler-Lagrange Equation and, 20, 21 Theorem 4.3 and, 21–24 Fermi levels, 257 Fermi-Dirac statistics, 257 FFM. See Friction force microscopy FIM. See Microscope, field ion FM. See Frequency modulation Forces. See also Coulomb forces; Pauli repulsion forces; Tip-sample forces; Van der Waals forces attractive, 48 capillary, 48–49 chemical binding, 50 elastic, 49–50 electrostatic/magnetic, 50 frequencies v., 85–90 friction, 50 ionic repulsion, 49 measurement of, 42 repulsive, 48, 62–63 Fourier coeYcients, 192, 195 Fourier N-fold transform, 126–127 Fourier optics, 244 aperture holes in, 270–271 free space propagation and, 272 image formation in, 270–272 thin lenses in, 271–272 Fourier series, 106, 279 Fourier transformations, 112, 132, 149, 268–269 Fraction-of-time (FOT), 104, 113. See also Linear time-variant; Nonstochastic approach deterministic, 110 deterministic linear systems, 148–151 deterministic/random linear systems, 146–190 deterministic/random systems, 146–148 history of, 108–109 parameters, 108 random, 110, 144
339
INDEX
random/LTV systems, 151–155 for signal analysis, 107–111 Free carrier absorption measurements, 241–242 angular beam apertures and, 284, 285 Fabry-Perot eVect and, 279–281 laser deflection internal and, 298 longitudinal averaging/sampling and, 277–279, 280 optical field distribution as, 276–277, 278 optical setup and, 283–286 sample alignment and, 285–286 spatial resolution in, 281–283 summary of, 290–291 surface recombination and, 286–289, 290, 291 virtual optical experiment and, 276–291 Free carriers, 229, 240. See also Plasma-optical eVects Frequency(ies), 55. See also resonance frequency carrier, 112–113 cycle, 105, 212–213 domains, 119, 135, 161–162 driving, 66 eigen, 55, 56 forces v., 85–90 lag-dependent cycle, 106, 121 LTV/output cycle, 165–176, 177 mapping functions, 149 Nth order (cumulant) lag-dependent cycle, 128 reduced-dimension (moment) lag-dependent cycle, 124 Frequency modulation (FM), 68–69 Frequency shift(s), 52, 70 calculations for, 78–82 experimental, 71–72 NC-AFM and, 70–72, 78–82 normalized, 81–82 origin of, 70–72
Fresnel formula, 269 Friction force microscopy (FFM), 45 Furstenberg, H., 108
G GACSs. See Generalized almost-cyclostationary signals Galerkin method, 261 Gardner, William A., 109 Gauss theorem, 259 Gaussian beam, 281–282 Gaussian wave, 277 GCSCFs. See Generalized cyclic spectral cumulant functions GCSMFs. See Generalized cyclic spectral moment functions GCTCFs. See Generalized cyclic temporal cumulant functions GCTMFs. See Generalized cyclic temporal moment functions Generalized almost-cyclostationary signals (GACSs), 103, 219. See also Linear time-variant acronyms/notations, 219 ambiguity functions of, 208 characterization of, 114–143 chirp signal and, 135–138 continuous-time, 114, 190–191, 194, 199–201, 202, 203 cycle frequencies and, 212–213 cyclic cumulants and, 127–133 equation proofs, 213–216 framework for, 109 input, 166–171, 172, 173, 174 introduction to, 104–114 LTV of, 113, 143–190 mean-square sense of time averages and, 211–212 nonuniformly sampled signal and, 138–140, 141, 142 processes, 105 result proofs, 216–218 sampling of, 190–201, 202, 203 second-order, 104, 204–206
