RF imperfections in high-rate wireless systems: impact and digital compensation

RF Imperfections in High-rate Wireless Systems RF Imperfections in High-rate Wireless Systems Impact and Digital Comp...

Author: Tim Schenk

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RF Imperfections in High-rate Wireless Systems

RF Imperfections in High-rate Wireless Systems Impact and Digital Compensation by

Tim Schenk Philips Research, Eindhoven The Netherlands

ABC

Library of Congress Control Number:2008920465

ISBN 978-1-4020-6902-4 (HB) ISBN 978-1-4020-6903-1 (e-book) Published by Springer, P.O. Box 17, 3300 AA Dordrecht, The Netherlands. www.springer.com

Printed on acid-free paper

All Rights Reserved ° c 2008 Springer Science + Business Media B.V. No part of this work may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without writte n permission from the Publisher, with the exception of any material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work.

Contents

Foreword Preface 1. INTRODUCTION

xi xiii 1

1.1

Wireless communications

1

1.2

OFDM

3

1.3

MIMO

4

1.4

RF transceiver impairments

6

1.5

Outline of the book

9

2. MULTIPLE-ANTENNA OFDM SYSTEMS

11

2.1

Introduction

11

2.2

Channel modelling 2.2.1 Multipath propagation 2.2.2 Stochastic channel model

11 12 16

2.3

System modelling 2.3.1 MIMO 2.3.2 OFDM 2.3.3 MIMO OFDM

18 19 22 26

2.4

Conclusions

29

3. DESIGN AND IMPLEMENTATION OF A MIMO OFDM SYSTEM

31

3.1

Introduction

31

3.2

Transmission format and preamble design 3.2.1 IEEE 802.11a 3.2.2 MIMO OFDM

32 32 34

vi

Contents

3.3

Frequency synchronisation 3.3.1 Inﬂuence of CFO 3.3.2 Algorithm description 3.3.3 Performance analysis 3.3.4 Numerical results 3.3.5 Summary

39 40 42 44 47 51

3.4

Channel estimation 3.4.1 Algorithm and performance analysis 3.4.2 Numerical results 3.4.3 Summary

52 52 58 62

3.5

Timing synchronisation 3.5.1 System description 3.5.2 Algorithm description 3.5.3 Numerical results 3.5.4 Summary and discussion

62 63 66 68 72

3.6

Multiple-antenna OFDM system implementation 3.6.1 Test system description 3.6.2 Baseband design 3.6.3 Measurement results 3.6.4 Comparison with simulation results

73 73 74 76 79

3.7

Conclusions

81

4. PHASE NOISE

83

4.1

Introduction

83

4.2

System and phase noise modelling 4.2.1 Oscillator modelling 4.2.2 Inﬂuence on signal model

84 85 88

4.3

Impact and distribution of the ICI term 4.3.1 System model 4.3.2 Bit-error probabilities 4.3.3 Properties of the ICI term 4.3.4 Simulation results 4.3.5 Summary

91 91 94 95 104 107

4.4

ICI-caused error term in MIMO OFDM 4.4.1 Transmitter phase noise 4.4.2 Receiver phase noise

108 109 110

Contents

4.4.3 Phase noise process 4.4.4 Numerical results 4.4.5 Summary and discussion 4.5 Compensation of the CPE 4.5.1 Maximum-likelihood estimator 4.5.2 MLE optimisation algorithm 4.5.3 Sub-optimal estimator 4.5.4 Cram´er-Rao lower bound 4.5.5 Numerical results 4.5.6 Summary 4.6 Compensation of the ICI 4.6.1 Suppression algorithm 4.6.2 Numerical results 4.6.3 Summary 4.7 Conclusions 5. IQ IMBALANCE 5.1 Introduction 5.2 System and IQ imbalance modelling 5.2.1 TX/RX front-end architecture 5.2.2 IQ mismatch 5.2.3 Inﬂuence on MIMO signal model 5.2.4 RX ﬁlter imbalance 5.3 Impact of IQ imbalance on system performance 5.3.1 Error in symbol detection 5.3.2 Probability of erroneous detection for M -QAM 5.3.3 Numerical results 5.3.4 Summary and discussion 5.4 Preamble based estimation and mitigation 5.4.1 Preamble design 5.4.2 TX IQ imbalance estimation 5.4.3 RX IQ imbalance estimation 5.4.4 TX and RX IQ imbalance estimation 5.4.5 Iterative TX and RX IQ imbalance estimation 5.4.6 Numerical results 5.4.7 Summary

vii 112 113 116 117 117 119 121 122 124 130 130 131 134 136 137 139 139 140 141 143 145 148 150 150 152 159 163 163 164 166 167 168 170 172 177

viii

Contents

5.5

Decision-directed mitigation 5.5.1 Adaptive ﬁlter based algorithm 5.5.2 Numerical results 5.6 Conclusions

178 178 180 182

6. NONLINEARITIES 6.1 Introduction 6.2 System and nonlinearities modelling 6.2.1 Nonlinearity modelling 6.2.2 Inﬂuence on signal model 6.3 Impact of nonlinearities on system performance 6.3.1 TX nonlinearities 6.3.2 RX nonlinearities 6.3.3 Numerical results 6.3.4 Summary and discussion 6.4 PAPR reduction by spatial shifting 6.4.1 Signal modelling 6.4.2 Spatial shifting 6.4.3 Combining SS and PS 6.4.4 Subcarrier grouping schemes 6.4.5 Side information/Transparency 6.4.6 Numerical results 6.4.7 Summary and discussion 6.5 RX-based correction of nonlinearities 6.5.1 Baseband amplitude clipping 6.5.2 Estimation of linear and nonlinear channel 6.5.3 Postdistortion 6.5.4 Iterative distortion removal 6.5.5 Simulation results 6.5.6 Summary 6.6 Conclusions

185 185 187 187 192 195 196 201 203 207 208 209 210 211 213 213 215 217 219 220 221 230 231 233 242 242

7. A GENERALISED ERROR MODEL 7.1 Introduction 7.2 Error model 7.2.1 TX and RX impairment model 7.2.2 Mapping of RF impairments 7.2.3 Nonstationarity of the impairments 7.2.4 Transmission structure

245 245 246 247 248 250 252

Contents

7.3

Performance evaluation 7.3.1 Preamble phase 7.3.2 Data phase 7.3.3 Probability of error 7.4 Numerical results 7.5 Conclusions

ix 252 253 253 254 257 261

8. DISCUSSION AND CONCLUSIONS 8.1 Summary and conclusions 8.1.1 Synchronisation for MIMO OFDM systems 8.1.2 Multiple-antenna OFDM proof of concept 8.1.3 Behavioural RF impairment modelling 8.1.4 Performance impact of front-end imperfections 8.1.5 Mitigation approaches for RF impairments 8.1.6 A generalised error model 8.2 Scope of future research 8.2.1 Inﬂuence of RF impairments 8.2.2 Mitigation approaches 8.2.3 Towards mm-wave systems

263 263 264 265 265 266 267 269 270 270 271 272

Glossary

275

Appendices A MSE in CFO estimation B MSE in channel estimation B.1 MSE in linear channel interpolation B.2 MSE in linear channel extrapolation C Measurement setup C.1 Baseband C.2 IF stages C.3 Antennas D Proof of Theorem 4.1 E Orthonormal polynomial basis

281 281 283 283 284 287 287 289 290 291 299

References

301

Index

315

Foreword

This book takes a modern, multidisciplinary view on radio system design: the advantages of digital signal processing are exploited to satisfy the ever increasing demands on better performing, ﬂexible radio frequency (RF) circuits. By accepting that analog circuits are inherently imperfect, but by searching for methods to mitigate and compensate for this, signiﬁcant steps can be made to improve the overall system solution. This is in contrast to the more traditional approach of designing RF circuits themselves to satisfy the demanding speciﬁcations set by current standards. Wireless communication has progressed dramatically in past years. Yet, the main research challenges changed focus to sustain this growth over many decades. In the early days, the main challenge of radio communications was to cover large distances. When in late 1980s, digital mobile communication emerged as a precursor to today’s mass-market cell phones, the priority shifted to achieving reliable communication over a diﬃcult, i.e., time-varying and frequency-selective mobile channel. About a decade or research in many institutes around the world has been devoted to this challenge, exploiting the opportunities given by increasing computational power in digital circuits. Today we face a diﬀerent paradigm: the technology for analog front-ends progress significantly slower than that of digital processors. So the imperfections of the RF have increasingly become the major bottleneck in our drive towards faster, yet more power-eﬃcient radio circuits. Since digital signal processing can mitigate the design constraints for analog RF circuits, it allows more cost-eﬀective and lower power consuming designs for the analog part. At the time of writing this book, algorithms for digital compensation, or digitally controlled calibration of the RF circuits start to be used in chipsets in the market. But the concepts addressed in this book

xii

Foreword

may not be limited to only oﬀering a proprietary advantage to smart implementers. It is not unlikely that future standards will increasingly be designed to anticipate for digital RF compensation, for instance by deﬁning new preambles or pilots that also allow receivers to track phase drifts, or to estimate I-Q imbalance. An important step made in this book is to bring together models describing analog circuits with insights of digital signal processing. Tractable but realistic models to evaluate the eﬀect of compensation measures are essential to support the design of eﬀective practical solutions. Right from the start the book follows a timely approach that coincides with clear direction taken in many modern systems, namely the use of multiple antennas to accommodate more radio traﬃc, to oﬀer better link throughput, and to reduce interference. Prof. Dr. Ir. Jean-Paul M.G. Linnartz Senior Director, Connectivity Systems and Networks, Philips Research Professor, Eindhoven University of Technology

Preface

Wireless connectivity has truly become ubiquitous over the last few years, as can be concluded from the increasing number of electronic devices that are being equipped with wireless communication capabilities. While several years ago the application of wireless was largely limited to mobile and portable telephones, the application range has currently been extended to a wide variety of consumer lifestyle areas. Examples of new products currently being equipped with wireless connectivity solutions are music players, game consoles, streaming audio and video solutions, computer peripherals and cameras. This causes wireless systems to move towards the commodity markets, where price pressure is high. This creates a drive for low-cost solutions. In parallel, there is a persistent demand for increasing performance of wireless solutions, since the same throughput and quality of service are demanded as that of the wired solutions they are replacing. To meet these requirements, wireless communication systems are continuously applying wider bandwidths, larger signal dynamics and higher carrier frequencies. This results in an ever increasing demand on the performance of radio frequency (RF) front-ends, which at the same time have to be low-cost and power-eﬃcient. Since the RF technology is, consequently, pushed to its operation boundaries, the intrinsic imperfections of the RF IC technology are more and more governing the system performance of wireless modems. Scope This book, therefore, presents a new vision on the design of wireless communication systems. It is motivated by the continuous functional scaling of digital IC technology, which allows for more and more functionality on a limited chip area. The presented approach is based on digital compensation algorithms for the RF impairments, which relieve

xiv

Preface

the speciﬁcations of the analogue front-end ICs, at the cost of only a limited increase in area of the digital baseband IC. To illustrate this approach, the book focuses on multiple-antenna orthogonal frequency division multiplexing (MIMO OFDM), which will be applied as the basis for the majority of near-future high-rate wireless systems. First the book introduces the basics of MIMO OFDM and presents the typically applied signal processing in implementations of such systems. Subsequently, several RF front-end impairments are treated, including phase noise, IQ imbalance and nonlinearities. The book presents baseband equivalent models for these RF impairments and, to provide an in-depth understanding of their impact, derives analytical performance results. These results are then used to design diﬀerent compensation approaches based on digital baseband processing. The book was written for wireless system designers, focussing on baseband and/or analogue front-end design, who want to familiarise with the digital compensation of RF imperfections. Another goal was to provide researchers in the ﬁeld of wireless communications or signal processing an overview of this emerging research topic. Acknowledgements My interest in the topic of RF impairments was initiated in the period 2001-2002, when I was performing the research for my M.Sc. thesis within Agere Systems in Nieuwegein, The Netherlands. There I observed several challenges in the development activities of Agere’s wireless local-area-network solutions, which could be attributed to the inherent imperfections of RF front-ends. Many valuable discussions with my colleagues at Agere and prof. Gert Brusaard resulted in the deﬁnition of a Ph.D. project focussing on digital compensation of RF front-end impairments in 2002. This project was carried out within the wireless systems research department of Agere Systems and the radio communications chair of the Eindhoven University of Technology (TU/e). I want to thank prof. Gert Brussaard, Jan Kruys and Bruce Tuch for arranging funding and providing the possibility to perform my research in an interesting environment. The publications and Ph.D. thesis resulting from this project are the basis for this book. I also thank my supervisors at Agere, dr. Allert van Zelst and dr. Xiao-Jiao Tao, and my supervisors at TU/e, dr. Peter Smulders and prof. Erik Fledderus, for the inspiring cooperation and their feedback on previous versions of the texts incorporated in this book. Parts of the work presented in this book resulted from valuable cooperations with Bas Driesen, Isabella Modonesi, Paul Mattheijssen, dr. Guido Dolmans,

Preface

xv

prof. Remco van der Hofstad, C´edric Dehos and dr. Dominique Morche. Their contributions are highly appreciated. Furthermore, I would like to acknowledge prof. Heidi Steendam, prof. Jaap Haartsen, prof. Remco van der Hofstad, prof. Leo Ligthart and Haibing Yang for their thorough reviews of my Ph.D. thesis, which was used as the basis for this book. I thank prof. Jean-Paul Linnartz for stimulating me to consider publishing this work as a book and for writing the foreword. The contributions, support and feedback of many colleagues at Agere and TU/e were greatly appreciated. Also, I acknowledge the support of my colleagues at Philips Research, which helped me to ﬁnish this book. Tim Schenk

Chapter 1 INTRODUCTION

1.1

Wireless communications

The unguided transmission of information using electromagnetic waves at radio frequency (RF) is often referred to as wireless communications, the ﬁrst demonstration of which took place at the end of the 19th century and is attributed to Hertz. The technology was, shortly thereafter, commercialised by, amongst others, Marconi in one of the ﬁrst wireless communication systems, i.e., wireless telegraphy. In the ﬁrst half of the 20th century the technology was developed further to enable more than the mere transmission of Morse code. This ﬁrst resulted in unidirectional radio broadcasting and several years later also in television broadcasting. The research into, and development of, bi-directional wireless communication systems was during, and shortly after, the Second World War mainly fuelled by police and military applications. In the 1940s and 1950s the ﬁrst commercial two-way communications products were introduced, including the popular Citizens’ Band radio. It took, however, until the 1970s for mobile cellular communications to be commercially deployed. While ﬁrst deployed as car-phone networks, due to the size of terminals, they were in ten years converted into networks with real portable handhelds. The popularity of these analogue systems, often referred to as ﬁrst generation (1G), was quickly surpassed with the introduction of digital communication systems, like GSM (global standard for mobile communications), in the 1990s [1]. The success of second generation (2G) systems like GSM stirred up research and development eﬀorts in the ﬁeld of digital wireless communications. The results of these activities over the last 20 years can

2

1 Introduction

application area

Broadcasting

WAN

DAB

EDGE

GSM GPRS

DVB -H

HSDPA HSUPA

UMTS

WiMAX

MAN

-T2

ZigBee

10 kb/s

100 kb/s

802.16 d/e

11n

UWB MB -OFDM

Bluetooth

1 Mb/s

OFDM MIMO OFDM

3G -LTE

IEEE 802.11b 11a/g

LAN

PAN

DVB-T

10 Mb/s

100 Mb/s

1 Gb/s

data rate

Figure 1.1.

The wireless communications landscape of today and the near future.

be observed from today’s diverse wireless communications landscape as schematically depicted in Fig. 1.1. This ﬁgure sketches, without the intention to be complete, the major wireless communication standards currently deployed or to be deployed in the near future. They are ordered by means of their data rate versus the application area. Standards still under development are given in dashed lines. The shading is used to indicate whether the systems are based on the multicarrier technique orthogonal frequency division multiplexing (OFDM) or OFDM combined with the multiple-antenna technique multiple-input multiple-output (MIMO), i.e., MIMO OFDM. Figure 1.1 shows that the digital broadcasting standards, i.e., digital audio broadcasting (DAB) and digital video broadcasting (DVB), are all based on OFDM technology. Currently, standardisation is ongoing for the next-generation terrestrial DVB (DVB-T2), which should enable the transmission of multichannel high-deﬁnition television (HDTV). The application of MIMO OFDM is considered a likely solution to fulﬁll this requirement [2]. For the mobile wide-area-network (WAN) communications standards, the long-term-evolution of 3G (3G-LTE), which is currently under development, is very likely to be based on MIMO OFDM [3]. For metropolitan-area-networks (MANs) the IEEE 802.16 standard provides an OFDM-based solution, which is often referred to as WiMAX. The evolution of this standard as IEEE 802.16d/e provides higher data

1.2 OFDM

3

rates, is based on MIMO and OFDM and enables a higher degree of mobility, so that the application area is shifted towards WAN. Wireless local-area-networks (LANs), as mainly deployed in oﬃce and home environments, are predominantly based on the OFDM standard IEEE 802.11a/g. This kind of systems is often referred to as Wi-Fi and achieves peak data rates up to 54 Mb/s. A high data rate extension of Wi-Fi is currently being standardised as IEEE 802.11n, which will also be based on MIMO OFDM. Also for personal-area-networks (PANs) OFDM is chosen as basis. An ultra wideband (UWB) design based on multi-band OFDM (MB-OFDM) was recently standardised as ECMA368 [4] and adopted as the basis for wireless USB systems. MB-OFDM is able to achieve maximum data rates of 480 Mb/s, while future extension will be able to achieve at least the double rate [5], for which MIMO OFDM is also considered a natural technology candidate. Overall, it can be concluded that throughout the application areas OFDM and MIMO are, and will be, forming the basis for high data rate systems. This justiﬁes a careful investigation of systems based on these techniques, the results of which are presented in this book. First, however, a short introduction of these two communication techniques is given in Sections 1.2 and 1.3.

1.2

OFDM

OFDM is a special case of frequency-division multiplexing (FDM). In FDM multiple baseband signals are multiplexed onto multiple carriers to create a composite transmission signal. The frequency spectrum of the resulting signal can schematically be depicted as in Fig. 1.2(a). At the receiver ﬁltering is applied to separate the parallel signals. When a single data signal is demultiplexed onto the diﬀerent carriers, these systems are also referred as multicarrier systems [6, 7]. It can be concluded from Fig. 1.2 that the conventional multicarrier technique is not very bandwidth eﬃcient since it has to apply appropriate separation between the signal bands to allow for receiver ﬁltering. Another disadvantage is that multiple RF carriers are required for up-conversion and multiple carriers and ﬁlters are needed for down-conversion. Both major drawbacks are solved in OFDM, which generates the diﬀerent subcarrier signals in an orthogonal way, as a result of which the total signal bandwidth is much smaller than for conventional FDM. Hence, the total bandwidth is reduced from B1 to B2 , see Fig. 1.2(b), which means more data can be transmitted in the same bandwidth using OFDM. Furthermore, the multicarrier signal can be generated in the baseband part of the transmitter and the signals are separated in the baseband part of receiver using the (inverse) discrete Fourier

4

1 Introduction B1

frequency (a) Conventional FDM

B2

frequency (b) OFDM

Figure 1.2.

Concept of OFDM.

transformation ((I)DFT) [8]. In that way only one RF carrier is necessary for up-conversion and only one RF carrier and one ﬁlter for downconversion. The technical basics of OFDM are further worked out in Section 2.3.2 of this book, which will show the ability of OFDM to deal with multipath channels. For the deﬁnition of the main objectives of this book, it is useful, however, to already mention two major drawbacks of the application of OFDM here: Accurate time/frequency synchronisation is required in OFDM systems to maintain the subcarrier orthogonality; synchronisation errors signiﬁcantly reduce system performance. The time-domain OFDM signal exhibits a high peak-to-average power ratio (PAPR), which results into a high sensitivity to nonlinearities in the transmission chain.

1.3

MIMO

MIMO systems are systems that apply multiple antennas at both TXand RX-side of the transmission link. The term multiple-input multipleoutput refers to the wireless channel which possesses multiple inputs, i.e., the TX antennas, and multiple outputs, i.e., the RX antennas. Basically MIMO applies the spatial dimension to improve performance and/or data rate of a single-user link, as is schematically depicted in Fig. 1.3. The conventional exploitation of the spatial dimension is shown in Fig. 1.3(a), where the diﬀerent TXs apply the same carrier frequency. Now due to path-loss eﬀects the coverage of one TX is limited to the

5

1.3 MIMO

RX2 TX1

TX2

RX

RX1 TX RX3 TX3

(a) Conventional spatial reuse

Figure 1.3.

(b) MIMO spatial use

Concept of MIMO.

sketched cell, in which the corresponding RX is located. By applying appropriate spatial separation between the diﬀerent TXs, the diﬀerent communications links can exist in parallel without interfering each other. This spatial reuse increases the total throughput of the network; in the depicted case it will be three times higher than the average single-link rate. In a MIMO link, as depicted in Fig. 1.3(b), the diﬀerent TX and RX antennas are all placed in a joint TX and RX terminal, respectively. In this case no simultaneous parallel transmissions exist as in Fig. 1.3(a), but the diﬀerent transmissions will interfere with each other (remember that they are applying the same carrier frequency for transmission). Consequently, the diﬀerent RX branches of the RX-terminal will receive linear combinations of the signals transmitted on the diﬀerent TX antennas. When the received signals are independent enough, the transmitted signals can be separated by applying processing to the received signals [9]. This independence of the received signals clearly depends on the MIMO channel, which can achieve this requirement when there exist diﬀerent propagation paths between the diﬀerent TX-RX antenna pairs. In this way, the combination of the MIMO channel and the signal processing eﬀectively creates spatially parallel transmissions and increases, next to the network throughput as for conventional spatial reuse, also the link data rate. Information theoretical research at the end of the 1990s showed that MIMO can improve the data rate by a factor equal to the minimum of the number of TX and RX antennas [10, 11]. With limited bandwidth

6

1 Introduction

allocated to speciﬁc wireless applications by regulations, MIMO techniques provide the potential to achieve high data rate extensions of previously deployed systems, without allocating new spectrum. Therefore, MIMO has since then grown to be one of the major research areas in wireless communications. The developed MIMO approaches can be basically split into two main groups: space time coding (STC) and space division multiplexing (SDM). The ﬁrst is used to increase the coverage area of the wireless network or to decrease power consumption of the terminals and was originally proposed in [12, 13]. In STC the codewords are in the transmitter spread over the diﬀerent TX branches. These systems can work in conﬁgurations where the TX has more antennas than the RX side. A good overview of STC techniques can be found in [14–16]. The second approach, SDM or layered space time, was originally proposed in [17, 18]. Here increase in link data rate is achieved by transmitting independent data streams on the diﬀerent TX branches, which obviously linearly increases the data rate with the number of TX branches, provided that the rank of the MIMO channel is high enough. At the receiver the streams are recovered by appropriate signal processing, a subject to which immense research eﬀort was/is focussed. For these techniques to work properly, generally, the number of RX branches has to be greater than or equal to the number of TX branches. A good overview of the diﬀerent SDM techniques can be found in [14, 16, 19]. The technical basics of MIMO are further worked out in Section 2.3.1 of this book. For the deﬁnition of the main objectives of this book, it is useful, however, to already mention two major drawbacks of the application of MIMO here: MIMO requires multiple TX/RX branches at both sides of the transmission link, which increases the cost of the solution. Since MIMO is an immature technique, the impact of synchronisation errors and system imperfections on system performance is largely unknown.

1.4

RF transceiver impairments

The implementation of a wireless communication transceiver based on the presented MIMO-concept, as depicted schematically in Fig. 1.4, is considered here. The ﬁgure depicts the physical layer (PHY) of the system, as under investigation in this work, which consist of a baseband section and multiple RF front-ends and multiple antennas connected to that. Since the baseband processing is generally digital and the RF frontend processing analogue, they are connected through an analogue-to-

7

1.4 RF transceiver impairments Digital AD/DA

Analogue

.. .

.. . AD/DA

Baseband Section

Figure 1.4.

RF Front - end 1

RF Front - end N

RF and baseband structure of the PHY of a general MIMO transceiver.

digital/digital-to-analogue (AD/DA) convertor, respectively. The baseband section of the PHY is also connected (in the left of the ﬁgure) to the higher layers, e.g., the medium-access-controller layer (MAC). The baseband part of the TX creates the modulated signals for transmission, which are up-converted to RF and ampliﬁed in the analogue front-end. The RX front-end ampliﬁes and down-converts the signals to baseband and the digital baseband part performs detection to the received signals to retrieve the transmitted signals. Although the signal processing in the digital baseband section of the PHY is several orders more complex compared to that of the RF front-end, most of the design eﬀort in system design is required for the RF part. This is explained by the fact that the front-ends have to operate at frequencies and with dynamic ranges that are several orders higher than that of the baseband part. The diﬃculty of meeting these boundary conditions makes tradeoﬀs between diﬀerent performance parameters like power consumption, linearity, gain, cost and noise level unavoidable [20]. Furthermore, process/implementation spread in RF integrated circuits (ICs) can be large, yielding a spread of the performance in the diﬀerent realisations of the same front-end. It is noted that in a MIMO system this can create spread between the diﬀerent front-ends used in one transceiver. While currently most wireless consumer products operate at carrier frequencies of up to 6 GHz, there is an increasing interest in the application of the mm-wave band for high-speed wireless communication purposes [21–23], since several GHz of bandwidth is available around 60 GHz for license-free use. Since current low-cost CMOS-based IC technology is enabling operation at these high frequencies, it is likely that next generation high speed wireless systems will exploit this frequency

8

1 Introduction

band. The process spread and impairments at these high frequencies, however, are much larger. As such, RF impairments will form a key design challenge for this kind of mm-wave systems. Although the RF front-end can thus greatly inﬂuence the system performance, it is generally modelled as almost ideal when evaluating the performance of a communication system. One of the reasons is that the exact modelling and simulation of the RF front-end is extremely involved and time-consuming. Therefore the inﬂuence of the front-ends is mostly modelled as an additive white Gaussian noise (AWGN) source in the RX. Depending on the design choices, design tradeoﬀs and process spread, however, other imperfections of the implementation can dominate the performance of the total system. It is therefore useful to apply baseband equivalent impairment models, which, instead of modelling the actual RF front-end, characterise the inﬂuence of its speciﬁc imperfections on the baseband signal. These models are then easily implemented in the baseband system simulation setup, but can also be applied in analytical studies. As such, the models can be used to derive the performance impact of a speciﬁc impairment and to calculate tolerable levels of the imperfections. Furthermore, they can be used as a basis for the design of digital baseband algorithms, which can be applied to compensate for the inﬂuence of the RF imperfections. For these behavioural baseband models to be of use for these two purposes, it is crucial that they both accurately model the non-idealities and exhibit a minimal number of parameters. A major advantage of these behavioural models is that they do not depend on the speciﬁc front-end design, but do have a direct link with the resulting performance measures of these designs. As such, a wide range of implementations can be evaluated using the same models. The impact of a certain imperfection depends on the applied transmission technology, i.e., MIMO OFDM in this work. Since several studies were performed previously focussing on impaired conventional singleinput single-output OFDM, conclusions from these studies can be used to make an informed decision on the key imperfections to be studied for a MIMO OFDM system. The ﬁrst two major impairments follow from the two drawbacks of OFDM mentioned in Section 1.2. First, loss of orthogonality due to phase noise, which was identiﬁed by several authors [24–26] as a major performance degradation factor. Secondly, the high PAPR of the OFDM time-domain signal creates a high performance sensitivity to nonlinearities, as identiﬁed by several authors [27–29]. Finally, the imbalance between the I and Q path (the real and imaginary signal path) in the RF front-end, creating limited matching, was identiﬁed as a major performance limiting imperfection in several papers [30–32]. In

1.5 Outline of the book

9

MIMO systems, additionally, the parameters of these impairments can vary per TX/RX branch. Since these RF imperfections exhibit a distinct behaviour, i.e., they do not merely induce additive white Gaussian RX noise, there is the possibility to compensate for their inﬂuence by means of digital signal processing, as was mentioned above. For that purpose, however, knowledge about the parameters of the impairments has to be gathered, which might require complex algorithms. Although the computational complexity of such algorithms might have been prohibitive before, the continuous functional scaling of digital IC technology allows for more and more functionality on a limited chip area. The digital compensation can be applied to relieve the speciﬁcations of the analogue front-end ICs, at the cost of only a limited increase in area of the digital baseband IC. Such approaches might especially be promising for system applying high system bandwidths and high carrier frequencies, where the design tradeoﬀs for the analogue part are more strenuous.

1.5

Outline of the book

Sections 1.1 to 1.4 show that MIMO OFDM will be the basis of the majority of (near) future high data rate wireless communication systems. It is, however, also shown that there are some drawbacks to the use of these techniques, which need careful investigation, i.e., synchronisation and RF front-end imperfections. That is why the content of this book is focussed on these two subjects. To reduce the inﬂuence of the RF impairments, mitigation approaches based on digital signal processing were shown to be promising [33–36]. The work is focussed on the major imperfections as identiﬁed above: phase noise, IQ imbalance and nonlinearities. Hence, the main subjects of this book are: synchronisation for MIMO OFDM-based wireless systems, the veriﬁcation of the MIMO OFDM concept by means of a test system, baseband equivalent RF impairment models, applicable in the MIMO OFDM context, the performance impact of RF imperfections in multiple antenna OFDM, mitigation algorithms/approaches for the inﬂuence of front-end impairments, a generalised error model mapping the impact of the key performance limiting non-idealities. The structure of this work is schematically depicted in Fig. 1.5. First the foundation of the book is laid down in Chapters 2 and 3. Chapter 2

10

1 Introduction Chapter 7: A generalised error model

Chapter 4: Phase Noise

Chapter 5: IQ imbalance

Chapter 6: Nonlinearities

Chapter 3: Design and implementation of a MIMO OFDM system Chapter 2: Multiple -antenna OFDM systems

Figure 1.5.

Structure of this book.

introduces a model for the MIMO multipath channel, which is used throughout this book. Subsequently, the basics of MIMO, OFDM and their combination, i.e., MIMO OFDM, are presented. Chapter 3 introduces algorithms for symbol timing, frequency synchronisation and channel estimation for these MIMO OFDM systems. It also presents an implementation of a test system applying the proposed algorithms, together with results from performance measurements. In the following chapters this basis is assumed and the focus is on RF front-end impairments. Chapters 4 to 6 treat the three identiﬁed major imperfections for MIMO OFDM, i.e., phase noise, IQ imbalance and nonlinearities, respectively. The organisation of these chapters comprises three parts: modelling of the imperfections and system, derivation of the impact of the impairments on system performance and the design of compensation approaches for the inﬂuence of the non-idealities. The diﬀerent impairments are brought together in Chapter 7, which proposes a generalised error model for their aggregate impact.

Chapter 2 MULTIPLE-ANTENNA OFDM SYSTEMS

2.1

Introduction

It was highlighted in the general introduction of this book that the application of multiple antennas in transmitter (TX) as well as receiver (RX) of wireless systems provides the possibility to exploit the degrees of freedom provided by the spatial multiple-input multiple-output (MIMO) wireless channel. These systems are generally named after the channel used for the transmission, i.e., MIMO systems. To overcome the frequency selectivity and time dispersion introduced by the wideband multipath propagation channel, the combination of MIMO with the multicarrier technique orthogonal frequency division multiplexing (OFDM) is considered in this book. It was shown in Chapter 1 that this combination, i.e., MIMO OFDM, is also considered as the basis for several next-generation high data rate wireless systems. In the next chapters diﬀerent aspects of such multiple-antenna OFDM systems are considered. Since the properties of the wireless channel, and the understanding hereof, are crucial for the design of this kind of systems, this chapter introduces a model for the wideband MIMO wireless channel. This is followed by a review of the basics of MIMO, OFDM and MIMO OFDM. This chapter derives system models for these systems, which will be used throughout this book.

2.2

Channel modelling

To design and evaluate wireless systems, it is important to have an accurate model of the channel experienced in the transmission from TX to RX. To that end, this section introduces a channel model for the MIMO wireless channel. First, in Section 2.2.1, the multipath channel

12

2 Multiple-antenna OFDM systems

is introduced, from which, subsequently, a time-discrete equivalent is derived. In Section 2.2.2, this model is used as the basis for a stochastic channel model, which is applied for design and evaluation purposes in the remainder of the book.

2.2.1

Multipath propagation

Consider the wireless transmission between the nt th TX and nr th RX of a MIMO system. When this system is deployed in a reﬂective environment, e.g., an indoor or microcellular environment, the transmitted signal will experience multipath propagation, as schematically depicted in Fig. 2.1. This implies that the RX receives signals via indirect paths, next to the signal component propagating through the direct path. These indirect paths can experience diﬀerent propagation eﬀects, e.g., reﬂections and diﬀractions. The signals propagating via these indirect paths will be delayed, attenuated and phase shifted relatively to the signal received via the direct path. To reveal the eﬀect of this multipath propagation on the received signals, it is useful to have an expression for the baseband equivalent of the transfer of the multipath channel at radio frequency fc . It can be shown that the impulse response of the channel between the nt th TX and nr th RX at time instant t and propagation delay τ is well modelled by [37, 38]:

Nm (t)

cnr nt (t, τ ) =

αn (t)e−jφn (t) δ(τ − τn (t)),

(2.1)

n=1

.. . TX nt

.. . RX nr

Figure 2.1. Multipath propagation between the nt th TX and nr th RX in a multipath environment.

13

2.2 Channel modelling

where the attenuation, phase shift and delay corresponding to the nth path are given by αn , φn and τn , respectively. The number of multipath components is given by Nm and j denotes the imaginary unit, i.e., j 2 = −1. Furthermore, δ(·) denotes the Dirac delta function. The phase shift φn (t) consists of a component due to free space propagation plus a component θn (t, τ ) due to other phase shifts experienced in the channel, e.g., by reﬂections. It can, thus, be expressed by φn (t) = 2πfc τn (t) + θn (t).

(2.2)

We note that all parameters in the channel model, αn (t), φn (t), τn (t), θn (t, τ ) and Nm (t), are expressed as a function of time, since the channel varies over time. This is caused by movement of either the TX, RX or the objects in their propagation environments. Although this timedependence is generally valid, we will, in the remainder of this work, assume a quasi-static channel. This means that we assume that for a transmission centred around time instant t0 , the channel impulse response cnr nt does not vary. It is noted that this is only valid when the transmission length is short as compared to the coherence time of the channel [37]. The quasi-static channel can at time instant t0 thus be expressed as cnr nt (t0 , τ ) = cnr nt (τ ) =

Nm

αn e−jφn δ(τ − τn ).

(2.3)

n=1

The expression for the channel impulse response in (2.3) allows us to deﬁne some relevant channel parameters, which will be used later on to characterise the channel. When we ﬁrst consider the power delay proﬁle (PDP), which deﬁnes the power of the channel impulse response as a function of the propagation delay, we ﬁnd that it is given by Pnr nt (τ ) =

Nm

|αn |2 δ(τ − τn ) =

n=1

Nm

αn2 δ(τ − τn ).

(2.4)

n=1

When we, subsequently, regard the ﬁrst moment of the PDP, i.e., the mean propagation delay, we ﬁnd that it is given by ∞ τ¯ =

−∞ ∞

N m

Pnr nt (τ )τ dτ

−∞

= Pnr nt (τ )dτ

αn2 τn

n=1 N m

n=1

.

(2.5)

αn2

The root-mean-square (rms) delay spread στ is often used as a measure for the time-dispersiveness of the channel and is given by the square root

14

2 Multiple-antenna OFDM systems

of the second moment of the PDP, which is given by ∞ στ2 =

−∞

N m

Pnr nt (τ )[τ − τ¯]2 dτ ∞ −∞

= Pnr nt (τ )dτ

n=1

αn2 [τn − τ¯]2 N m n=1

.

(2.6)

αn2

It is furthermore useful to derive the frequency-domain response of the channel, which is given by the Fourier transform of the impulse response in (2.3) and expressed as a function of frequency f by ∞ Hnr nt (f ) =

cnr nt (τ )e−j2πf τ dτ.

(2.7)

−∞

The multipath channel as expressed in (2.3) and (2.7) will only be observed by a system with unlimited bandwidth. Any practical system, however, will experience a time-discrete and bandlimited version of this channel model. Due to ﬁltering in both TX and RX and sampling in the receiver, the taps of the experienced channel will be equispaced weighted sums of the actual multipath components, as will be shown below. Let us, therefore, consider the relevant system parameters of the bandlimited link between the nt th TX and nr th RX, as illustrated in Fig. 2.2 [39]. When we deﬁne unt (nTs ) to be the discrete TX signal at sample instant n, we can observe in Fig. 2.2 that it is ﬁltered by the bandlimiting ﬁlter FTX and then transmitted through the multipath channel cnr nt (τ ). At the receiver, the signal experiences additive RX noise wnr (t) and is ﬁltered by the RX ﬁlter FRX . Subsequently, the signal is sampled at rate fs = 1/Ts yielding the sampled RX signal ynr (nTs ). This transmission chain allows us to express the RX signal as a function of the transmitted signal. It is given by ynr (nTs ) =

L max

gnr nt (lTs )unt ([n − l]Ts ) + vnr (nTs ),

(2.8)

l=Lmin

Figure 2.2. Block-diagram of a bandlimited communication link between the nt th TX and nr th RX.

15

2.2 Channel modelling

where vnr (nTs ) is the equivalent of the additive RX noise wnr (t) after RX ﬁltering and sampling. The sample index of the ﬁrst and last tap are given by Lmin = τ1 fs and Lmax = τNm fs , respectively. The ﬂoor operator · and ceil operator · round their argument to the closest lower and higher integer, respectively. The elements of the equispaced bandlimited channel, containing the TX and RX ﬁltering, are given by gnr nt (lTs ) = FRX (lTs ) cnr nt (lTs ) FTX (lTs ) =

Nm

αn e−jφn F(lTs − τn ),

(2.9)

n=1

where F(τ ) denotes the combined response of the TX and RX ﬁlter, i.e., F = FTX FRX , and denotes convolution. The convolution between a and b is deﬁned as t ab= a(τ )b(t − τ )dτ . (2.10) 0

ideal bandpass Here we assume the TX and RX ﬁlters FTX and FRX to be √ ﬁlters, i.e., their combined frequency response is given by Ts rect(f /fs ), where rect(·) is the rect function, which equals 1√for |x| < 1/2 and 0, otherwise. Since the inverse Fourier transform of Ts rect(f /fs ) is given s) by sinc(πtfs ) = sin(πtf πtfs , we can rewrite the bandlimited discrete-time impulse response as gnr nt (lTs ) =

Nm

αn e−jφn sinc (π[l − τn fs ]) .

(2.11)

n=1

It is clear from (2.11) that the taps of this impulse response are weighted sums of the original multipath components. Above we regarded the link between one TX/RX pair, but in a MIMO system an RX simultaneously receives signals originating from all Nt TXs. For the nr th RX branch the noiseless received signal can be expressed as ynr (nTs ) =

Nt L max

gnr nt (lTs )unt ([n − l]Ts )

nt =1 l=Lmin

=

Nt L−1

gnr nt (lTs )unt ([n − l]Ts ),

(2.12)

nt =1 l=0

where in the last step it was assumed that the delay of the ﬁrst arriving path is zero, i.e., Lmin = 0, and where L = Lmax − Lmin + 1. We will keep using this assumption throughout this book.

16

2 Multiple-antenna OFDM systems u(n)

z−1

z−1 G (1)

G (0)

···

z−1

u(n − L + 1)

G(L − 1) ···

y(n)

Figure 2.3.

Tapped delay line channel model.

When we want to simultaneously express (2.12) for all Nr branches, (2.12) can be written in matrix notation as a function of the sample index as y(n) =

L−1

G(l)u(n − l),

(2.13)

l=0

where y(n) = [y1 (n) y2 (n) . . . u(n) = [u1 (n) u2 (n) . . . ⎡ g11 (l) g12 (l) ⎢ g21 (l) g22 (l) ⎢ G(l) = ⎢ .. .. ⎣ . .

yNr (n)]T , uNt (n)]T , ... ... .. .

g1Nt (l) g2Nt (l) .. .

⎤ ⎥ ⎥ ⎥, ⎦

(2.14) (2.15)

(2.16)

gNr 1 (l) gNr 2 (l) . . . gNr Nt (l) and where T denotes the matrix transpose. It is clear from (2.13) that the transmission through the time-discrete equivalent channel model can, without loss of information, be modelled as a tapped delay line. This is schematically illustrated in Fig. 2.3, where z −1 denotes a delay of one sample period.

2.2.2

Stochastic channel model

For the time-discrete channel model, as presented in (2.13), to be useful for system design and evaluation, a wide range of realisations has to be generated, on which a model of the possible channels for the regarded environment can be based. For that purpose channel measurements [40] or ray tracing simulations [41] can be applied. Although these techniques can provide reliable results, they are cumbersome and time-consuming to perform. Especially since for every environment separate measurements/simulations have to be performed.

2.2 Channel modelling

17

Therefore, the use of a stochastic channel model was chosen for the research presented in this work. This approach enables the generation of representative channel realisations on the basis of only a few parameters characterising a certain environment. Here we will regard the PDP, path loss, rms delay spread, fading distribution and spatial correlation as suﬃcient parameters to accurately model the channel. If we regard (2.13) as the basis of the channel model, then the (nr ,nt )th element of G(l) is given by (2.17) gnr nt (l) = Pr Pd (l)βnr nt (l), where Pr models the average path loss of the MIMO channel, Pd (l) is the normalised PDP and βnr nt (l) models the fading of the channel taps. Note that Pr and Pd (l) are here deﬁned to be equal for all channel elements, since the path loss is a function of the distance, which is almost equal for all TX/RX pairs, and since the PDP is environment dependent. When we ﬁrst consider the normalised PDP, we can use the fact that signals that arrive with higher delays in general will have travelled longer paths and that their attenuation will, therefore, on average also be higher. Hence, it is reasonable, for both indoor and outdoor channels, to model the PDP as exponentially decaying as a function of the excess delay [42]. The time-discrete version of the normalised PDP can then be expressed as e−lTs /στ L−1 for στ > 0, −lTs /στ Pd (l) = (2.18) l=0 e 1 for στ = 0, where στ is the rms delay spread, which (like the PDP) is equal for all branches. The fading term βnr nt (l) is here modelled to have a standard complex normal distribution, i.e., CN (0, 1). For this fading model it is easily veriﬁed that the resulting phase and amplitude of βnr nt (l) are distributed according to a uniform and Rayleigh distribution, respectively. This model for the fast fading is commonly used for indoor and non-line-ofsight (NLOS) outdoor environments [37]. Another important parameter in MIMO channels is the correlation between the elements of the channel matrix G(l). It has been shown in several publications over the last few years that the spatial channel correlation can be split into correlation at the TX and correlation at the RX [43–45]. The main assumption here is that the correlation is dominated by propagation eﬀects that occur in the vicinity of the TX and RX. The channel matrix can then be expressed as T G(l) = RRX (l)Gi (l) RTX (l) , (2.19)

18

2 Multiple-antenna OFDM systems

where the elements of the matrix Gi (l) are independent identically distributed (i.i.d.) and generated according to (2.17). RTX (l) and RRX (l) model the TX and RX correlation matrices for the lth time lag of the channel, respectively. They are deﬁned as H RTX (l) = RTX (l) RTX (l)

T H = E gj∗ (l)gj∗ (l) (2.20) for j = 1, . . . , Nr , H RRX (l) = RRX (l) RRX (l)

T H for j = 1, . . . , Nt , = E g∗j (l)g∗j (l) (2.21) where here gj∗ (l) and g∗j (l) denote the jth row and jth column of G(l), respectively. Here H denotes the conjugate or Hermitian transpose and E(·) denotes the expected value. The correlation models can be found from measurements, but also in the literature diﬀerent correlation models have been proposed. Here we will use the approach presented in [19, pp. 42-52], where the correlation matrices are assumed to be equal for all taps of the impulse response, i.e., independent of l. Furthermore, this approach reduces the inﬂuence of spatial correlation to a single parameter, and the correlation matrices can be expressed as ⎡ ⎤ x −1 1 ρX ρ2X . . . ρN X ⎢ ⎥ .. . ⎢ ρX ⎥ 1 ρX . . . ⎢ ⎥ ⎢ ⎥ .. 2 RX = ⎢ ρ2 , (2.22) . ρX 1 ρX ⎥ X ⎢ ⎥ ⎢ ⎥ .. .. .. .. ⎣ . . . . ρX ⎦ Nx −1 2 ρX . . . ρX ρX 1 where X ∈ {TX, RX}, Nx ∈ {Nt , Nr } and 0 ≤ ρX ≤ 1 (see [19] for more details). It is easily veriﬁed that for ρX = 0 the correlation matrix RX equals the identity matrix and the channel elements are uncorrelated. For ρX = 1, in contrast, the channel elements are fully correlated. The stochastic channel model, as proposed in this section, will be applied for design and evaluation purposes throughout the remainder of this book.

2.3

System modelling

This section treats the basics of systems combing multiple-antenna techniques with orthogonal frequency division multiplexing (OFDM) and introduces a system model, which facilitates the derivations in the remainder of this book. Since MIMO was initially proposed for ﬂatfading channels, ﬁrst the application of multiple-antenna processing

19

2.3 System modelling

for narrowband systems is regarded in Section 2.3.1. Subsequently, Section 2.3.2 treats the basics of the multicarrier technique OFDM. The combination of these two techniques, i.e., MIMO OFDM, is then discussed in Section 2.3.3.

2.3.1

MIMO

Let us consider a system applying multiple antennas at both transmitter and receiver side of the wireless link, as schematically depicted in Fig. 2.4. The TX and RX apply Nt and Nr spatial branches, respectively. In the transmitter the vector containing the data bits, i.e., b, is, depending on the transmission scheme, encoded, punctured and interleaved. The resulting bit-stream is converted into Nt parallel streams by the serial-to-parallel convertor (S/P). The Nt streams are modulated to form the Nt ×1 transmit vector u. This baseband signal is up-converted to radio-frequency (RF) using the parallel TX RF front-ends, which are for now assumed to be ideal. It is noted that the front-ends convert all TX signal to one common carrier frequency fc . By the Nt antennas the RF signal are transmitted through the wireless MIMO channel, as described in Section 2.2. At the receiver the signal are obtained by the Nr co-located antennas. Subsequently, the signals are down-converted to baseband by the RX RF front-ends. The resulting Nr ×1 baseband signal vector is denoted here by y. MIMO processing is applied to the received signal vector ˜ is y to ﬁnd an estimate of the transmit vector u. This estimate u then demodulated, parallel-to-serial (P/S) converted, deinterleaved and ˜ the estimate of the transmitted data. decoded achieving b, When we assume a perfectly synchronised system, where the up- and down-conversion to/from RF are ideal, the received baseband signal can

u

b

Coding

Decoding

S/P

Nt .. .1

Nt .. .1 P/S

˜ b

Modulation

Nt .. . 1

TX RF Frontend

Nr Nt Demodu- . . . 1 MIMO . . . 1 detector lation ˜ u

Figure 2.4.

TX N t .. . TX 1

RX RF Frontend

y

Single carrier MIMO system model.

RX Nr .. . RX 1

20

2 Multiple-antenna OFDM systems

be expressed as in (2.13). Now we additionally assume that the channel can be regarded as frequency ﬂat over the system bandwidth, i.e., all spectral components of the transmitted signal experience approximately equal gain and linear phase. In this case, the length of the time-discrete baseband equivalent channel model L = 1 and the expression for the received signal at sample time n in (2.13) can be simpliﬁed to y(n) = G · u(n),

(2.23)

where G = G(0) is the Nr ×Nt complex channel transfer matrix. In any practical system, however, the system will experience thermal RX noise, which is well modelled by a zero-mean additive complex normal distributed term. The expression for the RX signal in (2.23) is then converted to y(n) = G · u(n) + v(n).

(2.24)

The MIMO processing (“MIMO detector” in Fig. 2.4) can now be designed such that it is optimal for the combination of coding, interleaving and modulation applied in the TX. In this way, a system can be designed that can optimally exploit the multiple branches to achieve higher data rates and/or higher performance. Furthermore, there is a trade-oﬀ between the complexity of the used MIMO processing technique and the achieved performance. The most basic MIMO detectors are based on least-squares estimation, a.k.a. zero-forcing (ZF), which basically multiply the received signal y with the inverse of the channel matrix. The estimate of transmitted TX stream after ZF processing is given by ˜ (n) = G† y(n) u = u(n) + G† v(n),

(2.25)

where † denotes the pseudo-inverse operation. The pseudo-inverse of matrix A is deﬁned as A† = (AH A)−1 AH .

(2.26)

We note that the estimate in (2.25) contains an error term due to the additive white Gaussian noise (AWGN) vector v. The advantage of ZF is that the complexity is low, but the disadvantage is that noiseenhancement can occur in fading channels, due to ill-conditioning of the pseudo-inverse of the channel matrix [46]. This problem does not occur in the minimum mean squared error (MMSE) based detector, which regularises the channel pseudo-inverse with a term depending on the

21

2.3 System modelling

signal-to-noise ratio (SNR). This results in performance improvements for low SNR values. This comes, however, at the cost of a small increase in complexity and the need for knowledge about the experienced SNR. In contrast to linear techniques like ZF and MMSE, which estimate all TX streams simultaneously, techniques based on successive interference cancellation (SIC) detect TX signal in a serial way. Generally a technique called optimal ordering is applied, which ﬁnds the TX streams with the lowest error variance. Then using a linear technique, like ZF or MMSE, this TX signal is detected. The detected signal is remodulated and multiplied with the channel matrix and then subtracted from the received signal. Subsequently, the optimal ordering is applied again for the remaining TX streams and the procedure is repeated until the last TX stream is detected. Note that in every step the order of the problem is reduced by one. The performance improvement of these techniques over the linear detectors is signiﬁcant, however, this comes at the cost of an increased complexity. A popular technique combining SIC with MMSE was proposed in [47] and named Vertical Bell Labs Layered Space-Time (V-BLAST). The theoretically optimum performance in terms of bit-error rate (BER) versus SNR is achieved by the maximum-likelihood detector (MLD). This method performs a likelihood search over all possible TX vectors u. The maximum-likelihood estimator of the TX vector is given by ˜ (n) = arg min y(n) − G{u}j u j

for j = 1, . . . M Nt

(2.27)

where {u}j denotes the jth out of M Nt possible TX vectors. M is the size of the constellation used for transmission of the signal and · denotes the vector (or L2 ) norm. Clearly, the complexity of this search grows exponentially with the number of TX branches Nt , which is the reason that the application of MLD only seems feasible for low M and Nt . Much research has been devoted to complexity reduction and performance enhancement of MIMO detection. These subjects are, however, beyond the scope of this book and the reader is referred to [14, 16, 19] for a more in depth study of the MIMO processing. Note that it was assumed in this section that the channel is perfectly known at the RX. In general this will not be the case and estimation of the MIMO channel matrix has to be carried out. This subject will be treated in Section 3.4.

