Differential Evolution in Electromagnetics (Adaptation, Learning, and Optimization, 4)

Anyong Qing and Ching Kwang Lee Differential Evolution in Electromagnetics Adaptation, Learning, and Optimization, Vo...

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Anyong Qing and Ching Kwang Lee Differential Evolution in Electromagnetics

Adaptation, Learning, and Optimization, Volume 4 Series Editor-in-Chief Meng-Hiot Lim Nanyang Technological University, Singapore E-mail: [email protected] Yew-Soon Ong Nanyang Technological University, Singapore E-mail: [email protected] Further volumes of this series can be found on our homepage: springer.com Vol. 1. Jingqiao Zhang and Arthur C. Sanderson Adaptive Differential Evolution, 2009 ISBN 978-3-642-01526-7 Vol. 2. Yoel Tenne and Chi-Keong Goh (Eds.) Computational Intelligence in Expensive Optimization Problems, 2010 ISBN 978-3-642-10700-9 Vol. 3. Ying-ping Chen (Ed.) Exploitation of Linkage Learning in Evolutionary Algorithms, 2010 ISBN 978-3-642-12833-2 Vol. 4. Anyong Qing and Ching Kwang Lee Differential Evolution in Electromagnetics, 2010 ISBN 978-3-642-12868-4

Anyong Qing and Ching Kwang Lee

Differential Evolution in Electromagnetics

123

Dr. Anyong Qing Temasek Laboratories National University of Singapore 5 Sports Dr 2 Singapore 117508 E-mail: [email protected]

Dr. Ching Kwang Lee School of Electrical and Electronic Engineering Nanyang Technological University Division of Communication Engineering S1-B1a-10, Nanyang Avenue Singapore 639798 E-mail: [email protected]

ISBN 978-3-642-12868-4

e-ISBN 978-3-642-12869-1

DOI 10.1007/978-3-642-12869-1 Adaptation, Learning, and Optimization

ISSN 1867-4534

Library of Congress Control Number: 2010926029 c 2010 Springer-Verlag Berlin Heidelberg

This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, speciﬁcally the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microﬁlm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a speciﬁc statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typeset & Cover Design: Scientiﬁc Publishing Services Pvt. Ltd., Chennai, India. Printed on acid-free paper 987654321 springer.com

Preface

1 Motivations After many years of development and applications, differential evolution has proven itself a very simple while very powerful stochastic global optimizer. Since its inception, it has been applied to solve problems in many scientific and engineering fields. Nowadays, our daily life relies heavily on electromagnetics. Differential evolution has played an essential role in many synthesis and design problems in electromagnetics. This book focuses on applications of differential evolution in electromagnetics to showcase the achievement of differential evolution and further boost its acceptance in electromagnetics community.

2 Layout This book is composed of two parts. Part one includes the first three chapters while the remaining five chapters belong to part two of this book. 2.1

Part One

This part focuses on a literature survey on differential evolution. As far as we know, it is by far the most extensive and exhaustive one. 2.1.1 Chapter 1 Chapter 1 gives details of the literature survey which covers publication collection, refining and analysis. It opens up with the purposes this literature survey aims to serve. Next, Platforms over which the literature survey is actually conducted are then discussed. Initial statistical results over these platforms are presented. After that, the refining process to remove irrelevant publications is discussed. Yearly outputs of formal publications with and without refining are presented. Result analysis, or publication classification, is then discussed. Topics according to which collected publications are clustered are suggested. In particular, theoretical studies on differential evolution are summarized. Finally, some future actions are discussed. We have noticed several misconceptions and misconducts on differential evolution through this literature survey. They are clearly pointed out at the end of this chapter.

VI

Preface

2.1.2 Chapter 2 Basics of differential evolution are presented in Chapter 2. It also serves the second part of this book so that repetition of description of differential evolution is avoided. A short history of differential evolution is first discussed. It covers its inception, early years until 1998 and years from 1998 onwards. Major events in early years and key milestones in years from 1998 onwards are highlighted. The basic framework of differential evolution is then explained by revisiting the originators’ inventory publication, followed by a description of the more generic classic differential evolution. Some prominent variants of the two fundamental evolutionary operations in differential evolution are presented. Next, dynamic differential evolution which was misunderstood and seriously underestimated before is briefly mentioned. Finally, essential features of differential evolution including both advantages and disadvantages are highlighted. It has to be pointed out that a state of the art review of differential evolution is not presented in this book due to tight time limit. Such a review will be part of a forming up encyclopedia of differential evolution. 2.1.3 Chapter 3 A retrospection of applications of differential evolution in electromagnetics in and before 2008 is presented in this Chapter. The coverage of the retrospection is clearly specified right at the beginning of this Chapter. The pioneering works of applications of differential evolution in electromagnetics are highlighted. Statistical results by both publication year and subject are presented. Detailed discussion of applications of differential evolution in specific subject is then given. Involved subjects include electromagnetic inverse problems, antenna arrays, microwave & RF engineering, antennas, electromagnetic structures, electromagnetic composite materials, frequency planning, radio network design, MIMO, radar, computational electromagnetics and electromagnetic compatibility. An outlook of applications of differential evolution in electromagnetics is also presented at the end of this Chapter. 2.2

Part Two

This part presents five new applications of differential evolution in differential evolution by different research groups. 2.2.1 Chapter 4 Reconstruction of two-dimensional dielectric cylinders by using differential evolution is presented in this Chapter. The efficiency of differential evolution has been numerically shown through various examples. In addition, the impact of initial guess on differential evolution is presented. The multiple signal classification is used to determine the number of cylinders, their approximate centers and approximate geometric dimensions while a least squares based method is used to generate an estimate of the permittivity of the cylinders. It has been shown that a proper choice of the initial guess can speed up the convergence of the optimization significantly.

Preface

VII

2.2.2 Chapter 5 Inspection of penetrable objects by using differential evolution together with a recently proposed iterative multiscaling approach is discussed in this Chapter. The solving procedure starts from a fixed test area and successively focuses on one or more "regions of interest" in order to determine the approximate shapes of the unknown objects. At each step of the minimization process, differential evolution is used to retrieve this support by minimizing a proper functional, which relates the measured scattered field data to the data numerically produced, at any iteration, by the current solution. Several new results are included concerning the reconstruction of inhomogeneous targets under various imaging conditions. The combined strategy has been proved to be quite effective in reconstructing complex dielectric cylinders such as hollow and E-shape cylinders in noisy environment. 2.2.3 Chapter 6 In this Chapter a flexible method for prediction of far-field radiated emissions is presented. It is a promising computational alternative to the expensive large semianechoic chambers necessary to perform electromagnetic compatibility far-field radiated emission measurements. In this method, the equipment under test is replaced by an equivalent set of infinitesimal dipoles (both electric and magnetic) distributed inside the volume occupied by the equipment under test which is determined from near-field measurements at a short distance of the equipment under test. A memetic metaheuristic technique combing genetic algorithms, differential evolution and downhill simplex method is used to determine the type, position, orientation and excitation current of each dipole of the equivalent set of dipoles. The information obtained from the equivalent dipole set is used to determine the radiation at the far-field, as well as to identify the radiating parts of the equipment. 2.2.4 Chapter 7 Differential evolution with Pareto tournaments (DEPT) was applied to address the multi-objective optimization of frequency assignment problem in two real-world GSM networks in this Chapter. Two performance indicators, hypervolume and coverage relation, are implemented to analyze results. Results are compared with those by other multi-objective metaheuristics. Final results show that fine-tuned DEPT outperforms both MO-VNS and MO-SVNS while performs worse than both GMO-SVNS and GMO-VNS, among which GMO-SVNS performs best. 2.2.5 Chapter 8 In this Chapter, differential evolution is combined with particle swarm optimization (PSO) and another evolutionary algorithm (EA) to create a novel hybrid algorithm, the PSO-EA-DEPSO. The alteration between PSO, PSO-EA, and DEPSO provides additional diversity to counteract premature convergence. This hybrid algorithm is then shown to outperform PSO, PSO-EA, and DEPSO when applied to wireless MIMO channel prediction.

VIII

Preface

3 Readership As its name indicates, this book is specially prepared for electromagnetic researchers facing optimization problems. It will be particularly attractive to researchers who have been frustrated by other optimization algorithms. This book is a premium resource for differential evolution community. People in this community will have a better understanding on differential evolution and its huge application potential. This book is also an ideal resource for evolutionary computation community. People in this community may find it helpful in presenting a more appropriate approach to conduct concerned literature survey and providing real engineering application examples.

Acknowledgement

First of all, I would take this opportunity to thank Prof. Hock Lim and Mr. Joseph Sing Kwong Ting, directors of Temasek Laboratories, National University of Singapore, for their support and encouragement of my study on differential evolution. The financial funding from Defence Science & Technology Agency, Singapore is greatly appreciated too. I would also like to thank Prof. Meng-Hiot Lim, series editor on evolutionary learning and optimization, and Dr. Thomas Ditzinger, senior editor within Springer Verlag responsible for this series, for their strong recommendation to publish this book. Thanks also go to Mr. Heather King for his carefulness and patience.

Contents

Contents 1

A Literature Survey on Differential Evolution…………………………...1 1.1 Motivations .............................................................................................1 1.1.1 Eliminating Inconsistencies .........................................................1 1.1.2 Crediting Original Contributions .................................................1 1.1.3 Knowing the State of the Art .......................................................1 1.1.4 Gaining Insight ............................................................................2 1.2 Platforms.................................................................................................2 1.2.1 Starting Point ...............................................................................2 1.2.2 Databases .....................................................................................3 1.2.3 Informal Online Resources and Tools .........................................4 1.3 Result Refining .......................................................................................5 1.3.1 Books ...........................................................................................5 1.3.2 Book Chapters .............................................................................6 1.3.3 Other Formal Publications ...........................................................6 1.3.4 Informal Notes .............................................................................7 1.4 Result Analysis .......................................................................................7 1.4.1 Theory of Differential Evolution .................................................7 1.4.2 Fundamentals of Differential Evolution ......................................8 1.4.3 Intrinsic Control Parameters ........................................................9 1.4.4 Evaluation of Differential Evolution............................................9 1.4.5 Applications of Differential Evolution ........................................9 1.4.6 Hybridization ...............................................................................9 1.5 Future Actions ......................................................................................10 1.5.1 Open Access ..............................................................................10 1.5.2 Future Update ............................................................................10 1.6 Misconceptions and Misconducts on Differential Evolution................10 References .............................................................................................................10

2

Basics of Differential Evolution…………………………………………..19 2.1 A Short History.....................................................................................19 2.1.1 Inception ....................................................................................19 2.1.2 Early Years ................................................................................20 2.1.2.1 Assessment..................................................................20 2.1.2.2 Reputation Building ....................................................20

XII

Contents

2.2

2.3

2.4 2.5 2.6

2.1.2.3 Applications ................................................................20 2.1.2.4 Promotion....................................................................21 2.1.2.5 Practical Advice ..........................................................21 2.1.2.6 Standardization ...........................................................22 2.1.2.7 More Adventures ........................................................22 2.1.3 Key Milestones in and after 1998 ..............................................22 The Foundational Differential Evolution Strategies .............................23 2.2.1 Notations....................................................................................23 2.2.2 Strategy Framework...................................................................24 2.2.2.1 Pseudo-code ................................................................24 2.2.2.2 Initialization ................................................................25 2.2.2.3 Differential Mutation ..................................................25 2.2.2.4 Crossover ....................................................................26 2.2.2.5 Selection......................................................................27 2.2.2.6 Termination Conditions ..............................................27 2.2.3 Intrinsic Control Parameters ......................................................27 Classic Differential Evolution ..............................................................28 2.3.1 Initialization ...............................................................................28 2.3.2 Differential Mutation .................................................................28 2.3.2.1 Current ........................................................................29 2.3.2.2 Best .............................................................................29 2.3.2.3 Better...........................................................................29 2.3.2.4 Random.......................................................................29 2.3.2.5 Mean ...........................................................................29 2.3.2.6 Best of Random ..........................................................29 2.3.2.7 Arithmetic Best ...........................................................29 2.3.2.8 Arithmetic Better ........................................................29 2.3.2.9 Arithmetic Random.....................................................29 2.3.2.10 Trigonometric .............................................................30 2.3.2.11 Directed.......................................................................30 2.3.3 Crossover ...................................................................................30 2.3.3.1 Binary Crossover ........................................................31 2.3.3.2 One-Point Crossover ...................................................32 2.3.3.3 Multi-point Crossover .................................................32 2.3.3.4 Arithmetic Crossover ..................................................32 2.3.3.5 Arithmetic One-Point Crossover.................................33 2.3.3.6 Arithmetic Multi-point Crossover...............................33 2.3.3.7 Arithmetic Binomial Crossover ..................................34 2.3.3.8 Arithmetic Exponential Crossover..............................35 Dynamic Differential Evolution ...........................................................36 State of the Art of Differential Evolution .............................................36 Essential Features of Differential Evolution.........................................37 2.6.1 Advantages ................................................................................37 2.6.1.1 Reliability....................................................................37 2.6.1.2 Efficiency....................................................................37

Contents

XIII

2.6.1.3 Simplicity....................................................................37 2.6.1.4 Robustness ..................................................................38 2.6.2 Disadvantages ............................................................................38 2.6.2.1 Efficiency....................................................................38 2.6.2.2 Incapability for Epistatic and Noisy Problems............38 References .............................................................................................................38 3

A Retrospective of Differential Evolution in Electromagnetics……….43 3.1 Introduction ..........................................................................................43 3.1.1 Coverage ....................................................................................43 3.1.2 Pioneering Works ......................................................................44 3.1.3 An Overview of Applications of Differential Evolution in Electromagnetics........................................................................44 3.1.3.1 Yearly Output .............................................................44 3.1.3.2 Output by Subject .......................................................44 3.2 Electromagnetic Inverse Problems .......................................................45 3.2.1 A Bird’s Eye View.....................................................................45 3.2.2 Further Classification.................................................................45 3.2.2.1 One-Dimensional Electromagnetic Inverse Problems .....................................................................46 3.2.2.2 Two-Dimensional Electromagnetic Inverse Problems .....................................................................46 3.2.2.3 Three-Dimensional Electromagnetic Inverse Problems .....................................................................47 3.3 Antenna Arrays.....................................................................................48 3.3.1 Conventional Antenna Arrays....................................................48 3.3.1.1 Ideal Antenna Arrays ..................................................48 3.3.1.2 Practical Antenna Arrays ............................................48 3.3.1.3 Phased Arrays .............................................................49 3.3.2 Time-Modulated Antenna Arrays ..............................................49 3.3.2.1 Ideal Antenna Arrays with Time Modulation .............49 3.3.2.2 Practical Antenna Arrays with Time Modulation .......49 3.3.2.3 Phased Antenna Arrays with Time Modulation..........50 3.3.3 Moving Phase Center Antenna Arrays.......................................50 3.4 Microwave and RF Engineering ...........................................................50 3.4.1 Design of Microwave and RF Devices ......................................50 3.4.1.1 Designing Microwave and RF Devices Using Differential Evolution .................................................51 3.4.1.2 Extracting Empirical Synthesis Formulas Using Differential Evolution .................................................51 3.4.2 Characterization of Microwave and RF Devices .......................51 3.4.2.1 Calibration of Measuring System for Characterizing Microwave and RF Devices ........................................51 3.4.2.2 Modeling of Microwave and RF Devices ...................52 3.5 Antennas ...............................................................................................52 3.5.1 Design of Antennas....................................................................52

XIV

Contents

3.5.1.1 Designing Antennas Using Differential Evolution .....52 3.5.1.2 Extracting Empirical Formulas for Synthesizing Antennas .....................................................................52 3.5.2 Measurement of Antennas .........................................................53 3.6 Electromagnetic Structures ...................................................................53 3.6.1 Plain Electromagnetic Structures ...............................................54 3.6.2 Frequency Selective Surfaces ....................................................55 3.7 Electromagnetic Composite Materials ..................................................56 3.7.1 Modeling of Electromagnetic Composite Materials ..................56 3.7.2 Retrieval of Effective Permittivity Tensor.................................56 3.8 Frequency Planning ..............................................................................57 3.9 Radio Network Design..........................................................................58 3.10 MIMO...................................................................................................58 3.11 Radar.....................................................................................................59 3.12 Computational Electromagnetics ..........................................................59 3.13 Electromagnetic Compatibility .............................................................60 3.14 Miscellaneous Applications ..................................................................60 3.15 An Outlook to Future Applications of Differentia Evolution in Electromagnetics...................................................................................60 References .............................................................................................................61

4

Application of Differential Evolution to a Two-Dimensional Inverse Scattering Problem …………………………......………………………...73 4.1 Introduction ..........................................................................................73 4.2 General Description of the Problem .....................................................74 4.2.1 Experimental Setup....................................................................74 4.2.2 The Optimization Problem.........................................................76 4.3 Mathematical Nature of the Optimization Problem and Differential Evolution ..............................................................................................76 4.4 Initial Guess ..........................................................................................77 4.4.1 Foldy-Lax Model of Scattering..................................................78 4.4.2 Multiple Signal Classification for Estimating the Scatterer Support.......................................................................................79 4.4.3 Least Square Based Method for Generating Initial Guess for the Relative Permittivity ..................................................................80 4.5 Numerical Results.................................................................................81 4.5.1 Measurement Setup....................................................................81 4.5.2 Control Parameters ....................................................................81 4.5.3 Numerical Example 1: A Single Cylinder .................................82 4.5.4 Numerical Example 2: Two Identical Cylinders........................87 4.5.5 Numerical Example 3: Two Different Cylinders .......................90 4.5.6 Numerical Example 4: Two Closely Located Identical Cylinders....................................................................................94 4.5.7 Numerical Example 5: Kite Cross-Section Cylinder .................97

Contents

XV

4.6 Conclusions ........................................................................................101 References ...........................................................................................................102 5

The Use of Differential Evolution for the Solution of Electromagnetic Inverse Scattering Problems…………………………………………….107 5.1 Introduction ........................................................................................107 5.2 Problem Formulation ..........................................................................108 5.2.1 The Inverse Scattering Formulation.........................................108 5.2.2 Discrete Setting........................................................................109 5.2.3 The Inverse Scattering Problem as an Optimization Problem....................................................................................110 5.3 The Iterative Multiscaling Approach ..................................................110 5.4 Numerical Results...............................................................................112 5.4.1 Off-Centered Dielectric Cylinder ............................................112 5.4.2 Off-Centered Dielectric Hollow Cylinder................................117 5.4.3 Centered Stratified Dielectric Square Cylinder........................121 5.4.4 Centered E-Shape Dielectric Cylinder .....................................126 5.5 Conclusions ........................................................................................129 References ...........................................................................................................129

6

Modeling of Electrically Large Equipment with Distributed Dipoles Using Metaheuristic Methods …………………………………………..133 6.1 Introduction ........................................................................................133 6.1.1 Near-Field to Far-Field Transformation ..................................133 6.1.2 Radiating Equipment Modeling with Prefixed Position Dipoles.....................................................................................134 6.1.3 Present Work ...........................................................................135 6.2 Electromagnetic Modeling of a Radiating Equipment with Distributed Infinitesimal Dipoles ..........................................................................135 6.2.1 Integral Equations for the Radiation of Electronic Equipment................................................................................136 6.2.2 Point-Matching Method with Dirac Delta Basis Functions .....137 6.2.3 Ground Plane in Semi-anechoic Chambers..............................137 6.3 Proposed Method for Near-Field to Far-Field Transformation...........138 6.3.1 Description of the Method .......................................................138 6.3.2 Optimization Problem..............................................................140 6.3.3 Source Identification................................................................140 6.4 Electromagnetic Optimization by Genetic Algorithms.......................140 6.4.1 EMOGA v1.0: Genetic Algorithm...........................................141 6.4.2 EMOGA 2.0: Metaheuristic Method .......................................142 6.4.2.1 Current Scaling .........................................................142 6.4.2.2 Correlation between Dipoles.....................................143 6.4.2.3 Memetization ............................................................143 6.5 Numerical Results...............................................................................144 6.5.1 Measurement Systems .............................................................144 6.5.1.1 General Measurement System ..................................144

XVI

Contents

6.5.1.2 Near-Field Measurement System..............................145 6.5.1.3 Far-Field Measurement System ................................147 6.5.2 Near-Field Results ...................................................................147 6.5.3 Far-Field Prediction ............................................................... `149 6.6 Conclusions ........................................................................................150 References ...........................................................................................................151 7

Application of Differential Evolution to a Multi-Objective Real-World Frequency Assignment Problem………………………………………...155 7.1 Introduction ........................................................................................155 7.2 Multi-objective FAP in a GSM Network............................................156 7.2.1 GSM Components and Frequency Planning ............................156 7.2.2 Interference Cost......................................................................157 7.2.3 Separation Cost ........................................................................158 7.3 Multi-objective Differential Evolution with Pareto Tournaments ......159 7.3.1 Algorithm Structure .................................................................159 7.3.2 Pareto Tournament...................................................................159 7.3.3 Problem Domain Knowledge...................................................160 7.4 Multi-objective Variable Neighborhood Search .................................160 7.4.1 Variable Neighborhood Search................................................160 7.4.2 Multi-objective Variable Neighborhood Search ......................161 7.4.3 Greedy Mutation ......................................................................162 7.4.4 Multi-objective Skewed Variable Neighborhood Search.........162 7.5 Experiments and Results.....................................................................163 7.5.1 Experimental Setup..................................................................163 7.5.1.1 Used GSM Instances.................................................163 7.5.1.2 Encoding ...................................................................165 7.5.1.3 Computational Facilities ...........................................165 7.5.1.4 Termination Conditions and Process Monitoring .....165 7.5.1.5 Confidence Building .................................................165 7.5.2 Methodology and Metrics ........................................................166 7.5.2.1 Hypervolume ............................................................166 7.5.2.2 Coverage Relation.....................................................166 7.5.3 Tuning of the DEPT Parameters ..............................................166 7.5.3.1 Population Size .........................................................167 7.5.3.2 Crossover Probability................................................168 7.5.3.3 Mutation Intensity.....................................................169 7.5.3.4 DEPT Scheme...........................................................170 7.5.3.5 Findings ....................................................................172 7.5.4 Empirical Results.....................................................................173 7.6 Conclusions ........................................................................................174 References ...........................................................................................................175 8

RNN Based MIMO Channel Prediction……………..............................177 8.1 Introduction ........................................................................................177 8.2 Received Signal Model.......................................................................178

Contents

XVII

8.2.1 Received Signal Model ............................................................178 8.2.2 Optimization Problem..............................................................179 8.3 Hybrid PSO-ES-DEPSO Training Algorithm.....................................179 8.4 MIMO Channel/Beam-Forming Models ............................................180 8.4.1 Channel Model.........................................................................180 8.4.2 Channel Estimation Model ......................................................182 8.4.3 MIMO Beam-Forming.............................................................182 8.5 Recurrent Neural Network for Channel Prediction.............................184 8.6 Training Procedure .............................................................................185 8.7 Numerical Results...............................................................................187 8.7.1 Algorithm Comparison ............................................................187 8.7.2 Robustness of PSO-ES-DEPSO Algorithm .............................188 8.7.3 Linear and Nonlinear Predictors with PSO-EA-DEPSO Algorithm.................................................................................191 8.7.4 Non-convexity of the Solution Space ......................................192 8.8 Performance Measures of RNN Predictors.........................................193 8.9 Conclusions ........................................................................................203 References ...........................................................................................................204 Index……….…………………………………………………………………...207

Chapter 1

A Literature Survey on Differential Evolution Anyong Qing

1

1.1 Motivations 1.1.1 Eliminating Inconsistencies It has been observed since 2004 that there are many inconsistent or even false claims prevailing in the community of differential evolution [1]. Two measures have been taken to clarify them. The first is a system level parametric study on differential evolution [1]-[4]. The second is the large scale literature survey mentioned here. It is one of the foundation stones of this book.

1.1.2 Crediting Original Contributions The academic society nowadays has become more and more utilitarian and impetuous. Many researchers dream a shortcut to their academic success. They tend to accept established view points especially those from topical review articles by leading researchers. Original publications are neglected that insufficient credits are given to originality. In some cases, they may not be aware that the original contributions are cited incorrectly [1]. Academic misconducts such as multiple submissions, exaggerated claims, or even plagiarism are not rare. It is one of the objectives of this survey to promote good academic conducts by locating and appropriately crediting original contributions.

1.1.3 Knowing the State of the Art It has been more than ten years since the inception of differential evolution. However, as far as we know, nobody else has done any comprehensive literature Anyong Qing Temasek Laboratories, National University of Singapore 5A, Engineering Dr 1 #06-09, Singapore 117411 e-mail: [email protected] 1

A. Qing and C.K. Lee: Differential Evolution in Electromagnetics, ALO 4, pp. 1–17. springerlink.com © Springer-Verlag Berlin Heidelberg 2010

2

A. Qing

survey on differential evolution. The state of the art of differential evolution is therefore not precisely known to interested researchers. This literature survey aims to fill this gap. It also serves to reveal the popularity of differential evolution.

1.1.4 Gaining Insight The literature survey involves not only literature collection but also literature analysis among which the latter is more important. Through the analysis, the following questions will be answered (a) (b) (c)

What is differential evolution? When is differential evolution used and why is it useful? When will differential evolution fail and why does it fail?

Answers to the above questions are crucial for potential applications of differential evolution. Insights gained may lead to future improvements on differential evolution.

1.2 Platforms In general, there are two platforms to look for publications on differential evolution. Although conventional publications printed on paper still play an important role, digital resources electronically available have been increasingly more preferred by both researchers and publishers. We have seen a quick transition from paper platform to digital platform within the last decade. It is noticed that differential evolution was proposed when paper platform was dominating [1], [5]-[15]. However, the dominance of paper platform does not last long. Both researchers and publishers have quickly realized the advantages of digital platform and have not hesitated to turn their attention to it. In this regard, digital platform is chosen as the main platform to carry out the literature survey. However, it is not the sole platform. Paper platform is also implemented whenever possible to supplement the digital platform so that missing of publications is minimized.

1.2.1 Starting Point At the initial stage, the literature survey is selective. Attention was focused on a bibliography [16] compiled by Prof. J. A. Lampinen which was posted online for open access. The bibliography itself was downloaded. Each and every publication included was also downloaded or copied whenever possible. The bibliography was expanded to include missing relevant publications appearing as references in available publications in the bibliography. Unfortunately, the bibliography has not been updated since it was last updated on Oct. 14, 2002. Consequently, it is very incomplete and can not stand in line with the state of the art of differential evolution.

1 A Literature Survey on Differential Evolution

3

1.2.2 Databases Many organizations have established their own databases which have been subscribed by most major libraries. Publications covered are usually formal, in another word, peer-reviewed and printed on paper. Nowadays, these databases go electronic for more exposure. They are also more timely updated. A publication from the following databases is counted if the keyword, differential evolution, appears in any field (title, abstract, keywords, text, and references) in the publication. Number of hits from different databases is shown in Table 1.1. (a) (b) (c) (d) (e) (f) (g) (h) (i)

Chinese Electronic Periodical Services Engineering Village 2 (EI) IEEE Explore Institute of Scientific and Technical Information of China ISI Web of Science (SCI) National Knowledge Infrastructure Scopus & ScienceDirect SpringerLink Wiley InterScience Table 1.1 Number of Hits from Different Databases year

IEEE

SCI

EI

InterScience

SpringerLink

Scopus

1995

0

7

9

0

1

4

1996

4

3

10

2

0

56

1997

4

9

19

2

7

56

1998

4

5

12

0

7

51

1999

8

18

45

7

7

71

2000

10

25

43

8

10

93

2001

8

23

50

4

11

112

2002

18

37

56

3

14

130

2003

23

60

96

8

15

206

2004

32

76

157

10

30

302

2005

55

132

247

11

43

381

2006

88

156

343

12

131

542

2007

116

258

528

26

143

696

2008

209

384

768

35

195

1132

4

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Please note that (a) The last search was conducted in October, 2009. To avoid any potential misleading to readers, partial search result for year 2009 is not presented here. (b) The search results may contain irrelevant publications on differential evolution equation, social differential evolution, cultural differential evolution, economical differential evolution, geological differential evolution, geographical differential evolution, genetic differential evolution, and so on. (c) Each and every database has its unique coverage. No database is exhaustive. (d) Usually, a database contains publications from the publisher owning the database. However, Scopus provided by Elsevier covers some non-Elsevier publications. Therefore, the number of hits on Scopus is the largest except for year 1995. In this sense, Scopus is more comprehensive. (e) The search is focused on publications in Chinese and English. Publications presented in other languages are not considered unless they are indexed in the above databases.

1.2.3 Informal Online Resources and Tools Besides the above electronic resources, many informal electronic resources scatter over the internet. Some of the covered publications are notes that have not been published in any formal publishing platforms such as books, journals, conference proceedings, technical reports, or theses. They are usually posted to the internet by either individual researchers or non-academic and/or non-profitable organizations for various purposes. Access is in general free. These resources can be reached with the help of free search engines such as Google, Yahoo, Microsoft Bing, or Ask (http://www.ask.com/). Alternatively, researchers may visit the websites where these resources are actually stored. Three of the most prominent websites are Google Scholar (http://scholar.google.com.sg/) provided by Google, Computer Science Bibliographies (http://liinwww.ira.uka.de/bibliography/index.html) maintained by AlfChristian Achilles and Paul Ortyl, and citeSeerX (http://citeseerx.ist.psu.edu/) provided by College of Information Sciences and Technology, Pennsylvania State University. The number of hits from these three websites is shown in Table 1.2. The search result is accurate as of October 22, 2009 and may similarly contain irrelevant publications. Likewise, partial search result for year 2009 is not presented here.

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Table 1.2 Number of Hits from Google Scholar, CSB, and citeSeerX

Google Scholar

CSB

citeSeerX

1995

6

9

6

1996

5

10

6

1997

6

6

6

1998

7

6

8

1999

38

18

9

2000

10

20

12

2001

12

21

17

2002

17

27

20

2003

24

32

22

2004

43

80

24

2005

67

80

24

2006

92

103

19

2007

138

167

9

2008

206

220

7

Due to the vast number of publications available from the internet, the search for relevant publications can only be done in a very restrictive way. Collection of publications here is accordingly far less than exhaustive.

1.3 Result Refining As mentioned before, publications found by the search may be irrelevant. Therefore, refining the survey result to eliminate irrelevant publications is compelling. This is done through tediously reading the reachable part of each and every found publication. Qualified publications are classified into four major categories: books, book chapters, other formal publications, and informal publications.

1.3.1 Books Book is a very important form of publications. 5 monographs on differential evolution [1], [17]-[20] have been published by now, among which the first book by the originators is introductory and well circulated.

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1.3.2 Book Chapters Some of the publications are presented as chapters in edited books. By now, at least one chapter is dedicated to differential evolution in at least 86 books [21]-[106] before (excluding) 2009.

1.3.3 Other Formal Publications Publications covered here include journal papers, presentations in conferences, symposiums, and workshops, degree theses, and technical reports. The number of qualified publications under this category is shown graphically in Fig. 1.1 where the ordinate is the year of publication while the number of qualified publications under this category is given beside the corresponding bar. Books and book chapters are excluded. Steady and accelerating growth has been observed since 1995, in which differential evolution was proposed. 2008

1029

2007

744

2006

536

2005

363

2004

236

2003

145

2002

94

2001

79

2000

57

1999

54

1998

21

1997

16

1996

7

1995

4

Fig. 1.1 Formal Publications on Differential Evolution

It has been noted that some publications do not actually involve a case study on differential evolution. Such publications do not contribute anything to differential evolution and are therefore unimportant to the differential evolution community. No further analysis on these publications is necessary. Remaining publications is shown graphically in Fig. 1.2. Similarly, books and book chapters are excluded. Please note that some publications may be wrongly treated as those without case studies on differential evolution because of availability.

1 A Literature Survey on Differential Evolution

7

638

2008 507

2007 365

2006 257

2005 154

2004 99

2003 68

2002

62

2001 41

2000

48

1999 1998

17

1997

15

1996

6

1995

4

Fig. 1.2 Formal Publications with Case Study on Differential Evolution

1.3.4 Informal Notes All qualified publications outside the above three categories are assembled under this category. There are 6 publications in this category as of October 22, 2009. Most of them are preprints or PowerPoint presentation notes posted to the internet by individual researchers. Bibliographic record is incomplete.

1.4 Result Analysis To make differential evolution benefit more existing researchers from active application fields and attract hesitating researchers from promising application fields, analysis on collected publications has to be conducted. The main goal of the result analysis is to look for practical usage advice for future applications and gain insight to further improve differential evolution. The analysis assembles publications on a specific topic so that researchers interested in the topic will not waste their time on irrelevant publications. At present, the analysis is focused on the following topics.

1.4.1 Theory of Differential Evolution Building a precise mathematical model for differential evolution has been posed as a challenge to the community as soon as differential evolution was proposed

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[1]. Such a solid mathematical foundation, if any, may hint desired revolutionary upgrading of differential evolution for higher reliability, better efficiency, and more robustness. However, it is yet to establish. Theoretical treatments on differential evolution in the history of differential evolution are very rare. The condition in rigorous mathematics under which differential evolution is sure to converge, the most fundamental question facing the differential evolution community or even the whole evolutionary computation community, is still pending for answer. Mathematical models for involved evolutionary operations have not been built. Interaction between evolutionary operations, intrinsic and non-intrinsic control parameters, and problems features has not been disclosed either. There is still a long way to go before differential evolution is fully appreciated. Nevertheless, some valuable pioneering efforts have been made to have a deeper understanding on differential evolution and may lead to the eventual theory of differential evolution. Studies by the originators are presented in their introductory monograph [17] and the latest book chapter by Price [19]. Other relevant studies are summarized in Table 1.3.

Table 1.3 Studies on Theory of Differential Evolution

focus

reference

evolution dynamics

[107]-[108]

stagnation

[109]-[111]

differential mutation

[112]-[115]

crossover

[116]-[117]

selection

[116]

parameter adaptation

[118]-[119]

Termination conditions

[120]-[125]

1.4.2 Fundamentals of Differential Evolution It is very critical for an applicant of differential evolution to have sufficient knowledge on differential evolution so that he or she can choose the most suitable differential evolution strategy and its corresponding intrinsic control parameter values. Otherwise, he or she may be confused by the huge number of differential evolution strategies and the infinite possibilities of setting intrinsic control parameter values. In the worst scenario, he or she may even be misled by past

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inappropriate usage of differential evolution and get an unnecessarily negative impression on differential evolution. An analysis on collected publications regarding fundamentals of differential evolution is in progress. Subjects in mind include evolution mechanism, encoding and decoding, initialization, differential mutation, crossover, selection, termination conditions, constraint handling, co-evolution, and so on.

1.4.3 Intrinsic Control Parameters Pleasant usage of differential evolution comes with not only the most suitable strategy but also appropriate setting of intrinsic control parameters. It has been well known that intrinsic control parameters of differential evolution play an essential role. Publications focusing on studying intrinsic control parameters will be assembled separately.

1.4.4 Evaluation of Differential Evolution It is a common practice to find out the advantages and disadvantages of an optimization algorithm through evaluation and comparison. Differential evolution has been evaluated by many researchers over various test bed and has earned its reputation in many comparative evaluations. Through specific evaluations, we may figure out a clearer picture on concerned component of differential evolution.

1.4.5 Applications of Differential Evolution Differential evolution will eventually be applied to solve practical application problems. Past applications provides valuable experience on usage of differential evolution. A preliminary attempt to classify qualified publications has been made to identify the fields in which differential evolution has been applied [1]. Such a preliminary classification did not cover publications in and after year 2008 and may be imprecise due to limited personal knowledge.

1.4.6 Hybridization It has been a well known fact that “for any algorithm, any elevated performance over one class of problems is offset by performance over another class” [126]. In another word, an optimization algorithm may outperform its counterparts over a specific class of problems. There is no exception for differential evolution. In accordance, many researchers have tried to hybridize different differential evolution strategies or differential evolution with other optimization algorithms. Analyzing different hybridization approaches is now in progress.

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1.5 Future Actions 1.5.1 Open Access We are currently managing a personal library covering different fields, among which differential evolution is one of the essential components. The personal library will be posted to internet for open access at appropriate time. It has to be seriously pointed out that although we have taken every possible measure to minimize missing publications, exhaustiveness of collection can not be claimed now and will not be claimed at any time in the future. Feedback from readers about any missing publications, be them formal or informal, is always welcome and appreciated. Missing publications notified by informants will be immediately integrated into the library. Moreover, this author will never claim absolute accuracy for the collection and analysis results. Besides limited personal knowledge, availability of collected publications may be the prime culprit. By chance, some title-only publications are collected during the search process. This author sincerely appeals to publishers, authors, and readers having such publications to share them as much as possible among the differential evolution community.

1.5.2 Future Update Finalizing for year 2009 will be carried out in early 2010. Update for years 2010 onwards is expected to take place on a yearly basis in order to make the survey result more accurate.

1.6 Misconceptions and Misconducts on Differential Evolution Differential evolution is one of the essential members of evolutionary algorithms. It shares some evolutionary operations and/or essential features with other evolutionary algorithms. However, it does not mean that it can be used interchangeably with other evolutionary algorithms. It is fundamentally different with other evolutionary algorithms in terms of evolution mechanism and evolutionary operations even if some evolutionary operations in differential evolution are identically named for historical reasons. It has been noticed that differential evolution has been mistermed as differential genetic algorithm [127] and has been regarded as variations of genetic algorithms [128]-[129], evolution strategies [130]. Such terms and/or classification are misleading.