340 Generalized almost-cyclostationary signals (GACSs) (Cont.) signal feature extractions and, 208–209, 210 signals, 135–140, 176–181 spectral parameters in, 125–127, 132–133 in strict sense, 116–119 temporal parameters in, 119–125, 127–132 time frequency of, 201–209, 210 time series, 114–115, 120–121 in wide sense, 119 Generalized cyclic autocorrelation functions. See Autocorrelation functions Generalized cyclic polyspectrum, 132 Generalized cyclic spectral cumulant functions (GCSCFs), 132–133 Generalized cyclic spectral moment functions (GCSMFs), 125–127 Generalized cyclic temporal cumulant functions (GCTCFs), 129, 132, 219 reduced-dimension, 130–131 theoretical, 134 Generalized cyclic temporal moment functions (GCTMFs), 122, 135, 219 Generalized multipole techniques, 261 Generic Contact, 8–11 Geometrical Optics in Isotropic Media. See Theorem 4.3 Geometrical ray devices, 227 Gerber, C., 42–43 Graphite, 77. See also Highly oriented pyrolithic graphite Grass fire concept, 4–5, 13–15. See also Skeletons
H Hamaker constant, 48 Hamiltonian canonical variables, 17, 20 Hamilton-Jacobi skeletons, 15–26
INDEX
Hamilton’s equations (Theorem 4.2), 292 comparisons to, 19 Eikonal equation and, 17–24 theorem, 18–19 HD. See Hydrodynamic model Heat flow equation, 257 Hertz approach, 49 Hexagonally packed intermediate (HPI), 59 Highly oriented pyrolithic graphite (HPOG) imaging, 76–78 substrates, 75–76 Hilbert space, 108 Hofstetter, E.M., 108 Holes, 255–256 Homeomorphism, 6–7 Homotopic Maps, 6 Homotopy, 6 comparisons, 7 preserving medial sets, 26–31 3D simple points, 27–28 2D simple points, 26–27 types, 6 Hooke’s law, 54 HPI. See Hexagonally packed intermediate HPOG. See Highly oriented pyrolithic graphite Hydrodynamic model (HD), 255 Hysteresis, 62, 63
I IGBTs. See Insulated bipolar transistors Impulse-repulse functions, 163–165 Impulse-response functions, 148, 150 lag product of, 156–157 system, 181 Inhomogenous media propagation direction’s boundary conditions and, 266–269
341
INDEX
propagator matrix and, 262–266 wave propagation and, 260–270, 271 Insulated bipolar transistors (IGBTs), 229, 243 samples, 279, 280 SOI, 249 Integral function (I1), 82 Intercorrelation functions, 157 Interferometer systems, 44 Interferometric techniques backside laser probing and, 311–320 diVerential backside laser probing and, 320–323 Mach-Zehnder interferometry and, 308–311 virtual optical experiments and, 308–324 Interferometry, 235 Mach-Zehnder, 250, 251, 308–311 Interior Point, 2 Intermittent contact regime, 63 Inverse-power laws, 80–81 Isometric isomorphism (Wold isomorphism), 108, 109 I-V. See Pulsed current-voltage measurements
J JKR. See Johnson, Kendall and Roberts model Johnson, Kendall and Roberts model (JKR), 49 Joule heats of electrons, 257 Jump-to-contacts, 53–54, 79
K Kac, M., 108 Kernel functions, 206–207 Kramers-Kronig relation, 231, 232 Kronecker delta, 107, 219
L Lagrangian methods, 17–18, 19 Langmuir-Blodgett film, 67–68 LAPTV. See Linear almostperiodically time-variant Laser(s) beam, continuous wave, 241 beam deflection, 43, 45–47 HeNe, 242 InGaAs, 242 probing, backside, 245–248 probing, diVerential backside, 248 probing, internal measurement techniques, 227, 229, 240–250, 251 reflectance thermometry, 239 Laser deflection measurements, internal, 243–245 detector responses and, 299–301 detector signals and, 295–296 experimental parameters for, 296 Fabry-Perot eVect and, 299, 300 free carrier absorption and, 298 image formation conditions for, 301–302 imaging lens’ projection in, 293, 294–295 of large temperature gradients, 302–307 low power dissipation and, 296–297 signals, 292–296 thin lens’ image formation in, 294 Laser devices optical output power of, 227 probing internal, 227, 229, 240–250, 251 Lattice temperature, 245 Lattice vibrations (thermo-optical eVect), 229 Legendre transformation, 18 Lennard-Jones model, 59, 82–83 Le´ skow, J., 108 Limit Point, 2