22

2.3.2

2 Multiple-antenna OFDM systems

OFDM

It can be concluded from (2.13) that when the symbol time is longer than the maximum time delay of the multipath channel, i.e., L > 1, leakage of neighbouring symbols in the desired symbol will occur. This eﬀect is often referred to as inter-symbol interference (ISI). The analogy in the frequency domain is that the channel can not be regarded as essentially frequency ﬂat over the system bandwidth. To overcome these eﬀects the use of multiple carriers was proposed. In these techniques the whole system bandwidth is subdivided into several parallel non-overlapping narrow subbands, so that the symbol time is increased. When this is implemented in the analogue domain it requires multiple carriers for frequency conversion and steep bandpass ﬁlters to separate the non-overlapping subchannels, which would result into a costly solution. Therefore, eﬃcient implementations in the digital part of the wireless system were proposed. The most applied version of these techniques is based on the discrete Fourier transform (DFT) and named orthogonal frequency division multiplexing (OFDM), the basics of which will be treated below. The concept of using the DFT as part of the digital modulation and demodulation in TX and RX part of the wireless system to achieve parallel data transmission was proposed in the early seventies by Weinstein and Ebert in their seminal paper [8]. Due to the properties of the DFT, the subchannels are shaped like sinc(x). An example of the spectra of three OFDM subcarriers is shown in Fig. 2.5, which shows that these spectra are partly overlapping, signiﬁcantly increasing the spectral efﬁciency as compared to conventional non-overlapping multicarrier systems, see Fig. 1.2. It is clear, however, from Fig. 2.5 that the separation of the carriers can not be carried out by bandpass ﬁltering. Therefore, baseband processing is applied which exploits the orthogonality property of the subcarriers. This property is apparent from Fig. 2.5, where at the maximum of one subcarrier all other carriers have a zero amplitude. The same OFDM signal is depicted in the time-domain in Fig. 2.6. From this ﬁgure it can be concluded that the symbol length contains {1, 2, 3} periods of the signal on the diﬀerent carriers, respectively. Here the number of periods depends on the position of the subcarrier within the OFDM spectrum. To increase the robustness of the OFDM system against ISI, caused by multipath propagation, the addition of a cyclic extension of the symbols was proposed in [8]. To that end the symbol length is prolonged for Ng samples with a guard interval, which basically preﬁxes a copy of the last Ng samples to the start of the OFDM symbol. When Ng is chosen suﬃciently large compared to the channel length, the ISI is contained in the cyclic preﬁx (CP) of the symbol. Since it is

23

amplitude

2.3 System modelling

frequency

amplitude

Figure 2.5.

Spectra of three subcarriers forming an OFDM signal.

Ng

Nc

time

Addition of CP

Figure 2.6. OFDM signal in the time-domain, showing the addition of the cyclic preﬁx (CP).

redundant information it can be disregarded at the RX, removing the inﬂuence of ISI. Since the addition of a CP decreases the eﬀective data rate of the system, the ratio between the number of carriers Nc , which is equal to the symbol length in samples, and the CP length Ng is an important design parameter. It must be chosen in a tradeoﬀ between ISI robustness and eﬀective data rate. The basic OFDM processing in both TX and RX is summarised in the block diagram of Fig. 2.7. First baseband processing is applied to the input bitstream, e.g., interleaving, channel coding, puncturing and mapping to complex symbols. The complex signal is then, after being demultiplexed (S/P), fed to the inverse DFT (IDFT), which converts the Nc ×1 signal vector s to the time domain. Subsequently, a CP of Ng samples is added to the signal, yielding the Ns ×1 TX baseband vector u, where the total symbol length equals Ns =Ng +Nc . Then the signal

24

2 Multiple-antenna OFDM systems s

S/P

u

0 . ..

IDFT

Nc − 1

P/S

.. .

0

Nc − 1 x

0 . ..

TX

Add CP P/S

TX RF Front -end

CP Removal, S/P

RX RF Front -end

RX

Nc − 1

DFT

.. .

0

Nc − 1

Figure 2.7.

y

OFDM system model.

is converted to the analogue domain and up-converted to RF fc and transmitted through the wireless (multipath) channel. The received signal is down-converted to baseband by the RF RX front-end, yielding the Ns ×1 vector y. The output of the analogue-todigital converter (ADC) is then passed to the RX baseband processing. This processing removes the CP, which annuls the inﬂuence of the ISI. The DFT processing then separates the signals on the diﬀerent carriers, yielding the Nc ×1 vector x. The multiplexed (P/S) data stream is then processed in the remaining RX processing, which removes the inﬂuence of the channel and applies decoding and deinterleaving. If we write out the OFDM processing in matrix notation, then the mth transmitted baseband symbol vector is given by um = ΘF−1 sm ,

(2.28)

where sm denotes the mth TX symbol vector before the IDFT-operation. The Nc ×Nc Fourier matrix is here denoted by F, and is deﬁned by ⎡ ⎤ 1 1 1 ... 1 ⎢ 1 W2 . . . W (Nc −1) ⎥ W1 ⎢ ⎥ ⎢ W2 W4 . . . W 2(Nc −1) ⎥ F=⎢ 1 (2.29) ⎥, ⎢ .. ⎥ .. .. .. .. ⎣ . ⎦ . . . . 1 W (Nc −1) W 2(Nc −1) . . . W (Nc −1)

2

2π ). The addition of a CP of Ng samples is modelled where W = exp(−j N c in (2.28) by multiplication with the Ns ×Nc matrix Θ, which is given by

Θ=

T 0 INg

T INc

,

(2.30)

25

2.3 System modelling

where IN denotes the N -dimensional identity matrix and 0 is the allzeros matrix. Using (2.13), the mth symbol vector of the received signal y is given by ˘ m + Gu ˆ m−1 + vm , ym = Gu (2.31) where the ﬁrst term models the desired component, the second term is the ISI term and the AWGN is modelled in the third term. The Ns ×Ns matrices modelling the inﬂuence of the multipath channel are given by ⎡

g(0) .. .

0

.. ⎢ . ⎢ ˘ =⎢ g(0) G ⎢ g(L − 1) . . . ⎢ . .. ⎣ .. . 0 g(L − 1) . . . g(0) ⎡ ⎢ 0 ˆ =⎢ G ⎢ ⎣ 0

g(L − 1) . . . g(1) .. .. . . 0 g(L − 1)

⎤ ⎥ ⎥ ⎥ ⎥, ⎥ ⎦

(2.32)

⎤ ⎥ ⎥ ⎥, ⎦

(2.33)

0

where g(l) was deﬁned in Section 2.2. We note that when the channel length is longer than the symbol length Ns , i.e., L > Ns , an ISI term from the (m − 2)th symbol should be added to (2.31). In practice this will, however, not be the case, since the CP is generally designed to be longer than the channel, i.e., Ng ≥ L. The mth received frequency-domain symbol vector can, subsequently, be expressed as xm = FΥym , (2.34) where F models the Fourier matrix as deﬁned in (2.29) and the Nc ×Ns matrix Υ models the removal of the CP and is deﬁned as Υ = [0 INc ] .

(2.35)

If we then substitute (2.28) and (2.31) into (2.34) and assume that L ≤ Ng , i.e., the ISI is fully contained within the GI, then we ﬁnd that ˘ m + Gu ˆ m−1 + vm xm = FΥ Gu −1 ˘ sm + FΥvm = FΥGΘF −1 ˘ = FΥGΘF sm + nm ,

(2.36)

26

2 Multiple-antenna OFDM systems

˘ is where nm is the frequency-domain AWGN. The Nc ×Nc matrix ΥGΘ a circulant matrix and given by ˘ =G ˘ +G ˆ ΥGΘ ⎡ g(0) 0 g(L − 1) . . . g(1) .. .. ⎢ .. .. . . ⎢ . . ⎢ ⎢ g(L − 2) . . . g(0) g(L − 1) =⎢ ⎢ g(L − 1) ... g(0) 0 ⎢ ⎢ . . .. .. ⎣ 0 g(L − 1) . . . g(0) 0

⎤ ⎥ ⎥ ⎥ ⎥ ⎥. ⎥ ⎥ ⎥ ⎦

(2.37) Here we can use the property that the eigenvectors of circulant matrices are given by the Fourier basis exp(jkl/Nc ) [48, pp. 141-142]. Consequently, (2.37) can be rewritten as ˘ = F−1 HF, ΥGΘ

(2.38)

˘ on its diwhere H is a diagonal matrix with the eigenvalues of ΥGΘ agonal. These eigenvalues are equal to the frequency response of the channel and the kth diagonal element of H is given by H(k) =

L−1

g(l)e−j

2πkl Nc

.

(2.39)

l=0

Using (2.38), the expression for the received signal in (2.36) can be rewritten as xm = Hsm + nm , (2.40) which can also be written as ⎤ ⎡ ⎡ H(0) xm (0) ⎥ ⎢ ⎢ .. .. ⎦=⎣ ⎣ . . xm (Nc − 1)

0

0

⎤⎡

sm (0) .. .

⎤

⎡

nm (0) .. .

⎤

⎥⎢ ⎥ ⎢ ⎥ ⎦⎣ ⎦+⎣ ⎦. H(Nc − 1) sm (Nc − 1) nm (Nc − 1) (2.41)

It is clear from (2.41) that, since H is diagonal for this perfectly synchronised case, the subcarriers can be considered as orthogonal. Hence, under the applied reasonable assumptions, i.e., L ≤ Ng and L ≤ Nc , all subcarriers experience a ﬂat fading channel.

2.3.3

MIMO OFDM

It was noted in Section 2.3.1 that MIMO was initially proposed for narrowband systems, and that this is why its application was in that

27

2.3 System modelling s Nt .. .1

Nt .. .1

Nt .. S/P

. 1 .. 0 .

u .. IDFT

. .. .

Nc − 1 Nt MIMO . .. . 1 detector . . . .. P/S .. 0 . Nc − 1 ˜s

x

Figure 2.8.

Nt Add CP . . . 1 P/S

TX RF Frontend

Nr .. Nr .1 CP .. 0 . DFT Removal, . 1 .. S/P Nc − 1 y

TX Nt .. . TX 1

RX Nr .. . RX 1

RX RF Frontend

MIMO OFDM system model.

section illustrated for a frequency ﬂat channel. The assumption that the channel impulse response length is short compared to the symbol length, i.e., L > 1, will no longer be valid when we want to apply MIMO to wideband systems to achieve high data rates. To retrieve the transmitted signal in such a wideband MIMO single-carrier (SC) system, a multi-tap time-space equaliser could be applied, see e.g. [49, 50], which, however, imposes a high computational burden. Therefore, the combination of MIMO with OFDM, i.e., MIMO OFDM, was proposed to overcome the problems with the frequency-selective channel. This is since OFDM subdivides the frequency-selective channel in parallel frequency ﬂat subchannels, as was shown in Section 2.3.2. For these subchannels the MIMO techniques discussed in Section 2.3.1 can be applied. A block diagram of such a multiple-antenna OFDM system is depicted in Fig. 2.8. The regarded system uses Nt TX and Nr RX branches and applies Nc subcarriers. In the TX, the Nt encoded, punctured, interleaved and modulated symbols streams are demultiplexed (S/P) to form the Nt Nc ×1 MIMO OFDM symbol vector s. This vector is transformed to the time domain using the IDFT and a CP is added for ISI robustness, yielding the Nt Ns ×1 baseband transmit vector u. The signals are subsequently up-converted to a common RF fc using the Nt RF front-ends and transmitted through the wireless MIMO multipath channel. At the RX, this signal is received on Nr RX antennas and downconverted to baseband yielding the Nr Ns ×1 baseband received signal vector y. After removal of the CP, the signal is transformed to the frequency domain by the DFT, yielding the Nr Nc ×1 frequency-domain RX vector x. Then, for every subcarrier, MIMO processing is applied to separate the signals originating from the diﬀerent TX branches. The

28

2 Multiple-antenna OFDM systems

resulting Nt Nc ×1 vector ˜s is subsequently multiplexed and afterwards demodulation, deinterleaving and decoding can be applied to yield an estimate of the transmitted data stream. When we combine the expressions obtained for MIMO and OFDM in the previous sections, then we ﬁnd that the mth transmitted symbol vector is given by um = (Θ ⊗ INt )(F−1 ⊗ INt )sm = (ΘF−1 ⊗ INt )sm ,

(2.42)

where ⊗ denotes the Kronecker or direct matrix product. The Kronecker product of the 2×2 matrices A and B is deﬁned by b11 b12 a11 B a12 B a11 a12 . (2.43) ⊗ = A⊗B = a21 B a22 B a21 a22 b21 b22 The TX vector sm is a concatenation of the MIMO signal vectors at the diﬀerent carriers and is deﬁned as sm = [sTm (0)

sTm (1)

...

sTm (Nc − 1)]T ,

(2.44)

where sm (k) is the Nt ×1 TX vector for the kth carrier during the mth symbol period. We note that for Nt = 1 (2.42) reduces to the expression for conventional OFDM in (2.28). In a similar way as the derivation of Section 2.3.2, it is found that the received frequency-domain baseband vector is given by ˘ xm = (F ⊗ INr )(Υ ⊗ INr )G(Θ ⊗ INt )(F−1 ⊗ INt )sm +(F ⊗ INr )(Υ ⊗ INr )vm = Hsm + nm ,

(2.45)

˘ ⊗ INt ) is a block-circulant matrix, which is given where (Υ ⊗ INr )G(Θ by ˘ ⊗ I Nt ) = (Υ ⊗ INr )G(Θ ⎡ G(0) 0 G(L − 1) . . . G(1) .. .. ⎢ .. .. . . ⎢ . . ⎢ ⎢ G(L − 2) . . . G(0) G(L − 1) ⎢ ⎢ G(L − 1) ... G(0) 0 ⎢ ⎢ . . .. .. ⎣ 0 G(L − 1) . . . G(0) 0

⎤ ⎥ ⎥ ⎥ ⎥ ⎥. ⎥ ⎥ ⎥ ⎦

(2.46)

29

2.4 Conclusions

Here G(l) denotes the Nr ×Nt MIMO matrix for the lth time lag of the channel impulse response, as deﬁned in (2.16). Similar to circulant matrices, block-circulant matrices can be diagonalised by the Fourier and inverse Fourier matrix [15, pp. 239-240], yielding a block-diagonal matrix. When we use this property to write out the expression for the Nr Nc ×Nt Nc frequency-domain channel matrix, we yield ⎡ ⎤ H(0) 0 ⎢ ⎥ .. H=⎣ (2.47) ⎦, . 0 H(Nc − 1) where the diagonal block element H(k) models the Nr ×Nt MIMO channel matrix for the kth carrier and is deﬁned by H(k) =

L−1

G(l)e−j

2πkl Nc

.

(2.48)

l=0

It can be understood from (2.45) that for Nt = Nr = 1, the notation simpliﬁes to that of a single-input single-output (SISO) OFDM systems in (2.40). It can, furthermore, be concluded that by the addition of OFDM to MIMO parallel frequency ﬂat subchannels are created. The received signal for the kth carrier can, thus, also be written as xm (k) = H(k)sm (k) + nm (k).

(2.49)

We note that (2.49) exhibits the same structure as (2.24) and that the single-carrier based MIMO processing can now be applied per subcarrier.

2.4

Conclusions

In this chapter a model for the wideband spatial multiple-input multiple-output (MIMO) channel was presented. The basics of, and systems models for, multiple-antenna and multicarrier systems are introduced. It is shown that by the use of MIMO in combination with OFDM the time-dispersive/frequency-selective channel can be converted into parallel frequency ﬂat MIMO channels, enabling the use of MIMO techniques originally proposed for narrowband/single-carrier MIMO systems. The notation introduced in this chapter will be used throughout this book.

Chapter 3 DESIGN AND IMPLEMENTATION OF A MIMO OFDM SYSTEM

3.1

Introduction

In the previous chapter perfect synchronisation of the wireless systems was assumed, which in practical systems will show not to be a valid assumption. Therefore, this chapter considers the design and implementation of a functional multiple-antenna OFDM based system. The main focus is on the digital signal processing based synchronisation techniques to be applied in a typical receiver of such a system. The following chapters will extend this baseline by additionally considering correction algorithms for RF impairments. For the illustration of this design process, we focus on the application of MIMO and OFDM to the ﬁeld of wireless local-area-networks (WLANs), the basics of which are treated in Section 3.2. It is shown how the multiple antenna system can be made backward compatible with already deployed single antenna WLAN stations based on the IEEE 802.11a standard [54]. Subsequently, synchronisation of the MIMO OFDM system is regarded in Sections 3.3 to 3.5. The algorithms are designed to exploit the transmission structures proposed in Section 3.2. First, Section 3.3 proposes a method for frequency synchronisation of the MIMO OFDM system for an unknown fading channel. Following that, the acquisition of an estimate of the MIMO channel matrix is treated for the diﬀerent transmission formats in Section 3.4. Finally, diﬀerent methods for timing synchronisation are introduced in Section 3.5. To illustrate the performance of a practical MIMO OFDM system in realistic propagation scenarios, an implementation of a 3 TXs and

c 2003, 2004, 2006 IEEE. Portions reprinted, with permission, from [51–53].

32

3 Design and implementation of a MIMO OFDM system

3 RXs system is presented in Section 3.6. For the synchronisation of the test system, the algorithms presented in Sections 3.3 to 3.5 were used. Finally, this chapter reports experimental throughput results of the presented system, which were obtained in an oﬃce environment.

3.2

Transmission format and preamble design

To apply MIMO OFDM in the design of a wireless system, more aspects than only the signal detection, as addressed in Section 2.3.1, have to be treated. Choices have to be made about the used (complex) modulation, e.g., M -QAM and M -PSK, the applied coding and interleaving method and scheme. Furthermore, the number of subcarriers Nc and the length of the CP Ng have to be found in a compromise between overhead ratio, complexity and ISI robustness. These subjects are, however, outside the scope of this book. Therefore, and since the ﬁrst application of this work is foreseen in WLANs, the system setup of the OFDMbased IEEE 802.11a standard [54] was chosen as the basis for the MIMO OFDM systems applied in many of the numerical studies in this book. Nonetheless, the proposed techniques in this book are more generally applicable and not limited to this speciﬁc setup. The relevant parameters of the IEEE 802.11a standard are summarised in Section 3.2.1. For a more in depth discussion of the IEEE 802.11a standard the reader is referred to [54–57]. Two other important aspects of the design of a realistic coherent detection based MIMO OFDM system are synchronisation and channel estimation. The former is split up into frequency and timing synchronisation, which will be treated in Section 3.3 and Section 3.5, respectively. The channel estimation, as discussed in Section 3.4, retrieves an estimate of the MIMO channel matrix at the RX, which is then used in the data detection. Generally these tasks are designed such that they rely on the availability of pilot carriers/symbols, i.e., subcarriers or symbols carrying predeﬁned symbols, which are known at the RX. Diﬀerent designs of pilot symbols, which enable synchronisation and channel estimation in a MIMO OFDM system, are proposed in Section 3.2.2. Due to scarcity of spectrum it is likely that MIMO OFDM systems will have to operate in a band that is already used by SISO systems. Section 3.2.2.1 will show that this imposes extra constraints on the packet structure, which will be taken into account in the design of the training sequences.

3.2.1

IEEE 802.11a

To show the extensions needed, due to MIMO, to the IEEE 802.11a physical layer (PHY) design, we here brieﬂy review its basics and frame

33

3.2 Transmission format and preamble design Table 3.1.

Summary of the IEEE 802.11a [54] parameters.

System Parameter

Paramater Value

Modulation Coding Type Coding Rate Coding generator polynomial Bandwidth Number of subcarriers Number of data subcarriers Number of pilot subcarriers OFDM Symbol duration Cyclic Preﬁx length Subcarrier spacing

BPSK, QPSK, 16-QAM, 64-QAM Convolutional 1/2, 2/3, 3/4 (133,171) 20 MHz 64 48 4 4 µs 800 ns 312.5 kHz

structure. The main parameters are summarised in Table 3.1, which shows that the data rate can be varied by changing the convolutional coding rate and the modulation size, yielding data rates varying from 6 up to 54 Mb/s, as shown in the upper half of Table 3.2. In the IEEE 802.11a design, 48 out of the 64 subcarriers, subdividing the 20 MHz signal bandwidth, carry data symbols and 4 subcarriers carry known symbols, i.e., pilot symbols, which are always modulated using BPSK. The remaining 12 carriers do not carry data. One of those subcarriers is located at DC and the other 11 are at the sides of the spectrum to minimise the out-of-band transmission. In this way, the need for steep analogue ﬁlters is prevented. A CP of 16 samples is used, yielding an overhead ratio due to the CP of 20%. Since IEEE 802.11a is based on unscheduled packet transmissions, the RX does not know in advance when it will receive a packet, how long this packet will be and when it will receive the following packet. Therefore, the RX has to detect the beginning of the packet and apply synchronisation and channel estimation at the beginning of every packet, since the TX and RX can be out of sync and since the channel can change between the reception of two packets. For that reason, a piece of known data is preﬁxed to the data part of the packet, i.e., the preamble. The structure of an IEEE 802.11a packet is depicted schematically in Fig. 3.1. The structure consists of 4 parts, where the ﬁrst 3 parts (20 µs) form the preamble and the last part (DATA) is the actual data transmitted in the packet. If we have a closer look at the preamble, then we see that the ﬁrst part (ST part) consists of a ten times repeated short training symbol (ST ) of 16 samples ( 800 ns). This part is in the RX used to detect

34

3 Design and implementation of a MIMO OFDM system 8µs TX

8µs

ST part LT part preamble

4µs SIG

ST ST ST ST ST ST ST ST ST ST CP LT

Figure 3.1.

DATA

LT

LT

CP

SIGNAL

802.11a PHY frame format.

the packet, adjust the automatic-gain-control (AGC) settings and derive a coarse estimate of the carrier frequency oﬀset (CFO). The second part is a twice repeated long training symbol (LT) of 64 samples, with an extended cyclic preﬁx (CP LT) of 32 samples, altogether denoted by the LT part. This part is used for channel estimation, ﬁne CFO estimation and symbol timing. The last part of the preamble is formed by the SIGNAL ﬁeld (SIG), which contains information about the transmission, including the packet length, applied modulation and coding rate.

3.2.2

MIMO OFDM

As mentioned above, numerical studies in this book are often based on a MIMO extension of the IEEE 802.11a standard, which uses the same OFDM parameters as were stated in Table 3.1. These parameters were chosen, since it was shown in Section 2.3.3 that no extra constraints were put on the OFDM design due to MIMO. Another more practical reason to keep the same OFDM parameters is that it enables the MIMO OFDM systems to communicate with the already deployed SISO systems in the same frequency band. This is often referred to as backward compatibility (BWC), a subject which is explained in more detail below.

3.2.2.1 Backward compatibility Since the spectrum is scarce, the proposed MIMO OFDM systems will have to operate in the same frequency band as the already deployed SISO systems, i.e., the IEEE 802.11a system for our test case. To enable eﬃcient co-existence, the MIMO systems have to exhibit some kind of BWC with these existing systems. Although BWC is possible on diﬀerent layers in the OSI stack, it was shown in [58] that it is achieved most eﬃciently on the PHY. That is why we will regard this kind of BWC in the following. The multiple-access scheme of IEEE 802.11a depends on the fact that a station (STA) does not access the wireless medium while another STA, or access point (AP), is transmitting. When a STA detects in the DATA part of the packet that the packet is intended for another STA or when

3.2 Transmission format and preamble design

35

the STA can not decode the packet, it will not transmit data for the time corresponding to the packet length it has detected from the SIGNAL ﬁeld. Hence, it is crucial that in a mixed-mode environment, where SISO as well as MIMO transmissions occur, the SISO terminals can decode the MIMO packets up to the end of the SIGNAL ﬁeld. In this way, the SISO STA can detect the packet duration and will not use the wireless medium for this time period and will, thus, not interfere the MIMO transmission. For the MIMO terminal, there is no problem in detecting and decoding the SISO transmission, as long as the OFDM structure is unchanged. The main requirement for BWC is thus that the SISO terminal can detect the SIGNAL ﬁeld of the MIMO transmission, which has to be equal to the SISO SIGNAL ﬁeld, so it knows the length of the MIMO transmission. This imposes extra constraints on the design of the preamble. It will be illustrated in the remainder of this section how these can be met with an eﬃcient preamble, which also enables synchronisation and MIMO channel estimation. To achieve BWC, the design of the ST part is not crucial, as long as it enables frame detection, coarse CFO estimation and the setting of the AGC. When we assume that the system uses a common oscillator, then it suﬃces to simultaneously transmit the ST part of the IEEE 802.11a standard, as depicted in Fig. 3.1, on all (or even a subset) of the TX branches. The legacy SISO station will now receive the sum of the channel impaired ST parts transmitted from the diﬀerent TX branches. In the remainder we will thus focus on the LT part and SIGNAL ﬁeld, since they were identiﬁed as the most crucial parts in creating a BWC MIMO frame format. The (frequency-domain) Nc Nr ×2 received signal matrix during the LT part of an IEEE 802.11a preamble transmission can be written as Xp = HSp + N, (3.1) where the subscript p refers to the preamble period. Furthermore, the transmitted frequency-domain LT-block for a single-antenna TX is deﬁned in [54], and is given by Sp = [1 1] ⊗ sp , which denotes two consecutive LT symbols sp . We recall that ⊗ denotes the Kronecker product as deﬁned in (2.43). The Nc Nr ×2 noise vector is given by N = [n1 n2 ], where nm denotes the noise vector during the mth LT symbol. From Xp we can draw an estimate of the channel matrix and determine the timing and frequency synchronisation as shown in Section 3.4, Section 3.5 and Section 3.3, respectively. Here we note that for the estimation of the MIMO channel, it is important that the subchannels from the different TX antennas to every RX antenna can be uniquely identiﬁed. To achieve that, the preambles on the diﬀerent TX antennas have to be

36

3 Design and implementation of a MIMO OFDM system TX1 TX2

Figure 3.2.

Sp

SIG

DATA 1 Sp

DATA 2

Time-multiplexed backward-compatible MIMO PHY frame format.

orthogonal and shift-orthogonal for at least the channel length [59]. For the frequency synchronisation a repetition is required, which is straightforwardly achieved when the IEEE 802.11a preamble is used as a basis of the MIMO preamble. In Sections 3.2.2.2 to 3.2.2.6, ﬁve preambles structures are proposed, which all meet the above mentioned prerequisites. The ﬁrst four, which are illustrated in Figs. 3.2 to 3.5, are also well applicable for BWC operation. The ﬁfth, which is not BWC, is illustrated in Fig. 3.6. The ﬁgures illustrate the frame structures for a system applying two spatial streams. We note that the SIGNAL ﬁeld depicted here can be followed by an extra MIMO SIGNAL ﬁeld, which holds extra information about the MIMO transmission, or by an extra ST part used for ﬁne tuning of the AGC settings [58]. This is, however, not considered here.

3.2.2.2 Time-Multiplexed The ﬁrst structure is the time-multiplexed (TM) preamble as depicted in Fig. 3.2. Here the LT symbols are sequentially send from the diﬀerent TXs. The TX that transmits the ﬁrst LT, also transmits the SIGNAL ﬁeld. A SISO RX observes a transmission originating from one TX branch and can thus detect the SIGNAL ﬁeld. If we write the LT part of the preamble as a function of the SISO LT block Sp , then we get Sp,TM = Sp ⊗ INt .

(3.2)

For this structure, (Nt − 1) LT blocks have to be transmitted after the SIGNAL ﬁeld. A clear disadvantage of this TM concept is that during the preamble phase the total TX power is Nt times smaller than during the other parts of the transmission, under the assumption that the output power of a TX branch is ﬁxed. This can be overcome by increasing the TX power of the TX branches during the preamble phase. This is, however, not considered here.

3.2.2.3 Time-Orthogonal Figure 3.3 illustrates the BWC use of time-orthogonal (TO) training. For this training the frame format on all TX branches is equal up to the end of the SIGNAL ﬁeld. This means that the SISO RX eﬀectively

37

3.2 Transmission format and preamble design TX1

Sp

SIG

Sp

DATA 1

TX2

Sp

SIG

− Sp

DATA 2

Figure 3.3.

Figure 3.4.

Time-orthogonal backward-compatible MIMO PHY frame format. TX1

Sp,e

SIGe

DATA 1

TX2

Sp,o

SIGo

DATA 2

Subcarrier-multiplexed backward-compatible MIMO PHY frame format.

sees and estimates a channel which is the sum of the channel elements corresponding to the diﬀerent TX branches. In this way, the legacy SISO station can decode the SIGNAL ﬁeld. Orthogonality, needed for the estimation of the MIMO channel, is achieved by using a WalshHadamard or Fourier matrix, which both are given by [1, 1; 1, −1] for the 2 TX case. We can rewrite the LT part of the preamble as Sp,TO = Sp ⊗ ΦNt ,

(3.3)

where ΦNt denotes the orthogonal matrix. When ΦNt equals the WalshHadamard matrix or the Fourier matrix, (2log2 (Nt ) − 1) or (Nt − 1) LT blocks after the SIGNAL ﬁeld are required, respectively.

3.2.2.4 Subcarrier-Multiplexed The BWC application of the subcarrier-multiplexed (SM) preamble is depicted in Fig. 3.4. Here the preamble symbols are transmitted simultaneously on all TXs, not increasing the overhead compared to a SISO system. On the set of subcarriers {nt −1, Nt +nt −1, . . . , Nc −Nt +nt −1}, TXnt transmits the original training symbol, where we assumed Nc to be a multiple of Nt . For the example in Fig. 3.4, this means that TX1 and TX2 transmit on the even (Sp,e ) and odd (Sp,o ) subcarriers, respectively. The corresponding carriers of the SIGNAL ﬁeld are transmitted by the same TX. The SISO RX sees and estimates the combined (interleaved) channel and can detect the SIGNAL ﬁeld. Note that we assume here that the SISO RX applies conventional per carrier channel estimation in the frequency domain. The LT part of the preamble can now be written as Sp,SM = [1, 1] ⊗ ((sp ◦ θ1 ) ⊗ e1 +(sp ◦ θ2 ) ⊗ e2 + . . . + (sp ◦ θNt ) ⊗ eNt ) , (3.4) where ◦ denotes element-wise multiplication. The element-wise multiplication of the N ×1 vectors x = [x1 , . . . , xN ]T and y = [y1 , . . . , yN ]T is

38

Figure 3.5.

deﬁned as

3 Design and implementation of a MIMO OFDM system TX1

Sp,1

SIG1

DATA 1

TX2

Sp,2

SIG2

DATA 2

Subcarrier-orthogonal backward-compatible MIMO PHY frame format.

⎡

⎤ ⎡ x1 ⎢ ⎥ ⎢ x ◦ y = ⎣ ... ⎦ ◦ ⎣ xN

⎤ ⎡ ⎤ y1 x1 y1 ⎥ .. ⎥ = ⎢ .. ⎦. . ⎦ ⎣ . yN xN yN

(3.5)

The Nc ×1 subcarrier interleaving vector for TXnt is here given by θnt = Cnt −1 {[ Nt , 0Nt −1 , Nt , 0Nt −1 , . . . , Nt , 0Nt −1 ]}T . (3.6) Here 0Nt denotes the 1×Nt all zero vector, the function Cm {·} cyclicly shifts its argument over m samples and em denotes the Nt ×1 vector em = [0m−1 , 1, 0Nt −m ]T ,

(3.7)

of which all elements are zero except the mth, which equals 1.

3.2.2.5 Subcarrier-Orthogonal The subcarrier-orthogonal (SO) preamble, as depicted in Fig. 3.5, also achieves orthogonality in the frequency domain. The long training symbols are multiplied with orthogonal vectors. When the SIGNAL ﬁeld is multiplied with the same vectors, the conventional SISO RX is able to detect the SIGNAL ﬁeld. We note that this concept also does not increase the overhead compared to a SISO transmission. When Fourier sequences are applied as orthogonal sequences, the transmitted LT part of the preamble is given by Sp,SO = [1, 1] ⊗ ((sp ◦ ζ1 ) ⊗ e1 + (sp ◦ ζ2 ) ⊗ e2 + . . . + (sp ◦ ζNt ) ⊗ eNt ) , (3.8) where ζnt equals the (1 + (nt − 1)Nc /Nt )th column of the Fourier matrix. It is noted that this preamble structure corresponds to what is sometimes referred to as a circular shift orthogonality preamble. This is because the training symbols for the nt th antenna are cyclicly shifted over (nt − 1)Nc /Nt samples. This method was previously shown in [59–62] to be optimal for MIMO channel estimation.

3.2.2.6 Frank-Zadoﬀ-Chu codes The preamble structures proposed in Sections 3.2.2.2 to 3.2.2.5 meet the prerequisites of being BWC and allowing for MIMO channel estimation and synchronisation. They are, however, not optimised to have

39

3.3 Frequency synchronisation Np TX1

DATA 1

TX2

DATA 2

Figure 3.6. Frame format for a system applying Frank-Zadoﬀ-Chu codes as the basis for the preamble.

a low overhead, since they are built up out of multiples of the IEEE 802.11a LT part. Furthermore, the peak-to-average power ratio (PAPR) for the SM and SO preamble can be undesirably increased compared to the 802.11a LT part. Chapter 6 will show that higher PAPR will result in worse performance due to nonlinearties in the system. It is, therefore, desirable to design a preamble with low PAPR. When we ignore the BWC requirements, we can design a more eﬃcient preamble structure, which also has a lower PAPR. Here we regard a preamble structure which consists of constant-envelope orthogonal codes with good periodic correlation properties, such as Frank-Zadoﬀ [63] or Chu [64] codes. The Np = N 2 code is a concatenation of the rows of the N -dimensional Fourier matrix, as deﬁned in (2.29). The periodic correlation function Ψ of the code sequence c = [c1 , c2 , ..., cNp ] is deﬁned as Np c{n+i}(modNp ) c∗n . (3.9) Ψ(i) = n=1

where x(mod N ) denotes x modulo N . For the Frank-Zadoﬀ-Chu codes, this periodic correlation yields ⎧ N ⎨ p |cn |2 = Np for i = 0, Ψ(i) = (3.10) n=1 ⎩ 0 for i = 0. The preamble is formed by a repetition of the code, or training symbol, with a diﬀerent cyclic shift applied to it for the diﬀerent transmit antennas. A typical value of the cyclic shift would be Np /Nt . Note that this cyclic shift has to be higher than the CIR length to preserve orthogonality. Altogether, this results in a preamble as depicted in Fig. 3.6 for a system with two TX antennas. Since the preamble is deﬁned in the time-domain, it will also be referred to as single-carrier (SC) preamble.

3.3

Frequency synchronisation

Frequency oﬀsets (FOs) in wireless systems are generally generated by both carrier frequency oﬀsets (CFOs), introduced by mismatch between

40

3 Design and implementation of a MIMO OFDM system

the local oscillators of the TX and RX, and Doppler shifts, caused by movements in the wireless channel. Since the inﬂuence of the latter is marginal for WLAN systems [65], we here focus on the CFO in such systems. Similar to a SISO OFDM system, a multi-antenna OFDM system is very sensitive to FOs, which introduces inter-carrier interference (ICI) [25, 66]. Accurate frequency synchronisation is thus essential for reliable reception of the transmitted data. Although CFO could be regarded as a front-end RF impairment and treated in one of the following chapters, it is included in this chapter since its solution, frequency synchronisation, forms a common task for almost any receiver. As such, frequency synchronisation can be regarded as a basic part of the design of a multiple-antenna OFDM system. Therefore, it ﬁts well in this chapter. Various carrier synchronisation schemes have been proposed for SISO OFDM systems. Some schemes rely on pilot or preamble data [66–68] and some use the inherent structure of the OFDM symbol in either frequency [32] or time domain [69]. For multiple antenna OFDM, dataaided schemes are proposed for receiver diversity and MIMO in [70] and [71], respectively. A blind method for receiver diversity combined with OFDM is proposed in [72]. This section elaborates on ideas from [70] and [71] resulting in a data-aided frequency synchronisation approach for MIMO OFDM systems. In addition to [71], we analytically show the tradeoﬀ between estimation accuracy on the one hand and preamble length and number of antennas on the other hand, which is facilitated by the design of a scalable preamble. Furthermore, we explore the performance gains in multipath environments by means of simulations. The outline of this section is as follows. First, in Section 3.3.1 the inﬂuence of CFO is studied. Then Section 3.3.2 presents a data-aided synchronisation technique, the performance of which is studied analytically in Section 3.3.3. Section 3.3.4, subsequently, compares these analytical ﬁndings with results from Monte Carlo simulations.

3.3.1

Inﬂuence of CFO

Consider a MIMO OFDM system with Nt TX and Nr RX antennas as described in Section 2.3.3. For convenience the expression for the mth Nr Nc ×1 received signal vector for a perfectly synchronised MIMO OFDM system in (2.45) is repeated here, and given by −1 ˘ ⊗ INt )sm + vm xm = (FΥ ⊗ INr ) G(ΘF = Hsm + nm .

(3.11)

41

3.3 Frequency synchronisation s

Nt .. .

u ..

1

0 . .. Nc − 1 Nr .. . 1 0. .. Nc − 1 x

IDFT

. .. .

.. DFT

. .. .

Add CP P/S

..

fc,RX

fc,TX

.

.. CP . Removal, S/P

Nt Nr .. .. . G . 1 RF 1

Freq Sync

..

..

..

.

.

.

··· y

v

t

Figure 3.7. Block diagram for a MIMO OFDM system experiencing CFO and applying frequency synchronisation.

When we now regard a MIMO OFDM system where the RF carriers in the TX and RX are not perfectly synchronised, as illustrated schematically in Fig. 3.7, the expression for the received symbol vector in (3.11) is no longer valid. The TX in the system of Fig. 3.7 uses one common RF oscillator at frequency fc,TX to up-convert the set of baseband signals. After transmission through the RF channel GRF , the baseband equivalent of which is given by G, the signals are down-converted at the RX using a common oscillator at RF fc,RX . When the diﬀerence between the frequency of the TX and RX oscillator is zero, i.e., ∆f = fc,RX − fc,TX = 0, the baseband RX signal is given by (3.11). When ∆f = 0, however, the received signal can be expressed as xm = (FΥ ⊗ INr )Em ym −1 ˘ = (FΥ ⊗ INr )Em G(ΘF ⊗ INt )sm + nm = (G m ⊗ INr ) Hsm + nm ,

(3.12)

where ym = tm + vm and Em = diag(em (0), em (1), . . . , em (Ns − 1)) ⊗ INr denotes the phase rotation due to the CFO. The phase rotation for the nth sample of the mth symbol is given by em (n) = exp (j2π∆f Ts (mNs + n)) ,

(3.13)

where we assumed the phase oﬀset to be zero at the start of the packet. ˘ From (3.12) we conclude that (Υ ⊗ INr )Em G(Θ ⊗ INt ) is not block circulant and can thus not be diagonalised by the DFT and IDFT operations. Instead we can rewrite the term as (G m ⊗ INr ) H, where the

42

3 Design and implementation of a MIMO OFDM system

Nc ×Nc matrix G m in (3.12) models the inﬂuence of the CFO on the received frequency-domain symbols and is given by [66] ⎛ ⎞ γ−1 . . . γ−(Nc −1) γ0 ⎜ γ1 γ0 . . . γ−(Nc −2) ⎟ ⎜ ⎟ Gm = ⎜ (3.14) ⎟, .. .. .. . . ⎝ ⎠ . . . . γNc −1 γNc −2 . . .

γ0

where γq =

π(Nc −1) 2πδ sin(π(δ − q)) ej Nc (δ−q) ej Nc (mNs +Ng ) . π Nc sin( Nc (δ − q))

(3.15)

Here the frequency oﬀset is normalised to the subcarrier spacing and denoted by δ = ∆f Nc Ts . We can conclude from (3.12) and (3.14) that the CFO has two eﬀects. The ﬁrst eﬀect is that the wanted carriers are multiplied with the elements on the diagonal of G m , i.e., γ0 . This causes a rotation and attenuation, which is common to all carriers. The second eﬀect is caused by the other elements of G m , i.e., for q = 0, that introduce cross leakage between the subcarriers. This inter-carrier interference (ICI) is, thus, a weighted sum of the signal carried on all other carriers, where the weight γq decreases with increasing subcarrier distance. This inﬂuence of CFO is illustrated schematically in Fig. 3.8, which depicts the amplitude spectra of three subcarriers. The ﬁgure shows that for perfect synchronisation (the dashed line) all other subcarriers have a zero crossing at the location of the maximum of another subcarrier, as was previously illustrated by Fig. 2.5. Now with a CFO of ∆f (the solid line) it can be observed that the wanted subcarrier is no longer sampled at its maximum value and that the other subcarriers have a non-zero contribution, yielding ICI.

3.3.2

Algorithm description

To reduce these eﬀects of CFO, accurate synchronisation is important, preferably before the reception of the data. Therefore the datapacket is preceded by a preamble, as was described in Section 3.2, which enables the estimation of the CFO. To regain the orthogonality between the MIMO OFDM carriers we have to correct for Em in (3.12) before applying the DFT operation, as shown in Fig. 3.7. Hence, the algorithm has to estimate Em and multiply the received time-domain sequence with E−1 m . From (3.13) it can be concluded that the only unknown parameter in Em is the CFO ∆f , or, equivalently, its normalised version δ. Diﬀerent estimation and correction methods for CFO have been proposed in the literature for conventional (SISO) OFDM systems [66, 69,

43

amplitude

3.3 Frequency synchronisation

frequency ∆f

Figure 3.8. Spectra of three subcarriers forming an OFDM signal under a frequency oﬀset of ∆f .

Np TX 1

Train 1

Train 1

Data 1

TX 2

Train 2 .. . Train Nt

Train 2 .. .

Data 2 .. .

TX Nt

Train Nt

Data Nt

Figure 3.9. Packet structure for a MIMO OFDM system applying the studied CFO estimation.

73]. The most attractive data-aided method was proposed by Moose in [66]. This method estimates δ using the phase of the complex correlation between two consecutive received versions of a repeated training symbol. Here we study an extension of the method of [66] for a MIMO OFDM system. Rather than estimating the CFO on every RX branch separately and then averaging over the diﬀerent estimates, we treat an algorithm that exploits the receiver diversity for the estimation without using knowledge of the MIMO channel. To that end a packet structure is introduced as illustrated schematically in Fig. 3.9. In this structure a training part of 2Np samples is preceding the data transmission. The training part consists of an once repeated preamble sequence, which has to enable MIMO channel estimation, since this preamble is also used for channel estimation, as described in Section 3.4. All preamble formats proposed in Section 3.2 exhibit this structure and, thus, enable the proposed synchronisation.

44

3 Design and implementation of a MIMO OFDM system

Let us now deﬁne the complex correlation Λ between two subsequently received frames of Np samples on the Nr RX antennas as n+Np −1

Λ(n) =

yH (m)y(m + Np ),

(3.16)

m=n

where y(n) denotes the Nr ×1 received signal vector during nth sample of the preamble period. Here Λ(n) denotes the sum of the correlation outputs for the diﬀerent RX branches. The estimate of the frequency oﬀset δ is then found in a similar way to [66] and given by ˜ p) Nc Ts θ(n Nc ∠Λ(np ) δ˜ = = , 2πTp 2πNp

(3.17)

where θ˜ = ∠Λ denotes the phase of the complex correlation Λ between two training symbols, fs denotes the sample frequency and Tp = Np /fs is the training symbol duration. The best instant to estimate this frequency oﬀset is found when |Λ| reaches its maximum [67], i.e., np = arg max |Λ(n)|. n

(3.18)

In a multipath environment this estimator achieves a maximum ratio combining (MRC) like performance, since the contribution of the RX branches to Λ is proportional to the received power on the diﬀerent branches. Clearly the RX branch with the highest instantaneous SNR, when assuming a ﬁxed noise ﬂoor, contributes most to the estimation of δ. This MRC-like performance decreases for lower SNR, since there the contributions are less correlated with the received power due to the inﬂuence of receiver noise. This receiver diversity performance is explored by means of simulation results in Section 3.3.4. Note that the maximum frequency oﬀset that can be estimated with this method is limited, since the angle θ˜ that can be estimated without phase ambiguity is limited to θmax = ±π. This relates to a maximum (normalised) frequency oﬀset of |δmax | = |θmax |Nc /2πNp = Nc /2Np . Larger ranges can be achieved by ﬁrst performing a coarse CFO estimation by preceding the preamble with training symbols with shorter length Np , as done in the IEEE 802.11a standard for SISO OFDM systems, see Section 3.2.1.

3.3.3

Performance analysis

As a measure for the performance of the introduced estimator we ﬁrst analytically study the variance of the estimate for a system experiencing

45

3.3 Frequency synchronisation

a fully orthogonal channel and additive white Gaussian noise (AWGN) at the receiver. The orthogonal channel for the kth carrier is deﬁned for Nr ≥ Nt and built up out of the ﬁrst Nt columns of the Nr -dimensional DFT matrix F, as deﬁned in (2.29). Note that the system experiences no fading and that the channel is frequency ﬂat. This channel will further be referred to as the AWGN channel . The correlation output (3.16) in such an AWGN environment can be written as n+Np −1

Λ(n) =

{t(m) + v(m)}H {t(m + Np ) + v(m + Np )},

(3.19)

m=n

since y(n) = t(n) + v(n). Here t(n) and v(n) are the Nr ×1 received signal and noise vector, respectively, as depicted in Fig. 3.7. The noise causes an error in the phase estimation of the correlation value Λ. The error in the estimation of the phase can be expressed by " −jθ Λ I e −1 θ˜ − θ = tan , (3.20) R (e−jθ Λ) where θ denotes the actual phase rotation due to CFO, and I(·) and R(·) output the imaginary and real part of the input, respectively. When we subsequently derive the normalised error in the CFO estimate ε = δ˜ − δ using (3.17), we ﬁnd that n+N p −1

I {t(m) + v(m)}H {t(m + Np ) + v(m + Np )}e−jθ

ε(n) =

Nc m=n 2πNp n+N p −1 R ({t(m) + v(m)}H {t(m + Np ) + v(m + Np )}e−jθ ) m=n n+N p −1

≈

Nc 2πNp

m=n

I vH (m)t(m) + tH (m)v(m + Np )e−jθ n+N p −1 m=n

,

(3.21)

t(m)2

where, in the second line of (3.21), we made an approximation for high SNR and used that t(n + Np ) = t(n)ejθ . It is easily found that the mean value of ε is zero and that, consequently, the estimator is unbiased. When we subsequently study the normalised error variance, it is shown in Appendix A that it is given by (A.6), which is repeated here for convenience Nc2 var {ε(n)} = E{ε2 (n)} ≈ . (3.22) (2π)2 Nr Np3 ℘

46

3 Design and implementation of a MIMO OFDM system

Here ℘ = σt2 /σv2 = Nt σu2 /σv2 = P/σv2 denotes the SNR per receive antenna and P is the total transmit power. The variance of the elements of u, t and v is given by σu2 , σt2 and σv2 , respectively. Since the proposed estimator is unbiased, the mean squared error (MSE) equals the variance of the estimation error and given by (3.22). It can be concluded from (3.22) that the MSE decreases linearly with the SNR and the number of receive antennas (Nr ) and cubically with the training symbol length (Np ). To check the optimality of the algorithm the Cram´er-Rao lower bound (CRLB) for CFO estimation in such a MIMO systems was derived by the author in [65], yielding var {ε(n)} ≥

Nc2 (2π)2 N

3

r Np ℘

.

(3.23)

The CRLB thus equals the theoretical value of the error variance in (3.22) for high SNR values, which means the estimator is equivalent to the maximum-likelihood estimator (MLE) for high SNRs, as was also concluded for the SISO version in [66]. Analytical results for the MSE as a function of the SNR are depicted in Fig. 3.10 for diﬀerent values of Np and Nr together with corresponding simulation results with perfect orthogonal AWGN channels. The number of TX antennas Nt was chosen to be equal to the number of RX antennas Nr and the CFO δ set to 0.2 in the simulations, although it is noted that the MSE does not depend on Nt and δ (when |δ| < Nc /2Np ) in the AWGN case. The single-carrier preamble based on Frank-Zadoﬀ-Chu codes, as described in Section 3.2.2.6, was applied. Other parameters are set to the values given in Table 3.1. It is clear from Fig. 3.10 that the theoretical MSE value is a good estimate of the variance for high SNR values, but underestimates the variance compared to simulation results for low SNRs. This is caused by the high SNR approximation in (3.21). The 6 dB decrease in variance expected from (3.22), when going from a 1×1 to a 4×4 conﬁguration, is also observable. More gain in variance is achieved when increasing the training symbol length Np : going from Np = 16 to Np = 36 gains 10.6 dB. For a given conﬁguration and SNR range, an optimal code length Np can now be chosen, using (3.22) and Fig. 3.10, yielding a compromise between accuracy, the maximum oﬀset that can be estimated and the preamble length.

47

3.3 Frequency synchronisation 10−1

Nr = 1, Np = 16 Nr = 2, Np = 16 Nr = 4, Np = 16 Nr = 1, Np = 36 Nr = 2, Np = 36 Nr = 4, Np = 36

10−2

MSE

10−3

10−4

10−5

10−6

0

5

10

15

20

25

Average SNR per RX antenna (dB)

Figure 3.10. Analytical (dashed lines) and simulated (solid lines) MSE of the CFO estimation for diﬀerent combinations of Np and Nr as a function of the SNR for AWGN channels.

3.3.4

Numerical results

To further evaluate and illustrate the performance of frequency synchronisation, simulations were performed with several antenna conﬁgurations and delay spreads. The CIRs of all channel elements are modelled using the channel model of Section 2.2. The channel is modelled to have no spatial correlation. For all simulations a Frank-Zadoﬀ-Chu code based preamble in accordance with Section 3.2.2.6 is applied, with a total length of 2Np = 2 · 64 samples. On every antenna, a CP of 32 samples is added. The preamble is followed by random uncoded BPSK OFDM data symbols. Other parameters are chosen according to the IEEE 802.11a standard, as presented in Table 3.1. The CFO δ was set to 0.2, and for all results 10,000 independent realisations were simulated. Figure 3.11 shows the normalised MSE of the CFO estimation as a function of the average SNR per RX antenna. The theoretical AWGN value of (3.22) is depicted together with results from Monte Carlo simulations with both AWGN channels and multipath channels with diﬀerent rms delay spreads (στ ). The ﬁgure depicts these results for the SISO and the 4×4 MIMO conﬁguration, i.e., a system with 4 TX- and 4 RXbranches.

48

3 Design and implementation of a MIMO OFDM system 10−1

σ τ = 10 ns σ τ = 50 ns σ τ = 100 ns AWGN AWGN Theo

10−2

MSE

10−3

10−4

10−5

10−6

10−7

0

5

10

15

20

Average SNR per RX antenna (dB)

Figure 3.11. MSE in CFO estimation for a 1×1 (solid lines) and a 4×4 (dashed lines) system from analysis and simulations with AWGN and multipath channels.

The simulations with multipath channels show a degradation in performance from the theoretical and the simulated AWGN case. The degradation is highest for the 10 ns case and decreases when the rms delay spread increases. This can be explained by the frequency diversity which is introduced at higher delay spreads. The degradation, however, is much smaller in the 4×4 case than in the SISO case. This can be explained by the fact that even at low delay spreads a diversity gain is achieved, i.e., spatial diversity, for the 4×4 system. We note that this gain would be smaller for spatially correlated channels. To further study the impact of the number of antennas, Fig. 3.12 depicts the gain in MSE of a MIMO system over a SISO system as a function of the rms delay spread for an average SNR of 20 dB. Clearly the highest gain is achieved at low delay spreads. For high rms delay spreads the improvement goes towards the improvement found for AWGN, i.e. 10 log10 (Nr ) dB. The decrease in improvement as a function of delay spread is caused by the increasing frequency diversity. When the frequency diversity is high, the addition of TX or RX diversity does not gain signiﬁcantly compared to the gain in case of an AWGN channel. The results also show that the gain of the MIMO conﬁgurations is more than 10 log10 (Nr ) dB higher than the corresponding case with only TX diversity, showing the inﬂuence of the mentioned MRC like performance.

49

3.3 Frequency synchronisation 15

Improvement wrt 1×1 (dB)

2×1 4×1 2×2 4×4 10

5

0

0

50

100 150 rms delay spread (ns)

200

250

Figure 3.12. Improvement in MSE for diﬀerent MIMO conﬁgurations compared to a SISO system.