References [1] Qing, A.: Differential Evolution: Fundamentals and Applications in Electrical Engineering. John Wiley, New York (2009) [2] Qing, A.: Dynamic differential evolution strategy and applications in electromagnetic inverse scattering problems. IEEE Trans. Geosci. Remote Sens. 44(1), 116– 125 (2006)

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[3] Qing, A.: A parametric study on differential evolution based on benchmark electromagnetic inverse scattering problem. In: 2007 IEEE Congress Evolutionary Computation, Singapore, September 25-28, pp. 1904–1909 (2007) [4] Qing, A.: A study on base vector for differential evolution. In: 2008 IEEE World Congress Computational Intelligence/2008 IEEE Congress Evolutionary Computation, Hong Kong, June 1-6, pp. 550–556 (2008) [5] Storn, R.: Modeling and Optimization of PET-Redundancy Assignment for MPEGSequences, Technical Report TR-95-018, International Computer Science Institute (May 1995) [6] Storn, R.: Differential Evolution Design of an IIR-Filter with Requirements for Magnitude and Group Delay, Technical Report TR-95-026, International Computer Science Institute (June 1995) [7] Storn, R., Price, K.V.: Differential Evolution - A Simple and Efficient Adaptive Scheme for Global Optimization over Continuous Spaces, Technical Report TR-95012, International Computer Science Insitute (Mar 1995) [8] Price, K.V.: Differential evolution: a fast and simple numerical optimizer. In: 1996 Biennial Conf. North American Fuzzy Information Processing Society, Berkeley, CA, June 19-22, pp. 524–527 (1996) [9] Storn, R.: Differential evolution design of an IIR-filter. In: 1996 IEEE Int. Conf. Evolutionary Computation, Nagoya, May 20-22, pp. 268–273 (1996) [10] Storn, R.: On the usage of differential evolution for function optimization. In: 1996 Biennial Conf. North American Fuzzy Information Processing Society, Berkeley, CA, June 19-22, pp. 519–523 (1996) [11] Storn, R.: System Design by Constraint Adaptation and Differential Evolution, Technical Report TR-96-039, International Computer Science Institute (November 1996) [12] Storn, R., Price, K.V.: Minimizing the real functions of the ICEC’96 contest by differential evolution. In: 1996 IEEE Int. Conf. Evolutionary Computation, Nagoya, May 20-22, pp. 842–844 (1996) [13] Price, K.V.: Differential evolution vs. the functions of the 2nd ICEO. In: 1997 IEEE Int. Conf. Evolutionary Computation, Indianapolis, IN, April 13-16, pp. 153–157 (1997) [14] Price, K., Storn, R.: Differential evolution: a simple evolution strategy for fast optimization. Dr. Dobb’s J. 22(4), 18–24, 78 (1997) [15] Storn, R., Price, K.V.: Differential evolution-a simple and efficient heuristic for global optimization over continuous spaces. J. Global Optimization 11(4), 341–359 (1997) [16] Lampinen, J.: A bibliography on differential evolution algorithm, Technical Report, Lappeenranta University of Technology, Department of Information Technology, Laboratory of Information Processing (2001) (last updated on October 14, 2002) available via internet, http://www2.lut.fi/~jlampine/debiblio.htm (accessed on October 12, 2009) [17] Price, K.V., Storn, R.M., Lampinen, J.A.: Differential Evolution: a Practical Approach to Global Optimization. Springer, Berlin (2005) [18] Feoktistov, V.: Differential Evolution: in Search of Solutions. Springer, Berlin (2006) [19] Chakraborty, U.K. (ed.): Advances in Differential Evolution. Springer, Berlin (2008)

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[20] Onwubolu, G.C., Davendra, D.: Differential Evolution: A Handbook for Global Permutation-Based Combinatorial Optimization-Studies in Computational Intelligence, vol. 175. Springer, Heidelberg (2009) [21] Corn, D., Dorigo, M., Glover, F. (eds.): New Ideas in Optimization. McGraw-Hill, London (1999) [22] Mastorakis, N.E. (ed.): Recent Advances in Circuits and Systems. World Scientific, Singapore (1998) [23] Topping, B.H.V. (ed.): Developments in computational mechanics with high performance computing. Civil-Comp Press, Edinburgh (1999) [24] Sincak, P., Vascak, J., Kvasnicka, V., Pospichal, J. (eds.): Intelligent Technologies Theory and Applications. IOS Press, Amsterdam (2002) [25] Huijsing, J.H., Steyaert, M., van Roermund, A. (eds.): Analog Circuit Design: Scalable Analog Circuit Design, High Speed D/A Converters, RF Power Amplifiers. Kluwer Academic Publishers, New York (2003) [26] Sarker, R., Mohammadian, M., Yao, X. (eds.): Evolutionary Optimization. Kluwer Academic Publishers, New York (2003) [27] Johnston, R.L. (ed.): Applications of Evolutionary Computation in ChemistryStructure & Bonding, vol. 110. Springer, Berlin (2004) [28] Onwubolu, G.C., Babu, B.V.: New Optimization Techniques in Engineering. Studies in Fuzziness and Soft Computing, vol. 141. Springer, Berlin (2004) [29] Zhong, J.J. (ed.): Biomanufacturing-Advances in Biochemical Engineering/Biotechnology, vol. 87. Springer, Berlin (2004) [30] Grigoras, D., Nicolau, A. (eds.): Concurrent information processing and computing. NATO science series, series III, Computer and systems sciences, vol. 195. IOS Press, Amsterdam (May 2005) [31] Hart, W.E., Krasnogor, N., Smith, J.E. (eds.): Recent Advances in Memetic Algorithms. Studies in Fuzziness and Soft Computing, vol. 166. Springer, Berlin (2005) [32] Hoffmann, F., Köppen, M., Klawonn, F., Roy, R. (eds.): Soft Computing: Methodologies and Applications-Advances in Soft Computing, vol. 32. Springer, Berlin (2005) [33] Palit, A.K., Popovic, D.: Computational Intelligence in Time Series Forecasting: Theory and Engineering Applications. Springer, Berlin (2005) [34] Pieruci, S. (ed.): Computer-Aided Chemical Engineering. Elsevier, Amsterdam (2005) [35] Tan, K.C., Khor, E.F., Lee, T.H.: Multiobjective Evolutionary Algorithms and Applications. Springer, Berlin (2005) [36] Abraham, A., de Baets, B., Köppen, M., Nickolay, B. (eds.): Applied Soft Computing Technologies: The Challenge of Complexity-Applied Soft Computing, vol. 34. Springer, Berlin (2006) [37] Abraham, A., Grosan, C., Ramos, V. (eds.): Stigmergic Optimization-Studies in Computational Intelligence, vol. 31. Springer, Berlin (2006) [38] Abraham, A., Grosan, C., Ramos, V. (eds.): Swarm Intelligence in Data Mining. Studies in Computational Intelligence, vol. 34. Springer, Berlin (2006) [39] Alba, E., Marti, R.: Metaheuristic Procedures for Training Neutral NetworksOperations Research/Computer Science Interfaces Series, vol. 36. Springer, Berlin (2006) [40] Brabazon, A., O’Neill, M.: Biologically Inspired Algorithms for Financial Modelling. Spriinger, Berlin (2006)

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[41] Burke, E.K., Kendall, G. (eds.): Search Methodologies: Introductory Tutorials in Optimization and Decision Support Techniques. Springer, Berlin (2006) [42] Caiti, A., Chapman, N.R., Hermand, J.P., Jesus, S.M. (eds.): Acoustic Sensing Techniques for the Shallow Water Environment: Inversion Methods and Experiments. Springer, Berlin (2006) [43] Castro-López, R., Fernández, F.V., Guerra-Vinuesa, O., Rodríguez-Vázquez, Á.: Reuse-Based Methodologies and Tools in the Design of Analog and Mixed-Signal Integrated Circuits. Springer, Berlin (2006) [44] Dzemyda, G., Šsltenis, V., Žilinskas, A. (eds.): Stochastic and Global Optimization. Springer, Berlin (2006) [45] Jin, Y. (ed.): Multi-Objective Machine Learning. Studies in Computational Intelligence, vol. 16. Springer, Berlin (2006) [46] Li, Z., Halang, W.A., Chen, G.: Integration of Fuzzy Logic and Chaos. Theory. Studies in Fuzziness and Soft Computing, vol. 187. Springer, Berlin (2006) [47] Liberti, L., Maculan, N. (eds.): Global Optimization: from Theory to Implementation-Nonconvex Optimization and Its Applications, vol. 84. Springer, Berlin (2006) [48] Liu, J., Jin, X., Tsui, K.C.: Autonomy Oriented Computing. Kluwer Academic Publishers, Bonston (2006) [49] Nedjah, N., Alba, E., de Macedo Mourelle, L. (eds.): Parallel Evolutionary Computations. Studies in Computational Intelligence, vol. 22. Springer, Berlin (2006) [50] Nedjah, N., de Macedo Mourelle, L. (eds.): Swarm Intelligent Systems. Studies in Computational Intelligence. Springer, Berlin (2006) [51] Pintér, J.D. (ed.): Global Optimization: Scientific and Engineering Case StudiesNonconvex Optimization and Its Applications, vol. 85. Springer, Berlin (2006) [52] Steyaert, M., van Roermund, A.H.M., Huijsing, J.H. (eds.): Analog Circuit Design. Springer, Berlin (2006) [53] Tiwari, A., Knowles, J., Avineri, E., Dahal, K., Roy, R. (eds.): Applications of Soft Computing: Recent Trends. Springer, Heidelberg (2006) [54] Wiak, S., Krawczyk, A., Trlep, M. (eds.): Computer Engineering in Applied Electromagnetism. Springer, Berlin (2006) [55] Zhang, H., Liu, D.: Fuzzy Modeling and Fuzzy Control. Birkhäuser, Boston (2006) [56] Zhang, G.Q., Van Driel, W.D., Fan, X.J. (eds.): Mechanics of Microelectronics. Springer, Berlin (2006) [57] Zomaya, A.Y. (ed.): Handbook of Nature-Inspired and Innovative Computing. Springer, Berlin (2006) [58] Zomaya, A.Y.: Parallel computing for bioinformatics and computational biology; models, enabling technologies, and case studies. John Wiley, New York (2006) [59] Chahl, J.S., Jain, L.C., Mizutani, A., Sato-Ilic, M. (eds.): Innovations in Intelligent Machines, vol. 1. Springer, Berlin (2007) [60] Cios, K.J., Pedrycz, W., Swiniarski, R.W., Kurgan, L.A.: Data Mining: A Knowledge Discovery Approach. Springer, Berlin (2007) [61] Corchado, E., Corchado, J.M., Abraham, A. (eds.): Innovations in Hybrid Intelligent Systems-Advances in Soft Computing, vol. 44. Springer, Berlin (2007) [62] Ebashi, S., Ohtsuki, I.: Regulatory Mechanisms of Striated Muscle ContractionAdvances in Experimental Medicine and Biology, vol. 592. Springer, Berlin (2007) [63] Grosan, C., Abraham, A., Ishibuchi, H. (eds.): Hybrid Evolutionary Algorithms. Studies in Computational Intelligence, vol. 75. Springer, Berlin (2007)

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[64] Jain, L.C., Palade, V., Srinivasan, D.: Advances in Evolutionary Computing for System Design. Studies in Computational Intelligence, vol. 66. Springer, Heidelberg (2007) [65] Kaburlasos, V.G., Ritter, G.X.: Computational Intelligence Based on Lattice Theory. Studies in Computational Intelligence, vol. 67. Springer, Berlin (2007) [66] Lobo, F.G., Lima, C.F., Michalewicz, Z. (eds.): Parameter Setting in Evolutionary Algorithms. Studies in Computational Intelligence, vol. 54. Springer, Berlin (2007) [67] Melin, P., Castillo, O., Ramírez, E.G., Kacprzyk, J., Pedrycz, W. (eds.): Analysis and Design of Intelligent Systems using Soft Computing Techniques-Advances in Soft Computing, vol. 41. Springer, Berlin (2007) [68] Nedjah, N., Abraham, A., de Macedo Mourelle, L. (eds.): Computational Intelligence in Information Assurance and Security. Studies in Computational Intelligence, vol. 57. Springer, Berlin (2007) [69] Nedjah, N., dos Santos Coelho, L., de Macedo Mourelle, L.: Mobile Robots: The Evolutionary Approach. Studies in Computational Intelligence, vol. 50. Springer, Berlin (2007) [70] Saad, A., Avineri, E., Dahal, K., Sarfraz, M., Roy, R.: Soft Computing in Industrial Applications: Recent and Emerging Methods and Techniques-Advances in Soft Computing, vol. 39. Springer, Berlin (2007) [71] Sobh, T., Elleithy, K., Mahmood, A., Karim, M.: Innovative Algorithms and Techniques in Automation, Industrial Electronics and Telecommunications. Springer, Berlin (2007) [72] Suri, J.S., Farag, A.A.: Deformable Models: Biomedical and Clinical Applications. Springer, Berlin (2007) [73] Törn, A., Žilinskas, J.: Models and Algorithms for Global Optimization: Essays Dedicated to Antanas Žilinskas on the Occasion of His 60th Birthday. Springer, Berlin (2007) [74] Valavanis, K.P. (ed.): Advances in Unmanned Aerial Vehicles: State of the Art and the Road to Autonomy. Springer, Berlin (2007) [75] Welcker, K.: Evolutionäre Algorithmen, Teubner (2007) [76] Yang, S., Ong, Y.S., Jin, Y.: Evolutionary Computation in Dynamic and Uncertain Environments. Studies in Computational Intelligence, vol. 51. Springer, Berlin (2007) [77] Abraham, A., Grosan, C., Pedrycz, W. (eds.): Engineering Evolutionary Intelligent Systems. Studies in Computational Intelligence, vol. 82. Springer, Berlin (2008) [78] Ao, S.I., Riger, B., Chen, S.S. (eds.): Advances in Computational Algorithms and Data Analysis. Lecture Notes Electrical Engineering, vol. 14. Springer, Berlin (2008) [79] Brabazon, A., O’Neill, M. (eds.): Natural Computing in Computational Finance. Studies in Computational Intelligence, vol. 100. Springer, Berlin (2008) [80] Castillo, O., Xu, L., Ao, S.I. (eds.): Trends in Intelligent Systems and Computer Engineering. Lecture Notes Electrical Engineering, vol. 6. Springer, Berlin (2008) [81] Chaturvedi, D.K.: Soft Computing: Techniques and Its Applications in Electrical Engineering. Studies in Computational Intelligence, vol. 103. Springer, Berlin (2008) [82] Cotta, C., Reich, S., Schaefer, R., Ligęza, A. (eds.): Knowledge-Driven Computing. Studies in Computational Intelligence, vol. 102. Springer, Berlin (2008) [83] Cotta, C., Seraux, M., Sörensen, K. (eds.): Adaptive and Multilevel MetaheuristicsStudies in Computational Intelligence, vol. 136. Springer, Heidelberg (2008)

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[84] Cotta, C., van Hemert, J. (eds.): Recent Advances in Evolutionary Computation for Combinatorial Optimization. Studies in Computational Intelligence, vol. 153. Springer, Berlin (2008) [85] Fulcher, J., Jain, L.C. (eds.): Computational Intelligence: a Compendium. Studies in Computational Intelligence, vol. 115. Springer, Berlin (2008) [86] Ghosh, A., Dehuri, S., Ghosh, S. (eds.): Multi-objective Evolutionary Algorithms for Knowledge Discovery from Databases. Studies in Computational Intelligence, vol. 98. Springer, Berlin (2008) [87] Grosse, C.U., Ohtsu, M. (eds.): Acoustic Emission Testing. Springer, Berlin (2008) [88] Kelemen, A., Abraham, A., Chen, Y. (eds.): Computational Intelligence in Bioinformatics. Studies in Computational Intelligence, vol. 94. Springer, Berlin (2008) [89] Kontoghiorghes, E.J., Rustem, B., Winker, P. (eds.): Computational Methods in Financial Engineering: Essays in Honour of Manfred Gilli. Springer, Berlin (2008) [90] Kramer, O.: Self-Adaptive Heuristics for Evolutionary Computation. Studies in Computational Intelligence, vol. 147. Springer, Berlin (2008) [91] Krasnogor, N., Nicosia, G., Pavone, M., Pelta, D. (eds.): Nature Inspired Cooperative Strategies for Optimization. Studies in Computational Intelligence, vol. 129. Springer, Berlin (2008) [92] Lee, K.Y., El-Sharkawi, M.A. (eds.): Modern Heuristic Optimization Techniques: Theory and Applications to Power Systems. Wiley-IEEE Press, New York (2008) [93] Liang, S. (ed.): Advances in Land Remote Sensing: System, Modeling, Inversion and Application. Springer, Berlin (2008) [94] Liu, Y., Sun, A., Loh, H.T., Lu, W.F., Lim, E.P. (eds.): Advances of Computational Intelligence in Industrial Systems. Studies in Computational Intelligence, vol. 116. Springer, Berlin (2008) [95] Prasad, B. (ed.): Soft Computing Applications in Industry. Studies in Fuzziness and Soft Computing, vol. 226. Springer, Berlin (2008) [96] Prasad, B. (ed.): Soft Computing Applications in Business. Studies in Fuzziness and Soft Computing, vol. 230. Springer, Berlin (2008) [97] Prokhorov, D. (ed.): Computational Intelligence in Automotive Applications. Studies in Computational Intelligence, vol. 132. Springer, Berlin (2008) [98] Riolo, R., Soule, T., Worzel, B. (eds.): Genetic Programming Theory and Practice, vol. 5. Springer, Berlin (2008) [99] Siarry, P., Michalewicz, Z. (eds.): Advances in Metaheuristics for Hard Optimization. Springer, Berlin (2008) [100] Smolinski, T.G., Milanova, M.G., Hassanien, A.E. (eds.): Applications of Computational Intelligence in Biology. Studies in Computational Intelligence, vol. 122. Springer, Berlin (2008) [101] Smolinski, T.G., Milanova, M.G.,, A.E.: Computational Intelligence in Biomedicine and Bioinformatics. Studies in Computational Intelligence, vol. 151. Springer, Berlin (2008) [102] Tizhoosh, H.R., Ventresca, M. (eds.): Oppositional Concepts in Computational Intelligence. Studies in Computational Intelligence, vol. 155. Springer, Berlin (2008) [103] Vakakis, A.F., Gendelman, O.V., Bergman, L.A., McFarland, D.M., Kerschen, G., Lee, Y.S.: Nonlinear Targeted Energy Transfer in Mechanical and Structural Systems II. Springer, Berlin (2008) [104] Wiak, S., Krawczyk, A., Dolezel, I. (eds.): Intelligent Computer Techniques in Applied Electromagnetics. Studies in Computational Intelligence, vol. 119. Springer, Berlin (2008)

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[105] Xhafa, F., Abraham, A. (eds.): Metaheuristics for Scheduling in Industrial and Manufacturing Applications. Studies in Computational Intelligence, vol. 128. Springer, Berlin (2008) [106] Yang, A., Shan, Y., Bui, L.T.: Success in Evolutionary Computation. Studies in Computational Intelligence, vol. 92. Springer, Berlin (2008) [107] Zhang, J., Sanderson, A.C.: An approximate Gaussian model of differential evolution with spherical fitness function. In: 2007 IEEE Congress Evolutionary Computation, Singapore, september 25-28, pp. 2220–2228 (2007) [108] Montgomery, J.: Differential evolution: Difference vectors and movement in solution space. In: IEEE Congress Evolutionary Computation, Trondheim, Norway, May 18-21, pp. 2833–2840 (2009) [109] Lampinen, J., Zelinka, I.: On stagnation of the differential evolution algorithm. In: 6th Int. Mendel Conf. Soft Computing, Brno, Czech Republic, June 7-9, pp. 76–83 (2000) [110] Sukov, A., Borisov, A.: A study of search technique in differential evolution. In: 7th Int. MENDEL Conf. Soft Computing, Brno, Czech Republic, June 6-8, pp. 144–148 (2001) [111] Tomislav, Š.: Improving convergence properties of the differential evolution algorithm. In: 8th Int. MENDEL Conf. Soft Computing, Brno, Czech Republic, June 57, pp. 80–86 (2002) [112] Ali, M.M.: Differential evolution with preferential crossover. European J. Operational Research 181(3), 1137–1147 (2007) [113] Sutton, A.M., Lunacek, M., Whitley, L.D.: Differential evolution and nonseparability: using selective pressure to focus search. In: 2007 Genetic Evolutionary Computation Conf., London, UK, July 7-11, pp. 1428–1435 (2007) [114] Zaharie, D.: Statistical properties of differential evolution and related random search algorithms. In: 18th Symp. Computational Statistics, Oporto, Portugal, August 2429, pp. 473–485 (2008) [115] Dasguptu, S., Das, S., Biswas, A., Abraham, A.: On stability and convergence of the population-dynamics in differential evolution. AI Communications 22(1), 1–20 (2009) [116] Zielinski, K., Laur, R.: Variants of differential evolution for multi-objective optimization. In: 2007 IEEE Symp. Computational Intelligence Multicriteria Decision Making, Honolulu, HI, April 1-5, pp. 91–98 (2007) [117] Zaharie, D.: A comparative analysis of crossover variants in differential evolution. In: Int. Multiconference Computer Science Information Technology, pp. 171–181 (2007) [118] Zaharie, D.: Parameter adaption in differential evolution by controlling the population diversity. In: 4th Int. Workshop Symbolic Numeric Algorithms Scientific Computing, Timi¸ soara, Romania, October 9-12, pp. 385–397 (2002) [119] Zaharie, D.: Control of population diversity and adaptation in differential evolution algorithms. In: 9th Int. Mendel Conf. Soft Computing, Brno, Czech Republic, June 2003, pp. 41–46 (2003) [120] Hajji, O., Brisset, S., Brochet, P.: A stop criterion to accelerate magnetic optimization process using genetic algorithms and finite element analysis. IEEE Trans. Magnetics 39(3 I), 1297–1300 (2003)

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[121] Zielinski, K., Peters, D., Laur, R.: Run time analysis regarding stopping criteria for differential evolution and particle swarm optimization. In: 1st Int. Conf. Experiments/Process/System Modelling/Simulation/Optimization, Athens, Greece, July 6-9 (2005) [122] Zielinski, K., Peters, D., Laur, R.: Stopping criteria for single-objective optimization. In: 3rd Int. Conf. Computational Intelligence Robotics Autonomous Systems, Singapore, December 13-16 (2005) [123] Zielinski, K., Laur, R.: Stopping criteria for constrained optimization with particle swarms. In: 2nd Int. Conf. Bioinspired Optimization Methods Applications, Ljubljana, Slovenia, October 9-10, pp. 45–54 (2006) [124] Zielinski, K., Weitkemper, P., Laur, R., Kammeyer, K.D.: Examination of stopping criteria for differential evolution based on a power allocation problem. In: 10th Int. Conf. Optimization Electrical Electronic Equipment, Brasov, Romania, May 18-19, pp. 149–156 (2006) [125] Zielinski, K., Laur, R.: Stopping criteria for a constrained single-objective particle swarm optimization algorithm. Informatics (Ljubljana) 31(1), 51–59 (2007) [126] Wolpert, D.H., Macready, W.G.: No free lunch theorems for optimization. IEEE Trans. Evolutionary Computation 1(1), 67–82 (1997) [127] Michael, C., McGraw, G.: Opportunism and Diversity in Automated Software Test Data Generation, Technical Report RSTR-003-97-13, version 1.3, RST Corporation, Sterling, VA, USA (December 8, 1997) [128] Masters, T., Land, W.: A new training algorithm for the general regression neural network. In: 1997 IEEE Int. Conf. Systems Man Cybernetics, Orlando, FL, October 12-15, vol. 3, pp. 1990–1994 (1997) [129] Engle, R.F., Manganelli, S.: CAViaR: conditional autoregressive value at risk by regression quantiles, UCSD Economics Discussion Paper 99-20, University of California, San Diego, Department of Economics (October 1999) [130] Cafolla, A.A.: A new stochastic optimisation strategy for quantitative analysis of core level photoemission data. Surface Science 402-404, 561–565 (1998)

Chapter 2

Basics of Differential Evolution Anyong Qing

1

2.1 A Short History 2.1.1 Inception Differential evolution was proposed by K.V. Price and R. Storn in 1995 [1]. At that time, Price was asked to solve the Chebyshev polynomial fitting problem [1]-[5] by Storn [2], [5]. Initially, he tried to solve it by using genetic annealing algorithm [6]. However, although he eventually found the solution to the 5-dimensional Chebyshev polynomial fitting problem by using genetic annealing algorithm, he was frustrated to notice that genetic annealing algorithm fails to fulfill the three requirements for a practical optimization technique: strong global search capability, fast convergence, and user friendliness. A breakthrough happened when Price came up with an innovative scheme for generating trial parameter vectors. In this scheme, a new parameter vector is generated by adding the weighted difference vector between two population members to a third member. Such a scheme was named as differential mutation and has been well known to be the crucial idea behind the success of differential evolution. The cornerstone for differential evolution was therefore laid. Price wrapped up his invention with other critical ideas: natural real code, arithmetic operations, mother-child competition and selection, and execution of evolutionary operations in the order of mutation-crossover-selection. Consequently, differential evolution, a very reliable, efficient, robust, and simple evolutionary algorithm was developed. Anyong Qing Temasek Laboratories, National University of Singapore 5A, Engineering Dr 1 #06-09, Singapore 117411 e-mail: [email protected] 1

A. Qing and C.K. Lee: Differential Evolution in Electromagnetics, ALO 4, pp. 19–42. springerlink.com © Springer-Verlag Berlin Heidelberg 2010

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2.1.2 Early Years 2.1.2.1 Assessment Evaluation is one of the essential parts of algorithm development and has to be conducted right after an optimization algorithm is developed. It is the undeniable responsibility of the algorithm’s originator(s) to evaluate the proposed optimization algorithm. During the evaluation process, the originator(s) may gain more insights behind the concerned new optimization algorithm and make further efforts to improve it. The first evaluation of differential evolution was reported in the founding publication [1]. DE/rand/1/exp and DE/current-to-best/1/exp were evaluated over a test bed containing 7 unconstrained toy functions and 2 constrained toy functions. All intrinsic control parameters are fixed by trial and error approach. Comparison with annealed Nelder & Mead strategy and adaptive simulated annealing was also presented. Price and Storn, the originators, published two successive performance evaluation reports in 1996 [7] and 1997 [8]. Different strategies of differential evolution were evaluated over larger test beds. It is interesting to note that differential evolution quickly came to the attention of other researchers [9]. Evaluation of differential evolution over a test bed of 15 functions was carried out. Comparison with a variety of methods was also made. Differential evolution solves all functions successfully. It converges most rapidly while optimizing 11 of the 15 functions. 2.1.2.2 Reputation Building Differential evolution proved itself by winning in the two International Contests on Evolutionary Optimization (ICEO) [2], [3], [5], [10] in 1996 and 1997. More importantly, it was the best evolutionary algorithm among all entries since the first two places were won by non-evolutionary algorithms. It finished 3rd in the 1st International Contest on Evolutionary Optimization held in Nagoya, Japan from May 20, 1996 to May 22, 1996 [2], [5]. Differential evolution was the best among all qualified entries in the 2nd International Contests on Evolutionary Optimization [3] although the actual contest was cancelled due to lack of valid entries. 2.1.2.3 Applications Each and every practical optimization algorithm has to be picked up by people working in practical fields. More importantly, it will receive further evaluation through practical applications and gain more insights to improve it to better suit requirements arising from outstanding problems. There is no exception for differential evolution. It was soon applied by both the originators [11]-[15] and other application engineers [16]-[29] to solve various practical problems. Benefiting fields in these early applications of differential evolution are summarized in Table 2.1.

2 Basics of Differential Evolution

21

Table 2.1 Fields Benefiting from Early Application of Differential Evolution application

reference

Priority encoding transmission (PET)-redundancy assignment

[11]

Design of an IIR-filter

[12], [13]

Design of a howling remover

[14]

Redesign a switched capacitor (SC)-filter suffering from parasitic capacitances

[15]

multisensor fusion

[16], [19]-[21]

Parameter estimation of a batch bioprocess

[17]

Parallel optimization

[18]

Training of general regression neural network

[22]

Software test data generation

[23]

Scheduling of core blowers

[24]-[25]

Image registration

[27]

Temperature control of a chemical reactor system

[28]-[29]

Selection of control policy of a robotic arm

[29]

2.1.2.4 Promotion To boost the awareness of differential evolution and expand the community, Rainer established a website [5]. Latest contents include (a) (b) (c) (d) (e)

History, basics and practical advice Codes Demos Applications Useful relevant links

The website is still one of the prime resources for differential evolution. 2.1.2.5 Practical Advice Differential evolution involves intrinsic control parameters. It has been realized right from the beginning that differential evolution is dependent on proper setting of these intrinsic control parameters. To convenience differential evolution applicants, some practical advices to choose intrinsic control parameter values are recommended by the originators [3], [5], [8], [14], [30]. The recommendations are (a) Choose a population 2 [30], 5 [8], 10 [5], [14], [30] times problem dimension, or larger [3]. (b) Choose mutation intensity from [0.5, 1] [5], [14].

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(c) Choose crossover probability considerably lower than 1 if convergent, otherwise choose it from [0.8, 1] [5], [8], [14]. However, in [30], it is suggested to choose crossover probability “as large as possible without causing the population to become devoid of diversity”. (d) Choose lower mutation intensity for larger population size and vice versa [14]. (e) Adjust mutation intensity and crossover probability in parallel [3]. (f) Increase population size as crossover probability increases [3]. Some usage rules regarding generating initial population [14], formulating objective function [14], monitoring evolution [14], and strategy selection [3] are also recommended by the originators. 2.1.2.6 Standardization Strategies of differential evolution are denoted by DE/x/y/z where x indicates how the differential mutation base is chosen, y≥1 is the number of vector differences added to the base vector, and z is the law which the number of parameters donated by the mutant follows [3], [8]. The notation was inked by the originators in 1997 [8]. 2.1.2.7 More Adventures DE/rand/1/exp and DE/target-to-best/1/exp were proposed in the founding publication of differential evolution [1]. Later on, Price and Storn went on to explore better strategies. DE/best/2/bin [7], DE/random-to-best/1/exp [13]-[14] (corrected as DE/target-to-best/1/exp in [3]), DE/best/1/exp [14], DE/best/2/exp [10], [14], DE/current/1/exp [15], DE/rand/1/bin [8], [30], and other variants [2], were subsequently developed. It is very interesting to note that the inherent defect of differential evolutional, namely, slow convergence, was soon observed [17], [22]. Proposals to hybridize differential evolution with deterministic local optimizers were timely put forward.

2.1.3 Key Milestones in and after 1998 Differential evolution turned into fast track in 1998. Its potential has been increasingly realized by more and more researchers from ever growing number of fields. Researchers with different background have worked together to know more about its mathematical foundation, develop new strategies, or solve more challenging problems. Key events happening in and after 1998 in the short history of differential evolution are summarized in Table 2.2.

2 Basics of Differential Evolution

23

Table 2.2 Key Milestones in and after 1998 year

event

source

1998

first modification to initialization

[31]

1998

first modification to differential mutation

[31]

1998

first thesis on differential evolution

[32]

1999

first differential evolution for integer optimization parameters

[33]

1999

initial idea of dynamic differential evolution

[4], [34]-[35]

1999

Pareto differential evolution for multi-objective optimization

[36]-[37]

1999

first multi-population differential evolution

[38]

1999

first adaptive differential evolution

[39]

2000

first application in electromagnetics

[40]

2000

first empirical study on differential evolution

[41]-[42]

2000

first approximation of objective and constraint functions

[43]

2001

bibliography on differential evolution

[44]

2001

generalized differential evolution for multi-objective optimization

[45]

2005

first book on differential evolution

[3]

2005

non-dominated sorting differential evolution

[46]

2005

first strategy adaptation differential evolution

[47]

2006

first special session on differential evolution in IEEE Congress on Evolutionary Computation

2006

opposition-based differential evolution

[48]-[49]

2006

first system-level parametric study

[35]

2007

first thematic study on crossover in differential evolution

[50]

2009

evolutionary crimes

[4]

2009-

special issue on differential evolution in IEEE Transactions on Evolutionary

2010

Computation

2.2 The Foundational Differential Evolution Strategies Two differential evolution strategies, DE/rand/1/exp and DE/target-to-best/1/exp, were proposed in the founding publication of differential evolution [1] to minimize a single objective function f(x) with N-dimensional real optimization parameters x. Details of these two foundational strategies are given here to entertain readers with basic features of differential evolution.

2.2.1 Notations The notations applied in [4] are strictly followed here. However, for the convenience of readers and self-completeness of this book, primary notations are

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summarized in Table 2.3. These notations will be consistently applied throughout this book unless specified otherwise. Secondary notations will be explained when it is mentioned for the first time. Table 2.3 Primary Notations notation

legend

f(x)

objective function to minimize

notation

legend

x

N-dimensional vector of optimization parameters

xj

the jth optimization parameter

pc

crossover probability

b jL

lower bound of xj

bjU

upper bound of xj

P

population

Np

population size

P0

initial population

Pn

population of generation n

pi

the ith individual in P

pn,i

the ith individual in Pn

pbest

the best individual in P

pn,best

the best individual in Pn

pworst

the worst individual in P

pn,worst

the worst individual in Pn

x

i

i j

i

vector of optimization parameters of p i

x i

n,i n,i

vector of optimization parameters of pn,i the jth optimization parameter in xn,i of pn,i

x

the jth optimization parameter in x of p

xj

vi

mutant for pi

vn+1,i

mutant for pn,i

xv,i

vector of optimization parameters of vi

xn+1,v,i

vector of optimization parameters of vn+1,i

xj

v,i i

v,i

the jth optimization parameter in x of v

i

i

xj

n+1,v,i n,i

the jth optimization parameter in xn+1,v,i of vn+1,i base for vn+1,i

b

base for v

b

xb,i

vector of optimization parameters of bi

xn,b,i

vector of optimization parameters of bn,i

xjb,i

the jth optimization parameter in xb,i of bi

xjn,b,i

the jth optimization parameter in xn,b,i of bn,i

the ith child

cn+1,i

the ith child of generation n + 1

vector of optimization parameters of c

xn+1,c,i

vector of optimization parameters of cn+1,i

xjc,i

the jth optimization parameter in xc,i of ci

xjn+1,c,i

F

mutation intensity

Fy

the yth mutation intensity

p1

index for donor 1 (one vector difference case)

p1y

index for donor 1 for the yth vector difference

p2

index for donor 2 (one vector difference case)

p2y

index for donor 2 for the yth vector difference

ci x

c,i

i

the jth optimization parameter in xn+1,c,i of cn+1,i

2.2.2 Strategy Framework Differential evolution optimizes f(x) with a population of Np individuals. It involves two stages, namely, initialization and evolution. Initialization generates initial population P0. Then the population evolves from one generation (Pn) to the next (Pn+1) until termination conditions are satisfied. While evolving from Pn to Pn+1, the three evolutionary operations, namely, differential mutation, crossover and selection, are executed in sequence. 2.2.2.1 Pseudo-code The Fortran-style pseudo-code of differential evolution is shown in Fig. 2.1.

2 Basics of Differential Evolution

25

Initialization n=0 do i = 1, Np generate p0,i evaluate f(x0,i) end do Evolution do while termination conditions are not satisfied n=n+1 do i = 1, Np differential mutation to obtain mutant vn+1,i crossover mutant vn+1,i with pn,i to deliver child cn+1,i evaluate child f(xn+1,c, i) selection to get individual pn+1,i end do end do

Fig. 2.1 Fortran-style Pseudo-code of Foundational Differential Evolution Strategies

2.2.2.2 Initialization Initialization generates the initial population P0 which contains Np individuals p0,i.

(

x 0j , i = b Lj + α ij bUj − b Lj

)

1≤ i ≤ Np, 1≤ j ≤ N

(1)

where the real random number, αji , is usually but not necessarily uniform in [0,1]. Alternatively, “in case a preliminary solution is available, the initial population is often generated by adding normally distributed random deviations to the nominal solution” x0 [1].

x

0, i j

⎧⎪ x 0j =⎨ 0 i ⎪⎩ x j + σ j

i =1 2 ≤ i ≤ N p −1

1≤ j ≤ N

(2)

where the real random number, σji , is usually but not necessarily “normally distributed”. 2.2.2.3 Differential Mutation Differential mutation generates a mutant vn+1,i for pn,i as follows

(

)

x n+1,v ,i = x n,b ,i + F x n, p1 − x n , p2 , 1 ≤ i ≠ p1 ≠ p2 ≤ N p n,i

The differential mutation base b is chosen in two different ways in [1].

(3)

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A Random bn,i is randomly chosen from Pn and is different with both donors p n , p1 and p n , p2 . B Target-to-best bn,i is a point on a line between pn,i and pn,best. It is generated through the following arithmetic recombination

(

x n ,b ,i = x n ,i + λ x n ,best − x n ,i

)

(4)

where λ is the recombination coefficient. Alternatively, by looking at the general formulation of differential mutation shown in (2.2) in [1], this form of differential mutation, target-to-best/1, can be regarded as a biased form of current/2 in which the current individual pn,i serves as differential mutation base, the first vector difference is xn,best-xn,i weighted by λ, and the second vector difference is x n , p1 − x n , p2 weighted by F. 2.2.2.4 Crossover Crossover delivers a child cn+1,i through mating vn+1,i with pn,i. Exponential crossover is applied in [1]. The Fortran-style pseudo-code for exponential crossover is given in Fig. 2.2. A starting point r (1≤r≤N) is first randomly chosen. xrn+1,c,i of cn+1,i is taken from xrn+1,v,i of vn+1,i. Parameters of cn+1,i after (in cyclic sense) r depends on a series of Bernoulli experiments of probability pc, a constant in [0,1]. vn+1,i will keep donating its parameters to cn+1,i until the Bernoulli experiment is unsuccessful or the crossover length L, i.e., the number of parameters of the child donated by the mutant, is already N - 1. The remaining parameters of cn+1,i come from pn,i. do j = 1, N xjn+1,c,i = xjn,i end do r = N * rand(0, 1) + 1 k=r

xkn+1,c,i = xkn+1,v,i L=1

E = rand(0, 1) do while (E pc and L < N -1) L=L+1 k = 1 + mod(k, N) xkn+1,c,i = xkn+1,v,i

E = rand(0, 1) end do

Fig. 2.2 Fortran-style Pseudo-code of Exponential Crossover for Differential Evolution

2 Basics of Differential Evolution

27

A demonstrative example is shown in Fig. 2.3. N = 8, r = 7 and the crossover length is 3.

xn,i

28.69

35.09

57.82

12.06

26.99

82.96

65.30

52.68

xn+1,v,i

15.23

16.22

78.33

68.12

32.88

67.55

99.28

85.86

1

0

15.23

35.09

Bernoulli experiments

xn+1,c,i

1 57.82

12.06

26.99

82.96

99.28

85.86

Fig. 2.3 Exponential Crossover

2.2.2.5 Selection The selection operation follows Darwin’s natural selection, or survival of the fittest [51]. Child cn+1,i competes with a predetermined individual in population Pn and replaces it if cn+1,i dominates its competitor. cn+1,i usually competes with pn,i [12] although it is not stated explicitly in [1]. Such a selection operation is mathematically expressed as

p

n +1, i

⎧⎪c n +1,i f (x n +1,c, i ) ≤ f (x n, i ) = ⎨ n ,i ⎪⎩p otherwise

(5)

Sometimes, the selection is conducted in a sense of stronger dominance [10], i.e.,

⎧⎪c n +1, i f (x n +1,c, i ) < f (x n, i ) p n +1,i = ⎨ n ,i ⎪⎩p otherwise

(6)

As far as we know, by now, there is no thematic study on the effect of these two selection schemes. 2.2.2.6 Termination Conditions Termination conditions are not specified in [1]. However, other early publications by the originators [8]-[10] suggest that limit of number of generations is implemented to terminate the evolution process. It is yet to clarify whether “objective met” [3] is implemented although it is mentioned in [1] that the best individual pn+1,best in the new population Pn+1 is updated at the end of each evolution loop.

2.2.3 Intrinsic Control Parameters The two differential evolution strategies in [1] share three intrinsic control parameters

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(1) Population size Np (2) Mutation intensity F (3) Crossover probability pc In general, DE/target-to-best/1/exp has one more intrinsic control parameter, the recombination coefficient λ. However, for simplicity, λ is usually chosen identical with F.

2.3 Classic Differential Evolution Besides the aforementioned foundational differential evolution strategies proposed in the originators’ founding publication on differential evolution [1], many more differential evolution strategies under the same umbrella of classic differential evolution have been put forward. All classic differential evolution strategies share the same framework as shown in Fig. 2.1 while distinguish themselves in terms of initialization, differential mutation, crossover, objective function evaluation, selection, and termination conditions.

2.3.1 Initialization The two general initialization schemes have already been described by the originators clearly in their founding publication on differential evolution [1]. However, Different differential evolution strategy may apply different probability distribution i i function to generate random numbers αj in (1) [52] or σj in (2) [31].

2.3.2 Differential Mutation According to the standard notation DE/x/y/z, the general formulation of differential mutation for classic differential evolution is [1]

(

x n +1,v ,i = x n ,b ,i + ∑ Fy x y ≥1

n , p1 y

−x

n , p2 y

),

1 ≤ i ≠ p1 y ≠ p2 y ≤ N p

(7)

It is interesting to note that Storn [53] uses normalized vector differences to generate mutant vn+1,i which is mathematically expressed as

x n+1,v ,i = x n,b ,i +

1 y

∑ F (x y

y ≥1

n , p1 y

−x

n , p2 y

),

1 ≤ i ≠ p1 y ≠ p2 y ≤ N p

(8)

Unless specified otherwise, we will stick to the general formulation shown in (7). Different differential mutation implements different differential mutation base n, p n, p bn,i and uses various number of vector differences x 1 y − x 2 y . Some of the most prominent differential mutation schemes are summarized here.

2 Basics of Differential Evolution

29

2.3.2.1 Current Individual pn,i serves as differential mutation base bn,i for vn+1,i. 2.3.2.2 Best Individual pn,best serves as differential mutation base bn,i for vn+1,i. 2.3.2.3 Better The differential mutation base bn,i for vn+1,i is randomly chosen from individuals dominating individual pn,i, i.e., d(bn,i, pn,i)=true where d(bn,i, pn,i) is the logic dominance function [4]. 2.3.2.4 Random The differential mutation base bn,i for vn+1,i is randomly chosen from Pn and is different with pn,i and all donors. 2.3.2.5 Mean The differential mutation base bn,i for vn+1,i is the geometrical center of Pn, i.e.,

x n ,b ,i =

Np

1 Np

∑x

n ,i

(9)

i =1

2.3.2.6 Best of Random The differential mutation base bn,i for vn+1,i is randomly chosen from Pn and dominates all donors which are also randomly chosen from Pn, i.e.,

(

d b n,i , p

n , p1 y

) = true

(

∩ d b n,i , p

n , p2 y

) = true

∀y

(10)

2.3.2.7 Arithmetic Best Arithmetic best is the synonym of target-to-best. 2.3.2.8 Arithmetic Better This scheme differs with arithmetic best by replacing pn,best in (4) with an individual dominating pn,i. A synonym for this scheme is target-to-better. 2.3.2.9 Arithmetic Random Likewise, in this scheme, pn,best in (4) is replaced by an individual randomly chosen from Pn. Its synonym is target-to-random.

30

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2.3.2.10 Trigonometric Trigonometric differential mutation was proposed by Fan and Lampinen in 2003 [54]. Mutant vn+1,i is generated as

∑ ( f (x ) − 3

n , pk

x

n+1, v ,i

=

x

n , p1

+x

n , p2

+x

3

n , p3

+

j =1

(

)

( ))(x

f x

(

n, p j

n, p j

− x n , pk

(

)

)

f x n , p1 + f x n , p2 + f x n , p3

)

k = mod( j ,3) + 1 (11)

where individuals p n , p1 , p n , p2 , and p n , p3 may be chosen from Pn in different ways such as current, best, better, and random as aforementioned. Equivalently, the trigonometric differential mutation can be regarded as a special differential mutation in which the center of p n , p1 , p n , p2 , and p n , p3 acts as differential mutation base and p n , p1 , p n , p2 , and p n , p3 donates equally to vector differences. It can be seen that the intrinsic control parameter F does not show up in trigonometric differential mutation which might be one of its most attractive features. 2.3.2.11 Directed Directed differential mutation was also proposed by Fan and Lampinen in 2003 [55]. Mutant vn+1,i is generated as

( (

)( )

3 ⎡ f x n , p1 ⎤ n , p1 x n +1,v ,i = x n , p1 + ∑ ⎢1 − x − x n , pi n , pi ⎥ x f i =2 ⎣ ⎦

)

(12)

where individual p n , p1 dominates individuals p n , p2 and p n , p3 . It can be seen from (12) that individual p n , p1 serves as both differential mutation base and donor. Similarly, the intrinsic control parameter F does not show up.

2.3.3 Crossover Crossover has been thought unessential for differential evolution [4]. It is even not applied in some differential evolution strategies [3], [56]. However, recent studies hint that its significance in differential evolution might be seriously underestimated [4]. Crossover has been extensively studied in genetic algorithms. Almost all crossover schemes there can be implemented in differential evolution straightforward or after minor adjustment. Besides the exponential crossover mentioned earlier, some of the crossover schemes commonly applied in differential evolution are summarized here. In most evolutionary algorithms, the child cn+1,i is required to be different from its parents. This convention is followed here although it is not absolutely necessary in differential evolution.

2 Basics of Differential Evolution

31

2.3.3.1 Binary Crossover Binary crossover may be one of the most common crossover schemes in differential evolution. In this scheme, as shown in Fig. 2.4, cn+1,i inherits an optimization parameter from either vn+1,i or pn,i according to the result of a Bernoulli experiment of crossover probability pc, where β is a real random number uniform in [0,1]. do j = 1, N xjn+1,c,i = xjn,i end do L=0 do j = 1, N ȕ = rand(0, 1) if (ȕ pc) then L=L+1 xjn+1,c,i = xjn+1,v,i end if end do if (L = 0) then r = N * rand(0, 1) + 1 xrn+1,c,i = xrn+1,v,i else if (L = N) then r = N * rand(0, 1) + 1 xrn+1,c,i = xrn,i end if

Fig. 2.4 Fortran-style Pseudo-code of Binomial Crossover

A demonstrative example is shown in Fig. 2.5. cn+1,i inherits parameters x2n+1,v,i, , x7n+1,v,i, and x8n+1,v,i from vn+1,i and x1n,i, x3n,i, x5n,i, x6n,i from pn,i. Therefore, x4 the crossover length is 4. n+1,v,i

xn,i

28.69

35.09

57.82

12.06

26.99

82.96

65.30

52.68

xn+1,v,i

15.23

16.22

78.33

68.12

32.88

67.55

99.28

85.86

0

1

0

1

0

0

1

1

28.69

16.22

57.82

68.12

26.99

82.96

99.28

85.86

Bernoulli experiments

xn+1,c,i

Fig. 2.5 Binomial Crossover

32

A. Qing

2.3.3.2 One-Point Crossover One-point crossover randomly selects a single crossover point r (1
x

n +1, c , i j

⎧⎪ x nj, i = ⎨ n +1, v ,i ⎪⎩ x j

j
r≤ j≤N

A demonstrative example is shown in Fig. 2.6 where r=4.

xn,i

28.69

35.09

57.82

12.06

26.99

82.96

65.30

52.68

xn+1,v,i

15.23

16.22

78.33

68.12

32.88

67.55

99.28

85.86

xn+1,c,i

28.69

35.09

57.82

68.12

32.88

67.55

99.28

85.86

Fig. 2.6 One-point Crossover

2.3.3.3 Multi-point Crossover For an M-point crossover, M crossover points rm (1
x

n +1, c , i j

⎧⎪ x nj , i m is odd rm −1 ≤ j < rm 1 ≤ m ≤ M = ⎨ n +1, v , i m is even ⎪⎩ x j

(14)

where r0=1 and rM+1=N+1. A demonstrative example for a two-point crossover is shown in Fig. 2.7 where r1=3 and r2=6.

xn,i

28.69

35.09

57.82

12.06

26.99

82.96

65.30

52.68

xn+1,v,i

15.23

16.22

78.33

68.12

32.88

67.55

99.28

85.86

xn+1,c,i

28.69

35.09

78.33

68.12

32.88

82.96

65.30

52.68

Fig. 2.7 Two-point Crossover

2.3.3.4 Arithmetic Crossover cn+1,i lies on a line between vn+1,i and pn,i. It is generated through the following linear combination between vn+1,i and pn,i.