342 Linear almost-periodically timevariant (LAPTV), 144, 219 filtering, 176–181 Linear time invariant (LTI), 219 Linear time-variant (LTV), 219 analysis of, 166–171, 172, 173 filtering, 144 FOT deterministic systems, 148–151, 171–176, 177 FOT deterministic/random linear systems and, 146–190 FOT deterministic/random systems and, 146–148 frequency domain/higher-order system characterization in, 161–162 of GACS, 113 introduction to, 143–146 multipath Doppler channels and, 183–188 output cycle frequencies and, 165–176, 177 output signal in, 162–165 product modulation and, 181–183 stochastic, 143–144 systems, 144–146 systems, FOT random, 151–155 time domain/higher-order system characterization in, 155–161 transformations/GACS, 143–190 LTI. See Linear time invariant LTV. See Linear time-variant
M Mach-Zehnder interferometry, 250, 251 techniques, 308–311 Maximal Inscribed Ball, 2, 4 Maxwell’s equations, 261 Mean-square continuous process, 105 sense of time averages, 211–212 Measurement technique(s) backside laser probing, 245–248 for C-V, 237
INDEX
development of, 238–239 device characterization methods as, 237–240 Fabry-Perot reflectivity, 248–250 Fabry-Perot transmission, 250 internal laser deflection, 243–245 internal laser probing, 240–250, 251 for I-V, 237 Mach-Zehnder interferometry and, 250, 251 OCVD as, 237 in virtual optical experiments, 233–251 Medial sets, 4–5 Eikonal equation and, 15–17 homotopy-preserving, 26–31 labeling, 30–31 3D, 30–31, 34–36 2D, 26–27, 32–33 Method of moments (MOM), 261 Microelectromechanical systems (MEMS), 228 Microelectronics, 226 Microscope, field ion (FIM), 95 Microscope, scanning tunnel (STM), 42 Microscopes at constant-amplitude mode, 79 UHV and, 73–75 Mirage eVects, internal/external, 239 Mo¨ bius strips, 7 Modified evaluation rule, 305–307 Modulators, acousto-optic (AOM), 246 Modulators, chirp, 151 MOM. See Method of moments Momentum, 17 MOS. See Semiconductors, metal-oxide MOSFET, 229, 235
N Nanotechnology, 42 Napolitano, A., 108
343
INDEX
NC-AFM. See Noncontact atomic force microscopy Newton’s first law, 54 Noncontact atomic force microscopy (NC-AFM) energy dissipation’s measurements and, 91–95 experimental applications of, 72–78 FM detection scheme, 68–79 frequency shift and, 70–72, 78–82 frequency shift’s calculation and, 78–82 integral function values and, 82 simulation of, 83–84 tip-sample interaction forces in, 90–91 UHV and, 73–75 Nonstochastic approach, 104, 113. See also Fraction-of-time history of, 108–109 parameters, 108 point on, 8 for signal analysis, 107–111 Nth-order wide sense cyclostationarity, 119
O Object modeling, 2 Object, 4 OCVD. See Open-circuit-voltagedecay measurements One-dimensional models (1D), 227 Open Ball, 2 Open Set, 2 Open-circuit-voltage-decay measurements (OCVD), 237 Optical absorption, 232 Optical beam testing, 227, 240 Optimization strategy fundamental principle of, 252, 274–275 general assumptions in, 275–276 Oscillation(s), 310 excitation mode and, 54
Fabry-Perot, 260 parameters, 60–61 Oscillator(s) harmonic, 54–57 resonant frequency of, 53, 56, 61–62 simple harmonic, 51
P Pauli repulsion forces, 48 Peltier-Thomson heat, 257 Phase(s) diagrams/amplitude, 56 distance curves v., 60–61 Q control and, 66 shift, 247, 248, 310, 314–320 Phase-locked loop (PLL), 70 Phonons, 234, 255 Plasma-optical eVects, 240. See also Free carriers absorption spectra and, 231 comparisons, 233–234 Drude theory and, 230–231 experimental results of, 232 Kramers-Kronig relation and, 231, 232 PLL. See Phase-locked loop Poisson’s equation, 214, 257 Probing backside laser, 245–248, 311–320 beam, 242, 276–277, 278 diVerential backside laser, 248, 320–323 internal laser, 227, 229, 240–250, 251 semiconductor samples, 227 Probing, optical, 239 detector responses with, 272–273 electrothermal device simulation and, 254–259 Fourier optics in, 270–272