The impact of the studied frequency synchronisation approach on the overall system performance is illustrated by means of an example in Fig. 3.13. It shows the raw bit-error rate (BER), i.e., without coding, for 1×1, 2×2 and 4×4 systems. All systems apply V-BLAST [47] as MIMO detection algorithm. Results are given for both perfect synchronisation and for the implementation of the above described synchronisation. The results without synchronisation are not depicted, since they showed ﬂooring at BER levels above 10−1 . The packet length is 16 µs, i.e. 4 MIMO OFDM symbols, independent of the MIMO conﬁguration. For the simulations, uncorrelated multipath channels with a rms delay spread of 50 ns are applied. To reveal the inﬂuence of the frequency synchronisation, the channel estimation is assumed perfect. It can be concluded from the results in Fig. 3.13 that the implementation of the frequency synchronisation causes some degradation in performance at low SNR, which can be explained by the poor estimation of the CFO at these SNR values. In the BER range of interest, i.e., BER values below 10−3 , however, the degradation compared to perfect synchronisation is very small. Subsequently, we study the performance for the diﬀerent backwardcompatible preamble formats as proposed in Section 3.2. We regard the MSE in the estimation of δ by a SISO system, which has to detect the

50

3 Design and implementation of a MIMO OFDM system 100 Freq Sync Implem Perfect Freq Sync 10−1

BER

10−2 1×1 10−3

4× 4

10−4

10−5 −10

2×2

0

10

20

30

40

Average SNR per RX antenna (dB)

Figure 3.13. Performance of a V-BLAST-based system applying perfect (dashed lines) and the proposed synchronisation (solid lines) for a 1×1, 2×2 and 4×4 conﬁguration.

SIGNAL ﬁeld, and by a MIMO system, which also has to decode the data part of the frame. For the SISO system we, therefore, regard the LT part up to the SIGNAL ﬁeld and for the MIMO system the whole LT part. Figure 3.14 presents MSE results from Monte Carlo simulations for a SISO and 4×4 system for rms delay spreads of 10 and 100 ns. The CRLB for CFO estimation, given by (3.23), is plotted for the SO and SM preamble as reference. It can be concluded from 3.14(a) that for the SISO system the SO and SM based CFO estimation approach the CRLB, which is explained by frequency diversity introduced by the preamble structure. The TO and TM preamble do not provide this diversity and, therefore, perform considerably worse. The TM approach performs 6 dB worse than the TO method, since 6 dB less energy is received. For the MIMO case the SO, SM and TM method achieve the plotted CRLB, since now also the spatial diversity is exploited. The TO method performs 6 dB better, since 6 dB more energy is received than in the other three cases. When the delay spread is increased further, it is clear from 3.14(b) that the MIMO curves do not change. The SISO curves, however, come closer to the SISO CRLB.

51

3.3 Frequency synchronisation 10−2

SISO

10−2 SISO 10−4

SM SO TO TM CRLB

MSE

MSE

10−4

SM SO TO TM CRLB

10−6

10−6

MIMO

MIMO 10−8

10−8 0

10 20 30 Average SNR per RX antenna (dB)

(a) rms delay spread στ = 10 ns

0

10 20 30 Average SNR per RX antenna (dB)

(b) rms delay spread στ = 100 ns

Figure 3.14. MSE in CFO estimation for the BWC preamble formats as a function of the SNR for a SISO (solid lines) and 4×4 (dashed lines) system.

Finally, the proposed studied algorithm is used as part of the implementation of the test system described in Section 3.6. That section also includes throughput measurement results of this system implementation in actual indoor channels.

3.3.5

Summary

A frequency synchronisation approach for a MIMO OFDM system was presented and analysed in this section. The CFO estimation algorithm is based on the use of the repetitive nature of the preambles proposed in Section 3.2. The analytical derivation of the accuracy shows that the MSE decreases linearly with the number of RX antennas and cubically with the training symbol length in an AWGN environment. In multipath environments a large increase in performance is achieved, compared to the SISO version of the algorithm, by the spatial diversity that is introduced by the MIMO structure. The suitability of the proposed synchronisation approach was shown through performance simulations of a MIMO OFDM WLAN system, which resulted in only a slightly higher BER than a perfectly synchronised system. It is shown that similar results are achieved for the BWC preamble structures, and it can thus be concluded that these are all well applicable for frequency synchronisation.

52

3 Design and implementation of a MIMO OFDM system

3.4

Channel estimation

Most MIMO detection schemes, as described in Section 2.3.1, require knowledge about the channel at the RX. As discussed in Section 3.2, we regard a system where the estimation of the channel transfer is enabled by preﬁxing a piece of known data to the data part of the transmission. Since the channel can be assumed quasi-static (see Section 2.2), the channel estimate derived in this preamble phase can be used for detection during the full length of the packet. The block diagram of the regarded MIMO OFDM system, applying channel estimation at the RX, is shown in Fig. 3.15.

3.4.1

Algorithm and performance analysis

In this section we regard estimation algorithms for the diﬀerent preamble formats and evaluate their performance.

3.4.1.1 Time-multiplexed Since for the TM preamble cases observations are available on all carriers for all TX-RX pairs, we propose the application of channel estimation in the frequency domain. For that purpose the CPs of the training symbols are removed and they are converted to the frequency domain. Here the estimate of the MIMO channel for the kth carrier is the found by least-squares estimation (LSE) [74]. The result is, using

s

u

Nt ..

..

. 1 .. 0 .

. .. .

IDFT

Add CP P/S

Nt ..

. 1

G

Nr ..

.

1

Nc −1 Nt ..

MIMO

.1 .. 0 .

detector

..

Nr ..

. .. .

Nc −1 ˜s

DFT

Channel Estimation

. ···

Nc −1

.

..

CP Removal, S/P y

x

..

Figure 3.15.

. 1 .. 0 .

..

v

.

Block diagram of a MIMO OFDM system applying channel estimation.

53

3.4 Channel estimation

(3.1), given by ˜ H(k) = Xp,TM (k)S†p,TM (k) = H(k) + N(k)S†p,TM (k),

(3.24)

where N(k)S†p,TM (k) is the error in the MIMO channel estimate for the kth carrier. The MSE in the estimation of the (nr , nt )th element of H(k) for the TM preamble can be derived and is given by #$ $2 '

& $% $ † 2 ˜ $ $ ETM = E |Hnr ,nt (k) − Hnr ,nt (k)| = E $ N(k)Sp,TM (k) nr ,nt $ #$ $ ' 2 $ nnr ,2nt −1 (k) + nnr ,2nt (k) $2 $ = σn , = E $$ (3.25) $ 2sp (k) 2σs2 where np,q (k) denotes the (p, q)th element of the Nr ×2Nt noise matrix N(k). Furthermore, the noise and preamble signal elements have variances σn2 and σs2 , respectively.

3.4.1.2 Time-orthogonal For the TO preamble, the estimates for the MIMO channel is also found by LSE, as described in (3.24). When Walsh-Hadamard sequences of length NH are used to create orthogonality, the MSE in channel estimation is given by #$ $2 '

& % $ $ ˜ nr ,n (k) − Hnr ,n (k)|2 = E $ N(k)S†p,TO (k) $ ETO = E |H t t $ $ ⎡$ $ ⎤ $ 2NH ±n (k) $2 σn2 $ $ ⎦ nr ,i . = E ⎣$ i=1 $ = $ 2NH sp (k) $ 2NH σs2

nr ,nt

(3.26)

Note that ETO is NH times smaller than the MSE achieved with the TM based preamble, i.e., ETM in (3.25), since the total received signal energy is NH times higher for the TO preamble. It is easily veriﬁed that when the NH -dimensional Fourier matrix is used to achieve time orthogonality, the same MSE as in (3.26) is achieved.

3.4.1.3 Subcarrier-multiplexed For the SM preamble observations are not available on all carriers, requiring interpolation and extrapolation to ﬁnd the channel response on the remaining carriers. On the carriers with observations, i.e., for k ∈ {nt − 1, Nt + nt − 1, . . . , Nc − Nt + nt − 1}, the estimated channel response between the nt th TX and nr th RX is found by LSE and given

54

3 Design and implementation of a MIMO OFDM system

by + xnr ,2 (k) nn ,1 (k) + nnr ,2 (k) ˜ nr ,n (k) = xnr ,1 (k) √ H = Hnr ,nt (k) + r √ . t 2 Nt sp (k) 2 Nt sp (k) (3.27) The MSE in the channel estimate is for these carriers found by a similar derivation as in (3.25). It is given by

2 ˜ nr ,n (k) − Hnr ,n (k)|2 = σn . ELS = E |εLS |2 = E |H (3.28) t t 2Nt σs2 It is noted that the signal power for the carriers of the SM preamble is given by Nt σs2 to keep the power transmitted on one branch constant. To ﬁnd the channel transfer for the other carriers, interpolation between the estimates on the subcarriers with pilots has to be applied. Diﬀerent interpolation methods can be applied [75], but here linear interpolation is regarded due to its low computational complexity. When we deﬁne i ∈ {1, . . . , Nt − 1} and a ∈ {0, . . . , Nc /Nt − 2}, the estimated channel response for carrier k = nt − 1 + aNt + i is given by [76] ˜ ˜ ˜ nr ,n (k) = (Nt − i)Hnr ,nt (k + Nt − i) + iHnr ,nt (k − i) , H t Nt

(3.29)

where we assume Nc to be a multiple of Nt . The MSE for these interpolated channel estimates is derived in Appendix B.1 and given by 2Nt − 1 σn2 5Nt − 1 Nt + 1 EIP = E |εIP |2 = + + R{R(Nt )} 3Nt 3Nt 6Nt2 σs2 −

Nt −1 Nt − i 4 R{R(i)}, (3.30) Nt Nt − 1 i=1

where R(a) denotes the correlation of the channel responses experienced at two carriers separated by a subcarrier spacings. For the regarded channel model with exponentially decaying PDP it was found in (B.6) that this correlation can be expressed as LTs

∗

R(a) = E [H(k)H (k + a)] =

Ts

e− στ ej2πa − 1 e στ − 1 Ts

e στ − e

j2πa L

LTs

e− στ − 1

,

(3.31)

where we recall that L, στ and Ts denote the channel length, rms delay spread and sample time, respectively. Note that the spatial channel index is omitted here for readability. For carriers at the sides of the spectrum interpolation is not possible, and thus extrapolation has to be applied. When linear extrapolation is

3.4 Channel estimation

55

applied the channel estimate for carrier k = Nc − Nt + nt − 1 + i is given by ˜ ˜ ˜ nr ,n (k) = −iHnr ,nt (k − Nt − i) + (Nt + i)Hnr ,nt (k − i) . H t Nt

(3.32)

for i = 1, . . . , Nt − nt . It is noted that for (3.32) the subcarrier index is higher than that of the set of pilot carrier numbers. The extension for carriers at the other side of the spectrum, i.e., with a subcarrier index lower than those in the set of the pilot carriers, is straightforward and omitted here. The average MSE of the channel estimate on these carriers is derived in Appendix B.2. The ﬁnal result, i.e., EEP = E[|εEP |2 ], is given by EEP =

8Nt − 1 σn2 11Nt − 1 5Nt − 1 + − R{R(Nt )} − , 3Nt 3Nt 6Nt2 σs2

(3.33)

where N t −1 i 2 Nt + i = R{R(Nt + i)} − R{R(i)}. Nt − 1 Nt Nt

(3.34)

i=1

The MSE averaged over all carriers forming the OFDM symbols can now easily be expressed as ESM =

ELS (Nc − (Nt − 1) − Nc /Nt )EIP (Nt − 1)EEP + + . Nt Nc Nc

(3.35)

It is noted that per subcarrier LS estimation, i.e., without inter- and extrapolation, is also possible for the SM based preamble. Then spatial shifted versions of Sp,SM have to be transmitted after the SIGNAL ﬁeld. The overhead ratio increases to a level equal to that of the TO and TM preamble, and the total MSE is then given by (3.28). Moreover, improved performance could be achieved when L < Nc , at the cost of increased complexity, by applying the estimation of the CIR in the timedomain and using the knowledge of the limited length of the CIR.

3.4.1.4 Subcarrier-orthogonal In the case of the SO preamble, as presented in Section 3.2.2.5, the transmitted preamble symbols on the nt th TX branch are given by the element-wise multiplication of the (1+(nt −1)L )th column of the Fourier matrix and the training symbol sp , where L = Nc /Nt . The Nc ×1 frequency-domain received signal vector on the nr th RX branch is then

56

3 Design and implementation of a MIMO OFDM system

found by averaging over the two consecutive received versions of the preamble vector and given by xnr =

Nt

diag(sp ◦ ζnt )F gnr ,nt +

nt =1

= Mgnr +

n1,nr + n2,nr 2

n1,nr + n2,nr , 2

(3.36)

where M is the Nc ×Nt L preamble matrix M = [diag(sp ◦ ζ1 )F , . . . , diag(sp ◦ ζNt )F ].

(3.37)

We recall that ◦ denotes element-wise multiplication and here the Nc ×L matrix F denotes the ﬁrst L columns of the Fourier matrix F. Also, gnr is an Nt L -dimensional vector given by gnr = [gnTr ,1 , . . . , gnTr ,Nt ]T .

(3.38)

Here gnr ,nt denotes the L ×1 impulse response vector of the channel between the nt th TX and nr th RX branch. Here, it is assumed that the maximum excess delay of the channel L ≤ L . The Nc ×1 vector nm,nr denotes the AWGN vector on the nr th RX branch during the mth symbol. The estimate of the impulse response vector corresponding to the nr th RX branch is then found by the Nt L ×1 vector ˜nr = M† xnr = gnr + M† g

n1,nr + n2,nr , 2

(3.39)

where we recall that † denotes the pseudo-inverse. The MSE in this estimation can now be expressed as [59] ( (2

1 1 ( ( † 2 ˆ nr ( E ˜ gnr − gnr = E (M n ESO,g = Nt L Nt L 1 † H †H ˆ ˆ M n tr M E n = n r nr Nt L σn2 σn2 = tr (MH M)−1 = , (3.40) 2Nt L 2Nc σs2 ˆ nr = (n1,nr + n2,nr )/2. where the averaged noise process is deﬁned as n The estimate for the Nc ×1 frequency-domain response of the channel between the nt th TX and nr th RX branch is then found from (3.39) by ˜ nr ,n = F g ˜nr ,nt , H t

(3.41)

57

3.4 Channel estimation

˜nr ,nt denotes the estimate of the CIR between the nt th TX and where g nr th RX branch. The error in the estimation of the channel response for the kth carrier is then given by ˜ nr ,n (k) − Hnr ,n (k) = εH (k) = H t t

−1 L

(˜ gnr ,nt (l) − gnr ,nt (l))e−j

2πkl Nc

l=0

=

−1 L

εg (l)e−j

2πkl Nc

,

(3.42)

l=0

where εg (l) = g˜nr ,nt (l) − gnr ,nt (l) denotes the error in the estimation of the lth element of the CIR. Using this, the MSE can be expressed as

L σn2 , ESO = E |εH (k)|2 = L ESO,g = 2Nc σs2

(3.43)

where ESO,g was found in (3.40). When the number of carriers is a multiple of the number of TX branches, L = Nc /Nt , the MSE in (3.43) can be expressed by σn2 ESO = . (3.44) 2Nt σs2

3.4.1.5 Frank-Zadoﬀ-Chu codes Here we will show that the FZC or SC based preamble, as presented in Section 3.2.2.6, is a special case of the SO preamble and that the same approach can be used for channel estimation and MSE evaluation. To that end, we regard a preamble for which the code length is equal to the number of subcarriers of the system, i.e., Np = N 2 = Nc . The frequency representation of the FZC preamble symbols for the ﬁrst TX branch can be expressed by Nc N −1 (i+1)N −1 2πkn 1 1 −j 2πkn N c = C(k) = √ cn e cn e−j Nc N Nc n=0 i=0 n=iN N −1 N −1 N −1 N −1 ) ) 1 1 −j 2πni −j 2πk(n+iN −j 2πk(n+iN N Nc c = cn+iN e = e N e N N

=

1 N

i=0 n=0 N −1 N −1 −j 2πkn −j 2πi(k+n) N N2

e

n=0

e

i=0 n=0

= e−j

{N −k(modN )}k N2

,

(3.45)

i=0

from which we conclude that C = [C(1), C(2), . . . , C(Nc )], i.e., the frequency-domain preamble vector, has constant modulus. Since the

58

3 Design and implementation of a MIMO OFDM system

cyclic shifting of this preamble for a samples, for the diﬀerent TX branches, corresponds to a multiplication with the (a + 1)th column of the Nc -dimensional Fourier matrix, it is easily veriﬁed that the FZC preamble is a special case of the SO preamble in (3.8), where sp is replaced by C. The preamble symbols transmitted on the nt th TX are thus given by C ◦ ζnt . From the above we can conclude that the channel can be estimated in an approach similar to the one presented in Section 3.4.1.4. Therefore, the MSE in channel estimation for the Nc -dimensional FZC codes based preamble also equals (3.44).

3.4.2

Numerical results

In this section the analytical derived MSE performance in MIMO channel estimation of the previous section is compared with results from Monte Carlo simulations. These simulations consider a MIMO extension of the IEEE 802.11a standard, the parameters of which were summarised in Table 3.1. To be able to compare these results with the results from the analytical performance study, all 64 subcarriers carried pilot symbols in the simulated scenario. The MSE (averaged over all carriers) is depicted in Fig. 3.16 as a function of the SNR for rms delay spreads of 50 ns (left ﬁgure) and 100 ns (right ﬁgure) for the diﬀerent preambles. The results from simulations are given by the lines and the analytical results are depicted by markers. Note that the results are depicted as a function of the average SNR per RX antenna, which is given by Nt σs2 /σn2 . The results for the SO and SC preamble are plotted together, since their performance was equal. It can be concluded from this ﬁgure that there is good agreement between the analytical results and the results from simulations. It is clear that the TO and SO methods always perform better than the TM method, since more signal energy is received. All three curves show no ﬂooring and do not depend on the rms delay spread, i.e., the curves are identical for στ =50 ns and στ =100 ns. For low SNR values, the SM method performs better than the methods based on the other preambles, since the interpolation results in noise averaging here. For high SNRs, ﬂooring occurs in the MSE curves for the SM method, since the interand extrapolation errors become dominant there. These errors increase when the delay spread increases. The ﬂooring of the MSE is further investigated in Fig. 3.17, where the level of the MSE ﬂoor at high SNR is depicted as a function of the rms delay spread for a 2×2 and a 4×4 system, where again linear interpolation is applied for channel estimation based on the SM preamble. The

59

3.4 Channel estimation στ = 50 ns

100

TM TO SO/SC SM

10−1

10−1

10−2 MSE

MSE

10−2

10−3

10−4

10−5

στ = 100 ns

100

10−3

10−4

0

20 Average SNR (dB)

40

10−5 0

20 Average SNR (dB)

40

Figure 3.16. Comparison of the MSE in channel estimation as a function of the SNR. Results are given for a 2×2 system applying diﬀerent preambles and experiencing a channel with rms delay spreads of 50 ns (left ﬁgure) and 100 ns (right ﬁgure).

analytical MSE results for the SM preamble, as found from (3.35), are depicted by square markers. Clearly, the MSE ﬂoor increases with increasing rms delay spread for the SM method. At high delay spreads, however, also MSE ﬂoors occur for the SO, TM and TO based estimation, which is explained by ISI between the preamble symbols. This ISI occurs for the TM and TO preamble when the channel length is longer than the training symbol length. For the SO preamble ISI occurs when the channel length is longer than Nc /Nt , since leakage between the diﬀerent estimated impulse responses occurs at that point. The ﬂooring for the TM and TO preamble does not depend on the MIMO conﬁguration, i.e., the curves for the 2×2 and 4×4 conﬁguration lie on top of each other. For the SM and SO preamble, however, it does and increases with increasing number of TX branches. For the SM case this is explained by the fact that more interpolation is necessary, since less observations per TX branch are available. Applying interpolation to estimate channel response is sensitive to timing oﬀsets from the ideal timing point [77]. These oﬀsets introduce phase rotations in the frequency domain, which increase with increasing oﬀset. When these phase rotations are too high, they can result in interpolation errors. The inﬂuence of these oﬀsets on the MSE ﬂoor of

60

3 Design and implementation of a MIMO OFDM system 100

SM SO/SC TM/TO

MSE ﬂoor

10−2

10−4

2× 2

10−6

10−8 101

4× 4

rms delay spread s τ (ns)

102

Figure 3.17. MSE ﬂoor in channel estimation for a 2×2 and a 4×4 system as a function of the rms delay spread.

the SM based channel estimation is given in Fig. 3.18 for a 2×2 and 4×4 system experiencing a channel with rms delay spreads of 50 and 100 ns. It can be concluded from this ﬁgure that the MSE ﬂoor increases when the timing oﬀset increases. When the delay spread or number of TX branches increases, the sensitivity to timing oﬀset increases. To test the inﬂuence of the error in the channel estimation on the ﬁnal system performance, BER simulations were performed for a 2×2 MIMO 802.11a extension, the results of which are given in Fig. 3.19. The simulations were carried out for data rates of 12 Mb/s (BPSK, rate 1/2) and 108 Mb/s (64-QAM, rate 3/4). The RX applies channel estimation using the proposed preambles. Frequency synchronisation is assumed perfect. ZF is used as MIMO estimation algorithm, which feeds softvalues to the Viterbi decoder. Results are given for a rms delay spread of 50 ns. The results in Fig. 3.19 show that for 12 Mb/s, the systems based on the SM, SO and TO preamble perform 1.5 dB worse, at a BER of 10−4 , than the ideal case with perfect channel knowledge. The TM performs another 1.5 dB worse. This can be explained by the fact that the MSE in channel estimation for the TM case is proportional larger than that

61

3.4 Channel estimation 100

2×2, 50 ns 2×2, 100 ns 4×4, 50 ns 4×4, 100 ns

MSE ﬂoor

10

−1

10−2

10−3

100

101 Timing oﬀset (samples)

Figure 3.18. MSE ﬂoor in channel estimation for the SM preamble for a 2×2 and a 4×4 system as a function of the timing oﬀsets for rms delay spreads of 50 ns and 100 ns. 100

ideal SM SO/SC

10−1

TO TM

108 Mbps

BER

10−2

12 Mbps

10−3

10−4

10−5

0

10

20

30

Average SNR per RX antenna (dB)

Figure 3.19. BER performance of a 2×2 system applying the diﬀerent preamble formats. Results are given for BPSK, rate 1/2 coding (12 Mb/s) and 64-QAM, rate 3/4 coding (108 Mb/s).

62

3 Design and implementation of a MIMO OFDM system

for the other preambles for this SNR range. For the 108 Mb/s we see the same diﬀerences at low SNRs, however, the SM-based method shows ﬂooring in the BER curve for high SNR values. This can be explained by the in Fig. 3.16 observed ﬂooring in the MSE curve for the channel estimation with this preamble. The ﬂooring in the BER curve means that the error in detection is no longer dominated by the AWGN, but by the error in channel estimation. Experimental results were obtained by implementing the TM-preamble and corresponding estimation approach in a test system, as described in Section 3.6. That section also includes throughput measurement results of this system implementation in actual indoor channels.

3.4.3

Summary

The data-aided estimation of the MIMO channel matrix, for use in coherent detection of the data part of the transmission, was treated in this section. To that end, diﬀerent packet structures as presented in Section 3.2.2 were considered. The least-squares estimator was applied for all preamble types and its MSE performance was derived analytically. These analytical results were compared with results from Monte Carlo simulations in Section 3.4.2. Overall it can be concluded that for low rms delay spreads, low number of TX branches and low timing oﬀsets, the SO/SC and SM method are very well applicable, since they introduce no extra overhead compared to the SISO preamble. When high delay spreads and high number of TX antennas should be supported, however, the TO and TM preamble are preferred, although they introduce a considerable overhead. More efﬁcient structures can, in this case, also be achieved by combining the preamble concepts, for instance SM with TO. A further decrease in overhead can be achieved by not implementing the LT-part after the SIGNAL ﬁeld as repeated symbols, but as separate symbols. This will, nevertheless, decrease the estimation performance.

3.5

Timing synchronisation

It was shown in Section 2.3.2 that the application of a suﬃciently long cyclic preﬁx (CP) in OFDM systems annuls the inﬂuence of inter-symbol interference (ISI) caused by multipath propagation. This CP, however, also signiﬁcantly decreases the eﬀective data rate of the system. It is, therefore, important that the ratio of the length of the CP and the number of carriers is minimised. One solution is to keep the CP length low compared to the channel impulse response (CIR) length. Then, however, ISI will possibly become the performance limiting factor and

63

3.5 Timing synchronisation

the placement of the DFT window within the stream of received OFDM symbols, here referred to as symbol timing or timing synchronisation, becomes important. Note that symbol timing is often referred to as ﬁne timing in previous literature, in contrast to coarse timing which indicates the packet detection. Several timing approaches for SISO OFDM have been proposed previously in the literature. Generally, they are based on maximisation of a timing measure which is found by either correlation between repeated dedicated training symbols, see e.g. [67] and [78], or correlation between the redundant parts in the data symbols, see e.g. [69]. The limited accuracy of these algorithms makes their applicability to systems with short CP lengths questionable. Therefore, this section discusses the use of techniques that are based on knowledge of the CIR, examples of which for SISO OFDM can be found in, e.g., [57, 79, 80]. Few publications treat the problem of symbol timing in multipleantenna OFDM systems, which indeed is diﬀerent from the SISO problem since the received signal is now a sum of diﬀerent signals propagating through diﬀerent channels. One solution for MIMO OFDM symbol timing can be found in [71], where the peak in the cross correlation of the received and transmitted preamble symbols is used as timing reference. In this section we also use the estimate of the MIMO CIR to determine the optimal placement of the DFT window. Instead of searching the dominant path, as done in [71], we propose a method that attempts to optimise the signal-to-ISI ratio (SIR) of the received symbols used for data detection. The outline of this section is as follows. First, Section 3.5.1 describes the considered MIMO OFDM system applying symbol timing and introduces the additionally required notation. Subsequently, Section 3.5.2 presents three algorithms for symbol timing. Finally, the performance of the diﬀerent symbol timing algorithms is illustrated and compared by the use of Monte Carlo simulations in Section 3.5.3.

3.5.1

System description

The block diagram of the baseband model for the considered MIMO OFDM system applying symbol timing is depicted in Fig. 3.20. For the regarded MIMO system, the Nr ×1 received signal vector for the nth sample time was deﬁned by (2.13), which is here repeated as (3.46) for convenience

y(n) =

L−1 l=0

G(l)u(n − l),

(3.46)

64

3 Design and implementation of a MIMO OFDM system s

v

u Nt .. .

..

1

0 . ..

IDFT

. .. .

Add CP P/S

Nt .. . 1

G

Nr .. .

..

1

.

Nc−1 ˜s ..

Nt

. 1 0 . ..

MIMO detector

Nc−1

y

x Nr .. .1 0 . ..

..

DFT

Nc−1

. .. .

CP Removal, S/P

.

··· Symbol Timing

Figure 3.20.

..

..

.

Baseband model for a MIMO OFDM system applying symbol timing.

where we, for now, assume a noiseless system. The elements of u are assumed to be i.i.d. and to have zero-mean and variance σu2 . For the following, it is useful to group these vectors per OFDM symbol, as was done previously in (2.31) for the OFDM system. Therefore, we introduce the following short notation for the nth sample of the mth OFDM symbol: ym (n) = y([m−1]Ns +n) and um (n) = u([m−1]Ns +n). Using this notation, the expression for the received signal vector in (3.46) can be rewritten for the nth sample of the mth received symbol vector as ym (n) =

n−1 l=0

L−n−1

G(l)um (n − l) +

G(l + n)um−1 (Ns − l),

(3.47)

l=0

for n = 1, . . . , Ns . Here we assumed that L ≤ Ns , i.e., that the ISI is only caused by the preceding symbol. We can conclude from (3.47) that the received signal contains samples from the regarded and the previous symbol, i.e., the ﬁrst term is the useful signal term and the second term is the ISI. The symbol timing, as designed in this section, now determines which Nc out of Ns samples of ym are used to determine the frequency-domain signal vector xm . This vector is then used by the MIMO detection to retrieve the transmitted data. If the symbol timing is chosen to be pnr , the Nc -dimensional vector input to the DFT for the nr th RX is given by ym,nr [pnr ] = [ym,nr (pnr ), . . . , ym,nr (pnr + Nc − 1)]T SIG ISI [p ] + ym,n [p ], = ym,n r nr r nr

(3.48)

65

3.5 Timing synchronisation

SIG ISI where the ym,n [p ] and ym,n [p ] are the desired and ISI vector, rer nr r nr spectively, and their structure is equal to that of (3.48). Their nth elements are given by

pnr +n−1 SIG

ym,nr [n, pnr ] =

gnr (l)um (pnr + n − l),

(3.49)

l=0

L−(pnr +n)−1 ISI

ym,nr [n, pnr ] =

gnr (l + pnr + n)um−1 (Ns − l),

(3.50)

l=0

for n = 0, . . . , Nc − 1, respectively. Here the nr th row of G is denoted by gnr . As a performance measure for the timing we now consider the power of the desired signal and ISI terms. Therefore we calculate the expected power value of the desired signal term for timing point p and a given CIR realisation G. We recall that we, throughout this work, apply a quasi-static assumption on the channel, i.e., the channel is assumed to be constant for a packet transmission. Since the DFT is applied per RX branch, it is derived here for the nr th RX branch. For a timing point p, the expected signal power is given by

$ H SIG SIG $gnr PnSIG (p) = E y [p] y [p] m,nr m,nr r 2

= N c σu

p

gnr (k)gnHr (k)

2

+ σu

k=0

N c +1

[Nc − k]gnr (k + p)gnHr (k + p),

k=1

(3.51) where the expectation in the ﬁrst line of (3.51) is calculated over the symbol index m, i.e., time, and where E[X|Y ] denotes the conditional expected value of X given Y . In a similar way, the expected value of the ISI power can be derived as

$ H ISI ISI $gnr (p) = E y [p] y [p] PnISI m,n m,n r r r =

2

L−1

gnr (k)gnHr (k) Nc σ u k=Nc +p

2

+ σu

N c −1

k gnr (k + p)gnHr (k + p). (3.52)

k=1

The expressions for the wanted signal and ISI power, in (3.51) and (3.52), are deﬁned for 1 ≤ p ≤ Ng . We recall that Ng denotes the length of the CP. Note that for p > Ng these expressions change slightly, since samples of the next symbol will be included in the DFT window, also

66

3 Design and implementation of a MIMO OFDM system

causing ISI. For these exact expressions, as applied in the simulations of Section 3.5.3, the reader is referred to [81]. Clearly the expressions for the expected power of the desired signal and ISI in (3.51) and (3.52) depend on the channel realisation. Knowledge of the MIMO CIR is, therefore, required to be able to apply these expressions for symbol timing. In a practical system, however, full knowledge of the channel is often not available and an estimate of the channel has to be acquired, a subject that was treated in Section 3.4. Although it was shown in that section that the diﬀerent channel estimation/preamble combinations will result in a diﬀerent performance, we can generally write the resulting estimate of the lth tap of the Nt ×Nr MIMO CIR matrix G(l) as ˜ G(l) = G(l) + ε(l),

(3.53)

where ε(l) is the estimation error in the lth tap of the CIR, the elements of which are here assumed to be zero-mean circularly symmetric complex Gaussian distributed with variance σε2 . The relation between this error variance and the SNR was studied in Section 3.4. The timing algorithms, designed in Sections 3.5.2, exploit the knowledge of the MIMO CIR to determine the symbol timing. Although only an estimate is available, we will assume perfect channel knowledge in algorithm design. The impact of this assumption is tested by a simulation study in Section 3.5.3.

3.5.2

Algorithm description

In this subsection we propose three timing algorithms: dominant path detection, SIR optimisation and a reduced complexity algorithm. They are presented in Sections 3.5.2.1, 3.5.2.2 and 3.5.2.3, respectively.

3.5.2.1 Dominant path detection The straightforward method to determine the symbol timing is to relate the timing to the path in the CIR with maximum power. This is based on the observation that paths in the CIR with the smallest delay will generally experience the lowest attenuation, which on average is valid when the channel exhibits an exponentially decaying PDP. Generally a shift of several samples is applied to the found timing point, to take into account possible smaller taps preceding the path with the maximum power in the CIR. RX-branch timing: When this technique is applied for a MIMO system to calculate the symbol timing for one of the RX branches, i.e.,

67

3.5 Timing synchronisation

nr , the estimated symbol timing point is given by pnr = arg max gnr (k)gnHr (k) − c + Ng , k

(3.54)

where c is the above mentioned shift. Furthermore, the constant Ng is added since the outcome of the arg max indicates the beginning of the OFDM symbol, rather than the beginning of the DFT window. This is also the case for (3.55), (3.58) and (3.59). It is noted that for the RX-branch timing, every RX branch uses a separate symbol timing. Joint timing: To make the processing in the MIMO RX less complex, a common symbol timing for the whole MIMO RX can be calculated, here referred to as joint timing. To that end, we consider all Nt Nr CIRs at the same time, and ﬁnd the maximum of the sum of their powers. The timing point is then found as N " r H gnr (k)gnr (k) − c + Ng . (3.55) p = arg max k

nr =1

A similar approach for the joint symbol timing was previously proposed by the authors of [71].

3.5.2.2 SIR optimisation A more optimal approach to symbol timing, in the case of perfect channel knowledge, is proposed here. It attempts to maximise the SIR at the input of the DFT, i.e., the method ﬁnds a timing point that, on average, minimises the amount of ISI and maximises the amount of signal falling into the DFT window. RX-branch timing: For this approach the symbol timing point for the nr th RX branch is found by ) pnr = arg max PnSIG (p) PnISI (p) , (3.56) r r p

where the expected signal and ISI power are given by (3.51) and (3.52), respectively. An alternative approach would be to maximise the desired signal power in (3.51) or to minimise the ISI power in (3.52). Since this is very similar to (3.56), it is not treated here. Joint timing: For the case of a joint symbol timing for the entire MIMO receiver, the ratio of the total signal power and total ISI power is calculated. The symbol timing point is then found by N " Nr r * SIG ISI p = arg max Pnr (p) Pnr (p) . (3.57) p

nr =1

nr =1

68

3 Design and implementation of a MIMO OFDM system

3.5.2.3 Reduced complexity algorithm A computationally less complex algorithm, i.e., requiring less operations for implementation, was previously proposed for SISO OFDM in [57, pp. 88-92] and [80], and ﬁnds the maximum of the convolution of the CIR powers with a rectangular window of length Ng , i.e., the length of the CP. In doing so, the algorithm attempts to maximise the amount of ISI within the CP, i.e., minimising the ISI within the DFT window. RX-branch timing: In the extension for a MIMO system, the resulting timing point for the nr th RX branch is found by

pnr = arg max p

Joint timing:

⎧ ⎨p+N g ⎩

k=p

⎫ ⎬ H gnr (k)gnr (k) + Ng . ⎭

(3.58)

The extension for joint timing is given by

p = arg max p

⎧ Nr p+N ⎨ g ⎩

nr =1 k=p

gnr (k)gnHr (k)

⎫ ⎬ ⎭

+ Ng .

(3.59)

This joint timing is similar to what was applied in the MIMO OFDM implementation discussed in [53].

3.5.3

Numerical results

To evaluate the performance of the proposed algorithms, Monte Carlo simulations were performed for a 4×4 MIMO extension of the IEEE 802.11a standard, see Section 3.2.1. In the simulations two channels types were applied: a single-cluster and a double-cluster channel, as schematically depicted in Fig. 3.21. The former relates to an indoor environment and the latter maps onto an outdoor scenario, e.g., a singlefrequency broadcasting system. For both channels the clusters are assumed to have exponentially decaying PDPs and the amplitude fading is assumed to be Rayleigh distributed. The normalised rms delay spread is deﬁned per cluster and denoted by στ . For the double-cluster model the distance between the start of the clusters is given by δτ . Furthermore, spatially uncorrelated MIMO channels were assumed. In the following we will derive the performance of the symbol timing as a function of the accuracy of the channel estimation. Therefore, we

69

3.5 Timing synchronisation δτ

l

l

(a) single-cluster

Figure 3.21.

(b) double-cluster

Schematic representation of the PDPs of the applied channel models.

deﬁne the CIR-to-CIR-error ratio (CCER) for the nr th RX branch as L−1 H gnr (l)gnr (l) E l=0 CCERnr = L−1 (3.60) , H E εnr (l)εnr (l) l=0

where we recall that εnr (l), as deﬁned in (3.53), denotes the estimation error of the lth tap of the CIR vector corresponding to the nr th RX antenna. In fact, (3.60) equals the inverse of the normalised mean squared error (MSE) in the estimation of the MIMO CIR for the nr th branch. The CCER averaged over the diﬀerent RX branches is then denoted by CCER. The higher the CCER is, the more accurate the CIR is known in the RX. As a ﬁrst measure of performance the MSE in the estimation of the symbol timing for the single-cluster channel with στ = Ng /2 is regarded. The symbol timing algorithms are regarded for one of the RX branches. The timing error is deﬁned as the diﬀerence between the estimated timing point and the timing that would result from (3.56) with ideal CIR knowledge, i.e., the timing point that results in the maximum achievable SIR. The results are depicted in Fig. 3.22 as a function of the CCER. Since high rms delay spreads are regarded in these simulations, the shift for the dominant path algorithm was always chosen to be c = 1. It can be concluded from Fig. 3.22 that, in terms of MSE, the dominant path search algorithm of (3.54) performs worst over the whole CCER range. For low CCER values the other two algorithms show a comparable performance. At high CCER values the SIR maximisation algorithm of (3.56) performs best. Although the reduced complexity algorithm of (3.58) performs best in the medium CCER range, it shows ﬂooring for the high CCER range. We note, however, that although the MSE in symbol timing gives a measure for the error in the timing, it does not show the degradation in performance due to this error. Therefore, the following ﬁgures will

70

3 Design and implementation of a MIMO OFDM system 103 SIR maximisation Dominant path Reduced compl.

102

MSE in timing

101

100

10−1

10−2

10−3 −5

0

5 10 CCER (dB)

15

20

Figure 3.22. MSE in symbol timing as a function of the CCER for the diﬀerent timing algorithms. Single-cluster channel, normalised rms delay spread στ = Ng /2.

report the achieved SIR with a certain timing algorithm as a function of the CCER. When the SIR has the same order of magnitude as, or is smaller than, the experienced SNR, the system performance will be limited by the ISI. In Fig. 3.23 the SIR performance is given for the single-cluster model for στ = Ng /4 and στ = Ng /2. The optimal SIR values obtained with perfect timing are depicted as dashed lines, at 28 dB and 18.3 dB, respectively. The maximum achievable performance is smaller for στ = Ng /2, since more ISI occurs due to the longer channel. It can be observed that the reduced complexity model achieves the bound for the lowest CCER value, closely followed by the SIR maximisation method. In addition, the results show that for the dominant path search method ﬂooring appears below the optimum for both cases, while it performs better than the SIR maximisation algorithm for very inaccurate channel estimates. To compare the performance of the joint and RX branch timing, the achievable SIRs are in Fig. 3.24 compared for a single-cluster channel with στ = 3Ng /8. It can be concluded that for the SIR maximisation the joint timing performs a little worse than the separate timing for low CCER values. For the reduced complexity and dominant path detector this is the other way around. This is explained by the fact that these techniques work best for an exponentially decaying channel, which

71

3.5 Timing synchronisation στ = Ng/2 30

28

28

26

26

24

24

22

22

SIR (dB)

SIR (dB)

στ = Ng/4 30

20 18

20 18

16

16

14

14

12

12

10

0

10

20

SIR maximisation Dominant path Reduced compl. Perfect timing

10

0

CCER (dB)

10 CCER (dB)

20

Figure 3.23. SIR as a function of the CCER for the diﬀerent timing algorithms. Single-cluster channel. 22

20

SIR (dB)

18

16

14 SIR maximisation Dominant path Reduced compl. Perfect timing

12

10 −5

0

5 10 CCER (dB)

15

20

Figure 3.24. SIR as a function of the CCER for the diﬀerent timing algorithms. Single-cluster channel, στ = 3Ng /8. RX-branch timing (solid lines) and joint timing (dashed lines).

72

3 Design and implementation of a MIMO OFDM system 15

SIR (dB)

14

13

12 SIR maximisation Dominant path Reduced compl. Perfect timing

11

10 −5

0

5

10

15

20

CCER (dB)

Figure 3.25. SIR as a function of the CCER for the diﬀerent timing algorithms. Double-cluster channel, στ = Ng /4, δτ = 9Ng /10.

is better achieved for the joint timing, since the fading and noise are averaged out here. Finally, Fig. 3.25 presents SIR results for the double-cluster model, where again per RX timing is considered. It is observed that the optimal performance is only achieved by the SIR maximisation method. The dominant path search method shows severe ﬂooring. From this result, and more results for the double-cluster model not shown here, it can be concluded that only the algorithm in (3.56) achieves the optimal performance for this kind of channels with two clusters and high clusters separation δτ . This can be explained by the fact that both other methods inherently assume that the power of the channel taps decreases as a function of delay, which is not the case for the regarded double-cluster channel. Experimental results were obtained by implementing the reduced complexity timing algorithm in a test system, as described in Section 3.6. That section also includes throughput measurement results of this system implementation in actual indoor channels.

3.5.4

Summary and discussion

The problem of symbol timing in multiple-antenna OFDM systems was considered in this section. Extensions of OFDM symbol timing

3.6 Multiple-antenna OFDM system implementation

73

methods were proposed for application in MIMO systems. Additionally, a novel SIR maximisation algorithm was presented. The performance of the diﬀerent algorithms was tested using simulations with a singleand double-cluster channel model. It can be concluded that the reduced complexity algorithm performs superior, especially for low CIR accuracy. The SIR maximisation method is, however, preferred when the channel is accurately known and when double-cluster channels occur. A possible extension to the symbol timing method proposed in this section would be the inclusion of the ICI in the SIR expressions, used for the timing metric. The SISO interference power expressions derived in [82] can form a good starting point.

3.6

Multiple-antenna OFDM system implementation

In the period 2001-2002, a MIMO test system has been built in Agere Systems in Nieuwegein, The Netherlands, with the goal to serve as a proof of concept, as a platform for algorithm development and to verify the performance of MIMO in realistic propagation scenarios. The test system was build to operate in the 5.x GHz ISM band and to simultaneously transceive 3 signals with bandwidths up to 20 MHz. The details of the system are treated in Section 3.6.1 and Appendix C. The baseband processing applied in both TX and RX of the system is described in Section 3.6.2. Then Section 3.6.3 presents the results from throughput measurements carried with the test system. In Section 3.6.4 these results are compared with results from Monte Carlo simulations.

3.6.1

Test system description

The MIMO test system is built up with in-house developed components. To access the hardware, two PCs are used: the TX and RX platform, respectively. Each PC contains three boards, where each single board consists of one entire TX or RX branch, resulting in a 3×3 MIMO system. Each board consists of a baseband part, an intermediate frequency (IF) part and a radio frequency (RF) front-end based on a 5.x GHz GaAs radio chip. A picture of the receiver equipment is given in Fig. 3.26. The TX baseband part is set up to send signals at zero-IF, whereas the RX down-converts the received signals to a low-IF frequency. In the baseband processing, this low-IF signal is digitally down-converted, using the so-called sampled-IF principle. This setup has been chosen to enable removal of the DC oﬀset and to overcome RX IQ imbalance.

74

Figure 3.26. branches.

3 Design and implementation of a MIMO OFDM system

Receiver equipment: a PC with 3 receiver boards, i.e., the 3 receiver

The baseband processing is built around two ﬁeld programmable gate arrays (FPGAs) per board. In the regarded setup, the FPGAs are mainly conﬁgured as memory banks. At the TX, waveforms are loaded into, and sent from, the memory banks and they are recorded at the RX. These recorded data are processed oﬄine with software written in MATLAB. This oﬄine digital processing allows for quick exploration of diﬀerent MIMO detection and synchronisation algorithms. By transmitting multiple packets, BER tests can be performed in order to evaluate diﬀerent MIMO system setups in real-life communication channels. For a more extended description of the test system the reader is referred to Appendix C.

3.6.2

Baseband design

The TX baseband signals are based on the IEEE 802.11a standard, the parameters of which were summarised in Table 3.1. The data rate is varied by changing the coding rate and modulation type. The coding and interleaving are applied per TX branch, i.e., per-antenna coding (PAC) [83]. Basically the binary input data is demultiplexed to Nt branches, which all apply the IEEE 802.11a TX processing, to form Nt independent data streams at the output of the TX. The TM preamble, as described in Section 3.2.2.2, is used to enable synchronisation and channel estimation at the RX. This approach was chosen since this preamble does not cause ﬂooring in the channel estimation for high rms delay spreads or due to residual CFO, see Fig. 3.16.

75

3.6 Multiple-antenna OFDM system implementation Table 3.2.

Data rates and packet lengths for the diﬀerent conﬁgurations

Throughput (Mb/s)

Antenna Setup

Modulation

Coding Rate

Packet length (Bytes)

6 9 12 18 24 36 48 54

1×1 1×1 1×1 1×1 1×1 1×1 1×1 1×1

BPSK BPSK QPSK QPSK 16-QAM 16-QAM 64-QAM 64-QAM

1/2 3/4 1/2 3/4 1/2 3/4 2/3 3/4

57 85.5 114 171 228 342 456 513

18 27 36 54 72 108 144 162

3×3 3×3 3×3 3×3 3×3 3×3 3×3 3×3

BPSK BSPK QPSK QPSK 16-QAM 16-QAM 64-QAM 64-QAM

1/2 3/4 1/2 3/4 1/2 3/4 2/3 3/4

171 256.5 342 513 684 1026 1368 1539

For this TM preamble the LT part of the IEEE 802.11a preamble is consecutively transmitted on the 3 TX branches. The ST part is omitted since frame detection was enabled by a trigger cable. Furthermore, since BWC was not an issue in these tests, also the SIGNAL ﬁeld was omitted. The remaining storage capacity of the memory of the TX, i.e., 19 MIMO OFDM data symbols, was always fully exploited, leading to diﬀerent data packet lengths for diﬀerent data rates. The tested data rates and corresponding packet lengths, i.e. the number of information bits, are given in Table 3.2. To obtain the average performance, 1000 packets were transmitted per rate per location. At the RX, the preamble is used to perform diﬀerent tasks. First the coarse timing is found using the maximum-normalised-correlation (MNC) criterion based algorithm, as proposed in [84]. This coarse timing algorithm is based on other correlation based timing algorithms presented in [78, 67], but was shown to exhibit superior performance for application in the considered MIMO OFDM system by the author in [65]. Then the CFO is estimated and removed using the algorithm introduced in Section 3.3. Subsequently, symbol timing is applied using the joint reduced complexity method as presented in Section 3.5. To reduce the computational complexity of the symbol timing algorithm the search window for the symbol timing is limited by the use of the coarse

76

3 Design and implementation of a MIMO OFDM system

timing. Channel estimation is applied using LS estimation, as described in Section 3.4. For the data part of the transmission ﬁrst the CPs are removed and then the signals are transformed to the frequency domain. On these signals MIMO detection is applied, which was PAC V-BLAST [83] for the results presented in this section. PAC V-BLAST is similar to conventional V-BLAST [47] except that the optimal ordering is now not applied per carrier, but over the whole signal bandwidth. This means that for one TX stream, all carriers are in parallel detected, enabling the decoding of this TX stream, since the signals are encoded per antenna. The detected symbols are then re-encoded and remodulated and subtracted from the received signals. The advantage of this technique is that the decisions in the detected streams are more reliable and, thus, reduce the error propagation. As described in Section 3.2.1 the IEEE 802.11a system uses 4 pilots carriers. These are used in the RX to compensate for the residual CFO and phase noise. For that purpose the sub-optimal algorithm with reduced computational complexity of Section 4.5.3 is applied.

3.6.3

Measurement results

The above described test system was used to carry out measurements in a wing on the third ﬂoor of the former Agere Systems building in Nieuwegein, The Netherlands, at the end of 2002. This building is a typical oﬃce environment with modular inner walls and concrete ﬂoors and side walls. Figure 3.27 depicts the ﬂoor plan of the wing (42 m × 12.7 m), which was used for the measurements. The ﬁgure also depicts the desks and metal cupboards. Furthermore, the RX and 9 diﬀerent TX positions and orientations that were used for the measurements are shown. The applied antennas were omnidirectional, see [85], but due to obstruction by the test system their eﬀective patterns covered only one

Figure 3.27. The ﬂoor plan of the wing (42 m × 12.7 m), where the measurements were performed. The locations and orientations of the RX and TXs are depicted.

77

3.6 Multiple-antenna OFDM system implementation Table 3.3. Position 1 2 3 4 5 6 7 8 9

Average SNR per RX antenna and rms delay spread for all positions. Average SNR (dB)

rms delay spread (ns)

24 24 24 23 25 24 22 14 26

98 97 100 104 91 100 109 131 100

half-plane. The antennas were aligned in the direction of the arrows in Fig. 3.27. The antenna were spaced two wavelengths apart in a linear array, as is shown in Fig. C.2. This spacing was chosen so that the width of the array was equal to the width of an laptop screen, i.e., one of the foreseen antenna locations in future products. The average output power of each TX branch was limited to 20 mW. For every TX location the average SNR per RX antenna and the rms delay spread were measured at the RX. The results are included in Table 3.3, where the presented values are found from averaging over the antennas and time. It is noted that these values were estimated at baseband level and, thus, include system inﬂuences. Since the channel experienced by the system includes the diﬀerent TX and RX ﬁlters, the estimated rms delay spread can be longer than the rms delay spread of the propagation channel. In the measurements the bit-error rate (BER) and packet-error rate (PER) performance are estimated for diﬀerent locations and diﬀerent rates. The measurement results are depicted in Fig. 3.28 to 3.30. Note that for certain rates no performance value is given; this corresponds to the fact that at these rates all 1000 packets were received correctly. Furthermore, note that the 1×1 measurements were only performed for locations 5, 6 and 7 and were obtained with the same test system by switching oﬀ two of the three boards at both TX and RX. From Fig. 3.28 to 3.30, it can be concluded that the performance of the 3×3 system setup is worse than the performance of its 1×1 counterpart with 1/3 of the throughput. To have a PER of 1%, it can be shown that, in the 1×1 case, a rate of 54, 36 and 24 Mb/s can be achieved for locations 5, 6 and 7, respectively. For the 3×3 case, these rates are respectively 108, 54 and 54 Mb/s, resulting in an average throughput enhancement of 1.92. For well-conditioned MIMO channels, i.e.,

78

3 Design and implementation of a MIMO OFDM system 100

100

10−1 10−1

10−3

PER

BER

10−2

10−2

10−4 10−3 loc. 2 loc. 6 loc. 8 loc. 9

10−5 10−6

18 36 54 72 108 144 162 Throughput (Mb/s)

loc. 2 loc. 6 loc. 8 loc. 9

10−4 18 36 54 72 108 144 162 Throughput (Mb/s)

Figure 3.28. Measurement results for the 3×3 conﬁguration for TX locations 2, 6, 8 and 9. For location 9, the measurements are compared with simulations (dotted line) with the same SNR (26 dB) and rms delay spread (100 ns).

100

100

10-1

10-1

10-3

PER

BER

10-2

10-4

10-5

10-6

Figure 3.29. 4, 5 and 7.