2 Basics of Differential Evolution

33

(

x nj +1, c, i = x nj +1, v,i + h x nj, i − x nj +1, v, i

)

1≤ j ≤ N

(15)

where h is the crossover intensity, an intrinsic control parameter accompanying arithmetic crossover. A demonstrative example for an arithmetic crossover is shown in Fig. 2.8 where h=0.1.

xn,i

28.69

35.09

57.82

12.06

26.99

82.96

65.30

52.68

xn+1,v,i

15.23

16.22

78.33

68.12

32.88

67.55

99.28

85.86

xn+1,c,i

16.576

18.107

76.279

62.514

32.291

69.091

95.891

82.542

Fig. 2.8 Arithmetic Crossover

2.3.3.5 Arithmetic One-Point Crossover As its name indicates, arithmetic one-point crossover is a hybrid crossover scheme hybridizing arithmetic crossover and one-point crossover. Child cn+1,i is generated as follows

x

n +1, c , i j

⎧⎪ x nj ,i j
(

)

r≤ j≤N

(16)

A demonstrative example for an arithmetic crossover is shown in Fig. 2.9 where r=4 and h=0.1.

xn,i

28.69

35.09

57.82

12.06

26.99

82.96

65.30

52.68

xn+1,v,i

15.23

16.22

78.33

68.12

32.88

67.55

99.28

85.86

xn+1,c,i

28.69

35.09

57.82

62.514

32.291

69.091

95.891

82.542

Fig. 2.9 Arithmetic One-point Crossover

2.3.3.6 Arithmetic Multi-point Crossover Likewise, arithmetic multi-point crossover is a hybrid crossover scheme hybridizing arithmetic crossover and multi-point crossover. Child cn+1,i is generated as follows

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A. Qing

⎧⎪ x nj, i m is odd x nj +1, c, i = ⎨ n +1, v ,i + h x nj ,i − x nj +1, v ,i ⎪⎩ x j

(

)

m is even

rm −1 ≤ j < rm 1 ≤ m ≤ M (17)

where r (1
xn,i

28.69

35.09

57.82

12.06

26.99

82.96

65.30

52.68

xn+1,v,i

15.23

16.22

78.33

68.12

32.88

67.55

99.28

85.86

xn+1,c,i

28.69

35.09

76.279

62.514

32.291

82.96

65.30

52.68

Fig. 2.10 Arithmetic One-point Crossover

2.3.3.7 Arithmetic Binomial Crossover This is another hybrid crossover scheme hybridizing arithmetic crossover and binomial crossover. Its FORTRAN-style pseudo-code is given in Fig. 2.11 while a demonstrative example is depicted in Fig. 2.12 where h=0.1. do j = 1, N xjn+1,c,i = xjn,i end do L=0 do j = 1, N ȕ = rand(0, 1) if (ȕ pc) then L=L+1 xjn+1,c,i = xj

n,i

+ h(xj

n,i

- xjn+1,v,i)

end if end do if (L = 0) then r = N * rand(0, 1) + 1 xrn+1,c,i = xr

n,i

+ h(xr

n,i

– xrn+1,v,i)

else if (L = N) then r = N * rand(0, 1) + 1 xrn+1,c,i = xrn,i end if

Fig. 2.11 FORTRAN-style Pseudo-code of Arithmetic Binomial Crossover

2 Basics of Differential Evolution

35

xn,i

28.69

35.09

57.82

12.06

26.99

82.96

65.30

52.68

xn+1,v,i

15.23

16.22

78.33

68.12

32.88

67.55

99.28

85.86

0

1

0

1

0

0

1

1

28.69

18.107

57.82

62.514

26.99

82.96

95.891

82.542

Bernoulli experiments

xn+1,c,i

Fig. 2.12 Arithmetic Binomial Crossover

2.3.3.8 Arithmetic Exponential Crossover This crossover scheme hybridizes arithmetic crossover and exponential crossover. Its FORTRAN-style pseudo-code is given in Fig. 2.13 while a demonstrative example is depicted in Fig. 2.14 where h=0.1. do j = 1, N xjn+1,c,i = xjn,i end do r = N * rand(0, 1) + 1 k=r xkn+1,c,i = xkn,i + h(xkn,i – xkn+1,v,i) L=1

E = rand(0, 1) do while (E pc and L < N -1) L=L+1 k = 1 + mod(k, N) xkn+1,c,i = xkn,i + h(xkn,i – xkn+1,v,i)

E = rand(0, 1) end do

Fig. 2.13 Fortran-style Pseudo-code of Arithmetic Exponential Crossover

xn,i

28.69

35.09

57.82

12.06

26.99

82.96

65.30

52.68

xn+1,v,i

15.23

16.22

78.33

68.12

32.88

67.55

99.28

85.86

1

0

16.576

35.09

Bernoulli experiments

xn+1,c,i

1 57.82

12.06

26.99

82.96

Fig. 2.14 Arithmetic Exponential Crossover

95.891

82.542

36

A. Qing

2.4 Dynamic Differential Evolution The first idea of dynamic differential evolution [4], [35] was mentioned by Price [34] although it is termed “single array” or “one-array” differential evolution instead of the present name. It is very unfortunate that it is claimed that “no dramatic difference in performance between the one- and two-array methods” which has been numerically proven incorrect [4]. The Fortran-style pseudo-code of dynamic differential evolution is shown in Fig. 2.15. Readers may easily find the difference between classic differential evolution in Fig. 2.2 and dynamic differential evolution here. Initialization n=0 do i = 1, Np generate pi evaluate f(xi) end do Evolution i=1 do while termination conditions are not satisfied n=n+1 differential mutation to obtain mutant vi crossover mutant vi with pi to deliver child ci evaluate child f(xc,i) selection to update individual pi extra selection to update best individual pbest i = mod(i, Np) + 1 end do Fig. 2.15 Fortran-style Pseudo-code of Dynamic Differential Evolution

For more details of dynamic differential, its features, advantages, and disadvantages, please refer to [4], [35], [57]-[58].

2.5 State of the Art of Differential Evolution Differential evolution has enjoyed fruitful development for more than a decade. Many great achievements have been inked. To know exactly the state of the art of differential evolution, several topical reviews are going on simultaneously now.

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37

Concerned topics include optimization parameters, differential evolution strategies, hybrid differential evolution, multi-objective differential evolution, coevolution differential evolution, evaluation of differential evolution, and intrinsic control parameters. Although preliminary classification on collected publications has been done, there is still a long way to go before completing these topical reviews so that inaccuracy in these topic reviews can be minimized. To avoid any potential misleading to readers, the incomplete topic reviews are not presented here. It is in this author’s mind to publish an encyclopedia of differential evolution covering every noticeable achievement in differential evolution in the near future. These topic reviews have been scheduled into the encyclopedia on differential evolution.

2.6 Essential Features of Differential Evolution Numerous strategies of differential evolution have been developed. Although each strategy has its unique characteristics, all differential evolution strategies share some essential features.

2.6.1 Advantages As a member of evolutionary algorithms, without exception, differential evolution shares all merits of evolutionary algorithms: strong global search ability, versatility to problem features, and little demand on good initial solution. However, it distinguishes itself from other evolutionary algorithms by behaving uniquely in many aspects. 2.6.1.1 Reliability The attraction of differential evolution comes mainly from its strong capability to locate global optimum with high probability. It has been well known among the differential evolution community. 2.6.1.2 Efficiency Another attraction of differential evolution is its efficiency. Both the originators and many other researchers have carried out extensive comparison with other evolutionary algorithms. Better efficiency is observed in most case studies. 2.6.1.3 Simplicity The simplicity of differential evolution involves two aspects. First of all, the algorithm itself is very simple which is very apparent from the above description, the pseudo-code in [2], [7], [10], and the C-style program in [34]. The other important fact that contributes significantly to the simplicity of differential evolution but is

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often neglected by researchers is the number of intrinsic control parameters. There are only three intrinsic control parameters in most implemented differential evolution strategies. As far as we know, it is one of the evolutionary algorithms with the fewest intrinsic control parameters. Undoubtedly, it is much easier to tune fewer intrinsic control parameters. 2.6.1.4 Robustness It is usually inevitable to tune intrinsic control parameters to suit a particular practical problem. It has been observed by both the originators and many other researchers that tuning differential evolution is far more relaxing, as pointed out before in section 1.2.5 in this chapter. The latest system level parametric study on differential evolution [1] gives more solid support to this advantage.

2.6.2 Disadvantages 2.6.2.1 Efficiency It is well known that evolutionary algorithms are less efficient than most deterministic algorithms provided that both are able to locate the global optimum. There is no exception for differential evolution even if it is one of the more efficient evolutionary algorithms. 2.6.2.2 Incapability for Epistatic and Noisy Problems It is pointed out that differential evolution may perform poorly on epistatic problems [3], [59] and noisy problems [60]. However, readers are reminded to accept such controversial claims [61] cautiously since they are not based on system level evaluation and comparison.

References [1] Storn, R., Price, K.V.: Differential Evolution - A Simple and Efficient Adaptive Scheme for Global Optimization over Continuous Spaces, Technical Report TR-95012, International Computer Science Insitute (March 1995) [2] Price, K.V.: Differential evolution vs. the functions of the 2nd ICEO. In: 1997 IEEE Int. Conf. Evolutionary Computation, Indianapolis, IN, April 13-16, pp. 153–157 (1997) [3] Price, K.V., Storn, R.M., Lampinen, J.A.: Differential Evolution: a Practical Approach to Global Optimization. Springer, Berlin (2005) [4] Qing, A.: Differential Evolution: Fundamentals and Applications in Electrical Engineering. John Wiley, New York (2009) [5] Storn, R.: Differential evolution (DE) for continuous function optimization (an algorithm by Kenneth Price and Rainer Storn) (2009), http://www.icsi.berkeley.edu/~storn/code.html (last accessed on October 23, 2009)

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[6] Price, K.V.: Genetic annealing. Dr. Bobb’s J. 19(10), 127–132 (1994) [7] Price, K.V.: Differential evolution: a fast and simple numerical optimizer. In: 1996 Biennial Conf. North American Fuzzy Information Processing Society, Berkeley, CA, June 19-22, pp. 524–527 (1996) [8] Storn, R., Price, K.V.: Differential evolution-a simple and efficient heuristic for global optimization over continuous spaces. J. Global Optimization 11(4), 341–359 (1997) [9] Brutovský, B., Ulicný, J., Miškovský, P.: Application of genetic algorithms based techniques in the theoretical analysis of molecular vibrations. In: 1st Int. Conf. Genetic Algorithms Occasion 130th Anniversary Mendel’s Laws in Brno, Brno, Czech Republic, September 26-28, pp. 29–33 (1995) [10] Storn, R., Price, K.V.: Minimizing the real functions of the ICEC 1996 contest by differential evolution. In: 1996 IEEE Int. Conf. Evolutionary Computation, Nagoya, May 20-22, pp. 842–844 (1996) [11] Storn, R.: Modeling and Optimization of PET-Redundancy Assignment for MPEGSequences, Technical Report TR-95-018, International Computer Science Institute (May 1995) [12] Storn, R.: Differential Evolution Design of an IIR-Filter with Requirements for Magnitude and Group Delay, Technical Report TR-95-026, International Computer Science Institute (June 1995) [13] Storn, R.: Differential evolution design of an IIR-filter. In: 1996 IEEE Int. Conf. Evolutionary Computation, Nagoya, May 20-22, pp. 268–273 (1996) [14] Storn, R.: On the usage of differential evolution for function optimization. In: 1996 Biennial Conf. North American Fuzzy Information Processing Society, Berkeley, CA, June 19-22, pp. 519–523 (1996) [15] Storn, R.: System Design by Constraint Adaptation and Differential Evolution, Technical Report TR-96-039, International Computer Science Institute (November 1996) [16] Joshi, R., Sanderson, A.C.: Multisensor fusion and model selection us-ing a minimal representation size framework. In: 1996 IEEE/SICE/RSJ Int. Conf. Multisensor Fusion Integration Intelligent Systems, Washington, DC, Deemeber 8-11, pp. 25–32 (1996) [17] Chiou, J.P., Wang, F.S.: Hybrid differential evolution for parameter estimation of a batch bioprocess. In: IEEE Int. Symp. Control Theory Applications, Singapore, July 29-30, pp. 171–174 (1997) [18] Fleiner, C.: Parallel Optimizations: Advanced Constructs and Compiler Optimizations for a Parallel, Object Oriented, Shared Memory Language Running on a Distributed System, Ph. D. Thesis, University of Fribourg (April 11, 1997) [19] Joshi, R., Sanderson, A.C.: Experimental studies on minimal representation multisensor fusion. In: 8th Int. Conf. Advanced Robotics, Monterey, CA, July 7-9, pp. 603– 610 (1997a) [20] Joshi, R., Sanderson, A.C.: Minimal representation multisensor fusion using differential evolution. In: 1997 IEEE Int. Symp. Computational Intelligence Robotics Automation, Monterey, CA, July 10-11, pp. 266–273 (1997b) [21] Joshi, R., Sanderson, A.C.: Multisensor fusion of touch and vision using minimal representation size. In: 1997 IEEE/RSJ Int. Conf. Intelligent Robots Systems, Grenoble, September 7-11, vol. 3, pp. v4–v5 (1997c) [22] Masters, T., Land, W.: A new training algorithm for the general regression neural network. In: 1997 IEEE Int. Conf. Systems Man Cybernetics, Orlando, FL, October 12-15, vol. 3, pp. 1990–1994 (1997)

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[23] Michael, C., McGraw, G.: Opportunism and Diversity in Automated Software Test Data Generation, Technical Report RSTR-003-97-13, ver-sion 1.3, RST Corporation, Sterling, VA, USA (December 8, 1997) [24] Rüttgers, M.: Differential Evolution: A Method for Optimization of Real Scheduling Problems, Technical Report TR-97-013, International Computer Science Institute (March 1997) [25] Rüttgers, M.: Design of a method for machine scheduling for core blowers in foundries. In: Reusch, B. (ed.) Fuzzy Days 1997. LNCS, vol. 1226, p. 572. Springer, Heidelberg (1997) [26] Rüttgers, M.: Design of a new algorithm for scheduling in parallel machine shops. In: 1997 5th European Congress Intelligent Techniques Soft Computing, Aachen, Germany, September 8-11, vol. 3, pp. 2182–2187 (1997) [27] Thomas, P., Vernon, D.: Image registration by differential evolution. In: 1st Irish Machine Vision Image Processing Conf., Magee College, University of Ulster, pp. 221– 225 (1997) [28] Wang, F.S., Chiou, J.P.: Differential evolution for dynamic optimization of differential algebraic systems. In: 1997 IEEE Int. Conf. Evolutionary Computation, Indianapolis, IN, April 13-16, pp. 531–536 (1997) [29] Wang, F.S., Chiou, J.P.: Optimal control and optimal time location problems of differential-algebraic systems by differential evolution. Industrial Engineering Chemistry Research 36(12), 5348–5357 (1997) [30] Price, K., Storn, R.: Differential evolution: a simple evolution strategy for fast optimization. Dr. Dobb’s J. 22(4), 18–24, 78 (1997) [31] Chang, T.T., Chang, H.C.: Application of differential evolution to passive shunt harmonic filter planning. In: 8th Int. Conf. Harmonics Quality Power, Athens, Greece, October 14-16, vol. 1, pp. 149–153 (1998) [32] Meyer, M.: Construction of a multi-purpose X-ray CCD detector and its implementation on a 4-circle kappa goniometer, Ph. D. Thesis, l’Université de Lausanne (1998) [33] Mastorakis, N.E. (ed.): Recent Advances in Circuits and Systems. World Scientific, Singapore (1998) [34] Corn, D., Dorigo, M., Glover, F. (eds.): New Ideas in Optimization. McGraw-Hill, London (1999) [35] Qing, A.: Dynamic differential evolution strategy and applications in electromagnetic inverse scattering problems. IEEE Trans. Geosci. Remote Sens. 44(1), 116–125 (2006) [36] Bergey, P.K.: An agent enhanced intelligent spreadsheet solver for multi-criteria decision making. In: 1999 Americas Conf. Information Systems, Milwaukee, August 13-15, pp. 966–968 (1999) [37] Chang, C.S., Xu, D.Y., Quek, H.B.: Pareto-optimal set based multiobjective tuning of fuzzy automatic train operation for mass transit system. IEE Proc. B-Electric Power Applications 146(5), 577–583 (1999) [38] Rigling, B.D., Moore, F.W.: Exploitation of sub-populations in evolution strategies for improved numerical optimization. In: 10th Midwest Artificial Intelligence Cognitive Science Conf., Bloomington, Indiana, April 23-25, pp. 80–88 (1999) [39] Lee, M.H., Han, C., Chang, K.S.: Dynamic optimization of a continuous polymer reactor using a modified differential evolution algorithm. Industrial Engineering Chemistry Research 38(12), 4825–4831 (1999)

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[40] Michalski, K.A.: Electromagnetic imaging of circular-cylindrical conductors and tunnels using a differential evolution algorithm. Microwave Optical Technology Letters 27(5), 330–334 (2000) [41] Babu, B.V., Chaturvedi, G.: Evolutionary computation strategy for optimization of an alkylation reaction. In: Int. Symp. 53rd Annual Session IIChE, Science City, Calcutta, December 18-21 (2000) [42] Babu, B.V., Munawar, S.A.: Differential evolution for the optimal design of heat exchangers. In: All India Seminar Chemical Engineering Progress Resource Development: A Vision 2010 Beyond, Orissa State Center, Bhuvaneshwar (March 13, 2000) [43] Pahner, U., Hameyer, K.: Adaptive coupling of differential evolution and multiquadrics approximation for the tuning of the optimization process. IEEE Trans. Magnetics 36(4), 1047–1051 (2000) [44] Lampinen, J.: A bibliography on differential evolution algorithm, Technical Report, Lappeenranta University of Technology, Department of Information Technology, Laboratory of Information Processing (2001) (last updated on October 14, 2002), available via internet, http://www2.lut.fi/~jlampine/debiblio.htm (accessed on October 12, 2009) [45] Lampinen, J.: Solving problems subject to multiple nonlinear constraints by the differential evolution. In: 7th Int. Conf. Soft Computing, Brno, Czech Republic, June 68, pp. 50–57 (2001) [46] Angira, R., Babu, B.V.: Non-dominated sorting differential evolution (NSDE): an extension of differential evolution for multi-objective optimization. In: 2nd Indian Int. Conf. Artificial Intelligence, Pune, India, December 20-22, pp. 1428–1443 (2005) [47] Qin, A.K., Suganthan, P.N.: Self-adaptive differential evolution algorithm for numerical optimization. In: 2005 IEEE Congress Evolutionary Computation, Edinburgh, UK, September 2-5, vol. 2, pp. 1785–1791 (2005) [48] Rahnamayan, S., Tizhoosh, H.R., Salama, M.M.A.: Opposition-based differential evolution for optimization of noisy problems. In: 2006 IEEE Congress Evolutionary Computation, Vancouver, Canada, July 16-21, pp. 1865–1872 (2006) [49] Rahnamayan, S., Tizhoosh, H.R., Salama, M.M.A.: Opposition-based differential evolution algorithms. In: 2006 IEEE Congress Evolutionary Computation, Vancouver, Canada, July 16-21, pp. 2010–2017 (2006) [50] Zaharie, D.: A comparative analysis of crossover variants in differential evolution. In: Int. Multiconference Computer Science Information Technology, pp. 171–181 (2007) [51] Lawson, K.: Darwin and Evolution for Kids: His Life and Ideas with 21 Activities. Chicago Review Press, Chicago (2003) [52] Chen, C.W., Chen, D.Z., Cao, G.Z.: An improved differential evolution algorithm in training and encoding prior knowledge into feedforward networks with application in chemistry. Chemometrics Intelligent Laboratory Systems 64(1), 27–43 (2002) [53] Chakraborty, U.K. (ed.): Advances in Differential Evolution. Springer, Berlin (March 2008) [54] Fan, H.Y., Lampinen, J.: A trigonometric mutation operation to differential evolution. J. Global Optimization 27, 105–129 (2003) [55] Fan, H.Y., Lampinen, J.: A directed mutation operation for the differential evolution algorithm. Int. J. Industrial Engineering-Theory Applications Practice 10(1), 6–15 (2003)

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[56] Fischer, M.M., Hlavackova-Schindler, K., Reismann, M.: An evolutionary mutationbased algorithm for weight training in neural networks for telecommunication flow modelling, Computational Intelligence Modelling, Control Automation. In: Evolutionary Computation and Fuzzy Logic for Intelligent Control, Knowledge Acquisition and Information Retrieval, Vienna, Austria, Febuary 17-19. Concurrent Systems Engineering Series, vol. 55, pp. 54–59 (1999) [57] Qing, A.: A parametric study on differential evolution based on benchmark electromagnetic inverse scattering problem. In: 2007 IEEE Congress Evolutionary Computation, Singapore, September 25-28, pp. 1904–1909 (2007) [58] Qing, A.: A study on base vector for differential evolution. In: 2008 IEEE World Congress Computational Intelligence/2008 IEEE Congress Evolutionary Computation, Hong Kong, June 1-6, pp. 550–556 (2008) [59] Lampinen, J., Zelinka, I.: Mixed variable non-linear optimization by differential evolution. In: 2nd Int. Prediction Conf., Zlin, Czech Republic, October 7-8, pp. 45–55 (1999) [60] Krink, T., Filipič, B., Fogel, G.B., Thomsen, R.: Noisy optimization problems - a particular challenge for differential evolution? In: 2004 IEEE Congress Evolutionary Computation, Portland, OR, June 19-23, vol. 1, pp. 332–339 (2004) [61] Bindal, A., Ierapetritou, M.G., Balakrishnan, S., Armaou, A., Makeev, A.G., Kevrekidis, I.G.: Equation-free, coarse-grained computational optimization using timesteppers. Chemical Engineering Science 61(2), 779–793 (2006)

Chapter 3

A Retrospective of Differential Evolution in Electromagnetics Anyong Qing

*

3.1 Introduction 3.1.1 Coverage The electromagnetic spectrum extends from below frequencies used for modern radio to gamma radiation at the short-wavelength end, covering wavelengths from thousands of kilometers down to a fraction of the size of an atom. The long wavelength limit is the size of the universe itself, while it is thought that the short wavelength limit is in the vicinity of the Planck length, although in principle the spectrum is infinite and continuous. Radio waves, microwaves, terahertz waves, infrared light, visible light, ultraviolet light, X-rays, and gamma rays are all kinds of electromagnetic waves. However, in this chapter, attention is focused on radio waves and microwaves since other electromagnetic waves are entertained by more specific subjects, for example, optics for light waves. In addition, electromagnetics is closely related with many other disciplines. Many inter-disciplinary fields have been increasingly created through mutual invasion between such disciplines and electromagnetics. However, in this chapter, an application of differential evolution will not be classified into the electromagnetics category unless it is applied to solve an electromagnetic problem. This chapter is based purely on the literature survey mentioned in Chapter 1 of this book. Please note that some of the collected publications are not cited here due to concern of language translation accuracy for non-English publications and/or classification accuracy for those publications whose full text is unavailable to this author at this moment. Similarly, to avoid any potential misleading to readers, partial result for year 2009 is not presented here. Anyong Qing Temasek Laboratories, National University of Singapore, 5A, Engineering Dr 1 #06-09, Singapore 117411 e-mail: [email protected] *

A. Qing and C.K. Lee: Differential Evolution in Electromagnetics, ALO 4, pp. 43–71. © Springer-Verlag Berlin Heidelberg 2010 springerlink.com

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3.1.2 Pioneering Works Applications of differential evolution in electromagnetics started in 2000. It was applied to solve electromagnetic inverse problems [1]-[2], optimize impedancematching tuners for water batch sterilization [3], and design RF low noise amplifier [4].

3.1.3 An Overview of Applications of Differential Evolution in Electromagnetics 3.1.3.1 Yearly Output Number of yearly publications on applications of differential evolution in electromagnetics is depicted in Fig. 3.1. It can be seen that there is a sharp jump from 4 in 2002 to 17 in 2003. Number of publications remains relatively stable around 20 since 2003.

20

2008

28

2007

15

2006

25

2005

2004

19

17

2003

2002

4

5

2001

2000

4

Fig. 3.1 Publications on Applications of Differential Evolution in Electromagnetics

3.1.3.2 Output by Subject Through result analysis, we have identified 12 major subjects in electromagnetics in which differential evolution has been applied. Number of publications on applications of differential evolution on these subjects is depicted in Fig. 3.2.

A Retrospective of Differential Evolution in Electromagnetics 3

miscellaneous electromagnetic compatibility

1

computational electromagnetics

1

radar

2

MIMO

2

radio network design

3

frequency planning

4

electromagnetic composite materials

4

electromagnetic structures

45

6

antennas

9

microwave & RF

12

antenna arrays electromagnetic inverse problems

34 46

Fig. 3.2 Publications on Applications of Differential Evolution on Specific Subjects in Electromagnetics

3.2 Electromagnetic Inverse Problems 3.2.1 A Bird’s Eye View The goal of electromagnetic inverse problems is to recover information on some inaccessible region from the scattered electromagnetic fields measured in the exterior region invasively [5]-[6]. It is intrinsically an optimization problem and has been studied by using many other optimization algorithms [5]-[6]. It can be seen from Fig. 3.2 that electromagnetic inverse problem is one of the most studied applications of differential evolution in electromagnetics due to its apparent prospect for practical applications. 46 publications [1]-[2], [7]-[50] (book chapters excluded) on this topic have been collected by the end of year 2008. Coincidently, electromagnetic inverse problem is one of the earliest applications of differential evolution in electromagnetics [1]-[2]. It is applied to locate a circular perfectly conducting cylinder and determine its size or radius.

3.2.2 Further Classification The concerned scatterer plays a central role in electromagnetic inverse problems. It is therefore a good choice to classify electromagnetic inverse problems according to the dimension of the involved scatterers. A scatterer can be one-, two-, or three-dimensional. Accordingly, electromagnetic inverse problems are further divided into three categories: one-, two-, and threedimensional electromagnetic inverse problems. Applications of differential evolution for one-, two-, and three-dimensional electromagnetic inverse problems are shown in Fig. 3.3.

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8

13 three-dimensional

one-dimensional

two-dimensional

24

Fig. 3.3 Applications of Differential Evolution for Electromagnetic Inverse Problems

3.2.2.1 One-dimensional Electromagnetic Inverse Problems One-dimensional electromagnetic inverse problems are much easier to solve. However, in practice, there is no one-dimensional scatterer. Nevertheless, a few scatterers can be approximated as one-dimensional if its property changes much more rapidly in one dimension while changes very little or remain unchanged to a large extent in the other two dimensions. One-dimensional electromagnetic inverse problems find their application in material characterization and some other similar applications. Accordingly, applications of differential evolution in one-dimensional electromagnetic inverse problems focus on this issue. Materials characterized by this approach are summarized in Table 3.1. Table 3.1 Materials Characterized by Using Differential Evolution

Materials

References

dielectric

[7], [10], [50]

magnetic

[15]

metamaterial

[17], [26]-[27], [39]

bianisotropic

[27], [34]

3.2.2.2 Two-dimensional Electromagnetic Inverse Problems The moderate computational complexity and closeness to real applications makes two-dimensional electromagnetic inverse problems the focus of researchers in this

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community. Without exception, solving two-dimensional electromagnetic inverse problems using differential evolution have been most extensively studied, as shown in Table 3.2. Table 3.2 Differential Evolution for Two-dimensional Electromagnetic Inverse Problems

Background

Electrical property

[1]-[2], [9], [13]-[14], [23]-[24], [35], [38],

PEC free space

References

[40]-[41], [49]

dielectric

[8], [37]

conductive

[12], [21], [28]-[29], [33]

half space

dielectric

[16], [22]

cylindrical

conductive

[11], [19], [44]

It should be pointed out that some of the two-dimensional electromagnetic inverse problems have even been proposed as benchmark application problems for evaluating evolutionary algorithms including differential evolution [35], [40]. 3.2.2.3 Three-dimensional Electromagnetic Inverse Problems Three-dimensional electromagnetic inverse problems are computationally much more formidable. Consequently, differential evolution is applied to solve threedimensional electromagnetic inverse problems much later. It was not applied to three-dimensional electromagnetic inverse problems until 2004. On the other hand, three-dimensional electromagnetic inverse problems pose the greatest attraction to application engineers because they are the closest to real applications. In accordance, with the help of rapidly advancing computing hardware, applications of differential evolution in three-dimensional electromagnetic inverse problems have been seen growing since 2004. Detailed classification of these publications is presented in Table 3.3. Table 3.3 Differential Evolution for Three-dimensional Electromagnetic Inverse Problems

Background

Electrical property

free space

half space

magnetic half space

References [18], [20], [31], [48]

PEC

[25], [32], [36], [47]

dielectric

[25], [32], [36], [47]

conductive

[45]-[46]

magnetic & conductive

[30], [42]-[43]

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3.3 Antenna Arrays Synthesis of antenna arrays is the second most studied application of differential evolution in electromagnetics owing to its great practical importance. As shown in Fig. 3.2, there are 40 publications in our collection before 2009 among which 25 publications talk about conventional antenna arrays, 14 publications deal with time-modulated antenna arrays, and 1 publication touches moving phase center antenna array.

3.3.1 Conventional Antenna Arrays 3.3.1.1 Ideal Antenna Arrays By arranging array elements properly, mutual coupling between adjacent elements in an antenna array is negligible for most practical antenna arrays. In this case, the principle of pattern multiplication [51] makes it possible to focus on array factor by replacing actual array elements with isotropic point sources. Such an approach significantly simplifies the synthesis process. Ideal linear antenna arrays are the simplest for antenna engineers. In accordance, synthesis of ideal linear arrays using differential evolution has been extensively studied [52]-[65]. Planar arrays provide additional variables to control and shape the pattern of an array. They are more versatile and can provide more symmetrical patterns with lower side lobes. In addition, they can be used to scan the main beam of the antenna toward any point in space. However, computing properties of planar arrays are more expensive that much fewer studies [53], [66] focus on this topic. 3.3.1.2 Practical Antenna Arrays The performance of a practical antenna array may deviate significantly from that of its ideal counterpart due to mutual coupling among elements [66]-[68]. Some countermeasures have been taken to address the effect of mutual coupling between array elements. A simpler approach [67]-[69] assumes that the principle of pattern multiplication still holds and that element currents change in amplitude but keep their shape. Under such assumptions, the actual array elements are replaced by unit-excited complex embedded element patterns [67]-[68], or active patterns [70], which can be obtained by either measurement, or full-wave simulation. Differential evolution can then be applied to determine the complex excitation coefficient for each array element so that the performance of the concerned antenna array can meet specified requirements [67]-[68], [70]. Synthesis accuracy of the above approach may be questionable when the involved assumptions are violated. For cases with stringent requirement on synthesis accuracy, an alternative approach can be implemented in which the antenna array is regarded as an integral system. Full-wave methods are applied to simulate the performance of the whole system. Therefore, mutual coupling between array elements is an inherent ingredient of the full-wave simulation

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results. Synthesis can then be achieved by applying differential evolution to tune the complex excitation coefficients and/or positions of array elements [71]. 3.3.1.3 Phased Arrays Phased arrays are electrically steerable that the physical array can be stationary [72]-[74]. This concept brings a lot of benefits. For example, it can eliminate all the headaches of a gimbal in a radar system or keep a phased array locked onto a satellite when the phased array is mounted on a moving platform. Differential evolution has also stretched to the field of phased arrays. It was applied to realize adaptive pattern nulling [75].

3.3.2 Time-Modulated Antenna Arrays It has been a tough challenge to antenna engineers for a long time to realize ultralow sidelobe level in the far-field pattern of antenna arrays. Although, in theory, it is achievable through tapering of excitation amplitude [51], actual constraints on electrical and mechanical tolerances usually make it impractical. Time modulation [76]-[77] provides a viable and flexible solution for this challenge. Introduction of the additional degree of freedom, time, may significantly reduce dynamic range ratios. Thus, the requirements on electrical and mechanical tolerances can be greatly relaxed [56]. Differential evolution also finds its application here. Besides optimizing excitation amplitudes and phases of array elements, it is also used to obtain optimal time sequence. 3.3.2.1 Ideal Antenna Arrays with Time Modulation As mentioned before, approximating a practical antenna array as an ideal one significantly simplifies the synthesis process. More importantly, degradation of performance of concerned antenna array may be negligible under certain situations. Therefore, synthesizing ideal antenna arrays with time modulation attracts the most interest. Time-modulated linear antenna arrays [56], [78]-[80] are studied first due to its simplicity. It has been demonstrated that differential evolution together with time modulation help suppress side band and reduce dynamic range ratios. The high computational cost of synthesizing planar antenna arrays does not stop antenna engineers to show their interest on them due to their unique advantages against linear antenna arrays. With the aid of time modulation and differential evolution, ultra-low sidelobe patterns can be achieved at significantly lower dynamic range ratios [82]-[85]. Additional benefits include suppressed inherent sideband patterns and reduced ripple levels [85]. 3.3.2.2 Practical Antenna Arrays with Time Modulation Likewise, mutual coupling among elements in a time-modulated antenna array may change the performance of a practical antenna array from that of its ideal

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counterpart. An approach [86]-[87] similar with that of unit-excited complex embedded element patterns [67]-[68], or active patterns [70], has been proposed to compensate mutual coupling in time-modulated linear antenna arrays. 3.3.2.3 Phased Antenna Arrays with Time Modulation Electrical steerability of phased arrays is very attractive to antenna engineers. Time modulation together with differential evolution is implemented to bring down the sidelobe level and lower the inherent sideband patterns [81].

3.3.3 Moving Phase Center Antenna Arrays Moving phase center antenna arrays [88] were proposed as an alternative approach against time modulation to achieve ultra-low sidelobe levels which is an extremely tough challenge to antenna engineers due to systematic error, random error, and so on. As the name indicates, moving phase center antenna arrays reduce the sidelobe levels of a phased antenna array by moving its phase center to Doppler shift sidelobe signals out of the radar receiver’s passband. An interesting feature of moving phase center antenna arrays is that the phase center motion is much faster than target. Thus, moving phase center antenna arrays poses promising potentials in airborne or space-borne radar. Differential evolution was applied to optimize the static excitation amplitude distribution of a moving phase center linear array to realize ultra-low sidelobe level within passband [89].

3.4 Microwave and RF Engineering The rocketing demand on wireless communication has created great market for microwave & RF devices. This demand is expected to keep on growing rapidly in at least the near future. Microwave & RF engineers have been working very hard to understand new market requirements and design new devices to meet the market demand. Differential evolution has been chosen as one of the tools to make it real. Unsurprisingly, it is one of the earliest fields in the short application history of differential evolution in electromagnetics. It is applied to design different RF circuits such as RF low noise amplifier [4]. What’s more, it is seen from Fig. 3.2 that applications of differential evolution in microwave & RF engineering forms the third largest category.

3.4.1 Design of Microwave and RF Devices There are two approaches to design microwave & RF devices. The first approach treats designing microwave & RF devices as a natural optimization problem while the second approach designs microwave & RF devices using empirical synthesis formulas. Differential evolution finds its applications in both approaches. In the first approach, differential evolution is implemented to solve the optimization problem. On the other hand, in the second approach, differential evolution is used to extract empirical synthesis formulas based on accumulated experimental or

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simulation data. Desirable microwave & RF devices are then synthesized using the extracted empirical synthesis formulas. 3.4.1.1 Designing Microwave and RF Devices Using Differential Evolution Besides the first attempt to design RF circuits mentioned earlier [4], various microwave & RF devices, for example, high performance downconverter mixer circuit [90], power amplifier [91]-[92], high-power millimeter-wave TM01-TE11 mode converter [93], and coaxial-waveguide adaptor [94], have been designed later using differential evolution. This approach is very versatile in terms of size and type of concerned microwave & RF devices. 3.4.1.2 Extracting Empirical Synthesis Formulas Using Differential Evolution Simulation of microwave & RF devices plays a critical role in the above approach to design microwave & RF devices. In many cases, accurate simulation of microwave & RF devices is very time-consuming, especially when parasitic effect comes into play [4]. Microwave & RF engineers have been struggling to develop efficient simulators for microwave & RF devices. Sometimes, efficiency instead of accuracy is placed at the first priority in some computer-aided design packages. Loss of accuracy can be compensated in later stages through refining the preliminary design. For well-established microwave & RF devices, a viable approach to have a quick preliminary design is to extract empirical synthesis formula, usually in closed form, from accumulated accurate simulation results or even real measurements and get the desirable design using the empirical synthesis formula. In recent years, differential evolution has been found helpful in extracting closed-form synthesis formulas for various microwave & RF devices. Closed-form synthesis formulas for coplanar strip lines, [95], multilayer homogeneous coupling structure [96]-[97], coplanar waveguide [98], and microcoplanar stripline [99], have been derived.

3.4.2 Characterization of Microwave and RF Devices Manufacturers and users of microwave & RF devices have to characterize the properties of concerned microwave & RF devices. The characterization process involves three stages: calibration of measuring system, measurement, and postmeasurement modeling. 3.4.2.1 Calibration of Measuring System for Characterizing Microwave and RF Devices Differential evolution has been applied to help calibrate the measuring system [100] for characterizing microwave & RF devices. It is implemented to identify the Y network parameters of an alternative four-port characterization system.

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3.4.2.2 Modeling of Microwave and RF Devices An empirical model for a microwave & RF device instead of discrete measurement data may give users an intuitive concept about its behavior. It may even make simulation of microwave & RF devices less stressful. Differential evolution also finds its role in building empirical model for microwave & RF devices. It is applied to model the nonlinearity of a laser diode implemented in a radio-over-fiber network [101].

3.5 Antennas Antenna is one of the most critical components in wireless communication systems. Antennas in a wireless communication system serve the same purpose that eyes serve to a human. The field of antennas is vigorous and dynamic. Antenna technology has evolved from a “cut and try” operation into a true engineering art. Many major advances have been witnessed in our daily life. However, many more issues and challenges are facing us today.

3.5.1 Design of Antennas Antenna design was considered a secondary issue in overall system design in the past. Today it plays a critical role. A good design can relax system requirements and improve overall system performance. Likewise, there are two approaches to design antennas: direct and indirect. For the direct approach, design of antennas is regarded as an inherent optimization. Optimization algorithms are applied to determine parameters of desirable antennas. On the other hand, for the indirect approach, these parameters are determined using empirical synthesis formulas between antenna parameters and properties. The empirical synthesis formulas are extracted from either numerical simulation results or experimental data. The extraction is also an inherent optimization problem which can be solved by various optimization algorithms. Differential evolution finds its applications in both approaches. 3.5.1.1 Designing Antennas Using Differential Evolution Here, an optimization problem can be constructed by defining one or more antenna properties-dependent objective functions in terms of concerned antenna parameters. Differential evolution can then be applied to solve the optimization problem. Antennas designed through this approach include microstrip antenna [94], helical antenna [94], horn antenna [94], and lens antenna [102]. 3.5.1.2 Extracting Empirical Formulas for Synthesizing Antennas The objective functions involved in designing antennas are closely related with antenna properties. Computation of antenna properties may be extremely

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expensive, especially when numerical methods based on rigorous electromagnetic theory are implemented to simulate previously intractable complex antennas accurately. Lengthy simulation has been and will continue to be one of the bottlenecks for antenna engineers to pick up this approach to design antennas. The widespread interest in antennas and the advances in computational technologies have contributed significantly to the maturity of antenna engineering. Numerous simulation results and experimental data have been accumulated. It is therefore possible to extract closed-form empirical synthesis formulas between antenna parameters and properties of desirable antennas. Synthesizing antennas beating specified requirements using the empirical synthesis formulas can be done straightforwardly and efficiently. For engineering design purpose, this might be more promising. The empirical synthesis formulas play an essential role in synthesizing desirable antennas here. The extraction of empirical synthesis formulas from numerical simulation results and/or experimental data accumulated is essentially an optimization problem. The proven power of differential evolution has attracted antenna engineers to derive empirical synthesis formulas for various antennas such as horn antennas [103] and microstrip antennas [104]-[108].

3.5.2 Measurement of Antennas Antenna measurement is one of the essential parts of antenna engineering. It serves to validate theoretical data and investigate antennas which cannot be investigated analytically because of their complex structural configuration and excitation method. In some antenna measurements, phase information is either unavailable or inaccurate. Such circumstances can be encountered at high frequencies and/or in poor mechanical probe positioning apparatus. Phaseless near field antenna measurements [109] may result in lower instrumentation costs at higher frequencies and more robust measurements at lower frequencies by decreasing the phase sensitivity and probe positioning errors. Therefore, it has attracted the attention of antenna researchers worldwide. One of the key issues facing antenna engineers working on phaseless near field antenna measurements is the phase retrieval from measured amplitude data which can be cast into an optimization problem. In general, the phase functional has a very strong nonlinear dependency on the amplitude data which makes the objective function to be minimized multi-modal. Differential evolution has been chosen to face this challenge because of its strong global search capability. It was applied to retrieve phase in measuring both non-scanned and scanned beam antennas [110].

3.6 Electromagnetic Structures An electromagnetic structure is a multilayered medium with or without periodic arrays embedded, as shown in Fig. 3.4. A plain electromagnetic structure [3], [111] does not have any periodic arrays embedded in the multilayered medium.

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On the other hand, a more commonly seen term for electromagnetic structures with one or more periodic arrays embedded in the multilayered medium is frequency selective surfaces [112]-[114].

z0 z1 z2

zn zN-1 zN

εh μh ε1 μ1

t1

ε2 μ2

t2

εi μi

εN μ N

tn

tN

εg μ g

Fig. 3.4 Electromagnetic Structure

3.6.1 Plain Electromagnetic Structures Plain electromagnetic structures are in widespread use throughout the world. They usually act as absorbers [115]-[117] to reduce or even avoid reflection of incident electromagnetic waves which is vital for high survivability and mission capability. Energy absorbed is converted to heat and may then be intentionally reused for other purposes, for example, sterilization [3]. The key to have a good plain electromagnetic structure is to choose appropriate comprising materials and configure them properly. Besides trial and error, many different approaches such as graphic [118], real-coded genetic algorithm [119], optimal control in minimal time [120], simulated annealing [120], binary genetic algorithm [121]-[125], to name just a few, have been proposed to design plain electromagnetic structures. It is noted that genetic algorithms have gained more and more momentum for this problem. It has been empirically demonstrated that genetic algorithms perform much worse than differential evolution [126]. In addition, in many practical designs, choice of materials is limited that mixed optimization parameters arise [111], [119], [121]-[125]. Although binary genetic algorithm can handle mixed optimization parameters, the tedious encoding and decoding operations cause loss of both efficiency and accuracy. Design of plain electromagnetic structures is one of the earliest applications of differential evolution in electromagnetics [3]. It is applied to tune stub susceptances of impedance-matching tuners for water batch sterilization so that power delivered to the water sample can be continuously maximized. The heating

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process can therefore be kept as short as possible. In addition, costly isolators for source protection become redundant. Differential evolution has been found very flexible and efficient to handle mixed optimization parameters [126]. It is applied to design a broadband planar microwave absorber [111]. Materials used come from an in-house developed real material database in which each material is represented by a unique index. Material indices as well as thickness are determined by using differential evolution. Within [2GHz, 16GHz], the frequency band of interest, the -10dB absorption band lies in [4.5GHz, >16.0GHz] for absolute bandwidth while the 10dB absorption band lies in [3.2GHz, 13.6GHz] for relative bandwidth. Moreover, the obtained absorber configurations are found very robust with fabrication errors.

3.6.2 Frequency Selective Surfaces Frequency selective surfaces [112]-[114] get this name because of the frequency filtering property. They are the counterpart in microwave engineering against filters, one of the fundamental devices in electric circuits. They have been widely applied in a multiplicity of very important engineering areas and have contributed significantly toward advancing our living standard. A frequency selective surface contains one or more periodic arrays of either patches or apertures embedded in multilayered medium. Each array in a frequency selective surface may have its own periodicities and elements [127]. Various array elements [112]-[113], [127]-[129], have been proposed for different applications. Analysis of frequency selective surfaces has been extensively studied. Numerous algorithms such as equivalent circuit method [130], periodic moment method [113], spectral-domain method [112], vector spectral-domain method [127], [129], finite element method [131], finite difference time domain method [132], T-matrix method [133], and many other more, have been proposed. On the contrary, engineers are still struggling for an accurate and efficient approach to design frequency selective surfaces. The problem is so complex that studies on it are far beyond maturity. By far, the tedious and costly trial and error approach is still popular. In the past decade, designing frequency selective surfaces using genetic algorithms [134]-[135] has been gaining momentum because of their power to synthesize complex frequency selective surfaces which are usually unachievable through trial and error or other conventional synthesis approaches. Unfortunately, the synthesis is often very time-consuming due to the poor efficiency of genetic algorithms. In addition, the binary encoding and decoding involved not only further lengthen the synthesis but also reduce synthesis resolution. Encouraged by the excellent performance of differential evolution over electromagnetic inverse problems, we have made the first trial to design frequency selective surfaces using differential evolution [136]-[139]. Unlike the pixel-wise approach commonly used by other designers of frequency selective surfaces using genetic algorithms, attention is focused on existing array elements and their corresponding array patterns. Differential evolution is implemented to tune the

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periodic arrays and/or multilayered medium. Frequency selective surfaces with staggered dipole array [136], [138] or double square loop array [137], [139] have been successfully designed within affordable time.