344 Probing, optical (Cont.) internal, 228 simulation steps in, 253–254 techniques, 251–274 Propagator matrix basic assumptions in, 262–263 computational variables/equations and, 263–266 inhomogenous media and, 262–266 Property 2.1 (Invariance under Translation and Rotation), 5 Property 2.2 (Reversibility), 6 Property 2.3 (Thickness), 6 Proposition 5.1, 27 Proposition 5.2 (2D end point), 28–29 Proposition 5.3 (3D end point), 29 Pulsed current-voltage measurements (I-V), 237
Q Q control, 56, 65–68 amplitude and, 66 damping and, 68 factors, 68 phase and, 66 Quate, C.F., 42–43
R R. See Recombination rate Ray equation, 21 RD-GCTCFs. See Reduced-dimension generalized cyclic temporal cumulant functions RD-GCTMFs. See Reduceddimension generalized cyclic temporal moment functions Recombination rate (R), 258 Reduced-dimension generalized cyclic temporal cumulant functions (RD-GCTCFs), 130–131, 219
INDEX
Reduced-dimension generalized cyclic temporal moment functions (RD-GCTMFs), 124, 219 Reflection(s), 290 beam, 261–262 coeYcients, 242, 249 multiple, 276 Refractive index absorption spectra and, 231 complex, 234, 254, 259 gradients of, 239, 244–245, 293 internal gradients of, 242 Kramers-Kronig relation and, 231, 232 modulations, 229–236, 259–260 plasma-optical eVect and, 230–234 real part of, 229, 232 of semiconductor, 239–240 temperature dependence of, 235–236, 239–240 Resonance frequency(ies) of cantilevers, 51–52 free, 64 of oscillators, 53, 56, 61–62 shifts, 62, 63 Resonators, Fabry-Perot, 232, 240, 248–249
S Sample alignment, 285–286 Scanning probe microscopy, 227 Scattering functions, 158 SCFs. See Spectral cumulant functions Self-excitations mode, 47 Semiconductor(s) devices, 311 physics, 226, 229 refractive index of, 239–240 samples, 227 Semiconductors, metal-oxide (MOS), 303 Sheets, simple, 31
INDEX
Signal(s). See also specific types analysis for nonstochastic approach, 107–111 analysis/FOT, 107–111 CTMF of output, 186 CTMF of sampled, 196–197, 199–201, 202, 203 deflection, 243–245 detector, 275, 295–296 LTV/output, 162–165 measurement, 314–320 phase shift, 247, 248, 310, 314–320 processing, 246–247 scaled deflection, 244 Silicon air interface, 269 nitride, 43 Simulators DESSISISE, 253, 255 general-purpose device, 253–254, 255 Skeleton(s), 4, 36. See also Central sets; Grass fire concept; Medial sets; Three-dimensional models; Two-dimensional models algorithms for, 2 average outward flux and, 24–26 computation techniques for, 13–15 cut locus and, 2 definitions of, 2–5 global properties/structure of, 5–7 Hamilton-Jacobi, 15–26 local structure of, 7–13 mechanics/optics and, 15–26 nonterminal point in, 14 peeling process in, 14 properties of, 2–13 simple point in, 14
345 thinning in, 14 Skeleton Transform, 4–5 SMFs. See Spectral moment functions Space charge regions, 290 discretization in, 258–259 Hilbert, 108 kX–Z-, 266–268 modulations, 254 real, 267, 268–269 Spaced-frequency spaced-time correlation function, 158 Spatial resolution, 281–283 Spectral cumulant functions (SCFs), 132–133 Spectral moment functions (SMFs), 219 Nth order, 125–126 system, 161 Spectroscopy, 41. See also specific modes curves, 61 photothermal deflection, 239, 245 3D force, 88–90 Sphere(s), 8 3D and, 7 of curvature, 5, 9 at ridge point, 9, 10 at turning point, 9, 10 at umbilic point, 9, 10 Spooner, C.M., 109 Static mode, 44 Steinhaus, H., 108 STM. See Microscope, scanning tunnel Stochastic processes, 162–163, 188 Stochastic systems, 144, 162–165 Surface(s), 8. See also Spheres recombination, 286–289, 290, 291 simple, 31