10-2 loc. 1 loc. 3 loc. 4 loc. 5 loc. 7 54 72 108 144 162 Throughput (Mb/s)

10-3

loc. 1 loc. 3 loc. 4 loc. 5 loc. 7 54 72 108 144 162 Throughput (Mb/s)

Measurement results for the 3×3 conﬁguration for TX locations 1, 3,

3.6 Multiple-antenna OFDM system implementation 100

10−1

100 loc. 5 loc. 6 loc. 7

loc. 5 loc. 6 loc. 7

10−2

10−1

10−3

PER

BER

79

10−4

10−2

10−5

10−6

Figure 3.30. and 7.

24 36 48 54 Throughput (Mb/s)

10−3

24 36 48 54 Throughput (Mb/s)

Measurement results for the 1×1 conﬁguration for TX locations 5, 6

channels with i.i.d. channel elements, the theoretical throughput would be expected to be higher than that. Two arguments can explain why the measured MIMO channels do not provide the expected throughput improvement.: ﬁrstly the environment does not provide enough scattering, leading to ill-conditioned MIMO channels, and secondly mutual coupling between the branches at the transmitter and the receiver leads to performance degradation. Since our system was not shielded very well, as can be seen from Fig. 3.26, the last point is most likely resulting in the highest performance loss in our case.

3.6.4

Comparison with simulation results

To verify the results from the measurements presented above, PER simulations were performed for the 3×3 rates 72, 108, 144 and 162 Mb/s (see Table 3.2). Location 9 was used for this comparison (see Fig. 3.27), where an rms delay spread and average SNR per RX antenna of about 100 ns and 26 dB, respectively, were observed. These parameters were used in simulations over 10,000 packets to ﬁnd the PER, the result of which is depicted in Fig. 3.31. Ideal synchronisation and i.i.d. channel elements were assumed in these simulations. To show the inﬂuence of channel knowledge on the system performance, the simulations for 72

80

3 Design and implementation of a MIMO OFDM system 100

PER

10−1

10−2

10−3 10

72 Mb/ s, perfect CSI 72 Mb/s 108 Mb/ s 144 Mb/s 162 Mb/ s 54 Mb/s, 1× 1

15 20 25 30 Average SNR per RX antenna (dB)

35

Figure 3.31. PER simulation results for a 1×1 and 3×3 conﬁguration, with perfect CSI at the RX (dotted line) and with channel estimation (solid lines), and for diﬀerent throughputs.

Mb/s were also performed with perfect channel state information (CSI) at the RX. From these simulations results, it can be concluded that, as expected, the performance deteriorates when the data rate is increased. Furthermore, the channel estimation approach results in a loss of more than 4 dB. A diﬀerent preamble or more advanced channel estimation algorithm might reduce this loss. Furthermore, it can be observed that the 3×3 curves fall oﬀ faster than the 1×1 curve (54 Mb/s, 64-QAM, rate 3/4 and a packet length of 513 bytes), such that at high SNRs, most of the higher MIMO rates outperform the SISO 54 Mb/s rate, even having longer packets. This can be explained by the fact that the MIMO conﬁgurations beneﬁt from both the frequency and spatial diversity, due to the used interleaver structure. To compare the performance results from the measurements with the simulation results for location 9, the PER results from simulation for an SNR of 26 dB are regarded. These results are plotted as a function of throughput in Fig. 3.28 by the dotted lines. It can be concluded that the performance of the test system in a real propagation environment and with the implementation is worse than the performance of the idealised simulations. This can most likely be explained by system degradations that are not taken into account in the simulations, such as mutual coupling, residual IQ imbalance, errors in the frequency oﬀset estimation,

3.7 Conclusions

81

non-linearities, phase noise and quantisation. Another explanation could be the assumption of i.i.d. channel elements in the simulations.

3.7

Conclusions

In this chapter the design and implementation of a practical multipleantenna OFDM based system was studied. The main focus was on the digital signal processing based synchronisation techniques to be applied in a typical receiver of such a system. This was illustrated for application in wireless local-area-networks (WLANs). Diﬀerent possible transmission structures were presented, which enable backward-compatible (BWC) operation with already deployed single-antenna WLAN devices in the same frequency band. Since OFDM systems were previously shown to be very much aﬀected by carrier frequency oﬀsets, a frequency synchronisation approach for MIMO OFDM systems was presented and analysed. The method exploits the repetitive nature of the proposed preambles. It was shown that the MSE decreases linearly with the number of RX antennas and cubically with the training symbol length in an AWGN environment. In multipath environments a large increase in performance is achieved, compared to the SISO version of the algorithm, by the spatial diversity that is introduced by MIMO. The suitability of the proposed synchronisation approach was shown through performance simulations of a MIMO OFDM WLAN system, which resulted in only a slightly higher BER than that of a perfectly synchronised system. The regarded MIMO systems apply coherent detection and, thus, require an estimate of the channel transfer at the RX to apply MIMO detection. To that end, channel estimation for the diﬀerent packet structure were studied in this chapter. It can be concluded from the performance results that for low delay spreads, low number of TX branches and low timing oﬀsets the SO/SC and SM methods are very well applicable, since they introduce no extra overhead compared to the SISO preamble. When high delay spreads and high number of TX antennas should be supported, however, the TO and TM preamble are preferred, although they introduce a considerable overhead. More eﬃcient structures can in this case also be achieved by combining the diﬀerent preamble concepts, for instance SM with TO. The third synchronisation task, i.e., timing synchronisation, was regarded for a demanding scenario, where the cyclic preﬁx is short compared to the channel length. For this case inter-symbol interference occurs and the timing has to be designed to minimise its inﬂuence. Based on this concept, three types of algorithms for MIMO OFDM symbol timing were regarded, all for a RX-branch mode, where timing is applied

82

3 Design and implementation of a MIMO OFDM system

separately for every RX branch, and a joint mode, where the same timing is applied for all RX branches. From a numerical performance analysis it is concluded that the performance of the reduced complexity algorithm is superior, especially for low CIR accuracy. The SIR maximisation method is, however, preferred when the channel is accurately known and when double-cluster channels occur. To prove the applicability of the combination of MIMO with OFDM for WLAN systems and to test the performance of the proposed algorithms in a real WLAN propagation scenario, a MIMO OFDM test system with 3 TXs and 3 RXs was built. This chapter presented the details of the design and showed results from throughput measurements with this system setup in a typical oﬃce environment. It was concluded from the results that the measurements performed with the test system showed a slightly worse performance than the idealised simulation results. The explanation is that in the simulations, system imperfections such as phase noise, IQ imbalance and quantisation as well as ill-conditioned MIMO channels were not taken into account. The latter can be caused by either mutual coupling in the TX or RX or by correlation in the propagation channel. Overall, it can be concluded from the measurements that the implementation of a 3×3 MIMO OFDM system achieved about a two times higher throughput than its 1×1 counterpart at a given range, although theoretically a higher throughput gain was expected.

Chapter 4 PHASE NOISE

4.1

Introduction

In the previous chapters, the combination of multiple antenna techniques with orthogonal frequency division multiplexing (OFDM) was shown to be a promising basis for next generation high data rate systems. The comparison between simulation and measurement results in Section 3.6.3, however, shows that a fair diﬀerence occurs between the theoretical and practical system performance. The discussion of the results highlights that imperfections in the system implementation could be one of the causes of this diﬀerence. This, and previous contributions in the literature about the inﬂuence of front-end impairments in singleinput single-output (SISO) OFDM, motivates a closer investigation of these imperfections. The imperfection of the radio frequency (RF) oscillator, which originates from thermal noise, was identiﬁed as one of the major performance limiting factors of SISO OFDM systems in previous studies, see e.g. [25, 26, 33, 89–95]. Since, in general, the disturbance of the amplitude of the oscillator output is marginal [96], most inﬂuence of the oscillator imperfection is noticeable in random deviation of the frequency of the oscillator output. These frequency deviations are often modelled as a random excess phase, and therefore referred to as phase noise. Phase noise (PN) will more and more appear to be a factor limiting performance of OFDM systems, when low-cost implementations or systems with high carrier frequencies [21] are considered, since it is in those cases harder to produce an oscillator with suﬃcient stability [97].

c 2004, 2005, 2007 IEEE. Portions reprinted, with permission, from [86–88].

84

4 Phase noise

Therefore, this chapter treats the inﬂuence and suppression of PN in OFDM systems in general and in multiple-input multiple-output (MIMO) OFDM systems speciﬁcally. First, a model for phase noise caused by an imperfect oscillator is developed in Section 4.2.1. Subsequently, the inﬂuence of PN in MIMO OFDM systems is treated in Section 4.2.2. Here the PN is modelled in both the transmitter (TX) and receiver (RX) part of the wireless system. It is shown how the inﬂuence can be split up into a rotational and an additive part, where the latter also involves inﬂuence of the channel. The additive contribution of PN, often referred to as inter-carrier interference (ICI), is further studied in Section 4.3, where its limit distribution is derived. It is shown that the commonly assumed complex Gaussian distribution does not model the ICI term correctly and significantly underestimates the inﬂuence of PN in OFDM systems. The error caused by the ICI in detection of the transmitted data in a zero-forcing (ZF) based MIMO OFDM system is investigated in Section 4.4. It is shown that the performance largely depends on the ratio of the number of TX and RX branches. Section 4.5 subsequently proposes an estimation and compensation approach for the rotational error due to phase noise. First an estimation algorithm based on maximum-likelihood theory is introduced. To improve the applicability of the suppression, the section additionally proposes a sub-optimal algorithm with reduced computational complexity. Section 4.5.4 derives an analytical lower bound on the performance of the estimators. Section 4.5.5 then proceeds to compare the analytical performance with results from simulations, where additional results from BER simulations are analysed regarding the performance of a multipleantenna OFDM system, implementing both suppression algorithms. The suppression of the additive ICI term is treated in Section 4.6. For this purpose a decision-directed iterative algorithm is studied. In the considered approach an estimate of the ICI is made using an initial detection of the transmitted data symbols. This estimated ICI is, subsequently, subtracted from the received signal and a redetection of the data symbols is applied. The performance of this suppression approach is studied in Section 4.6.2. Finally, conclusions are drawn in Section 4.7.

4.2

System and phase noise modelling

Let us consider a MIMO OFDM system applying Nt TX branches and Nr RX branches, as described in Section 2.3. For convenience the MIMO OFDM system build-up is shortly repeated here to show the inﬂuence of

85

4.2 System and phase noise modelling s

Nt .. .

aTX(t)

u ..

1

0 . ..

.. .

IDFT

Nt

. Add CP P/S

..

···

.1

Nc−1 Nt .. . 1 0 . ..

Nc−1 ˜s

MIMO detector

..

Nr

GRF ..

. 1 0 . ..

. .. .

DFT

Nr .. CP .1 Removal, S/P

Nc−1 x

Figure 4.1.

··· v

y

··· aRX(t)

Block diagram of a MIMO OFDM system.

imperfect oscillators. Consider Fig. 4.1 which depicts the system model for a MIMO system, now including the up- and down-conversion. In this scheme, the time-domain Nt Ns ×1 transmit vector u, as deﬁned in (2.42), is up-converted to radio frequency (RF) fc using a common oscillator for all MIMO branches, here denoted as aTX (t). Subsequently, the RF signal is fed through the wireless MIMO channel, here denoted by GRF . Note that the baseband equivalent of GRF is denoted by G, as previously deﬁned in Chapter 2. In the RX, the signal is downconverted to baseband using the common oscillator aRX (t), yielding the time-domain signal y, from which the transmitted signal is estimated as described in Section 2.3.

4.2.1

Oscillator modelling

The oscillator processes in Fig. 4.1 can be written as aTX (t) = ej{2πfc t+θTX (t)} , −j{2πfc t−θRX (t)}

aRX (t) = e

(4.1) ,

(4.2)

where θTX (t) and θRX (t) model the TX and RX phase noise, respectively. In the remainder we will assume that the oscillators used for up- and down-conversion to RF are built as free-running oscillators. For this kind of oscillators it was found [98] that the phase error θ(t) becomes, asymptotically as t → ∞, a Brownian motion (Wiener process). In the remainder we assume that t is large enough for this model to be valid. The variance of the phase noise process θ(t) increases linearly with time and at rate c, which depends on the quality of the oscillator, i.e., σθ2 = E[θ2 (t)] = ct. The phase noise process is, thus, given by √ θ(t) = cB(t), (4.3)

86

4 Phase noise

where B(t) is a standard Brownian motion. Due to the properties of the Brownian motion [99], the parameter c is suﬃcient to model the stochastic phase noise process. Since c is not a well-known parameter for the characterisation of an oscillator, we here calculate the power spectral density (PSD) of the oscillator process a(t). For that purpose we ﬁrst calculate the autocorrelation of a(t), which is given by Raa (t, t + τ ) = E [a∗ (t)a(t + τ )] = e− 2 c|τ | ej2πfc τ , 1

where we used the identity that for X ∼ N (0, σ 2 ), / . 1 2 . E [exp(jX)] = exp − E X 2

(4.4)

(4.5)

It can be concluded from (4.4) that, although that the phase noise process θ(t) is nonstationary, the oscillator process a(t) is indeed stationary for t → ∞. The PSD (deﬁned for −∞ < f < ∞) is found by applying the Fourier transform to (4.4), the result of which is given by Sa (ω) = F (Raa (t, t + τ )) ∞ = Raa (t, t + τ )e−jωτ dτ = −∞

c/2 , (∆ω) + (c/2)2 2

(4.6)

where ω = 2πf , ∆ω = 2π (f − fc ) and F (·) denotes the Fourier transform here. The single-sided power spectral density (deﬁned for 0 ≤ f < ∞) is, by deﬁnition, given by Sa,ss (ω) = 2Sa (ω) =

c , (∆ω) + (c/2)2 2

(4.7)

Similar expressions are found in [96], [98] and [100]. The PSD of the oscillator process is depicted schematically in Fig. 4.2. Several observations can be made from the PSD in (4.6) and Fig. 4.2. The spectrum exhibits the well-known Lorentzian shape, with the 1/f 2 fall-oﬀ at high frequencies. Furthermore, the total oscillator power does not depend on c, since integration over the total frequency band results in 12 . A commonly used measure for the characterisation of a PSD as in Fig. 4.2 is the single sideband −3 dB bandwidth. From (4.6) it is easily found that the −3 dB bandwidth of the Lorentzian spectrum is given by β=

c . 4π

(4.8)

87

4.2 System and phase noise modelling

10 log10(4/c)

Sa,ss

-20 dB/decade

fc + β

fc

f (log scale)

Figure 4.2. Single-sided representation of the power spectral density of the oscillator process centred around fc .

It can now be concluded that for the regarded model the quality of the oscillator is given by one parameter, i.e., β (or similarly c). When β increases, the variance of the phase noise process increases faster with time and the PSD of the oscillator process becomes wider. When we, subsequently, for our purpose want to consider Fig. 4.1 as the representation of our discrete time equivalent baseband model, the oscillator processes for the nth sample time can be written as aTX (n) = ejθTX (n) ,

(4.9)

aRX (n) = ejθRX (n) ,

(4.10)

where θX (n) = θX (nTs ) for X ∈ {TX, RX} and n ∈ {0, 1, 2, ...}. Now, √ again the phase noise process is given by θX (n) = cX B(n) and using the property B(0) = 0, we can rewrite the PN process as θX (n + 1) =

√

cX B(n + 1) =

√

cX B(n) + εX (n) =

n

εX (i),

(4.11)

i=0

for n ∈ {0, 1, 2, . . .}. Here εX (n) is an i.i.d. Gaussian random variable with zero-mean and variance σε2 = cX Ts . The variance of the PN process θX (n) is then given by E θX2 (n) = ncX Ts , (4.12) which grows linearly with the sample index n, revealing the nonstationary behaviour of the PN process. Note, however, that the oscillator

88

4 Phase noise

process is stationary, as can be concluded from (4.4) and (4.6). The regarded model is for free running carriers. An extension of these models can, however, be made for oscillators applying a phase-locked loop, see for example [101].

4.2.2

Inﬂuence on signal model

We, subsequently, study the inﬂuence of the phase noise on the signal model of the MIMO system. To that end, we will use the signal model as deﬁned in Section 2.3.3. For convenience the expression for the received signal vector during the mth symbol period in (2.45) is repeated here xm = (F ⊗ INr )(Υ ⊗ INr )ym ,

(4.13)

where we recall that F is the Fourier matrix, Υ denotes the removal of the guard interval (GI) and ym is the received baseband time-domain signal. In a system, as depicted in Fig. 4.1, experiencing PN, the received signal is given by ˘ TX,m ⊗IN )(Θ⊗IN )(F−1 ⊗IN )sm +vm , (4.14) ym = (ERX,m ⊗INr )G(E t t t ˘ models the quasi-static time-domain where, as deﬁned in Section 2.2, G channel matrix, Θ denotes the addition of the GI and vm is the additive RX noise. Furthermore, ETX,m and ERX,m model the PN during the mth symbol in the TX and RX, respectively, and are given by EX,m = diag{aX,m (0), aX,m (1), . . . , aX,m (Ns − 1)},

(4.15)

where aX,m (n) = aX (mNs + n) and where X ∈ {TX, RX}. Substituting (4.14) into (4.13) then results in ˘ TX,m ⊗ IN )(ΘF−1 ⊗ IN )sm + vm ) xm = (FΥ ⊗ INr )((ERX,m ⊗ INr )G(E t t = (G RX,m ⊗ INr )H(G TX,m ⊗ INt )sm + nm , (4.16) where we recall that H denotes the block-diagonal frequency-domain channel matrix, which is found by applying the IDFT and DFT operation to the block-circulant matrix ˘ (Υ ⊗ INr )G(Θ ⊗ INt ).

(4.17)

In Section 2.3.3 it was shown that, for the case with perfect synchronisation and thus no PN, this block diagonality property could be exploited by applying MIMO processing per subcarrier. Since G TX,m and G RX,m are non-diagonal matrices for the case of PN, it can be concluded that the term between the DFT and IDFT operator, i.e., ˘ TX,m ⊗ IN )(Θ ⊗ IN ), (Υ ⊗ INr )(ERX,m ⊗ INr )G(E t t

(4.18)

89

4.2 System and phase noise modelling

is not block circulant and, consequently, does not have the Fourier bases as eigenvectors. Hence, (4.18) can not be rewritten as a block-diagonal matrix preceded and followed by a Fourier matrix. As a result, intercarrier interference (ICI) will occur. The second line in (4.16) is found by transforming the channel to the frequency domain and rewriting the inﬂuence of the PN to G TX,m = FΥETX,m ΘF−1 , G RX,m = FΥERX,m ΘF−1 . The Nc ×Nc matrix G X,m is given by ⎡ X X X γ1,m · · · γN γ0,m c −1,m X X X ⎢ γ−1,m γ · · · γ 0,m Nc −2,m ⎢ G X,m = ⎢ .. .. .. .. ⎣ . . . . X X X γ . . . γ γ−N 0,m −Nc +2,m c +1,m

(4.19) (4.20) ⎤ ⎥ ⎥ ⎥, ⎦

(4.21)

where again X ∈ {TX, RX} and where the (k, l)th element of G X,m is given by Nc −1 2π{k−l}i 1 X ejθX,m (Ng +i) e−j Nc . (4.22) γk−l,m = Nc i=0

X X = 1 and γk−l,m = 0 for Note that for a system not experiencing PN γ0,m k = l. Consequently, G TX and G RX reduce to identity matrices and all carriers are orthogonal. All elements on the diagonal of G TX and G RX are equal, i.e., γ0TX and γ0RX , respectively, and have approximately unity amplitude for moderate PN. Since they are located on the diagonal, they cause a rotation of the wanted signals. As this rotation is equal for all carriers, it is often referred to as the common phase error (CPE). The other elements in G TX and G RX cause interference among carriers, i.e., the inter-carrier interference (ICI). We use this property to rewrite (4.16) to

xm

γ0,m 0 12 3 RX TX = γ0,m γ0,m Hsm + ξm + nm ,

(4.23)

where ξm represents the ICI contribution for the mth symbol and can be written as ξm = (ϕRX,m ⊗ INr )H(ϕTX,m ⊗ INt )sm RX TX + γ0,m H(ϕTX,m ⊗ INt )sm + γ0,m (ϕRX,m ⊗ INr )Hsm . (4.24) TX Here ϕTX,m = G TX,m − γ0,m INt Nc and ϕRX,m have the same structure as ϕTX,m . Thus, the ﬁrst term in (4.23) is the desired signal times some

90 1.5

1.5

1

1

0.5

0.5

0

Q

Q

4 Phase noise

0

−0.5

−0.5

−1

−1

−1.5 −1.5

−1

−0.5

0 I

(a) Low β

Figure 4.3.

0.5

1

1.5

−1.5 −1.5

−1

−0.5

0 I

0.5

1

1.5

(b) High β

Inﬂuence of PN on the reception of QPSK symbols.

common rotation γ0,m . The second term models the ICI, as worked out in (4.24). It is important to note that this term contains the complex channel matrix and will thus exhibit properties of the channel. The last term in (4.23) models the additive white Gaussian noise (AWGN). To illustrate the inﬂuence of PN on the received constellation points of a transmitted QPSK modulation, a system with 64 carriers experiencing PN and no multipath propagation was simulated, the results of which are depicted in Fig. 4.3. Simulations were preferred above measurements here, and throughout this chapter, since simulations allow us to conveniently vary β and to isolate the inﬂuence of the PN. For the simulation of Fig. 4.3 we considered a system experiencing no AWGN, such that we only see the inﬂuence of the PN. The constellation plot for several consecutive received symbols for a low corner frequency β is depicted in Fig. 4.3(a). The two eﬀects of PN, rotational and additive, are observable, although the rotational error is most pronounced. When the corner frequency of the PN PSD β is increased, then the additive error becomes more prominent, as is shown in Fig. 4.3(b). We can conclude from these results that the CPE and the ICI are caused by the lower and higher frequencies in de PN PSD, respectively. This conﬁrms ﬁndings in [102], where it is shown that frequency components in the PN PSD < fs /Nc contribute to the CPE and frequency components > fs /Nc to the ICI.

4.3 Impact and distribution of the ICI term

4.3

91

Impact and distribution of the ICI term

We concluded from (4.23) that the eﬀect of phase noise on the received MIMO signal is two-fold. The inﬂuence of the ﬁrst eﬀect, the CPE, can be eﬀectively removed, as will be shown in Section 4.5 of this book. The additive ICI term, however, is harder to remove, as will be discussed in Section 4.6. Hence, the ICI will appear to be the main performance limiting factor in PN-impaired systems. This justiﬁes a careful investigation of the inﬂuence of the ICI term on system performance. The properties of the ICI term were previously studied by several authors, see e.g. [25, 26, 33, 89, 93]. Many of these papers, implicitly or explicitly, assume that the ICI term is complex Gaussian distributed, due to the central limit theorem. This is, however, as also noted by the authors of [94, 95], an approximation which is only valid for some combinations of PN and subcarrier spacings. The authors of [94] show that the complex Gaussian approximation only holds for fast PN, i.e., PN which changes fast compared to the symbol time (for consequently high β-values), and that error probabilities calculated under this Gaussian assumption are very inaccurate for other types of PN. The authors of [95] conﬁrm these ﬁndings with numerical results from Monte Carlo simulations, which show that good agreement with the Gaussian distribution is only achieved when the ratio of the −3 dB bandwidth of the PSD of the oscillator spectrum β and the subcarrier spacing is close to or larger than 1. It is noted, however, that these values of PN correspond to such a severe system performance degradation, that for practical systems this ratio always will be chosen to be much smaller than 1. In this section we extend previous work by analytically studying the distribution of the PN-caused ICI term in OFDM systems. We will derive a limit distribution for the ICI, which will be shown to exhibit thicker tails than the complex Gaussian distribution with the same variance. Subsequently, we will derive the tail probabilities of the ICI term for a system applying QPSK modulation. We show that previous approaches from literature, based on the Gaussian assumptions, signiﬁcantly underestimate these probabilities. These analytical results are validated by simulation results.

4.3.1

System model

In this section a SISO system is regarded, although the extension to MIMO systems is straightforward. Furthermore, there is no multipath channel to bring about the inﬂuence of only the phase noise. The block diagram of such a system is given in Fig. 4.4.

92

4 Phase noise u

s 0 . ..

.. .

IDFT

y

aTX(t)

Add CP P/S

Nc−1

x CP Removal, S/P

.. .

DFT

Nc−1

aRX(t) v

Figure 4.4.

0 . ..

Block diagram of the regarded SISO OFDM system.

For this system the expression for the received signal of (4.16) can be simpliﬁed to xm = FΥ(ERX,m ETX,m ΘF−1 sm + vm ) = FΥEm ΘF−1 sm + nm = γ0,m sm + ξm + nm ,

(4.25)

where Em is given by Em = diag{aRX,m (0)aTX,m (0), . . . , aRX,m (Ns − 1)aTX,m (Ns − 1)} = diag{ejθm (0) , ejθm (1) , . . . , ejθm (Ns −1) },

(4.26)

and where θm (n) = θTX,m (n) + θRX,m (n). The kth element of the ICI vector for the mth symbol ξm is given by ξm (k) =

N c −1

γk−l,m sm (l),

(4.27)

l=0,l=k

where γk−l,m was previously deﬁned in (4.22). The increments of the joint TX and RX phase noise can also be assumed to be independent, as in Section 4.2.1, so that θ(n + 1) = θ(n) + ε(n + 1) =

n+1

ε(i),

(4.28)

i=1

for n ∈ {0, 1, 2, . . .} and where θ(0) = 0, ε(n) ∼ N 0, σε2 and σε2 = 4πβTs . Since β is generally small compared to the subcarrier spacing 1/(Nc Ts ), it is useful to deﬁne βn = βNc Ts , which is the normalised version of β. Subsequently, we can rewrite σε2 as σε2 =

4πβn σ2 = , Nc Nc

where σ 2 = 4πβn is independent of Nc .

(4.29)

93

4.3 Impact and distribution of the ICI term

Using these properties of the PN, the expression for the ICI in (4.27) can be rewritten as: . / Nc −1 N c −1 1 2π(k − l)n sm (l) exp (jθ(mNs + Ng + n)) exp −j Nc Nc n=0 l=0,l=k ⎛ ⎞ . / mNs +Ng +n Nc −1 N c −1 2π(k − l)n 1 ⎝ ⎠ sm (l) exp j ε(i) exp −j = Nc Nc n=0 i=1 l=0,l=k ⎛ ⎞ . / mNs +Ng +n Nc −1 N c −1 2π(k − l)n χm ⎝ ⎠ sm (l) exp j ε(i) exp −j = Nc Nc n=0 i=mNs +Ng l=0,l=k 4 5 . / Nc −1 N n c −1 2π(k − l)n χm sm (l) exp j ε (i) exp −j , (4.30) = Nc Nc

ξm (k)=

n=0

l=0,l=k

i=0

where

⎛ χm = exp ⎝j

mNs +Ng −1

⎞

ε(i)⎠ .

(4.31)

i=1

Here the elements of ε (i) are i.i.d. according to N 0, σε2 and, since Nc −1 c −1 c −1 {ε (i)}N = {ε(mNs + Ng + i)}N is independent of i=0 i=0 , {ε (i)}i=0 mNs +Ng −1 . {ε(i)}i=1 In the remainder of this section we will, without loss of generality, regard the subcarrier k = 0 in symbol m = 0. When we then omit the subcarrier and symbol indeces for brevity, (4.30) reduces to 4 n 5 . / Nc −1 N c −1 χ0 2πnl ξ0 (0) = ξ = s(l) exp j ε (i) exp j . (4.32) Nc Nc l=1

n=0

i=0

Ng −1 ε(i) is independent of all other The random variable χ0 = exp j i=1 terms appearing in (4.32), and has norm 1 and, thus, gives rise to a rotation in the complex plane of the random variable 4 n 5 . / Nc −1 N c −1 1 2πnl s(l) exp j ε (i) exp j . (4.33) Nc Nc l=1

n=0

i=0

For convenience, we will, therefore, in the sequel take χ0 = 1 and study (4.33), where we keep in mind that for the ICI, a random and independent rotation should be performed in the end.

94

4.3.2

4 Phase noise

Bit-error probabilities

We recall that the received signal vector after the DFT-processing xm is given by (4.25). Before detection is applied to this signal, ﬁrst the CPE γ0,m is removed. When this is assumed to be done perfectly, the resulting signal for the kth carrier is given by x = s + ξ + n = s + ν,

(4.34)

which is input to the decision device. Here n and ν denotes the AWGN and additive error term, respectively. Note that the symbol and carrier indices are omitted in (4.34) for brevity. As a measure of performance of the system the symbol-error probability (SEP) can be considered, which is the probability that the output of the decision device for x is incorrect. The SEP can be found by evaluating the probability of erroneous detection conditioned on the transmitted symbol for each of the symbols in the regarded modulation. The SEP can then be expressed as Ps =

M 1 1 − P x ∈ Rm |s = Sm , M

(4.35)

m=1

where Rm is the decision region corresponding to the symbol Sm out of the modulation alphabet S with M elements. In a similar way the bit-error probability (BEP) can be evaluated, which can be expressed as 4 5 M B 1 1 1− (4.36) Pb = P x ∈ Rm,b |s = Sm , M B m=1

b=1

where Rm,b is the decision region corresponding to the bth bit of symbol Sm . Clearly, (4.35) and (4.36) largely depend on the distribution of the error term ν. When ν consists of a complex normal distributed error source, i.e., when ν = n, the expressions for (4.35) and (4.36) are well studied for diﬀerent kind of modulations, see e.g. [38, 103]. For a system experiencing ICI due to PN, however, this is not the case, which justiﬁes the study of the distribution of the term ξ in (4.33), as presented in Section 4.3.3. To further illustrate the dependence of the BEP on the distribution of the error term ν, we further work out (4.36) as an example for the QPSK or 4-QAM modulation, for which M = 4 and B = 2. It is easily veriﬁed that [88] % & % & % & % & P νI < − √12 + P νI > √12 + P νR < − √12 + P νR > √12 PbQPSK = , 4 (4.37)

4.3 Impact and distribution of the ICI term

95

where νI and νR denote the imaginary and real part of the error term ν, respectively. In a noiseless scenario, νI and νR can be replaced by ξI and ξR in (4.37), respectively. Here ξI and ξR denote the imaginary and real part of the ICI term ξ, respectively. The ﬁnal expression for the BEP in (4.37) clearly illustrates the dependence of the system performance on the tail probabilities of the error term, i.e., the tail probabilities of the ICI term in a PN-impaired system. The foregoing justiﬁes a careful study of the tail probabilities of ξ, since they fully determine the system performance, as illustrated by (4.37). This study is presented in Section 4.3.3.3.

4.3.3

Properties of the ICI term

With the aim of deriving a statistical characterisation of the error term in detection of the transmitted symbols in a PN-impaired system, we will, in this section, study the ICI term in (4.33) 4 n 5 . / Nc −1 N c −1 2πln 1 s(l) exp j ε (i) exp j . (4.38) ξ= Nc Nc l=1

n=0

i=0

In Section 4.3.3.1, we will study the convergence of ξ when Nc → ∞. In Section 4.3.3.2, we will study the distribution of the limit of ξ when σ is small, i.e., when the interference is quite small, and in Section 4.3.3.3, we will qualitatively study the tails of the distribution when σ is small. Finally, in Section 4.3.4 we will investigate the BEP performance of a system impaired by PN by means of simulations and show how the derived distribution of ξ can be used to accurately predict the BEP performance.

4.3.3.1 Convergence of the ICI term for large Nc Before stating the main convergence result, we need some notation and a key observation. We let {B(t)}t≥0 be a standard Brownian motion. Then 6 . / . /7Nc −1 i+1 i c −1 d {ε (i)}N = σ B , (4.39) − B i=0 Nc Nc i=0 d

where X = Y when the random variables X and Y have the same distribution. Therefore, we have that 4 n . / . /5 N N c −1 c −1 2πln i + 1 i d 1 ξ= s(l) ej Nc exp j σ B −B . Nc Nc Nc n=0 i=0 l=1 (4.40)

96

4 Phase noise

% &

% & i − B , and will idenIn the sequel, we will use ε (i) = σ B i+1 Nc Nc tify the limit of the right-hand side of (4.40), which, for convenience, we again write as ξ. We also deﬁne σs(l) 1 ζ= ejσB(t) [1 − ej2πlt ]dB(t), (4.41) 2πl 0 l∈Z:l=0

which will turn out to be the limit of ξ when Nc → ∞. Intermezzo Some words of caution are necessary here. The integral on the right-hand side of (4.41) is a so-called stochastic integral, which can be deﬁned properly. In fact, the construction of such integral uses limits of the form in (4.40), and these ideas can be used to prove rigorously that the right-hand side of (4.40) converges to (4.41), as we will perform in more detail in Appendix D. Before continuing with the analysis, we list some properties of stochastic integrals that we will rely on. Firstly, we will use the rules that, for functions f, g : [0, 1] × R → R 1 1 such that 0 E[|f (t, B(t))|2 ]dt, 0 E[|g(t, B(t))|2 ]dt < ∞, we have that

1 f (t, B(t))dB(t) = 0, (4.42) E 0

and 1 1

1 f (t, B(t))dB(t) g(t, B(t))dB(t) = E f (t, B(t))g(t, B(t)) dt. E 0

0

0

(4.43)

Secondly, we will use for any function f : [0, 1] → R, we have that 1 f (t)dB(t) (4.44) 0

1 has a normal distribution with mean zero and variance 0 f 2 (t)dt. References for these statements can be found in [104, Chapter 13] or [99]. We will prove the following main result:

Theorem 4.1 When Nc → ∞, for any σ > 0 ﬁxed, ξ in the right-hand side of (4.40) converges in probability to ζ, deﬁned in (4.41). The signiﬁcance of Theorem 4.1 lies in the fact that it proves that the ICI converges when the number of subcarriers grows large, and it identiﬁes the limit explicitly. In the sequel of this section, we make use of this explicit limit ζ in (4.41) to derive properties of the ICI. We will also

4.3 Impact and distribution of the ICI term

97

present simulations that show that the distribution of ξ is already close to the distribution of ζ when Nc equals 64, thus showing that the limit result in Theorem 4.1 is also of practical use. The proof of Theorem 4.1 is deferred to Appendix D.

Simulation results To illustrate the convergence for large number of subcarriers in Theorem 4.1, we have carried out Monte Carlo simulations. In these simulations an OFDM system experiencing PN was simulated, where the PN was modelled as in (4.28). From the results of these simulations the ICI term ξ, as expressed by (4.38) in the analysis above, was evaluated. The evaluation was carried out for QPSK modulation and a system for which all carriers were containing data symbols. As in the analytical derivations, no multipath channel was simulated. For all results 105 independent experiments were performed. First Fig. 4.5 depicts the empirical cumulative distribution function (ECDF), denoted by F(x) in√the ﬁgure, of the normalised real part of the ICI variable, i.e., of R{ξ}/ Nc , as found from these simulations. The results are given for σε2 = 10−3 . For a sample frequency of fs = 1/Ts , this means that the corner frequency of the PN spectrum β is given by 8.1·10−2 times the subcarrier spacing fs /Nc . For an IEEE 802.11a based system [54], where the subcarrier spacing fs /Nc equals 312.5 kHz, this corresponds to a β of 25.3 kHz. Results are depicted in the ﬁgure for systems applying diﬀerent number of carriers, i.e., for Nc equals 64, 128, 256 and 512. From the results in Fig. 4.5 we can observe the convergence in distribution for high number of subcarriers Nc , as stated in the main result of Theorem 4.1 and proven in Appendix D. It is concluded that already for a moderate number of 64 subcarrier convergence seems to be reached, since all curves lie on top of each other. This shows that although the limit distribution was derived for high Nc , it is already applicable for moderate values of Nc . For the imaginary part of the ICI term (4.38) the same convergence occurs. The real part of the normalised ICI term is further studied in Fig. 4.6. It depicts the result of Fig. 4.5 for Nc = 512 subcarriers again, but now in a normal probability plot. This ﬁgure shows the distribution of the ICI together with the corresponding normal distribution, i.e., the normal distribution with the same mean and variance. The scaling of the plot is such that a normal distribution would be depicted as a straight line. It can be concluded from Fig. 4.6 that the ICI is clearly not normally distributed and that, since the curve is above and below the straight line in the left and right part of the ﬁgure, respectively, the ICI distribution

98

4 Phase noise 1

Nc = Nc = Nc = Nc =

0.9 0.8

64 128 256 512

0.7

F(x)

0.6 0.5 0.4 0.3 0.2 0.1 0 −0.04

−0.03

−0.02

−0.01

0

0.01

0.02

0.03

0.04

x ξ Figure 4.5. The ECDF of the real part of the normalised ICI term, i.e., √N , found c from Monte Carlo simulations of an OFDM system experiencing PN. The results are depicted for systems with diﬀerent number of subcarriers Nc . The PN process was modelled according to (4.28) with a variance equal to σε2 = 10−3 .

0.999 0.997 0.99 0.98 0.95 0.90

√ R{ ξ }/ N c (4.38) Corr. normal distr.

F(x)

0.75 0.50 0.25 0.10 0.05 0.02 0.01 0.003 0.001

−5

0 x

5 x 10−3

Figure 4.6. Normal probability plot of the real part of the normalised ICI term, i.e., √ξ , found from Monte Carlo simulations of an OFDM system experiencing PN. The Nc results are depicted for a system with Nc = 512 subcarriers. The PN process was modelled according to (4.28) with a variance equal to σε2 = 10−3 .

99

4.3 Impact and distribution of the ICI term

has thicker tails than the normal distribution. Similar results were found for the imaginary part of the ICI. This result endorses the analytical results obtained in this section.

4.3.3.2 Approximation of the ICI for small σ In this section, we will investigate the distribution of the limit σs(l) 1 ζ= ejσB(t) [1 − ej2πlt ]dB(t). (4.45) 2πl 0 l∈Z:l=0

Since the expression for the limit ζ is a stochastic integral, it is diﬃcult to apply it for analyses. Therefore we here regard the special case where σ is quite small. When this is the case, it is reasonable to assume that we can replace ejσB(t) in (4.45) by 1. This leads to σs(l) 1 ζ ≈ [1 − ej2πlt ]dB(t) 2πl 0 l∈Z:l=0 1 σs(l) B(1) − = ej2πlt dB(t) . (4.46) 2πl 0 l∈Z:l=0

We note that Z = B(1) is standard normally distributed, so that, with X=

s(l) , 2πl

(4.47)

l∈Z:l=0

we arrive at ζ ≈ σXZ −

σs(l) 1 ej2πlt dB(t). 2πl 0

(4.48)

l∈Z:l=0

1 Next, since t → ej2πlt is deterministic, we have that 0 ej2πlt dB(t) has a complex normal distribution. Furthermore, since, for |k| = |l|, E

1

cos (2πlt)dB(t) 0

1 0

cos (2πkt)dB(t) 1 cos (2πlt) cos (2πkt)dt = 0 , (4.49) = 0

√ 1 we have that, with Z(l) = − 2 0 cos (2πlt)dB(t), the sequence {Z(l)}∞ l=1 is a sequence of i.i.d. standard normal random variables, where we are using that a vector of normal random variables is independent when all covariances are equal to zero.

100

4 Phase noise

√ 1 Similarly, we can see that, for l ≥ 1, Z (l) = − 2 0 sin (2πlt)dB(t) are i.i.d. standard normal variables. All these random variables are 1 independent of Z = B(1) = 0 dB(t), since 1 1 1 cos (2πlt)dB(t) 1dB(t) = cos (2πlt)dt = 0. (4.50) E 0

0

0

Finally, using a similar argument, also {Z(l)}l≥1 and {Z (l)}l≥1 are independent. We conclude that 1 1 ζ ≈ ζa = σ ZX1 + (Y+,1 − Y−,2 ) + iσ ZX2 + (Y+,2 + Y−,1 ) , (4.51) 2 2 where we deﬁne, with s(l) = R(l) + jI(l) for l > 0, and s(l) = R (l) + jI (l) for l < 0, X1 =

∞ 1 R(l) + R (l) , 2π l

(4.52)

l=1

∞ 1 I(l) + I (l) X2 = , 2π l l=1 ∞

1 Y±,1 = √ 2π

Z(l)[R(l) ± R (l)] , l

l=1 ∞

1 Y±,2 = − √ 2π

l=1

Z (l)[I(l) ± I (l)] . l

(4.53) (4.54) (4.55)

Note that for the BEP, we have to look at the probability that ζ is larger than a constant, say 1. This we can do when σ tends to zero, to investigate the BEP when the interference decreases. We will investigate such probabilities in more detail in Section 4.3.3.3.

Simulation results In the following we will numerically study ζa , as deﬁned by (4.51), which is the approximation of the limit distribution of the ICI term ζ, as deﬁned in (4.45), for small values of σε . To that end, Monte Carlo simulations were performed, in which, again, an OFDM system experiencing PN was simulated, where the PN was modelled as in (4.28). From the results of these simulations, the ICI term ξ, as expressed by (4.38), was found. The evaluation was carried out for QPSK modulation and a system with Nc = 512 subcarriers that all contained data symbols. Furthermore, ζa was simulated using (4.51), also for QPSK modulation.

101

4.3 Impact and distribution of the ICI term 1

1

R{ζa} / Nc R{ξ}/ Nc

0.9

1

R{ζa} / Nc R{ξ}/ Nc

0.9

0.8

0.8

0.7

0.7

0.7

0.6

0.6

0.6

0.5

F(x)

0.8

F(x)

F(x)

0.9

0.5

0.5

0.4

0.4

0.4

0.3

0.3

0.3

0.2

0.2

0.2

0.1

0.1

0.1

0 −0.15 −0.1 −0.05

0

0.05

x

(a) σε2 = 10−2

0.1 0.15

0

−0.05

0

x

(b) σε2 = 10−3

0.05

R{ζa} / Nc R{ξ}/ Nc

0 −0.015 −0.01−0.005 0

0.005 0.01 0.015

x

(c) σε2 = 10−4

ξ Figure 4.7. The ECDF of the real part of the normalised ICI term, i.e., √N (dashed c lines) and normalised real part of the approximation of the limit distribution for small σε , i.e., √ζNa c (solid lines). The results are depicted for Nc = 512 subcarriers and varying σε2 .

The results from these simulations for the normalised real part of the two resulting distributions are presented in Fig. 4.7, by means of their empirical cumulative distribution function (ECDF). The results are given in Fig. 4.7(a), Fig. 4.7(b) and Fig. 4.7(c) for σε2 equal to 10−2 , 10−3 and 10−4 , respectively. This corresponds to a −3 dB oscillator bandwidth β of 8.1 · 10−1 , 8.1 · 10−2 and 8.1 · 10−3 times the subcarrier spacing fs /Nc , respectively. For all simulation results in this section, we have performed 105 independent experiments. It can be concluded from Fig. 4.7 that the resemblance of the two distributions increases with decreasing β, which is as expected since ζa was derived under the assumption of small σ and, thus, small β. It is concluded that for the considered system already for σε2 = 10−3 reasonable agreement between the CDFs seems to be achieved. This indicates that the approximate limit distribution ζa in (4.51) correctly models the distribution of the ICI for these values of σε2 . The ECDF of results from similar simulations, but now for σε2 = 10−4 , are given in Fig. 4.8. Here the ECDF is depicted on a logarithmic scale, which enables us to study the tails of the distributions. The ﬁgure, again, depicts the simulated real part of the normalised ICI of (4.38), the real part of the approximation of the limit distribution of the ICI term of (4.51), but now also the corresponding normal distribution, i.e., with the same mean and variance as the other variables. In Fig. 4.8, we have performed 106 experiments for each result.

102

4 Phase noise 100

√ R{ξ}/ Nc Normal√distr. R{ζa}/ Nc

F(x)

10−1

10−2

10−3

10−4 −0.02

−0.015

−0.01 −0.005 x

0

0.005

0.01

ξ Figure 4.8. The ECDF of the real part of normalised ICI term √N (dotted line), c as expressed in (4.38), and of the normalised approximation of the limit distribution √ζa (solid line), as expressed in (4.51), and of the corresponding normal distribution Nc (dashed line) with the same mean and variance. Results are given for Nc = 512 subcarriers, QPSK modulation and a PN variance of σε2 = 10−4 .

It can be concluded from Fig. 4.8 that the limit distribution ζa well approximates the ICI, even in the tails of the distribution. Furthermore, it is found that the Gaussian distribution shows lower tail probabilities, and has a faster fall oﬀ. For instance, the probability that the real part of the normalised ICI is smaller or equal than −0.01 is approximately 2 · 10−3 , i.e., P(ξ ≤ −0.01) ≈ 2 · 10−3 . This is correctly predicted by the proposed limit distribution. The corresponding Gaussian approximation of the ICI, however, predicts the probability to be approximately 2·10−4 , which underestimates it by about a factor 10.

4.3.3.3 Tail probabilities In this section, we will analytically show that the tail probabilities of the random variables in Section 4.3.3.2 are diﬀerent from the ones for a Gaussian approximation, what was already numerically illustrated in Fig. 4.8. Tail probabilities are important in the ICI case since the BEP can be rewritten in terms of the tail probabilities, as was elucidated for

4.3 Impact and distribution of the ICI term

103

QPSK in Section 4.3.2. Therefore, a system with smaller tail probabilities performs better than a system with larger tail probabilities. We present a qualitative analysis of such tail probabilities in order to show that the usual Gaussian assumptions lead to an underestimation for the BEPs. This will be substantiated further by the results from appropriate BEP simulations in Section 4.3.4. We will start by computing the ﬁrst two moments of the random variables appearing in (4.51): E[ZX1 ] = E[Y±,1 ] = 0,

(4.56)

while, writing var(R1 ) = σR2 , E[(ZX1 )2 ] = E[Z 2 ]E[X12 ] = σ 2 var(X1 ) ∞ 1 var(R1 ) σ 2 σR2 = 2σ 2 = , (4.57) = 2σ 2 var(R1 ) 2 2 π l 6 3 l=1

and 2 ] E[Y±,1

∞ 1 σ 2 σR2 σ 2 var(R1 ) = . = σ var(R1 ) = π 2 l2 6 6 2

(4.58)

l=1

Using further that E[ZX1 Y±,1 ] = E[ZX1 Y±,2 ] = 0, the random variable ξR = σZX1 + σ2 (Y+,1 − Y−,2 ), which signiﬁes the real part of the ICI, has mean zero and variance 1 2 2 var(ξR ) = σ 2 E[(ZX1 )2 ] + E[Y+,1 ] + E[Y−,2 ] /4 . 1 1 5σ 2 σR2 + = . (4.59) = σ 2 σR2 3 12 12 Therefore, the usual Gaussian assumptions lead to a tail estimate of the form / . %y& 6y 2 P(ξR > y) ≈ Q (4.60) = exp − 2 2 [1 + o(1)] . σ 5σ σR In the explanation below, we will assume that R(l) and I(l) are ±1 with equal probability, for which σR2 = 1. For this, we ﬁx an M , and we investigate the probability that R(l) = R (l) = 1 for l ≤ M , while R(l)+R (l) ≥ 0 and Y+,1 + Y−,2 ≥ 0. By symmetry of the random l>M l variables involved, this probability is at least 41M · 12 · 12 . Also, when the above is true, then X1 ≥

M 1 1 = log M [1 + o(1)]. πl π l=1

(4.61)

104

4 Phase noise

Therefore, in order for σZX1 ≥ y to hold, we only need that Z≥

πy , σ log M

which has probability . / . / πy π2 y2 Q ∼ exp − 2 [1 + o(1)] . σ log M 2σ log2 M

(4.62)

(4.63)

Therefore, for every M ≥ 1 and y > 0, . / π2 y2 1 1 [1 + o(1)] . (4.64) P(ξR > y) ≥ M · exp − 2 4 4 2σ log2 M y2 y2 1 For example, for M = σ2 log ) , 4 y , we obtain that 4M +1 = exp o( 2 2 ( ) σ log M so that

σ

4

5 π2 y2 P(ξR > y) ≥ exp − 2 [1 + o(1)] . 8σ log2 ( σy )

(4.65)

The tail in (4.65) is much larger than the ones in (4.60), so that Gaussian assumptions, as formulated in (4.60), lead to a systematic underestimation of the tails, and therefore of the BEPs. Indeed, when y = 1 and σ is quite small, the exponent has become a factor 48 log2 ( σ1 )/5π 2 smaller, which is substantial for σ small. This can be clearly seen in the results from simulations in Fig. 4.6, where we see thicker tails of the ICI distribution compared to a normal random variable with the same variance. This is even more clear from the ECDF depicted on a logarithmic scale in Fig. 4.8. A similar analysis could be carried out for the imaginary part of the ICI ξI , yielding similar conclusions about the tail probabilities.

4.3.4

Simulation results

To conﬁrm these analytical ﬁndings, we have carried out BEP simulations. An IEEE 802.11a-like system was simulated, see Section 3.2.1. In the system, however, all 64 subcarriers contain data symbols, which for Fig. 4.9 are taken from the 16-QAM modulation alphabet and for Fig. 4.10 from both the 16-QAM and 64-QAM modulation alphabet. A system without coding and not experiencing a multipath channel was simulated. The packet length in the simulations was equal to 10 symbols. Figure 4.9 shows BEP results for a system impaired by both AWGN and PN. The BEP is depicted versus the signal-to-noise ratio (SNR) for diﬀerent values of β, i.e., the −3 dB bandwidth of the LO power spectral density. The PN is generated according to the model in (4.28) and the CPE is perfectly removed, leaving the ICI as the only inﬂuence of the

105

4.3 Impact and distribution of the ICI term 100

β β β β

10−1

= = = =

700 Hz 1000 Hz 2000 Hz 5000 Hz

BEP

10−2

10−3

10−4

10−5 10−6 0

5

10

15

20

25

30

35

Average SNR (dB)

Figure 4.9. BEP results from Monte Carlo simulations with an IEEE 802.11a-like system applying 16-QAM modulation. Results are included for: a system experiencing PN as modelled by (4.28) for diﬀerent values of β and applying perfect CPE correction (dashed lines), a system modelling the resulting ICI as an equivalent complex Gaussian process (dash-dot lines) and a system modelling the resulting ICI according to the small σ approximation of the limit distribution, as deﬁned by (4.51) (solid lines).

PN. Results from these simulations are given by the dashed lines in the ﬁgure. The simulated βs vary from 700 to 5000 Hz, which corresponds to σε2 values varying from 1.4π · 10−4 to π · 10−3 . These results are depicted together with results from simulations where the inﬂuence of the ICI is modelled as an additional zero-mean complex Gaussian noise source at the receiver, i.e., the commonly used approach in previous literature, where the variance of the noise equals that of the actual ICI. The results of these simulations are given by the dash-dot lines. Finally, results are included from simulations where the inﬂuence of the ICI is modelled using the small σ approximation of the proposed model for ICI, i.e., ζa , as deﬁned by (4.51). These results are depicted by solid lines. It can be concluded from Fig. 4.9 that the BEP is severely underestimated using the Gaussian approximation for the ICI term. Especially for small values of β, i.e., small σ values, the diﬀerences between the actual and predicted BEP by the Gaussian model are very large, which conﬁrms our analytical ﬁndings on tail probabilities in the previous section.

106

4 Phase noise

For instance for β = 2000 Hz and an SNR of 30 dB, the BEP is underestimated by a factor of 20 using the Gaussian approach. For lower values of β this diﬀerence is even higher. The resulting BEPs using the approximate of the proposed limit distribution, in contrast, closely resemble those of the actual PN-impaired system. It is noted that this limit distribution was found under the assumption of a large number of carriers, but, judging the results, it already holds for a system with a modest number of 64 subcarriers. Results depicted in Fig. 4.10 are obtained from similar simulations carried out for a system experiencing no additive RX noise, i.e., the system is only impaired by PN. Results from BEP simulations, for the same three cases as for Fig. 4.9, are depicted versus β for both the 16-QAM (in solid lines) and 64-QAM (in dashed lines) modulation. These results can be interpreted as the high SNR ﬂooring of the BEP curve for a certain β, i.e., the maximum achievable BEP performance for a certain level of PN.