3.7 Electromagnetic Composite Materials Materials have been playing an increasingly important role in our society. Recent technological advancements are largely indebted to the development of better materials. The demand for unique electromagnetic properties has been growing rapidly and consistently in numerous practical fields such as aerospace, communication, remote sensing, defense, food, medical, power and transportation industries, and many more. However, material scientists and application engineers have been more and more often frustrated to notice that conventional electromagnetic materials cannot break the performance barriers to meet practical demands for specific electromagnetic properties. In another word, material properties demanded by practical electromagnetic applications cannot be observed in conventional electromagnetic materials. Incapability of conventional electromagnetic materials has led to wide spread interest in electromagnetic composite materials. Material scientists tailor materials through combining metals, ceramics, polymers, and semiconductors in various configurations to realize desirable electromagnetic properties for concerned electromagnetic applications.

3.7.1 Modeling of Electromagnetic Composite Materials The first step in developing electromagnetic composite materials with desirable electromagnetic properties is to understand the interaction among composing ingredients. Differential evolution has joined this effort. An accurate and efficient model to predict electromagnetic properties of electromagnetic composite materials with simple inclusions, the TCQ model [140]-[142], which is based on T-matrix method, configurational averaging technique, and quasi-crystalline approximation, has been improved by using differential evolution. More exactly, differential evolution is used to determine the effective wave numbers of the concerned electromagnetic composite material that correspond to eigen value of 1 of the coefficient matrix in the governing equation for the TCQ model. With the help of differential evolution, the relationship between effective wave number of composite materials, volume concentration, wave propagation direction, size and shape of inclusion particles have been numerically studied. Effective anisotropy in electromagnetic composite materials with aligned spheroidal inclusions has been numerically confirmed for the first time [141]-[142].

3.7.2 Retrieval of Effective Permittivity Tensor Permittivity and permeability are more fundamental electromagnetic properties for electromagnetic materials than wave number. This observation also applies to

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electromagnetic composite materials since electromagnetic composite materials belong to the family of electromagnetic materials. There is a one-to-one mapping between permittivity, permeability and wave number for isotropic electromagnetic materials. Any of the three quantities can be uniquely determined if the other two quantities are known. For example, for isotropic non-magnetic material, it is well known that k = ε r k0 regardless of propagation direction, where k is the wave number in the non-magnetic material of relative permittivity εr and k0 is the wave number in free space. Therefore, the knowledge of k is equivalent to the more fundamental knowledge of relative permittivity. Unfortunately, this is not true for anisotropic electromagnetic materials whose electromagnetic properties are represented by permittivity tensor and/or permeability tensor. The relationship between wave number, permittivity tensor, and permeability tensor is highly nonlinear and may even be non-unique. It is by no means easy to retrieve permittivity tensor and/or permeability tensor from wave number of anisotropic electromagnetic materials. As mentioned above, it has been numerically confirmed that electromagnetic composite materials with aligned spheroidal inclusions are effectively anisotropic. It is therefore necessary to retrieve the effective permittivity tensor to have a deeper understanding on such effectively anisotropic composite materials. Differential evolution plays a critical role in such studies [143]. The wave numbers computed using the retrieved permittivity tensor agree well with those from the TCQ model.

3.8 Frequency Planning Frequency is a limited natural resource. It is reused as often as possible in order to maximize capacity and minimize investment. At the same time, radio interferences arising from reusing frequencies have to be avoided or minimized to satisfy national, regional, and international electromagnetic compatibility regulations. Frequency planning, or frequency assignment [144], tries to assign a limited number of available frequencies to a set of base stations, or cells, subject to a set of specified constraints. The difficulty of this problem lies with the serious shortage of available frequencies. Usually, the number of demanded frequencies by the base stations is far beyond the available ones that frequencies have to be reused. However, neighboring frequencies should not be assigned to nearby base stations so that they will not interfere with each other. Frequency planning is performed to meet the needs of, and would normally be developed in association with, a communications plan. It is a very important and demanding problem in telecommunications, especially in GSM networks. Emerging wireless communication makes effective frequency planning more and more imperative. It is arguably the most challenging and time-consuming task in designing a mobile network. Frequency planning is one of the latest fields intruded by differential evolution [145][146][147][148]. Various differential evolution strategies [145], [148], as well as hybrid differential evolution [147], have been applied to solve real-world

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frequency planning problems [144], [147]-[148]. Comparison with other optimization algorithm has also been conducted. It is very interesting to note that it is even implemented to study the effect of intrinsic control parameters [145], [148].

3.9 Radio Network Design Radio network design aims to maximize coverage while minimizing the number of base station transmitters by locating transmitters appropriately. It is another very important issue in mobile communication and often accompanies frequency planning. Radio network design is a natural optimization problem. One of the essential features of this problem is the non-uniqueness of its solutions. There are a “very high amount of possible solutions” [149]. Another important feature of this problem is its binary optimization parameters. Radio network design is one of the applications targeted by differential evolution very recently [149]-[151]. Results show that it is possible to obtain a satisfactory set of locations to cover a maximum area for a determined number of base station transmitters. Comparison with other algorithms such as random search [149], population-based incremental learning [150]-[151], simulated annealing [150]-[151], and cross-generational elitist selection, heterogeneous recombination (by incest prevention), and cataclysmic mutation [150]-[151], has been carried out. It has been noticed that differential evolution is very efficient. However, there is still plenty of room to improve its final solution. Nevertheless, it has to be pointed out that the intrinsic control parameters for all competing algorithms are not carefully tuned to their optimal values that the presented results and corresponding conclusions may be misleading due to the potential commitment of evolutionary crimes [126].

3.10 MIMO Soaring demand on high speed data transmission arising from wireless communication has imposed serious challenge to channel capacity especially in rich multipath environments. However, the conventional single-input singleoutput communication system is subject to the Shannon limit. Use of multiple antennas at both ends of wireless links has emerged as a promising solution to break the bottleneck. The multi-input multi-output communication system offers some important advantages such as higher bit-rate and increased capacity. This provides new opportunities as well as challenges for differential evolution. Deploying extra antennas will inevitably increase system complexity and cost. It is therefore one of the most fundamental problems in a MIMO system to have an optimal deployment of antennas. This problem becomes more complicated when implementing antennas at base stations costs differently with that at receivers [152]. Differential evolution is used to derive an empirical formulation for the optimum ratio of number of antennas at base station to that at mobile under low signal-to-noise condition [153].

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To take full advantage of MIMO system for better spectral efficiency and higher reliability, it is necessary for the transmitters and receivers to have knowledge of the channel coefficients. Unfortunately, this is an unrealistic assumption for most MIMO systems. The channel coefficients have to be estimated or predicted. This provides another opportunity for differential evolution. It is hybridized with particle swarm optimization to train a recurrent neural network for MIMO channel prediction [154].

3.11 Radar Radar is the invention that changed the world [155]. It operates by radiating electromagnetic energy and detecting the echo returned from reflecting objects which provides information about the target such as range, angular location, size, shape, and even trajectory of moving targets. Although it was originally developed to satisfy the needs of the military for surveillance and weapon control, it has seen many significant civil applications. Radar researchers have also taken advantage of differential evolution to facilitate their studies [156]-[157]. As early as in 2003, a hybrid differential evolution which integrates differential evolution with Newton’s method was applied to estimate the target motion parameters [156]. More accurate result than ever was obtained. Later in 2008 [157], both differential evolution and memetic differential evolution was used to design polyphase code for spread spectrum radar. The memetic differential evolution hybridizes differential evolution with a gradient-based local search algorithm with a dynamic step adaptation procedure. Differential evolution was compared with other algorithms such as evolutionary programming, particle swarm optimization, tabu search, variable neighborhood search, and genetic algorithm. Results obtained by differential evolution are comparable with those by evolutionary programming and particle swarm and are better than those by tabu search, variable neighborhood search, and genetic algorithm.

3.12 Computational Electromagnetics Computational electromagnetics (CEM) [158] aims to solve Maxwell’s equations using various numerical algorithms. It has evolved rapidly in the past decades to stand as an important tool for electrical engineers to solve radiation and scattering from complex objects that cannot be handled before. However, although tremendous progress has been witnessed after many years’ fruitful study, there is still a long way to go before it is brought to the same confidence level as that taken by circuit simulation and enjoys the same pervasiveness in engineering design as does circuit simulation. Differential evolution has also been implemented to help CEM researchers to realize their dream. It is used to trace the optimal higher-order Whitney element to enhance convergence for the vector finite element modeling of microwaves and antennas [159].

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3.13 Electromagnetic Compatibility Electromagnetic compatibility (EMC) [160] is one of the emerging branches of electrical engineering which studies the unintentional generation, propagation and reception of electromagnetic energy to properly operate different equipments that use electromagnetic phenomena in the same electromagnetic environment and avoid mutual interference. There are two key issues in EMC: emission and susceptibility. Emission issues are related to the unwanted generation of electromagnetic energy by some source, and to the countermeasures which should be taken in order to reduce such generation and to avoid the escape of any remaining energies into the external environment. Susceptibility or immunity issues, on the other hand, refer to the correct operation of electrical equipment, referred to as the victim, in the presence of unplanned electromagnetic disturbances. Most radiated emission tests in large semi-anechoic chambers according to EMC standards are very expensive. A cheaper alternate is to predict the far-field radiated emissions in the conditions of the standards using a model of the radiating equipment extracted from near-field measurements performed at shorter distances. Differential evolution has been applied to extract a dipole model of radiating equipment under test (EUT) [161]. More exactly, differential evolution is applied to determine the position, orientation and excitation current of each dipole of the equivalent set of infinitesimal dipoles distributed inside a volume enclosing the EUT. The approach has been validated by real measurements in anechoic and semi-anechoic chambers.

3.14 Miscellaneous Applications In addition to the aforementioned applications of differential evolution in electromagnetics, differential evolution has also been applied to help choose free dipole for electromagnetic formation flight [162], optimize source pulses and detection templates in ultrawide-band radio systems [163], built a prediction model for rain attenuation in a terrestrial point-to-point line of sight link at 97 GHz [164].

3.15 An Outlook to Future Applications of Differential Evolution in Electromagnetics Differential evolution has shown great promise in electromagnetics because of its natural advantages over deterministic optimization algorithms for complex optimization problems in electromagnetics. It has also been known that it outperforms some other stochastic algorithms including evolutionary algorithms [126]. However, application of differential evolution in electromagnetics is seriously lagging behind both fundamental study on differential evolution and application study in other fields such as chemistry, as revealed by the literature survey

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mentioned in Chapter 1 of this book. Moreover, it seems that application of differential evolution in electromagnetics is losing its momentum comparing to the rapid growth of interest in differential evolution. This may be due partially to the lack of awareness of differential evolution among electromagnetic researchers and misconceptions on differential evolution due to evolutionary crimes [126]. There are still lots of existing complex optimization problems in electromagtics waiting for solution. For example, three-dimensional electromagnetic inverse problems, efficient and accurate antenna design based on rigorous electromagnetic theory, antenna array design with full consideration of element properties and mutual coupling between elements, multi-objective microwave circuit design based on multi-scale full-wave simulation, design of frequency selective surfaces comprising of complicated array elements, and many more. What’s more, many more complex optimization problems in electromagtics arising from computational electromagnetics, bioelectromagnetics, electromagnetic compability, and so on, are coming to the surface. These emerging problems provide both challenges and opportunities for differential evolution.

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[131] Eibert, T.F., Volakis, J.L.: Fast spectral domain algorithm for hybrid finite element/boundary integral modelling of doubly periodic structures. IEE Proc. HMicrowaves Antennas Propagation 147(5), 329–334 (2000) [132] Ko, W., Mittra, R.: Implementation of Floquet boundary condition in FDTD for FSS analysis. In: 1993 IEEE APS Int. Symp., Ann Arbor, MI, June 28-July 2, vol. 1, pp. 14–17 (1993) [133] Gurel, L., Chew, W.C.: Recursive T-matrix algorithms for the solution of electromagnetic scattering from strip and patch geometries. IEEE Trans. Antennas Propagat. 41(1), 91–99 (1993) [134] Mittra, R.: Genetic algorithm: the last word for solving all of your design problems? In: 1997 IEEE AP-S Int. Symp., Montreal, Que. Canada, July 13-18, vol. 3, pp. 1672–1675 (1997) [135] Weile, D.S., Michielssen, E.: The use of domain decomposision genetic algorithms exploiting model reduction for the design of frequency selective surfaces. Computer Methods Applied Mechanics Engineering 186, 439–458 (2000) [136] Luo, X.F., Lee, C.K., Qing, A.: Design of frequency selective surfaces (FSS) using differential evolution strategy (DES). In: 27th ESA Antenna Technology Workshop on Innovative Periodic Antennas: Electromagnetic Bandgap, Left handed materials, Fractal and Frequency Selective Surfaces, Santiago de Compostela, Spain, March 911, pp. 201–207 (2004) [137] Luo, X.F., Teo, P.T., Qing, A., Lee, C.K.: Design of double-square-loop frequency selective surfaces using differential evolution strategy coupled with equivalent circuit model. In: 4th Int. conf. Microwave Millimeter Wave Technology, Beijing, China, August 18-21, pp. 94–97 (2004) [138] Luo, X.F., Qing, A., Lee, C.K.: On the application of differential evolution strategy to the design of frequency selective surfaces. Int. J. RF Microwave Computer-Aided Engineering 15(2), 173–180 (2005) [139] Luo, X.F., Teo, P.T., Qing, A., Lee, C.K.: Design of double-square-loop frequency selective surfaces using differential evolution strategy coupled with equivalent circuit model. Microwave Optical Technology Letters 44(2), 159–162 (2005) [140] Qing, A., Xu, X., Gan, Y.B.: Effective wave number of composite materials with oriented randomly distributed inclusions. In: 2003 IEEE AP-S Int. Symp., Columbus, Ohio, June 22-27, vol. 4, pp. 655–658 (2003) [141] Qing, A., Xu, X., Gan, Y.B.: Effective wave number of composite materials with randomly distributed inclusions. In: Int. Conf. Materials Advanced Technologies, Singapore, December 7-12, pp. 44–47 (2003) [142] Qing, A., Xu, X., Gan, Y.B.: Anisotropy of composite materials with inclusion with orientation preference. IEEE Trans. Antennas Propagation 53(2), 737–744 (2005) [143] Qing, A., Xu, X., Gan, Y.B.: Effective permittivity tensor of composite material with aligned spheroidal inclusions. J. Electromagnetic Waves Applications 18(7), 899–910 (2004) [144] Eisenblätter, A., Koster, A.: FAP Web – A Website about Frequency Assignment Problems, http://fap.zib.de/ (last accessed on December 28, 2009) [145] Maximiano, M.D.S., Vega-Rodriguez, M.A., Gomez-Pulido, J.A., Sánchez-Pérez, J.M.: Solving the frequency assignment problem with differential evolution. In: 15th Int. Conf. Software Telecommunications Computer Networks, Split, Croatia, September 27-29, pp. 119–123 (2007)

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[146] Bernardino, E., Bernardino, A., Pérez, J.M.S., Pulido, J.A.G., Rodríguez, M.A.V.: Solving the frequency assignment problem using genetic algorithms, evolutionary simulated annealing and differential evolution. In: IASTED Int. Conf. Software Engineering, pp. 330–335 (2008) [147] Chaves-Gonzalez, J.M., Maximiano, M.D., Vega-Rodriguez, M.A., Gómez-Pulido, J.A., Sánchez-Pérez, J.M.: Comparing hybrid versions of SS and DE to solve a realistic FAP problem. In: Corchado, E., Abraham, A., Pedrycz, W. (eds.) HAIS 2008. LNCS (LNAI), vol. 5271, pp. 257–264. Springer, Heidelberg (2008) [148] Maximiano, M.D.S., Vega-Rodríguez, M.A., Gómez-Pulido, J.A., Sánchez-Pérez, J.M.: Analysis of parameter settings for differential evolution algorithm to solve a real-world frequency assignment problem in GSM networks. In: 2nd Int. Conf. Advanced Engineering Computing Applications Sciences, Valencia, Spain, September 29-October 4, pp. 77–82 (2008) [149] Priem Mendes, S., Gómez Pulido, J.A., Vega Rodríguez, M.A., Jaraíz Simón, M.D., Sánchez Pérez, J.M.: A differential evolution based algorithm to optimize the radio network design problem. In: 2nd IEEE Int. Conf. e-Science Grid Computing, Amsterdam, The Netherlands, December 2006, pp. 119–125 (2006) [150] Vega-Rodriguez, M.A., Gomez-Pulido, J.A., Alba, E., Vega-Perez, D., PriemMendes, S.: Using omnidirectional BTS and different evolutionary approaches to solve the RND problem. In: Moreno Díaz, R., Pichler, F., Quesada Arencibia, A. (eds.) EUROCAST 2007. LNCS, vol. 4739, pp. 853–860. Springer, Heidelberg (2007) [151] Vega-Rodriguez, M.A., Gomez-Pulido, J.A., Alba, E., Vega-Perez, D., PriemMendes, S., Molina, G.: Evaluation of different metaheuristics solving the RND problem. In: Giacobini, M. (ed.) EvoWorkshops 2007. LNCS, vol. 4448, pp. 101– 110. Springer, Heidelberg (2007) [152] Du, J., Li, Y.: Optimization of antenna configuration for MIMO systems. IEEE Trans. Communications 53(9), 1451–1454 (2005) [153] Develi, I.: Determination of optimum antenna number ratio based on differential evolution for MIMO systems under low SNR conditions. Wireless Personal Communications 43(4), 1667–1673 (2007) [154] Potter, C., Venayagamoorthy, G.K., Kosbar, K.: MIMO beam-forming with neural network channel prediction trained by a novel PSO-EA-DEPSO algorithm. In: 2008 IEEE World Congress Computational Intelligence/2008 IEEE Int. Joint Conf. Neural Networks, Hong Kong, June 1-6, pp. 3338–3344 (2008) [155] Buderi, R.: The Invention That Changed the World: How a Small Group of Radar Pioneers Won the Second World War and Launched a Technological Revolution. Simon & Schuster, New York (1996) [156] Wei, G., Wu, S., Mao, E.: Differential evolution for target motion parameter estimation. In: 2003 IEEE Int. Conf. Neural Networks Signal Processing, Nanjing, China, December 14-17, vol. 1, pp. 563–566 (2003) [157] Perez-Bellido, A.M., Salcedo-Sanz, S., Ortiz-Garcia, E.G., Portilla-Figueras, J.A., Lopez-Ferreras, F.: A comparison of memetic algorithms for the spread spectrum radar polyphase codes design problem. Engineering Applications Artificial Intelligence 21(8), 1233–1238 (2008) [158] Chew, W.C., Jin, J.M., Michielssen, E., Song, J. (eds.): Fast and Efficient Algorithms in Computational Electromagnetics. Artech House, Boston (2001)

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[159] Rekanos, I.T., Yioultsis, T.V.: Convergence enhancement for the vector finite element modeling of microwaves and antennas via differential evolution. AEU Int. J. Electronics Communications 60(6), 428–434 (2006) [160] Paul, C.R.: Introduction to Electromagnetic Compatibility, 2nd edn. John Wiley, New York (2006) [161] Regué, J.R., Ribó, M., Gomila, J., Pérez, A., Martin, A.: Modeling of radiating equipment by distributed dipoles using metaheuristic methods. In: IEEE Int. Symp. Electromagnetic Compatibility, Chicago, IL, August 8-12, vol. 2, pp. 596–601 (2005) [162] Electromagnetic Formation Flight, NRO DII Final Review, (August 29, 2003) [163] Shan, D.M., Chen, Z.N., Wu, X.H.: Signal optimization for UWB radio systems. IEEE Trans. Antennas Propagation 53(7), 2178–2184 (2005) [164] Develi, I.: Differential evolution based prediction of rain attenuation over a LOS terrestrial links situated in the southern United Kingdom. Radio Science 42(3), Art. No. RS3011 (June 2007)

Chapter 4

Application of Differential Evolution to a Two-Dimensional Inverse Scattering Problem Krishna Agarwal, Xudong Chen, and Yu Zhong

*

4.1 Introduction Inverse scattering problems [1]-[2] are of great importance in non-destructive and non-invasive evaluation applications. Typically, the region of investigation is inaccessible and has to be evaluated using different approaches including electromagnetic waves. In such scenarios, the region is illuminated by electromagnetic waves from various directions and the electromagnetic fields scattered by objects in the region are measured at various receivers. The electrical and geometric properties of objects present inside the region are then reconstructed using the measured scattered electromagnetic fields. In recent years, many researchers [3]-[28] have started using differential evolution in the field of inverse scattering problems. Qing has been working towards designing differential evolution strategies that are suitable for electromagnetic problems [5], [7], [14], [19], [23], [26]. Use of differential evolution for two-dimensional inverse scattering problems has been considered in [3]-[5], [7]-[8], [13]-[14], [19]-[20], [27]-[28]. Chen et al have used differential evolution to reconstruct the electromagnetic properties of anisotropic and bianisotropic scatterers [9], [15], [18]. Detection of unexploded ordnances using differential evolution was reported in [10], [16]-[17], [24]. Application of differential evolution in electrical impedance tomography has been reported in [6], [11]-[12], [22]. In this chapter, we consider a two-dimensional inverse scattering problem and present the application of differential evolution for this example. In addition, the impact of the initial guess on the optimization algorithm is presented. Among Krishna Agarwal · Xudong Chen · Yu Zhong Department of Electrical and Computer Engineering, National University of Singapore, Singapore – 117576 e-mail: [email protected], [email protected], [email protected] *

A. Qing and C.K. Lee: Differential Evolution in Electromagnetics, ALO 4, pp. 73–105. © Springer-Verlag Berlin Heidelberg 2010 springerlink.com

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various qualitative methods [29]-[43], [46]-[47] useful for generating initial guess, multiple signal classification [36], [38], [42]-[43] for estimating scatterer profile and a least squares based retrieval method [39], [42]-[43], [48] for estimating the relative permittivity are illustrated.

4.2 General Description of the Problem 4.2.1 Experimental Setup The problem investigates a region that typically contains M cylinders infinitely extending in the z direction as shown in Fig. 4.1.

Fig. 4.1 Domain of Investigation

The background medium is assumed to be free space. The cylinders are assumed to be homogeneous and dielectric. The relative permittivity of cylinder m is represented using εm. As shown in Fig. 4.1, the position of cylinder m can be represented using its local origin, Om, while its contour is represented using local shape functions [2], [5], [49]-[50]. The relative permittivity, position, and contour of the cylinders are unknown. The solution to the problem would be a correct estimate of the relative permittivity, position and contour of each cylinder that is present in the domain. To solve the problem, Ns number of sources and Nd number of detectors are placed around the domain as shown in Fig. 4.2. The locations of the sources and the detectors are represented by rs, 1≤s≤Ns and rd, 1≤d≤Nd, respectively.

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Fig. 4.2 Measurement Setup

In the considered problem, the sources are electric current line sources, and hence the incident wave is a transverse magnetic (TM) wave. Because the cylinders are infinite in the z direction and are dielectric in nature, the scattered field will also be transverse magnetic in nature. During the measurements, one source illuminates the domain at a time and all of the Nd detectors receive electric field. This electric field received at the detectors has two components: the electric field that is radiated directly from the source and the electric field that is scattered by the cylinders present in the region of investigation. Since the electric field incident due to radiation from the source is known, the scattered field,

E zsca (rd ,rs )zˆ , can be found easily and is considered the

measured quantity. The expression of the scattered electric field

E zsca (rd ,rs )zˆ in terms of

cylinders’ parameters is given using an integral equation as below:

E zsca (rd , rs ) =

−ωμ 0 4

∑ ∫ J (r′, r )H (k M

m=1

m

s

(1) 0

0

rd − r′ )dr′

(1)

where, k0 is the free space wave number, H0(1)(·) denotes the Hankel function of the first kind of the zeroth order, and Jm(r', rs) represents the z directed electric current induced on point r' present on the mth cylinder, when the source located at rs is radiating. The definitions of other terms are as commonly used in electromagnetics. It should be noted that the time harmonic term e-iωt is implicitly present and suppressed for simplicity.

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A multistatic response matrix Esca of size Nd × Ns can be formed, such that its

(d, s)th element is E z (rd , rs ) . This multistatic response matrix is typically used for solving the inverse scattering problem. sca

4.2.2 The Optimization Problem In the following description, cubic B-splines local shape functions [2], [5], [50] have been used. Accordingly, let the contour of the m th cylinder be represented using Nm number of control points, rnm, 1≤n≤Nm, defined with respect to the local origin. Thus, for each cylinder, (Nm + 3) parameters have to be retrieved. The inverse problem shall involve the retrieval of

⎛M ⎞ ⎜ ∑ N m + 3M ⎟ parameters, ⎝ m=1 ⎠

x = {O m , ε m , rnm | 1 ≤ n ≤ N m ,1 ≤ m ≤ M } . Here, the notation x is used to denote the vector containing the optimization parameters and is referred to as the optimization vector. A cost function can be defined as a function of the optimization vector x as below:

f (x ) = E sca − E(x ) E sca

(2)

where ║·║ denotes the Frobenius norm of a matrix and E(x) is the multistatic response matrix computed for the optimization vector x using a forward computation method. In this chapter, the method of moments has been used as the forward computation method [51]. It is obvious that the error function f(x) is very small if the vector x is close to the actual parameters of the cylinders present in the region. Other cost functions may be used or additional functions like penalty functions may be employed. However, this point is not in the scope of the present chapter.

4.3 Mathematical Nature of the Optimization Problem and Differential Evolution It is a well known fact that inverse scattering problems are often ill-posed [22], [37], [52]. The primary reasons for the ill-posedness of inverse scattering problems are listed below: 1.

Data insufficiency: the information that can be collected from multistatic measurements as described above is limited and usually not enough to reconstruct the region completely. This may happen due to two reasons. Sometimes, full aspect of the region of investigation is not available, thus reducing the information that can be collected. Full aspect or not, the number of measurements are always finite due to obvious practical reasons. Often, the number of measurements is far less than the number of unknowns.

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2.

3.

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Non-uniqueness: For extended scatterers (scatterers that are not small in comparison to the wavelength used for measurements), the mapping from the scatterers’ properties to the measured field is non-injective. Instability: The presence of noise in the measurements makes the solutions of the inverse scattering problem unstable.

The inverse scattering problems are further complicated due to the fact that the measured fields are often non-linear functions of the parameters to be retrieved. This fact is evident in Eq. (1) since the induced current, Jm(r', rs), is a non-linear expression of the permittivity of the cylinder εm and incorporates the multiple scattering effect. Due to these reasons, the solution space of inverse scattering problems may have various local minima and unstable solutions. The nature of the problem makes the selection of the globally optimal solutions difficult. In such situation, evolutionary algorithms like differential evolution [53]-[54] provide important and practically useful tools for optimization. The advantage of differential evolution for inverse scattering problems over evolutionary algorithms like genetic algorithm has been proven in [5], [23] and various works thereafter. We succinctly mention here the features of differential evolution that make it suitable for the inverse scattering problems [53]-[54]: 1.

2.

Since the solution space is non-linear and has multiple local minima, traditional gradient based methods cannot be used directly for solving the problem. This is because the performance of gradient based algorithms is highly dependent on the initial guess. Further, computing the gradients for such problems is sometimes difficult. In such scenario, the random exploratory schemes that begin with multiple starting points and use multiple search directions are of great advantage. As opposed to gradient based methods, exploratory methods are generally very random and unguided. They may typically require a lot of iterations and computational resources to converge to the global minimum. Further, the population size required by such methods is typically very large, making the optimization computationally intensive. Thus, given a good starting point, a guided updating scheme is more efficient for reaching global minima.

Differential evolution is a very good combination of the exploratory features typically demonstrated by evolutionary algorithms and a random but guided updating scheme that ensures fast convergence [53]-[54]. These features make it suitable for the inverse scattering problems.

4.4 Initial Guess In practice, no prior information is known about the required solution. Typically, even the number of cylinders is not known. Despite the strengths of differential evolution, it is very difficult to solve the problem without any prior information. Therefore most approaches assume some kind of a priori information, known as initial guess.

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In the problem being considered, the initial guess typically consists of one or more of the following: • • •

number of cylinders, approximate centers of the cylinders, or search range for εm, and other optimization parameters, such as rnm.

The advantages of such initial guess are obvious. For example, if the number of cylinders is not known, typical approaches use sufficiently large number of cylinders and hope that the optimization will reduce the number of extra cylinders by eventually shrinking their sizes to be negligibly small. This is not only computationally expensive, but also unreasonably optimistic. If the centers of the cylinders, Om, are known, the dimension of the optimization vector reduces by 2M. Such reduction is very valuable to the optimization process. Providing a tighter constraint for the rest of the parameters is greatly beneficial for quick convergence. If these constraints are chosen correctly, part of the local minima may be excluded, thus increasing the chance of correct convergence. There are many qualitative methods that can be used to generate the initial guess. Some of the examples include multiple signal classification (MUSIC), decomposition of the time reversal operator (DORT), linear sampling method (LSM), and factorization method (FM). These methods are useful for determining the scatterer support, or the approximate profile of the scatterer in the space. These methods can be used to determine the number of cylinders, their approximate centers and some useful bounds on rnm. Here, multiple signal classification is presented as a tool to generate the initial guess. After determining the support, a least squares based method can be used to generate an estimate of the relative permittivity. Before introducing the above mentioned methods for generating the initial guess, a mathematical formulation of the forward scattering problem known as the FoldyLax model [42]-[46] of scattering is presented. This formulation serves two purposes. First, it is an indispensable formulation that will be used in the multiple signal classification as well as in the least squares based method. Second, it provides another perspective of the problem at hand, in terms of the physics involved.

4.4.1 Foldy-Lax Model of Scattering Let the cross-sections of the cylinders be divided into a total number of Np small square pixels, each of dimension a which is very small as compared to wavelength λ. Let the centers of the pixels be represented as rp. Then, Eq. (1) can be written as below using the Foldy-Lax model of multiple scattering:

E zsca (rd , rs ) =

−ωμ0 4

∑ J (r , r )H (k p

s

∀p

(1) 0

0

rd − rp

)

(3)

The expression for J(rp, rs) is as below:

⎡ J (rp , rs ) = ξ p ⎢ E zinc (rp , rs ) − ωμ4 0 ⎣

∑ J (r′ , r )H (k

p′≠ p

p

s

(1) 0

0

)

⎤ r p − r ′p ⎥ ⎦

(4)

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where ξp represents the electric polarization at the pth pixel, which is given in terms of the relative permittivity, εm(rp), at the pixel as:

ξ p = −iωa 2ε 0 [ε m (rp ) − 1]

(5)

E zinc (rp , rs ) is the z directed electric field incident on the pth pixel from

Further,

the source located at rs. The second term on the right hand side of Eq. (4) represents the multiple scattering among the pixels. Considering all the measurements, Eqs. (3) and (4) can be written in matrix form as below:

E sca = G sca ⋅ J

(6)

J = ξ ⋅ (Einc + G mut ⋅ J )

(7)

where •

Gsca is a Nd × Np-dimensional matrix that represents the Green’s function and maps the currents induced on the pixels to the scattered field measured at the detectors. The (d, p)th element of Gsca is given by

• • •

− ωμ 0 4

(

)

H 0(1) k0 rd − rp .

J is a Np × Ns-dimensional matrix representing the electric current induced at the various pixels due to the various incidences. The (p, s)th element of J is J(rp, rs). ξ is a Np × Np-dimensional diagonal matrix, in which the pth diagonal element is ξp. Einc is a Np × Ns-dimensional matrix representing the electric field incident from the sources at the various pixels. The (p, s)th element of Einc is given by

E zinc (rp , rs ) .

•

Gmut is a Np × Np-dimensional matrix, representing the Green’s function and maps the currents induced on the other pixels to the scattered field received at a pixel. The (p, p') term of Gmut is given by

− ωμ 0 4

(

)

H 0(1) k 0 rp − rp′ and the

diagonal elements of Gmut are zero.

4.4.2 Multiple Signal Classification for Estimating the Scatterer Support Let G(r) be a Nd-dimensional vector, whose dth element represents the Green’s function

− ωμ 0 4

H 0(1) (k0 rd − r ) , where r belongs to the region of interest. Then,

using Eq. (6), the range, S0, of the multistatic response matrix, Esca, is spanned by the vectors G rp ; ∀p . On the other hand, using the singular value decomposition of

{ ( ) }

Esca, such that Esca · vl=σlul and E*sca · ul=σlvl, where the superscript * denotes the Hermitian, the noise subspace Snoise, that is orthogonal to the range S0, is spanned by

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the vectors {ul; σl = 0} [55]. Considering the orthogonality of S0 and Snoise, it is evident that G(rp) is orthogonal to {ul; σl = 0} [42]-[43]. Mathematically, | ul*· G(rp)| = 0 for σl = 0. Thus, a pseudospectrum that generates very high values (poles) at rp can be formed as below: 2 Φ(r ) = log10 ⎡⎢ u*l ⋅ G (r ) ;σ l = 0⎤⎥ ⎣ ⎦

−1 2

(8)

where the overhead bar stands for average of the quantity under it. In the noisy scenario, the trailing singular values σl are not equal to zero. The trailing singular values rather take small non-zero values below the noise threshold. In such case, the noise subspace Snoise is spanned by the left singular vectors whose corresponding singular values are very small [38], [43]. After plotting the MUSIC pseudospectrum, the number of scatterers and their approximate distributions can be determined heuristically. Alternatively, the scatterer support can be estimated by choosing the points for whom Ф(r) is above a chosen threshold Фth. After generating an estimate of the scatterer support, an estimate of the permittivity can be obtained using the least squares based method, which is discussed below.

4.4.3 Least Square Based Method for Generating Initial Guess for the Relative Permittivity This method employs two stages of least squares based pseudoinverse in order to generate an estimate of the relative permittivity of the estimated scatterer support. For the ease of reference, the points that are estimated to be belonging to the scatterer support be denoted as rq. Then, the matrices used in Eqs. (6) and (7) correspond to rq instead of rp. For the estimated support, the matrices that are unknown are J and ξ. The matrix J can be computed as below:

J = G +sca ⋅ E sca

(9)

where the superscript + denotes the matrix pseudoinverse based on the least squares error. After computing J, using Eq. (7), ξ can be expressed as ξ = J · Z+, where Z = Einc + Gmut · J. However, since ξ is a diagonal matrix, the qth element of ξ, ξq can be computed by using:

ξ q = J q ⋅ Z +q

(10)

where Jq and Zq are the qth row vectors in J and Z respectively. The value of the relative permittivity εq can then be calculated using Eq. (5). Since the cylinders considered are homogeneous, the average value of εq can be used as the initial guess for the optimization.

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4.5 Numerical Results 4.5.1 Measurement Setup Five examples have been considered. In all the examples, the region of investigation is a square region centered at the origin and measuring 3 λ on each side. The region is illuminated by 8 sources, while 32 detectors are used to detect the scattered electric field. The sources and detectors are placed uniformly along a circle of radius 2.5λ centered at the origin. Thus, the sth source is placed at an angle θ s = 2π s 8 at a distance 2.5λ from the origin and the dth detector is placed

at an angle θd = 2πd 832 at a distance 2.5λ from the origin. The measured electric fields are corrupted by additive white Gaussian noise of 10%.

4.5.2 Control Parameters The control parameters for dynamic differential evolution scheme are chosen based on the parametric study presented in [23]. In this study, the advantage of the dynamic differential evolution [19] has been clearly demonstrated in terms of the sensitivity of the performance to the control parameters. It was also demonstrated that a small population size is usually sufficient for the optimization problem. The performance graphs presented a guideline for choosing the mutation intensity and the cross-over probability. Following this parametric study, the population size for all the numerical examples in this chapter is twice the number of optimization parameters. The mutation intensity is 0.6 and the crossover probability is 0.9. The termination of the optimization is subject to any of the three conditions stated below: 1) f(x) < fth , where fth is a threshold value, usually determined by the level of noise in the measurement. Since all the examples in this chapter are corrupted by 10% additive white Gaussian noise, the value of fth is chosen to be 0.1. 2) There is no improvement in the optimization process over 30 iterations, meaning that neither the average of f(x) for the current population is better than the average of f(x) for the population before 30 iterations, nor the f(xbest) of the current population is better than the f(xbest)of the population before 30 iterations. 3) The number of iterations has exceeded 1000. In the numerical examples, if an initial guess for rnm is not used, the search region for rnm is constrained as 0.1λ≤ rnm ≤1λ. If an initial guess rini is used, the search region for rnm is constrained as rini - 0.2λ ≤ rnm ≤ rini + 0.2λ. Similarly, if an initial guess for εm is not used, the search region for εm is constrained as 1≤ εm ≤10. If an initial guess εini is used, the search region for εm is constrained as max(1, εini – 3) ≤ εm ≤ εini + 3. Initial guess for the number of cylinders (M') and their approximate centers generated using the multiple signal classification is used in all the examples.

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4.5.3 Numerical Example 1: A Single Cylinder In this example, the region contains only one cylinder of radius 0.5λ centered at the origin. The relative permittivity of the cylinder is 2. The experimental setup is shown in Fig. 4.3.

Fig. 4.3 A Single Cylinder

The result of the multiple signal classification method is shown in Fig. 4.4. It is evident that the region contains one cylinder only, i.e. M'=1. The center of the cylinder can be chosen approximately as O1(0, 0).

Fig. 4.4 Pseudospectrum Formed Using Multiple Signal Classification

A threshold of Фth = -0.6 is applied on the pseudospectrum for estimating the scatterer support. The estimated scatterer support is shown in Fig. 4.5.

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Fig. 4.5 Scatterer Support Estimated (white) Using Multiple Signal Classification (Фth = -0.6)

Further, a reasonable estimate of rnm is about 0.55λ. Subsequently, the estimated scatterer support is used to generate an initial guess for the relative permittivity using the least squares method. The computed estimate of the relative permittivity is εini = 3.1116. Since M'=1 and the center of the cylinder is estimated, the number of optimization parameters is N + 1, where N is the number of control points. Here, N = 6 has been chosen. We present two sets of results for this example in order to understand the impact of the initial guess on the optimization. The first set of results does not use the estimates of rnm and εini. Fig. 4.6 shows the convergence profile of the optimization problem for a typical run. The blue solid line plots f (xbest) for various iterations while the red dashed line plots average of f (x) for various iterations.

Fig. 4.6 Convergence of the Optimization Algorithm (initial guess has been used for M' and Om)

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Fig. 4.7 shows the relative permittivity of xbest for various iterations. The algorithm terminates at the 1000th iteration due to the third termination condition with εbest = 1.979.

Fig. 4.7 Relative Permittivity of xbest for Various Iterations (initial guess has been used for M' and Om)

The contours of the cylinders corresponding to the xbest (dashed lines) for the first iteration and the 1000th iteration are shown in Figs. 4.8 and 4.9 respectively. The original profile of the cylinder is also plotted for comparison.

Fig. 4.8 Contour of the Cylinder Corresponding to xbest (dashed lines) of the Initial Population (initial guess has been used for M' and Om)

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Fig. 4.9 Contour of the Cylinder Corresponding to xbest (dashed lines) of the Final Population (initial guess has been used for M' and Om)

The second set of results uses the estimates of rnm and εini. Fig. 4.10 shows the convergence profile of the optimization problem for a typical run. As compared to Fig. 4.6, it is evident that the optimization converges much faster (282 iterations).

Fig. 4.10 Convergence of the Optimization Algorithm (initial guess has been used for M', Om, rnm and εini)

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Fig. 4.11 shows the relative permittivity of the xbest for various iterations. The algorithm terminates at the 282nd iteration due to the first termination condition with εbest = 1.99.

Fig. 4.11 Relative Permittivity of xbest for Various Iterations (initial guess has been used for M', Om, rnm and εini)

The contours of the cylinders corresponding to the xbest (dashed lines) for the first iteration and the 282nd iteration are shown in Figs. 4.12 and 4.13 respectively.

Fig. 4.12 Contour of the Cylinder Corresponding to xbest (dashed lines) of the Initial Population (initial guess has been used for M', Om, rnm and εini)

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Fig. 4.13 Contour of the Cylinder Corresponding to xbest (dashed lines) of the Final Population (initial guess has been used for M', Om, rnm and εini)

4.5.4 Numerical Example 2: Two Identical Cylinders The region contains two circular cylinders placed at (-0.8λ, 0) and (0.8λ, 0). The radii and the relative permittivities for both cylinders are 0.3λ and 2 respectively. The geometry is shown in Fig. 4.14.

Fig. 4.14 Two Identical Cylinders

The result of multiple signal classification method is shown in Fig. 4.15. It is evident that the region contains two cylinders, i.e. M' = 2. The centers of the cylinders can be chosen approximately as O1(-0.825λ, 0) and O2(0.825λ, 0).

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Fig. 4.15 Pseudospectrum Formed Using the Multiple Signal Classification

A threshold of Фth = -0.35 is applied on the pseudospectrum for estimating the scatterer support. The estimated scatterer support is shown in Fig. 4.16. Initial guess for rnm is taken to be 0.35λ. Subsequently, the estimated scatterer support is used to generate an initial guess for the relative permittivity using the least squares method. The computed estimate of the relative permittivity is εini = 3.7708.

Fig. 4.16 Scatterer Support Estimated (white) Using Multiple Signal Classification (Фth = -0.35)

Since M' = 2 and the centers of the cylinders are estimated, the number of optimization parameters is 2(N+1), where N is the number of control points for each cylinder. Here, N = 6 has been chosen. Fig. 4.17 shows the convergence profile of the optimization problem for a typical run. The blue solid line plots f (xbest) for various iterations while the red dashed line plots average of f (x) for various iterations.

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Fig. 4.17 Convergence of the Optimization Algorithm (initial guess has been used for M', Om, rnm and εini)

Fig. 4.18 shows the relative permittivity of the xbest for various iterations. The algorithm terminates at the 225th iteration due to the first termination condition. The retrieved relative permittivity is 2.035 and 2.064 respectively.

Fig. 4.18 Relative Permittivity of xbest for Various Iterations (initial guess has been used for M', Om, rnm and εini)

The contours of the cylinders corresponding to the xbest (dashed lines) for the first iteration and the 225th iteration are shown in Figs. 4.19 and 4.20 respectively. The original profile of the cylinder is also plotted for comparison.

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Fig. 4.19 Contours of the Cylinders Corresponding to xbest (dashed lines) of the Initial Population (initial guess has been used for M', Om, rnm and εini)

Fig. 4.20 Contours of the Cylinders Corresponding to xbest (dashed lines) of the Final Population (initial guess has been used for M', Om, rnm and εini)

4.5.5 Numerical Example 3: Two Different Cylinders The region now contains two circular cylinders placed at (-0.8λ, 0) and (0.8λ, 0). The radius for both the cylinders is 0.3λ. However, the relative permittivity of the cylinders are 2 and 3 respectively. The geometry is shown in Fig. 4.21.

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Fig. 4.21 Two Different Cylinders

The result of multiple signal classification method is shown in Fig. 4.22. It is noticeable that the pseudospectrum corresponding to the cylinder with higher permittivity (right side) is more spread out than the other. However, it cannot be concluded whether the wider pattern is due to the higher permittivity or the larger size of the cylinder. Nevertheless, the presence of two cylinders is unarguable, i.e. M' = 2.