346
T Tapping mode. See also Amplitude modulation AFM, 53, 56, 59–60, 61, 62, 64 in air/liquids, 58–59, 60 of Langmuir-Blodgett film, 67–68 Taylor expansion, 259, 269, 280 TD. See Thermodynamic model Temperature distributions, 249 gradients, large, 302–307 lattice, 245 profiles, 240–241, 249–250 Temperature dependence of absorption coeYcient, 235 of refractive index, 235–236, 239–240 Temporal cumulant functions (TCFs), 219 Nth-order, 127–128 Temporal moment functions (TMFs), 119, 219 discrete, 197 system, 156 valid, 193–194 Theorem 4.1. See Euler-Lagrange Equation Theorem 4.2. See Hamilton’s equations Theorem 4.3 (Geometrical Optics in Isotropic Media), 21–24 Theorem 5.1 (Malandain et al., 1993), 28, 30 Thermodynamic model (TD) state variables in, 255–256 subsystems, 255 Thermometry Fabry-Perot, 240 laser reflectance, 239 Thermo-optical eVect, 229, 234. See also Lattice vibrations Thinning average outward flux ordered, 28–29
INDEX
in skeletons, 14 topology-preserved, 29–30 3D end point. See Proposition 5.3 Three-dimensional models (3D), 1–2. See also Definition 2.16; Definition 2.17; Proposition 5.3 analysis/manipulation of, 2 central sets, 9, 10–11 force spectroscopy, 89–90 medial sets, 30–31 simple points, 27–28 skeletal points’ classification in, 8–13 spheres and, 7 structure of, 6 Time, 119 -dependent modulations, 254 discretization in, 258–259 domain/LTV system characterization, 155–161 series/GACS, 114–115, 120–121 Time frequency distributions, 201–209 Time-selective fading channels, 188 Tip-sample force(s) in AFM, 47–50 in NC-AFM, 90–91 relationships, 60–61 spring, 51 TMFs. See Temporal moment functions Topology, concept of, 3 Transistors. See also Insulated bipolar transistors bipolar pnp, 229 internal pnp, 229 2D end point. See Proposition 5.2 Two-dimensional models (2D), 1. See also Proposition 5.2 analysis/manipulation of, 2 branchpoint in, 7 classes, 7 endpoint of curve in, 7 first-order equations, 17 interior curve point in, 7
347
INDEX
medial sets, 26–27, 32–33 simple points, 26–27 skeletal points’ classification in, 13 structure of, 5
U UHV. See Ultrahigh vacuum Ultrahigh vacuum (UHV), 68, 71–72 microscopes and, 73–75 NC-AFM and, 73–75 Unit step function, 219 University of Hamburg, 73 Urbanik, K., 108
V Van der Waals forces causes of, 48 interactions of, 49, 90 weak, 76 Vector transposition (VT), 219 Virtual optical experiment(s). See also Probing, optical absorption spectra and, 231 backside laser probing as, 245–248 conclusions, 324–326 for C-V, 237 development of, 238–239 device characterization methods in, 237–240 Drude theory and, 230–231 Fabry-Perot reflectivity as, 248–250 Fabry-Perot transmission as, 250 free carrier absorption measurements and, 276–291 interferometric techniques and, 308–324 internal laser deflection and, 243–245, 291–308
internal laser probing as, 240–250, 251 introduction to, 226–229 for I-V, 237 Kramers-Kronig relation and, 231, 232 Mach-Zehnder interferometry and, 250, 251 numerical analysis advantages of, 228 OCVD as, 237 optimization strategy and, 252, 274–276 refractive index’s modulation in, 229–236, 259–260 Virtual Reality Modeling Language (VRML), 35 Voronoi graphs, 14 VRML. See Virtual Reality Modeling Language VT. See Vector transposition
W Wave(s) continuous, 241 emerging, 254 Gaussian, 277 guide model, 262 infrared electromagnetic, 229 lengths, 236 propagation, 260–270, 271 reflected, 270 standing, 276 transmitted, 270 Wiener, Norbert, 108 Wigner-Ville distributions, 207–209, 210 Wold isomorphism. See Isometric isomorphism
Z Zadeh’s bifrequency functions, 148