100

10−1

Actual System Gaussian approx. Small σ approx. ζa

BEP

10−2

10−3

10−4

10−5 102

103

104

105

β (Hz)

Figure 4.10. BEP results from Monte Carlo simulations with an IEEE 802.11a-like system applying 16-QAM (solid lines) and 64-QAM (dashed lines) modulation. Results are included for: a system experiencing PN as modelled by (4.28) for diﬀerent values of β and applying perfect CPE correction (“Actual System”), a system modelling the resulting ICI as an equivalent complex Gaussian process (“Gaussian approx.”) and a system modelling the resulting ICI according to the small σ approximation of the limit distribution, as deﬁned by (4.51) (“Small σ approx. ζa ”).

4.3 Impact and distribution of the ICI term

107

It can be concluded from Fig. 4.10 that for both modulation formats the Gaussian approximation of the ICI only provides reliable BEP results for very high values of β, where the BEP is so high that reliable data transfer is almost not possible. This conﬁrms ﬁndings previously presented in [94] and [95], where simulations were used to show the validity of the Gaussian model for the ICI only for very large values of β. The BEPs found from simulations with ζa , the approximate expression for our limit distribution ζ, on the other hand, show good agreement with the BEPs found from simulations with the actual PN-impaired system. It is noted though that a small discrepancy occurs at high values of β, where σ becomes high, since the approximate results of Section 4.3.3.2 were derived for small σ. The results are, however, even for these high β values reasonable. For these high values of β the actual evaluation of ζ in (4.41) will obtain better results. Overall it can be concluded from the simulation results in this section that, although derived for large Nc and small σ, the approximate limit distribution ζa can be well applied to ﬁnd the BEP of a PN-impaired system. It is noted, furthermore, that simulations were carried out for a system with a low number of carriers, i.e., Nc = 64, and that the applicability of our results will even be better for systems with higher number of subcarriers like, e.g., WiMAX, DVB and DAB, since for this kind of systems Nc is much larger and the typically permissible σ is considerably lower.

4.3.5

Summary

In most of the previous contributions in the literature, the ICI was assumed to be distributed according to a zero-mean complex Gaussian distribution. In this section, however, it was shown that this assumption is not valid and the limit distribution of the ICI term for large number of subcarriers is derived. This distribution is shown to exhibit thicker tails than the Gaussian distribution with the same mean and variance. In an analysis of the tail probabilities these ﬁndings were conﬁrmed and it was shown that bit-error probabilities are severely underestimated when the Gaussian approximation for the ICI term is used. Results from a numerical study show the validity of the limit distribution, obtained under the assumption of a large number of subcarriers, already for a modest number of subcarriers. Furthermore, they show that for small values of the PN variance, the limit distribution very well resembles the ICI distribution. Finally, it is shown that the tail probabilities are severely underestimated by the corresponding Gaussian distribution.

108

4 Phase noise

The results of this section can be used by system designers to better specify the level of tolerable PN, for an OFDM system to achieve a certain bit-error probability. This section shows that applying the Gaussian approximation would lead to a serious underspeciﬁcation of the local oscillator.

4.4

ICI-caused error term in MIMO OFDM

As was shown in Section 4.3, the inﬂuence of TX- and RX-induced phase noise is equivalent for single-antenna OFDM systems not experiencing a fading channel. It was anticipated from (4.23), however, that for a MIMO OFDM system in a fading channel the inﬂuence of TX and RX phase noise on the system performance would be diﬀerent. It can be concluded from (4.23) that the inﬂuence of TX and RX PN on the common phase error (CPE) is equal. In contrast, the structure of the ICI term in (4.24) reveals speciﬁc diﬀerences between TX and RX PN. Therefore, this section compares the inﬂuence of TX- and RX-caused ICI on the performance of a zero-forcing (ZF) based MIMO receiver. The performance is found by analytically deriving the power of the ICI-induced errors in the estimation of the transmitted symbols. This analysis, which ﬁrst appeared in [87], is carried out for two extreme fading cases, i.e., frequency ﬂat Rayleigh fading and per subcarrier independent Rayleigh fading channels. The results can be used to derive the probability of error of MIMO OFDM systems experiencing PN. Results for more realistic fading situations, with partial correlation between the channel elements for the subcarriers, are bounded by the results for these two cases. For these fading conditions, namely, the channel can be rewritten as a sum of the here considered channels, i.e., the frequency ﬂat Rayleigh faded and the subcarrier independent Rayleigh faded channel. For convenience we repeat (4.23) here, i.e., the expression for the Nc Nr ×1 received signal vector for the mth OFDM symbol in a PNimpaired system xm = γ0,m Hsm + ξm + nm ,

(4.66)

where γ0,m denotes the CPE, H is the Nc Nr ×Nc Nt block-diagonal channel matrix, sm is the Nc Nt ×1 transmitted signal vector, ξm is the Nc Nr ×1 ICI vector and nm is the Nc Nr ×1 noise vector. We recall that when the number of carriers is chosen to be large in comparison to the size of the channel dispersion, the MIMO processing can be applied per subcarrier, since the channel experienced at a subcarrier can be considered as essentially being frequency ﬂat. When ZF is considered as MIMO processing, as introduced in Section 2.3.1, the received signal vector xm is multiplied with the pseudo-inverse of

4.4 ICI-caused error term in MIMO OFDM

109

the estimated channel matrix to retrieve estimates of the transmitted symbols. We note that this matrix is a block-diagonal matrix with the pseudo-inverses of the MIMO channel experienced on the diﬀerent carriers on its diagonal. The channel matrix is here assumed to be perfectly known at the RX. The estimate of the transmitted signal ˜sm is then given by ˜sm = H† xm = γ0,m sm + Ξm + H† nm ,

(4.67)

†

where we recall that denotes the pseudo-inverse. The error in detection due to the ICI term is given by Ξm = H† ξm RX = H† (ϕRX,m ⊗ INr )H(ϕTX,m ⊗ INt )sm + γ0,m (ϕTX,m ⊗ INt )sm TX +γ0,m H† (ϕRX,m ⊗ INr )Hsm

(4.68)

The form in which the TX and RX terms appear in (4.68) indicates that TX and RX PN-induced ICI inﬂuence the performance of the estimator diﬀerently. To compare their inﬂuence, they are analysed separately in the following. For brevity the symbol index m is omitted in the remainder of this section.

4.4.1

Transmitter phase noise

For a system only experiencing TX PN and where, thus, the RX oscillator is ideal, i.e., G RX = INc , the error term in the estimated TX signal Ξ only contains the ICI induced error due to TX PN and is given by the Nc Nt × 1 vector Ξ = (ϕTX ⊗ INt )s.

(4.69)

The Nt ×1 error vector for the kth carrier is given by Ξ(k) =

Nc

TX γk−l s(l),

(4.70)

l=1,l=k TX was deﬁned in (4.22) and s(k) denotes the Nt ×1 TX vector where γk−l for the kth subcarrier. The elements of s are i.i.d. with zero-mean and a variance of σs2 . The power of this error term, averaged over Nt antennas, for the kth carrier is then given by 1 1 H PΞ(k) = tr E Ξ(k)ΞH (k) = E Ξ (k)Ξ(k) Nt Nt Nc

$ $ TX $2 E $γk−l , (4.71) = σs2

l=1,l=k

110

4 Phase noise

where H denotes the conjugate transpose and tr{·} calculates the trace of its argument.

4.4.2

Receiver phase noise

For a system only experiencing RX PN and where, thus, the TX oscillator is ideal, i.e., G TX = INc , the error term in the estimated RX signal Ξ only contains the ICI induced error due to RX PN and is given by the Nc Nt × 1 vector Ξ = H† (ϕRX ⊗ INr ) Hs.

(4.72)

The Nt ×1 error vector for the kth carrier is given by Ξ(k) = H† (k)

Nc

RX γk−l H(l)s(l).

(4.73)

l=1,l=k

We note that this term, in contrast to the TX PN case in (4.70), does contain inﬂuence of the MIMO channel matrix. The average power of Ξ(k) is given by 1 H E Ξ (k)Ξ(k) Nt ⎡⎛ ⎞H⎛ ⎞⎤ N N c c 1 ⎢⎝ † ⎥ RX RX E ⎣ H (k) γk−l H(l)s(l)⎠⎝H† (k) γk−l H(l)s(l)⎠⎦ = Nt

PΞ(k) =

l=1,l=k

=

l=1,l=k

Nc

$ $ H 1 H RX $2 E s (l)HH (l)H† (k)H† (k)H(l)s(l) E $γk−l Nt l=1,l=k

=

Nc

$ $ H σs2 H RX $2 E $γk−l tr E H (l)H† (k)H† (k)H(l) . (4.74) Nt l=1,l=k

The average power of the error term of estimated symbols depends on the interaction between channel elements at diﬀerent subcarrier locations. Therefore, the channel conditions will inﬂuence the power of the error term. To quantify the impact, the analysis is carried out for two special cases: Case 1, where the channel is ﬂat Rayleigh faded over the whole system bandwidth; and Case 2, where all subcarriers experience independent Rayleigh faded channel elements.

111

4.4 ICI-caused error term in MIMO OFDM

4.4.2.1 Case 1 - Flat fading For the ﬂat fading case, we have H(k) = H(l), so that (4.74) can be simpliﬁed to Nc

$ $ 2 RX $2 E $γk−l PΞ(k) = σs . (4.75) l=1,l=k

When comparing (4.75) to (4.71), it is clear that for ﬂat fading there is no diﬀerence between the impact of TX and RX PN.

4.4.2.2 Case 2 - Independent Fading For the independent fading case, the elements of H(k) and H(l) are i.i.d. according to the zero-mean, unit variance complex Gaussian distribution. Using these channel properties, (4.74) can be rewritten to PΞ(k)

Nc

$ $ † σs2 RX $2 H H $ = E γk−l tr E H(k)H (k) E H(l)H (l) Nt l=1,l=k

Nc

$ $ † σs2 RX $2 tr E H(k)HH (k) Nt σh2 INr E $γk−l = Nt l=1,l=k

Nc

$ $ † H RX $2 = σs tr E H(k)H (k) E $γk−l , 2

(4.76)

l=1,l=k

where σh2 denotes the variance of the channel elements and equals unity † for our channel model. Furthermore, the matrix H(k)HH (k) is distributed according to the complex inverse Wishart distribution [105]. For the special case of Nr > Nt , it has been shown that [105] † Nt H . (4.77) tr E H(k)H (k) = Nr − Nt When we apply this, (4.76) can be rewritten as PΞ(k) =

Nc

$ $ Nt RX $2 , σs2 E $γk−l Nr − Nt

for Nr > Nt .

(4.78)

l=1,l=k

When comparing the result of (4.78) with the one for TX PN in (4.71), we can conclude that when Nr > 2Nt , the RX PN has less impact than TX PN for independent Rayleigh faded channels. When Nt < Nr < 2Nt , the RX PN has greater impact than TX PN. For example, for a 2 TX and 3 RX system, Nt /(Nr − Nt ) = 2, RX PN leads to a twice greater average power level of the error term.

112

4 Phase noise

4.4.3

Phase noise process

The average power of Ξ(k), as derived above, depends on the PN process through γq . Here we assume the phase noise to be modelled as a free running oscillator, as deﬁned in Section n−14.2.1. We recall that the phase noise process is given by θ (n) = X i=0 ε(n) , where ε(n) ∼ N 0, σε2 and σε2 = 4πβTs . Applying the properties of the PN process, E[|γq |2 ] can be calculated. To this end, γq , as deﬁned in (4.22), is rewritten for q ∈ {−Nc + 1, . . . , −1, 1, . . . , Nc − 1} using the small angle approximation, which is valid when the −3 dB linewidth β is small as compared to the subcarrier spacing 1/(Ts Nc ). When, furthermore, the order of summation is changed, γq is given by γq =

Nc −1 2πqn 1 ejθX (Ng +n) e−j Nc Nc

χ0 ≈ Nc =

n=0 N c −1

jχ0 Nc

−j 2πqn N

e

n=0 N n c −1 n=0 i=0

c

Nc −1 2πqn jχ0 + θX (n)e−j Nc Nc n=0

ε(i)e−j

2πqn Nc

=

Nc −1 N c −1 2πqn jχ0 ε(i) e−j Nc , (4.79) Nc i=0

n=i

Ng ε(l) . Using the i.i.d. property of the elements where χ0 = exp j l=0 of ε(n), the expected value of the power of γq can be rewritten to Nc −1 N c −1 c −1 2πqm 2 N |χ0 |2 −j 2πqn Nc E |γq |2 = E ε (i) e ej Nc 2 Nc i=0 n=i m=i % πqi & πqi 2 j −j Nc −1 e Nc − e Nc σ2 = ε2 % πq &2 πq Nc n=0 ej Nc − e−j Nc % & N c −1 σε2 sin2 qnπ Nc 2πβTs n=0 % & = % &. = qπ 2 qπ 2 Nc sin Nc Nc sin2 N c

(4.80)

Substituting expression (4.80) into (4.71), for the case of TX PN, the expression for the average power of the error term due to ICI on the kth carrier can be written as . / Nc 2σs2 πβTs −2 (k − l)π PΞ(k) = sin . (4.81) Nc Nc l=1,l=k

113

4.4 ICI-caused error term in MIMO OFDM

In most systems the number of carriers is chosen to be a power of 2, i.e., Nc = 2M , due to ease of implementation of the discrete Fourier transform for this number of carriers. When this is assumed, the sum in (4.81) can be rewritten to 2 M

−2

.

sin

l=1,l=k

(k − l)π 2M

/ =

M −1 2

=

.

sin

l=1

2 = 1+ +2 2 sin (π/4) = 1+4+8

−2

−2 −1 2M

22M − 1 = 3

/

−2 −1 6 2M

=1+2

−2

sin

.

πl

−1 −1 2M

sin−2

l=1 −2

.

sin

l=1

l=1 N2 c

πl 2M

πl 2M

/ + cos

−2

.

.

πl 2M

πl 2M

/

/7

/

2M −1

−1 . 3

(4.82)

Substituting (4.82) into (4.81) results in the following expression for the average power of the error term Ξ(k): PΞ(k) =

2σs2 πβTs (Nc2 − 1) . 3Nc

(4.83)

We note that this expression no longer depends on the carrier index k. For a large number of carriers (4.83) is well approximated by 23 σs2 πβTs Nc . Similarly, the expressions for the average ICI power in (4.75) and (4.78) for RX PN can be further simpliﬁed when the PN model of Section 4.2.1 is assumed. The approximation of PΞ(k) for a large number of carriers for Case 1, i.e., ﬂat Rayleigh fading, is given by 2 PΞ(k) ≈ σs2 πβTs Nc 3

(4.84)

and for Case 2 with Nr > Nt , i.e., for a system experiencing independent Rayleigh fading, it is given by PΞ(k) ≈

4.4.4

2Nt σ 2 πβTs Nc . 3(Nr − Nt ) s

(4.85)

Numerical results

The analytical results of Section 4.4.3 are in Fig. 4.11 compared with simulation results from Monte Carlo simulations. In these simulations the sample time Ts was 50 ns, the number of subcarriers Nc = 64, 64-QAM modulation was applied and σs2 = 1. The PN is modelled as

114

4 Phase noise 100

PΞ(k)

10−1

10−2

TX PN, analy RX PN, 2×3, analy RX PN, 2×8, analy TX PN, 2×3, sim RX PN, 2×3, sim TX PN, 2×8, sim RX PN, 2×8, sim

10−3

10−4

101

102

103 b (Hz)

104

105

Figure 4.11. Average power of the error in estimated symbols due to ICI as a function of the PN corner frequency β, for σs2 = 1 and 64-QAM modulation. The channel is modelled as independent Rayleigh faded per subcarrier. Analytical (“analy”) and simulation (“sim”) results are depicted.

described in Section 4.2.1. In Fig. 4.11 the average ICI-induced estimation error is plotted as a function of β, the −3 dB bandwith of the Lorentzian power spectral density of the PN. The results are given for diﬀerent combinations of number of TX and RX antennas. Every subcarrier experienced an i.i.d. Rayleigh faded channel in the simulations. It is clear from Fig. 4.11 that there is good agreement between the analytical results from Section 4.4.3 and the simulation results. The discrepancy at high β values can be explained by the fact that the small angle approximation is no longer valid. Here β is no longer small compared to the subcarrier spacing, which is 312.5 kHz for the regarded scenario. Figure 4.12 shows the BER as a function of the average SNR per RX antenna for diﬀerent MIMO conﬁgurations. Five curves show the inﬂuence of TX PN for a corner frequency of the PN PSD of β = 1000 Hz. A ﬂat Rayleigh fading channel is used for simulations. For three curves the corresponding curves without PN are also plotted. It is clear from Fig. 4.12 that all MIMO conﬁguration reach the same error ﬂoor, i.e., 5.8 · 10−3 , independently of Nt and Nr . This eﬀect was explained in Section 4.4.1.

115

4.4 ICI-caused error term in MIMO OFDM 100 2× 2 2× 3 2× 4 3× 4 4× 4

BER

10-1

10-2

10-3

0

10

20

30

40

50

Average SNR per RX antenna (dB)

Figure 4.12. BER performance as a function of average SNR for diﬀerent MIMO OFDM conﬁgurations without PN (dashed curves) and with TX PN for β = 1000 Hz (solid curves) for a ﬂat Rayleigh fading channel. 64-QAM modulation is used. 100

2× 2 2× 3 2× 4 2× 8

BER

10−1

10−2

10−3

10−4 0

10

20

30

40

50

Average SNR per RX antenna (dB)

Figure 4.13. BER performance as a function of average SNR for diﬀerent MIMO OFDM conﬁgurations with TX PN (solid curves) and RX PN (dashed curves) for β = 1000 Hz for an independent Rayleigh fading channel. 64-QAM modulation is used.

116

4 Phase noise

Figure 4.13 compares the BER performance of diﬀerent MIMO OFDM systems experiencing either TX (solid curves) or RX (dashed curves) PN. For these simulations the channel was independently Rayleigh faded and the PN was characterised by β = 1000 Hz. It is clear from Fig. 4.13 that, again, all curves for TX PN result in the same BER ﬂoor. The ﬂoor, however, varies for RX PN, and clearly depends on the number of receiver antennas Nr . For 2 and 3 RX antennas, the performance is worse for the RX PN. For Nr = 4, the performance of the systems with TX and RX PN are almost similar, although there is a small oﬀset, which might be addressed to the diﬀerence in distribution of the error terms. For 8 RX antennas, the ﬂoor for RX PN is lower than for TX PN. These numerical results conﬁrm the analytical results of Section 4.4.2.

4.4.5

Summary and discussion

In this section the inﬂuence of TX and RX phase noise on signal detection in MIMO OFDM systems was investigated and shown to be diﬀerent. An expression is found for the power of the error term after zero-forcing (ZF) estimation caused by the ICI term. For TX PN, the power of this error term is equal for all MIMO conﬁgurations. The power of the error term for RX PN is equal to that of the TX case for a ﬂat Rayleigh faded channel. For independent Rayleigh faded channels, however, the RX PN is concluded to have less impact than TX PN when Nr > 2Nt and more impact when Nt < Nr < 2Nt . Simulation results conﬁrm the results from the analytical study. The observed diﬀerences between the inﬂuence of TX and RX phase noise in systems experiencing fading channels can intuitively be understood as follows. The ICI term eﬀectively creates an extra TX or RX additive noise source for the TX and RX phase noise, respectively. Since the TX noise source occurs in front of the propagation channel, it will result in high SNR ﬂooring in the BER curves according to the AWGN BER performance, i.e., independent of the MIMO conﬁguration. Therefore, also the power of the resulting error term is independent of the dimensions of the channel matrix. For RX phase noise, in contrast, the ICI source occurs behind the fading MIMO channel. As such, its inﬂuence is similar to that of the commonly studied additive Gaussian RX noise source, i.e., the BER performance (and thus the BER ﬂooring) depends on the MIMO conﬁguration. Consequently, the power of the resulting detection error depends on the dimensions of the MIMO channel, i.e., the number of TX and RX branches. For the special case of ﬂat fading, the ZF MIMO detection removes the inﬂuence of the channel in the RX phase noise caused ICI term. As such, its inﬂuence is eﬀectively equal to that of a TX noise source.

4.5 Compensation of the CPE

4.5

117

Compensation of the CPE

Phase noise primarily jeopardises the performance of an OFDM system through the CPE, since it rotates the complex constellation points towards the decision boundaries, increasing the probability of an erroneous detection. Due to its additive character and zero-mean distribution, the ICI is less destructive. Therefore, it is important to estimate and correct for the CPE, to suppress a large part of the inﬂuence of the PN. We conclude from (4.22) that the CPE changes on a symbol-by-symbol basis. It, therefore, has to be estimated and corrected for every MIMO OFDM symbol separately. Two methods to do this are pilot-aided and decision-directed estimation. In the ﬁrst approach, known symbols are transmitted on certain carriers to enable estimation, i.e., pilot carriers. A clear disadvantage of this method is the induced overhead. For decision-directed estimation, the estimated symbols on all the carriers from detection without CPE correction are used as a basis to estimate the CPE. The major disadvantages of this method are the processing delay, complexity and the low accuracy for large values of CPE. Since most OFDM systems apply pilot carriers in their frame format, the approach presented here is based on pilot-aided CPE estimation. We denote the set of P pilot carrier numbers as P = {p1 , p2 , . . . , pP }. From (4.23) it is clear that the received signal, or observation vector, on pilot carrier p (p ∈ P) during the reception of the mth MIMO OFDM is given by xm (p) = γ0,m H(p)sm (p) + ξm (p) + nm (p). (4.86) We assume that the channel matrix H is known at the receiver, which is a valid assumption since a system based on coherent detection would apply channel training. Furthermore, the symbols sm (p) are pilots and thus known at the receiver. This enables the estimation of γ0,m and its phase ϑm .

4.5.1

Maximum-likelihood estimator

Suppression approaches proposed for SISO OFDM [89, 91, 106], which inherently assume independently and identically distributed (i.i.d.) noise terms, are not directly applicable to the MIMO case, since the ICI can exhibit spatial correlation. This can be observed from the expression for the ICI in (4.24) which contains the complex channel matrix. This channel might exhibit spatial correlation due to either the propagation channel or mutual coupling in the TX/RX chain including antennas and RF/analog front-ends. Consequently, the ICI term will also exhibit correlation and its elements are not i.i.d. This aﬀects the performance of

118

4 Phase noise

estimation and detection approaches, which are based on the assumption of i.i.d. white Gaussian noise terms. Therefore, and to understand the best attainable performance, an exact maximum-likelihood estimator (MLE) is developed in this section, as originally proposed in [107, 86]. The MLE was chosen for its well-known asymptotic optimality. Similarly to the derivation in [108], we assume the observation noise zm (p) = xm (p) − γ0,m H(p)sm (p) = ξm (p) + nm (p)

(4.87)

is multivariate complex normally distributed with the unknown Nr ×Nr covariance matrix Ω, i.e., zm (p) ∼ CN (0, Ω) for any p ∈ P. Note that the ICI term is diﬀerent than for the non-fading channel case studied in Section 4.3, since the channel is here modelled to be Rayleigh faded. For this case, it is more likely that the ICI term, as expressed by (4.24), will resemble a complex normally distributed variable, since the channel elements are also complex normally distributed. Furthermore, it is assumed that the estimation errors on the diﬀerent pilot carriers are uncorrelated over frequency, which can be achieved by spacing the pilot carriers far enough apart, but such that the observation noise on the same pilot carriers on the diﬀerent RX antennas can be correlated. As shown in Section 4.2.2, γ0,m is a function of the symbol index m and thus changes over time. Also the observation noise changes over time, but its covariance matrix stays constant, since the channel is quasistatic. We exploit this fact by simultaneously estimating the vector γ0 = [γ0,1 , γ0,2 , . . . , γ0,M ]T , where M is the number of symbols in the regarded packet. The joint probability density function, conditioned on all unknown parameters, is given by [109] ⎛ ⎞ M −P M N r π −1 ⎠ (4.88) p(z|γ0 , Ω) = exp ⎝− zH m (p)Ω zm (p) . det(Ω)P M m=1 p∈P

When we now deﬁne the matrix Zm = [zm (p1 ), zm (p2 ), . . . , zm (pP )]H of size P ×Nr , the joint probability density function of (4.88) can be written as 4 M 5 π −P M Nr −1 H exp − tr(Zm Ω Zm ) . (4.89) p(Z|γ0 , Ω) = det(Ω)P M m=1

The MLE is then obtained by maximising ln (p(Z|γ0 , Ω)), i.e., the loglikelihood (LL) function, which is given by L(γ0 , Ω) = C1 + P M ln(det(Ω−1 )) −

M m=1

tr Zm Ω−1 ZH m ,

(4.90)

119

4.5 Compensation of the CPE

where C1 denotes a constant. The LL function is maximised over γ0 and Ω. Setting the partial derivative of the LLfunction with respect to Ω ˜ = M (ZH Zm )/ (P M ) [110]. to zero gives the conditional estimate Ω m m=1 When this is substituted in (4.90), the conditional LL function is given by 5" 4 M ˜ = C2 − P M ln det L(γ0 , Ω) , (4.91) ZH m Zm m=1

where C2 denotes a constant. Maximising this log-likelihood function then boils down to minimising ⎛ ⎞ M ⎠ Φ(γ0 ) = det ⎝ zm (p)zH (4.92) m (p) . m=1 p∈P

The elements of γ0 have approximately unity amplitude and, thus, represent phase rotations. Therefore, they can be characterised by their phases ϑ = [ϑ1 , ϑ2 , . . . , ϑM ]T . The estimation of ϑ is preferred over estimation of γ0 , since ϑ only contains real variables. That is why the remainder of this section will focus on the estimation of ϑ.

4.5.2

MLE optimisation algorithm

The minimisation of the determinant criterion (4.92), or cost function Φ(ϑ), can be implemented using several optimisation techniques. An iterative technique, based on the Gauss-Newton algorithm was chosen here, since it is well established for problems like (4.92). The algorithm can, for this linear case, be summarised as follows: 1 Set the iteration number i to 0 and select a feasible initial estimate for ϑ(0) . 2 Stop if the convergence criterion is reached. 3 Calculate the gradient vector d(i) and the Hessian matrix H(i) , i.e., the matrix of second derivatives. 4 Solving the search direction vector δ (i) from H(i) δ (i) = −d(i) . 5 Set ϑ(i+1) = ϑ(i) + δ (k) and i = i + 1. Return to Step 2. Here y (i) denotes the vector y during the ith iteration. In Step 3 the gradient d and the Hessian H have to be calculated. To derive these, we rewrite the cost function Φ as a function of ϑ, by substituting γ0 in (4.92) by cos(ϑ) + j sin(ϑ), resulting in Φ(ϑ) = det(E),

(4.93)

120

4 Phase noise

where E =A+

M

Bm cos(ϑm ) +

m=1

M

Cm sin(ϑm ),

(4.94)

m=1

⎧ M ⎪ H ⎪ A = xm (p)xH ⎪ m (p) + ym (p)ym (p) ⎪ ⎪ m=1 p∈P ⎨ H (p) + y (p)xH (p) xm (p)ym Bm = − m m ⎪ p∈P ⎪ ⎪ ⎪ H (p) − y (p)xH (p) ⎪ xm (p)ym m ⎩ Cm = j m

(4.95)

p∈P

and ym (p) = H(p)sm (p). From (4.93) the mth element of the gradient vector d = ∂Φ(ϑ)/∂ϑ is found to be dm =

Nr

˘k , det E

(4.96)

k=1

˘ k is identical to E, as deﬁned above, but the kth row is replaced where E by the kth row of −Bm sin(ϑm ) + Cm cos(ϑm ). From (4.93) the (q, r)th entry of the Hessian matrix is found to be ⎧ N N r r ⎪ ˇ k,l , for q = r ⎪ det E ⎨ 2 ∂ Φ(ϑ) k=1 l=1 = (4.97) Hq,r = % & Nr Nr ⎪ ∂ϑq ∂ϑr ⎪ ˜ ⎩ det Ek,l , for q = r k=1 l=1

˘ k , except that the lth row is replaced by the ˇ k,l is identical to E where E lth row of 6 −Bm sin(ϑm ) + Cm cos(ϑm ), for l = k (4.98) −Bm cos(ϑm ) − Cm sin(ϑm ), for l = k ˜ k,l is also identical to E ˘ k , except that the lth row is replaced by the E lth row of 6 −Bm sin(ϑm ) + Cm cos(ϑm ), for l = k (4.99) 0, for l = k Another important part of the optimisation algorithm is the convergence criterion used in Step 2. Since the cost function was found to be quadratic from visual inspection, a relative simple convergence criterion was anticipated to suﬃce. Therefore, the relative increment of the parameters compared to the previous iteration was chosen as convergence

4.5 Compensation of the CPE

measure, which is given by ( ( ( ϑ(i) − ϑ(i−1) ( ( ( ( ≤ , ( ( ( ϑ(i)

121

(4.100)

where is the tolerance level. The choice of depends on the required accuracy.

4.5.3

Sub-optimal estimator

Since the MLE procedure proposed in Section 4.5.2 consists of an iterative process and since every iteration includes the computation of a gradient and Hessian, the computational complexity of the implementation will be high. This solution to the estimation problem might, therefore, not be very cost-eﬀective. Therefore, a sub-optimal algorithm is studied, where we diverge from the original constraints and assume the AWGN term is the dominant source of observation noise zm (p) in (4.87), i.e., the covariance matrix Ω in (4.88) can be approximated by a diagonal matrix. The determinant of Ω then reduces to the product of its elements and maximising the LL function equals maximising the exponent term in (4.89). This is achieved by minimising zH m,P zm,P Φ(γ0,m ) = tr Zm Ω−1 ZH m = σz2 1 = 2 xm,P − γ˜0,m HP sm,P 2 , σz

(4.101)

where we assumed the estimation noise term also to be i.i.d. over space. Consequently, Ω = σz2 INr , where σz2 is the variance of the observation noise. The error vector on the collection of pilot carriers is deﬁned as zm,P = xm,P − γ0,m HP sm,P = ξm,P + nm,P .

(4.102)

Here zm,P is given by the P Nr ×1 vector [zTm (p1 ), zTm (p2 ), . . . , zTm (pP )]T , which is a concatenation of the error vectors zm at the diﬀerent pilot carriers. The vectors xm,P , sm,P and nm,P have a similar build-up as a concatenation. The pth block-diagonal element of the P Nr ×P Nt blockdiagonal channel matrix HP is given by the Nr ×Nt matrix H(p). Then, under this widely used, but more strict, i.i.d white Gaussian distributed noise assumption, the MLE reduces to the least-squares estimator (LSE). The well-known solution of the LS problem is given by [74] −1 H γ˜0,m = A†m,P xm,P = AH Am,P xm,P , m,P Am,P

(4.103)

122

4 Phase noise

where Am,P = HP sm,P and A† denotes the pseudo-inverse of the matrix A. The estimate of the phase error for the mth symbol, i.e., ϑ˜m , is then found by calculating the phase of γ˜0,m . We recall that the channel is quasi-static, hence, if the pilot tones in the packet are equal for the consecutive OFDM symbols, it is suﬃcient to calculate the pseudo-inverse A†m,P only once per packet. Since the LSE does not include an iteration its complexity will be much lower than that of the MLE optimisation. It is noted that the 1×1 version of this reduced-complexity algorithm is equal to the one proposed in [91].

4.5.4

Cram´ er-Rao lower bound

In this section we derive the Cram´er-Rao lower bound (CRLB) [74], which puts a lower bound on the variance of unbiased estimators. This enables us to compare the performance of the CPE estimators derived in Section 4.5.1 and Section 4.5.3 to the maximum achievable performance. In this performance analysis we will focus on the case where the PN is only located in the RX. The CRLB bound on the estimation of ϑm is given by [74] var(ϑ˜m ) ≥ I−1 (ϑ) mm , (4.104) where X−1 denotes the inverse of the matrix X and the (k, l)th element of the Fisher information matrix I(ϑ) is given by 2 ∂ ln (p(z|ϑ)) [I(ϑ)]kl = −E , (4.105) ∂ϑk ∂ϑl where ϑm denotes the mth element of the vector ϑ. The likelihood function was deﬁned in (4.88) and the LL function is then found to be L(ϑ) = C − P M det (Ω) −

M

−1 zH m (p)Ω zm (p),

(4.106)

m=1 p∈P

where zm (p) = xm (p) − exp(jϑm )Hp sm (p). It is then easily found that the non-zero elements of the Fisher information matrix lie on the diagional and that the mth diagonal element is given by ⎡ ⎤ 2 ∂ L(ϑ) H −1 ⎦ −E sH (4.107) = 2E ⎣ m (p)H (p)Ω H(p)sm (p) . ∂ϑ2m p∈P

Since the covariance matrix Ω is equal for all OFDM symbols, the symbol index m is omitted in the remainder for readability. For the pth

123

4.5 Compensation of the CPE

pilot the covariance matrix is given by

Ω = E (ξ(p) + n(p)) (ξ(p) + n(p))H = E ξ(p)ξH (p) + E n(p)nH (p) = Ωξ + σn2 I,

(4.108)

where we denote Ωξ = E ξ(p)ξ H (p) and E n(p)nH (p) = σn2 I. Since we regard the case of RX PN only, the expression for ICI in (4.24) can be simpliﬁed to ξ(p) = (ϕRX (p) ⊗ INr ) Hs, where ϕRX (p) denotes the pth row of ϕRX . The covariance matrix of the ICI is then given by Ωξ = E ξ(p)ξ H (p)

= E (ϕRX (p) ⊗ INr ) HssH HH (ϕRX (p) ⊗ INr )H ⎡ ⎤ Nc (4.109) = σs2 E ⎣ |γp−i |2 ⎦ R, i=1,i=p

where E ssH = σs2 I and where the correlation matrix R= E H(k)HH (k) , which is assumed to be subcarrier independent. It was shown in Section 4.4 that for a PN process as described in Section 4.2.1, the variance of the ICI ⎤ ⎡ Nc σξ2 = σs2 E ⎣ (4.110) |γp−i |2 ⎦ i=1,i=p

is well approximated by (4.83), for reasonable small PN and for Nc equal to a power of 2. The total covariance matrix is then given by Ω = σn2 INr + σξ2 R = σn2 INr + = U σn2 INr + σξ2 Λ UH ,

2σs2 πβT (Nc2 − 1) R 3Nc (4.111)

where U is a unitary matrix and Λ contains the eigenvalues of R on its diagonal and is given by diag(λ1 , . . . , λNr ). The inverse of (4.111) is H given by UΛ−1 tot U , where 2 2 2 2 (4.112) Λ−1 tot = diag 1/(σξ λ1 + σn ), ..., 1/(σξ λNr + σn ) . The diagonal elements of the Fisher information matrix, for a given channel matrix H(p), are now given by substituting this expression into

124

4 Phase noise

(4.107), which yields: ⎡ ⎤ $ $ 2 L(ϑ) $ $ H −1 H ⎦ −E ∂ ∂ϑ sH $H(p) = 2E ⎣ 2 m (p)H (p)UΛtot U H(p)sm (p)$H(p)

m

p∈P

= 2σs2

H H tr UΛ−1 tot U H(p)H (p) .

(4.113)

p∈P

When we, subsequently, take the expected value over the channel H(p), we get

2 L(ϑ) H H = 2σs2 −E ∂ ∂ϑ tr UΛ−1 2 tot U E H(p)H (p) m

p∈P Nr 2 Λ = 2P σ = 2P σs2 tr Λ−1 tot s

λi . (4.114) + σn2

σ2λ i=1 ξ i

The CRLB on the estimation of ϑm , averaged over the fading channel, is then given by var(ϑ˜m ) ≥

2P σ 2 s

1 Nr

λi 2 i=1 σξ2 λi +σn

.

(4.115)

We can simplify this expression for two extreme cases: a low SNR region and a high SNR region. For the low SNR region the additive Gaussian RX noise is dominant and the CRLB is given by var(ϑ˜m ) ≥

1 , 2P Nr ℘

(4.116)

where ℘ is the SNR per receiver branch and given by Nt σs2 /σn2 and where r we used that, by deﬁnition, N i=1 λi = Nt Nr . For the high SNR region, where the ICI is the dominant contributor in z, the CRLB is given by var(ϑ˜m ) ≥

4.5.5

πβT (Nc2 − 1) . 3P Nr Nc

(4.117)

Numerical results

Monte Carlo simulations were carried out to evaluate the performance of the estimators proposed in this section and to compare the performance to the lowerbound derived in Section 4.5.4. Furthermore, a goal was to test the performance of a MIMO OFDM system experiencing

4.5 Compensation of the CPE

125

PN and applying the proposed suppression techniques. As a test case an uncoded MIMO OFDM system with 64 carriers in a bandwidth of 20 MHz was studied, as described in Section 3.2, but now all 64 carriers are used. The guard interval was assumed to be longer than the channel length, so no inter-symbol interference occurred. 10 MIMO OFDM symbols are transmitted per packet and there are 10 pilots in every symbol. The channel elements for the diﬀerent subcarriers are modelled to be i.i.d. and the spatial correlation matrix of (2.22) was applied. The PN was modelled using the model in Section 4.2.1, and was, like in Section 4.5.4, only modelled in the RX. The corner frequency of the PN oscillator process β was varied for the diﬀerent simulations, since systems experiencing high spatial correlation can sustain less PN than system experiencing an uncorrelated channel. In the stopping criterium of (4.100), the tolerance level was chosen to be = 10−8 . Convergence was on average reached in four steps. Figure 4.14 depicts the empirical cumulative distribution function (ECDF), denoted by F(x), of the error in the estimation of the phase of the CPE term, ϑ = ∠γ0 , for both the MLE of Subsection 4.5.1 and LSE of Subsection 4.5.3 for a 2×2 system. The corner frequency of the PN oscillator process β = 300 Hz. The results are shown for no spatial correlation (ρ = 0) and for a correlation between the two branches of 0.9 at both TX and RX. In this simulation, no white Gaussian receiver noise is added, to show the inﬂuence of the ICI on the performance of the estimation. It is clear from the results in Fig. 4.14 that the ML and LS estimator have the same performance in spatially uncorrelated channels, since the curves are on top of each other. Furthermore, it is observed that the estimation is unbiased. When the spatial correlation is increased to 0.9, the MLE performance does not degrade, while the performance of the LSE does. The mean squared error (MSE) (in radians2 ) in the estimation of the CPE ϑ is given in Fig. 4.15 for a 2×2 system experiencing an uncorrelated channel (ρ = 0). Results are only depicted for the MLE, since results were equal for the LSE and MLE for this uncorrelated case. Phase noise is modelled with diﬀerent corner frequencies of the PN PSD. The corresponding CRLB (4.115), as derived in Section 4.5.4, is depicted for the low and high SNR case, i.e., (4.116) and (4.117), respectively. It can be concluded from Fig. 4.15 that the estimators achieve the CRLB. The performance at low SNR decreases linearly with the SNR, while at high SNR it is inversely proportional to the corner frequency of the PN PSD β.

126

4 Phase noise 1 0.9 0.8

MLE, MLE, LSE, LSE,

ρ ρ ρ ρ

=0 = 0.9 =0 = 0.9

0.7 F(J˜ − J )

0.6 0.5 0.4 0.3 0.2 0.1 0 −0.05

0 ˜ J − J (radians)

0.05

Figure 4.14. ECDF of the error in the estimation of the CPE ϑ for the MLE and the LSE in a 2×2 system. Results are depicted for a spatially uncorrelated (ρ=0) and a highly correlated (ρ=0.9) case. The system experiences phase noise with β = 300 Hz. 10−2

MSE (radians2)

10−3

b = 100 Hz b = 200 Hz b = 300 Hz b = 500 Hz b = 1000 Hz CRLB, high SNR CRLB, low SNR

10−4

10−5 0

10 20 30 40 Average SNR per RX antenna (dB)

50

Figure 4.15. MSE in the estimation of the CPE ϑ by the MLE for a 2×2 system in a spatially uncorrelated channel for diﬀerent corner frequencies of the PN PSD β.

127

4.5 Compensation of the CPE

MSE (radians2)

10−2 2× 4 2× 6 2× 8 CRLB, low SNR CRLB, high SNR

10−3

10−4

10−5 0

10

20

30

40

50

Average SNR per RX antenna (dB)

Figure 4.16. MSE in the estimation of the CPE ϑ by the MLE for a 2 TX system in an uncorrelated channel. The corner frequency of the PN PSD β equals 500 Hz. The number of RX antennas equals 4, 6 and 8.

Results for other MIMO conﬁgurations are given in Fig. 4.16, where the number of RX antennas, i.e., Nr , is varied. Again only the results for the MLE are depicted, since the results are equal for the LSE in this uncorrelated case. The PN is modelled with a corner frequency of β = 500 Hz. Again the CRLB is depicted for low and high SNR values. Fig. 4.16 shows that the estimators also attain the CRLB for MIMO conﬁgurations with more RX branches. It can be concluded that the MSE decreases linearly with the number of RX branches, for both the low and high SNR region. The performance of the two estimators in correlated fading channels is shown in Fig. 4.17 as a function of the SNR. This ﬁgure depicts the MSE as a function of the SNR for a 2×2 system experiencing PN with β = 200 Hz. The channel correlation is varied. The results for the LSE are given in dashed lines and for the MLE in solid lines. Again the CRLB is depicted for low and high SNR. It is observed in Fig. 4.17 that the performance of the LSE and MLE is equal for the uncorrelated case, i.e., the MSE curves lie on top of each other. This agrees with the observations made from Fig. 4.14 and

128

4 Phase noise 10−2 r=0 r = 0.7 r = 0.8 r = 0.9 CRLB

MSE (radians2)

10−3

10−4

10−5

0

10

20

30

40

50

60

Average SNR per RX antenna (dB)

Figure 4.17. MSE in the estimation of the CPE ϑ by the MLE (solid lines) and LSE (dashed lines) for a 2×2 system experiencing RX PN. The corner frequency of the PN PSD β = 200 Hz. The channel correlation ρ is varied.

the assumptions which lie at the basis of the design of the estimators. Both estimators approach the CRLB for this case. When the correlation increases, the performance of the LSE decreases at high SNR. This is explained by the correlation in the ICI. At low SNR, where the additive receiver noise is dominant, the performance of the LSE and MLE is equal. Finally, simulations were carried out to test the inﬂuence of the errors in CPE estimation on the bit-error rate (BER) performance of the regarded MIMO OFDM system. For these simulations a channel correlation of ρ = 0 and ρ = 0.5 was used, since the MSE was in Fig. 4.17 shown to depend on the channel correlation. 64-QAM was used as modulation, since it has tight CPE speciﬁcations, i.e., the detection of the outer points of this modulation scheme results in errors for small phase rotations. For these simulations, packets of 10 OFDM symbols were used. The BER as a function of the SNR is depicted in Fig. 4.18 for diﬀerent values of β for a 2×4 system. Curves are given for a system applying ideal CPE correction (with perfect knowledge of the CPE) and for systems applying correction based on the MLE and LSE CPE estimator.

129

4.5 Compensation of the CPE 100 r=0 10−1

r = 0.5

BER

10−2

10−3

no PN no corr ideal corr MLE corr LSE corr

10−4

10−5

0

r=0

10 20 30 40 Average SNR per RX antenna (dB)

50

Figure 4.18. BER as a function of the SNR for an uncoded 2×4 system applying 64-QAM modulation. The channel correlation ρ ∈ {0, 0.5}. PN with β = 200 Hz (solid lines) and β = 300 Hz (dashed lines). Results are depicted for a system with correction with perfect CPE knowledge and correction based on the MLE and LSE.

Furthermore, the curves for a system not experiencing PN and for a system not applying CPE correction are given as reference. Note that the required SNR values are high due to the channel correlation and since no coding is applied. It can be concluded from Fig. 4.18 that considerable improvement in BER performance is achieved compared to a system not applying CPE correction. For all regarded cases, there is no visible diﬀerence in BER performance between system applying perfect, MLE or LSE based CPE correction. The ﬂooring at high SNR of the BER curve is explained by the ICI which becomes dominant after CPE correction. For the regarded test cases the remaining CPE after correction is thus not important for the overall BER. We conclude that the additive Gaussian receiver noise is dominant in the BER performance at low SNR and that the PNinduced ICI is dominant in the BER performance at high SNR. We can, furthermore, conclude that for the considered MIMO OFDM system the computationally low complex LSE based correction is well applicable.

130

4 Phase noise

That is why the LSE based correction was implementation in the test system described in Section 3.6.

4.5.6

Summary

In this section an estimator, based on maximum-likelihood theory, for the PN-caused CPE in MIMO OFDM systems was derived. The estimator was shown to be equivalent to optimising a concise determinant criterion. Implementation of the MLE using a Gauss-Newton based iterative solution was presented. Since the computational complexity of the MLE procedure was high, a sub-optimal algorithm was studied, based on LSE. Monte Carlo simulations were performed and the Cram´er-Rao lower bound was derived to understand the performance of the estimators. The results show that in case of uncorrelated MIMO channels the LSE and MLE have similar performance and attain the CRLB. In case of both spatial correlation and high SNR the MLE performs better than the LSE in terms of the mean squared error. Finally, results from BER simulations reveal that the diﬀerence between systems applying the LSE or MLE of CPE can not be observed for the scenarios considered. Hence, it can be concluded that for the considered MIMO OFDM systems, the computationally less complex LSE is well applicable for CPE correction.

4.6

Compensation of the ICI

It was noted in Section 4.5 that, generally, the CPE has the most signiﬁcant impact on the ﬁnal system performance. It is clear from Sections 4.3 and 4.4 and the numerical results of Section 4.5, however, that after the removal of the CPE, the ICI can still considerably aﬀect the system performance. The level of impact is, obviously, dependent on the phase noise process, i.e., the eﬀect of the ICI grows with increasing −3 dB bandwidth of the oscillator, denoted by β. The above motivates a more careful look at possibilities for the suppression of the inﬂuence of the ICI term. Recently, some contributions are appearing in literature on the subject of ICI compensation [111–114]. They are all based on the principle of decision-directed ICI estimation. In this section we will review the basics of this approach. To ease explanation, we will focus on a system only impaired by RX phase noise, although a similar technique can also be applied for TX, or combined TX/RX, phase noise.

131

4.6 Compensation of the ICI s

Nt .. . 1 0 . ..

u .. IDFT

Nc−1

.. .

Slicing

Nt .. . 1 0. ..

D (˜s )

···

Nc−1 ˜s

..

.

MIMO detector

..

. .. .

. . . Add CP . . G . . ERX . . P/S

Nr . 1 0. ..

Nc−1

.. .

v ···

.. PN Comp ···

..

. .. .

DFT ···

x

ICI est.

. .. .

.. CP . Removal, S/P y

.. .

Figure 4.19. Baseband model for a MIMO OFDM system experiencing RX PN and applying CPE and ICI compensation.

4.6.1

Suppression algorithm

We consider a MIMO OFDM system, as schematically depicted in Fig. 4.19, where the receiver applies compensation for both the CPE and ICI caused by receiver phase noise. We recall that the expression for the mth symbol of the phase noise impaired received MIMO OFDM signal is given by (4.23), which is here repeated for convenience as (4.118). xm = γ0,m Hsm + ξm + nm .

(4.118)

In this expression the ICI term is given by ξm = (ϕm ⊗ INr )Hsm ,

(4.119)

ϕm = G m − γ0,m INr Nc ,

(4.120)

where and the (k, l)th element of G m is given by γk−l,m , as deﬁned in (4.22). Since pilot-aided estimation of all ICI parameters would impose a large amount of overhead, we consider an iterative decision-directed approach. In this approach, ﬁrst an initial estimate of the transmitted symbols is made. This is done by ﬁrst compensating for the CPE, e.g., by the use of the approach presented in Section 4.5, and then applying MIMO detection. For simplicity we assume here that the CPE γ0,m and MIMO channel H are perfectly estimated/known at the receiver. The estimate

132

4 Phase noise

of the TX signal, for ZF-based MIMO processing, in this ﬁrst iteration is then given by ˜sm,(1) = (γ0,m H)† xm = sm + εm,(1) ,

(4.121)

where the error term in the ﬁrst iteration is denoted by εm,(1) and given by εm,(1) = (γ0,m H)† (ξm + nm ). (4.122) When, subsequently, detection is made on the estimated signal vector ˜sm,(1) , e.g., by the use of hard slicing, we yield the vector D(˜sm,(1) ). Ideally, D(˜sm,(1) ) would equal s, however, due to the error term it is likely it contains detection errors. Therefore, an iteration is made to decrease the inﬂuence of this error term. A ﬁrst, rather straightforward, approach is to estimate the error κm = (ξm + nm ) in the received signal, consisting of noise and ICI, and subtracting that from the received signal xm before the detection of the signal in the next iteration. The estimate of the error κm can be found using ˜ m,(1) = xm − γ0,m HD(˜sm,(1) ). κ

(4.123)

The disadvantage of this approach is, however, that subcarriers that were already correctly detected get improved performance, but that almost no performance improvement is achieved for highly distorted subcarriers. For these subcarriers the symbols were likely incorrectly detected in the ﬁrst iteration and, hence, also the estimate of κm will probably be very inaccurate for these subcarriers. The main problem of this approach is, thus, that for a reliable estimate of κm,(1) ﬁrst a reliable estimate of sm is required on the same subcarrier. Hence, the overall performance gains of this approach are very marginal. Alternatively, one could estimate the ICI term by exploiting the structure of the matrix ϕm , i.e., by using the correlation between the ICI terms experienced at the diﬀerent subcarriers. When we denote the resulting estimate of the ICI term during the (p−1)th iteration as ξ˜m,(p−1) , an estimate of the data symbol in the pth iteration can be found as ˜sm,(p) = (γ0,m H)† (xm − ξ˜m,(p−1) ).

(4.124)

With the aim of estimating the ICI vector ξm , we ﬁrst regard the estimation of the elements of ϕm . For this purpose we rewrite (4.118), using γk = γ−k(modNc ) , to xm = Bm γm + nm

(4.125)

γm = [γ0,m , . . . , γNc −1,m ]T ,

(4.126)

where

133

4.6 Compensation of the ICI

and where the Nc Nr × Nc matrix ⎡ am (0) am (1) · · · ⎢ a (1) am (2) · · · m ⎢ Bm = ⎢ .. .. ⎣ . . am (Nc − 1) am (0) · · ·

am (Nc − 1) am (0) .. .

⎤ ⎥ ⎥ ⎥, ⎦

(4.127)

am (Nc − 2)

with am (k) = H(k)sm (k). When we use the estimate of sm (k) during the pth iteration, found using (4.124), we can derive an estimate of am (k), given by H(k)˜sm,(p) (k). ˜ m,(p) , can then be used to The resulting estimate of Bm , denoted by B estimate the set of PN parameters {γm,k } for k ∈ {0, . . . , Nc − 1}. Different estimation methods are possible, but if we for simplicity regard least-squares estimation we ﬁnd ˜† ˜m,(p) = B γ m,(p) xm .