Fig. 4.22 Pseudospectrum Formed Using the Multiple Signal Classification

The centers of the cylinders can be chosen approximately as O1(-0.825λ, 0) and O2(0.825λ, 0). A threshold of Фth = -0.45 is applied on the pseudospectrum for estimating the scatterer support. The estimated scatterer support is shown in Fig. 4.23.

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Fig. 4.23 Scatterer Support Estimated (shown in white) Using Multiple Signal Classification (Фth = -0.45)

Initial guess for rnm is taken to be 0.45λ, such that the constraint on rnm, [0.25λ, 0.65λ], covers both cylinders in the estimated support approximately. Subsequently, the estimated scatterer support is used to generate an initial guess for the relative permittivity using the least squares method. The computed estimate of the relative permittivity is εini = 3.1617. Since M' = 2 and the centers of the cylinders are estimated, the number of optimization parameters is 2(N+1), where N is the number of control points for each cylinder. Here, N = 6 has been chosen. Fig. 4.24 shows the convergence profile of the optimization problem for a typical run. The blue solid line plots f(xbest) for various iterations while the red dashed line plots average of f (x) for various iterations.

Fig. 4.24 Convergence of the Optimization Algorithm (initial guess has been used for M', Om, rnm and εini)

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Fig. 4.25 shows the relative permittivity of the xbest for various iterations. The algorithm terminates at the 635th iteration due to the second termination condition. The retrieved relative permittivity is 1.969 and 2.877 respectively.

Fig. 4.25 Relative Permittivity of xbest for Various Iterations (initial guess has been used for M', Om, rnm and εini)

The contours of the cylinders corresponding to the xbest (dashed lines) for the first iteration and the 635th iteration are shown in Figs. 4.26 and 4.27 respectively. The original profile of the cylinder is also plotted for comparison.

Fig. 4.26 Contours of the Cylinders Corresponding to xbest (dashed lines) of the Initial Population (initial guess has been used for M', Om, rnm and εini)

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Fig. 4.27 Contours of the Cylinders Corresponding to xbest (dashed lines) of the Final Population (initial guess has been used for M', Om, rnm and εini)

It can be seen that this example took more number of iterations for convergence as compared to the previous examples. Furthermore, the algorithm terminates due to no improvement in the cost function over thirty iterations. The value of f (xbest) at the end of the last iteration is 0.1103 . Such convergence pattern is reasonable because this inverse problem is more complicated than the previous examples due to the dissimilarity between the permittivity of the cylinders.

4.5.6 Numerical Example 4: Two Closely Located Identical Cylinders The region contains two circular cylinders placed at (-0.5λ, 0) and (0.5λ, 0). The radii and the relative permittivity for both cylinders are 0.3λ and 2 respectively. The geometry is shown in 4.28.

Fig. 4.28 Two Closely Located Identical Cylinders

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This example is similar to the example 2. However, the cylinders are now closer to each other. Due to their proximity, the coupling between them is higher (mutual multiple scattering), which makes this inverse problem comparatively more difficult than the example 2. The result of multiple signal classification method is shown in 4.29. Though the cylinders are close to each other, it is still clearly evident that the region contains two cylinders, i.e. M' = 2. The centers of the cylinders can be chosen approximately as O1(-0.4λ, 0) and O2 (0.45λ , 0) .

Fig. 4.29 Pseudospectrum Formed Using the Multiple Signal Classification

A threshold of Фth = -0.4 is applied on the pseudospectrum for estimating the scatterer support. The estimated scatterer support is shown in Fig. 4.30. Initial guess for rnm is taken to be 0.4λ. Subsequently, the estimated scatterer support is used to generate an initial guess for the relative permittivity using the least squares method. The computed estimate of the relative permittivity is εini = 4.33. It is noticeable that the estimated relative permittivity is higher than all the previous examples. This is due to the stronger coupling between the cylinders.

Fig. 4.30 Scatterer Support Estimated (shown in white) Using Multiple Signal Classification (Фth = -0.4)

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Since M' = 2 and the centers of the cylinders are estimated, the number of optimization parameters is 2(N+1), where N is the number of control points for each cylinder. Here, N =6 has been chosen. Fig. 4.31 shows the convergence profile of the optimization problem for a typical run. The blue solid line plots f(xbest) for various iterations while the red dashed line plots average of f (x) for various iterations.

Fig. 4.31 Convergence of the Optimization Algorithm (initial guess has been used for M', Om, rnm and εini)

Fig. 4.32 shows the relative permittivity of the xbest for various iterations. The algorithm terminates at the 430th iteration due to the first termination condition. The retrieved relative permittivity is 2.032 and 1.977 respectively.

Fig. 4.32 Relative Permittivity of xbest for Various Iterations (initial guess has been used for M', Om, rnm and εini)

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The contours of the cylinders corresponding to the xbest (dashed lines) for the first iteration and the 430th iteration are shown in Figs. 4.33 and 4.34 respectively. The original profile of the cylinder is also plotted for comparison.

Fig. 4.33 Contours of the Cylinders Corresponding to xbest (dashed lines) of the Initial Population (initial guess has been used for M', Om, rnm and εini)

Fig. 4.34 Contours of the Cylinders Corresponding to xbest (dashed lines) of the Final Population (initial guess has been used for M', Om, rnm and εini)

4.5.7 Numerical Example 5: Kite Cross-Section Cylinder The domain now contains a cylinder placed at the origin with a cross-section commonly known as the kite profile. Geometrically, this profile can be represented as below:

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r (θ ) = {0.75 sin θ ,0.5[cosθ + 0.65 sin (2θ ) − 0.65]} 0 ≤ θ ≤ 2π (11) The geometry is shown in Fig. 4.35. This profile is difficult to reconstruct due to the concavity of the structure and the sharp change in curvatures at the two arms of the kite.

Fig. 4.35 Kite Cross-section Cylinder

The result of multiple signal classification method is shown in Fig. 4.36. Though the cross section of the cylinder is not represented very accurately, it is still easy to recognize that the region contains one cylinder, i.e. M' = 1. Further, it suggests that reconstruction of this cylinder shall require more number of control points. The center of the cylinder can be chosen approximately as O1(0, 0).

Fig. 4.36 Pseudospectrum Formed Using the Multiple Signal Classification

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A threshold of Фth = -0.6 is applied on the pseudospectrum for estimating the scatterer support. The estimated scatterer support is shown in Fig. 4.37. Initial guess for rnm is taken to be 0.7λ. Subsequently, the estimated scatterer support is used to generate an initial guess for the relative permittivity using the least squares method. The computed estimate of the relative permittivity is εini = 3.6204.

Fig. 4.37 Scatterer Support Estimated (shown in white) Using Multiple Signal Classification (Фth = -0.6)

As discussed above, the reconstruction of this cylinder shall require more control points. Thus, N =16 is used for optimization. Fig. 4.38 shows the convergence profile of the optimization problem for a typical run. The blue solid line plots f (xbest) for various iterations while the red dashed line plots average of f (x) for various iterations.

Fig. 4.38 Convergence of the Optimization Algorithm (initial guess has been used for M', Om, rnm and εini)

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Fig. 4.39 shows the relative permittivity of the xbest for various iterations. The algorithm terminates at the 1000th iteration due to the third termination condition. At the end of the 1000th iteration, f(xbest)=0.151 and εbest = 2.008.

Fig. 4.39 Relative Permittivity of xbest for Various Iterations (initial guess has been used for M', Om, rnm and εini)

The contours of the cylinders corresponding to the xbest (dashed lines) for the first iteration and the 1000th iteration are shown in Figs. 4.40 and 4.41 respectively. The original profile of the cylinder is also plotted for comparison.

Fig. 4.40 Contour of the Cylinder Corresponding to xbest (dashed lines) of the Initial Population (initial guess has been used for M', Om, rnm and εini)

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Fig. 4.41 Contour of the Cylinder Corresponding to xbest (dashed lines) of the Final Population (initial guess has been used for M', Om, rnm and εini)

4.6 Conclusions This chapter presents the application of the differential evolution for the inverse scattering problems. Though the differential evolution has been introduced much later than other evolutionary methods, it has already found various applications in inverse scattering problems. This is due to the suitability of the differential evolution for the non-linear ill-posed nature of the inverse scattering problems. Among the varieties of inverse scattering problems, reconstruction of the twodimensional dielectric cylinders has been presented in this chapter as an example. Various numerical examples are considered for this purpose. The Efficiency of the differential evolution has been shown for all these examples. Special attention is drawn to the generation of initial guess for the optimization algorithm. Use of the multiple signal classification and a least squares based method has been demonstrated for this purpose. The multiple signal classification is used for estimating the profile of the scatterers. This is helpful for determining the number of cylinders, their approximate centers and approximate geometric dimensions. The least squares based method is used to generate an estimate of the permittivity of the cylinders. The impact of the initial guess on the convergence of the optimization algorithm has been demonstrated using example 1. It has been shown that a proper choice of the initial guess can speed up the convergence of the optimization significantly. The chosen inverse scattering problem is only one among the vast range of inverse scattering problems. Most methods that have been developed to solve inverse scattering problems involve optimization. Though the formulation of the optimization problem, the cost function, and the optimization parameters may change from one method to another, most of these methods can be easily cast into an optimization scheme involving the differential evolution. Further, for specific methods, it is possible to combine the differential evolution and gradient based

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optimization for improving the overall optimization scheme. Since the differential evolution is parallelizable and has significant computational advantages over other evolutionary algorithms like the genetic algorithm, with advances in the computer engineering, it may be possible to use differential evolution for some real time applications in future. Thus, there is a large scope for the differential evolution in the inverse scattering problems and the use of the differential evolution in the inverse scattering community is expected to increase manifold.

References [1] Qing, A., Lee, C.K., Jen, L.: Electromagnetic inverse scattering of two-dimensional perfectly conducting objects by real-coded genetic algorithm. IEEE Trans. Geoscience Remote Sensing 39(3), 665–676 (2001) [2] Qing, A., Gan, Y.B.: Electromagnetic inverse problems. In: Chang, K. (ed.) Encyclopedia of RF and Microwave Engineering, vol. 2, pp. 1200–1216. John Wiley, New York (2005) [3] Michalski, K.A.: Electromagnetic imaging of circular-cylindrical conductors and tunnels using a differential evolution algorithm. Microwave Optical Technology Letters 27(5), 330–334 (2000) [4] Michalski, K.A.: Electromagnetic imaging of elliptical-cylindrical conductors and tunnels using a differential evolution algorithm. Microwave Optical Technology Letters 28(3), 164–169 (2001) [5] Qing, A.: Electromagnetic imaging of two-dimensional perfectly conducting cylinders with transverse electric scattered field. IEEE Trans. Antennas Propagation 50(12), 1786–1794 (2002) [6] Li, Y., Rao, L., He, R., Xu, G., Wu, Q., Ge, M., Yan, W.: Image reconstruction of EIT using differential evolution algorithm. In: 25th IEEE Annual Int. Conf. Engineering Medicine Biology Society, September 17-21, vol. 2, pp. 1011–1014 (2003) [7] Qing, A.: Electromagnetic inverse scattering of multiple two-dimensional perfectly conducting objects by the differential evolution strategy. IEEE Trans. Antennas Propagation 51(6), 1251–1262 (2003) [8] Caorsi, S., Massa, A., Pastorino, M., Raffetto, M., Randazzo, A.: Microwave imaging of cylindrical inhomogeneities based on an analytical forward solver and multiple illuminations. In: IEEE Int. Workshop Imaging Systems Techniques, Stresa, Italy, May 14, pp. 100–105 (2004) [9] Chen, X., Grzegorczyk, T.M., Wu, B.I., Pacheco Jr., J., Kong, J.A.: Robust method to retrieve the constitutive effective parameters of metamaterials. Physical Review E 70(1), art. no. 016608 (July 2004) [10] Chen, X., O’Neill, K., Barrowes, B.E., Grzegorczyk, T.M., Kong, J.A.: Application of a spheroidal-mode approach and a differential evolution algorithm for inversion of magneto-quasistatic data in UXO discrimination. Inverse Problems 20(6), s27–s40 (2004) [11] Li, Y., Rao, L., He, R., Xu, G., Guo, X., Yan, W., Wang, L., Yang, S.: Three EIT approaches for static imaging of head. In: Annual Int. Conf. IEEE Engineering Medicine Biology Society, San Francisco, CA, September 1-5, vol. 1, pp. 578–581 (2004)

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[12] Li, Y., Rao, L.Y., He, R.J., Xu, G.Z., Wu, Q., Yan, W.L., Dong, G.Y., Yang, Q.X.: A novel combination method of electrical impedance-tomography inverse problem for brain imaging. In: 11th IEEE Biennial Conf. Electromagnetic Field Computation, Seoul, Korea, June 6-9, pp. 1848–1851 (2004) [13] Massa, A., Pastorino, M., Randazzo, A.: Reconstruction of two-dimensional buried objects by a differential evolution method. Inverse Problems 20(6), S135–S150 (2004) [14] Qing, A.: Electromagnetic inverse scattering of multiple perfectly conducting cylinders by differential evolution strategy with individuals in groups (GDES). IEEE Trans. Antennas Propagation 52(5), 1223–1229 (2004) [15] Chen, X., Wu, B.I., Kong, J.A., Grzegorczyk, T.M.: Retrieval of the effective constitutive parameters of bianisotropic metamaterials. Phys. Rev. E 71(4), 46610 (2005) [16] Shubitidze, F., O’Neill, K., Shamatava, I., Sun, K., Paulsen, K.: Analyzing multi-axis data versus scalar data for UXO discrimination. SPIE, vol. 5794, pp. 336–345 (2005) [17] Shubitidze, F., O’Neill, K., Shamatava, I., Sun, K., Paulsen, K.: Combined differential evolution and surface magnetic charge model algorithm for discrimination of UXO from non-UXO items: simple and general inversions. SPIE, vol. 5794, pp. 346–357 (2005) [18] Chen, X., Grzegorczyk, T.M., Kong, J.A.: Optimization approach to the retrieval of the constitutive parameters of slab of genral bianisotropic medium. Progress Electromagnetics Research 60, 1–18 (2006) [19] Qing, A.: Dynamic differential evolution strategy and applications in electromagnetic inverse scattering problems. IEEE Trans. Geoscience Remote Sensing 44(1), 116–125 (2006) [20] Agarwal, K., Chen, X.: Application of differential evolution in 2-dimensional electromagnetic inverse problems. In: 2007 IEEE Congress Evolutionary Computation, Singapore, September 25-28, pp. 4305–4312 (2007) [21] Bachorec, T., Jirku, T., Dedkova, J.: New numerical technique for non-destructive testing of the conductive materials. In: Progress Electromagnetics Research Symp., Beijing, China, March 26-30, pp. 976–980 (2007) [22] Pastorino, M.: Stochastic optimization methods applied to microwave imaging: A review. IEEE Trans. Antennas Propagation 55(3), 538–548 (2007) [23] Qing, A.: A parametric study on differential evolution based on benchmark electromagnetic inverse scattering problem. In: IEEE Congress Evolutionary Computation, Singapore, September 25-28, pp. 1904–1909 (2007) [24] Shubitidze, F., O’Neill, K., Barrowes, B.E., Shamatava, I., Fernandez, J.P., Sun, K., Paulsen, K.K.: Application of the normalized surface magnetic charge model to UXO discrimination in cases with overlapping signals. J. Applied Geophysics 61(3-4), 292–303 (2007) [25] Breard, A., Perrusson, G., Lesselier, D.: Hybrid differential evolution and retrieval of buried spheres in subsoil. IEEE Geoscience Remote Sensing Letters 5(4), 788–792 (2008) [26] Qing, A.: A study on base vector for differential evolution. In: IEEE Congress Evolutionary Computation, Hong Kong, China, June 1-6, pp. 550–556 (2008) [27] Rekanos, I.T.: Shape reconstruction of a perfectly conducting scatterer using differential evolution and particle swarm optimization. IEEE Trans. Geoscience Remote Sensing 46(7), 1967–1974 (2008)

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[28] Semnani, A., Kamyab, M., Rekanos, I.T.: Reconstruction of one-dimensional dielectric scatterers using differential evolution and particle swarm optimization. IEEE Geoscience Remote Sensing Letters 6(4), 671–675 (2009) [29] Belkebir, K., Bonnard, S., Pezin, F., Sabouroux, P., Saillard, M.: Validation of 2D inverse scattering algorithms from multi-frequency experimental data. J. Electromagnetic Waves Applications 14(12), 1637–1667 (2000) [30] Cheney, M.: The linear sampling method and the MUSIC algorithm. Inverse Problems 17(4), 591–595 (2001) [31] Kirsch, A.: The MUSIC algorithm and the factorization method in inverse scattering theory for inhomogeneous media. Inverse Problems 18(4), 1025–1040 (2002) [32] Marklein, R., Mayer, K., Hannemann, R., Krylow, T., Balasubramanian, K., Langenberg, K.J., Schmitz, V.: Linear and nonlinear inversion algorithms applied in nondestructive evaluation. Inverse Problems 18(6), 1733–1759 (2002) [33] Colton, D., Haddar, H., Piana, M.: The linear sampling method in inverse electromagnetic scattering theory. Inverse Problems 19(6), S105–S137 (2003) [34] Dubois, A., Belkebir, K., Saillard, M.: Localization and characterization of twodimensional targets buried in a cluttered environment. Inverse Problems 20(6), S63– S79 (2004) [35] Kirsch, A.: The factorization method for Maxwell’s equations. Inverse Problems 20(6), S117–S134 (2004) [36] Ammari, H., Iakovleva, E., Hyeonbae, K.B.: Reconstruction of a small inclusion in a two-dimensional open waveguide. SIAM J. Applied Mathematics 65(6), 2107–2127 (2005) [37] Cakoni, F.: Recent developments in the qualitative approach to inverse electromagnetic scattering theory. J. Computational Applied Mathematics 204(2), 242–255 (2007) [38] Devaney, A.J.: Time reversal imaging of obscured targets from multistatic data. IEEE Trans. Antennas Propagation 53(5), 1600–1610 (2005) [39] Devaney, A.J., Marengo, E.A., Gruber, F.K.: Time-reversal-based imaging and inverse scattering of multiply scattering point targets. J. Acoustical Society America 118(5), 3129–3138 (2005) [40] Cakoni, F., Colton, D.: Qualitative Methods in Inverse Scattering Theory: an Introduction. Springer, Berlin (2006) [41] Catapano, I., Crocco, L., D’Urso, M., Isernia, T.: On the effect of support estimation and of a new model in 2-D inverse scattering problems. IEEE Trans. Antennas Propagation 55(6), 1895–1899 (2007) [42] Zhong, Y., Chen, X.: MUSIC imaging and electromagnetic inverse scattering of multiple-scattering small anisotropic spheres. IEEE Trans. Antennas Propagation 55(12), 3542–3549 (2007) [43] Agarwal, K., Chen, X.: Applicability of MUSIC-Type imaging in two-dimensional electromagnetic inverse problems. IEEE Trans. Antennas Propagation 56(10), 3217– 3223 (2008) [44] Lax, M.: Multiple scattering of waves. Reviews of Modern Physics 23, 287–310 (1951) [45] Foldy, L.L.: The multiple scattering of waves: 1. General theory of isotropic scattering by randomly distributed scatterers. Physical Review 67, 107–119 (1945)

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[46] Chen, X., Agarwal, K.: MUSIC algorithm for two-dimensional inverse problems with special characteristics of cylinders. IEEE Trans. Antennas Propagation 56(6), 1808– 1812 (2008) [47] Catapano, I., Crocco, L.: An imaging method for concealed targets. IEEE Trans. Geoscience Remote Sensing 47(5), 1301–1309 (2009) [48] Chen, X., Zhong, Y.: A robust noniterative method for obtaining scattering strengths of multiply scattering point targets. J. Acoustical Society America 122(3), 1325–1327 (2007) [49] Chiu, C.C., Chen, W.T.: Electromagnetic imaging for an imperfectly conducting cylinder by the genetic algorithm. IEEE Trans. Microwave Theory Techniques 48(11), 1901–1905 (2000) [50] Qing, A.: Microwave imaging of parallel perfectly conducting cylinders with transverse electric scattering data. J. Electromagnetic Waves Applications 15(5), 665– 685 (2001) [51] Peterson, A.F., Ray, S.L., Mittra, R.: Computational Methods for Electromagnetics. IEEE Press, New York (1998) [52] Colton, D.L., Kress, R.: Inverse Acoustic and Electromagnetic Scattering Theory, 2nd edn. Springer, New York (1998) [53] Price, K.V., Storn, R.M., Lampinen, J.A.: Differential Evolution: a Practical Approach to Global Optimization. Springer, Berlin (2005) [54] Qing, A.: Differential Evolution: Fundamentals and Applications in Electrical Engineering. John Wiley, New York (2009) [55] Horn, R.A., Johnson, C.R.: Matrix Analysis. Cambridge University Press, Cambridge (1985)

Chapter 5

The Use of Differential Evolution for the Solution of Electromagnetic Inverse Scattering Problems A. Donelli , A. Massa, G. Oliveri, M. Pastorino, and A. Randazzo *

Abstract. Inspection of penetrable objects by using differential evolution together with a recently proposed iterative multiscaling approach is discussed in this Chapter. Several new results are included concerning the reconstruction of inhomogeneous targets under various imaging conditions.

5.1 Introduction Electromagnetic inverse scattering problem [1]-[4] arises in several important diagnostic applications e.g., nondestructive testing and evaluation, geophysical prospecting and medical imaging, in which measured samples of the scattered electric field are used as input data to retrieve physical and geometrical properties of unknown targets under test. High-frequency electromagnetic approaches are good candidates for these tasks, due to their specific features, e.g., the capability of penetrating dielectric materials and retrieving the dielectric properties of the medium by using a nonionizing and safe radiation [5]-[22]. Solving electromagnetic inverse scattering problem remains a challenge due to ill-posedness and nonlinearity. It is usually recast into an optimization problem. Due to the ill-posedness, false solutions can be obtained, which correspond to local minima in the search space [1]-[2]. For this reason, global optimization approaches have been widely adopted as solving procedures [23], among which A. Donelli · A. Massa · G. Oliveri Department of Information Engineering and Computer Science, University of Trento, Italy e-mail: [email protected], [email protected], [email protected] *

M. Pastorino · A. Randazzo Department of Biophysical and Electronic Engineering, University of Genoa, Italy e-mail: [email protected], [email protected] A. Qing and C.K. Lee: Differential Evolution in Electromagnetics, ALO 4, pp. 107–131. © Springer-Verlag Berlin Heidelberg 2010 springerlink.com

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differential evolution [24]-[28] has been successfully applied for retrieving the profiles of perfect electric conducting bodies and to determine the distributions of complex permittivity of dielectric targets [27], [29]-[39]. In this chapter, the application of differential evolution for the solution of electromagnetic problems in imaging is discussed [2]. In particular, an inverse scattering based technique for inspecting penetrable bodies is considered [1]. The iterative multiscaling approach [40]-[42] is applied to avoid dealing with a large scale optimization process that would require unrealistic computational resources. The solving procedure starts from a fixed test area and successively focuses on one or more "regions of interest" in order to determine the approximate shapes of the unknown objects. At each step of the minimization process, differential evolution is used to retrieve this support by minimizing a proper functional, which relates the measured scattered field data to the data numerically produced, at any iteration, by the current solution. New and original reconstruction results that allow one to draw some further conclusions about the effectiveness of this powerful global optimization method are presented.

5.2 Problem Formulation 5.2.1 The Inverse Scattering Formulation Let us consider inhomogeneous dielectric targets occupying a space region Ω bounded by ∂Ω. The external medium is infinite, lossless, and homogeneous. The scalar electromagnetic inverse scattering problem is governed by the following two operator equations [43]

Ae3 D w = Ψ s (m )

(1)

Ao3 D w = Ψ i (o )

(2)

where w denotes the polarization current density [A/m2], Ψs(m) is the scattered field (copolarized scalar component) inside the measurement domain, and Ψi(o) is the incident field inside the space region occupied by the objects, which it is a known quantity everywhere. The two operators Ae3D and Ao3D are scalar integral Fredholm operators of the first and second kind [1], respectively, having as kernels the Green's function for free space [44] − jkR

Γ3 D = − e4πR

(3)

where R the distance between a point of the domain and a point of the range of the considered operator. For infinite cylinders (i.e., objects that are uniform along a coordinate, say, the z coordinate), the corresponding scattering equations can be obtained by applying the following integration to Eqs. (1) and (2) and, consequently, to Eq. (3)

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∫ (⋅)dz ∞

(4)

−∞

The results are two operator equations, analogous to Eqs. (1) and (2), where the superscript "3D" is substituted by the "2D" and the kernels are now given by [44]

Γ 2 D = − 4j H 0(2 ) (kρ )

(5)

or, in the asymptotic case (far field measurements)

Γ 2 D = − ηω

− jkb 8πρ

e − jkρ

(6)

where H0(2)(·) is the Hankel function of the second kind of zeroth order, kb is the wavenumber of the propagation medium and ρ plays, in the 2D case, the role of R in the 3D case. Inverting Eqs. (1) and (2) is not an easy task. The problem related to Eq. (1) is highly ill posed, and both are nonlinear operators if we consider that [2]

w = τΨ t (o )

(7)

t(o)

where Ψ is the total field inside the object space and τ is the scattering potential, which indicates the contrast between the object and free space in terms of the physical parameters. Outside the object, it results τ =0 (8) Therefore, a "perfect" reconstruction of τ is sufficient to define, contemporaneously, the position, the external shape and the internal structure of the target. This fact also gives reason of the claimed powerful features of inverse scattering based imaging.

5.2.2 Discrete Setting In the discrete setting, Eqs. (1) and (2) are replaced by matrix equations:

A 3e D ⋅ w = Ψ s ( m )

(9)

A 3o D ⋅ w = Ψ i (o )

(10)

where w is an array containing the N coefficients of the expansion of w in a set of basis function un; Ψi(o) is an array of dimensions N containing the values of Ψi(o) weighted by proper weighting functions, Ψs(m) is an array of dimensions M containing the values of Ψs(m) weighted by proper weighting functions. The elements of the two matrices in Eqs. (9) and (10) are essentially the values of Ae3Dun and Ao3Dun, weighted by the same weighting functions. In turn, according to Eq. (7), w can be written as

w = T ⋅ Ψ t (o )

(11)

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where T is a diagonal matrix of size N × N containing the values of the scattering potential and Ψt(o) is an array of dimensions N containing the values of Ψt(o) weighted by proper weighting functions.

5.2.3 The Inverse Scattering Problem as an Optimization Problem Until the introduction of the use of stochastic optimization methods in imaging, it was very difficult to solve the above inverse scattering problem without significant approximations, since local optimization methods can be trapped in local minima if a very good estimate of the solution is not available. To apply stochastic optimization methods, a functional is usually constructed. Although there are a lot of different possible ways to construct this functional, the simplest considered expression is the following

[

]

Φ T, Ψ t (o ) = m1 A 3e D ⋅ T ⋅ Ψ t (o ) − Ψ s ( m ) + m2 A 3o D ⋅ T ⋅ Ψ t (o ) − Ψ i (o ) 2

2

(12)

where m1 and m2 are regularization constants. The nonlinear functional in Eq. (12) can be modified by introducing multiview, multi-illumination, and multifrequency information. In those cases, summations can be included in Eq. (12) after a proper indexing of the quantities that change at each view, illumination, operating frequency. To simplify notation, this extension is not considered here (although the results reported in this chapter have been obtained under such conditions, as it will be detailed in Section 5.6). The structure of Eq. (12) makes evident the suitability of choosing a stochastic optimization method (as well as a parallel processing, not discussed here). The main problem has a numerical nature, since good spatial resolutions require high values of N. This results in a large optimization problem, difficult to be handled with reasonable computational resources. There are essentially two possible ways to simplify the problem. The first one is to use a reduced parameterization, accepting a "model approximation." The second one is to progressively reduce the test area by refining the "object domain."

5.3 The Iterative Multiscaling Approach The idea of focusing the search in some subdomains of the object space has been followed from very long time ago [45], but only recently, some effective approaches have been proposed. One of those successful methods is a multiscaling approach, proposed for the first time in [40] and further studied in other papers [41]-[42]. This iterative method is the one adopted in the following. Essentially, the approach considers different levels of resolution on the basis of a synthetic zoom procedure, allowing for an efficient use of the "limited amount of the information content of inverse scattering data" and guaranteeing a sufficient resolution level in the retrieved image of the investigation domain. Beginning

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from a coarse representation of the investigation domain, the method iteratively defines a sub-gridding of the area where the scatterers are located. By exploiting the “knowledge” of the scenario under test, which was acquired at the previous steps, the procedure estimates the locations and space occupations of the unknown scatterers. The process terminates when a “stationary” reconstruction is reached. In more details, the developed method includes the following two procedures, which are iteratively repeated at each step: • •

A clustering procedure, which aims at defining the number of Q scatterers in the investigation domain and, consequently, the regions where the synthetic zoom has to be performed. A retrieval procedure, which aims at reconstructing the dielectric profile in each region defined by the clustering procedure.

The iterative approach also requires a termination procedure, which is responsible for stopping the multistep procedure when a “stationary” reconstruction is achieved. The clustering procedure assumes a grey-level representation of the reconstructed dielectric profile of the scenario under test at a given level of the multiscaling approach. Firstly, the pixel representation of the estimated profile is binarized by a thresholding operation. The original image is then segmented into some regions, namely, the object regions and the background. Successively, a noise filtering is performed in order to eliminate some artifacts. The supports of the scatterers are then clearly defined. At this point the object detection is performed by applying differential evolution to Eq. (12). As previously mentioned, the clustering and retrieval procedures are iteratively applied until a requested resolution level is reached. According to this multiresolution strategy, a higher resolution level is adopted only for the reduced investigation domains, where the clustering procedure has estimated the presence of one or more objects. In this way, high reconstruction accuracy is obtained without resorting to a large scale optimization problem. Following [41], the multiresolution approach is iterated until a “stationary condition” for the iterative procedure is achieved (being S the final step) in each of the regions defined by the clustering procedure. To this end, the following conditions must occur. •

The number of regions identified by the clustering procedure is stationary, i.e. s

1 s

•

∑Q r =1

s

− Qr ≤ q

(13)

where q is a fixed threshold and Qk is the number of regions of interest at the kth step. The "qualitative" dimensions of the retrieved scatterers are stable, i.e.,

min

h =1,...,Q s

l h , s +1 − l h ,s l h ,s +1

< lth

(14)

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where l = x, y, and lth is a linear threshold value, which is valid for the linear dimensions l of all the Qs scatterers "located" by the clustering procedure at the resolution level s.

5.4 Numerical Results In the following, numerical results obtained by using differential evolution approach combined with the iterative multiscaling approach discussed in this Chapter (in the following denoted as IMSA-strategy for sake of brevity) are reported. In particular, DE/rand/1/bin has been considered in the present implementation. At each step of the iterative multiscaling method, the differential evolution procedure is stopped when a maximum number of iterations, kmax, is reached. A two-dimensional cylindrical geometry, with transverse-magnetic illumination conditions, has been considered and comparisons with the reconstructions obtained by using a bare differential evolution algorithm [27] are provided. In all the simulations, the investigation area is a square domain of side LDI, which, in the inversion phase, has been discretized into Ntot sub-domains with a discretization satisfying the Hagmann-Gandhi-Durney criterion [46]. The unknown objects have been illuminated by using a set of V plane waves with frequency f0 and impinging from different directions uniformly distributed in the range [0, 2π]. For any illumination, the scattered electric field data have been collected in M measurement points equally spaced on a circle of radius rO. The electric field data have been synthetically generated by using a numerical code based on the Method of Moments [47]. Moreover, in order to avoid the socalled “inverse crime,” a different discretization of the investigation domain has been used for the direct procedure. In order to quantitatively evaluate the reconstruction accuracy, the following error figures have been introduced:

1 χj = Nj

⎧ τ ( xi , yi ) − τ actual (xi , yi ) ⎫ ⎬ × 100 τ actual ( xi , yi ) i =1 ⎩ ⎭ Nj

∑⎨

(15)

where τ and τactual are the reconstructed and actual values of the scattering potential. In Eq. (15), the subscript j = tot means that the error is calculated on the whole investigation domain, j = int denotes that only the subdomains corresponding to the regions occupied by the objects are considered in the summation (and, consequently, Nint is the number of cells occupied by the object), and j = ext means that the error is computed only on the subdomains composing the void “background” of the investigation area (being Next the corresponding number of subdomains.)

5.4.1 Off-Centered Dielectric Cylinder In the first experiment, a single homogeneous off-centered dielectric cylinder with square cross section of side Dscatt = 3λ0/10, where λ0 is the wavelength in

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free-space, has been considered. The 2D gray map of the profile is shown in Fig. 5.1 while the 3D view of the profile is plotted in Fig. 5.2.

Fig. 5.1 2D Gray Map of a Homogeneous Off-center Dielectric Cylinder

Fig. 5.2 3D View of a Homogeneous Off-center Dielectric Cylinder

The cylinder was centered at xb = yb = λ0/4 and it is characterized by a scattering potential τscatt = 1.5 + j0.0. The parameters of the simulation are: LDI = λ0, rO = λ0, V = 4, M = 10, f0 = 10 GHz, Ntot = 10 × 10, Npar = 1000, Npop = 200, F = 0.5, cr = 0.8, and kmax = 2000. Noiseless reconstruction is considered first. Figs. 5.3 and 5.4 draw the gray map of the reconstructed real part of the scattering potential by bare differential evolution and the proposed method at a scale level S = 3 respectively. The corresponding 3D views are plotted in Figs. 5.5 and 5.6 respectively.

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Fig. 5.3 Gray Map of the Reconstructed Real Part of the Scattering Potential of the Off-centered Dielectric Cylinder Obtained by Bare Differential Evolution

Fig. 5.4 Gray Map of the Reconstructed Real Part of the Scattering Potential of the Off-centered Dielectric Cylinder Obtained by the Proposed Approach

Fig. 5.5 3D View of the Reconstructed Real Part of the Scattering Potential of the Offcentered Dielectric Cylinder Obtained by Bare Differential Evolution

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Fig. 5.6 3D View of the Reconstructed Real Part of the Scattering Potential of the Offcentered Dielectric Cylinder Obtained by the Proposed Approach

As can be seen, the proposed method provides a better reconstruction of the area under investigation. This fact is further confirmed by looking at the horizontal and vertical cuts of the reconstructed real part of the scattering potential as shown in Figs. 5.7 and 5.8 respectively.

Fig. 5.7 Horizontal Cuts of the Reconstructed Real Part of the Scattering Potential of the Off-centered Dielectric Cylinder

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Fig. 5.8 Vertical Cuts of the Reconstructed Real Part of the Scattering Potential of the Offcentered Dielectric Cylinder

Quantitative comparison is given in Table 5.1 which provides the values of the error figures. The advantage of the proposed approach can be seen clearly from the presented lower error figures.

Table 5.1 Error Figures for Noiseless Reconstruction of the Off-centered Dielectric Hollow Cylinder

DE IMSA-DE (S=3)

χtot

χint

χext

1.95 0.91

11.01 1.28

2.31 0.11

Finally, the convergence behavior of the two approaches for reconstructing the off-centered dielectric cylinder using noiseless measuring data is given in Fig. 5.9. As can be seen, in the bare differential evolution approach, the search for the optimum solution stagnates (i.e., the algorithm is not able to generate a solution better than the current optimum). On the contrary, the multiscaling method is able to converge to a correct solution.

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Fig. 5.9 Convergence Behavior for Reconstructing the Off-centered Dielectric Cylinder

5.4.2 Off-Centered Dielectric Hollow Cylinder The second profile as shown in Figs. 5.10 and 5.11 is an off-centered and hollowed square dielectric cylinder. The scatterer is defined by the following parameters: side of the inner cylinder Linner = 0.4λ0, side of the outer cylinder Louter = 1.2λ0, center of cylinder xb = yb = -0.2 λ0, scattering potential of the inner cylinder τinner = 0.0 + j0.0, and scattering potential of the outer cylinder τouter = 0.5 + j0.0.

Fig. 5.10 2D Gray Map of the Off-centered Dielectric Hollow Cylinder

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Fig. 5.11 3D View of the Off-centered Dielectric Hollow Cylinder

The simulation has been performed by using the following parameters: f0 = 6 GHz, LDI = 2.4λ0, rO = 1.6λ0, V = 4, M = 21, Ntot = 12 × 12, Npop = 100, F = 0.5, cr = 0.8, kmax = 2000. The resulting number of unknowns is Npar = 1296. The numerically computed electric field data has been corrupted by a Gaussian noise with SNR = 20 dB. Figs. 5.12 and 5.13 draw the gray map of the reconstructed real part of the scattering potential by bare differential evolution and the proposed method at a scale level S = 3 respectively. The corresponding 3D views are plotted in Figs. 5.14 and 5.15 respectively.

Fig. 5.12 Gray Map of the Reconstructed Real Part of the Scattering Potential of the Off-centered Dielectric Hollow Cylinder Obtained by Bare Differential Evolution

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Fig. 5.13 Gray Map of the Reconstructed Real Part of the Scattering Potential of the Off-centered Dielectric Hollow Cylinder Obtained by the Proposed Approach

Fig. 5.14 3D View of the Reconstructed Real Part of the Scattering Potential of the Off-centered Dielectric Hollow Cylinder Obtained by Bare Differential Evolution

Fig. 5.15 3D View of the Reconstructed Real Part of the Scattering Potential of the Off-centered Dielectric Hollow Cylinder Obtained by the Proposed Approach

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Moreover, the horizontal and vertical cuts of the reconstructed real part of the scattering potential are shown in Figs. 5.16 and 5.17 respectively.

Fig. 5.16 Horizontal Cuts of the Reconstructed Real Part of the Scattering Potential of the Off-centered Dielectric Hollow Cylinder

Fig. 5.17 Vertical Cuts of the Reconstructed Real Part of the Scattering Potential of the Offcentered Dielectric Hollow Cylinder

Obviously, the new approach provides better results than bare differential evolution, which is further confirmed by the error figures reported in Table 5.2 where the optimum scale level Sopt = 2 is applied for the proposed approach.

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Table 5.2 Error Figures for Reconstruction of the Off-centered Dielectric Cylinder

DE IMSA-DE

χtot

χint

χext

4.45 2.65

9.14 3.77

1.53 0.018

Finally, the convergence behavior of the two approaches is given in Fig. 5.18. As can be seen, in this case, too, the bare differential evolution stagnates.

Fig. 5.18 Convergence Behavior for Reconstructing the Off-centered Dielectric Hollow Cylinder

5.4.3 Centered Stratified Dielectric Square Cylinder The third example is a stratified square cylinder as shown in Figs. 5.19 and 5.20. The parameters of the scatterer are: side of the inner cylinder Linner = 0.4λ0, side of the outer cylinder Louter = 1.2λ0, center of the cylinder xb = yb = 0.0, scattering potential of the inner cylinder τinner = 2.0 + j0.0, and scattering potential of the outer cylinder τouter = 0.5 + j0.0. Other parameters are the same as those of the second example.

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Fig. 5.19 2D Gray Map of the Centered Stratified Dielectric Square Cylinder

Fig. 5.20 3D View of the Centered Stratified Dielectric Square Cylinder

Figs. 5.21 and 5.22 draw the gray map of the reconstructed real part of the scattering potential by bare differential evolution and the proposed approach respectively. The corresponding 3D views are plotted in Figs. 5.23 and 5.24 respectively.

Fig. 5.21 Gray Map of the Reconstructed Real Part of the Scattering Potential of the Centered Stratified Dielectric Square Cylinder Obtained by Bare Differential Evolution

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Fig. 5.22 Gray Map of the Reconstructed Real Part of the Scattering Potential of the Centered Stratified Dielectric Square Cylinder Obtained by the Proposed Approach

Fig. 5.23 3D View of the Reconstructed Real Part of the Scattering Potential of the Centered Stratified Dielectric Square Cylinder Obtained by Bare Differential Evolution

Fig. 5.24 3D View of the Reconstructed Real Part of the Scattering Potential of the Centered Stratified Dielectric Square Cylinder Obtained by the Proposed Approach

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In addition, the horizontal and vertical cuts of the reconstructed real part of the scattering potential are shown in Figs. 5.25 and 5.26 respectively.

Fig. 5.25 Horizontal Cuts of the Reconstructed Real Part of the Scattering Potential of the Centered Stratified Dielectric Square Cylinder

Fig. 5.26 Vertical Cuts of the Reconstructed Real Part of the Scattering Potential of the Centered Stratified Dielectric Square Cylinder

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The error figures are tabulated in Table 5.3. It can be seen that differential evolution is unable to produce a good solution. On the contrary, the proposed approach provides a quite good reconstruction of the object. Table 5.3 Error Figures for Reconstruction of the Centered Stratified Dielectric Square Cylinder

DE IMSA-DE

χtot

χint

χext

7.75 5.88

9.81 6.83

1.67 0.0018

Finally, the convergence behavior of the two approaches is given in Fig. 5.27. It is evident that the bare differential evolution algorithm remains trapped in a suboptimal region of the search space, whereas the multiscale approach is able to reach a correct solution.

Fig. 5.27 Convergence Behavior for Reconstructing the Centered Stratified Dielectric Square Cylinder

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5.4.4 Centered E-Shape Dielectric Cylinder The last example is an E-shaped cylinder as shown in Figs. 5.28 and 5.29. The parameters of the scatterer are: height of the “E” H = 1.0λ0, width of the “E” T = 1.2λ0, center of the cylinder xb = 0.0 and yb = -0.5, scattering potential τscatt = 0.5 + j0.0. Other parameters are the same as those of the second example.

Fig. 5.28 2D Gray Map of the E-shape Dielectric Cylinder

Fig. 5.29 3D View of the E-shape Dielectric Cylinder

Figs. 5.30 and 5.31 draw the gray map of the reconstructed real part of the scattering potential by bare differential evolution and the proposed approach respectively. The corresponding 3D views are plotted in Figs. 5.32 and 5.33 respectively.

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Fig. 5.30 Gray Map of the Reconstructed Real Part of the Scattering Potential of the E-shape Dielectric Cylinder Obtained by Bare Differential Evolution

Fig. 5.31 Gray Map of the Reconstructed Real Part of the Scattering Potential of the E-shape Dielectric Cylinder Obtained by the Proposed Approach

Fig. 5.32 3D View of the Reconstructed Real Part of the Scattering Potential of the E-shape Dielectric Cylinder Obtained by Bare Differential Evolution

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Fig. 5.33 3D View of the Reconstructed Real Part of the Scattering Potential of the E-shape Dielectric Cylinder Obtained by the Proposed Approach

The error figures are tabulated in Table 5.4. Likewise, differential evolution is not able to produce a meaningful reconstruction while the solution provided by the proposed approach is still quite good. Table 5.4 Error Figures for Reconstruction of the E-shape Dielectric Cylinder

DE IMSA-DE

χtot

χint

χext

8.51 5.13

10.08 7.11

2.07 0.24

Fig. 5.34 Convergence Behavior for Reconstructing the E-shape Dielectric Cylinder

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Finally, the convergence behavior of the two approaches is given in Fig. 5.34. As for the previously reported cases, the multiscale strategy is able to reach a quite good solution, whereas the bare differential evolution falls into stagnation.

5.5 Conclusions Inspecting penetrable targets using differential evolution together with iterative multiscaling approach is considered in this Chapter. The combined strategy has been proved to be quite effective in reconstructing complex dielectric cylinders such as hollow and E-shape cylinders in noisy environment. Consequently, the high computational load usually associated with stochastic optimization approaches is significantly reduced.

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[13] Tijhuis, G., Belkebir, K., Litman, A.C.S., de Hon, B.P.: Theoretical and computational aspects of 2-D inverse profiling. IEEE Trans. Geoscience Remote Sensing 39(6), 1316–1330 (2001) [14] Abubakar, A., van den Berg, P.M., Mallorqui, J.J.: Imaging of biomedical data using a multiplicative regularized contrast source inversion method. IEEE Trans. Microwave Theory Techniques 50(7), 1761–1771 (2002) [15] Bond, E.J., Li, X., Hagness, S.C., Van Veen, B.D.: Microwave imaging via spacetime beamforming for early detection of breast cancer. IEEE Trans. Antennas Propagation 51(8), 1690–1705 (2003) [16] Cui, T.J., Aydiner, A.A., Chew, W.C., Wright, D.L., Smith, D.V.: Three-dimensional imaging of buried objects in very lossy earth by inversion of VETEM data. IEEE Trans. Geoscience Remote Sensing 41(10), 2197–2210 (2003) [17] Fang, Q., Meaney, P.M., Geimer, S.D., Streltsov, A.V., Paulsen, K.D.: Microwave image reconstruction from 3-D fields coupled to 2-D parameter estimation. IEEE Trans. Medical Imaging 23(4), 475–484 (2004) [18] Zhang, Z.Q., Liu, Q.H.: Three-dimensional nonlinear image reconstruction for microwave biomedical imaging. IEEE Trans. Biomedical Engineering 51(3), 544– 548 (2004) [19] Belkebir, K., Saillard, M.: Testing inversion algorithms against experimental data: Inhomogeneous targets. Inverse Problems 21(6), S1–S4 (2005) [20] Abubakar, A., Habashy, T.M.: Nonlinear inversion of multi-frequency microwave Fresnel data using the multiplicative regularized contrast source inversion. Progress Electromagnetics Research 62, 193–201 (2006) [21] Solimene, R., Soldovieri, F., Prisco, G., Pierri, R.: Three-dimensional microwave tomography by a 2-D slice-based reconstruction algorithm. IEEE Geoscience Remote Sensing Letters 4(4), 556–560 (2007) [22] Arunachalam, K., Udpa, L., Udpa, S.S.: A computational investigation of microwave breast imaging using deformable reflector. IEEE Trans. Biomedical Engineering 55(2), 554–562 (2008) [23] Pastorino, M.: Stochastic optimization methods applied to microwave imaging: A review. IEEE Trans. Antennas Propagation 55(3), 538–548 (2007) [24] Storn, R., Price, K.V.: Differential Evolution - A Simple and Efficient Adaptive Scheme for Global Optimization over Continuous Spaces, Technical Report TR-95012, International Computer Science Insitute (March 1995) [25] Storn, R., Price, K.V.: Differential evolution-a simple and efficient heuristic for global optimization over continuous spaces. J. Global Optimization 11(4), 341–359 (1997) [26] Price, K.V., Storn, R.M., Lampinen, J.A.: Differential Evolution: a Practical Approach to Global Optimization. Springer, Berlin (2005) [27] Chakraborty, U.K. (ed.): Advances in Differential Evolution. Springer, Berlin (2008) [28] Qing, A.: Differential Evolution: Fundamentals and Applications in Electrical Engineering. John Wiley & IEEE, New York (2009) [29] Michalski, K.A.: Electromagnetic imaging of circular-cylindrical conductors and tunnels using a differential evolution algorithm. Microwave Optical Technology Letters 27(5), 330–334 (2000) [30] Michalski, K.A.: Electromagnetic imaging of elliptical-cylindrical conductors and tunnels using a differential evolution algorithm. Microwave Optical Technology Letters 28(3), 164–169 (2001)

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Chapter 6 Modeling of Electrically Large Equipment with Distributed Dipoles Using Metaheuristic Methods Joan Ramon Regué, Miquel Ribó, and José Gomila 1

6.1 Introduction 6.1.1 Near-Field to Far-Field Transformation The reference environment for measurement of radiated emissions is the open area test site (OATS). The far field of a radiating element is of great interest [1], [2] since it is independent of distance. However, in an OATS environment, it is hardly possible to separate interferences generated by the equipment under test (EUT) from external ones. Semi-anechoic chambers have to be used instead. For large radiating elements, the minimal distance [1], [3]-[5] needed to measure the far field is large, especially at low frequencies. Therefore, large costly semi-anechoic chambers which are only affordable to very few big laboratories are required. A cheaper alternative based on near-field to far-field transformations, as shown in Fig. 6.1, makes it possible to use small chambers to measure near field at short distances and predict the far field at a larger distance.

Near-field measurements

EUT

(a)

Far-field computation EUT model (b)

Fig. 6.1 Measurements of Radiated Emissions (a) near-field measurements in a small anechoic chamber (b) far-field computation Joan Ramon Regué . Miquel Ribó . José Gomila Grup de Recerca en Electromagnetisme i Comunicacions (GRECO) Ls Salle, Universitat Ramon Llull (URL), Barcelona, Spain e-mail: [email protected]

1

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The near-field to far-field transformation technique [3], [6]-[11] has been successfully used in many antenna radiation-pattern measurement systems [12]. From an electromagnetic point of view, intentional radiators like antennas and nonintentional radiators such as equipments submitted to electromagnetic-compatibility tests are equivalent. Therefore, such near-field measurement systems are applicable to make electromagnetic-compatibility measurements [13]-[25]. Some researchers have adapted the near-field to far-field transformation technique used in antenna measurements for electromagnetic compatibility measurements [15]-[16]. Other people have studied methods specifically focused on electromagnetic compatibility (EMC) [17]-[20]. However, difficulties arise in such electromagnetic-compatibility measurements: • • •

Lower working frequencies Different measurement environments Non-controlled emissions

6.1.2 Radiating Equipment Modeling with Prefixed Position Dipoles Any radiation source of finite dimensions can be substituted by an equivalent expansion in multipolar moments, which will produce the same field values at any point exterior to the volume containing all sources [26]. When a radiating source is electrically small at the radiation frequency, this expansion in multipolar moments can be simplified by considering only its first terms: the dipolar moments. In this way, an electrically small source can be characterized by six equivalent dipoles: three orthogonal electric dipoles and three orthogonal magnetic dipoles. Once the magnitudes and the phases of the currents of these equivalent dipoles are found, the free space and OATS radiation characteristics, as well as the total radiated power, can then be easily calculated [27]-[33]. Some approaches have been proposed to break the size limit [34]-[36]. In [34] and [35], the equipment is modeled as an equivalent set of dipoles distributed in several positions on the surface or in the interior of the EUT, whose positions are empirically prefixed by the user [34], or determined by optimization techniques [35]. The excitations of the equivalent dipoles are found by a least-square method. In [36], the EUT is modeled with a minimum number of twelve infinitesimal dipoles, placed at the six sides of a fictitious cube surrounding the equipment. The excitations of the equivalent dipoles are found using a genetic algorithm. In [37], the equipment is modeled with an isotropic-radiator mesh placed in the interior of a cube surrounding the EUT. The excitations are found using a neural network. Some other works [7], [9] prearrange dipoles on a closed surface that wraps all sources up. Consequently, the number of dipoles used must be very large. Moreover, taking into account that they model sources by currents on a surface surrounding them, they do not provide any information of the inner structure of the sources.

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6.1.3 Present Work In this chapter, a prediction method for radiated emissions in far field is explained [38]-[41]. This technique replaces the EUT by an equivalent set of infinitesimal dipoles (both electric and magnetic) distributed inside the volume occupied by the EUT. The information used to find this equivalent set of dipoles is near-field measurements at a short distance of the EUT. A metaheuristic technique based on a genetic algorithm and differential evolution is used to determine the parameters (type, position, orientation and exciting current) of each dipole of the equivalent set of dipoles. The equivalent set of dipoles is then used to determine the far-field radiation and identify the radiating parts of equipments. The modeling of an EUT through distributed infinitesimal dipoles enables the identification of the radiating parts of the equipment and allows the prediction of its far-field radiation in free space or in an OATS. This approach introduces several advantages as compared to traditional ones: •

•

•

•

•

The near-field measurement system can be very versatile. This method is equally valid for electric- and magnetic-field measurements, as well as for a combination of components of both fields. It also enables measurements with phase or just amplitude information Points over which measurements are carried out are not restricted to a prearranged configuration. Measurements with spherical, cylindrical, planar or any other geometry can be performed, provided there are enough measurement points and are sufficiently well distributed to describe the field behavior The environment where near-field measurements can be performed can be anechoic, semi-anechoic or even some others. Far-field predictions can be performed in a different environment to that where near-field measurements were performed, which enables, for example, the transformation from anechoic measurements into semi-anechoic ones, or vice versa The infinitesimal dipoles of the equivalent set are placed by the algorithm at the positions of the main sources of radiation of the EUT. This fact provides information about the inner structure of the EUT [38], and allows the identification of the parts of the EUT which need to be redesigned in order to reduce electromagnetic interference (EMI) The metaheuristic algorithm required to find the equivalent dipole set is very versatile, since it can be used with different spatial configurations of measurement points, measured variables or measurement environment without any significant modification

6.2 Electromagnetic Modeling of a Radiating Equipment with Distributed Infinitesimal Dipoles Radiating equipments are modeled as bodies over which electric and magnetic currents flow. The currents can be approximated by infinitesimal electric and

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magnetic dipoles. A possible way to implement this approximation would be to create a regular mesh of infinitesimal dipoles. Such a model could be highly precise. However, it would be computationally very costly. In electromagnetic compatibility practice, only a few printed circuit traces, cables, loops or slots contribute in a significant way to the total radiation of the equipment. Therefore, in this chapter, a model formed by a much reduced number of infinitesimal dipoles is implemented to model the main radiation sources of the equipment.

6.2.1 Integral Equations for the Radiation of Electronic Equipment Previous research models radiating electronic equipments by electric [9] or magnetic currents [7] only. However, for generality, in this chapter, we will formulate the electromagnetic fields radiated by a general radiating body, as shown in Fig. 6.2, in terms of the equivalent electric and magnetic currents that flow in it [38], since different elements that generate electromagnetic radiation in electronic equipment can be more efficiently modeled with one of the two kinds of currents.

J, M (a)

(b)

Fig. 6.2 Generic Radiating Equipment and Its Equivalent Model (a) generic radiating equipment (b) equivalent model with electric and magnetic currents which generates the same electromagnetic radiation

Electromagnetic fields generated in free space by electric-current volumetricdensity J and magnetic-current volumetric-density M are [42]: (∇ ⋅ A ) ∇ × F E = − jωA − j ∇ωμε − ε

(1)

(∇ ⋅ F ) ∇ × A H = − jωF − j ∇ωμε + μ

(2)

where ε and μ are the permittivity and permeability of free space respectively, ω is the angular frequency,

A(r ) = F(r ) =

ε

μ

∫ J(r )

4π V

∫ M(r )

4π V

e

e

− jk r −r ′

r −r′

− jk r −r ′

r −r′

dr′

(3)

dr′

(4)

k = ω με , r' is a source point and r is the field point.

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6.2.2 Point-Matching Method with Dirac Delta Basis Functions Electric- and magnetic-current densities can be expanded as a linear combination of a set of basis functions Jn and Mm: N

J (r ) ≈ ∑ α n J n (r )

(5)

n =1

M

M (r ) ≈ ∑ β m M m (r )

(6)

m =1

Since we intend to model the radiating equipment with electric and magnetic infinitesimal dipoles, Dirac delta basis functions are used:

J n (r ) = δ (r − rn )e n

(7)

M m (r ) = δ (r − rm )m m

(8)

where rn and rm are the positions of Jn and Mm, en and mm are the excitation currents of Jn and Mm which produce electric fields En(r, rn, en) and Em(r, rm, mm) respectively. In accordance, we have: N

M

n =1

m =1

E(r ) ≈ ∑ α nE n (r, rn , e n ) + ∑ β mE m (r, rm , m m )

(9)

6.2.3 Ground Plane in Semi-anechoic Chambers Anechoic chambers create a virtual free space. However, as shown in Fig. 6.3, the ground plane plays a role in semi-anechoic chambers.

EUT

(a)

EUT

(b)

Fig. 6.3 Radiation in Anechoic and Semi-anechoic Chamber (a) in an anechoic chamber, only the direct radiation is detected (b) in a semi-anechoic chamber, the combination of direct and reflected (on the ground plane) radiation is detected

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The simplest way to deal with ground plane is to regard it as a perfect conductor. Image theory [42] is therefore applicable. As shown in Fig. 6.4, the electromagnetic field generated by the EUT is therefore a superposition of the field generated by the original dipole set and the field generated by the image dipole set.

EUT Ground plane

Ground plane

(a)

(b)

(c)

Fig. 6.4 Effect of Ground Plane (a) EUT over a ground plane (b) EUT model over a ground plane (c) EUT model using image theory

6.3 Proposed Method for Near-Field to Far-Field Transformation In this section we describe a method to identify sources that generate radiated interferences and predict far-field radiation from near-field measurements [38]-[40]. Our goal here is to get a set of dipoles, {(αn, rn, en)} n=1,…,N and {(βm, rm, mm)} m=1,…,M, that produce the same electromagnetic field as that measured in near-field measurements.

6.3.1 Description of the Method We consider the method as shown in Fig. 6.5. An EUT radiates electromagnetic interference upon which near-field electromagnetic-radiation measurements can be performed in a small anechoic or semi-anechoic chamber. We intend to find an equivalent set of N+M infinitesimal electric and magnetic dipoles that radiate the same near fields as those radiated by the original EUT. The process starts from a generic model for the EUT composed of a set of infinitesimal dipoles distributed in the volume occupied by the equipment. Nearfield measurements of the EUT in anechoic chambers or semi-anechoic ones from which we can find an equivalent set of infinitesimal electric and magnetic dipoles are first carried out. A metaheuristic optimization algorithm is applied to find out parameters for the equivalent dipole set which radiates the same near field as that measured at the measurement points. Once the EUT’s equivalent model has been found, it can be used for two different tasks. The first task is the extraction of the information of the radiating parts in the EUT. In this way, the EUT main radiation sources will be identified, based on the position, the current, the orientation and the type of the dipoles of the equivalent set. The second task is the calculation of the field that the EUT would radiate in other circumstances and at any point of the space. It will be particularly interesting, from the point of view of electromagnetic compatibility, to find the field level that would be measured in the radiatedemission tests.

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EUT (a)

Near-field meas

Near-field meas

EUT EUT (b2)

(b1)

Same near-field values

Same near-field values

(c2)

(c1)

(d) Computed far-field values

(e1)

Computed far-field values

(e2)

Fig. 6.5 Process of the Proposed Method (a) EUT generating EMI (b1) near-field measurements in a small anechoic chamber (b2) near-field measurements in a small semianechoic chamber (c1) equivalent system which radiates the same near-field values in an anechoic environment (c2) equivalent system which radiates the same near-field values in a semi-anechoic environment (d) equivalent EUT model from which radiating sources can be identified (e1) calculation of the far-field radiation in free space (e2) calculation of the farfield radiation in an OATS

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6.3.2 Optimization Problem The problem of finding the equivalent dipole set that radiates the same near field can easily be cast into a minimization problem by defining the following objective function based on least squares:

f (x ) = ∑ γ p E(rp ) − ∑ α n E n (rp , rn , e n ) + ∑ β m Em (rp , rm , m m ) P

N

M

p =1

n =1

m=1

2

(10)

where x={(αn, rn, en)} n=1,…,N {(βm, rm, mm)} m=1,…,M is the vector of optimization parameters, γp is the weight for measurement point rp, which is necessary in certain occasions, and P is the total number of measurement points. f(x) is nonlinear and multidimensional. Conventional nonlinear multidimensional minimization algorithms [43], such as downhill or conjugategradient methods, can easily fall into local minima. Therefore, it is necessary to use a global search method, such as a metaheuristic one based on genetic algorithms [44].

6.3.3 Source Identification During the design and prototyping phase of electronic equipment, it is very important to locate the parts that generate higher radiation levels so that appropriate actions can be taken to reduce them. Since dipoles of the equivalent set tend to be placed by the algorithm on the main EUT radiation sources (whence the name of “distributed dipoles”), we will identify the interference-generating parts of the EUT from information on type of dipole and its excitation current, position and orientation. This method simultaneously identifies the radiating parts of the equipment and performs near-field to far-field transformation. Therefore, it saves a lot of time since these two processes were performed separately so far.

6.4 Electromagnetic Optimization by Genetic Algorithms Electromagnetic optimization by genetic algorithms (EMOGA) is a metaheuristic optimization algorithm [45] used to solve the problem presented in this chapter [38], [40], namely, to obtain near-field to far-field transformation and to identify radiation sources. It has evolved to a second version

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which includes new operators such as correlation, current scaling, and differential evolution [49], [50].

6.4.1 EMOGA v1.0: Genetic Algorithm EMOGA v1.0 is a hybrid algorithm which hybridizes a simple genetic algorithm [44], [46] with a final local search method (a downhill simplex method [43] or a direct search method [47]-[48]). Its block diagram is shown in Fig. 6.6. In particular, evolutionary operations including [46] roulette-wheel selection, single-point crossover, perturbation mutation, and elitism are implemented.

Start

Initialization

Mutation

Evaluation

Crossover

Selection

No

Converged?

Yes Local search

End

Fig. 6.6 Block Diagram of EMOGA v1.0

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6.4.2 EMOGA v2.0: Metaheuristic Method EMOGA v1.0 has evolved to the more reliable and efficient EMOGA v2.0 to suit our problem in a better way by introducing new operations, such as current scaling, dipole correlation, memetization and differential evolution [49], [50]. Its block diagram is given in Fig. 6.7.

Start

Initialization

Mutation

Current scaling

Evaluation

Crossover

Correlation

Selection

Memetization

No

Differential

Converged?

Yes Local search

End

Fig. 6.7 Block Diagram of EMOGA v2.0

6.4.2.1 Current Scaling In measurements of radiated emissions, the most important issue is to find the farfield radiation maximum. To guarantee a solution compatible with this, one strategy is to scale the field radiated by the equivalent set of dipoles to the measured near-field maximum. This can be achieved by scaling the currents of all dipoles of the equivalent set by the quotient between the maximal measured near field and the maximal near field generated by the equivalent set of dipoles before scaling the current. Current scaling improves the behavior of the algorithm in two aspects: •

Since current scaling fixes the maximal level of the field, it is also determining the order of magnitude of the maximum of the dipole excitation.

Modeling of Electrically Large Equipment with Distributed Dipoles

•

143

The knowledge of the order of magnitude of dipole currents reduces the search space, which is well known to be advantageous. Current scaling also helps to reduce error. Regardless of level difference from the field generated by the equivalent set to the measured one, the average error after current scaling tends to diminish. Having the maximal measured and calculated field equal places the two sets of field values in the same dynamic range, and, therefore, makes the calculated error more meaningful.

6.4.2.2 Correlation between Dipoles Let us assume a dipole set S1=(D1, D2, D3, …, DN) formed by N dipoles that generates a given radiation. Another set S2=(D3, D 1, D 2, …, DN) with exactly the same dipoles but arranged in a different order, will present exactly the same radiation as that of S1. In fact, there will be N! arrangements of dipoles (and therefore sets of dipoles) that will generate exactly the same radiation. This fact may confuse the algorithm, particularly at the crossover phase. In order to solve this problem, we use a technique which reorders dipoles before crossover. This reordering is performed in a way that minimizes the Euclidean distance between the two parent individuals selected for crossover, that is to say, we look for the dipole order with the maximal correlation between both individuals. In order to find the correct disposition of dipoles, all (N2+N)/2 Euclidean distances that can be generated combining the N dipoles have to be calculated. A distance matrix d accordingly is generated as

ª d11 d12 «d « 21 d 22 d = « d 31 d 32 « # « # «¬d N 1 d N 2

d13 d 23 d 33 # dN3

" " " % "

d1N º d 2 N »» d3 N » » # » d NN »¼

(11)

where dij = dji. The combination that minimizes the distance of the set of dipoles is then found using a backtracking algorithm. 6.4.2.3 Memetization Memetization [49] is a technique that applies a local search algorithm, for example, a downhill simplex method [43] in this case, to all individuals of the population. Application of local search on all individuals places them on their

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nearest local minima. However, it has to be pointed out that it is very costly from a computational point of view.

6.5 Numerical Results In this section we will find the model of an UHF transmitter using the proposed approach. More examples can be found in [38]-[41], [51].

6.5.1 Measurement Systems 6.5.1.1 General Measurement System The measurements of this example were performed in compliance electromagnetic compatibility measurement chambers for radiated emissions. Therefore, we had to work with data acquisition systems in cylindrical geometry using mast and turn table, the only one permitted by the measurement infrastructure. The near-field measurements were performed in a semi-anechoic chamber of 3 meters with the floor covered with absorbers in order to make it anechoic. The far-field measurements were performed in a semi-anechoic chamber of 10 meters. The measuring system, as shown in Fig. 6.8, is based on a probe capable of measuring electric fields in three orthogonal axes. This probe transmits the measured field in each axis to an optic-electric transducer which provides a

EXTERIOR OF THE MEASUREMENT ENVIRONMENT MEASUREMENT SYSTEM COMPUTER AND CONTROL BUSES

INTERIOR OF THE MEASUREMENT ENVIRONMENT (ANECHOIC CHAMBER) OPTIC-ELECTRIC TRANSDUCER

OPTICAL FIBER LINK FIELD PROBE

EUT

COAXIAL CABLE

NETWORK ANALYZER

Fig. 6.8 Near-field Measurement Setup (for far-field measurements, the setup is the same, but placed in a larger semi-anechoic chamber)

Modeling of Electrically Large Equipment with Distributed Dipoles

145

voltage proportional to the instantaneous field detected by the probe. The output voltage of the transducer is injected to a network analyzer which makes it possible to simultaneously measure two axes of the probe (vertical and horizontal polarization) in order to obtain the magnitude of each polarization and their relative phase. A computer is in charge of controlling the system and storing measurement data. We performed measurements of a radiofrequency transmitter in the UHF band, transmitting at 430 MHz, as shown in Fig. 6.9.

Fig. 6.9 Transmitter Disposition

6.5.1.2 Near-Field Measurement System Measurements of the spurious radiation at 430 MHz generated by the transmitter were performed in an anechoic chamber, as shown in Fig. 6.10.

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Fig. 6.10 Near-field Measurement Setup

It has to be pointed out that much information about spatial behavior of the radiation of the equipment is lost in a cylindrical coordinate measurement system, as shown in Fig. 6.11. This fact affects the algorithm convergence. To overcome this problem, a technique using near-field measurements performed at two different distances were applied: field levels on two concentric cylinders, of radii 1.1 m and 2 m respectively, were used. BLIND CONE WHERE RADIATION PATTERN INFORMATION IS LOST

EUT

Fig. 6.11 Problem of Cylindrical Measurement System

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6.5.1.3 Far-Field Measurement System In order to validate the far-field levels predicted by the algorithm, far-field measurements were performed in a semi-anechoic chamber of 10 meters, as shown in Fig. 6.12. In this case, we eliminated the pedestal placed under the equipment in near-field measurements.

Fig. 6.12 Far-field Measurement Setup in Semi-anechoic Chamber

6.5.2 Near-Field Results The near-field measurements at 1.1 m and 2 m away from the EUT were used by the EMOGA algorithm to find an equivalent set of five infinitesimal dipoles, as tabulated in Table 6.1. In it, en = (Ix, Iy, Iz)ejø or mn = (Ix, Iy, Iz)ejø, depending on the type of dipole (electric or magnetic), and rn = (x, y, z). The average error between the fields measured and those generated by the equivalent set was 1.39 dB. Table 6.1 Dipoles of the Equivalent Set Obtained by EMOGA type

Ix(ȝA)

Iy(ȝA)

Iz(ȝA)

ø(rad)

x(cm)

y(cm)

z(cm)

D1

m

-105.41

-560.92

1081.70

2.66

15.05

11.53

221.25

D2

m

23.24

224.32

585.36

1.31

25.82

1.18

227.18

D3

m

333.57

-274.01

-60.82

3.13

1.48

17.63

249.54

D4

m

-111.30

-166.73

353.54

2.28

-6.62

17.50

166.80

D5

e

116.70

-210.01

193.23

4.66

6.35

-6.42

-31.00

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The three most important dipoles (D1, D2 and D3) are placed near to the transmitter’s position and to the RF cable that connects it with the load, while D4 is in the area where there is only the power-supply cable. All of them are magnetic dipoles. In Figs. 6.13 and 6.14 we compare the measurements performed in near-field (1.1 m and 2 m away, respectively) with the field generated by the equivalent set of dipoles in the same measurement points. We can see that they are highly similar, both concerning the pattern and the electric field maxima, which occur in the horizontal component (Eø).

(a) measured Eø

(b) Eø of equivalent set

(c) measured Ez

(d) Ez of equivalent set

Fig. 6.13 Pattern Measured and Generated by the Equivalent Set in Near Field (1.1 m)

Modeling of Electrically Large Equipment with Distributed Dipoles

(a) measured Eø

149

(b) Eø of equivalent set

(c) measured Ez

(d) Ez of equivalent set

Fig. 6.14 Pattern Measured and Generated by the Equivalent Set in Near Field (2 m)

6.5.3 Far-Field Prediction Since the pedestal placed under the EUT in near-field measurements is removed, the original equivalent set of dipoles has to be modified by subtracting the height of the platform (150 cm) from the z coordinate of the dipoles. The resulting dipoles that fall below z = 0 zero are deleted, since in the semi-anechoic case they are under the ground plane. The new equivalent set is shown in Table 6.2. Table 6.2 Dipoles of the Equivalent Set for Far Field type

Ix(ȝA)

Iy(ȝA)

Iz(ȝA)

ø(rad)

x(cm)

y(cm)

z(cm)

D1

m

-105.41

-560.92

1081.70

2.66

15.05

11.53

71.25

D2

m

23.24

224.32

585.36

1.31

25.82

1.18

77.18

D3

m

333.57

-274.01

-60.82

3.13

1.48

17.63

99.54

D4

m

-111.30

-166.73

353.54

2.28

-6.62

17.50

16.80

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The measured transmitter far-field (10 m away) radiation in a semi-anechoic environment and that predicted by the equivalent set of dipoles is given in Fig. 6.15. As can be seen, prediction and measurement are in very good agreement (mainly in the much more energetic horizontal Eø component), both concerning the shape of the radiated field and its maximum value. The error in the maximum field value (which occurs for the horizontal Eø component) is of only 1.3 dB.

(a) measured Eø

(c) measured Ez

(b) Eø of equivalent set

(d) Ez of equivalent set

Fig. 6.15 Patterns Measured and Predicted by the Equivalent Set in Far Field in a Semi-anechoic Environment (10 m)

6.6 Conclusions A metaheuristic algorithm featuring differential evolution has been described for performing both near-field to far-field transformation and source identification in electromagnetic-compatibility radiated-emission measurements. The method is very flexible since it allows inhomogeneous data as its input (different fieldcomponent measurements performed at points of space not necessarily distributed

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over planar, cylindrical or spherical surfaces). The method is a promising computational alternative to the expensive large semi-anechoic chambers necessary to perform EMC far-field radiated emission measurements. It also proves that non-deterministic algorithms are a valid alternative to solve some kinds of complicated computational-electromagnetics problems.

References [1] IEEE Standard Test Procedures for Antennas, ANSI/IEEE Std 149-1979 (1979) [2] Evans, G.E.: Antenna Measurement Techniques. Artech House, Boston (1990) [3] Yaghjian, A.D.: An overview of near-field antenna measurements. IEEE Trans. Antennas Propagation 34(1), 30–45 (1986) [4] IEEE Standard Definitions of Terms for Antennas, IEEE Std 145-1993. IEEE, Los Alamitos (1993) [5] Balanis, C.A.: Antenna Theory: Analysis and Design. John Wiley & Sons, New York (1997) [6] Johnson, R.C., Ecker, H.A., Hollis, J.S.: Determination of far-field antenna patterns from near-field measurements. Proc. IEEE 61(12), 1668–1694 (1973) [7] Petre, P., Sarkar, T.K.: Planar near-field to far-field transformation using an equivalent magnetic current approach. IEEE Trans. Antennas Propagation 40(11), 1348–1356 (1992) [8] Bucci, O.M., D’Elia, G., Migliore, M.D.: An effective near-field far-field transformation technique from truncated and inaccurate amplitude-only data. IEEE Trans. Antennas Propagation 47(9), 1377–1385 (1999) [9] Sarkar, T.K., Taaghol, A.: Near-field to near/far-field transformation for arbitrary near-field geometry utilizing an equivalent electric current and MoM. IEEE Trans. Antennas Propagation 47(3), 566–573 (1999) [10] Las-Heras, F., Sarkar, T.K.: Radial field retrieval in spherical scanning for current reconstruction and NF-FF transformation. IEEE Trans. Antennas Propagation 50(6), 866–874 (2002) [11] Pérez, J.R., Basterrechea, J.: Antenna far-fields from planar acquisition using microgenetic algorithms. Electronic Letters 39(13), 948–949 (2003) [12] Slater, D.: Near-Field Antenna Measurements. Artech House, Boston (1991) [13] Bolomey, J.C.: Rapid near-field techniques for EMC applications. In: 10th Int. Zurich Symp. Technical Exhibition EMC, Zurich, Switzerland, March 9-11, pp. 55–60 (1993) [14] Bolomey, J.C.: Introduction to near-field techniques for emc applications: State of the art and prospective. In: 2001 IEEE EMC Int. Symp., Montréal, Canada, August 1317, vol. 1, p. 356 (2001) [15] Bucci, O.M., D’Elia, G., Migliore, M.D., Pierri, R.: Phaseless near-field measurements. In: 2001 IEEE EMC Int. Symp., Montréal, Canada, August 13-17, vol. 1, pp. 389–394 (2001) [16] Sarkar, T.K., Las-Heras, F.: A direct optimization approach for source reconstruction NFFF using amplitude only data. In: 2001 IEEE EMC Int. Symp., Montréal, Canada, August 13-17, vol. 1, pp. 399–402 (2001)

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[17] Laroussi, R., Costache, G.I.: Novel technique for radiated far-field predictions from near-field measurements. In: 1992 Asia-Pacific Microwave Conf., Adelaide, Australia, August 11-13, vol. 2, pp. 791–794 (1992) [18] Laroussi, R., Costache, G.I.: Far-field predictions from near-field measurements using an exact integral equation solution. IEEE Trans. Electromagnetic Compatibility 36(3), 189–195 (1994) [19] Vandenbosch, G., Van de Capelle, A.: Simulation of an EMC reference site in a nearfield antenna range. In: 10th Int. Zurich Symp. Technical Exhibition EMC, Zurich, Switzerland, March 9-11, pp. 61–66 (1993) [20] Cerri, G., Leo, R.D., Mariani, V., Martiri, M.: Radiated emission estimation by measurement of magnetic near fields. In: 12th Int. Symp. Technical Exhibition EMC, Zurich, Switzerland, Febuary 18-20, pp. 447–452 (1997) [21] Fourestié, B., Altman, Z., Wiart, J., Bolomey, J.C., Brouaye, F.: A novel near-field measurement facility for random emissions. In: 2001 IEEE EMC Int. Symp., Montréal, Canada, August 13-17, vol. 1, pp. 378–382 (2001) [22] Fourestié, B., Altman, Z., Bolomey, J.C., Wiart, J., Brouaye, F.: Statistical modal analysis applied to near-field measurements of random emissions. IEEE Trans. Antennas Propagation 50(12), 1803–1812 (2002) [23] Nadeau, B., Laurin, J.J.: Extrapolations using vectorial planar near-field measurements for EMC applications. In: 1998 IEEE Int. Symp. Electromagnetic Compatibility, Denver, Colorado, August 23-28, vol. 2, pp. 924–928 (1998) [24] Laurin, J.J.: Near-field characterization of PCBs for radiated emissions prediction. In: 1993 IEEE Int. Symp. Electromagnetic Compatibility, Dallas, TX, August 9-13, vol. 2, pp. 322–326 (1993) [25] Laurin, J.J., Ouardhiri, Z., Colinas, J.: Near-field imaging of radiated emission sources on printed-circuit boards. In: 2001 IEEE EMC Int. Symp., Montréal, Canada, August 13-17, vol. 1, pp. 368–373 (2001) [26] Jackson, J.D.: Classical electrodynamics, 3rd edn. John Wiley & Sons, New York (1999) [27] Sreenivasiah, I., Chang, D.C., Ma, M.T.: Emission characteristics of electrically small radiating sources from tests inside a TEM cell. IEEE Trans. Electromagnetic Compatibility 23(3), 113–121 (1981) [28] Koepke, G.H., Ma, M.T.: A new method for determining the emission characteristics of an unknown interference source. In: 5th Int. Zurich Symp. Technical Exhibition EMC, Zurich, Switzerland, March 8-10, pp. 35–40 (1983) [29] Wilson, P., Hansen, D., Koenigstein, D.: Simulating open area test site emission measurements based on data obtained in a novel broadband TEM cell. In: IEEE National Symp. Electromagnetic Compatibility, Denver, Colorado, May 23-25, pp. 171–177 (1989) [30] Kanda, M., Hill, D.A.: A three-loop method for determining the radiation characteristics of an electrically small source. IEEE Trans. Electromagnetic Compatibility 34(1), 1–3 (1992) [31] Wilson, P.: On simulating oats near-field emission measurements via GTEM cell measurements. In: IEEE Int. Symp. Electromagnetic Compatibility, Dallas, TX, August 9-13, pp. 53–57 (1993) [32] Wilson, P.: On correlating TEM cell and OATS emission measurements. IEEE Trans. Electromagnetic Compatibility 37(1), 1–16 (1995)

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Chapter 7 Application of Differential Evolution to a Multi-Objective Real-World Frequency Assignment Problem Marisa Silva Maximiano, Miguel A. Vega-Rodríguez, Juan A. Gómez-Pulido, and Juan M. Sánchez-Pérez 1

7.1 Introduction Frequency spectrum is one of the scarcest resources for any mobile operator. Frequencies have to be reused throughout the network. Consequently, interferences may occur and some separation constraints may be violated. Frequency assignment problem (FAP) aims to use effectively the available frequency spectrum to minimize interferences by carefully allocating available frequencies to existing base stations [1]. It is a very demanding problem in telecommunications, especially in GSM networks [2], even though it is very timeconsuming. It is one of the most fundamental problems in mobile communications planning. A good FAP solution leads to better network quality and increased capacity without sacrificing quality of service (QoS) for all users of the mobile network. Marisa Silva Maximiano Polytechnic Institute of Leiria, Department of Informatic Engineering, School of Technology and Management, Campus 2 - Morro do Lena - Alto do Vieiro, Apartado 4163, 2411-901 Leiria, Portugal e-mail: [email protected] 1

Miguel A. Vega-Rodríguez . Juan A. Gómez-Pulido . Juan M. Sánchez-Pérez University of Extremadura, Department of Technologies of Computers and Communications, Escuela Politécnica. Campus Universitario s/n. 10071, Cáceres, Spain e-mail: {mavega,jangomez,sanperez}@unex.es A. Qing and C.K. Lee: Differential Evolution in Electromagnetics, ALO 4, pp. 155–176. © Springer-Verlag Berlin Heidelberg 2010 springerlink.com

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FAP is a multiobjective NP-hard optimization problem with two conflicting objectives: interference cost and separation cost. In the past few years, different interactive and non-interactive algorithms [3] such as heuristic algorithms, graph theory and evolutionary algorithms, have been applied to solve FAP. In this Chapter, classic differential evolution [4]-[8] has been adapted for multiobjective FAP using Pareto tournaments (DEPT). Pareto tournaments are used whenever it is necessary to select the prevailing solution between two competing solutions. In our study we have used two distinct test scenarios that represent real-world instances of GSM networks. We have comparatively assessed the algorithm against trajectory-based algorithms using two complementary indicators: the hypervolume indicator [9] and the coverage relation [10]. The organization of the paper is as follows. In Section 2, we present the multiobjective FAP in a GSM network and its mathematical formulation. In Section 3 the DEPT algorithm is explained. Section 4 explains others multiobjective algorithms, namely MO-VNS, MO-SVNS and their Greedy variants. Numerical results are presented in Section 5. Finally, we offer some conclusions and outline of future work in the last section.

7.2 Multi-objective FAP in a GSM Network In the following we first provide a brief description of the GSM system components as well as the definition of the frequency assignment problem in the GSM context. Next, we describe the mathematical model used to address the realworld instances of FAP. At the end we describe the configuration of the two instances used.

7.2.1 GSM Components and Frequency Planning A GSM system contains many different components. In fact, as shown in Fig. 7.1, a GSM system can be divided into three major systems: network and switching system (NSS), base station system (BSS), and operation and maintenance SubSystem (OSS). NSS is responsible for performing call processing and subscriber-related functions. It includes several functional units, as shown in Fig. 7.1: HLR, MSC, VLR, AUC, EIR. All radio-related functions are performed in the BSS subsystem. It is composed of base station controllers (BSCs) and base transceiver stations (BTSs). The main function of BTS is to handle the radio interface to the mobile station (MS). BTS is the radio transceivers needed in the network to serve each cell in the network.

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Fig. 7.1 GSM Architecture

Mobile communication is dependent on a radio link between a user’s mobile terminal and the base stations (BTS) on the network. After the placement of the base stations, the next fundamental phase is the selection and configuration of their antennas. The configuration is performed in order to provide a desired network coverage. The most relevant components in a GSM infrastructure are essentially BTSs, inside which a set of transceivers (TRXs) have been installed, either broadcast control channel (BCCH) or traffic channel (TCH). A base station typically serves three different sectors, or cells. The number of TRXs to be installed inside each sector is fixed and it depends on the traffic demand the sector has to support. Each transceiver uses a frequency slot, also called channel. Nearby transceivers have to use different channels. An operator has to reuse his channels multiple times to operate enormous amounts of transceiver in the network due to limited radio spectrum. Unfortunately, significant interferences between transmitters using the same channel (co-channel) or an adjacent channel may occur. Consequently, service quality of the GSM network degrades. FAP tries to properly allocate a frequency to each and every TRX of the network to minimize such undesired interferences. Two distinct goals that correspond to the interferences cost and the separation cost have to be considered simultaneously.

7.2.2 Interference Cost Interferences occur when frequencies are reused by several TRXs. It is usually represented by an interference matrix M [11] whose element Mij represents the degradation of network quality due to co-channel interference, or adjacent channel interference [12]. Co-channel interference occurs when sectors i and j operate at the same frequency while adjacent channel interference happens when sectors i and j operate on adjacent channels.

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Two values, μij and σij, representing mean and standard deviation respectively of a Gaussian probability distribution, are used to quantify the interference on the GSM network. Higher μij value corresponds to lower interference level, or better communication quality. Interference cost is accordingly defined as [14].

C I (p ) = ∑

∑ C (p, u, t ) sig

(1)

t∈T u∈T ,u ≠t

where p is a FAP solution, T is a set of TRXs,

⎧K ⎪ ⎪Cc Csig (p, u, t ) = ⎨ ⎪Ca ⎪ ⎩0

st = su , p(t ) − p(u ) < 2

(μ (μ

st su

,σ s t s u

st su

,σ st su

) )

st ≠ su , μ st s u > 0, p(t ) − p(u ) = 0 st ≠ su , μ st su > 0, p(t ) − p(u ) = 1

(2)

otherwise

K is a very large value defined in the configuration files of the network which makes it undesirable to allocate the same or adjacent frequencies to TRXs that are installed in the same sector, st is a sector in which transceiver t is installed, and su is the other sector in which transceiver u is installed, Cc is co-channel interferences and Ca is adjacent-channel interferences, p(t) is the frequency assigned to transceiver t, and p(u) is the frequency assigned to transceiver u. In this paper, we follow the setting in [13], i.e., K=100.000, signaling threshold CSH=6dB, adjacent channel interference rejection CACR =18dB. CSH is the minimum quality signalling threshold, that is, if the carrier-to-interference ratio is lower than this threshold there will be a degradation of the communication quality. On the other hand, CACR is a hardware specific constant that measures the ability of the receiver to receive the wanted signal in the presence of an unwanted signal at an adjacent channel. For more detailed explanation about the mathematical formulation, please refer to [14].

7.2.3 Separation Cost Technical limitations in the construction of sectors and BTSs mean that certain combinations of TRX channel are not permitted. These constraints include (1) Site Channel Separation Any pair of frequencies at a site (BTS) must be separated by a certain fixed amount, typically 2 channels for a large problem. If a BTS uses high power TRXs, its channel separation should be larger. Violation of this separation regulation involves a cost Csite. In our case, this cost is the violated number of site channel separations. (2) Sector Channel Separation This is similar to the previous one, but at sector level. In conclusion, any pair of frequencies at a sector must be separated by a certain fixed amount, typically 3 channels for a large problem. It is important to observe that sector channel separation generally is larger than site channel separation, due to shorter distances

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between involved TRXs. Similarly, violation of sector channel separation regulation involves a cost Csector. Likewise, this cost is the number of violated sector channel separations. In total, the separation cost is

CS (p ) = Csite (p ) + Csec tor (p )

(3)

7.3 Multi-objective Differential Evolution with Pareto Tournaments Multi-objective differential evolution with Pareto tournaments [15] is based on classic differential evolution [4][5][6][7][8]. It incorporates Pareto tournaments to address multi-objective FAP.

7.3.1 Algorithm Structure The pseudocode for DEPT algorithm is given in Fig. 7.2 where Np is population size. It can be seen that the algorithm structure look almost identical with that of classic differential evolution given in Fig. 2.1 of this book, except that new individual pt+1,i is generated through tournament between ct+1,i and pt,i. t=0 initialize population Pt evaluation population Pt while time limit is not reached do for i = 1, Np create a child ct+1,i evaluate child ct+1,i Pareto tournament between ct+1,i and pt,i to get individual pt+1,i end for t=t+1 end while Fig. 7.2 Pseudocode for DEPT Algorithm

7.3.2 Pareto Tournament Pareto tournament [16] is implemented in DEPT to select individuals that survive to the next generation. An individual has a fitness value that is a scalar value to be minimized. In multi-objective scenario, this fitness value is defined as i i fitness i = N p N isDo min ated + N do min ates

(4)

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where NiisDominated is the number of individuals that dominate individual i while Nidominates is the number of individuals that are dominated by individual i. Child ct+1,i joins population Pt to compute NiisDominated and Nidominates, and accordingly fitness, for both pt,i and ct+1,i. A tournament between pt,i and ct+1,i is then implemented based on their fitness value. 1. 2. 3. 4. 5.

The individual with smaller fitness wins. In case of a tie, the winner is determined according to their relevant dominance. If pt,i and ct+1,i continue to tie, the winner will be the one having smaller interference cost. If pt,i and ct+1,i tie again, the winner will be the one having smaller the separation cost. If tie happens again, pt,i survives.

7.3.3 Problem Domain Knowledge Whenever possible, knowledge about the problem domain has to be incorporated to create valid solutions. In our practice, an operation is performed to assure that a frequency is assigned to a TRX installed in a sector only if this frequency does not produce adjacent-channel or co-channel interferences within this same sector. It is triggered when a frequency value is to be replaced by a new value. The set of available frequencies are maintained for every sector, and are continuously updated. At the beginning, valid frequencies are available to all sectors. After that, when a new frequency f is assigned to a TRX, f as well as its respective adjacent frequencies f + 1 and f – 1, is removed from the set of frequencies available to the sector in which the TRX is installed. Meanwhile, the older frequency value reappears in the set.