(4.128)

The estimate of ϕm can then straightforwardly be constructed from the elements of the estimate of γm using (4.119) and (4.21). An estimate of the ICI vector experienced during the pth iteration is then found as ˜m,(p) ⊗ INr )HD{˜sm,(p) }, ξ˜m,(p) = (ϕ

(4.129)

where D{·} denotes the hard slicing operation. An estimate of the transmitted data symbols in the (p + 1)th iteration is, subsequently, found using (4.124). The advantage of this solution is that subcarriers with high reliability and, hence, correct detection in the pth iteration contribute to improving the estimate of the other subcarriers in the (p+1)th iteration. Improvements to the algorithm can be made, however, if we exploit the observation from (4.80) that the power of γk−l,m is proportional to 1/ sin2 ((k − l)π/Nc ). Consequently, the main contributions to the ICI occur for small values of |q| = |k − l|, i.e., the signiﬁcant ICI terms are due to elements of ϕm close to the diagonal. Hence, estimation of a subvector of γm will be suﬃcient to determine the major contributions to the ICI. The errors in the estimates of the other parameters of γm might, in contrast, create errors in the estimates of the ICI vector and thus decrease the performance of the detection algorithm. This approach will thus increase the performance and decrease the required computational complexity. Let us now consider this subvector approach, where we only estimate the ICI coeﬃcients {γq,m } for |q| ≤ N . To this end, we ﬁrst deﬁne the (2N + 1) × 1 subvector = [γ0,m , . . . , γN,m , γNc −N,m , . . . , γNc −1,m ]T , γm

(4.130)

134

4 Phase noise

and the Nc Nr × (2N + 1) submatrix Bm , which is given by the columns during the {1, . . . , N + 1, Nc − N + 1, . . . , Nc } of Bm . An estimate γm pth iteration is then found by ˜ † ˜m,(p) γ =B m,(p) xm .

(4.131)

Similarly to (4.129), now an estimate of the ICI vector can be derived, where the estimate of ϕm only contains non-zero elements for the coeﬃ . Using (4.124), subsequently, an estimate of the cients contained in γm TX data symbols for the (p + 1)th iteration can be found. Further improvements can be made by only incorporating the subcarriers with high SNR in the ICI estimation process, since it is likely for these subcarriers that D(˜sm,(p) ) indeed equals sm . Equivalently, the log-likelihood values, in case of soft value decoding, could be applied to select the most reliable symbols to be used in the ICI estimation procedure.

4.6.2

Numerical results

To illustrate the performance of the suppression approach introduced in Section 4.6.1, Monte Carlo simulations were performed. As a test case the same system as simulated in Section 4.5.5 was considered, i.e., an uncoded MIMO OFDM system applying 64 subcarriers, 20 MHz bandwidth, 10 MIMO OFDM symbols per packet and 10 pilots per OFDM symbol. The channel was modelled to have i.i.d. Rayleigh faded elements for the diﬀerent subcarriers. The phase noise was modelled in the RX, based on the model discussed in Section 4.2.1. In the simulations a 2×2 MIMO system was considered. For the ICI compensation algorithm three iterations per MIMO OFDM symbol were applied. Figure 4.20 depicts results for a system applying QPSK modulation. The −3 dB bandwidth of the simulated oscillator, β, was equal to 400 Hz. The ﬁgure displays bit-error rate (BER) results for a system without phase noise (“no PN”), a system not applying correction for the phase noise (“no corr”), a system applying the LSE-based CPE correction of Section 4.5 (“CPE corr”), a systems applying the ICI compensation as proposed in this section (“ICI comp, N =..”) and a system applying ICI compensation with perfect knowledge of a part of ϕm (“ideal ICI comp”). For the studied ICI compensation method the results are displayed for N = 1, 3 and 6. The results for the system with perfect knowledge of ϕm are depicted for N = 6. The latter results can be considered as a lowerbound on the BER that could be achieved when applying the ICI compensation algorithm for that value of N . It can be concluded from Fig. 4.20 that for the regarded PN scenario considerable degradation occurs compared to a system not experiencing

135

4.6 Compensation of the ICI 100

10−1

BER

10−2

no PN

10−3

no corr CPE corr ICI comp, N=1 ICI comp, N=3 ICI comp, N=6

10−4

10−5

ideal ICI comp 0

10 20 30 40 Average SNR per RX antenna (dB)

50

Figure 4.20. BER as a function of the SNR for an uncoded 2×2 system applying QPSK modulation and diﬀerent phase noise compensation approaches. Results are depicted for β = 400 Hz.

PN. After correction for the LS estimated CPE, still a considerable performance degradation remains. Already for N = 1, the ICI compensation algorithm achieves an improvement in performance. When more elements of ϕm are estimated, i.e., for N = 3, a further increase in performance is achieved. When too many elements are taken into account, however, the accuracy of the estimate of the ICI vector decreases and the gain in performance achieved by the algorithm decreases again, as illustrated by the curve for N = 6. The lowerbound on the achievable BER for the algorithm with N = 6 is depicted by the dashed line, which is the performance of the ICI compensation with perfect knowledge of {γq,m } with |q| ≤ 6. The remaining diﬀerence with the curve for a system without PN can be attributed to 1) detection errors in the initial iteration and the resulting error propagation and 2) model mismatch due to contributions of elements of ϕm with |q| > 6. The gap between this curve and those for the actual algorithm shows that there is room for improvement of the algorithms, e.g., by the methods proposed at the end of Section 4.6.1. Figure 4.21 depicts similar results as Fig. 4.20, but now for 16-QAM modulation and two PN scenarios, i.e., for β = 400 Hz and β = 1000 Hz. It can be concluded from the ﬁgure that 16-QAM is more sensitive to PN

136

4 Phase noise 100

BER

10−1

10−2

no PN no corr ICI comp, N=1

10−3

ICI comp, N=6 ideal ICI comp 10−4

0

10 20 30 Average SNR per RX antenna (dB)

40

Figure 4.21. BER as a function of the SNR for an uncoded 2×2 system applying 16-QAM modulation and diﬀerent phase noise compensation approaches. Results are depicted for β = 400 Hz (solid lines) and β = 1000 Hz (dashed lines).

than QPSK, which is as expected from the previous analyses in this chapter. Also, the degradation is more severe for β = 1000 Hz, which is as expected from Section 4.3. Again the CPE/ICI compensation algorithm achieves a signiﬁcant performance increase. The resulting performance is better for β = 400 Hz than for β = 1000 Hz, which is also observed for the lowerbound with ideal {γq,m }-knowledge, again for N = 6. The observed diﬀerence can be attributed to the increased number of detection errors in the ﬁrst iteration for increased β. Moreover, for higher β values there is an increased contribution from {γq,m } with high |q|. Hence, to capture these contributions the algorithm should increase N . Since this might reduce the accuracy of the estimation, however, a tradeoﬀ in the choice of N has te be made when applying the algorithm.

4.6.3

Summary

This section reviewed the possibility to suppress the inﬂuence of the phase noise caused ICI term using digital signal processing in the baseband part of the receiver. To this end, a decision-directed approach was studied, which estimates the data symbols in an iterative manner. Using the data symbol estimates from the previous iteration, an estimate of

4.7 Conclusions

137

the ICI vector is derived. This vector is subtracted from the received signal vector, the result of which is then used for data detection. Monte Carlo simulations were performed to derive the performance of the compensation approach. It was shown that a considerable performance improvement, compared to a system not applying compensation, can be achieved by applying the ICI suppression algorithm in combination with the approach of Section 4.5. The results suggest that the number of terms to be taken into account in the ICI estimation should be tuned to the experienced phase noise process. Also, it is expected that an additional gain in performance can be achieved by applying more advanced processing, e.g., by the use of coding and spreading in the transmitter and the use of the log-likelihood values of the detected bits in the receiver. Additional challenges lie in the reduction of the complexity of the compensation approach.

4.7

Conclusions

The inﬂuence, estimation and compensation of phase noise (PN) in multiple-antenna OFDM systems were treated in this chapter. First a model was introduced for the phase noise of a free running oscillator, which was shown to exhibit a Lorentzian power spectral density. Subsequently, the inﬂuence of both transmitter (TX) and receiver (RX) PN on the received signal was studied. This revealed that the inﬂuence is twofold: a rotation common to all carriers occurs, referred to as common phase error (CPE), and an additive term due to leakage between the subcarriers, referred to as inter-carrier interference (ICI). The ICI term was in most of the previous contributions assumed to be zero-mean complex Gaussian distributed. In this chapter, however, we proved this assumption does not hold. Moreover, the limit distribution for a large number of subcarriers was derived analytically, which was numerically shown to hold even for a moderate number of carriers. This distribution was shown to exhibit thicker tails than the Gaussian distribution with the same mean and variance. In an investigation of the tail probabilities it was proven that the probabilities of bit error are severely underestimated when the Gaussian approximation for the ICI term is used. Furthermore, this chapter investigated the inﬂuence of TX and RX PN on the error in detection for a ZF-based MIMO OFDM systems. An analytical expression for the power of the ICI-induced error term was found and shown to diﬀer for TX and RX-caused PN. For TX PN the power of this error term is equal for all MIMO conﬁgurations. The power of the error term for RX PN is equal to that for the TX case in the case of a ﬂat Rayleigh faded channel. For the case of per subcarrier independent

138

4 Phase noise

Rayleigh faded channels, however, the RX PN is concluded to have less impact than TX PN when Nr > 2Nt and more when Nt < Nr < 2Nt . This chapter also proposed a compensation approach for the CPE term. The proposed estimator is based on maximum-likelihood estimation (MLE) and exploits the correlation in the wireless MIMO channel. An implementation of the MLE using a Gauss-Newton based iterative solution was proposed. Since the computational complexity of the MLE procedure was high, a sub-optimal algorithm was studied, based on leastsquares estimation (LSE). Results from a performance evaluation showed that for uncorrelated MIMO channels the LSE and MLE perform similarly, but that for spatially correlated MIMO channels the MLE obtains an improved estimation performance. Results from a bit-error rate (BER) study showed that the diﬀerences between the system performance obtained with LSE and MLE are only marginal and that, thus, the LSE-based compensation approach is well applicable for CPE correction in a MIMO OFDM system. Finally, the possibilities to achieve performance improvements using an ICI compensation approach were studied. To this end, an iterative algorithm was considered, in which the ICI is estimated using initial estimates of the data symbols. After subtraction of this ICI estimate from the received signal, a new detection of the data symbols is made. It is shown that the best performance is achieved by estimation of only the dominant contributions to the ICI. Results from BER simulations show that the algorithm indeed achieves a considerable performance improvement. Due to the decision-directed nature of the compensation method, however, the full potential of the approach is not leveraged. Hence, further improvement of the performance of the compensation algorithm remains a topic for future research.

Chapter 5 IQ IMBALANCE

5.1

Introduction

More and more devices are being equipped with wireless capabilities, causing wireless systems to move towards the commodity market. This means that the price pressure on wireless products is high, creating a drive for low-cost solutions. This is especially an issue for multiple antenna systems, since they require multiple radio frequency (RF) frontends. Furthermore, the growing number of wireless standards forces for ﬂexible solutions, which can support several standards. The concept of direct-conversion [118] for frequency translation is promising to fulﬁll these requirements, since it does not need external intermediate frequency (IF) ﬁlters and image rejection ﬁlters [119]. Instead, the image rejection is provided by the signal processing in the in-phase (I) and quadrature (Q) arm. Therefore, this architecture opens the door to monolithic integration of the analogue front-end and, thus, low-cost implementations. The direct-conversion transceiver architecture, also referred to as homodyne or zero-IF architecture, however, has some disadvantages compared to more conventionally used heterodyne architectures. These disadvantages include DC oﬀset through self-mixing, 1/f -noise and the more severe IQ mismatch. This chapter will focus on the latter impairment which is caused by mutual diﬀerences in the used components for frequency translation. These diﬀerences result in a phase and/or amplitude imbalance between the I and Q signals, an eﬀect which we will

c 2006, 2007 IEEE. Portions reprinted, with permission, from [115–117].

140

5 IQ imbalance

refer to as IQ imbalance. Overall, it can be shown that IQ imbalance results in a limited image rejection [36, 120, 121]. In this chapter, we will consider the inﬂuence and mitigation of IQ mismatch in both the transmitter (TX) and receiver (RX) front-ends. First, Section 5.2 introduces the homodyne transceiver structure and shows the inﬂuence of IQ mismatch on the transmitted and received signals. Subsequently, a homodyne-based multiple-input multiple-output (MIMO) OFDM system is regarded, where the inﬂuence of TX and RX IQ imbalance on the received symbols is shown. The section regards both frequency-independent IQ imbalance and imbalance in the RX low-pass ﬁlters (LPFs), which has a frequency-selective behaviour. The signal model introduced in Section 5.2 is used in Section 5.3 to ﬁnd the impact of IQ imbalance on the system performance of a MIMO OFDM system. The section regards two cases: a system only experiencing TX IQ imbalance and a system only experiencing RX IQ imbalance. For these cases the section derives analytical expressions for the probability of symbol error for systems applying a rectangular M -QAM modulation. Additionally, these analytical results are compared with results from Monte Carlo simulations. To reduce the performance degradation due to IQ imbalance, Section 5.4 introduces a data-aided estimation and compensation approach for frequency-independent TX and RX IQ imbalance. For this purpose, a preamble design is proposed in Section 5.4.1, which is exploited in the estimation algorithms presented in Sections 5.4.2 to 5.4.5. These algorithms jointly estimate the channel and IQ imbalance matrices. It is shown through a numerical study that the proposed compensation approaches are well applicable and signiﬁcantly suppress the inﬂuence of IQ mismatch. Subsequently, Section 5.5 presents a decision-directed based mitigation approach for the frequency-selective IQ imbalance. The technique is based on adaptive ﬁltering, where the ﬁlter weights are updated using decisions on the transmitted data symbols. Results from simulations show that the proposed technique yields considerable performance improvements compared to a system applying no compensation of the inﬂuence of IQ mismatch. Finally, conclusions are drawn in Section 5.6.

5.2

System and IQ imbalance modelling

Consider a MIMO OFDM system applying Nt TX branches and Nr RX branches, as described in Section 2.3.3 and illustrated in Fig. 2.8. It was shown in that section that the Nt -dimensional baseband transmit vector u is input to the Nt TX RF front-ends, which are used to

141

5.2 System and IQ imbalance modelling

up-convert the signals to be transmitted through the wireless channel. At the receiver, the signals obtained at the Nr antennas are downconverted using the Nr RX RF front-ends, yielding the Nr -dimensional complex baseband vector y. In this section we will now regard the architectures used for frequency translation in the TX and RX. More speciﬁcally, we will focus on the direct-conversion based architecture, which, as highlighted in the introduction of this chapter, enables the monolithic integration of the RF front-end. Subsequently, we will derive the inﬂuence of phase and amplitude mismatch between the I and Q path on the received MIMO OFDM signal vectors. Throughout this section we will apply, for notational convenience, the time continuous notation t, also for the digital time-discrete signals.

5.2.1

TX/RX front-end architecture

We ﬁrst consider the up-conversion of the baseband signal unt (t) in the nt th TX branch, as illustrated schematically in Fig. 5.1. The real and imaginary part of the digital baseband signal unr (t) are passed through the digital-to-analogue convertors (DACs). The signal is then up-converted to radio frequency (RF) fc , using the quadrature mixing structure illustrated in the ﬁgure. The RF signal is, before transmission through the channel, fed through the power ampliﬁer (PA), which, in the analyses of this chapter, we will assume to be perfect with unity gain. In the case of perfect matching between the I and Q branch, the local oscillator (LO) signal multiplying the Q signal aQ (t) is the 90◦ phase shifted version of the LO signal multiplying the I signal aI (t). They can be expressed as aQ (t) = sin(ωc t), aI (t) = cos(ωc t).

(5.1) (5.2)

aI(t)

un (t) t

R(·)

TX Baseband Modem

DAC

+

DAC

−

PA

u RF,n (t) t

I(·) Digital Baseband

Figure 5.1.

aQ(t)

Block diagram of a homodyne transmitter.

142

5 IQ imbalance bI(t)

yRF,nr(t)

ynr(t)

LNA

LPF I

ADC

LPF Q

ADC j Digital Baseband

bQ(t)

Figure 5.2.

RX Baseband Modem

Block diagram of a homodyne receiver.

Using these expressions, the RF TX signal for the nt th branch can be expressed as uRF,nt (t) = 2 (R{unt (t)} cos(ωc t) − I{unt (t)} sin(ωc t)) = unt (t)ejωc t + u∗nt (t)e−jωc t ,

(5.3)

where ωc = 2πfc and where R{·} and I{·} give the real and imaginary part of their arguments. The factor 2 is added for notational convenience. At the nr th RX branch, as illustrated in Fig. 5.2, the received RF signal yRF,nr (t) is ﬁrst ampliﬁed by a low-noise ampliﬁer (LNA), which we will, in the analyses of this chapter, assume to be perfect with unity gain. Quadrature down-mixing is done again by two 90◦ phase shifted LO signals at RF fc . Low-pass ﬁltering is applied in both branches to remove higher order modulation products. Both signals are then passed through the analogue-to-digital convertors (ADCs) and combined to form the baseband signal ynr (t), which is input to the baseband RX modem. In the case of perfect matching between the I and Q branch, the LO signals multiplying the I and Q signal again diﬀer by a 90◦ phase shift. They can be expressed as bQ (t) = − sin(ωc t), bI (t) = cos(ωc t).

(5.4) (5.5)

From (5.3) it can be concluded that the received RF signal on the nr th RX branch is given by yRF,nr (t) = ynr (t)ejωc t + yn∗ r (t)e−jωc t .

(5.6)

Using (5.4), (5.5) and (5.6) the baseband RX signals can be found to be given by ynr (t) = yI,nr (t) + jyQ,nr (t), (5.7)

143

5.2 System and IQ imbalance modelling

where, as we deﬁne LPF{·} to be the low-pass ﬁltering operation, yI,nr (t) = LPF{bI (t)yRF,nr (t)} = LPF{cos(ωc t)yRF,nr (t)} 1 = LPF{ynr (t)(1 + ej2ωc t ) + yn∗ r (t)(1 + e−j2ωc t )} 2 = R{ynr (t)},

(5.8)

and yQ,nr (t) = LPF{bQ (t)yRF,nr (t)} = LPF{− sin(ωc t)yRF,nr (t)} j = LPF{ynr (t)(ej2ωc t − 1) + yn∗ r (t)(1 − e−j2ωc t )} 2 (5.9) = I{ynr (t)}.

5.2.2

IQ mismatch

The results in Section 5.2.1 show that for a system with ideal matching the baseband signals are perfectly up-converted in the TX and that the image signal centred around −fc is perfectly rejected in the downconversion. In any practical system, however, perfect matching between the I and Q branch of the quadrature TX/RX is not possible due to limited accuracy in the implementation of the RF front-end. This will result in phase and amplitude mismatch between the I and Q branch, the inﬂuence of which will be shown below. Several stages in the transceiver structure can contribute to the IQ mismatch, e.g., errors in the nominal 90◦ phase shift between the LO signals used for up- and down-conversion of the I and Q signals and the diﬀerence in amplitude transfer of the total I and Q arms. These imbalances are generally modelled as phase and/or amplitude errors in the LO signal used for up- and down-conversion, which can be veriﬁed to be equivalent to modelling these imbalances in the signal path. The imbalances can be modelled either symmetrical or asymmetrical. In the symmetrical method, each arm (I and Q) experiences half of the phase and amplitude errors, see e.g. [20, 122]. In the asymmetrical method, the I branch is modelled to be ideal and the errors are modelled in the Q branch, see e.g. [36, 123]. It is noted that it is easily veriﬁed that these two methods are equivalent. We will use the asymmetrical model for the further analyses in this chapter. For this model the imbalanced LO signals used for up-conversion are given by aQ (t) = gT sin(ωc t + φT ),

(5.10)

aI (t) = cos(ωc t),

(5.11)

144

5 IQ imbalance

where gT and φT model the TX gain and phase mismatch, respectively. We can conclude from (5.1) and (5.2) that for perfect matching, these imbalance parameters are given by gT = 1 and φT = 0. The TX RF signal on the nt th branch can then be expressed as uRF,nt (t) = 2 (R{unt (t)} cos(ωc t) − I{unt (t)}gT sin(ωc t + φT )) = ejωc t (R{unt (t)} + jgT ejφT I{unt (t)}) +e−jωc t (R{unt (t)} − jgT e−jφT I{unt (t)}).

(5.12)

When we deﬁne the coeﬃcients G1 and G2 , given by G1 = (1 + gT ejφT )/2, G2 = (1 − gT e−jφT )/2,

(5.13) (5.14)

respectively, uRF,nt (t) can be written as uRF,nt (t) = G1 unt (t) + G∗2 u∗nt (t) ejωc t + G∗1 u∗nt (t) + G2 unt (t) e−jωc t . (5.15) It is noted that for perfect TX matching G1 = 1 and G2 = 0 and that (5.15) reduces to (5.3). When we subsequently regard the imbalance in the RX, the imbalanced LO signals used for down-conversion are given by bQ (t) = −gR sin(ωc t + φR ), bI (t) = cos(ωc t),

(5.16) (5.17)

where gR and φR model the RX gain and phase mismatch, respectively. Note that we can conclude from (5.4) and (5.5) that when there is ideal matching, these imbalance parameters are given by gR = 1 and φR = 0. Down-conversion of the RF RX signal, as expressed by (5.6), then yields yˆnr (t) = yˆI,nr (t) + j yˆQ,nr (t) = LPF{cos(ωc t)yRF,nr (t)} + jLPF{−gR sin(ωc t + φR )yRF,nr (t)} = R{ynr (t)} + jI{gR e−jφR ynr (t)} = K1 ynr (t) + K2 yn∗ r (t),

(5.18)

where the coeﬃcients K1 and K2 are given by K1 = (1 + gR e−jφR )/2, K2 = (1 − gR ejφR )/2,

(5.19) (5.20)

respectively. Again, for perfect matching we ﬁnd that K1 = 1 and K2 = 0. For that case (5.18) reduces to (5.7).

5.2 System and IQ imbalance modelling

5.2.3

145

Inﬂuence on MIMO signal model

Now that we have deﬁned a model for the imbalance and have derived its inﬂuence on the up- and down-conversion in Section 5.2.2, we will study the inﬂuence of IQ mismatch on the detected symbols in a multiple-antenna OFDM system here. We will assume here, and throughout this chapter, perfect timing and frequency synchronisation. Moreover, we omit the inﬂuence of noise for now to ease the notation and explanation. We recall (2.42), which deﬁnes the Nt Ns ×1 TX baseband for the mth MIMO OFDM symbol: um = (Θ ⊗ INt )(F−1 ⊗ INt )sm .

(5.21)

If we up-convert this signal to RF in the imbalanced quadrature TX, we ﬁnd, using (5.15), that the Nt ×1 vector RF TX vector is given by uRF,m (t) = (G1 u(t) + G∗2 u∗ (t)) ejωc t +(G∗1 u∗ (t) + G2 u(t)) e−jωc t , (5.22) for t ∈ {(m−1)Ns Ts , . . . , (mNs −1)Ts } and where u(((m−1)Ns +n)Ts ) = um (n). The diagonal imbalance matrices in (5.22) are deﬁned by G1 = (I + gT ejφT )/2, G2 = I − G∗1 = (I − gT e−jφT )/2,

(5.23) (5.24)

where I denotes the identity matrix and where φT = diag{φT,1 , φT,2 , . . . , φT,Nt }, gT = diag{gT,1 , gT,2 , . . . , gT,Nt }.

(5.25) (5.26)

We note that (5.23) and (5.24) contain diﬀerent imbalance values for the diﬀerent TX streams, since it is likely that the diﬀerent TX branches will exhibit unequal mismatches. The up-converted signal is then transmitted through the wireless channel. The received Nr ×1 signal vector at the output of the channel is given by yRF (t) = GRF uRF (t), (5.27) where GRF (τ ) is the Nr ×Nt RF MIMO channel impulse response for time lag τ and denotes convolution. Note that the sample-spaced baseband equivalent of GRF (τ ) is more commonly used in this book, and denoted by G(l). After down-conversion with the imbalanced quadrature RX, the received baseband signal is given by ˆ (t) = K1 y(t) + K2 y∗ (t). y

(5.28)

146

5 IQ imbalance

The diagonal imbalance matrices in (5.28) are given by K1 = (I + gR e−jφR )/2, K2 = I − K∗1 = (I − gR ejφR )/2,

(5.29) (5.30)

φR = diag{φR,1 , φR,2 , . . . , φR,Nr }, gR = diag{gR,1 , gR,2 , . . . , gR,Nr }.

(5.31) (5.32)

where

When we, for convenience, repeat the expression for the Nr Nc ×1 received frequency-domain vector in (2.45) here ˆ m = (F ⊗ INr )(Υ ⊗ INr )ˆ x ym ,

(5.33)

we can rewrite the expression for the kth carrier using (5.27) and (5.28) as ˆ m (k) = x

N c −1

ˆ m (Ng + n)e−j y

n=0 N c −1

= K1

2πnk Nc

ym (Ng + n)e−j

2πnk Nc

+ K2

n=0

= K1 H(k)

N c −1

∗ ym (Ng + n)ej

2πnk Nc

n=0 N c −1

ˆ m (Ng + n)e−j u

2πnk Nc

n=0 ∗

+K2 H (−k)

N c −1

ˆ ∗m (Ng + n)ej u

2πnk Nc

.

(5.34)

n=0

In (5.34), the Nr ×Nt MIMO channel on the kth subcarrier is denoted by H(k), where the ideal down-converted version of uRF,m is given by ˆ (t) = G1 u(t) + G∗2 u∗ (t). u

(5.35)

Using (5.21), (5.22) and (5.35), we can rewrite (5.34) as ˆ m (k) = {K1 H(k)G1 + K2 H∗ (−k)G2 } sm (k) x + {K2 H∗ (−k)G∗1 + K1 H(k)G∗2 } s∗m (−k),

(5.36)

for k ∈ {−K, . . . , −1, 1, . . . , K}. Note that, for notational convenience, the carriers are numbered diﬀerently from the previous chapters. It can be concluded from (5.36) that the received frequency-domain vector during the mth symbol on the kth subcarrier is given by the TX

147

5.2 System and IQ imbalance modelling

(a) TX baseband

Figure 5.3.

(b) RX RF

(c) RX baseband

The inﬂuence of RX IQ imbalance on the reception of an OFDM signal.

vector on that subcarrier, i.e., sm (k), times a complex matrix plus the complex conjugate of the TX vector on the −kth carrier, i.e., s∗m (−k), times another complex matrix. The latter component is often referred to as the mirror signal, since the subcarrier is located at the same distance from, but at the other side of, the DC-carrier. In case of perfect TX matching G1 = I and G2 = 0 and in case of ideal RX matching K1 = I and K2 = 0. Hence, when there is perfect TX and RX matching, (5.36) reduces to H(k)sm (k), i.e., the transmitted symbol vector times the MIMO channel matrix at that subcarrier. The inﬂuence of IQ imbalance in a system experiencing a frequencyselective channel is shown schematically in Fig. 5.3 [124], where we assume ideal up-conversion. The transmitted baseband signal is shown in Fig. 5.3(a), where two subcarriers (−k and k) are highlighted. These subcarriers have the same separation from DC. The signal is (ideally) upconverted to RF and transmitted through the frequency-selective channel, resulting in the received RF signal depicted in Fig. 5.3(b). It is clear that subcarrier −k is more attenuated by the channel than subcarrier k. Subsequently, the RX signal is down-converted to baseband using the homodyne structure of Fig. 5.2. Since this structure exhibits IQ mismatch, the mirror signal is not fully rejected, and mixes down into the regarded baseband channel. This is illustrated in Fig. 5.3(c), which shows that carrier k experiences a contribution of the signal received on the mirror carrier −k and vice versa. We note that the same behaviour can be observed for a system experiencing a ﬂat-fading channel, however, it that case the magnitudes of the desired signals and leakage contributions are not frequency dependent. To show the eﬀect of the IQ imbalance on the reception of an OFDM signal more clearly, a noiseless single-input single-output (SISO) system applying 16-QAM modulation is regarded. The system does not experience a multipath channel and has a 10% amplitude and 5◦ phase imbalance between the I and Q branches of the RX. The received signal

148

5 IQ imbalance 1.5 1

Q

0.5 0 −0.5 −1 −1.5 −1.5

Figure 5.4.

−1

−0.5

0 I

0.5

1

1.5

Inﬂuence of IQ imbalance on the noiseless reception of 16-QAM symbols.

is depicted in Fig. 5.4, which shows that the transmitted 16-QAM points are distorted by an additive rotated 16-QAM constellation of lower amplitude. This is due to the leakage of the mirror carrier −k, where the rotation and reduced amplitude are due to the imbalance parameter K2 . Furthermore, one can observe a small rotation and a small decrease in amplitude due to the multiplication of the desired signal (on carrier k) with imbalance parameter K1 .

5.2.4

RX ﬁlter imbalance

It can be concluded from (5.36) that the IQ imbalance matrices do not vary as a function of the subcarrier index k, i.e., the modelled IQ mismatch has a frequency-independent (FI) behaviour. Although this will, generally, be the main source of IQ imbalance, other sources of IQ imbalance can occur in the transceiver structure, which do have a frequency-selective (FS) behaviour. This FS IQ imbalance occurs, e.g., due to mismatches between ﬁlters and diﬀerences in group delay in the I and Q branches. Although FS IQ imbalance may be caused in other parts in the transmission link, it is most apparent due to mismatches of the cut-oﬀ frequency, ripple and group delay of the RX LPFs in the I and Q arm of the RX [125, 126]. Therefore, we will focus on the eﬀect of this mismatch here. To investigate the inﬂuence of imbalance between the I and Q branch LPF (“LPF I” and “LPF Q” in Fig. 5.2), we ﬁrst consider a case where there are no other sources of IQ imbalance.

5.2 System and IQ imbalance modelling

149

The received baseband signal vector can then be expressed as ˆ (t) = fI (τ ) R{y(t)} + jfQ (τ ) I{y(t)} y fI (τ ) − fQ (τ ) fI (τ ) + fQ (τ ) y(t) + y∗ (t). = 2 2

(5.37)

where fI (τ ) and fQ (τ ) model the impulse response of the LPF in-phase and quadrature arm, respectively. Note that it is assumed here that the LPF ﬁlters do fully suppress the term at the double frequency. ˆ (t) is transformed to the frequency domain by the When the signal y OFDM RX processing of (5.33), then the received signal on the kth subcarrier is given by F I (k) + F Q (k) H(k)sm (k) 2 ∗ F I (−k) − F ∗Q (−k) ∗ H (−k)s∗m (−k) + 2 = K1 (k)H(k)sm (k) + K2 (−k)H∗ (−k)s∗m (−k),

xm (k) =

(5.38)

where the Nr ×Nr diagonal matrices F I (k) and F Q (k) model the frequency response of the LPFs for the kth carrier for the I and Q branches, respectively. They are given by the DFT of fI (τ ) and fQ (τ ), respectively. Here we used that F I (k) = F ∗I (−k) and F Q (k) = F ∗Q (−k), since F I and F Q are the frequency responses of real ﬁlters. It is clear from (5.38) that when there is no mismatch, i.e., F I (k) = F Q (k) ∀k, there is no leakage of the mirror carrier into the desired signal. Furthermore, it can be concluded from (5.38) that the IQ matrices K1 and K2 are a function of the subcarrier index, hence, exhibit FS behaviour. When we, subsequently, regard a more realistic scenario of a system that experiences a combination of FI IQ imbalance and ﬁlter imbalance, we can ﬁnd the received frequency-domain signal vector by combining (5.36) and (5.38). The results for the kth carrier during the mth symbol is then given by ˆ 1 (k)H(k)G ˆ ˆ ˆ∗ xm (k) = K 1 + K2 (−k)H (−k)G2 sm (k) ∗ ∗ ˆ 2 (−k)H ˆ ∗ (−k)G∗ + K ˆ 1 (k)H(k)G ˆ + K 1 2 sm (−k), (5.39) where ˆ 1 (k) = (I + F −1 (k)F Q (k)gR e−jφR )/2, K I ∗ ˆ ˆ K2 (k) = I − K (k),

(5.40)

ˆ H(k) = F I (k)H(k).

(5.42)

1

(5.41)

150

5 IQ imbalance

We conclude from (5.39) that the structure of the received frequencydomain signal in the case of both FI IQ imbalance and imbalance between the RX LPF ﬁlters is very similar to the expression for the case with only FI IQ imbalance in (5.36). The diﬀerence, however, is that the RX IQ imbalance matrices are a function of the subcarrier index k.

5.3

Impact of IQ imbalance on system performance

In the previous section, the inﬂuence of TX and RX IQ imbalance on the received signal vector in a multiple-antenna OFDM system was investigated. Here this is extended to show the inﬂuence on the detection of the transmitted symbols. For that purpose Section 5.3.1 studies the error in coherent detection of the transmitted MIMO vector using zeroforcing (ZF) processing. These results are in Section 5.3.2 applied to derive the probability of symbol error for a transmitted M -QAM modulated signal. Section 5.3.3, subsequently, compares the analytical results with results from a numerical study. For all studies in this section FI IQ imbalance is assumed, but the results can be easily extended to the more general case of FS IQ imbalance.

5.3.1

Error in symbol detection

In the foregoing the additive RX noise was omitted to simplify notation. Any practical system will, however, experience this noise term, which changes the expression for the kth carrier of the mth received symbol in (5.36) to xm (k) = K1 H(k)G1 + K2 H∗ (−k)G2 sm (k) + K1 nm (k) + K2 H∗ (−k)G∗1 + K1 H(k)G∗2 s∗m (−k) + K2 n∗m (−k), (5.43) where nm (k) denotes the Nr ×1 additive noise vector for the kth subcarrier during the mth symbol. We recall that sm (−k) denotes the mirror TX signal, i.e., the signal transmitted on carrier −k. Note that the noise term is also inﬂuenced by the IQ imbalance, since its major source, i.e., the LNA, will be located in front of the down-mixing in any conventional architecture. When we assume perfect channel knowledge at the RX, the channel estimate is given by the transfer from the TX baseband signals to the RX baseband signals and can be written as ˜ H(k) = K1 H(k)G1 + K2 H∗ (−k)G2 ,

(5.44)

where it is noted that the channel estimate now includes the inﬂuence of both TX and RX IQ imbalance. When this transfer is used for

151

5.3 Impact of IQ imbalance on system performance

ZF-based MIMO processing, the Nt ×1 estimated TX signal vector for the kth subcarrier is found by ˜ † (k)xm (k) ˜sm (k) = H = sm (k) + ε(k),

(5.45)

where we recall that † denotes the pseudo-inverse and where it is easily veriﬁed that the error term is given by ˜ † (k) K2 H∗ (−k)G∗ + K1 H(k)G∗ s∗ (−k) ε(k) = H 1 2 m ˜ † (k) K1 nm (k) + K2 n∗ (−k) . (5.46) +H m

We can conclude from (5.46) that the error term includes contributions of the signal term on the mirror carrier −k and of the noise terms corresponding to subcarrier k and −k. When we study the inﬂuence of TX and RX IQ imbalance separately, we ﬁnd that for a system only experiencing TX IQ imbalance, the error term in (5.46) is given by † † εT (k) = H(k)G1 H(k)G∗2 s∗m (−k) + H(k)G1 nm (k) = G†1 H† (k)H(k)G∗2 s∗m (−k) + (H(k)G1 )† nm (k) = Ge s∗m (−k) + (H(k)G1 )† nm (k),

(5.47)

where we used that K1 = I and K2 = 0 and where Ge is a diagonal matrix given by jφT )(I + gT ejφT )−1 . Ge = G†1 G∗2 = G−1 1 (I − G1 ) = (I − gT e

(5.48)

For RX IQ imbalance, the error term in (5.46) is given by † εR (k) = K1 H(k) K2 H∗ (−k) s∗m (−k) + K2 n∗m (−k) + K1 nm (k) = H† (k)K†1 K2 (H∗ (−k)s∗m (−k) + n∗m (−k)) + H† (k)nm (k) = H† (k)Ke H∗ (−k)s∗m (−k) + H† (k) (Ke n∗m (−k) + nm (k)) ,(5.49) where we used that G1 = I and G2 = 0 and where Ke is a diagonal matrix given by ∗ jφR )(I + gR e−jφR )−1 . Ke = K†1 K2 = K−1 1 (I − K1 ) = (I − gR e

(5.50)

When comparing the error term due to TX IQ imbalance in (5.47) to that of RX IQ imbalance in (5.49), we conclude that the inﬂuence of noise in the error term is almost equal, although εR (k) also exhibits a small noise contribution from the mirror. Considering the inﬂuence of

152

5 IQ imbalance

the mirror leakage we can see that the term Ge s∗m (−k) in εT (k) will be of ﬁnite size for given Ge , but that H† (k)Ke H∗ (−k)s∗m (−k) in εR (k) can become inﬁnite large for given Ke , since it contains the multiplication of the channel matrix for carrier −k with the inverse channel matrix for carrier k. When, through fading, the elements of H(−k) are large and the elements in H(k) are small, the error term will become large. Since errors in detection are caused by the tails of the error distribution, we can conclude qualitatively that in faded channels the inﬂuence of RX IQ imbalance will be larger than that of a comparable TX IQ imbalance.

5.3.2

Probability of erroneous detection for M -QAM

Although Section 5.3.1 provides a qualitative distinction between the inﬂuence of TX and RX IQ imbalance, a quantitative study is required to provide a more precise understanding of the impact on system performance. To that end, we will study the inﬂuence of the error vector ε(k) in (5.46) on the detection of the transmitted vector sm (k). We calculate the probability of erroneous detection of transmit signals chosen from a rectangular M -dimensional quadrature amplitude modulation (M -QAM) constellation. The inﬂuence is derived separately for TX and RX IQ imbalance in Sections 5.3.2.1 and 5.3.2.2, respectively. For both cases we will derive the probability of error in three steps. First the distribution of the eﬀective SNR is derived. Subsequently, the probability of symbol error in detection of s for a given SNR value is calculated. Finally, the SER expressions are averaged over the SNR distribution.

5.3.2.1 TX IQ imbalance Let us ﬁrst consider the case of TX IQ imbalance, where the error term εT is deﬁned by (5.47). In our approach to calculate the probability of error, we will ﬁrst consider the case for which the signal on the mirror carrier is given. We will treat this as a (known) translation of the wanted signal term, and later on we will average over all possible realisations of the mirror signal. When we assume that the estimated signal on carrier k in symbol m is translated by the known Nt ×1 vector dm (k) = Ge s∗m (−k),

(5.51)

the error in the estimated translated TX signal is found by combining (5.47) and (5.51), yielding εT (k) = εT (k) − dm (k) = (H(k)G1 )† nm (k) = H† (k)nm (k),

(5.52)

153

5.3 Impact of IQ imbalance on system performance

where we assumed the IQ imbalance to be equal on all TX branches and where nm (k) ∼ CN (0, (σn2 /|G1 |2 )I). We note that it can be veriﬁed that the real and imaginary part of the elements of nm (k) are i.i.d., since the elements of nm (k) are i.i.d. Note also that µT = E εT (k) = 0, (5.53) since the mean of the elements of nm (k) equals zero, and that the covariance matrix of the error term (5.52) is given by $ ΩT = E εT (k){εT (k)}H $G1 , H(k)

$ † H$ = E H† (k)nm (k)nH m (k){H (k)} G1 , H(k) =

−1 σn2 H H (k)H(k) , |G1 |2

(5.54)

$ where E X$Y denotes the conditional expected value of X given Y. The eﬀective SNR for the nt th branch and the kth carrier can, subsequently, be found using (5.54) and is given by ℘nt (k) =

|G1 |2 σs2

σn2 (HH (k)H(k))−1

,

(5.55)

nt nt

where [A]mm denotes the mth diagonal element of the matrix A and the covariance matrix of sm equals σs2 I. When the channel matrix H(k) has i.i.d. complex Gaussian entries, often referred to as Rayleigh fading, ℘nt (k) is chi-square distributed with 2R = 2(Nr − Nt + 1) degrees of freedom [127]. The probability density function (pdf) of ℘nt (k) is then given by / . (ρ/℘0 )R−1 ρ p℘n (k) (ρ) = , (5.56) exp − t ℘0 (R − 1)! ℘0 where ℘0 is the average SNR, given by ℘0 = |G1 |2

σs2 . σn2

(5.57)

Now we have found the distribution of the SNR, we can proceed to the second step of the derivation, in which we derive the probability of erroneous detection for a given SNR. In the remainder, we will consider one carrier in one of the TX streams and, therefore, omit the subcarrier and branch index, i.e., we abbreviate ℘nt (k) = ℘. Since the statistics are equal for all subcarriers and TX streams, the resulting probability of error can easily be generalised for all carriers later on.

154

5 IQ imbalance

If we ﬁrst consider the probability of erroneous detection of rectangular M -QAM symbols for a given SNR ℘ and a given translation d = dR + jdI , we can use that the real and imaginary part of the estimated symbols are independent. Consequently, the probability of erroneous √ detection of rectangular M -QAM symbols can √ be derived using two M -dimensional pulse-amplitude-modulation ( M -PAM) signals, one for the real and one for the imaginary part. The probability of symbol error is then given by % &% & Ped,M -QAM,Es = 1 − 1 − Ped,R√M -PAM,Es /2 1 − Ped,I√M -PAM,Es /2 , (5.58) where Ped,R√M -PAM,Es /2 and Ped,I√M -PAM,Es /2 denote the probability of error √ for the d-translated M -PAM modulation for the real and imaginary part of the M -QAM constellation, respectively. The power of these PAM signals is half of that of the QAM symbols, the spectral density of which is denoted by Es . We recall that the translation due to TX IQ imbalance was deﬁned in (5.51). The inﬂuence of a translation of dR on the 4-PAM constellation is schematically depicted in Fig. 5.5. The original 4-PAM constellation is depicted in Fig. 5.5(a), where the decision boundaries are depicted by dashed lines. The transmitted symbols are depicted by black dots. The distances to the decision boundaries are Es /5, i.e., when the absolute value of the error in symbol estimation is larger than Es /5 a symbol error occurs. Subsequently, we consider the dR -translated constellation, as caused by IQ-imbalance, which is schematically depicted in Fig. 5.5(b). In this ﬁgure, the originally transmitted symbols are depicted by dashed white dots and the translated symbols are depicted by black dots. It is clear

−

0

4 E 5 s

4 E 5 s

(a) Original constellation

dR

−

dR

4 E 5 s

dR

0

dR

4 E 5 s

(b) dR -translated constellation

Figure 5.5.

Inﬂuence of translation on the 4-PAM constellation.

5.3 Impact of IQ imbalance on system performance

155

that the distances to the decision boundaries are changed. For half of the cases the distances now equal Es /5 − dR and the other half they equal Es /5 + dR , where the smallest one will impose the dominant error probability. When we now generalise the observations from √ Fig. 5.5, we ﬁnd that for the real part the probability of error of an M -PAM system is given by a1 P |ε | > a2 − dR + P |ε | > a2 + dR Ped,R√M -PAM,Es = 24 4 5 4 55 a2 − dR a2 + dR = a1 Q +Q , (5.59) N0 /2 N0 /2 9 √ √ 3Es ∗ where a1 = ( M − 1)/ M , a2 = M −1 , dR = R{Ge s (−k)}, Q(x) denotes the Q-function or complementary cumulative normal distribution function and N0 denotes the noise power spectral density. A similar derivation can be done for Ped,I√M -PAM,Es , yielding the same result as (5.59), by replacing dR by dI = I{Ge s∗ (−k)}. Substituting this into (5.58) yields √ √ Ped,M -QAM = 1 − 1 − c1 U(c2 ℘, 2dR ) + U(c2 ℘, − 2dR ) √ √ · 1 − c1 U(c2 ℘, 2dI ) + U(c2 ℘, − 2dI ) , (5.60) & %√ √ b , c1 = 1 − 1/ M , c2 = 3/(M − 1) and a − √N where U(a, b) = Q 0 the SNR is deﬁned as ℘ = Es /N0 . For high SNR-values, (5.60) is well approximated by : : / . // . . 2 2 √ √ d |dR | +Q c2 ℘ − |dI | . (5.61) Pe,M -QAM ≈ c1 Q c2 ℘ − N0 N0 We note that (5.60) was calculated for a given s(−k), but that we aim at the averaged SER over all possible realisations of s(−k). Since the d values originate from the M -QAM constellation, the probability of the occurrence of the diﬀerent symbols has a discrete uniform distribution. The average probability of error is thus given by Peav ,M -QAM =

M 1 dq Pe,M -QAM , M

(5.62)

q=1

where dm is the mirror interference term, which is a multiplication of the (given) Ge and the mth symbol out of the M -QAM modulation alphabet.

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5 IQ imbalance

When we work out the above for 4-QAM, i.e., QPSK, modulation, we ﬁnd that the possible values of the translation are given by √ √ Es (±1 ± j) G∗2 Es (±1 ± j) √ √ d q = Ge = G1 2 2 √ jφ T Es (±1 ± j) 1 − gT e √ = , (5.63) 1 + gT ejφT 2 all with equal probability. It is then veriﬁed that due to symmetry dq Pe,M -QAM is equal for all q ∈ {1, . . . , 4} and that the average probability of error in (5.62) is given by . / 1 av Pe,4-QAM = 1 − 1 − [V (℘, gS ) + V (℘, −gS )] 2 . / 1 · 1 − [V (℘, gD ) + V (℘, −gD )] 2 1 = [V (℘, gS ) + V (℘, −gS ) + V (℘, gD ) + V (℘, −gD )] 2 1 − [V (℘, gS ) + V (℘, −gS )] [V (℘, gD ) + V (℘, −gD )] , (5.64) 4 where we have deﬁned √ V(℘, x) = Q ( ℘(1 − x)) , gS = gR + gI , gD = gR − gI , 1 − gT2 , 1 + gT2 + 2gT cos(φT ) −2gT sin(φT ) . gI = I{Ge } = 1 + gT2 + 2gT cos(φT )

gR = R{Ge } =

(5.65) (5.66) (5.67) (5.68) (5.69)

The expression in (5.64) can be approximated for high SNR values by Peav ,4-QAM ≈

1 √ √ [Q ( ℘(1 − gS )) + Q ( ℘(1 − gD )) 2 √ √ +Q ( ℘(1 + gS )) + Q ( ℘(1 + gD ))] .

(5.70)

It can be concluded from (5.70) and Fig. 5.5 that the TX IQ imbalance will result in a shifting of the SER curves. Flooring will, however, only occur for IQ imbalance values where the translated constellation points are located outside the correct decision region, i.e., when the factor mul√ tiplying ℘ is smaller than or equal to 0.

5.3 Impact of IQ imbalance on system performance

157

For even higher SNR values the last two terms in (5.70) can also be omitted. The expressions in (5.64) and (5.70) provide the SER for the nt th branch and kth subcarrier of a system experiencing TX IQ imbalance, but no fading channel. Hence, the probability of error was calculated above for a given SNR ℘. To calculate the average SER for faded channels, we have to integrate these SER expressions over the distribution of the SNR in (5.56). The SER for the nt th branch and kth subcarrier of an uncoded system is then found by ∞ Pe = Peav (5.71) ,M -QAM (ρ)p℘ (ρ)dρ, 0

where the subcarrier and branch index were omitted for readability. The ﬁnal expression for a MIMO OFDM system experiencing TX IQ imbalance and an i.i.d. Rayleigh faded channel is then found by substituting (5.56) and (5.62) into (5.71) and then averaging over all subcarriers and TX branches. A closed form expression can be found using a similar approach as we will use for RX IQ imbalance in Section 5.3.2.2, yielding an expression based on hypergeometric functions. For the 4-QAM example, the SER for a fading channel can be found by substituting (5.56) and either (5.70) or (5.64) into (5.71). The ﬁnal average SER expression is then found by averaging over all subcarriers and TX branches.

5.3.2.2 RX IQ imbalance Let us, subsequently, consider the case of RX IQ imbalance for which the error vector εR is given by (5.49). The mean of this error term is given by µR = E [εR (k)] = 0, (5.72) since the elements of s∗m (−k) and nm (k) have zero mean. The covariance matrix of the error vector for the kth carrier is given by $ $ ΩR = E εR (k)εH R (k) Ke , H(k)

† H = E H† (k)Ke H∗ (−k)s∗m (−k)sTm (−k)HT (−k)KH {H (k)} e

† H +E H† (k)Ke n∗m (−k)nTm (−k)KH {H (k)} e

† H +E H† (k)nm (k)nH (k){H (k)} m −1 † H = (Nt σs2 + σn2 )H† (k)Ke KH + σn2 HH (k)H(k) e {H (k)} −1 = |Ke |2 Nt σs2 + (1 + |Ke |2 )σn2 HH (k)H(k) , (5.73)

158

5 IQ imbalance

where we assumed H(−k) to be independent of H(k), which was shown in [124] to be a reasonable assumption for a practical system experiencing indoor multipath channels. Furthermore, we assumed in (5.73) that the eﬀect of IQ imbalance on all RX branches was equal, i.e., Ke = Ke I. The eﬀective SNR for the nt th branch and the kth carrier, is then found from (5.73) and given by ℘nt (k) =

σs2

(|Ke |2 Nt σs2 + (1 + |Ke |2 )σn2 ) (HH (k)H(k))−1

.

(5.74)

nt nt

Note again that, when the channel matrix H(k) has i.i.d. complex Gaussian entries, ℘nt (k) is chi-square distributed with 2R = 2(Nr −Nt +1) degrees of freedom and that its pdf is given by (5.56). The average SNR is here given by ℘0 =

σs2 . |Ke |2 Nt σs2 + (1 + |Ke |2 )σn2

(5.75)

The SER for the nt th branch and kth subcarrier of an uncoded system is then found by Pe =

∞

Pe,M -QAM (ρ)p℘ (ρ)dρ,

(5.76)

0

where the Pe,M -QAM (℘) denotes the SER for an M -QAM constellation, an approximation of which is given by √ Pe,M -QAM (℘) = c3 Q( c2 ℘).

(5.77)

√ In this expression c3 = 4(1 − 1/ M ) and we recall that c2 = 3/(M − 1). The average SER is now found by averaging Pe over the diﬀerent subcarriers and branches. The average (approximate) bit-error rate (BER) is then found by dividing the SER by log2 (M ). To derive a closed form expression for (5.76), we use an alternative representation for the Gaussian Q-function [103, p.71], given by 1 Q(x) = π

π/2 0

.

x2 exp − 2 sin2 ϕ

/ dϕ

for

x ≥ 0.

(5.78)

By substituting this expression and (5.56) into (5.76), working out one integral and by change of integration variable, we ﬁnd that the probability

5.3 Impact of IQ imbalance on system performance

159

of symbol error is given by .

/ c2 ℘ exp − dϕp℘ (ρ)dρ 2 sin2 ϕ 0 0 . / c2 ℘0 −R c3 π/2 dϕ 1+ = π 0 2 sin2 ϕ . / √ c3 (1 − ℘0 c4 ) 1 1 3 −c2 ℘0 = ,R + ; ; , 2 F1 2 2 2 2 2

c3 Pe = π

9

∞ π/2

(5.79)

(R− 1 )!