7.4 Multi-objective Variable Neighborhood Search 7.4.1 Variable Neighborhood Search Variable neighborhood search (VNS) [17]–[18] is a metaheuristic based on trajectories. It starts from an initial solution and performs several changes of neighborhood within a local search. The algorithm increases neighborhood size (k parameter) when the search does not move forward. Following the notations in [17], denote Nk, 1≤k≤Nk, as a finite set of preselected neighborhood structures, and Nk(S) the set of solutions in the kth neighborhood of current solution S. Several neighborhoods Nk can be produced from the solution space of S. VNS gradually applies a mutation operator to generate a new solution S' from S, conducted according to the type of environment used in each iteration Nk(S). If this solution S' improves the current solution S, the next iteration will start again to use the first environment E1. Otherwise, the algorithm will go to the next neighborhood environment.

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The mutation operator was implemented using specific information from the domain of the problem. It assigns a random frequency value for a TRX from unused frequencies of the sector in which the TRX is installed. The set of available frequencies of the specific sector is updated accordingly.

7.4.2 Multi-objective Variable Neighborhood Search The multi-objective variable neighborhood search (MO-VNS) is developed to solve multi-objective optimization problems by implementing dominance concept in VNS to select better solution. The pseudocode of MO-VNS is given in Fig. 7.3. It starts from a random initial solution. Neighborhood solutions are generated by mutation until no further improvements are possible. It also changes the environment when the obtained solution is worse than the current solution.

ParetoSolutionsĸ0 while time limit is not reached do SĸgenerateSolution(at random) if S dominates any Pareto solution in ParetoSolutions add S into ParetoSolutions remove solutions in ParetoSolutions dominated by S end if kĸ1 while k kmax do SƍĸmutationInEnvironmentk(S) if Sƍ dominates any Pareto solution in ParetoSolutions add Sƍ into ParetoSolutions remove solutions in ParetoSolutions dominated by Sƍ end if if Sƍ dominates S SĸSƍ kĸ1 else kĸ k+1 end if end while end while return paretoSolutions

Fig. 7.3 Pseudocode for Multi-objective Variable Neighborhood Search

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At the beginning, the set of pareto solutions is empty. It will contain all nondominated solutions that represent the best solutions found so far. The size of the set of pareto solutions is not always the same because it changes along the run. In every iteration, a comparison between existent pareto solutions and the candidate solution is performed to see if it is a better solution.

7.4.3 Greedy Mutation The greedy mutation, as shown in Fig. 7.4, considers all potential neighborhood solutions. At the beginning, the sector in which a TRX is installed and the corresponding unused frequencies are identified. Then, the best solution in terms of dominance is chosen from all possible solutions by testing all frequencies available to that TRX. for TRXid = 0 to allTRXInNetwork do if randomNumber() mutation probability do sectorĸgetSector(TRXid) unusedFreqĸgetUnusedFreq4Sector(sector) populationĸgenerateSolution(unusedFreq) computeDominance(population) bestĸselectBestSolution(population) end if end for return best

Fig. 7.4 Pseudocode for Greedy Mutation

This procedure uses a temporary population that only contains the candidate neighborhood solutions. The individuals inside are eliminated right after the mutation operation.

7.4.4 Multi-objective Skewed Variable Neighborhood Search The multiobjective skewed variable neighbourhood search (MO-SVNS), as shown in Fig. 7.5 where 0≤α≤1.0 is a quality magnitude parameter, implements Pareto dominance in skewed variable neighbourhood search (SVNS) [17]–[18] to solve multi-objective optimization problems. It restarts from another initial solution to explore remote areas when it is lost in a local optimum which may be far from the global optimum. However, unlike MO-VNS, it is more receptive in accepting a new solution, even if it is slightly

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worse than the previous solution. It also increases the size of neighborhood when the search does not move forward. ParetoSolutionsĸ0 while time limit is not reached do SĸgenerateSolution(at random) if S dominates any Pareto solution in ParetoSolutions add S into ParetoSolutions remove solutions in ParetoSolutions dominated by S end if kĸ1 while k kmax do SƍĸmutationInEnvironmentk(S) if Sƍ dominates any Pareto solution in ParetoSolutions add Sƍ into ParetoSolutions remove solutions in ParetoSolutions dominated by Sƍ end if if Sƍ dominates S+Į*distance(S, Sƍ) SĸSƍ kĸ1 else kĸ k+1 end if end while end while return paretoSolutions

Fig. 7.5 Pseudocode for Multi-objective Skewed Variable Neighborhood Search

The concept behind is to keep the best found solution and quantifys the distance between two distinct solutions S and S′ using a bi-dimensional distance function that represents the two different goals to be minimized. The distance function is the difference between the lower cost value of S and S′ and the highest cost value of all solutions inside the set of Pareto Solutions.

7.5 Experiments and Results 7.5.1 Experimental Setup 7.5.1.1 Used GSM Instances Two different real-world FAP instances currently operating in Seattle and Denver were used. These instances have different sizes among which the Denver instance is larger and therefore most complex to tackle.

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Fig. 7.6 Topology of the Denver GSM Instance

Fig. 7.7 Topology of the Seattle GSM Instance

The Denver instance has 2612 TRXs, installed in 711 sectors distributed in 334 BTSs. Every TRX has only 18 valid frequencies to be assigned to it. The Seattle instance is smaller. It has 970 TRXs installed in 1180 sectors distributed in 503

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BTSs. Each TRX has 15 valid frequencies to be assigned to it. The range of valid frequencies may differ from TRXs. Figs. 7.6 and 7.7 show the network topology for each of the instances. In these figures, every triangle represents a sectorized antenna in which the TRXs operate. 7.5.1.2 Encoding A solution is coded as a vector of integers. As an example, Fig. 7.8 shows the representation of a solution for the Denver instance. Each gene corresponds to the frequency value assigned to the TRX. In our experiments scenario, the range of valid frequencies for different TRXs can be different. Therefore, for every TRX it is kept a set of valid frequencies. In terms of memory consumption, store all this information becomes an important overhead.

Fig. 7.8 Solution Encoding

7.5.1.3 Computational Facilities All experiments have been carried out under exactly the same conditions: PC with 2.66Ghz Intel Pentium processor and 3GB RAM running Windows XP operating system. The code was developed under the .NET framework 3.5 with C# language through object-oriented programming (OOP) approach. 7.5.1.4 Termination Conditions and Process Monitoring An execution is performed during 30 minutes. The minimum and average fitness and objective values of all solutions in the Pareto front are saved every 2 minutes. At the end of each run, the final result is also saved. 7.5.1.5 Confidence Building In order to provide results with a statistical confidence, 30 independent runs are implemented for each experiment.

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7.5.2 Methodology and Metrics Evaluating the performance of multi-objective algorithms is far from being a trivial task since there are multiple solutions instead of a good solution. We have incorporated two complementary measurements besides common statistical comparisons to assess solution quality: hypervolume indicator that defines the volume of the objective space dominated by a Pareto front, and coverage relation [10] to determine best Pareto front. Both are applied to all non-dominated solutions as comparison criterion for assessing Pareto optimal solutions. 7.5.2.1 Hypervolume The hypervolume indicator [9] is truly useful because it rewards convergence towards the Pareto front. It defines the measure of the region simultaneously dominated by every Pareto front point and bounded above by a reference point. In our case, the hypervolume represents the two-dimensional area dominated by the Pareto front. The Pareto front with higher hypervolume value is a better configuration plan for FAP. It is necessary to define a search space by specifying upper bound and lower bound. In our practice, the upper bound is reference point. The upper bound was (300000, 2500) for the Denver instance and (70000, 200) for the Seattle instance. The lower bound points were (130000, 1200) and (4000, 15) respectively. 7.5.2.2 Coverage Relation Coverage relation [9] was used as an additional performance indicator. In terms of dominance, solution x1 covers solution x2, if x1 dominates or ties with x2. Given two sets of non-dominated solutions, we compute the fraction of a set of solutions covered by the other set of solutions. A higher percentage of coverage means a better solution set.

7.5.3 Tuning of the DEPT Parameters DEPT has many variants and involve three intrinsic control parameters. The tuning process aims to find the best variant and its corresponding optimal intrinsic control parameter values. It starts from DE/Rand/2/Bin with CR=0.2 and F=0.1, the best parameters based on a previous work with a mono-objective version of FAP [19][20]. The findings are applied in the final experiments. 10 instead of 30 independent runs are implemented for the tuning process. In addition, we have considered three different time limits, 120, 600, and 1800 seconds. Statistical analysis over minimum, average and standard deviation of the two objective functions and the fitness value has been performed. The tuning is performed in the following order: 1. 2. 3. 4.

population size crossover probability mutation intensity differential evolution strategies

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7.5.3.1 Population Size Population sizes tested are 10, 25, 50, 75 and 100. Figs. 7.9 and 7.10 show the corresponding Pareto fronts. In general, higher NP value leads to better

Fig. 7.9 Pareto Fronts of Experiments against Population Size Using the Denver Instance

Fig. 7.10 Pareto Fronts of Experiments against Population Size Using the Seattle Instance

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hypervolume. NP=75 works best for both instances, with 24.8% of hypervolume for the Denver instance and 23.8% for Seattle instance. 7.5.3.2 Crossover Probability The next experiment was focused on crossover probability (CR). The CR parameter normally uses values between [0, 1]. Our first set of values was 0.1, 0.25, 0.5, 0.75, and 0.9. Results are tabulated in Table 7.1 where IC means Interference Cost, SC stands for Separation Cost and HV represents the HyperVolume. We clearly identified that the best solutions were founded using CR of 0.1. A CR=0.1 has higher mean hypervolume and smaller standard deviation value (1.1%).

Table 7.1 Results for Experiments against Crossover Probability (CR)

CR IC Seattle SC HV IC Denver SC HV

0.1

0.25

0.5

0.75

0.9

Best

42306.3

45582.5

47664.6

47516.7

48379.7

Avg

46524.7

48284.3

50018.4

51134.9

51037.7

Std

3344.8

1888.3

1886.2

1975.1

2542.1

Best

64

84

91

97

91

Avg

100.0

108.4

117.0

111.6

116.9

Std

24.2

15.4

17.0

9.4

22.3

(%)

28.3

22.0

18.9

18.0

18.3

Best

229511.2

233629.2

235176.8

236098.1

237040.9

Avg

236070.5

238991.0

240475.4

240831.8

241193.0

Std

6148.3

3854.8

3891.8

3318.2

3535.5

Best

1644

1695

1719

1729

1722

Avg

1750.6

1758.1

1773.7

1784.5

1776.9

Std

78.6

40.0

32.8

37.9

39.2

(%)

26.7

23.9

22.6

22.1

22.0

27.5

22.9

20.8

20.0

20.1

Mean HV (%)

Inspired by the above finding, we further investigate if smaller crossover probability is better. The second set of crossover probability, 0.01, 0.02, 0.03, 0.05, 0.07, and 0.09, is therefore tested. Results are tabulated in Table 7.2. Hypervolume is the highest at 0.01.

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Table 7.2 Results for Experiments against Crossover Probability (CR) CR Best

0.01 34882.4

0.02 37774.2

0.03 39066.0

0.05 40693.5

0.07 41746.3

0.09 42604.6 46763.2

Avg

37677.0

41320.4

42985.6

45208.6

45989.1

Std

2300.1

2833.5

3112.7

3724.5

3510.9

3589.9

Best

32

36

42

47

55

59

Avg

73.8

76.9

82.3

85.5

91.0

95.1

Std

25.1

25.6

26.7

26.2

25.7

24.6

(%)

46.4

40.9

37.5

33.9

31.1

29.2

Best

226637.1

226734.7

225774.4

226621.7

228018.0

228558.5

Avg

232924.9

233724.5

233053.9

234546.1

236359.9

236690.3

Std

3175.0

4329.7

5646.1

5947.0

6402.0

5624.7

Best

1634

1625

1606

1612

1628

1640

Avg

1713.6

1717.5

1702.3

1712.6

1712.6

1728.2

Std

41.1

56.0

64.3

67.3

58.8

69.6

(%) HV Mean HV (%)

28.3

28.4

29.4

28.8

27.8

27.1

37.3

34.6

33.4

31.4

29.5

28.2

IC Seattle SC HV IC Denver SC

7.5.3.3 Mutation Intensity The tested mutation intensity values are 0.1, 0.25, 0.5, 0.75 and 0.9. Results are shown in Table 7.3. Higher F is more favorable. Table 7.3 Results for Experiments against Mutation Intensity (F) F

IC

Seattle SC

HV

IC

Denver SC

HV

0.1

0.25

0.5

0.75

0.9

Best

34704.0

34621.4

34882.7

34654.1

34311.6

Avg

37737.7

38029.6

38434.7

37772.9

37602.5

Std

2384.5

2603.3

2755.5

2363.1

2452.8

Best

33

28

27

31

29

Avg

72.6

68.2

65.1

73.4

71.7

Std

24.2

25.3

23.7

25.9

26.2

(%)

46.3

47.6

47.5

46.7

47.6

Best

225994.0

226297.3

225867.5

225872.1

225865.6

Avg

232824.8

233180.0

233830.9

232124.7

231808.6

Std

3370.1

3542.1

3768.6

3627.9

3528.9

Best

1628

1609

1605

1611

1615

Avg

1715.4

1697.1

1700.4

1701.4

1699.0

Std

43.0

47.2

47.2

45.9

45.7

(%)

28.6

29.1

29.3

29.3

29.2

37.5

38.3

38.4

38.0

38.4

Mean HV (%)

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7.5.3.4 DEPT Scheme DEPT schemes differ themselves in terms of base of differential mutation, number of difference vectors, and crossover schemes. Different DEPT scheme may behave differently. The tested DEPT schemes are grouped according to the implemented crossover scheme. Results for DEPT schemes involving binomial crossover are given in Table 7.4 while results for DEPT schemes applying exponential crossover are shown in Table 7.5.

Table 7.4 Results for Experiments against DEPT Schemes Using Binomial Crossover (1: rand/1/bin, 2: rand/2/bin, 3: rand/3/bin, 4: best/1/bin, 5: best/2/bin, 6: best/3/bin, 7: best-torandom/1/bin) DEPT Scheme Seattle

IC

2

3

4

5

6

7

Best

34316.6

34366.2

34929.5

35379.8

34859.8

35135.6

34317.6

Avg

37842.2

37607.5

37933.2

38521.6

38066.1

38215.4

37375.5

Std

2115.5

2447.2

2188.0

2317.5

2360.1

2310.4

2269.5

Best

32

29

33

29

32

31

30

Avg

70.1

71.6

73.6

71.6

73.9

71.5

71.4

Std

24.1

26.1

24.4

25.0

26.0

24.8

26.3

HV

(%)

46.8

47.6

45.9

46.2

46.1

46.2

47.6

IC

Best

227092.5

226454.4

226562.1

226177.4

226618.3

226795.7

226983.0

Avg

233811.1

232203.9

232800.7

232957.6

232110.4

232356.9

233092.7

Std

3791.5

3350.2

3540.5

3826.0

3005.9

3192.4

3471.3

Best

1610

1612

1622

1624

1618

1618

1607

Avg

1704.6

1710.4

1711.7

1702.7

1705.9

1716.0

1705.6

Std

48.1

48.6

47.2

45.9

44.2

48.7

42.2

(%)

28.8

29.0

28.6

28.8

28.8

28.7

28.9

37.8

38.3

37.3

37.5

37.5

37.4

38.2

SC

Denver

1

SC

HV

Mean HV (%)

It is apparent that DEPT schemes using exponential crossover outperform those using binomial crossover. It is further noticed that DE/rand2Best/1/exp is the most profitable for both instances. In terms of differential mutation base, it can be seen from Figs. 7.11-7.14 that random base is more beneficial. These results are much clearer from results by rand/*/bin. However, as shown in Fig. 7.14, in the Seattle instance, DE/RandToBest/1/Exp clearly outperforms all other approaches.

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Table 7.5 Results for Experiments against DEPT Schemes Using Exponential Crossover (1: rand/1/exp, 2: rand/2/exp, 3: rand/3/exp, 4: best/1/exp, 5: best/2/exp, 6: best/3/exp, 7: best-to-random/1/exp) 1 Seattle

IC

SC

Denver

2

3

4

5

6

7

Best

20034.4

20560.0

19954.5

20400.1

19287.6

20220.7

18123.0

Avg

26083.2

26006.0

26062.3

26383.4

26047.1

26121.7

23897.4

Std

2179.6

2136.4

2350.2

2298.0

2511.8

2371.0

2144.3

Best

25

25

22

27

21

23

27

Avg

71.7

71.2

70.3

73.2

74.0

72.0

70.9

Std

23.6

22.6

23.6

23.3

25.1

24.8

23.7

HV

(%)

67.1

66.9

68.2

65.9

68.7

67.5

69.4

IC

Best

200990.1

202740.2

203580.0

201437.5

201981.7

202988.4

203283.1

Avg

226315.2

228028.3

226350.3

226367.9

226860.6

225976.8

227214.2

Std

7531.0

10216.0

6738.6

6571.7

6359.7

6592.8

6411.9

Best

1427

1455

1438

1433

1448

1443

1444

Avg

1661.2

1586.2

1661.2

1659.3

1654.0

1661.8

1661.4

Std

65.1

73.0

61.3

60.0

62.4

54.9

51.6

(%)

43.9

43.4

42.8

43.6

43.0

42.9

42.7

55.5

55.1

55.5

54.8

55.9

55.2

56.0

SC

HV

Mean HV (%)

DEPT Schemes Using Random Base

DEPT Schemes Using Best Base

Fig. 7.11 Pareto Fronts Obtained Using */*/bin over the Denver Instance

The results obtained when comparing the random schemes against the schemes using the best-so-far approach, do not express a so clear difference between the two approaches. Indeed, in Figs. 7.12 and 7.14 that are showing the Pareto Fronts for the schemes using the exponential crossover it is possible to identify that the differences between the two approaches are smaller, especially for the Denver instance Fig. 7.12.

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DEPT Schemes Using Random Base

DEPT Schemes Using Best Base

Fig. 7.12 Pareto Fronts Obtained Using DEPT Schemes */*/exp over the Denver Instance

DEPT Schemes Using Random Base

DEPT Schemes Using Best Base

Fig. 7.13 Pareto Fronts Obtained Using DEPT Schemes */*/bin over the Seattle Instance

DEPT Schemes Using Random Base

DEPT Schemes Using Best Base

Fig. 7.14 Pareto Fronts Obtained Using DEPT Schemes */*/exp over the Seattle Instance

7.5.3.5 Findings According to the above tuning, DE/rand-to-best/1/exp is chosen. The corresponding optimal intrinsic control parameter values are: NP=75; CR=0.01; F=0.9.

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7.5.4 Empirical Results Now, DEPT is compared with MO-VNS [21], MO-SVNS [21], greedy MO-VNS (GMO-VNS), and greedy MO-SVNS (GMO-SVNS) where α=0.4 for MO-SVNS and greedy MO-SVNS. Hypervolumes are tabulated in Table 7.6 while the final Pareto fronts by different algorithms are plotted in Figs. 7.15 and 7.16. DEPT clearly outperforms MO-VNS and MO-SVNS. It is also seen that the greedy mutation in GMO-VNS and GMO-SVNS significantly improves MO-VNS and MO-SVNS. Table 7.6 Hypervolumes Obtained by Different Algorithms

DEPT

MO-VNS

MO-SVNS

GMO-VNS

GMO-SVNS

Denver

72.3

22.4

22.4

75.0

81.1

Seattle

43.4

17.9

18.8

57.5

56.9

Fig. 7.15 Pareto Front for the Denver Instance

Now, let’s have a look at the coverage relation which is shown in Table 7.7. It can be seen that GMO-VNS is 100% better than MO-VNS and GMO-SVNS is 100% better than MO-SVNS. Furthermore, it appears that GMO-SVNS is better than GMO-VNS, which fully agrees with our previous results [21]. Moreover, we see that both GMO-SVNS and GMO-VNS are in general better than DEPT.

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Fig. 7.16 Pareto Front for the Seattle Instance

Table 7.7 Percentage of Non-dominated Solutions Obtained by Algorithms B Covered by Non-dominated Solutions Obtained by algorithm A

Algorithm A DEPT

MO-VNS

MO-SVNS

GMO-VNS

GMO-SVNS

Algorithm B MO-VNS MO-SVNS GMO-VNS GMO-SVNS DEPT MO-SVNS GMO-VNS GMO-SVNS DEPT MO-VNS GMO-VNS GMO-SVNS DEPT MO-VNS MO-SVNS GMO-SVNS DEPT MO-VNS MO-SVNS GMO-VNS

Denver 100 100 0 0 0 75.0 0 0 0 57.1 0 0 100 100 100 4.2 100 100 100 80.0

Seattle 100 100 0 12.5 0 63.6 0 0 0 55.6 0 0 24.1 100 100 12.5 10.1 100 100 20.0

mean 100 100 0 6.3 0 69.3 0 0 0 56.3 0 0 62.0 100 100 8.3 55.1 100 100 50.0

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7.6 Conclusions In this paper we have tackled real-world frequency assignment problem for GSM mobile networks through a multi-objective optimization approach. Differential evolution with Pareto tournaments have been proposed to solve the multiobjective optimization problem. Comparative study against MO-VNS, MO-SVNS, GMO-VNS, and GMO-SVNS over two real-world instances has been conducted. Final results show that fine-tuned DEPT outperforms both MO-VNS and MOSVNS while performs worse than both GMO-SVNS and GMO-VNS, among which GMO-SVNS performs best.

Acknowledgement This work was partially funded by the Spanish Ministry of Science and Innovation and FEDER under the contract TIN2008-06491-C04-04 (the M* project). Thanks also to the Polytechnic Institute of Leiria, for the economic support offered to Marisa Maximiano to make this research.

References [1] Eisenblätter, A.: Frequency assignment in GSM networks: Models, heuristics and lower bounds, Ph.D. Thesis, Technische Universität Berlin (June 2001) [2] GSM Association, GSM World, http://www.gsmworld.com/newsroom/market-data/ market_data_summary.htm (last accessed January 2010) [3] Shirazi, S.A.G., Amindavar, H.: Fixed Channel assignment using new dynamic programming approach in cellular radio networks. Computers Electrical Engineering 31(4-5), 303–333 (2005) [4] Storn, R., Price, K.V.: Differential Evolution - A Simple and Efficient Adaptive Scheme for Global Optimization over Continuous Spaces, Technical Report TR-95012, International Computer Science Insitute (March 1995) [5] Storn, R., Price, K.V.: Differential evolution-a simple and efficient heuristic for global optimization over continuous spaces. J. Global Optimization 11(4), 341–359 (1997) [6] Price, K.V., Storn, R.M., Lampinen, J.A.: Differential Evolution: a Practical Approach to Global Optimization. Springer, Berlin (2005) [7] Qing, A.: Differential Evolution: Fundamentals and Applications in Electrical Engineering. John Wiley, New York (2009) [8] Storn, R.: Differential evolution (DE) for continuous function optimization (an algorithm by Kenneth Price and Rainer Storn), http://www.icsi.berkeley.edu/~storn/code.html (last accessed October 23, 2009) [9] Fonseca, C.M., Paquete, L., López-Ibáñez, M.: An improved dimension-sweep algorithm for the hypervolume indicator. In: IEEE Congress Evolutionary Computation, Vancouver, BC, Canada, July 16-21, pp. 1157–1163 (2006)

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[10] Zitzler, E., Thiele, L.: Multiobjective optimization using evolutionary algorithms - A comparative case study. In: 5th Int. Conf. Parallel Problem Solving Nature, Amsterdam, The Netherlands, September 27-30, pp. 292–304 (1998) [11] Kuurne, A.M.J.: On GSM mobile measurement based interference matrix generation. In: IEEE 55th Vehicular Technology Conf., Birmingham, AL, May 6-9, vol. 4, pp. 1965–1969 (2002) [12] Luna, F., Estébanez, C., León, C., Chaves-González, J.M., Alba, E., Aler, R., Segura, C., Vega-Rodríguez, M.A., Nebro, A.J., Valls, J.M., Miranda, G., Gómez-Pulido, G.A.: Metaheuristics for solving a real-world frequency assignment problem in GSM networks. In: Genetic evolutionary computation Conf., Atlanta, GA, July 12-16, pp. 1579–1586 (2008) [13] Mishra, A.R.: Fundamentals of Cellular Network Planning and Optimisation: 2G/2.5G/3G...Evolution to 4G. John Wiley, New York (2004) [14] Luna, F., Blum, C., Alba, E., Nebro, A.J.: ACO vs EAs for solving a real-world frequency assignment problem in GSM networks. In: Genetic Evolutionary Computation Conf., London, UK, July 7-11, pp. 94–101 (2007) [15] Maximiano, M., Vega-Rodríguez, M.A., Gómez-Pulido, J.A., Sánchez-Pérez, J.M.: Parameter analysis for differential evolution with Pareto tournaments in a multiobjective FAP. In: Corchado, E., Yin, H. (eds.) IDEAL 2009. LNCS, vol. 5788, pp. 799–806. Springer, Heidelberg (2009) [16] Weicker, N., Szabo, G., Weicker, K., Widmayer, P.: Evolutionary multiobjective optimization for base station transmitter placement with frequency assignment. IEEE Trans. Evolutionary Computation 7(2), 189–203 (2003) [17] Hansen, P., Mladenovic, N.: Variable neighborhood search: Principles and applications. European J. Operational Research 130(3), 449–467 (2001) [18] Hansen, P., Mladenovic, N., Pérez, J.A.M.: Variable neighbourhood search. Computers Operations Research 24(11), 1097–1100 (1997) [19] Maximiano, M., Vega-Rodríguez, M.A., Gómez-Pulido, J.A., Sánchez-Pérez, J.M.: A hybrid differential evolution algorithm to solve a real-world frequency assignment problem. In: Int. Multiconference Computer Science Information Technology, Polskie Towarzystwo Informatyczne, Wisla, Poland, October 20-22, pp. 201–205 (2008) [20] Maximiano, M., Vega-Rodríguez, M.A., Gómez-Pulido, J.A., Sánchez-Pérez, J.M.: Analysis of parameter settings for differential evolution algorithm to solve a realworld frequency assignment problem in GSM networks. In: 2nd Int. Conf. Advanced Engineering Computing Applications Sciences, Valencia, Spain, September 29October 4, pp. 77–82 (2008) [21] Maximiano, M., Vega-Rodríguez, M.A., Gómez-Pulido, J.A., Sánchez-Pérez, J.M.: Multiobjective frequency assignment problem using the MO-VNS and MO-SVNS algorithms. In: World Congress Nature Biologically Inspired Computing, Coimbatore, India, December 9-11, pp. 221–226 (2009)

Chapter 8

RNN Based MIMO Channel Prediction Chris Potter

*

Abstract. In this work, differential evolution (DE) is combined with particle swarm optimization (PSO) and another evolutionary algorithm (EA) to create a novel hybrid PSO-EA-DEPSO algorithm. The alteration between PSO, PSO-EA, and DEPSO provides additional diversity to counteract premature convergence. This hybrid algorithm is then shown to outperform PSO, PSO-EA, and DEPSO when applied to wireless MIMO channel prediction.

8.1 Introduction Multiple-input multiple-output (MIMO) wireless communication systems have been shown to provide significant gains in both spectral efficiency and reliability compared to traditional single-input single-output systems [1]-[2]. These results, however, are based on the assumption that the transmitter and/or receiver have perfect knowledge of the channel state information (CSI). One possible alternative is to estimate the channel at the receiver and send the channel state information back to the transmitter. The performance using this approach suffers when the transmitted CSI has become outdated due to channel fluctuation. It may be possible to reduce this effect by sending back a prediction of the CSI. Linear predictors [3] have been used for narrow-band prediction in [4]-[6]. One drawback is their difficulty in estimating the correlation coefficients of the channel in the presence of received data that has undergone non-linear distortions [7]. One solution to this problem are neural networks, which if trained properly, are well suited for non-linearities since they are equipped with arbitrary activation functions. A multi-layer perceptron (MLP) neural network predictor was utilized in [8], while a hybrid network was employed in [9]-[10]. In this work, a recurrent neural network (RNN) is chosen for narrow-band channel prediction. Their advantage over feed-forward neural network predictors Chris Potter Dynetics Inc e-mail: [email protected] *

A. Qing and C.K. Lee: Differential Evolution in Electromagnetics, ALO 4, pp. 177–206. © Springer-Verlag Berlin Heidelberg 2010 springerlink.com

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is the potential to learn temporal statistics from previous neural network outputs. In [7], a RNN was trained online by the extended Kalman filter and real-time recurrent learning algorithms for narrow-band channel prediction. A disadvantage of these online training algorithms is the necessity of an accurate measurement of the channel prediction error (which requires full CSI knowledge). In contrast, a RNN trained off-line by PSO-EA was utilized in [11] for time series prediction. All of the prediction techniques mentioned up to this point have been for SISO systems. In this work, the main focus will be on wireless systems with multiple antennas at both the transmitter and receiver. Although particle swarm optimization (PSO) [12]-[13] utilizes the personal and global bests of the swarm when updating the particles, stagnation around local extremum can occur. One solution to this problem is to combine PSO with other evolutionary algorithms (EA) such as differential evolution (DE) [14]-[18]. This in theory will provide additional diversity to the swarm through mutation, crossover, and selection. Examples of these hybrid algorithms include PSO-EA [11] and DEPSO [19]. In this work, a new hybrid PSO-EA-DEPSO training algorithm is proposed. This algorithm is shown to outperform PSO, PSO-EA, and DEPSO as well as the Levinson-Durbin algorithm with respect to mean squared error (MSE) when applied to the problem of multiple-input multiple-output (MIMO) wireless channel prediction. The robustness of this new algorithm is illustrated by varying the temporal and spatial correlation of the MIMO channel after the weights have been trained. To analysis how the MIMO RNN predictor trained by the new PSO-EADEPSO algorithm impacts the performance on the wireless link, new expressions for the received SNR, array gain, and probability of error for a MIMO beamforming system when the transmitter and receiver only have access to a channel prediction are derived. This work abstains from the common assumptions that the channel prediction error is Gaussian and/or independent of the true CSI. An analysis of these expressions reveal that the array gain decreases with SNR and is larger for spatially correlated channels. Also, the diversity gain is shown to match the perfect CSI case up until the channel prediction error saturates the received signal. The reduction of the average probability of error before saturation is strictly due to coding gain.

8.2 Received Signal Model 8.2.1 Received Signal Model A MIMO wireless flat fading baseband communication system with Nt transmitting antennas and Nr receiving antennas is modeled at discrete time k by

y (k ) = H (k ) ⋅ x(k ) + n(k )

(1)

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179

where y(k) is an Nr-dimensional vector containing received signals, H(k)={hmn(k)} is an Nr× Nt channel matrix describing the complex channel gain between the mth receiving antenna and nth transmitting antenna, x(k) is an Nt-dimensional vector containing transmitted symbol xi(k), 1≤i≤ Nt, of constellation C, and n(k) is an Nrdimensional vector of complex white noise with mean zero and variance variance No, i.e.,

ni ~ CN(0, N o ) 1 ≤ i ≤ N r iid

(2)

Denoting Pt and T respectively as the total transmitted power and symbol period, the transmitted symbols must satisfy

E x(k ) 2 ≤ PtT 2

where E(·) and

(3)

2

⋅ 2 denote the expectation operator and two norm, respectively.

It is assumed the MIMO channel can induce non-linear distortions such as low noise amplification and non-linear scattering on the transmitted signal. These distortions are lumped into the mapping f: C → C where C is the set of complex numbers. For virtually any wireless communications application, the received signal must remain finite and thus it is assumed that f is bounded on C.

8.2.2 Optimization Problem The purpose here is to predict the channel state information (CSI) that minimizes the mean squared error between the predicted and measured channel. This can be cast into an minimization problem by defining the following objective function

C (k ) = E hˆmn (k ) − hmn (k ) where

[

2

1 ≤ m ≤ N r ,1 ≤ n ≤ N t

(4)

]

hˆmn (k ) = F hˆmn (k ),K, hˆmn (k − N p ) is the prediction, hˆmn (k − 1) , …,

hˆmn (k − N p ) are the Np most recent estimates, and F maps the estimates to the

prediction. Since it was only required that f be bounded, the possibility that the derivative of f may not exist requires the use of training algorithms that do not rely on gradient information.

8.3 Hybrid PSO-ES-DEPSO Training Algorithm In this work, a new algorithm is proposed that is a hybrid version of PSO, EA, and DEPSO. The block diagram is displayed in Fig. 8.1.

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Fig. 8.1 Hybrid PSO-EA-DEPSO Training Algorithm

The motivation behind the algorithm is to alternate between PSO, EA, and DEPSO to continually provide diversity for the particles/parents. PSO is implemented for one fourth of the total iterations to converge quickly on the potential solution. PSO-EA and DEPSO then alternate for the remaining iterations to prevent premature convergence, since DEPSO can be more favorable in dense solution spaces where a small change in the personal best could lead the swarm to a lower fitness solution, while PSO-EA may be desirable in sparse solution spaces when the swarm needs a “nudge” that is independent of any personal/global bests to migrate from a local minimum in search of solutions with better fitness.

8.4 MIMO Channel/Beam-Forming Models 8.4.1 Channel Model The MIMO sub-channels are represented by [20]

[

]

I Q (k ) + jg mn (k ) 1 ≤ m ≤ N r ,1 ≤ n ≤ N t g mn (k ) = f g mn

(5)

where f, as mentioned previously, is arbitrary but bounded on C, M

I (k ) = 2 M ∑ cos[φn + 2πf d kTs cosα n ] g mn n =1

(6)

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181

is the in-phase component, M

Q (k ) = 2 M ∑ cos[ψ n + 2πf d kTs sin α n ] g mn

(7)

n =1

is the quadrature component, Ts is the sampling period, fd is the maximum doppler frequency, and

αn =

2πn −π +θ 4M

(8)

øn, ψn and θ are U[-π, π], where U[a, b] is a uniform random variable between a and b. The system is assumed to be operating in an urban environment in which the receiver is spatially uncorrelated due to scattering from adjacent objects. The overall channel is thus modeled by

H (k ) = G (k ) ⋅ ΦTX (k )

(9)

where

ΦTX =

∑ E[h( ) (k )h( ) (k )] Nr

1 Nr

r =1

H r;

(10)

r;

h(r;) denotes the rth row of H(k), the superscript H stands for Hermitian operator. These correlation matrices are dependent on many parameters, including the angle of arrival, transmitting and receiving antenna distances, and angular spread [21]. Since the focus of this work will not be emphasized on any particular antenna geometry, spatial generality will be maintained and a Toeplitz structure will be considered

ΦTX

⎡ 1 ⎢ ⎢ γt = ⎢ γ t4 ⎢ ⎢ M ⎢γ ( N t −1)2 ⎣ t

γ t4 L γ t( N −1) M 1 γt O γt 1 O γ t4 O O O γt L γ t4 γ t 1 γt

t

2

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

(11)

where γt is the transmit spatial correlation factor. It should be noted that the simplicity of this model does not prevent accurate approximations for a variety of antenna configurations [22]. Stacking the columns of H(k) into the Nt Nr-dimensional column vector vec[H(k)], it follows that

vec[H (k )] ~ CN (0, Ψ )

(12)

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where

Ψ = ΦTX ⊗ Φ RX

(13)

⊗ denoting the kronecker product [23]. This form will prove useful when deriving the array and diversity gains in Section 8.

8.4.2 Channel Estimation Model For a meaningful prediction, it is necessary for the RNN to learn the statistics of the fading process. This can be accomplished by supplying the RNN with previously obtained channel estimates. Following the same procedure as those in [24]-[26], the channel is written in terms of its minimum mean squared error

H (k ) by

(MMSE) estimate

H (k ) = H (k ) + W (k ) where

(14)

[

]

H (k ) and W(k) are uncorrelated with hmn (k ) ~ CN 0,σ h2 (k ) and

[

]

wmn (k ) ~ CN 0,σ w2 (k ) . The channel estimation error between the mth receiver and nth transmitter is thus

σ w2 (k ) = σ h2 (k ) − σ h2 (k )

(15)

8.4.3 MIMO Beam-Forming When channel state information is available at both the transmitter and receiver, spatial diversity can be achieved. Suppose that the transmitter sends the symbol x(k) across all antennas. To maintain equal transmit power with respect to the SISO case, it is required that

E x(k ) = PtT = Es 2

where Es is the average symbol energy [27]. x (k ) = Next, the symbol is pre-processed by ~ H is then post-processed by u (k) to yield

(16)

v (k )x(k ) . The received symbol

~ y (k ) = u H (k ) ⋅ y (k ) = u H (k ) ⋅ H(k ) ⋅ ~ x (k ) + u H (k ) ⋅ n(k )

(17)

The average received energy

E~ y (k ) = Es u H (k ) ⋅ H (k ) ⋅ v (k ) + σ n2 2

2

(18)

is now maximized with respect to u(k) and v(k), subject to the constraints

u(k ) 2 = v(k ) 2 = 1 . 2

2

RNN Based MIMO Channel Prediction

183

Defining the Lagrangian as

[

]

[

]

Λ(u, v ) = u H (k ) ⋅ H(k ) ⋅ v(k ) − λ1 (k ) u(k ) 2 − 1 − λ2 (k ) v(k ) 2 − 1 (19) 2

2

2

where λ1(k), λ2(k) ∈ C, and taking the derivative yields

∇u * ( k )Λ(u, v ) = H (k ) ⋅ v (k ) ⋅ v H (k ) ⋅ H H (k ) ⋅ u(k ) − λ1 (k )u(k )

(20)

∇ λ* (k )Λ(u, v ) = u H (k ) ⋅ u(k ) − 1

(21)

∇ v * (k )Λ(u, v ) = H H (k ) ⋅ u(k ) ⋅ u H (k ) ⋅ H (k ) ⋅ v(k ) − λ2 (k )v(k )

(22)

∇ λ* (k )Λ (u, v ) = v H (k ) ⋅ v(k ) − 1

(23)

1

2

Setting these to zero, it follows immediately that

H(k ) ⋅ v(k ) =

λ1 (k )u(k )

(24)

v (k ) ⋅ H H (k ) ⋅ u(k ) H

u H (k ) ⋅ u(k ) = 1 H H (k ) ⋅ u(k ) =

(25)

λ2 (k )v(k )

(26)

u (k ) ⋅ H (k ) ⋅ v (k ) H

v H (k ) ⋅ v (k ) = 1

(27) H

H

Multiplying the left of Eqs. (24) and (26) by u (k) and v (k), respectively, yields

λ1 (k ) = v H (k ) ⋅ H H (k ) ⋅ u(k ) ⋅ u H (k ) ⋅ H(k ) ⋅ v(k ) = λ2 (k )

(28)

Inserting λ1(k) and λ2(k) into Eqs. (24) and (26) yields

H(k ) ⋅ v(k ) = u H (k ) ⋅ H(k ) ⋅ v(k )u(k ) = σ (k )u(k )

(29)

H H (k ) ⋅ u(k ) = v H (k ) ⋅ H H (k ) ⋅ u(k )v(k ) = σ (k )v(k )

where

σ (k ) = λ1 (k ) = λ2 (k )

This shows that Eq. (10) is maximized when u(k) and v(k) are chosen to be the left and right singular vectors, respectively, that correspond to the maximum singular value. The received symbol when u(k)= u1(k) and v(k)= v1(k), the left and right singular vectors corresponding to the maximum singular value σ H ( k ) , 1

respectively, is

u1H (k ) ⋅ y (k ) = σ H 1 (k )x(k ) + u1H (k ) ⋅ n(k )

(30)

184

C. Potter

This signal processing technique is known throughout the literature as MIMO beam-forming or dominant eigenmode transmission. Letting one can easily show that

n~(k ) = u1H (k ) ⋅ n(k ) ,

E n~(k ) = N o 2

(31)

and remains white. The average received energy is

E~ y (k ) = Esσ H2 1 (k ) + σ n2 2

(32)

When the transmitter and receiver only have a prediction of the channel matrix

ˆ (k ) = U ˆ (k ) ⋅ D ˆ (k ) ⋅ V ˆ H (k ) , the received symbol is H

ˆ (k ) ⋅ D ˆ (k ) ⋅ V ˆ H (k ) ⋅ vˆ (k )x(k ) + u H (k ) ⋅ n (k ) (33) uˆ 1H (k ) ⋅ y (k ) = u1H (k ) ⋅ U 1 1 To supply the transmitter with the dominant right singular vector for preprocessing, a feedback link must be established. Let td be defined as the total delay arising from processing and transmission latencies. A constraint on this delay to keep the CSI from becoming stale is [28]

td = 0.423 f d

(34)

where fd is the maximum doppler frequency. This seems discouraging, since for a maximum doppler frequency of 100 Hz, the total delay must be less than 4.23 ms. However, with the aid of accurate doppler estimation [29]-[30], this delay constraint can be relaxed. For example, assuming a worst case approximation error of 0.5 Hz, the maximum delay increases to 846 ms, a tolerable value for most wireless latencies [31].

8.5 Recurrent Neural Network for Channel Prediction Recurrent neural networks are fundamentally different from their feedforward counterparts in that they possess an internal state of what has previously been been processed by the network. This state gives the network the ability to better describe data possessing temporal correlations than feed-forward networks. An analogous statement is that a feed-forward neural network, by the universal approximation theorem, can represent any function defined on a fixed input space; however, a recurrent neural network can utilize its internal state to represent data that is temporally correlated over potentially unbounded input spaces. The recurrent neural network used for prediction is shown in Fig. 8.2.

RNN Based MIMO Channel Prediction

185

z−1

• •

z−1 d1(k−1)

d1(k)

+ •

dϑ(k−1)

•

• •

+

s(k )

dϑ(k)

•

s(k − N p )

•

A Fig. 8.2 Recurrent Neural Network Channel Predictor

The output of the activation functions are

⎤ ⎡ N p +ϑ d j (k ) = φ j ⎢ ∑ a ji ri (k )⎥ 1 ≤ j ≤ ϑ ⎦⎥ ⎣⎢ i =1 where r(k) =[d1(k-1), …,

(35)

dϑ (k − 1) , s(k), …, s(k-Np)]T is the RNN input and

øj(x)=tanh(x), 1≤j≤ ϑ , are the non-linear activation functions. The RNN predictor is implemented in two stages. During the first stage, d1(k-1), … dϑ (k − 1) =0. The outputs are then fed back to form r(k). After the second stage, the RNN prediction

hˆmn (k + 1) is set equal to d1(k). This process is

repeated for all NrNt channel coefficients. Unless otherwise stated, Np=5 and ϑ =2 is chosen. This yields Nw= ϑ (Np+ ϑ )=14 neural network weights which are used to construct the 2×7 matrix A={aji}.

8.6 Training Procedure The RNN in Fig. 8.2 is trained off-line with data generated from Eq. (10). To be specific, let Ntrain denote the number of training samples. Then for each 0≤k≤Ntrain, the training data can be generated using the aforementioned channel models in following sequence of steps. • •

Choose the number of transmit antennas, Nt, and receive antennas, Nr. Choose the spatial correlation factor γt.

186

• • • •

C. Potter

Use γt and Nt to generate ФTX. Choose fd and Ts to generate at discrete time k, the sub-channels gmnI(k) and gmnQ(k) for 0≤m≤Nr, 0≤n≤Nt , in Eqs. (6) and (7), respectively. Perform the non-linear mapping induced by f to obtain gmn(k) in Eq. (5) and form the matrix G(k)={gmn(k)}. Calculate H(k) using Eq. (9).

For each sub-channel, a RNN whose dynamics are described in the previous section, is trained with s(k-i)=h(k-i), 0≤i≤Np-1 on iteration η with fitness function

C (η ) =

∑ [h (k ) − hˆ (k ,η )]

N train 1 N train

A mn

A mn

2

(36)

k =1

where A={I, Q}. For each iteration, P sets of RNN weights are updated by a training algorithm chosen by the user and are given fitness values determined by C(η). When a specified number of iterations have occurred or the fitness function has reached a desired value, the RNN weights are frozen and brought online. At this point, the RNN does not know the channel coefficients and must instead use the estimates for channel prediction. This is done by setting s(k-i)= h (k-i), 0≤i≤Np-1. This method of training the RNN offline is less restrictive than online training [7], [32]-[33], where it is assumed that the instantaneous value of the error (and hence full channel knowledge) is known at the receiver. In addition, a more robust training method than that in [34] is proposed in this study, where the RNN weights had to be retrained whenever the channel changed. Unless otherwise specified, the parameters used for training are tabulated in Table 8.1. The value of Nw was given in Section 4 and is used to calculate τ. The remaining parameters were found on a trial and error basis and generally speaking are dependent on the optimization problem at hand. Table 8.1 Parameter Values for Training Algorithms

parameter

value

description

Nw

14

Number of RNN weights

Vmax

2

Maximum PSO velocity

Xmax

4

Maximum PSO position

w

0.8

PSO Inertia Weight

c1

1

PSO Cognitive Weight

c2

1.5

PSO Social Weight

P

40

Number of PSO Particles

pc

0.5

Crossover Probability for DEPSO

L

7

DEPSO parameter

τ

0.3265

EA Parameter

RNN Based MIMO Channel Prediction

187

8.7 Numerical Results At this point, it is illustrative to present a few examples. First, the new PSO-EADEPSO algorithm is compared with PSO, PSO-EA, and DEPSO algorithms. In the second example, the robustness of the training algorithm is investigated by varying fdTs, the normalized doppler frequency. In the third and final example, a RNN is compared to a linear predictor using the PSO-EA-DEPSO algorithm.

8.7.1 Algorithm Comparison To provide a comparison of the training algorithms described in the previous section, a 2×2 (Nr×Nt) spatially uncorrelated MIMO channel with f(x)=x, Ntrain=100, γt=0, σw2(k)=0.001, and normalized doppler frequency fdTs=0.05 is predicted using PSO, DEPSO, PSO-EA, and the new hybrid PSO-EA-DEPSO algorithm. For each iteration and training algorithm, the fitness values corresponding to the global best of h11(k) for 50 random trials are collected. The fitness values are plotted for the I and Q components in Figs. 8.3 and 8.4 respectively.

Fig. 8.3 Mean squared error of in-phase component when fdTs=0.05

188

C. Potter

Fig. 8.4 Mean squared error of quadrature component when fdTs=0.05

For the first 50 iterations, the proposed hybrid algorithm is operating in PSO mode and thus the performance resembles that of PSO. For the remaining iterations, the hybrid algorithm uses its diversity obtained by alternating between DEPSO and PSO-EA to outperform the competition.

8.7.2 Robustness of PSO-ES-DEPSO Algorithm Although accurate training of the weights is important, the robustness to different channel conditions is also critical for the RNN predictor. To investigate this, the weights are trained with the new hybrid PSO-EA-DEPSO training algorithm when f(x)=x, Ntrain=100, γt=0, σw2(k)=0.01, and normalized doppler frequency fdTs=0.1 are brought online to predict a channel with the same parameters except a time varying fdTs. The in-phase (I) and quadrature (Q) channel coefficients along with their predictions are illustrated in Figs. 8.5 and 8.6 respectively.

RNN Based MIMO Channel Prediction

189

Fig. 8.5 In-phase channel coefficients for varying fdTs

Fig. 8.6 Quadrature channel coefficients for varying fdTs

Next, the effect of spatial correlation is investigated by training the weights with f(x)=x, Ntrain=100, γt=0, σw2(k)=0.01, normalized doppler frequency fdTs=0.1 and bringing them online for channel prediction with the same parameters except γt=0.7. The I and Q components are plotted in Figs. 8.7 and 8.8 respectively. The

190

C. Potter

accuracy of these predictions verify the robustness of the RNN equipped with the new PSO-EA-DEPSO training algorithm.

Fig. 8.7 In-phase channel coefficients for γt=0.7

Fig. 8.8 Quadrature channel coefficients for γt=0.7

RNN Based MIMO Channel Prediction

191

8.7.3 Linear and Nonlinear Predictors with PSO-EA-DEPSO Algorithm The strength of RNNs and neural networks in general is their ability to describe non-linearities. These can be present in the transmitter/receiver hardware due to devices such as low noise amplifiers (LNAs) and automatic gain controllers (AGCs), or in the wireless environment when non-linear scatterers distort the electromagnetic signal. A byproduct of these non-linearities is that the signal can become non-stationary, which greatly complicates the prediction problem. To exemplify this, a RNN using the new hybrid algorithm is compared to a fifth order feed-forward linear predictor using the Levinson-Durbin Algorithm [35] for f(x)=exp(-x), Ntrain=100, γt=0, fdTs=0.5, and σw2(k)=0.01. Observing Figs. 8.9 and 8.10, the RNN outperforms the linear predictor. This example suggests the RNN trained with the PSO-EA-DEPSO algorithm is capable of predicting certain non-linear, non-stationary channels better than the Levinson-Durbin linear predictor.

Fig. 8.9 In-phase channel coefficients for non-linear channel

192

C. Potter

Fig. 8.10 Quadrature channel coefficients for non-linear channel

8.7.4 Non-convexity of the Solution Space Now that the performance and robustness of the new hybrid algorithm has been established, the (non-)convexity of the optimization problem is investigated. For ten random trials, the RNN weights obtained during training and the corresponding fitness values are displayed in Table 8.2 when f(x)=x, Ntrain=100, γt=0, σw2(k)=0.01, and fdTs=0.1. Table 8.2 RNN weights trained by the new PSO-EA-DEPSO algorithm and their corresponding fitness values for ten random trials fitness

weights

1

2

3

4

5

6

7

8

9

10

0.0199

0.0249

0.0062

0.0744

0.0246

0.0529

0.0051

0.0103

0.0433

0.0057

-0.1852

-0.2938

0.0988

-2.3740

-1.3550

-0.7193

-0.0915

-2.9857

-3.0606

-0.9186

-0.0481

-0.0695

0.5724

-0.5708

-0.8356

-0.3019

0.2574

2.5382

-0.2169

1.6185

-0.2177

-0.6067

-0.2857

-0.9908

1.9431

1.3150

-0.9999

1.9331

-0.8632

0.9503

1.5102

2.4664

1.6808

-0.2108

-3.0000

-2.2981

3.0000

-3.0000

2.1963

-0.2512

-2.1005

-3.0403

-2.9904

2.8505

-0.7334

0.6472

-3.0000

-1.0616

-2.9998

-3.0000

-0.3749

0.1797

0.7507

-2.2392

2.2325

-0.7102

-0.5636

3.0000

3.0002

2.5535

2.5595

2.3267

2.3462

-0.3983

1.0659

2.3951

3.0000

0.5107

-1.8604

0.9262

5.8845

3.0412

-3.0000

-1.4910

2.2044

3.0129

-1.7209

-3.0000

-2.2399

-1.5449

2.1256

-2.8989

-3.0000

2.6533

3.0000

3.0463

-3.0000

3.0000

3.0015

-3.0000

2.4032

-2.2344

-1.6517

3.0411

-0.4324

-0.7710

-0.6671

1.1447

1.2529

-1.0727

-6.0000

2.7577

3.0000

-4.0465

-0.5025

-0.3501

1.0829

-1.6065

-0.2251

3.0000

0.4291

-1.7064

0.6771

0.9490

2.1018

1.8101

2.0453

-0.7452

-0.2497

-2.6953

3.0298

3.0116

-3.0000

1.7369

-0.0355

0.1754

-3.0000

0.9632

-3.0001

0.0360

3.0480

-3.0000

-0.1994

-1.1739

-2.9869

-3.0000

-0.7356

1.7151

2.1385

1.2633

RNN Based MIMO Channel Prediction

193

A quick inspection of the RNN weights indicate for each run, the swarm has found a distinct local minimum, which suggests the optimization problem is nonconvex. The fact that each run yields a different local minimum also demonstrates the difficulty of this optimization problem. The proposed hybrid algorithm in this scenario seeks to find the “best” local minimum by exploring its diversity through alternation of the PSO-EA and DEPSO algorithms.

8.8 Performance Measures of RNN Predictors Now that good training performance has been established, the impact of prediction error at the receiver is investigated. Up to this point it has been a common assumption in the literature to assume that the prediction error is Gaussian and/or independent of the true CSI. A contribution of this work is to derive new performance measures that do not rely on these assumptions. The approach taken will start with a new approximation for the array gain followed by a new upper bound on the probability of error. This bound will show the effect of prediction error on the diversity and coding gains before saturation. These two expressions also relate known parameters (e.g. number of antennas, spatial correlation, SNR, etc.) with performance and hence aid in the analysis. This is followed by new tight approximations for the array gain and probability of error that are based on parameter estimation. All of these measures are dependent on the received SNR, which will be the starting point of the derivation. With the aid of Eqs. (16) and (30), it follows that the instantaneous received SNR for the MIMO RNN beam-former when the receiver has perfect CSI is

ρ bf (k ) = ρσ H2 (k )

(37)

1

where

ρ=Es/No

is

the

SNR

of

a

SISO

AWGN

channel.

Writing

vˆ 1 (k ) = v1 (k ) + Δv1 (k ) and uˆ 1 (k ) = u1 (k ) + Δu1 (k ) the received SNR for

the MIMO RNN predictor is

ρbf (k ) = σ H2 (k ) [β (k ) + ρ −1 ]

(38)

1

where

β (k ) = σ H (k )v1H (k ) ⋅ Δv1 (k ) + σ H (k )Δu1H (k ) ⋅ u1 (k ) + Δu1H (k ) ⋅ H (k ) ⋅ Δv1 (k ) (39) 2

1

1

represents the effective noise due to channel prediction error.

Remark 8.1. When perfect CSI is available at the transmitter and receiver, β=0 and Eq. (38) agrees with Eq. (5.48) in [36]. The correlation between

σ H2

1

(k ) and β(k) requires their joint probability density

function (pdf) to calculate exact expressions for the array gain and probability of error. The approach taken in this work will be the derivation of accurate approximations that conserve this dependence. Before proceeding to these results, however, it is first noted in Eq. (39) that for small prediction errors

σ H2

1

(k ) and

194

C. Potter

β(k) are approximately uncorrelated. This observation leads to a new closed form approximation for the array gain in the low prediction error regime.

Theorem 8.2. The array gain for a MIMO RNN beam-former with low channel prediction error is approximated by.

[

η = E[ρbf (k )] = Nt N r α +

(

)

Nt + N r 2 3 N t N r +1

](1 − α ) (

μ β ( k ) + ρ −1

(μ β (

)

2

+ 2σ β2 ( k ) ( k )

k) +ρ

)

−1 3

(40)

where Nt

α=

∑ λΨ j − N t j =1

(41)

N t ( N r −1)

Proof. Representing Eq. (38) by its Taylor series expansion about the means

[

η = μσ

2 H1 ( k )

]

, μβ (k ) it follows that [15]

[

]

E ρbf (k ) = E{12 [γ 1 (k ) + γ 2 (k ) + γ 3 (k )]} where

[

γ 1 (k ) = σ H2

[

γ 2 (k ) = 2 σ H2

1 (k )

1 (k )

− μσ 2

H1 ( k )

]

2 ∂ 2 ρ bf (k )

- μσ 2

H1 ( k )

][β (k ) − μ

[

∂ σ H2 1 ( k )

(43)

]

2

]

∂ 2 ρ bf ( k ) β ( k ) ∂σ 2 H1 ( k ) ∂β ( k )

γ 3 (k ) = [β (k ) − μ β (k ) ]2 ∂ [β (k )]

(42)

∂ 2 ρ bf ( k )

(44) (45)

2

Computing these derivatives yields

γ 1 (k ) = 0 γ 2 (k ) = −

(46)

⎤ ⎡ 2 ⎢σ H2 1 ( k ) − μ 2 ⎥ β ( k )− μ β ( k ) σ H1 ( k ) ⎦ ⎥ ⎣⎢

[

] (47)

[μ β ( ) + ρ ]

−1 2

k

γ 3 (k ) = [β (k ) − μ β ( k ) ]2

2 σ H2 1 ( k )

[μ β (

k)

+ ρ −1

]

(48)

3

Inserting Eqs. (46)-(48) into Eq. (42) and taking the expectation yields

[

]

E ρ bf (k ) =

μ

σ2

H1 ( k )

μ β (k ) + ρ

−1

−

[

cov σ H2 1 ( k ) , β ( k )

[μ β ( ) + ρ ] k

−1 2

]

+

σ β2 ( k ) μ

σ2

H1 ( k ) −1 3

[μ β ( ) + ρ ] k

(49)

RNN Based MIMO Channel Prediction

195

σ H2

Assuming the prediction error is low enough that

1

( k ) and β(k) are

approximately uncorrelated, it immediately follows that

[

]

E ρbf (k ) = μσ 2 The pdf of

σ H2 (k ) 1

H1 ( k )

[μ β ( ) + ρ ] + 2σ β ( ) [μ β ( ) + ρ ] −1 2

k

2

k

−1 3

k

(50)

was extended from the diversity equation of correlated

branches in a SIMO system [38] to MIMO systems in [39] with mean

μσ

2 H1 ( k )

[

= Nt N r α +

(

)

Nt + Nr 2 3 N t N r +1

](1 - α )

(51)

where Nt

α=

∑ λΨ j − N t j =1

(52)

N t ( N r −1)

which was to be proven.

Remark 8.3. As the accuracy of the prediction improves, which η→ μσ 2

H1 ( k )

μ β ( k ) , σ β2 (k ) →0, from

, which is in agreement with [36].

The probability of error was approximated for SISO systems in [27] by

⎡ Pe (k ) = LQ ⎢ ⎣

ρ h ( k ) d m2 2 2

⎤ ⎥ ⎦

(53)

where Q(·) is the Gaussian Q function [40] while L and dm are respectively the number of the nearest neighbors and minimum Euclidean distance in the normalized constellation. Inserting Eq. (38) into Eq. (53), the average probability of error for a MIMO beam-forming system is

⎧ ⎡ E[Pe (k )] = LE ⎨Q ⎢ ⎩ ⎣

0.5σ H2 1 ( k ) d m2

β ( k )+ ρ

−1

⎤⎫ ⎥⎦ ⎬ ⎭

(54)

Observing this expression, the average probability will behave similar to the perfect CSI case until the noise floor becomes saturated by β(k). This is mathematically justified with the following result.

Theorem 8.4. The average probability of error before saturation due to channel prediction error for a MIMO RNN predictor has diversity gain and coding gain with probability p described by

Gd = N t r [Φ RX (k )]

(55)

196

C. Potter

Gc = where

d m2

{

δ L det [Φ RX ( k )]N r

(56)

}

1 Gd

δ = 4[ρβ p min (N t , N r ) + 1]

(57)

and βp(k) represents the 100pth percentile of β(k).

Proof. Noting first that

H (k ) F = 2

r [H i (k )]

∑σ i =0

2 H i (k )

≤ σ H2 1 (k )r [H (k )] ≤ σ H2 1 (k ) min( Nt , N r )

(58)

The beam-forming SNR is lower-bounded with probability p by

ρbf (k ) ≥

ρ H (k ) F ρ min ( N t , N r )β p +1 2

(59)

Substituting Eq. (59) into Eq. (53) and applying the Chernoff bound [40], the probability of error is bounded by

Pe (k ) ≤ L exp⎛⎜ − ⎝

ρ H (k ) δ

Thus

d m2

⎞ ⎟ ⎠

(60)

( )

E[Pe (k )] ≤ LM H (k ) 2 − F

2 F

ρd m2 δ

(61)

where Mx(s)=E(esx) is the moment generating function (MGF). This has been computed in [36] as r (Ψ )

M x (s ) = ∏ 1+ s1λΨ

(62)

i

i =1

Inserting this into Eq. (54) yields r (Ψ )

E[Pe (k )] ≤ L∏

i =1 1+

1 2 ρd m δ

(63)

λ Ψi

Applying the approximation (1+x)-1≈x-1 for large x, the average probability of error can be written as

E[Pe (k )] ≤ L =L

[( )] ρd m2 δ

[( )] ρd m2 δ

− r (Φ TX ) N r

− r (Ψ ) r (Ψ )

∏λ i =1

−1 Ψi

=L

det[ΦTX (k )]

−Nr

[( )] ρd m2 δ

− r (Ψ )

det[Φ RX (k ) ⊗ ΦTX (k )]

⎧⎪ ρd m2 =⎨ ⎪⎩ δ L det (ΦTX )− N r

[

−1

⎫⎪ 1 [r (Φ TX ) N r ] ⎬ ⎪⎭

]

− r (Φ TX ) N r

(64)

RNN Based MIMO Channel Prediction

197

which implies that for small prediction errors (i.e. before the noise floor due to channel prediction error saturates the signal)

Gc =

{

δ [L det (Φ σX

d m2

)]− N r }

1 [ N r r ( ΦσX

Gd = N t r (Φ RX )

)]

(65) (66)



the desired result.

Remark 8.5. If only μβ(k) and σ β(k) are available, the Bienaymé-Chebyshev inequality can be employed to Theorem 8.4 to obtain with probability p=1-N-2

β (k ) ≤ Nσ β (k ) + μ β (k )

(67)

Theorem 8.4 suggests that until saturation, the probability of error will maintain the same diversity order as the perfect CSI case. The penalty instilled by the prediction error is a loss in coding gain. Before illustrating this, new tight approximations for the average probability of error and array gain are derived. Noting through Figs. 8.11 and 8.12 that the cumulative density function (c.d.f.) of ρbf0.5(k) is accurately fitted by a gamma distribution with scale parameter θ and shape parameter κ, the probability of error is accurately approximated by the following result.

Fig. 8.11 Comparison of simulated and fitted c.d.f.'s for a MIMO spatially uncorrelated channel

198

C. Potter

Fig. 8.12 c.d.f.'s for a MIMO spatially correlated ( γ t

= 0.7 ) channel

Theorem 8.6. The average probability of error for a MIMO RNN predictor before saturation is tightly approximated by

E[Pe (k )] ≈ 1.135 where

Τ1 =1 F1

(

L 4π θ κ Γ (k )d m

( ) [ d m2 4

) (

; ; ( d 1θ )2 −1 F1

k −1 1 2 2

(

m

−κ 2

dm 2

Γ ( k 2−1 )Τ1 + Γ( k2 )Τ2

(

; ; 12 0.99 +

k −1 1 2 2

(

Τ2 =(0.99d m + θ1 )1 F1 κ2 , 32 , 12 0.99 + dm1θ

) )− 2

(,

1 κ θ 1 1 2

F

))

1 2 d mθ 3 2

]

(68)

(69)

, (d 1θ )2 m

)

(70)

Proof. Approximating the Gaussian Q function by [41]

[1 − e Q(x ) ≈

− ( Ax )

2

]e

−x2 2

(71)

B 2π x

with A=1.98 and B=1.135 while recalling the pdf of a Gamma distribution, the probability of error is

E[Pe (k )] ≈

1

B 4π θ Γ (k )d m κ

∫

∞

0

[1 − e

− ( Ad m x )

2

]e

− ( d m x )2 4 κ − 2 − x θ

x

e

dx

(72)

RNN Based MIMO Channel Prediction

199

Making the substitution y=x2 and utilizing [42], the result follows after several manipulations.

Remark 8.7. The array gain for a MIMO RNN predictor is accurately approximated by

η ≈ θ 2κ (κ + 1)

(73)

Proof. This result follows immediately after performing the transformation y=x2 and calculating the mean. These two results provide closed form expressions that depend only on the scale and shape parameters of the Gamma distribution which can be found via maximum likelihood estimation. Figs. 8.13 and 8.14 illustrate the parameter values used for this work.

Fig. 8.13 Parameters of fitted Gamma distribution for a MIMO spatially uncorrelated channel

The array gain is investigated along with its approximations from Theorem 8.2 and Remark 8.7 for a 2×2 MIMO RNN predictor trained with f(x)=x, fdTs=0.1, Ntrain=200, γt=0. The weights are brought online for the prediction of a channel using the same parameter values in Fig. 8.15 for various channel estimation errors. The same parameters values are used with the exception of γt=0.7 to consider the effects of spatial correlation in Fig. 8.16. For both scenarios, the approximation due to Remark 8.7 is superior. The array gain is slightly larger for the spatially correlated case which is clear from Eq. (51).

200

C. Potter

Fig. 8.14 Parameters of fitted Gamma distribution for a MIMO spatially correlated (γt=0.7) channel

Fig. 8.15 Array gain for a 2×2 MIMO beam-forming fast fading uncorrelated channel

RNN Based MIMO Channel Prediction

201

Fig. 8.16 Array gain for a 2×2 MIMO beam-forming fast fading spatially correlated (γt=0.7) channel

Fig. 8.17 Average Probability of error for a 2×2 MIMO beam-forming fast fading spatially uncorrelated channel

202

C. Potter

Fig. 8.18 Average Probability of error for a 2×2 MIMO beam-forming fast fading spatially correlated (γt=0.7) channel

Fig. 8.19 Average Probability of error comparison for a 2×2 MIMO beam-forming fast fading spatially uncorrelated channel

RNN Based MIMO Channel Prediction

203

Fig. 8.20 Average Probability of error for a 2×2 MIMO beam-forming fast fading spatially correlated (γt=0.7) channel

The average probability of error was simulated for a 2×2 MIMO system when f(x)=x, fdTs=0.1, γt=0 using 8×105 BPSK symbols with symbol period T=Ts and is plotted versus the theoretical average probability of error given by Eq. (54) for various channel estimation errors in Fig. 8.17 and with γt=0.7 in Fig. 8.18. For both cases, the RNN was trained with f(x)=x, γt=0, fdTs=0.1, with σw2=0.001. As proposed in Theorem 8.4, an increase in the prediction error leads to a decrease in coding gain but maintains full diversity gain until saturation. Next, the approximation due to Theorem 8.6 is plotted for comparison in Figs. 8.19 and 8.20 and is seen to be in good agreement with both the simulated probability of error and (54) until saturation.

8.9 Conclusions A recurrent neural network trained off-line by a novel PSO-EA-DEPSO was used to predict a MIMO channel. This training algorithm was shown to be superior to PSO, PSO-EA, and DEPSO for different fast fading scenarios. The RNN predictor was then shown to outperform a linear predictor trained by the Levinson-Durbin algorithm. New expressions for the received SNR, array gain, average probability of error, and diversity order for the MIMO RNN predictor were then derived. These expressions differed from numerous works in that the prediction error was not assumed to be independent of the actual CSI and/or Gaussian. The array gain

204

C. Potter

for spatially correlated systems was shown to be slightly higher. It was verified through simulation that increasing the prediction error caused a loss in coding gain but still were able to achieve full diversity gain up until saturation.

References [1] Telatar, E.: Capacity of multi-antenna gaussian channels. AT&T Bell Laboratories Internal Technical Memorandum (June 1995) [2] Foschini, G., Gans, M.: On limits of wireless communications in a fading environment when using multiple antennas. Wireless Personal Communications 6(3), 311–335 (1998) [3] Papoulis, A., Pillai, S.: Probability, Random Variables, and Stochastic Processes. McGraw-Hill Science, New York (1990) [4] Huang, J., Winters, J.: Sinusoidal modeling and prediction of fast fading processes. In: Global Telecommunications Conf., Sydney, Australia, November 8-12, pp. 892– 897 (1998) [5] Andersen, J., Jensen, J., Jensen, S., Frederiksen, F.: Prediction of future fading based on past measurements. In: IEEE Vehicular Technology Conf., Amsterdam, The Netherlands, September 19-22, pp. 151–155 (1999) [6] Duel-Hallen, A., Hu, S., Hallen, H.: Long-range prediction of fading signals: enabling adaptive transmission for mobile radio channels. IEEE Signal Processing Magazine 17(3), 62–75 (2000) [7] Liu, W., Yang, L.L., Hanzo, L.: Recurrent neural network based narrowband channel prediction. In: IEEE Vehicular Technology Conf., Melbourne, Australia, May 7-10, vol. 5, pp. 2173–2177 (2006) [8] Visweswaran, B., Kiran, T.: Channel prediction based power control in w-cdma systems. In: 1st Int. Conf. 3G Mobile Communications Technologies, London, UK, March 27-29, pp. 41–45 (2000) [9] Gao, X., Tanskanen, J., Ovaska, S.: Comparison of linear and neural network-based power prediction schemes for ds/cdma systems. In: IEEE Vehicular Technology Conf., Atlanta, GA, April 28-May 1, pp. 61–65 (1996) [10] Gao, X., Tanskanen, J., Ovaska, S.: Power prediction in mobile communication systems using an optimal neural network structure. IEEE Trans. Neural Networks 8(6), 1446–1455 (1997) [11] Cai, X., Zhang, N., Venayagamoorthy, G.K., Wunsch, D.C.: Time series prediction with recurrent neural networks trained by a hybrid pso-ea algorithm. Elsevier J. Neurocomputing 70(13-15), 2342–2353 (2007) [12] Kennedy, J., Eberhart, R.C.: Particle swarm optimization. In: IEEE Int. Conf. Neural Networks, Perth, WA, November 27 - December 1, vol. 4, pp. 1942–1948 (1995) [13] Kennedy, J., Eberhart, R., Shi, Y.: Swarm Intelligence. Morgan Kaufmann Publishers, San Francisco (2001) [14] Storn, R., Price, K.V.: Differential Evolution - A Simple and Efficient Adaptive Scheme for Global Optimization over Continuous Spaces, Technical Report TR-95012, International Computer Science Insitute (March 1995) [15] Storn, R., Price, K.V.: Differential evolution-a simple and efficient heuristic for global optimization over continuous spaces. J. Global Optimization 11(4), 341–359 (1997)

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[16] Mayer, D., Kinghorn, B., Archer, A.: Differential evolution-an easy and efficient evolutionary algorithm for model optimisation. Elsevier J. Agricultural Systems 83(3), 315–328 (2005) [17] Price, K.V., Storn, R.M., Lampinen, J.A.: Differential Evolution: a Practical Approach to Global Optimization. Springer, Berlin (2005) [18] Qing, A.: Differential Evolution: Fundamentals and Applications in Electrical Engineering. John Wiley & IEEE, New York (2009) [19] Xu, R., Venayagamoorthy, G.K., Wunsch, D.C.: Modeling of gene regulatory networks with hybrid differential evolution and particle swarm optimization. Elsevier J. Neural Networks 21(8), 917–927 (2007) [20] Zheng, Y., Xiao, C.: Improved models for the generation of multiple uncorrelated Rayleigh fading waveforms. IEEE Communications Letters 6(6), 256–258 (2002) [21] Shiu, D., Goschini, G., Gans, M., Kahn, J.: Fading correlation and its effect on the capacity of multi-element antenna systems. IEEE Trans. Communications 48(3), 502– 513 (2000) [22] van Zelst, A., Hammerschmidt, J.: A single coefficient spatial correlation model for multiple-input multiple-output (mimo) radio channels. In: General Assembly Int. Union Radio Science, Maastricht, The Netherlands, August 17-24 (2002) [23] Ortega, J.M.: Matrix Theory-A Second Course. Plenum, New York (1987) [24] Yoo, T., Goldsmith, A.: Capacity of fading mimo channels with channel estimation error. In: IEEE Int. Conf. Communications, Paris, France, June 20-24, vol. 2, pp. 808–813 (2004) [25] Yoo, T., Yoon, E., Goldsmith, A.: Mimo capacity with channel uncertainty: Does feedback help? In: IEEE Global Telecommunications Conf., Dallas, TX, November 29-December 3, vol. 1, pp. 96–100 (2004) [26] Yoo, T., Goldsmith, A.: Capacity and power allocation for fading mimo channels with channel estimation error. IEEE Trans. Information Theory 52(5), 2203–2214 (2006) [27] Proakis, J.: Digital Communications. McGraw-Hill, New York (1995) [28] Rappaport, T.S.: Wireless Communications: Principles and Practice. Prentice Hall, Upper Saddle River (2002) [29] Rice, S.: Statistical properties of a sine wave plus random noise. Bell System Technical J. 27(1), 109–157 (1948) [30] Sampath, A., Holtzman, J.: Estimation of maximum doppler frequency for handoff decisions. In: IEEE Vehicular Technology Conf., Secaucus, NJ, May 18-20, pp. 859– 862 (1993) [31] Bitran, Y.: Broadband data, video, voice, and mobile convergence-extending the triple play. Texas Instruments White Paper (2004) [32] Jakes, W.: Microwave Mobile Communications. IEEE Press, New York (1993) [33] Choi, J., Bouchard, M., Yeap, T.: Decision feedback recurrent neural equalization with fast convergence rate. IEEE Trans. Neural Networks 16(3), 699–708 (2005) [34] Potter, C., Venayagamoorthy, G.K., Kosbar, K.: MIMO beam-forming with neural network channel prediction trained by a novel PSO-EA-DEPSO algorithm. In: 2008 IEEE World Congress Computational Intelligence/2008 IEEE Int. Joint Conf. Neural Networks, Hong Kong, June 1-6, pp. 3338–3344 (2008) [35] Jacson, L.: Digital Filters and Signal Processing. Kluwer, Norwell (1989) [36] Paulraj, A., Nabar, R., Gore, D.: Introduction to Space-Time Wireless Communications. Cambridge University, Cambridge (2003)

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[37] Bain, L., Engelhardt, M.: Introduction to Probability and Mathematical Statistics. Brooks/Cole, Pacific Grove (1992) [38] Lee, W.: Mobile communication engineering. McGraw Hill, New York (1982) [39] Fu, J., Taniguchi, T., Karasawa, Y.: The largest eigenvalue characteristics for mimo channel with spatial correlation. J. Global Optimization 6(2), 109–133 (1995) [40] Goldsmith, A.: Wireless Communications. Cambridge University Press, New York (2005) [41] Karagiannidis, G., Lioumpas, A.: An improved approximation for the gaussian q function. IEEE Communications Letters 11(8), 644–646 (2007) [42] Gradshteyn, S., Ryzhik, I., Jeffrey, A.: Table of Integrals, Series, and Products, 5th edn. Academic, San Diego (2007)

Index

A Absorber 54, 64 absorption band 55 academic misconducts 1 active pattern 48, 50 adaptive simulated annealing 20 adjacent channel 157, 158, 160 adjacent channel interference rejection 158 advantage 37, 49, 116, 135, 177 algorithm 19, 20, 73, 112, 134, 151, 177 anechoic chamber 60, 145 anisotropic electromagnetic material 57 annealed Nelder & Mead strategy 20 antenna 48, 49, 134, 157, 165, 178, 181 antenna array 48, 49, 50, 61 application problem 9, 47 arithmetic crossover 32, 33, 34, 35 arithmetic multi-point crossover 33 arithmetic one-point crossover 33, 34 arithmetic operation 19 arithmetic recombination 26 arithmetic two-point crossover 34 array element 48, 49, 55, 61 array factor 48 array gain 178, 193, 194, 197, 199, 203 artifacts 11 Ask 4 available frequency 155 average symbol energy 182 B base transceiver stations 156 base vector 22 basis functions 137 benchmark application problems 47 Bernoulli experiments 26, 27, 35 best solution 162, 168 bibliography 2, 4

binary crossover 31 bioelectromagnetics 61 binomial crossover 34, 170, broadband planar microwave absorber 55 BTS 156 C calibration 51 candidate solution 162 case study 6, 7 cell 57, 112, 156, 157 channel 58, 59, 157-160, 177-206 channel coefficient 59, 185, 186, 188 channel interference 157, 158, 160 channel state information 177, 179, 182 Chebyshev polynomial fitting problem 19 Chinese 3, 4 Chinese Electronic Periodical Services 3 citeSeerX 4, 5 classic differential evolution 28, 36, 156, 159 classification 9, 10, 37, 43, 45, 74, 79, 98, 101 clustering procedure 111, 112 co-channel 157, 158, 160 co-evolution 9 co-evolution differential evolution 37 comparative evaluation 9 comparison 9, 20, 38, 58, 77, 84, 89, 112, 162, 187 complex permittivity 108 computational electromagnetics 45, 59,61, 151 Computer Science Bibliographies 4 constraint 9, 23, 49, contrast 109, 178 contrast source 129 control parameters 9, 20. 27, 58, 81, 166 conventional antenna array 48

208

Index

convergence 19, 22, 59, 77, 83, 166 cost function 76, 94, 109 coverage 4, 43, 58, 156 coverage relation 156, 166 crossover 8, 9, 81, 141, 166, 186 crossover intensity 33 crossover length 26 crossover probability 22, 81, 166, 186 cultural differential evolution 4 cut and try 52 D data insufficiency 76 database 3, 55 DE 20, 22, 125, 128, 177, 178 DE particle swarm optimization Decoding 9, 54 degradation of network quality 157 Denver 163. 166 DEPT 156, 166, 170 DEPT scheme 170 Defect 22 deterministic optimization algorithms 60 diagnostic applications 107 diagonal matrix 79, 110 dielectric materials 107 differential evolution 1, 19, 43, 73, 107, 135 differential evolution equation 4 differential evolution strategy 8, 28 differential evolution with Pareto tournament 159 differential genetic algorithm 10 differential mutation 8, 9, 19, 170 differential mutation base 22, 29, 170 digital resources 2 digital platform 2 directed differential mutation 30 disadvantage 9, 38, 178 distance function 163 diversity 22, 177 dominance 2, 19, 160 donor 24, 30 Doppler double square loop array 56 dynamic differential evolution 36, 81 dynamic range ratios 49 E economic differential evolution 4 effective anisotropy 56

effective wave numbers 56 efficiency 37, 51, 177 electromagnetic compatibility 57, 60, 134 electromagnetic composite materials 56, 57 electromagnetic fields 45, 73, 136 electromagnetic formation flight 60 electromagnetic inverse problem 44, 45, 47, 55 electromagnetic materials 56, 57 electromagnetic energy 59, 60 electromagnetic spectrum 43 electromagnetic structure 53 electromagnetic theory 53, 61 electromagnetic waves 43, 54, 73 electronic resources 4 Elsevier 4 emission 60, 133, 142, 144 encoding 9, 54, 165 Engineering Village 2 3 English 4, 43 epistatic problems 38 equivalent circuit method 55 error figures 112, 116, 125 estimation 182 evaluation 9, 20, 73, 107 evolution mechanism 9, 10 evolution strategy 8, 28 evolutionary algorithm 10, 19, 47, 77, 156, 177 evolutionary crimes 58, 61 evolutionary operation 8, 10, 24, 141 evolutionary programming 59 excitation amplitude 49, 50 experiments 26, 163, 166 exponential crossover 26, 35, 170, 171 external medium 108 F FAP 155, 156, 159, 161 FAP solution 155, 158 feed forward 177, 184, 191 filters 55 finite difference time domain method 55 finite element method 55 fitness 159, 160, 165, 180 Foldy-Lax model 78 frequency 55, 187 frequency assignment 57, 155 frequency assignment problem 155 frequency planning 57

Index

209

frequency selective surfaces 54, 55 frequency spectrum 155 frequency value 160, 165 functional 53, 108, 110, 156 G generation 24, 58, 101, 159 genetic algorithm 10, 30, 54, 55, 77, 134 genetic annealing algorithm 19 genetic differential evolution 4 geographical differential evolution 4 geological differential evolution 4 geometrical center 29 geometrical properties 107 geophysical prospecting 107 global optimum 37, 38, 162 global search 19, 37, 53, 140 GMO-SVNS 173, 175 GMO-VNS 173, 175 Google 4 Google Scholar 4 graphic 4, 54 Greedy mutation 162, 173 Greedy variants 156 Green’s function 79, 108 grey-level representation 111 GSM 57, 155 GSM networks 57, 155, 156 H Hankel function 75, 109 helical antenna 52 history 8, 19, 22, 50 horn antenna 52, 53 hybrid differential evolution 37, 57, 59 hybridization 9 hypervolume 156, 166 I IEEE Explore 3 ill-posedness 76, 107 imaging 107, 109 impedance-matching tuner 54 inception 1, 19 incident field 108 infinite cylinders 108 initial guess 73, 77 initial population 22, 24, 25 initial solution 37, 160, 161, 162

initialization 9, 28 insight 2, 7, 20 instability 77 Institute of Scientific and Technical Information of China 3 Integral Fredholm operators 108 interference cost 156, 157, 158 interference matrix 157 interferences 57, 133, 138, 157, 160 International Contests on Evolutionary Optimization 20 intrinsic control parameters 8, 20, 58, 166 inverse scattering 10, 73, 107 inverse scattering problems 10, 73, 77, 101, 107 ISI Web of Science 3 isotropic electromagnetic material 57 isotropic point sources 48 iterative multiscaling approach 107, 108, 110, 112, 129 K Kronecker product 182 L language 4, 43, 165 large problem 158 laser diode 52 least Squares 74, 78, 134, 140 lens antenna 52 Levinson-Durbin algorithm 178, 191, 203 limit of number of generations 27 limited radio spectrum 157 linear antenna array 48, 49, 50 literature survey 1, 2, 43 local minimum 144, 180, 193 local optimizer 22 local optimum 162 local shape functions 74, 76 logic dominance function 29 lower bound 166, 196 M mathematical formulation 78, 156, 158 mathematical model 7, 8, 156 Maxwell’s equations 59 mean 29 mean squared error 178, 179, 182 measurement 53, 81, 144

210

Index

measurement domain 108 measurement points 112, 135, 138, 140 medical imaging 170 memetic differential evolution 59 metaheuristics 133, 135 method of moment 76, 112 Microsoft Bing 4 microstrip antenna 53 microwave circuit 61 microwave & RF 50, 51, 52 microwave tomography 130 milestone 22 MIMO 58, 177 minimization process 108 misconception 10, 61 mixed optimization parameters 54, 55 MO-SVNS 156 MO-VNS 156, 161, 175 mobile communications 155 mobile stations 156 model approximation 110 moment generating function 196 mother-child competition 19 moving phase center antenna array 48, 50 multi-layer perceptron 177 multi-objective 37, 61, 155, 159, 166 multi-objective differential evolution 37, 159 multi-objective FAP 156, 159 multi-objective optimization 161, 162, 174 multi-objective skewed variable neighborhood search 162 multi-point crossover (or M-point crossover) 32, 33 multilayered medium 53 multiple-input multiple-output 177, 178 multiple signal classification 74, 78, 79, 81, 101 multiresolution strategy 111 multistatic response matrix 76, 79 mutant 22, 26, 30 mutation 9, 19, 58, 160, 169, 178 mutation intensity 21, 22, 28, 81, 166 mutual coupling 48, 49, 61 N National Knowledge Infrastructure 3 natural real code 19, natural selection 27 network 51, 59, 134, 155 network quality 155, 157

Newton’s method 59 Noise filtering 111 noisy problems 38 non-intrinsic control parameters 8 non-uniqueness 58, 77 nondestructive testing 107 nonlinear operator 109 nonlinearity 52, 107 normalized vector difference 28 notation 22, 23 numerical algorithms 59 O objective function 22, 23, 52, 140, 166, 179 objective function evaluation 28 objective space 166 one-point crossover 32, 33, 141 operator 78, 108, 141, 155, 157, 179, 181 optimal control in minimal time 54 optimization 31, 52, 59, 77, 177, 193 optimization algorithm 9, 20, optimization parameters 23, 54, 76, 101, 140 optimization problem 45, 50, 52, 76, 107, 140, 156, 179, 193 optimization vector 76, 78 originality 1 P paper platform 2 parallel processing 110 parameterization 110 penetrable bodies 108 parasitic effect 51 Pareto 156, 159, 165, 173, 175 Pareto front 165, 166, 167, 171, 173 Pareto solution 162, 163 Pareto tournament 156, 159, 175 particle swarm optimization 59, 177 pattern nulling 49 perfect electric conductor 108 performance 9, 20, 36, 48, 77, 166, 177 performance indicator 166 periodic arrays 53, 54, 55, 56 periodic moment method 55 permeability 56, 57, 136 permeability tensor 57 permittivity 57, 74, 80, 108, 136 permittivity tensor 56, 57

Index

211

personal library 10 phased antenna array 50 phase center 48, 50 phaseless near field antenna measurement 53 pixel-wise approach 55 plain electromagnetic structure 53, 54 planar antenna array 49 Planck length 43 plane wave 112 platform 2, 49, 149 polarization current 108 population 19, 21, 25, 77, 81, 143, 162 population size 22, 28, 77, 81, 159, 166, 167 population-based incremental learning 58 practical advice 21 practical usage advice 7 prediction 59, 60, 135, 150, 177-206 principle of pattern multiplication 48 probability distribution function 28 probability of error 178, 193, 195, 196, 203 problem feature 37 propagation medium 109 pseudocode 159, 161 PSO evolutionary algorithm 177, 188 PSO-EA-DEPSO algorithm 177, 187, 191 Q Q function 195, 198 QoS 155 Qualitative methods 74, 78 quality of service 155 R radar 45, 49, 59 radio network design 45, 58 random frequency 161 random search 58 real material database 55 real-world 57, 155, 156, 163 real-world instances 156, 175 recombination coefficient 26, 28 recurrent neural networks 184 relative permittivity 57, 74 reliability 8, 37, 59, 177 reputation 9, 20 resolution 55, 110 retrieval procedure 111

RF circuits 50, 51 RF low noise amplifier 44, 50 ripple 49 robustness 8, 38, 178 S sampling period 181 scanned beam antennas 53 scatterer 45, 46, 73, 77, 78, 101, 111, 126, 191 scattered electric field 75, 81, 107, 112 scattering equation 108 scattering potential 109, 110, 112, 113 ScienceDirect 3 Scopus 3, 4 search engines 4 search space 107, 125, 143, 166 Seattle 163, 166, 168, 172 sector 157, 160, , 162, 164 sector channel 158, 159 sector channel separation 158, 159 selection 9, 19, 22, 27 semi-anechoic chambers 60, 133, 137, 151 separation constraints 155 separation cost 156, 157, 158, 159, 160, 168 Shannon limit 58 shape 48, 56, 59, 74, 76, 108, 150, 197 side lobe 48 signal-to-noise ratio 58 signaling threshold 158 simplicity 28, 37, 49, 75, 181 simulated annealing 20, 54, 58 single-input single-output 177 singular vector 80, 183, 184 singular value 79, 80, 183 site channel separation 158 social differential evolution 4 spectral-domain method 55 SpringerLink 3 staggered dipole array 56 standard deviation 158 state of the art 1, 2, 36 statistical confidence 165 stochastic algorithms 60 stochastic optimization method 110 survival of the fittest 27 susceptibility 60 SVNS 156, 162, 175 symbol period 177, 203 synthesis 48, 49, 51, 55

212

Index

system level evaluation 38 system level parametric study 1, 38 T T-matrix method 55, 56 tabu search 59 tapering 49 TCQ model 56, 57 technical limitations 158 telecommunications 57, 155 termination 9, 24, 28, 36, 81, 84, 111, 165 termination conditions 9, 24, 27, 36, 165 termination procedure 111 test bed 9, 20 theory of differential evolution 7 thresholding operation 111 time modulation 49, 50 time-modulated antenna array 48, 49 time sequence 49 topical review 1, 36, 37 topology 165 total field 109 toy function 20 traffic demand 157 transceiver 156, 157, 158 transmit power 182 transverse magnetic 75, 112 trial and error 20, 54, 55, 186 trial parameter vector 19 trigonometric differential mutation 30

TRX 157, 158, 160, 161, 165 two norm 179 U undesired interferences 157 upper bound 166, 193 usage 7, 9, 22 user friendliness 19 V variable interferences variable neighborhood search 59, 160-163 vector difference 22, 24, 26, 28, 30 vector spectral-domain method 55 VNS 156, 160, 161 W wave number 56, 57, 75 website 4, 21 weighted difference vector 19 weighting function 109, 110 Wiley Interscience 3 wireless 50, 52, 57, 177-179, 184, 191 wireless communication 50, 52, 57, 177, 179 Y Yahoo 4

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