2c2 2 where c4 = π (R−1)! and 2 F1 denotes the hypergeometric function [128]. For the special case where the number of TX and RX branches is equal, i.e., R = 1, we can use the ﬁndings of [127] for AWGN distorted ZF-based MIMO systems, to rewrite (5.79) to / . : c 2 ℘0 c3 Pe = . (5.80) 1− 2 2 + c2 ℘0

5.3.3

Numerical results

The analytical results derived in Section 5.3.2 are here compared with results from Monte Carlo simulations. All simulations are carried out for an IEEE 802.11a-like MIMO system, as described in Section 3.2. For these simulations all 64 subcarriers contain data symbols and no coding is applied. It is assumed that the MIMO transfer is perfectly estimated, yielding the estimate expressed in (5.44). All ﬁgures presented in this section depict the analytical results by lines and the results from simulations by markers. First the inﬂuence of TX IQ imbalance in a non-fading channel is studied. A SISO system applying 64-QAM modulation experiencing different values of IQ imbalance is simulated. The SER results are reported in Fig. 5.6 together with the analytical results found using (5.62). The results are depicted as a function of the average SNR per RX antenna, which is given by Nt σs2 /σn2 , i.e., σs2 /σn2 for the regarded SISO system. We note that this deﬁnition was chosen to allow for a fair comparison between the diﬀerent antenna conﬁgurations. For a given SNR and noise power, the power per TX branch is scaled by 1/Nt , such that the total TX power is independent of the number of TX antennas. It can be concluded from Fig. 5.6 that the analytical results accurately predict the SER results found from simulations, proving the applicability of the results derived in (5.62). It can furthermore be concluded that considerable degradation only occurs for very high values of IQ imbalance. Additionally, although degradation in SER performance occurs,

160

5 IQ imbalance 100

10−1

SER

10−2

10−3

10−4

{1.00, 0◦} {1.05, 3◦} {1.10, 3◦} {1.10, 5◦} {1.10, 10◦} {1.15, 10◦}

10−5

10−6

0

10 20 30 40 50 Average SNR per RX antenna (dB)

60

Figure 5.6. SER performance of a system experiencing TX IQ imbalance and an AWGN channel. Results are depicted for an 1×1 64-QAM system. Analytical results of (5.62). The ﬁgure legend reports the IQ imbalance parameters {gT , φT }.

ﬂooring only seems to occur for gT = 1.15 and φT = 10◦ , where the translation d is such that some of the 64-QAM constellation points are shifted outside their decision regions. For these cases, the coeﬃcient √ multiplying the ℘ in at least one of the Q-functions forming the SER expression is smaller than or equal to 0. Similar results are depicted in Fig. 5.7, but now for a system applying a 4-QAM modulation. The results from simulations are compared with the analytical results of (5.64) and (5.70), which are the derived exact SER and high SNR approximation of the SER, respectively. The results presented in Fig. 5.7 allow the conclusion that both the exact as well as the approximate analytical expressions for the SER correctly predict the system performance. A small disparity between the curves is only visible for SNR values below 5 dB. It is furthermore clear from the ﬁgure that SER degradation due to TX IQ imbalance only occurs for very high imbalance values and that no ﬂooring occurs for the regarded values of IQ imbalance. The SER results for diﬀerent MIMO systems experiencing TX IQ imbalance and a fading channel are depicted in Fig. 5.8. The ﬁgure compares the analytical results of (5.71) with simulation results for 1×1, 2×2 and 2×4 4-QAM system. In these simulations independent fading over the antennas and subcarriers was simulated. We observe in Fig. 5.8

161

5.3 Impact of IQ imbalance on system performance 100

10−1

SER

10−2

10−3

10−4 {1.00, 0◦} {1.20, 20◦} {1.30, 30◦} {1.40, 40◦} {1.50, 50◦}

−5

10

10−6

0

5 10 15 20 Average SNR per RX antenna (dB)

25

Figure 5.7. SER performance of a system experiencing TX IQ imbalance and an AWGN channel. Results are depicted for an 1×1 4-QAM system. The exact analytical results of (5.64) (solid lines) and approximate SER results of (5.70) (dashed lines) are depicted. The ﬁgure legend reports {gT , φT }. 100

{1, 0◦} {1.30, 30◦} {1.50, 50◦}

SER

10−1

10−2

10−3

10− 4 0

5

10

15

20

25

30

Average SNR per RX antenna (dB)

Figure 5.8. SER performance of a system experiencing TX IQ imbalance and a Rayleigh fading channel. Results are depicted for an 1×1 (solid lines), 2×2 (dashed lines) and 2×4 (dash-dot lines) 4-QAM system. IQ imbalance is equal on TX branches. The ﬁgure legend reports {gT , φT }.

162

5 IQ imbalance

that also for the Rayleigh fading channel the analytical expressions accurately predict the error performance. Furthermore, we conclude that the inﬂuence of the TX IQ imbalance on the performance of the symmetric MIMO systems is small. At a SER of 10−2 , the degradation for these systems is 0.3 dB and 3.8 dB for gT = 1.30, φT = 30◦ and gT = 1.50, φT = 50◦ , respectively. For the 2×4 system this is 0.9 dB and 6 dB, respectively. We note, however, that in none of the cases ﬂooring in the SER performance occurs. Furthermore, although the IQ imbalance of gT = 1.30, φT = 30◦ is very high, the degradation in SER performance is below 1 dB at a SER of 10−2 for all the studied systems. Results of simulations testing the inﬂuence of RX IQ imbalance are depicted in Fig. 5.9 together with the corresponding analytical results of (5.79). Results are depicted for an 1×1 and 2×2 64-QAM and a 2×2 4-QAM system. We conclude from the results in Fig. 5.9 that also for RX IQ imbalance in MIMO OFDM systems the analytical SER expressions derived in Section 5.3.2 are well applicable. When comparing the results with those in Fig. 5.8, we conclude that the degradation due to the much lower RX IQ imbalance is considerably higher than that of TX IQ imbalance, 100

{1, 0 ◦ } {1.01 , 1 ◦ } {1.01 , 3 ◦ }

SER

10−1

10−2

10−3

10−4 10

20

30

40

50

60

Average SNR per RX antenna (dB)

Figure 5.9. SER performance of systems experiencing RX IQ imbalance and a Rayleigh fading channel. Results are depicted for a 1×1 64-QAM (solid lines), 2×2 64-QAM (dashed lines) and 2×2 4-QAM (dash-dot lines) system. Analytical results of (5.79) are in lines and simulation results are depicted by markers. The ﬁgure legend reports {gR , φR }.

5.4 Preamble based estimation and mitigation

163

as was already qualitatively concluded in Section 5.3.1. It is noted, furthermore, that small values of RX IQ imbalance, in contrast to TX IQ imbalance, already cause ﬂooring in the SER performance, which can be explained by the nature of the error term.

5.3.4

Summary and discussion

The impact of IQ imbalance on the performance of a multiple-antenna OFDM system was studied in this section. For this purpose, a system experiencing only frequency-independent (FI) TX IQ imbalance and a system experiencing only FI RX IQ imbalance were investigated. The error in the estimation of the TX vector was derived for both cases in Section 5.3.1, which allowed for the qualitative conclusion that in case of fading channels the inﬂuence of RX IQ imbalance will be greater than that of TX IQ imbalance. To further substantiate this conclusion, an analytical study of the probability of erroneous detection of transmitted M -QAM symbols was performed in Section 5.3.2. The results were in Section 5.3.3 compared with results from a simulation study, conﬁrming that the derived analytical expressions can be used to accurately predict the symbol-error rate of a MIMO OFDM system experiencing IQ imbalance for a wide range of imbalance parameters. The observed diﬀerences between the inﬂuence of TX and RX IQ imbalance can intuitively be understood as follows. For both cases the IQ imbalance results in leakage of the signal on the mirror subcarrier. For the TX IQ imbalance, the leakage consist of, scaled and rotated, M -QAM symbols, see (5.47). In detection this can be considered equal to shifting of the decision boundaries. For the RX IQ imbalance, the leakage consist of M -QAM symbols multiplied with the channel matrix, see (5.49). For a Rayleigh fading channel, the leakage can be approximated as an additional Gaussian RX noise term. Hence, the inﬂuence is similar to that of the commonly studied additive Gaussian RX noise source, i.e., the SER performance (and thus the SER ﬂooring) depends on the MIMO conﬁguration.

5.4

Preamble based estimation and mitigation

In the previous sections, it was shown that IQ imbalance considerably aﬀects system performance of MIMO OFDM systems. The occurrence of IQ imbalance could be avoided by putting stringent speciﬁcations on the IQ matching of the transceiver, which, however, yields very challenging design tasks. Therefore, we here propose the estimation and compensation of the inﬂuence of the IQ imbalance in the baseband part of the RX. In this section, we focus on data-aided estimation and compensation of

164

5 IQ imbalance

the FI IQ imbalance, since the data-aided estimation of the FS imbalance parameters was foreseen to require too much pilot data to be eﬃcient. An approach for compensation of the inﬂuence of FS IQ imbalance is presented in Section 5.5. Correction techniques for RX-caused FI IQ imbalance in SISO systems were previously proposed in e.g. [30, 122, 123, 129], while a correction approach for TX-caused IQ imbalance in SISO systems was proposed in [130]. The authors of [125] propose a compensation approach for the combined inﬂuence of TX and RX IQ imbalance in SISO systems. Recently, the compensation of IQ imbalance in multiple-antenna OFDM systems was addressed in [131, 132]. The approach presented in this section extends previous work by considering both TX and RX IQ imbalance in multiple-antenna OFDM systems, where every TX/RX branch can experience a diﬀerent IQ imbalance. Furthermore, not only compensation of the IQ imbalance is proposed, like in [125, 131, 132], but also estimation of the imbalance parameters, which enables the possible compensation of IQ mismatch in adaptive RF front-ends. The presented techniques exploit that the IQ imbalance parameters are, in contrast to the wireless channel parameters, time-invariant. Section 5.4.1 presents, based on the preamble structures introduced in Chapter 2, a preamble design which enables the simultaneous estimation of the IQ imbalance and channel parameters. The estimation algorithms for systems experiencing TX or RX IQ imbalance are subsequently introduced in Sections 5.4.2 and 5.4.3, respectively. Section 5.4.5 presents an algorithm for systems experiencing both TX and RX IQ imbalance. Finally, the performance of the algorithms is evaluated using a simulation study, the results of which are presented in Section 5.4.6. Throughout this section, we will assume that the system only experiences FI IQ imbalance.

5.4.1

Preamble design

To enable the estimation of both the MIMO channel transfer H and the IQ imbalance matrices G1 , G2 , K1 and K2 , we propose a data-aided approach enabled by the transmission of a preamble. In Chapter 2 it was explained that in packet-based transmissions the actual data is often preceded by several training symbols. While the packet structures proposed in Section 3.2.2 were designed to enable both frequency synchronisation and acquisition of the wideband MIMO channel matrix, here we, additionally, want to enable the estimation of the imbalance parameters.

165

5.4 Preamble based estimation and mitigation

To enable this, the preamble, again, has to be orthogonal and shiftorthogonal for at least the channel length, which is required for reliable channel estimation. Since IQ imbalance generates leakage from carrier −k into the received signal on carrier k, as concluded from (5.36), the preamble symbols, moreover, have to possess orthogonality between these subcarriers. The latter can be achieved by the use of WalshHadamard codes and requires two OFDM symbols to be transmitted consecutively, i.e., sp,1 and sp,2 . To create orthogonality the symbols on the kth carrier of sp,1 and sp,2 are given by for all k , sp,1 (k) ∈ {−1, 1} 6 sp,1 (k) for k ∈ {1, 2, . . . , K}, sp,2 (k) = −sp,1 (k) for k ∈ {−K, −K + 1, . . . , −1},

(5.81) (5.82)

respectively. The requirements for channel estimation can now be met by combining the above proposed symbols with the orthogonal structures proposed in Sections 3.2.2.2 to 3.2.2.6, where now Sp = [sp,1 , sp,2 ]. Note that the deﬁnition of (5.81) and (5.82) allows us to design the preambles such that the time-domain versions of sp,1 and sp,2 exhibit a minimal peak-to-average power ratio (PAPR). Chapter 6 will show that lower PAPR will result in less nonlinear distortion and, thus, a better channel estimation performance. Although multiple structures are possible, we will in the numerical results of this chapter use the time-orthogonal (TO) structure. The transmission format, based on the TO preamble of Section 3.2.2.3, is schematically depicted in Fig. 5.10 for a MIMO system with two transmit branches. When this preamble is applied for per subcarrier MIMO least-squares channel estimation, as described in (3.24), two joint wireless channel/IQ imbalance transfer matrices are found for every carrier, i.e., C+ (k) and C− (k). These are obtained when the signs of the training symbols on carrier k and −k are equal and diﬀerent, respectively. It can be veriﬁed that due to the deﬁnition of (5.81) and (5.82) both estimates are obtained for every subcarrier, when a preamble as deﬁned above is applied. When we leave out the noise terms in our notation, the estimated combined channel and IQ imbalance transfer matrices are given by C+ (k) = K1 H(k) (G1 + G∗2 ) + K2 H∗ (−k) (G∗1 + G2 ) , C− (k) = K1 H(k) (G1 − G∗2 ) + K2 H∗ (−k) (G2 − G∗1 ) . sp,1 sp,2 sp,1 sp,2 TX2 sp,1 sp,2 −sp,1 −sp,2 TX1

Figure 5.10.

DATA 1 DATA 2

Transmission format for a system with 2 TX branches.

(5.83) (5.84)

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5 IQ imbalance

From (5.83) and (5.84) we can, subsequently, ﬁnd the channel and FI IQ imbalance parameters using the algorithms presented in Sections 5.4.2 to 5.4.5.

5.4.2

TX IQ imbalance estimation

When we ﬁrst consider a system only impaired by TX IQ imbalance, i.e., K1 = I and K2 = 0, the expressions for the estimated transfer matrices in (5.83) and (5.84) reduce to C+ (k) = H(k), C− (k) = H(k) (G1 − G∗2 ) = H(k)gT exp(jφT ),

(5.85) (5.86)

where we used the property G1 + G∗2 = I. Using (5.85) and (5.86) we can estimate the diagonal imbalance matrices φT and gT for the kth carrier as % & ˜T (k) = arctan I{C†+ (k)C− (k)}R{C†+ (k)C− (k)}−1 , φ (5.87) 9 g˜T (k) = R{C†+ (k)C− (k)}2 + I{C†+ (k)C− (k)}2 , (5.88) where x ˜ denotes the estimate of parameter x. Improved estimates of these imbalance parameters are obtained by averaging over the frequency index k, which exploits the frequency independence of the IQ imbalance. A further improvement is achieved by exploiting the property that the IQ imbalance is time invariant. This can be done by averaging the imbalance parameters with those found in the previous P packets. This averaging over time and frequency yields the improved estimates ¯T = φ

P 1 2KP

˜T,p (k), φ

(5.89)

g˜T,p (k),

(5.90)

p=1 1≤|k|≤K

P 1 g¯T = 2KP

p=1 1≤|k|≤K

˜T,p (k) and g˜T,p (k) denote the estimates for the kth carrier during where φ the pth packet. Now that the imbalance parameters have been estimated, the MIMO channel matrix for the kth carrier can be found by ¯T )) C+ (k) + C− (k)(¯ gT exp(j φ ˜ H(k) = 2

−1

,

(5.91)

5.4 Preamble based estimation and mitigation

167

where this estimate reduces the inﬂuence of noise on the channel estimation error, compared to just using the estimate C+ (k) in (5.85). ¯T , g¯T and H(k), ˜ The estimated parameters, i.e., φ are used during the data detection phase to correct the received signals for the IQ imbalance and to detect the transmitted symbols. This can be done by feeding back the estimated IQ imbalance parameters to the TX and correcting for their inﬂuence in either the baseband or RF front-end. Here, however, we regard compensation in the RX baseband. To this end we rewrite the expression for the RX signal vectors when experiencing TX IQ imbalance in (5.36) as ˆ m (k) x H(k) 0 G1 G∗2 sm (k) = ˆ ∗m (−k) x s∗m (−k) 0 H∗ (−k) G2 G∗1 H(k)G1 sm (k) H(k)G∗2 = . (5.92) s∗m (−k) H∗ (−k)G2 H∗ (−k)G∗1 30 1 2 U(k) ˜ An estimate of the matrix U(k), denoted by U(k), is constructed using ¯ ˜ φT , g¯T and H(k). The transmitted data signal can then be retrieved as ˜sm (k) ˆ m (k) x † ˜ = U (k) . (5.93) ˜s∗m (−k) ˆ ∗m (−k) x

5.4.3

RX IQ imbalance estimation

For a system only experiencing RX IQ mismatch, i.e., G1 = I and G2 = 0, the expressions for the estimated transfer matrices (5.83) and (5.84) reduce to C+ (k) = K1 H(k) + K2 H∗ (−k), C− (k) = K1 H(k) − K2 H∗ (−k).

(5.94) (5.95)

We subsequently deﬁne Cs (k) = (C+ (k) + C− (k))/2, Cd (k) = (C+ (k) − C− (k))/2, Q(k) = (Cs (k) − C∗s (−k)) (Cs (k) + C∗d (−k))† ,

(5.96) (5.97) (5.98)

and since Q(k) can be written as gR exp(−jφR ), the imbalance parameters for the kth carrier are found by % & ˜R (k) = −∠Q(k) = − arctan I{Q(k)}R{Q(k)}−1 , (5.99) φ (5.100) g˜R (k) = |Q(k)| = R{Q(k)}2 + I{Q(k)}2 .

168

5 IQ imbalance

Again, the estimates can be improved by averaging over the estimates obtained at the 2K subcarriers and P previous packets, as done for the ¯R , g¯R , Q ¯R ), ¯ = g¯R exp(−j φ TX IQ imbalance in (5.89) and (5.90), yielding φ ¯ ¯ K1 and K2 . The estimate of the MIMO channel matrix for the kth subcarrier is then found to be ¯ −1 (Cs (k) − C∗ (−k)) (Cs (k) + C∗d (−k)) + Q d ˜ H(k) = , (5.101) 2 which provides us with the parameters to estimate. An estimate of the transmitted data signal can now be found using (5.93), where the estimate of U(k) for RX IQ imbalance is given by ˜ ¯ ∗H ˜ ∗ (−k) ¯ 1 H(k) K K 2 ˜ . (5.102) U(k) = ¯ ∗ ˜ ∗ ¯ ∗ H(−k) ˜ K2 H (k) K 1

5.4.4

TX and RX IQ imbalance estimation

When the MIMO OFDM system experiences both TX and RX IQ imbalance, the estimation problem can not be solved directly, as was done in Sections 5.4.2 and 5.4.3. Therefore, (5.83) and (5.84) are simpliﬁed by making the following approximations G1 ± G∗2 ≈ G1

± G∗1 + G2 ≈ ±G∗1 ,

and

(5.103)

which are valid for small values of TX IQ imbalance. The expressions for the estimated transfer matrices in (5.83) and (5.84) then reduce to ˆ + (k) = K1 H(k)G1 + K2 H∗ (−k)G∗ , C 1 ˆ − (k) = K1 H(k)G1 − K2 H∗ (−k)G∗ . C 1

(5.104) (5.105)

Applying these approximate expressions, the RX imbalance parameters can be estimated. If we deﬁne ˆ s (k) = (C ˆ + (k) + C ˆ − (k))/2 = K1 H(k)G1 , C (5.106) ∗ ∗ ˆ d (k) = (C ˆ + (k) − C ˆ − (k))/2 = K2 H (−k)G , C (5.107) 1

we ﬁnd for the kth subcarrier ˆ s (k) − C ˆ ∗ (−k))(C ˆ s (k) + C ˆ ∗ (−k))† Q(k) = (C d

= gR exp(−jφR ).

d

(5.108)

The RX phase and amplitude imbalance matrices are then easily found from (5.108) as % & ˜R (k) = − arctan I{Q(k)}R{Q(k)}−1 , φ (5.109) (5.110) g˜R (k) = R{Q(k)}2 + I{Q(k)}2 ,

5.4 Preamble based estimation and mitigation

169

respectively. Improved estimates of the imbalance parameters can be obtained by averaging over time and frequency as in (5.89) and (5.90), ¯R and Q ¯R ). ¯ = g¯R exp(−j φ yielding g¯R , φ Following this, we can go back to the original expressions in (5.83) and (5.84). Using these expressions and the estimated RX IQ imbalance parameters, the estimate of the MIMO channel matrix for the kth carrier is found to be given by ¯ ∗ (C+ (k)+C∗ (−k))+C+ (k)−C∗ (−k)), (5.111) ˜ ¯ Q ¯ ∗ )−1 (Q H(k) = (Q+ + + ¯R ). ¯ = g¯R exp(−j φ where Q Finally, to estimate the TX IQ imbalance parameters, we rewrite (5.84) for carrier k and its mirror −k in matrix notation as C− (k) K1 H(k) K2 H∗ (−k) gT ejφT = . (5.112) C− (−k) K1 H(−k) K2 H∗ (k) −gT e−jφT 2 30 1 T(k)

From (5.112) it is then easily found that ' # ˜ M1,k C− (k) g˜T ej φT † ˜ = T (k) , = ˜ C− (−k) M2,k −˜ gT e−j φT

(5.113)

˜ where T(k) is constructed from the estimated channel and RX IQ parameters, derived in the previous steps. From (5.113) the TX IQ imbalance parameters for the kth subcarrier are estimated as & % ˜T (k) = arctan I{M1 (k) − M∗ (k)}R{M1 (k) − M∗ (k)}−1 , (5.114) φ 2 2 R{M1 (k) − M∗2 (k)}2 + I{M1 (k) − M∗2 (k)}2 g˜T (k) = . (5.115) 2 ¯T and g¯T , Again, improved estimates of the imbalance parameters, i.e., φ are obtained by averaging the imbalance estimates over the subcarriers and the previous P packets. An estimate of the transmitted data signal can now be found using (5.93). The estimate of U(k) is found from the estimated parameters above as ˜ ˜ A(k) B(k) ˜ U(k) = ˜ ∗ (5.116) ˜ ∗ (−k) , B (−k) A where ¯ 2H ¯ 2, ˜ ¯1 +K ˜ ∗ (−k)G ˜ ¯ 1 H(k) G A(k) = K ∗ ∗ ¯ +K ¯ 1 H(k) ˜ ¯ 2H ˜ (−k)G ˜ ¯ ∗. B(k) = K G 1 2

(5.117) (5.118)

170

5.4.5

5 IQ imbalance

Iterative TX and RX IQ imbalance estimation

The data-aided approach proposed in Section 5.4.4 is based on the assumption that the TX IQ imbalance is relatively small, which limits the performance in the estimation of the TX/RX imbalance and channel parameters. Although this can be partly overcome by averaging over the diﬀerent subcarriers and packets, the approximations of (5.103) create a bias in the estimates, which increases with increasing TX imbalance. This will be shown in the numerical results of Section 5.4.6. Therefore, this section proposes an improved technique, which applies the algorithm of Section 5.4.4 only for the ﬁrst received packet. For all other packets the estimates of gT , φT , gR and φR from the preceding packet are used to solve the estimation problem. Note that these parameters are system parameters that, in contrast to the wireless channel H, can be assumed to be constant over a large number of packets. The procedure is schematically depicted in the ﬂow diagram of Fig. 5.11. For an improved estimate of the MIMO channel matrix for the kth carrier experienced during the (p + 1)th packet transmission (for p ≥ 1), we can exploit that the estimated transfer matrices on carriers k and −k

{C+,(p+1), C−,(p+1)}

Estimation H(p+1) ˜ H (p+1)

b˜(p) Estimation gT,(p+1) & f T,(p+1) ˜(p) α

b˜(p+1) Estimation gR,(p+1) & f R,(p+1)

p=p+1

Figure 5.11.

Flow diagram for the iterative estimation of TX and RX IQ imbalance.

5.4 Preamble based estimation and mitigation

171

are given by (5.83) and (5.84), to yield:

∗ ∗ ˜ ˜ †(p) {B2,(p+1) β˜(p) + B3,(p+1) }{β˜(p) + β˜(p) }† H (p+1) (k) = α ˜ (p) + α ˜ ∗(p) }† {α ˜ ∗(p) B1,(p+1) + B2,(p+1) } /2, (5.119) +{α where x(p) denotes x during the pth packet and where we have deﬁned B1 = C+ (k) + C∗+ (−k) = H(k) + H∗ (−k), (5.120) ∗ ∗ ∗ (5.121) B2 = C+ (k) − C+ (−k) = αH(k) − α H (−k), ∗ ∗ ∗ ∗ B3 = C− (k) + C− (−k) = αH(k)β + α H (−k)β , (5.122) α = gR e−jφR , β = gT e−jφT .

(5.123) (5.124)

Subsequently, estimates of the TX IQ imbalance parameters can be found using (5.114) and (5.115), where we use the MIMO channel estimate found in (5.119) and the RX IQ imbalance estimated during the ˜ (p+1) (k). The new estimates of the TX IQ papth packet to construct T rameters, after averaging over the subcarriers, are then used to construct β˜(p+1) . Following that, the estimates of the RX IQ imbalance parameters on the kth carrier are found by ˜ 1,(p+1) (k) K ˜ 2,(p+1) (k)] = C(p+1) (k)Y† [K (p+1) (k),

(5.125)

where we have deﬁned C(p) (k) = C+,(p) (k) C−,(p) (k) C+,(p) (−k) C−,(p) (−k) , (5.126) ' # ˜ (p) (k)β(p) ˜ (p) (−k)β(p) ˜ (p) (−k) H ˜ (p) (k) H H H . Y(p) (k) = ˜ ∗ (−k)β ∗ ˜ ∗ (k)β ∗ ˜ ∗ (−k) −H ˜∗ −H H (p) (p) (p) H(p) (k) (p) (p) (5.127) From (5.125) we construct ˜ 1,(p+1) (k) − K ˜∗ ˜ (p+1) (k) = K α 2,(p+1) (k), which is used to ﬁnd the imbalance matrices % & ˜R (k) = − arctan I{α ˜ (p+1) (k)}R{α ˜ (p+1) (k)}−1 , φ 9 ˜ (p+1) (k)}2 + I{α ˜ (p+1) (k)}2 . g˜R (k) = R{α

(5.128)

(5.129) (5.130)

Finally, averaging these estimates over the diﬀerent subcarriers provides all estimates for the (p+1)th packet. The transmitted data symbols are retrieved in the same way as in Section 5.4.4.

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5 IQ imbalance

To reduce the inﬂuence of noise on the estimation error of the IQ imbalance parameters, averaging over diﬀerent packets as in Section 5.4.4 can be applied. This is, however, only beneﬁcial when the errors in the estimates for the pth packet do not dominate the errors in the estimates for the (p + 1)th packet. From simulations this was found to be valid for p > 10.

5.4.6

Numerical results

Monte Carlo simulations were performed to test the performance of the estimation approaches for FI IQ imbalance presented in Sections 5.4.2 to 5.4.5. This section presents results from this numerical study, which was carried out for a 2×2 extension of the IEEE 802.11a standard, as presented in Section 3.2. The TO preamble, as introduced in Section 5.4.1, was used for IQ imbalance and channel estimation. The channel is modelled as quasi-static, i.e., the channel is constant over the length of the packet, but generated independently for the diﬀerent packets. The channel was modelled as described in Section 2.2 with a rms delay spread of 50 ns and spatially independent channel elements. Figure 5.12(a) depicts results for the mean squared error (MSE) in the estimation of the TX IQ imbalance parameters by the algorithm proposed in Section 5.4.2. The system experienced TX IQ imbalance with φT = diag{3◦ , −3◦ } and gT = diag{1.1, 0.9} and no RX IQ imbalance. Averaging over P received packets is applied to improve the performance.

102

102

P=1 P = 10 P = 100

100

MSE

MSE

100

10−2

10−4

10−6

P=1 P = 10 P = 100

10−2

10−4

0

10 20 30 40 Average SNR per RX antenna (dB)

(a) TX IQ imbalance estimation

10−6

0

10 20 30 40 Average SNR per RX antenna (dB)

(b) RX IQ imbalance estimation

Figure 5.12. MSE in the estimation of the diagonal elements of φ (solid lines) and g (dashed lines).

5.4 Preamble based estimation and mitigation

173

It can be concluded from 5.12(a) that the MSE (in degrees2 ) in the estimation of φT decreases linearly (on log-scale) with SNR. When the number of packets is increased, the MSE decreases even further. Regarding the estimation of gT , it can be concluded that for P = 1 the MSE decreases linearly with SNR. When averaging of the IQ parameters over P packets is applied, the MSE only improves linearly as a function of P for high SNR. For low SNR a bias in the estimation seems to limit the performance. Similar MSE results are depicted in 5.12(b), but now for a system only experiencing RX IQ imbalance and applying the estimation approach of Section 5.4.3. The imbalance parameters φR = diag{3◦ , −3◦ } and gR = diag{1.1, 0.9} were used for the simulations. The MSE results in 5.12(b) are very similar to those in 5.12(a). The bias in the estimation of the RX amplitude imbalance seems to limit the performance less for than the case of TX imbalance. MSE results for a system impaired by both TX and RX IQ imbalance (φT = diag{1◦ , −1◦ }, φR = diag{3◦ , −3◦ }, gT = diag{1.05, 0.95} and gR = diag{1.1, 0.9}) and applying the joint TX and RX imbalance estimation of Section 5.4.4 are depicted in Fig. 5.13. It can be concluded from this ﬁgure that the MSE results for the estimation of all imbalance matrices show ﬂooring at high SNR values due to the approximations made in (5.103). The MSE is greatly reduced by averaging over P = 100 packets, however, the ﬂooring remains. To reveal the improvement of the iterative technique of Section 5.4.5, simulations were carried out to compare the MSE performance of both algorithms for TX/RX IQ imbalance estimation. Figure 5.14 reports the results for the MSE in the estimation of the diagonal elements of K1 (in solid lines) and G1 (in dashed lines) and the elements of H (in dotted lines) as a function of the SNR per RX branch. The imbalance parameters were chosen equal on all TX and RX branches, i.e., gT = gR = {1.1, 1.1} and φT = φR = {3◦ , 3◦ } and a rms delay spread of 100 ns was used, which corresponds to a case with high IQ imbalance and high frequency fading. The results are given for the ﬁrst packet (“original”) and for the methods of Section 5.4.4 (“averaging”) and Section 5.4.5 (“iterative”) after P = 20 packets. We conclude from Fig. 5.14 that the MSE curves for the estimates of K1 and H after the ﬁrst packet show ﬂooring, which is improved by the method of Section 5.4.4 by averaging over 20 packets. Note, again, that only the IQ imbalance parameters are averaged, not the channel estimates. The ﬂooring at high SNR, however, remains. In contrast, the method of Section 5.4.5 does solve the ﬂooring at high SNR, i.e., the MSE of K1 and H fall oﬀ linearly with SNR here. At low SNR, however,

174

5 IQ imbalance 104

100

gT gR

φT φR 103

MSE (degrees2)

10−1

MSE

10−2

10−3

10−4

10−5

102

101

100 10−1

0

20 SNR (dB)

40

10−2

0

20 SNR (dB)

40

Figure 5.13. MSE in the estimation of TX and RX IQ imbalance parameters for P =1 (solid lines) and P =100 (dashed lines) applying the joint TX and RX IQ imbalance estimation of Section 5.4.4.

100 original averaging iterative

10−1

MSE

10−2

10−3 10−4 10−5 10−6 10

20

30

40

50

Average SNR per RX antenna (dB)

Figure 5.14. MSE in the estimation of the imbalance parameters K1 (solid lines), G1 (dashed lines) and H (dotted lines). Results are given for the ﬁrst packet (“original”) and for P = 20 packets (“averaging” and “iterative”).

175

5.4 Preamble based estimation and mitigation

a small increase in the MSE of K1 occurs, compared to the averaging method. The MSE in the estimation of G1 gains less from the averaging algorithm, which can be attributed to a bias in the estimation. The MSE results presented above reveal the performance of the estimators. However, we are also interested in the performance of a system using these estimates to correct for the IQ imbalance. Therefore BER simulations were carried out, the results of which are presented in Fig. 5.15 to Fig. 5.17. The simulations were done for a 2×2 space division multiplexing system applying ZF estimation with 64-QAM modulation and no coding. The imbalance estimation and compensation approaches of Sections 5.4.3 to 5.4.5 are applied. The correction for the channel and IQ imbalance is applied in the RX baseband. Reference curves are given for a system not applying IQ imbalance compensation, and for a system not impaired by IQ imbalance, but applying channel estimation based on the proposed preamble. We conclude from Fig. 5.15 that the impact of RX IQ imbalance without compensation is severe and limits the system performance signiﬁcantly. When correction of the IQ imbalance based on the approach of Section 5.4.3 is applied, the BER performance resembles that of a system without IQ imbalance. Averaging over more packets does not increase 100

BER

10−1

10−2

10−3

10−4

IQ imb., no cmp. No IQ imb. P=1 P = 100

0

10 20 30 40 Average SNR per RX antenna (dB)

50

Figure 5.15. BER performance of a 2×2 ZF-based system in detection of uncoded 64-QAM modulated data for a system experiencing RX IQ imbalance (φR = diag{3◦ , −3◦ } and gR = diag{1.1, 0.9}). The estimation approaches of Sections 5.4.3 (solid lines) and 5.4.4 (dashed lines) are applied.

176

5 IQ imbalance

the performance further, since the inﬂuence of the noise is dominant over that of the remaining IQ imbalance after compensation. When the correction based on the joint TX/RQ IQ imbalance estimation approach of Section 5.4.4 is applied, the BER performance is a little improved when the number packets is increased, however, a diﬀerence of 1.5 dB compared to the ideal curve remains, due to the higher channel estimation errors for this approach. For a system experiencing both TX and RX IQ imbalance, results are depicted in Fig. 5.16, from which we conclude that also here considerable improvement in BER performance is achieved by the approach of Section 5.4.4. More packets are required, however, to achieve a similar performance to that in Fig. 5.15. Again a diﬀerence of 1.5 dB compared to the ideal case (without IQ imbalance) remains, even for a high number of packets P . Results for systems applying other MIMO detectors and/or coding are similar, but shifted to another (lower) SNR range. Finally, Fig. 5.17 compares the BER performance of systems applying the algorithms of Section 5.4.4 and Section 5.4.5. The channel and IQ imbalance parameters were equal to those of the simulations for Fig. 5.14. As a reference, the BER of a system not experiencing IQ imbalance

100

BER

10−1

10−2

IQ imb., no cmp. No IQ imb. P =1 P = 10 P = 50 P = 100

10−3

10−4

0

10 20 30 40 Average SNR per RX antenna (dB)

50

Figure 5.16. BER performance of a 2×2 ZF-based system in detection of uncoded 64-QAM modulated data for a system experiencing TX and RX IQ imbalance (φT = diag{1◦ , −1◦ }, φR = diag{3◦ , −3◦ }, gT = diag{1.05, 0.95} and gR = diag{1.1, 0.9}). The estimation approach of Section 5.4.4 is applied.

177

5.4 Preamble based estimation and mitigation 100 ideal P=1 P = 10 P = 20 P = 50

BER

10−1

10−2

10−3 0

10 20 30 40 Average SNR per RX antenna (dB)

50

Figure 5.17. BER performance of a 2×2 ZF-based system in detection of uncoded 64-QAM modulated data for a system experiencing TX and RX IQ imbalance (gT = gR = diag{1.1, 1.1} and φT = φR = diag{3◦ , 3◦ }). The estimation approaches of Sections 5.4.4 (solid lines) and 5.4.5 (dashed lines) are applied.

(ideal ), but applying channel estimation with the same preamble, is given. It is clear from Fig. 5.17 that the BER improves with increasing P for both methods. The improvements in BER performance are, however, larger for the method proposed in Section 5.4.5. The BER curve for the iterative estimation comes closer to the reference BER curve at low SNR than that of the averaging approach. Furthermore, at high SNR, the curves for the iterative method show lower ﬂooring. When we combine these results with the results found in Fig. 5.14, we can conclude that the improvement in BER performance for the approach of Section 5.4.5 can be attributed to the improved estimates of K1 and H.

5.4.7

Summary

In this section data-aided estimation and compensation approaches were proposed for MIMO OFDM systems experiencing frequency-independent (FI) TX and/or RX IQ imbalance. For a system experiencing both TX and RX IQ imbalance two estimators were presented, one of which applies the IQ imbalance estimates from previous packets. All approaches exploit that the imbalance parameters are time invariant and frequency independent. A numerical performance study revealed the MSE performance of the diﬀerent estimation methods. From a numerical

178

5 IQ imbalance

BER study it is concluded that the introduced estimation and compensation approaches provide a considerable improvement in performance compared to a system applying no compensation for the inﬂuence of IQ mismatch.

5.5

Decision-directed mitigation

Data-aided estimation of the frequency-selective (FS) IQ imbalance parameters would require a large amount of pilot data and thus impose a prohibitive overhead. Therefore, this section proposes a decisiondirected compensation approach. In this method, the IQ imbalance parameters are not explicitly estimated, but an adaptive ﬁlter (AF) matrix is applied, one for every subcarrier pair, to retrieve the transmitted data. The ﬁlter weights are adapted such that the ﬁlter compensates for the combined inﬂuence of the MIMO channel matrix, the FI IQ imbalance and the RX ﬁlter imbalance and retrieves the transmitted data. The design of the adaptive ﬁlter is proposed in Section 5.5.1. The performance of the algorithm is tested with simulations, the results of which are reported in Section 5.5.2.

5.5.1

Adaptive ﬁlter based algorithm

For the design of the AF, it is useful to rewrite the expressions for the RX signals on the kth and −kth carrier in this case of both FI IQ imbalance and RX LPF imbalance, (5.39), to ˆ m (k) sm (k) x A(k) B(k) = , (5.131) ˆ ∗m (−k) B∗ (−k) A∗ (−k) x s∗m (−k) 2 2 30 1 30 1 ˘ m (k) x

˘ sm (k)

where ˆ 1 (k)H(k)G ˆ ˆ ˆ∗ A(k) = K 1 + K2 (−k)H (−k)G2 , ∗ ˆ 2 (−k)H ˆ ∗ (−k)G∗ + K ˆ 1 (k)H(k)G ˆ B(k) = K . 1

2

(5.132) (5.133)

The data can now be recovered by applying the 2Nt ×2Nr ﬁlter matrix ˆ m (k), yielding the estimates of the Wm (k) to the received data vector x transmitted symbols ˆ m (k) ˜sm (k) x = Wm (k) . (5.134) ˜s∗m (−k) ˆ ∗m (−k) x A block diagram of the decision-directed updating of the MIMO weighting matrix Wm (k) is depicted in Fig. 5.18. Here D{·} gives the hard decision based slicing estimate of its input and em (k) denotes the error vector for the kth carrier in the mth symbol.

179

5.5 Decision-directed mitigation x ˆ m(k) ∗ (−k) x ˆm

D{s˜m(−k)}

˜s m(k) Wm(k)

Slicing

∗ (−k) ˜s m

− e m(k) Update algorithm

e m(−k)

∗ (−k)} ˜m D{s

+ −

+

Figure 5.18. Block diagram of the decision-directed adaptive MIMO ﬁlter. D{·} gives the hard decision based slicing estimate of its input.

Diﬀerent approaches can be chosen for the adaption of the ﬁlter matrix Wm (k). Since the adaptation is decision-directed, it requires data symbols to converge. During this adaptation the data is detected with a higher probability of error. Hence, a high convergence speed of the AF is crucial to achieve a reasonable system performance. Therefore, the exponentially weighted recursive least-squares (RLS) algorithm [133] was chosen, since its convergence rate is generally an order of magnitude better than the more commonly applied least-mean-squares (LMS) algorithm. It is noted, however, that the RLS algorithm imposes a higher computational burden than the LMS algorithm. The initial ﬁlter matrix W0 (k) is found by using the estimates of the FI IQ imbalance and MIMO channel matrices from the data-aided estimation of Section 5.4. The resulting ﬁlter matrix is given by W0 (k) =

˜ ˜ A(k) B(k) ∗ ˜ ˜ B (−k) A∗ (−k)

† ,

(5.135)

where ˜ 2H ˜ 2, ˜ ˜1 +K ˜ ∗ (−k)G ˜ ˜ 1 H(k) G A(k) = K ∗ ∗ ˜ ˜ 2H ˜ +K ˜ 1 H(k) ˜ (−k)G ˜ ˜ ∗. B(k) = K G 1 2 The adaptation process can be summarised as follows [133]: P0 (k) = δ −1 I.

(5.136) (5.137)

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5 IQ imbalance

For the symbols m = 1, 2, . . . , M Πm (k) km (k) em (k) H Wm (k)

= = = =

Pm−1 (k)˘ xm (k) −1 ˘H Πm (k)(λ + x m (k)Πm (k)) D{Wm−1 (k)˘ xm (k)} − Wm−1 (k)˘ xm (k) H H Wm−1 (k) + km (k)em (k)

Pm (k) = λ−1 Pm−1 (k) − λ−1 km (k)˘ xH m (k)Pm−1 (k), The regularisation parameter δ has to be chosen inversely proportional to the SNR and the exponential weighting factor λ is chosen to be smaller than 1. It is noted that, when the AF is applied, the ﬁrst data symbols of the packet are detected with a higher BER than those located further in the packet, since the AF needs several symbols to converge. The reliability of detection of the ﬁrst symbols can be increased by redetecting the ﬁrst few symbols, as soon as the ﬁlter weights are converged. This, however, comes at the cost of additional complexity. Alternatively, the ﬁrst few symbols of a packet could be made more robust by the use of a lower order modulation or by the use of a more robust coding scheme. This will, however, increase the packet size and requires standardisation of the scheme.

5.5.2

Numerical results

The performance of the proposed decision-directed approach was evaluated using Monte Carlo simulations. Again a 2×2 MIMO system applying 64-QAM modulation was simulated. Now, however, a non-fading perfect orthogonal (AWGN) MIMO channel was applied. Figure 5.19 depicts results for a system only experiencing FI IQ imbalance (in solid lines) and a system experiencing both FI and FS IQ imbalance (in dashed lines). The FI IQ imbalance parameters were equal to those applied for the simulations for Fig. 5.14. The FS IQ imbalance was modelled in a similar way as in [125]. The LPFs in the I paths were modelled using a 6th order Chebyshev Type 1 ﬁlter with a ripple of 1 dB and a pass band of 0.9 times the sample frequency. The parameters of the LPFs in the Q paths were slightly diﬀerent, to create the mismatch. The ripple was set to 1.05 dB and the pass band to 0.905 times the sample frequency. The exponential weighting factor λ of the RLS ﬁlter was chosen to be 0.99 for the results presented in this section. The ﬁlter was initialised with the estimates from the iterative FI IQ imbalance estimation technique of Section 5.4.5, acquired after P = 50 packets. The results in Fig. 5.19 show that compensation with the estimates from the iterative approach of Section 5.4.5, indicated by “IQ imb., init.

181

5.5 Decision-directed mitigation 10−1

BER

10−2

10−3

10−4 14

no IQ imb. IQ imb., no cmp. IQ imb., init. cmp. IQ imb., AF no IQ imb., AF 16

18

20

22

24

26

Average SNR per RX antenna (dB)

Figure 5.19. BER performance of a 2×2 system applying 64-QAM modulation and no coding. gT = gR = diag{1.1, 1.1}, φT = φR = diag{3◦ , 3◦ }, packet length is 500 symbols and adaptive ﬁlter (AF) initialisation based on P = 50 packets. Results for a system experiencing FI IQ imbalance (solid lines) and for a system experiencing both FI and FS IQ imbalance (dashed lines). 10−1

BER

10−2

no cmp. init. cmp. AF, 5 symb. AF, 50 symb. AF, 100 symb. AF, 500 symb. AF, 1000 symb.

10−3

10−4 15

20

25

30

Average SNR per RX antenna (dB)

Figure 5.20. BER performance of a 2×2 system applying 64-QAM modulation and no coding. gT = gR = diag{1.1, 1.1}, φT = φR = diag{3◦ , 3◦ } and FS IQ imbalance. Results for diﬀerent packet lengths, AF initialisation based on P = 50 packets.

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5 IQ imbalance

cmp.”, does increase the performance of the system, compared to the case with IQ imbalance and no compensation, indicated by “IQ imb., no cmp”. A large performance degradation compared to the reference curve for a system not experiencing IQ imbalance, indicated by “no IQ imb.”, however, still exists. When the proposed AF is applied, indicated by “IQ imb., AF”, the BER performance improves beyond the curve without IQ imbalance. This can be explained by the fact that the AF also improves the estimates of the MIMO channel matrices. The eﬀect of this improvement in the channel estimates is illustrated by the results for a system that does not experience IQ imbalance, but does apply the AF of Section 5.5.1, indicated by “no IQ imb., AF”. For the system experiencing both types of IQ imbalance, the performance is slightly worse than that of a system only experiencing FI IQ imbalance. The convergence of the AF is tested in simulations where the packet length was varied. Results of these simulations are depicted in Fig. 5.20. The same channel, imbalance and ﬁlter parameters were used as for the simulations of Fig. 5.19. The curves for a system not applying compensation (“no cmp.”) and for a system not applying the AF (“init. comp.”), i.e., a system which only corrects for the FI IQ imbalance estimated in the preamble phase, are given in the ﬁgure as reference. We conclude from the results in Fig. 5.20 that for a packet length of 5 symbols the gain compared to the initial correction is small. When the packet length is increased, however, the performance improves largely compared to the system only applying correction based on the initial estimates.

5.6

Conclusions

The inﬂuence, estimation and compensation of IQ mismatch in directconversion based multiple-antenna OFDM systems were treated in this chapter. First, a model for the transceiver structure was introduced, which was used to derive the inﬂuence of both frequency-independent (FI) IQ imbalance and RX low-pass ﬁlter imbalance on the received signal. It is concluded that IQ imbalance in MIMO OFDM results in scaling and rotation of the signals and in signal leakage from the DCmirrored subcarrier. Subsequently, the inﬂuence of IQ imbalance on symbol detection was studied, where TX and RX IQ imbalance were regarded separately. From the expressions for the errors it was qualitatively reasoned that, in case of fading channels, the inﬂuence of RX IQ imbalance would be greater than that of TX IQ imbalance. This conclusion was further substantiated by the derivation of analytical symbol-error rate (SER) expressions for a system applying M -QAM modulation. The resulting compact SER

5.6 Conclusions

183

expressions were shown, using simulations, to accurately predict the performance of such an impaired system for a wide range of imbalance parameters. To reduce the impact of IQ imbalance on system performance, diﬀerent data-aided estimation and compensation approaches were proposed for the case of TX, RX and TX/RX FI IQ imbalance. The derived estimation algorithms use a preamble based on Walsh-Hadamard sequences and take advantage of the time invariance and frequency independence of the mismatch parameters. The estimation performance of the different methods was studied in simulations revealing the mean squared error. The eﬀectiveness of the proposed compensation was proven by a numerical bit-error rate study, which revealed that a considerable performance improvement, compared to a system not applying compensation, is achieved by applying these IQ mismatch mitigation algorithms. Finally, this chapter presented a compensation method for frequencyselective (FS) IQ imbalance. This method is based on an adaptive MIMO ﬁlter, which estimates the transmitted MIMO signal vector transmitted for a subcarrier pair. The ﬁlter is initialised using estimates obtained from the data-aided FI estimation and is, in subsequent steps, updated in a decision-directed way. From a numerical study it is concluded that this method is able to signiﬁcantly reduce the inﬂuence of both FS and FI IQ imbalance.

Chapter 6 NONLINEARITIES

6.1

Introduction

The transfer of the system and channel experienced during a wireless transmission is commonly modelled as linear. Although this approach seems valid for the propagation channel, as discussed in Section 2.2, several elements in the transmitter (TX) and receiver (RX) analogue front-end can exhibit a nonlinear transfer. The major sources of nonlinearity are generally the analogue-to-digital (A/D) and digitalto-analogue (D/A) converters, mixers and ampliﬁers, i.e., the power ampliﬁer (PA) in the TX and the low-noise ampliﬁer (LNA) in the RX. The inﬂuence of these nonlinearities is negligible in conventional systems applying constant modulus signals, making the generally applied linear approximation valid. The time-domain signals in OFDM-based systems, however, exhibit a high peak-to-average power ratio (PAPR), which can easily result in clipping and nonlinear distortion in the diﬀerent parts of the RF front-end. Two conventional solutions to overcome this problem are either to use highly linear components for the front-end or to apply a large input power backoﬀ (BO) when feeding the signals to the nonlinear components, as such ensuring that the signals experience the linear part of the transfer of the components. The former solution has the disadvantage that the cost of the total front-end will increase. The latter solution is ineﬃcient in terms of power consumption, since the power eﬃciency of the used components decreases rapidly with increasing BO.

c 2006, 2007 IEEE. Portions reprinted, with permission, from [134–136].

186

6 Nonlinearities

Since a MIMO OFDM solution will require multiple RF front-ends in both TX and RX of the system, the disadvantages are increased compared to conventional OFDM systems. Hence, a power- and costeﬀective solution has to either allow for a certain level of nonlinearity or to compensate for it at another stage in the transmission chain. Moreover, all MIMO transmission chains will exhibit their own nonlinear transfer, making previous analyses and compensation approaches, for single-input single-output (SISO) OFDM, not directly applicable for the considered system. This chapter, therefore, treats the inﬂuence of nonlinearities in MIMO OFDM systems. First Section 6.2 reviews several models for nonlinearities and shows the inﬂuence of nonlinearities on the signal model of an OFDM system. The inﬂuence on the probability of erroneous symbol detection is investigated in Section 6.3. Previous contributions in the literature are limited to single-antenna OFDM systems experiencing TX nonlinearities and often assume a non-fading channel. Novel symbolerror rate (SER) expressions are presented in this chapter for the inﬂuence of both TX- and RX-caused nonlinearities on the SER performance of a MIMO OFDM applying M -QAM modulation for AWGN as well as Rayleigh fading channels. It is shown that the results are applicable for diﬀerent memoryless nonlinearities. To overcome the inﬂuence of nonlinearities on system performance, Section 6.4 introduces a technique for PAPR reduction in space division multiplexing based systems named spatial shifting. In this technique the extra degree of freedom provided by MIMO is exploited to reduce the PAPR. It is based on the rearrangement of the TX vector in such a way that subparts are transmitted on those TX branches that result in the lowest overall PAPR. It is shown that a signiﬁcant reduction in PAPR can be achieved by this technique. Furthermore, it is shown that even higher PAPR reduction can be achieved by combining spatial shifting with PAPR reduction techniques designed for single-antenna systems. Following this, two methods for RX-based correction of the errors caused by nonlinearities are presented in Section 6.5. For both methods, the TX signals is allowed to be distorted to such a level that the spectral regrowth outside the signal bandwidth is limited to imposed TX spectral mask. At the RX an estimate of the linear as well as an estimate of the nonlinear channel are acquired, which are used in the two proposed compensation approaches: correction of the nonlinearities by postdistortion or by the use of an iterative nonlinear distortion removal method. For channel estimation an eﬃcient preamble structure is proposed, which consists of a constant-modulus and a high PAPR part. We

187

6.2 System and nonlinearities modelling

conclude from simulation results that the inﬂuence of the nonlinearities is signiﬁcantly reduced by both methods. Finally, conclusions are drawn in Section 6.6.

6.2

System and nonlinearities modelling

Let us now consider a MIMO OFDM system applying Nt TX branches and Nr RX branches as described in Section 2.3. For convenience the MIMO OFDM system build-up is shortly repeated here to show the inﬂuence of nonlinearities in the TX and RX front-ends. Consider Fig. 6.1 which depicts the block diagram of a MIMO OFDM system experiencing both TX and RX nonlinearities, in the ﬁgure modelled as GTX and GRX , respectively. In this ﬁgure the Nt Ns ×1 time-domain transmit vector u, as deﬁned in (2.42), ﬁrst experiences the TX nonlinearity GTX before being transmitted through the wireless channel, here modelled by its baseband equivalent G. In the RX the signal is passed through the RX nonlinearity modelled by GRX and experiences the (time-domain) additive noise v. The resulting Nr Ns ×1 time-domain RX vector is denoted by y.

6.2.1

Nonlinearity modelling

In this section, we will describe several baseband equivalent models, as previously presented in the literature, for the modelling of nonlinearities occurring in a wireless communication system. Since, in general, the power ampliﬁer (PA) is the major source of nonlinearity in the transmission chain, most presented models were originally introduced for PAs. We will here, and throughout this chapter, consider the nonlinearities to be memoryless, which is a reasonable assumption for moderate

s

Nt .. .

ud

u ..

1

0 . ..

IDFT

. .. .

Nt Add CP P/S

..

.1

GTX 1 · · · Nt

Nc−1 ..

Nt

. 1 0 . .. Nc−1 ˜s

MIMO detector

..

Nr

. 1 0 . ..

Nc−1 x

G ..

DFT

. .. .

Nr .. CP .1 Removal, S/P y

1 · · · Nr GRX ··· v t d

t

Figure 6.1. Block diagram of a MIMO OFDM system experiencing nonlinearities in both TX and RX.

188

6 Nonlinearities

bandwidths. For more information on nonlinearities with memory the reader is referred to [137–139]. The applications of simpliﬁed nonlinearity models based on Wiener and Hammerstein models seems a promising approach in this case [140]. Also, we will limit our analyses of nonlinearities to their inﬂuence within the signal bandwidth. The spectral regrowth is not considered in this chapter. First consider the general input-output relation of a nonlinearity. The input to the nonlinearity is generally an amplitude and phase modulated bandpass signal, which in complex baseband representation is denoted by uin (t) = A(t)ejφ(t) . (6.1) The resulting signal at the output of the nonlinear function g(·) can be expressed by uout (t) = g(uin (t)) = gA (A(t))ej(φ(t)+gφ (A(t))) ,

(6.2)

where gA (·) and gφ (·) model the amplitude-to-amplitude (AM-AM) and amplitude-to-phase (AM-PM) transfer, respectively. For the considered MIMO OFDM system, as depicted in Fig. 6.1, the nonlinearities are modelled by GTX and GRX . The input-output relation of these blocks is given by GTX −→ uout,nt (t) = gTX,nt (uin,nt (t)) GRX −→ yout,nr (t) = gRX,nr (yin,nr (t))

(6.3) (6.4)

for nt = 1, . . . , Nt and nr = 1, . . . , Nr . The functions gTX,nt (·) and gRX,nr (·) model the nonlinearity for the nt th TX and nr th RX branch, respectively. They can represent any memoryless nonlinear function, some examples of which are given in the next subsections. Sections 6.2.1.1 to 6.2.1.4 will introduce diﬀerent commonly used models for nonlinearities to relate (6.2) to previously proposed models in the literature. It is noted once again that most of these models were originally proposed for ampliﬁers, but can be applied more generally. Furthermore, since we are interested in the nonlinear characteristic, rather than the actual ampliﬁcation, the small signal gain of the models is normalised.

6.2.1.1 Ideal clipping ampliﬁer First we regard the model for the ideal clipping ampliﬁer. This model exhibits a linear AM-AM transfer up to a certain input level, where the output power is limited, an eﬀect which is often referred to as clipping. For a normalised small-scale gain and a saturation amplitude level of

189

6.2 System and nonlinearities modelling

A0 =

√

P0 , the AM-AM transfer can be expressed as 6 A for A ≤ A0 , gA (A) = A0 for A > A0 ,

(6.5)

where we omitted the time index t for readability. The AM-PM conversion gφ for this ampliﬁer model is zero. The nonlinear model in (6.5) can, for instance, be experienced when a signal is fed through the combination of a PA and an ideal predistorter [141], which can compensate for the nonlinear transfer of the PA up to the saturation level. It can, moreover, be experienced when deliberate clipping is applied in the baseband part of the transmitter to reduce the PAPR of the signal input to the RF front-end.

6.2.1.2 Travelling wave tube ampliﬁer The model treated in this section was originally proposed in [142] for travelling wave tube ampliﬁers (TWTA) and exhibits both AM-AM and AM-PM distortion, which are deﬁned as χA A , 1 + κA A2 χφ A2 , gφ (A) = 1 + κφ A2

gA (A) =

(6.6) (6.7)

respectively. In [142] the parameters of the model are chosen to be χA = 1, κA = 0.25, χφ = 0.26 and κφ = 0.25, which results in a normalised small signal gain.

6.2.1.3 Solid-state ampliﬁer The model of [143] was originally proposed for solid-state ampliﬁers (SSA) and exhibits only AM-AM distortion, which is deﬁned as A

gA (A) = .

1+

A A0

1 2p / 2p

,

(6.8)

with p > 0. The parameter A0 , again, denotes the clipping level and p determines the smoothness of the nonlinear function. For large p the model converges to the ideal clipping ampliﬁer transfer in (6.5), while for small p the transfer is more smooth. Typical values of p are in the range of 1 to 3 [57]. An example of the transfer of the SSA model is given in Fig. 6.2.

190

6 Nonlinearities

6.2.1.4 Polynomial nonlinearity model A more general nonlinear model is the polynomial model, which simultaneously models AM-AM and AM-PM distortion as g(u) = u

N −1

βn+1 |u| = e n

n=0

jφ

N −1

βn+1 An+1 .

(6.9)

n=0

where the model parameters βn are complex parameters. For the special case of only AM-AM distortion these parameters are real. The AM-AM distortion model is then given by gA (A) =

N −1

βn+1 An+1 .

(6.10)

n=0

It was shown in [144] that only the odd orders of the polynomial contribute to the in-band distortion, and that the out-of-band spectral regrowth is caused by the even orders in the polynomial. Since the nonlinearities, which occur in the RF front-end, are in general smooth, only a limited number of orders contributes to the distortion. Now, since we want to study the inﬂuence of RF front-end nonlinearities in the signal band, it is useful to deﬁne the following special case of (6.10), which takes into account the odd orders up to the 5th: gA (A) = A(1 + β3 A2 + β5 A4 ),

(6.11)

and where the ﬁrst order coeﬃcient β1 = 1, since we are assuming a normalised small signal gain. It is, subsequently, useful to relate the parameters in the model of (6.11), i.e., β3 and β5 , to measures of nonlinearity commonly used in RF design and measurement. This can be done by means of gain compression or intermodulation. In the ﬁrst approach the 1-dB compression point is often used, which is deﬁned as the input signal level that causes the small-signal gain to drop by 1 dB [20]. Here we will, however, regard the intermodulation distortion, which is more often applied as a measure for nonlinearity. It is commonly found using a two-tone measurement, where two signals with a diﬀerent frequency are applied as input to the nonlinear system, yielding, due to mixing, frequency components which are not harmonics of the input signals. The derived performance measures are the third and ﬁfth order interception points (IP3 and IP5 ), which are deﬁned to be at the intersection of the powers of the fundamentals with that of the third/ﬁfth order, in a graph plotted as a function of the input amplitude. The corresponding input amplitude

191

6.2 System and nonlinearities modelling

values, referred to as input IP3 (AIIP3 ) and input IP5 (AIIP5 ), can be related to the parameters in (6.11) using a similar derivation as in [20, 141], yielding 4 , 3A2IIP3 8 |β5 | = . 5A4IIP5 |β3 | =

(6.12) (6.13)

To show the equivalence between the above reviewed nonlinearity models, we derive the Taylor series expansions of the diﬀerent AM-AM models under the assumption of small A. For the TWTA model of Section 6.2.1.2, this yields gA (A) = χA A − χA κA A3 + χA κ2A A5 + o(A7 ).

(6.14)

The third and ﬁfth order interception points can then be related to the TWTA AM-AM model using (6.12) and (6.13). For a normalised small signal gain, i.e., χA = 1, this yields : 4 , (6.15) AIIP3 = 3κA ; 8 . (6.16) AIIP5 = 4 5κ2A For the SSA model of Section 6.2.1.3 with p ∈ { 12 , 1, 32 , . . .} the result of the series expansion is given by gA (A) = A −

1

A2p+1 2pA2p 0

+

2p + 1 2(2p)2 A4p 0

A4p+1 + o(A6p+1 ).

(6.17)

It is then easily veriﬁed that, for p = 1, (6.17) reduces to gA (A) = A −

1 3 3 5 A + A + o(A7 ), 2 2A0 8A40

(6.18)

for which the third and ﬁfth order interception points are deﬁned by : 8 A0 , (6.19) AIIP3 = 3 : 4 64 A0 . (6.20) AIIP5 = 3 Here we stress the equivalency of the nonlinearity models presented above, i.e., the TWTA and SSA model map to the polynomial model.

192

6 Nonlinearities 3

amplitude out

2.5 2 1.5 1

p=1 p=2 p=3

0.5 0

0

2

4

6

8

amplitude in Figure 6.2. Amplitude-amplitude transfer of the SSA nonlinearity model (6.8). Results are depicted for diﬀerent values of the smoothness factor p and for clipping level A0 = 3 (solid lines) and A0 = 2 (dashed lines). Small and large amplitude approximations are given in dotted lines.

The special case for the polynomial model in (6.11) directly maps to (6.14) and (6.18) for the TWTA model and SSA model with p = 1, respectively. To illustrate the AM-AM transfer of the considered nonlinearity models, Fig. 6.2 depicts the transfer of the SSA nonlinearity model of (6.8) for diﬀerent values of the smoothness parameter p and for two diﬀerent clipping levels A0 . It can be concluded from the ﬁgure that when p is increased the curve more and more resembles the response of the ideal clipping ampliﬁer of Section 6.2.1.1, and that the response for low input amplitudes becomes more linear. Furthermore, the inﬂuence of A0 is obvious from the ﬁgure.

6.2.2

Inﬂuence on signal model

In this section we study the inﬂuence of nonlinearities on the signal model of an OFDM system. To that end, we will frequently use Bussgang’s theorem [145–147], which states that for a zero-mean Gaussian input to a nonlinearity, the corresponding output signal is given by an attenuated version of the input signal plus a statistically independent Gaussian error term. When we ﬁrst consider a single antenna signal and apply the previously deﬁned input-output relation (6.2), the distorted signal at the output of the nonlinearity can be written as [28]: ud = g(u) = αu + d,

(6.21)

6.2 System and nonlinearities modelling

193

where d represents the distortion noise term which is statistically independent of u, as a result of which E{ud∗ } = 0. When we now rewrite u in polar coordinates, as in (6.1), the expression for the nonlinear distorted signal in (6.21) can be rewritten as ud = g(u) = gA (A)ej(φ+gφ (A)) = g(A)ejφ .

(6.22)

The parameter α in (6.21) can be found by relating the cross-correlation of ud and u to the autocorrelation of u as E{u∗ ud } = αE{u∗ u}.

(6.23)

Hence, E{u∗ g(u)} E{Ae−jφ g(A)ejφ } E{u∗ ud } = = E{u∗ u} 2σ 2 2σ 2 ∞ 1 = Ag(A)p(A)dA, 2σ 2 0

α =

(6.24)

where by deﬁnition E{u∗ u} = σu2 = 2σ 2 and p(A) denotes the pdf of the signal envelope term A, which is Rayleigh distributed when u is complex Gaussian distributed, and is deﬁned as . / A A2 p(A) = 2 exp − 2 . (6.25) σ 2σ Bussgangs’ theorem is deﬁned for zero-mean Gaussian processes, but is shown to also be applicable for OFDM signal with large number of subcarriers Nc [29]. This can be proved using the central limit theorem [147, pp. 214-221], which states that, under certain general conditions, the sum of Nc independent random variables approaches a normal distribution. When we, subsequently, regard the N th order polynomial nonlinearity model of (6.9), we can derive α using (6.24). We will use that ∞ An p(A)dA = 2n/2 σ n Γ(1 + n/2), (6.26) 0

where Γ(·) denotes the Gamma function. It is, subsequently, found that α for this polynomial nonlinearity can be expressed as α=

N −1 n=0

βn+1 2n/2 σ n Γ(2 + n/2).

(6.27)

194

6 Nonlinearities

In a similar way we can calculate the variance of d, which we will here denote by σd2 . It is easily veriﬁed, using the above, that σd2 = E{u∗d ud } − |α|2 E{u∗ u} ∞ = |g(A)|2 p(A)dA − 2|α|2 σ 2 ,

(6.28)

0

where we used that E{ud∗ } = 0. Subsequently, we will use (6.28) to calculate the variance of the distortion term for the polynomial model of (6.9). Consequently, we can apply that $ $2 N 2N $ $ $ n$ βn A $ = γn An , (6.29) $ $ $ n=1

where γn =

n=2

n−1

∗ , β˘m β˘n−m

(6.30)

m=1

6

and β˘m =

βm for 1 ≤ m ≤ N + 1, 0 for m > N + 1.

(6.31)

Substituting (6.9), (6.26), (6.27) and (6.29) into (6.28) then yields the expression for the variance of the distortion term 2

σd =

2N n=2

=

2N n=2

γn

∞

An p(A)dA − 2|α|2 σ 2

0

$2 $N −1 $ $ $ $ γn 2n/2 σ n Γ(1 + n/2)−2σ 2 $ βn+1 2n/2 σ n Γ(2 + n/2)$, (6.32) $ $ n=0

which is a function of the parameters of the polynomial nonlinearity model and the variance of the input signal u. The mapping of the TWTA and SSA nonlinearity models onto the polynomials model was derived in the previous section by means of series expansion. Therefore, the calculation of α and σd2 for these nonlinearity models is straightforward using (6.27) and (6.32). For the ideal clipping ampliﬁer model of Section 6.2.1.1, it is easily veriﬁed that α is given by A0 ∞ 1 A0 2 α = A p(A)dA + Ap(A)dA 2σ 2 0 2σ 2 A0 . . / / √ A0 πA0 A2 , (6.33) = 1 − exp − 02 + √ Q 2σ σ 2σ

195

6.3 Impact of nonlinearities on system performance

0

0

−0.5

−0.5

−1

−1 −1

−0.5

0

0.5

I

(a) IBO = 11 dB

1

Q

1 0.5

Q

1 0.5

Q

1 0.5

0

−0.5 −1 −1

−0.5

0

0.5

1

−1

−0.5

(b) IBO = 5 dB

0

0.5

1

I

I

(c) IBO = 3 dB

Figure 6.3. The inﬂuence of the SSA nonlinearity on the reception of a 16-QAM based IEEE 802.11a transmission.

where we recall that Q(x) denotes the Q-function or complementary cumulative normal distribution function. For this clipping ampliﬁer the variance of the distortion term d is found using (6.28), yielding ∞ A0 2 2 2 σd = A p(A)dA + A0 p(A)dA − 2|α|2 σ 2 0 A . 02 // . A = 2σ 2 1 − α2 − exp − 02 . (6.34) 2σ The inﬂuence of the SSA nonlinearity on the received symbols as described above is illustrated in Fig. 6.3. Here the received constellation points in an IEEE 802.11a-based SISO OFDM system experiencing a nonlinearity modelled by the SSA model of Section 6.2.1.3 with p = 1 are depicted. Diﬀerent values of input backoﬀ (IBO) are applied, where the IBO is deﬁned as P0 A2 IBO = 2 = 02 . (6.35) σu 2σ In the simulations the system did neither experience a fading channel nor additive receiver noise. The transmission consisted of 16-QAM symbols. It can be concluded from the ﬁgure that the lower the IBO, the higher the inﬂuence of the nonlinearities on the estimated symbols is. The behaviour predicted by (6.21) is clear from the ﬁgures: the power of the distortion term d increases with decreasing IBO and the mean of the detected symbols for a certain transmitted symbol is scaled, as predicted by the multiplication by α.

6.3

Impact of nonlinearities on system performance

The inﬂuence of nonlinearities on the OFDM signal model was discussed in Section 6.2.2. Although these results reveal the inﬂuence of

196

6 Nonlinearities

nonlinearities on the received OFDM signal, they do not uncover the performance impact of this impairment in a MIMO OFDM system. To that end, this section studies the symbol-error rate (SER) performance of a multiple-antenna OFDM system applying zero-forcing (ZF) detection. As in the previous chapters, we separately regard the inﬂuence of TX- and RX-caused nonlinearities in Sections 6.3.1 and 6.3.2, respectively. The derived analytical results are in Section 6.3.3 compared with results from simulations. The inﬂuence of nonlinearities on the performance of conventional SISO OFDM systems was earlier treated by several authors [27–29, 148, 149]. The work presented in [27–29] focusses a system experiencing an AWGN channel and TX nonlinearities. The work in [148] is limited to TX-based clipping for an OFDM system experiencing a wireless channel. Finally, the inﬂuence of TX clipping in a system experiencing either an AWGN or Rayleigh fading channel is treated in [149]. The authors of [149], however, limit their analyses to the derivation of channel capacity and eﬀective SNR. The approach chosen in this section, as originally presented in [134], extends the work presented in these papers by regarding the inﬂuence of both TX- and RX-caused nonlinearities on the SER performance of a MIMO OFDM system applying M -QAM modulation. In this analytical study, we will both regard AWGN and Rayleigh fading channels. The results derived in this section are applicable for the diﬀerent presented memoryless nonlinearities, the similarity of which was shown in Section 6.2.2. The results will reveal a performance dependence on the location of the nonlinearities, i.e., the resulting SER ﬂooring at high SNR values is for TX- and RX-caused nonlinearities independent and dependent of the MIMO conﬁguration, respectively.

6.3.1

TX nonlinearities

To derive the SER of a MIMO OFDM system experiencing TX nonlinearities, we will ﬁrst extend the signal model derived in Section 6.2.2 to the MIMO case. Using (6.21), the Nt Ns ×1 output of the nonlinearity can be written in vector notation as ud = g(u) = (INs ⊗ α)u + d,

(6.36)

where the Nt ×Nt diagonal matrix α is given by diag{α1 , . . . , αNt }. Using (6.36) we, subsequently, ﬁnd that the Nr Nc ×1 received frequencydomain signal vector is given by x = H(INc ⊗ α)s + He + n,

(6.37)

197

6.3 Impact of nonlinearities on system performance

where e denotes the frequency-domain equivalent of d. Therefore, it can be concluded that the elements of e belong to a zero-mean process with a variance of σe2 , which is equal to σd2 , i.e., the variance of the time-domain distortion noise, as deﬁned by (6.28). When ZF-based MIMO processing is applied to the received signal x in (6.37), we get ˜s = H† x = (INc ⊗ α)s + e + H† n = ˆs + εT ,

(6.38)

where we assumed perfect channel knowledge. Here ˆs = (INc ⊗ α)s denotes the scaled TX vector and εT = e+H† n denotes the error vector. The error probability can now be derived in three steps. First the distribution of the eﬀective SNR is derived. Subsequently, the probability of symbol error in detection of ˆs for a given SNR value is calculated. Finally, the SER expressions are averaged over the SNR distribution. The mean of the Nt ×1 error vector εT for the kth carrier is given by µ = E [εT (k)] = 0,

(6.39)

since the elements of n(k) and e(k) possess zero mean. The Nt ×Nt covariance matrix of the error in εT (k) is given by $ $ Ω = E εT (k)εH T (k) H(k) −1 = σe2 + σn2 HH (k)H(k) . (6.40) The eﬀective SNR for the kth carrier of the nt th TX branch is then from (6.40) found to be given by ℘nt (k) =

℘t,nt (k)℘r,nt (k) , ℘t,nt (k) + ℘r,nt (k)

(6.41)

where we have deﬁned the TX and RX SNR as ℘t,nt (k) =

σs2 , σe2

℘r,nt (k) =

σs2

σn2 (HH (k)H(k))−1

(6.42) ,

(6.43)

nt nt

respectively. We recall that [A]mm denotes the mth diagonal element of matrix A and that the covariance matrix of sm equals σs2 I. When the channel matrix H(k) has i.i.d. complex Gaussian entries, ℘r,nt (k) is chi-square distributed with 2R = 2(Nr −Nt +1) degrees of freedom. The pdf of ℘r,nt (k) is then given by . / ρ (ρ/℘0 )R−1 exp − p℘r,nt (k) (ρ) = , (6.44) ℘0 (R − 1)! ℘0

198

6 Nonlinearities

where ℘0 is the average SNR, given by ℘0 =

σs2 . σn2

(6.45)

When we, subsequently, want to calculate the probability of error of a system applying a rectangular M -QAM constellation and experiencing TX nonlinearity, we can use the observation from (6.38) that the estimated signal is given by a scaled version of the TX signal, denoted by ˆs, plus some error vector εT . In the remainder we will assume that the diagonal elements of α are real variables, which corresponds to the case of an AM-AM nonlinearity. This assumption is made, since for most ampliﬁers the inﬂuence of the AM-AM distortion is dominant over the inﬂuence of the AM-PM distortion. In a approach similar to that used for IQ imbalance in Section 5.3.2, we√will derive the SER for the M -QAM constellation based on that of a M -PAM constellation. Since the real and imaginary part of the estimated symbols are independent, for a given SNR ℘ and scaling α and a zero-mean circularly symmetric complex Gaussian noise term, we can separately study the probability of error for the real and imaginary part of the signal. Since, moreover, the inﬂuence of α is identical for the real and imaginary part of the constellation, the expression for the SER can be written as 2 Peα,M -QAM,Es = 1 − 1 − Peα,√M -PAM,Es /2 , (6.46) where Peα,√M -PAM,Es /2 denotes the probability of error for the α-scaled √ M -PAM modulation with half the signal power of the corresponding QAM symbols. As an example, the impact of the α-scaling on the 4-PAM constellation is schematically depicted in Fig. 6.4. The original 4-PAM constellation is depicted in Fig. 6.4(a), which is a repetition of Fig. 5.5(a), where we recall that the dashed lines depict the decision boundaries and that the transmitted symbols are depicted by black dots. Note again that the distance of the mean symbol realisation to the decision boundary is Es /5. This means that a symbol error occurs when the magnitude of the error in the symbol estimation is larger than Es /5. Subsequently, we regard the α-scaled constellation, as caused by nonlinearities, which is schematically depicted in Fig. 6.4(b). In this ﬁgure, the originally transmitted symbols are depicted by dashed white dots and the scaled symbols are depicted by black dots. It is clear that the distances to the decision boundaries are changed, but diﬀerently from the IQ imbalance case, the shifts depend on the transmitted symbols.

199

6.3 Impact of nonlinearities on system performance

s3

s2

s1 −

0

4 E 5 s

s4 4 E 5 s

(a) Original constellation

δ1

αs1 −

δ3

δ2

αs2 4 E 5 s

0

αs3

δ4

αs4 4 E 5 s

(b) α-scaled constellation

Figure 6.4.

Inﬂuence of scaling on the 4-PAM constellation.

It can be concluded that the eﬀect is more severe for the outer constellation points, i.e., for s1 and s4 in the ﬁgure. It is readily concluded from Fig. 6.4 that the shift for the nth constellation point sn is given by δn = (α − 1)sn . The distance to the decision boundaries are changed by δn , i.e., for half of the cases they are increased by δn and for the other half of the cases they are decreased by δ n . For example for s2 , the distance to the left decision boundary at − 4Es /5 is increased by |(α − 1)s2 | and the distance to the right decision boundary at 0 is decreased by |(α − 1)s2 |. If we now additionally use that, by deﬁnition, the nth point of the N -PAM constellation is given by : 3Es (2n − 1 − N ), (6.47) sn = N2 − 1 we can derive the probability of symbol error for the α-scaled N -PAM constellation. We ﬁnd that it is given by 1 (P (|ε| > a1 − δn ) + P (|ε| > a1 + δn )) 2N n=1 −P (|ε| > a1 + δ1 ) − P (|ε| > a1 − δN ) , (6.48) 9 s where, by deﬁnition, a1 = N3E 2 −1 and N

Peα,N -PAM,Es =

δn = αsn − sn = (α − 1)a1 (2n − 1 − N ).

(6.49)

We note that the last two terms in (6.48) are caused by the outer points in the PAM constellation, which only have a decision boundary at one side.

200

6 Nonlinearities

√ For most used M -QAM modulations N = M is a multiple of 2. In these cases, symmetry occurs in (6.48) and (6.49), which we exploit to simplify (6.48) to 1 P (|ε| > a1 (1 − (1 − α)(2n − 1))) = N N/2

Peα,N -PAM,Es

n=1

+P (|ε| > a1 (1 + (1 − α)(2n − 1))) −P (|ε| > a1 (1 + (1 − α)(N − 1))) . (6.50)

By substituting (6.50) into (6.46), we then ﬁnd that √

.

Peα,M -QAM,Es

2 =1− 1− √ M

6 M /2 −√ √ Q βn c1 ℘ + Q βn+ c1 ℘ n=1

−Q

%

&7/2

√ + β√ c1 ℘ M /2

, (6.51)

where we have deﬁned c1 = 3/(M − 1), βn− = 1 − (1 − α)(2n − 1), βn+ = 1 + (1 − α)(2n − 1),

(6.52) (6.53) (6.54)

and the SNR is deﬁned as ℘ = Es /N0 . When we work out (6.51) for the special case of 4-QAM modulation, we ﬁnd √ Peα,4-QAM = 1 − (1 − Q (α ℘))2 √ √ = 2Q (α ℘) − [Q (α ℘)]2 , (6.55) which for large ℘ can be approximated as √ Peα,4-QAM ≈ 2Q (α ℘) .

(6.56)

Now, using the expression for the SNR ℘ in (6.41), and when we for readability omit the subcarrier and TX branch index, we can calculate the SER in a Rayleigh fading channel, which yields . / ∞ ℘t ρ α Pe = Pe,M -QAM (6.57) p℘r (ρ)dρ. ℘t + ρ 0 For some special cases we can derive a closed form solution for (6.57). For high values of ℘0 in (6.45), we can make the following approximation ℘ t ℘r σ2 ≈ ℘t = s2 . ℘t + ℘r σe

(6.58)

6.3 Impact of nonlinearities on system performance

201

Using this result, the SER in (6.57) is well approximated as Pe ≈ Peα,M -QAM (℘t ),

(6.59)

which for 4-QAM can be approximated by substituting (6.56) into (6.59), √ yielding 2Q(α ℘t ). We note that (6.59) predicts the ﬂooring in the SER curves as a function of the SNR. For low values of ℘0 , we can make the following approximation ℘t ℘r ≈ ℘r . ℘t + ℘r The SER expression in (6.57) is then well approximated by ∞ Pe ≈ Peα,M -QAM (ρ) p℘r (ρ)dρ.

(6.60)

(6.61)

0

When we substitute (6.56) into (6.61), and use the result of (5.79), we ﬁnd that for the low SNR region the SER approximation for a 4-QAM system in a Rayleigh fading channel is given by / . √ 2(1 − ℘0 c2 ) 1 1 3 −α2 ℘0 Pe ≈ , R + ; ; (6.62) F 2 1 2 2 2 2 2 9 (R− 21 )! where c2 = α π2 (R−1)! and we recall that 2 F1 (·) denotes the hypergeometric function [128]. The ﬁnal expression for the average SER is now found by averaging (6.57) over all TX branches and all subcarriers.

6.3.2

RX nonlinearities

When the nonlinearity is experienced in the front-end of the RX, we can write the Nr Ns ×1 received time-domain baseband vector (without the inﬂuence of AWGN) as td = g(t) = (INs ⊗ α)t + d,

(6.63)

as schematically depicted in Fig. 6.1. Here α denotes the Nr ×Nr diagonal matrix given by diag{α1 , . . . , αNr } and d is the nonlinear distortion noise. The Nr Nc ×1 received signal vector in the frequency domain can then be expressed as x = (INc ⊗ α)Hs + e + n,

(6.64)

where the Nr Nc ×1 vectors e and n are the frequency-domain equivalents of the distortion noise vector d and additive RX noise v, respectively.

202

6 Nonlinearities

Note that here, and in the schematic model of Fig. 6.1, the additive RX noise is modelled behind the nonlinearity, for notational convenience. It is noted, however, that it equivalently could have been modelled in front of the nonlinearity, since the output of the nonlinearity due to Gaussian noise input signal would again be a Gaussian signal, as found from Bussgang’s theorem in (6.21). The deﬁnition of the SNR would, however, diﬀer slightly for that case. When ZF-based MIMO processing is applied with perfect channel knowledge, the estimated Nt Nc ×1 TX vector is given by ˜s = H† x = (INc ⊗ α)s + H† (e + n) = ˆs + εR ,

(6.65)

where the scaled TX signal is given by ˆs = (INc ⊗α)s and the error vector is deﬁned as εR = H† (e + n). Note that we here made the assumption that the diﬀerent RX branches exhibit the same αnt , i.e., α = αINr . This is, e.g., achieved when the nonlinearities on the diﬀerent branches are equal. The mean of the Nt ×1 error vector for the kth carrier is given by µ = E [εR (k)] = 0,

(6.66)

since the elements of n(k) and e(k) possess zero mean. The Nt ×Nt covariance matrix of the error vector in (6.65) for the kth carrier is given by $ $ Ω = E εR (k)εH R (k) H(k) −1 , (6.67) = (σe2 + σn2 ) HH (k)H(k) where we recall that σe2 = σd2 , as derived for diﬀerent nonlinearities in Section 6.2.2. The eﬀective SNR for the kth carrier of the nt th TX branch is then given by σs2

. (6.68) ℘nt (k) = (σe2 + σn2 ) (HH (k)H(k))−1 nt nt

When the experienced channel exhibits complex Gaussian fading, ℘nt (k) is distributed according to the chi-square distribution with 2R degrees of freedom, the pdf of which is given by (6.44). Here the average SNR ℘0 is given by σ2 (6.69) ℘0 = 2 s 2 . σe + σn

6.3 Impact of nonlinearities on system performance

203

For a given SNR the probability of erroneous symbol detection of an M -QAM symbol in ˆs(k) is given by Peα,M -QAM , which was found to be given by (6.51). Using the expression for the eﬀective SNR ℘ in (6.68) for a complex Gaussian fading channel, and when we omit the subcarrier and TX branch index, we ﬁnd the resulting SER is given by Pe = 0

∞

Peα,M -QAM (ρ)p℘ (ρ)dρ.

(6.70)

For the special case of 4-QAM modulation, (6.70) can be approximated by substituting (6.56) into (6.70), which yields, similarly to (6.62),

∞

∞

√ 2Q (α ρ) p℘ (ρ)dρ 0 0 . / √ 2(1 − ℘0 c2 ) 1 1 3 −α2 ℘0 F = , R + ; ; . 2 1 2 2 2 2 2

Pe =

6.3.3

Peα,4-QAM (ρ)p℘ (ρ)dρ

≈

(6.71)

Numerical results

The analytical results derived in Sections 6.3.1 and 6.3.2 are here compared with results from Monte Carlo simulations. All simulations are carried out for a MIMO OFDM system comprising Nc = 1024 carriers. For these simulations all subcarriers contain 16-QAM data symbols and no coding is applied. The MIMO channel is modelled to exhibit spatial and (per subcarrier) frequency-independent Rayleigh fading. It is assumed that the MIMO channel matrix is perfectly estimated at the RX. All ﬁgures presented in this section depict the analytical results by lines and the results from simulations by markers. The idealised ampliﬁer of (6.5) was modelled as nonlinearity, for which diﬀerent values of IBO, as deﬁned by (6.35), were applied. The IBO values were, however, always chosen to be equal on all TX/RX branches. The ﬁrst results in Fig. 6.5 are for a 1×1 system experiencing an AWGN channel. For this case it is not relevant whether the clipping ampliﬁer is located in TX or RX, since it is clear from Fig. 6.1 that the nonlinearities are at the same position in the transmission chain. In Fig. 6.5 the results from simulations are compared with the theoretical results of (6.51). Results for IBO values of 6, 5, 4 and 3 dB are depicted together with the reference curve for a system not experiencing the clipping ampliﬁer. The results are depicted as a function of the average SNR for the case no nonlinearity is experienced, which is given by σs2 /σn2 for the SISO system. We note that the eﬀective SNR ℘ to be substituted into (6.51), to ﬁnd the theoretical results, is given by (6.41) and equals Nt σs2 /(σd2 + σn2 ).

204

6 Nonlinearities 100 10−1

SER

10−2 10−3 10−4 10−5

no clipping IBO = 6 dB IBO = 5 dB IBO = 4 dB IBO = 3 dB

10−6 10−7

0

10

20 Average SNR (dB)

30

40

Figure 6.5. SER performance of a 16-QAM SISO OFDM system experiencing TX clipping and an AWGN channel. Analytical results of (6.51) are in lines and simulation results are depicted by markers. 100 no clipping IBO = 6 dB IBO = 3 dB IBO = 2 dB

SER

10−1

10−2

10−3

10−4 0

10

20 30 Average SNR (dB)

40

Figure 6.6. SER performance of a system experiencing TX clipping and a Rayleigh fading channel. Results are depicted for a 1×4 (dashed lines) and a 2×4 (solid lines) system. Analytical results of (6.57) are in lines and simulation results are depicted by markers.

205

6.3 Impact of nonlinearities on system performance 100

no clipping IBO = 3 dB IBO = 6 dB

10−1

SER

10−2

10−3

10−4

10−5

0

10

20

30

40

Average SNR (dB)

Figure 6.7. SER performance of a system experiencing RX clipping and a Rayleigh fading channel. Results are depicted for a 1×4 (dashed lines) and a 2×4 (solid lines) system. Analytical results of (6.70) are in lines and simulation results are depicted by markers.

We conclude from Fig. 6.5 that there is good agreement between the analytical results and the results from Monte Carlo simulations. This proves the applicability of the derived theoretical results. It can be concluded that for an IBO of 6 dB the degradation compared to the non-distorted signal is small. Severe ﬂooring, however, occurs for IBO values of 4 and 3 dB. The SER results for a 1×4 (dashed lines) and a 2×4 (solid lines) system experiencing a Rayleigh fading channel and TX nonlinearities are depicted in Fig. 6.6, again as a function of the average SNR per RX antenna Nt σs2 /σn2 . The ﬁgure compares the analytical results of (6.57) with results from Monte Carlo simulations for diﬀerent values of IBO. It can be concluded that also for TX nonlinearities and a Rayleigh fading channel there is a good agreement between the analytical and simulation results. For low SNR values the curves are bounded by (6.61), which is equal to the SER performance for the reference system, i.e., not experiencing the nonlinearity. At high SNR values the nonlinear-caused distortion noise becomes dominant and ﬂooring occurs which can be approximated by (6.59). This is clear from Fig. 6.6, where the diﬀerent MIMO conﬁgurations experiencing the same nonlinearity, reveal the

206

6 Nonlinearities

same SER ﬂoor. This is explained by (6.59), which does not depend on the channel or MIMO conﬁguration. For both the 1×4 and 2×4 system the performance impact of clipping with an IBO of 6 dB is limited, while for lower IBO values the inﬂuence is severe. The results for the same MIMO conﬁgurations and a Rayleigh fading channel, but now for a system experiencing RX nonlinearities, are depicted in Fig. 6.7. It can be concluded from the ﬁgure that also for RX clipping good agreement between the simulation results and the analytical results of (6.70) is observed. In contrast to the TX nonlinearity case, the ﬂooring for high SNR values does depend on the fading channel/MIMO conﬁguration here. This agrees with the theoretical results and can be concluded from the eﬀective SNR expression in (6.68). Although the performance impact of clipping with IBO = 6 dB is small for the 1×4 system within the regarded SER range, it is very pronounced for the 2×4 system; for IBO = 3 dB the ﬂooring is more than 20 times higher for the 2×4 system. Finally, Fig. 6.8 compares the impact of TX (in solid lines) and RX nonlinearities (in dashed lines) on the system performance. The results 100

SER

10−1

10−2

10−3

10−4

1×1 1×4 2×2 2×4 0

10

20

30

40

50

Average SNR (dB)

Figure 6.8. Comparison of the impact of TX and RX nonlinearities on the SER performance of diﬀerent MIMO conﬁgurations. The nonlinearities are modelled as clipping ampliﬁers with IBO = 3 dB. The analytical results for TX nonlinearities (6.57) (solid lines) and for RX nonlinearities (6.70) (dashed lines) are depicted together.

6.3 Impact of nonlinearities on system performance

207

are obtained from the analytical SER expressions in (6.57) and (6.70), respectively. Diﬀerent system conﬁgurations are considered and a Rayleigh fading channel is assumed. The nonlinearities are modelled as clipping ampliﬁers with IBO = 3 dB. Again, it can be concluded that the SER ﬂooring at high SNR for TX nonlinearities does not depend on the MIMO conﬁguration, while it does for RX nonlinearities. Hence, a system experiencing RX nonlinearities beneﬁts from the spatial diversity provided by the MIMO channel, so the inﬂuence of the nonlinearity is reduced by using a larger number of RX antennas. For symmetric systems, i.e., Nt = Nr , TX nonlinearities have less performance impact than RX nonlinearities.

6.3.4

Summary and discussion

The impact of nonlinearities on the SER performance of a MIMO OFDM system applying M -QAM modulation was studied in this section. For this purpose, a system only experiencing TX nonlinearities and a system only experiencing RX nonlinearities were investigated. Analytical expressions for the SER in Rayleigh fading channel conditions were derived in this section, which show good agreement with results from a simulation study. It can be concluded from the results that, for high SNR values, the TX nonlinearities cause performance ﬂoors, which are independent of the MIMO conﬁguration. For RX nonlinearities, however, the performance in this high SNR region depends on the MIMO conﬁguration. The observed diﬀerences between the inﬂuence of TX and RX nonlinearities in systems experiencing a fading channel can intuitively be understood using Bussgang’s theorem, which showed that the nonlinearities cause a scaling of the signal and introduce an additive error term. The inﬂuence of the scaling is similar for the TX and RX nonlinearities. The additive distortion noise term, however, eﬀectively creates an additional TX or RX noise source for the TX and RX nonlinearities, respectively. Since the TX noise source occurs in front of the propagation channel, it will result in ﬂooring in the SER curves according to the AWGN SER performance, i.e., independent of the MIMO conﬁguration. The RX distortion noise source due to nonlinearities, however, occurs behind the propagation channel. As such, its inﬂuence is similar to that of the commonly studied additive Gaussian RX noise source, i.e., the SER performance (and in this case thus the SER ﬂooring) depends on the MIMO conﬁguration.

208

6.4

6 Nonlinearities

PAPR reduction by spatial shifting

We conclude from the previous section that nonlinearities in the RF front-end largely inﬂuence the performance of (MIMO) OFDM systems. This can be explained by the fact that the considered system combines signals with a high peak-to-average power ratio (PAPR) with a front-end that inherently contains nonlinear components. The solutions for the mitigation of the eﬀects thus lie in two areas: compensation/prevention of the nonlinearities and reduction of the PAPR of the signals. In Section 6.5 we will propose a solution based on nonlinearity compensation, while in this section we will regard PAPR reduction of the signals. Several baseband methods have been proposed in literature to reduce the PAPR of single-input single-output (SISO) OFDM transmissions. Some methods try to reduce the PAPR by deliberately clipping the signal in the TX baseband, which, however, can result in unwanted spectral regrowth. When instead of clipping, peak windowing is applied, see e.g. [150, 151], this disadvantage can largely be overcome, however, at the cost of a larger distortion within the signal bandwidth. Other techniques apply coding on top of the OFDM-processing to reduces the PAPR, see e.g. [152, 153]. The disadvantage of this technique is that the codes often do not achieve the same error correction performance as codes speciﬁcally designed for this purpose and can thus result in decreased data rate. Yet another approach is to apply diﬀerent phase shifts to the data on the diﬀerent carriers such that the time-domain signal has a minimum PAPR, see e.g. [154–157], or to apply diﬀerent signal representations for the same data and choosing the one with the lowest PAPR, see e.g. [158]. Although these techniques achieve a high PAPR reduction, extra computational complexity is required and (a small amount of) overhead is introduced, since the information which phase shifts/signal representation has been selected, often referred to as side information (SI), has to be transmitted to the RX. The application of the techniques of [154] and [158] to MIMO OFDM was studied in [159] and [160]. The authors compare the results for the case of applying the techniques per TX branch separately with a simpliﬁed approach, where the optimisation is carried out jointly over all TX branches. Although the latter requires the transmission of less SI, the achieved PAPR reduction is also smaller. Here we present a MIMO OFDM speciﬁc PAPR reduction technique. It is named spatial shifting (SS) and originally proposed in [161, 135]. In this technique the extra degree of freedom provided by MIMO is exploited to reduce the PAPR. It is based on the rearrangement of the TX vector in such a way that the PAPR is minimised. Additionally, this

6.4 PAPR reduction by spatial shifting

209

section presents several extension to the concept of SS and evaluates the performance. The outline of this section is as follows. First, the signal model is introduced in Section 6.4.1. Subsequently, Section 6.4.2 presents the SS-based PAPR reduction technique. To further increase the PAPR reduction performance of SS, the combination of SS with the phase shifting (PS) procedure of [154] is proposed in Section 6.4.3. Section 6.4.4 regards the application of these PAPR reduction approaches for diﬀerent subcarrier grouping schemes. To reduce the overhead, the application of SS in a transparent mode, i.e., without the transmission of SI, is introduced in Section 6.4.5. The performance of the presented algorithms is evaluated in Section 6.4.6 by means of Monte Carlo simulations. Finally, Section 6.4.7 presents the conclusions of this section.

6.4.1

Signal modelling

Consider a MIMO OFDM system applying Nt TX branches and Nc subcarriers. For notational convenience we will deﬁne the Nc Nt ×1 transmit vector corresponding to the mth time-domain symbol diﬀerently than in Section 2.3.3. The only diﬀerence, however, lies in the ordering of the elements of the TX vector, which is here given by ςm = INt ⊗ F−1 (6.72) Nc sm , where the Nt -dimensional identity matrix is denoted by INt , FNc represents the Nc -dimensional discrete Fourier transform (DFT) matrix, the (k, l)th element of which is given by exp(−j2π Nklc ). The coded, interleaved and (complex) modulated data symbols are contained in the Nc Nt ×1 frequency-domain vector sm . The vector ςm , and similarly sm , is given by a stacking of the OFDM vectors of the signals on the diﬀerent TX branches, which can be expressed as T T T T ςm = ςm,1 , ςm,2 , . . . , ςm,N , (6.73) t where ςm,nt denotes the Nc ×1 subvector for the nt th branch. Subsequently, the PAPR of the mth symbol on the nt th TX branch can be deﬁned as ∗ max ςm,nt ◦ ςm,n t , PAPRm,nt = (6.74) ∗ µ ςm,nt ◦ ςm,n t where we recall that ∗ and ◦ denote complex conjugation and elementwise multiplication, respectively. The functions µ(y) and max(y) produce the average and maximum value of the input vector y as their output, respectively.

210

6.4.2

6 Nonlinearities

Spatial shifting

In the here proposed PAPR reduction technique spatial shifting, the data carriers of the mth OFDM symbol are subdivided into P disjoint subcarrier groups, referred to as partial transmit sequences (PTSs), similarly to what was proposed in [154] for conventional OFDM. The pth PTS for the nt th TX branch can be deﬁned as the Nc ×1 vector s(p) (6.75) m,nt = Ψp sm,nt , where Ψp is a diagonal Nc ×Nc matrix, where the diagonal elements are 1 for carriers belonging to subcarrier group p and all other elements are zero. The construction of Ψp is treated in Section 6.4.4. The resulting time-domain pth PTS is then given by (p) (p) = F−1 ςm,n Nc sm,nt . t

(6.76)

The vector containing the pth PTS for all TX streams is denoted by

(p) T (p) T (p) T T (p) ςm = ςm,1 , ςm,2 , . . . , ςm,Nt . (6.77) The original time-domain transmit vector is then simply found by a summation over the P PTSs, resulting in ςm,nt =

P

(p) ςm,n t

=

F−1 Nc

p=1

P

Ψp sm,nt ,

(6.78)

p=1

where, by deﬁnition, Pp=1 Ψp = INc . Alternative transmit vectors representing the same data can now be constructed by mutually interchanging the corresponding PTSs between the diﬀerent TX branches, provided that the same subcarrier grouping is applied on all branches. This exchanging of the PTSs between the spatial streams is here referred to as spatial shifting (SS), and schematically depicted in Fig. 6.9. The mth transmit vector for the pth PTS (6.77) after SS can be written as

T T T T (p) (p) (p) (p) ςm,cp = ςm,c , ς , . . . , ς , (6.79) m,cp,2 m,cp,N p,1 t

where the SS vector cp = [cp,1 , . . . , cp,Nt ] is found by cyclicly shifting of the vector [1, . . . , Nt ]. The total Nc Nt ×1 MIMO OFDM vector after SS is then given by ς˜m,c =

P p=1

(p)

ςm,cp ,

(6.80)

211

6.4 PAPR reduction by spatial shifting

where the SS vector for all PTSs is given by c = [c1 , . . . , cP ]. In the case of P PTSs and Nt TX branches, there are NtP −1 unique realisations of ς˜m,c . When we would alternatively generate the SS vector cp = [cp,1 , . . . , cp,Nt ] by reshuﬄing of the vector [1, . . . , Nt ], there are (Nt !)P −1 unique realisations of ς˜m,c . Although this would yield a larger PAPR reduction, it also increases the complexity of the algorithm and is, therefore, not considered in the following. To achieve a low PAPR on all TX branches, we now want to select the transmit vector ς˜m,c that exhibits the lowest peak power averaged over the TX branches. When we apply this as the criterion to determine ˜m , we ﬁnd that the the SS vector for the mth MIMO OFDM symbol c selected SS vector is given by ⎛ ˜m = arg min ⎝ c c

Nt

⎞

∗ ⎠, max ς˜m,c,nt ◦ ς˜m,c,n t

(6.81)

nt =1

where ς˜m,c,nt is the nt th Nc ×1 subvector of ς˜m,c and where arg min(.) produces the argument for which the expression is minimised. For convenience the SS procedure is summarised schematically in Fig. 6.9. First subcarrier grouping is carried out to form P PTSs, equal on all Nt TX branches. Subsequently, the PTSs are transformed to the time domain using the IDFT, where the optimisation algorithm of (6.81) is applied to ﬁnd the optimal SS vector. This vector is used to apply the SS and, afterwards, the PTSs are summed to form the ﬁnal TX vector. We note that the variables bnt ,p , as depicted in the ﬁgure, all equal 1 for the SS-algorithm, i.e., they are only used in the phase shifting algorithm.

6.4.3

Combining SS and PS

When the above treated SS technique is combined with phase shifting (PS), as originally proposed for SISO systems in [154], a further improvement in PAPR performance can be achieved. In this combined technique, the mutually (spatially) interchanged PTSs are, before being summed, multiplied by a set of phase shifts, for the pth PTS denoted by bp = [b1,p , . . . , bNt ,p ]. The mth transmit vector for the pth PTS after SS and PS can be written as

T (p) (p) (p) ςm,bp ,cp = b1,p ςm,c , . . . , bNt ,p ςm,c p,1 p,N

t

T T

.

(6.82)

212

6 Nonlinearities (1)

sm,1

IDFT 2

...

... IDFT Nt

ς˜m,1

b1,1

...

1

...

...

... IDFT 2

...

P

...

...

...

IDFT Nt

...

...

(P)

sm,Nt

IDFT 1

ς˜m,N

t

...

(P) sm,2

spatial shifting

(P)

sm,1

...

Spatial Interleaving Subcarrier Grouping

sm

(1)

sm,Nt

IDFT 1

spatial shifting

(1)

sm,2

b1,P

... bNt,1 ...

bN ,P t

Optimisation

Figure 6.9.

Block diagram of a MIMO OFDM TX combining SS and PS.

The resulting MIMO OFDM vector is then found by summation of the vectors for the diﬀerent PTSs in (6.82) and given by ς˜m,b,c =

P

(p)

ςm,bp ,cp ,

(6.83)

p=1

where the PS vector for all PTSs is given by b = [b1 , . . . , bP ]. The number of possible phases is chosen as a compromise between complexity and achievable PAPR reduction. Low complexity implementations are found for two or four phases, for which bnt ,p ∈ {−1, 1} and bnt ,p ∈ {±1, ±j}, respectively, since for these cases no multiplications have to be performed, merely changing of the sign and interchanging of the real and imaginary part. The PS and SS vectors for the mth symbol are selected as the combination that minimises the peak power averaged over the TX branches and is given by ⎞ ⎛ Nt ∗ ˜ m, c ⎠. ˜m } = arg min ⎝ (6.84) {b max ς˜m,b,c,nt ◦ ς˜m,b,c,n t {b,c}

nt =1

The combined SS/PS technique, as described above, is summarised in the block diagram of Fig. 6.9. A computationally less complex, sub-optimal solution is found by applying the same PS for all TX antennas, as was also proposed in [159]. The resulting PS vector b then only contains P elements and the

213

6.4 PAPR reduction by spatial shifting

resulting MIMO OFDM vector is given by ς˜m,b,c =

P

(p)

bp ςm,cp .

(6.85)

p=1

The optimal b and c are again found by the use of (6.84).

6.4.4

Subcarrier grouping schemes

The authors of [155] studied the inﬂuence of diﬀerent grouping schemes on the performance of PS for single-antenna OFDM. In this section we regard the inﬂuence of the subcarrier grouping schemes on the performance of SS and SS/PS. The considered grouping schemes are: i) grouping of neighbouring carriers (“NC”), ii) subcarrier interleaved (“SInt”) grouping, iii) pseudo-random (“PR”) grouping. For the ﬁrst two schemes the subcarrier grouping matrix Ψp in (6.75) can be written as NC:

Ψp = diag(0(p−1) Nc , 1 Nc , 0(P −p) Nc ), P

P

(6.86)

P

SInt: Ψp = diag(1 Nc ⊗ [0p−1 , 1, 0P −p ]),

(6.87)

P

for p = 1, . . . , P , respectively. Here 0L and 1L are the 1×L all-zeros and all-ones vector, respectively. Note that we here assumed Nc to be a multiple of P . For the PR grouping the carrier grouping matrix Ψp is constructed in a random method, however, with the following two constraints: i) Nc /P of the diagonal elements have to equal 1 and the others have to equal 0, ii) the set of P matrices must be constructed such that Pp=1 Ψp = INc . The grouping schemes are illustrated in Fig. 6.10 for P = 2 groups and Nc = 16 subcarriers. The ﬁrst and second row contain the nonzero elements of Ψ1 and Ψ2 , respectively. The eﬀect of the diﬀerent subcarrier grouping schemes on the PAPR reduction performance of SS and SS/PS is studied in Section 6.4.6.

6.4.5

Side information/Transparency

It can be concluded from (6.81) and (6.84) that the above proposed techniques have to be applied per MIMO OFDM symbol, i.e., for every m, to perform optimally. For the RX to correct for the SS and PS applied at the TX, SI has to be transmitted containing the information ˜ m and c ˜m were chosen. For a high number of which PS and SS vectors b PTSs, TX antennas or phase shifts the overhead clearly might become

214

6 Nonlinearities NC

PR

SInt

1

1

1

Ψ1 0

1

89

16

1

0

1

3

5

7

9

11 13 15

1

0

123

89

11 1314

1

Ψ2 0

1

89

16

0

2

4

6

8

10 12 14 16

0

4567

10 12

1516

Figure 6.10. Subcarrier grouping schemes illustrated for Nc = 16 subcarriers and P = 2 groups.

high. To avoid this, the number of possible phase shifts and PTSs can be minimised or, alternatively, the optimisation of (6.81) and (6.84) can be carried out jointly for a number of K symbols. The chosen PS and SS vectors are then given by ⎛ ˜ c ˜} = arg min ⎝ {b, {b,c}

Nt K

⎞

∗ ⎠. max ς˜m,b,c,nt ◦ ς˜m,b,c,n t

(6.88)

m=1 nt =1

Although this will clearly lead to less reduction in PAPR, it will result in a solution which is computationally less complex and has an overhead which is K times lower. For some systems, however, it might be beneﬁcial to design a fully transparent solution, where the RX needs no information about the applied SS and PS vector. Such a solution is found when the same SS/PS vectors {b, c} are applied to all symbols within a MIMO OFDM packet, thus also to the part of the transmission used for MIMO channel estimation. The inﬂuence of the SS/PS applied at the TX is now included in the eﬀective MIMO channel estimate found at the RX. When the RX compensates for this eﬀective MIMO channel, the SS/PS is also removed. From the viewpoint of the RX, the SS method can be seen as eﬀectively interchanging the columns of the physical MIMO channel matrix and the PS method can be seen as phase shifting of the physical MIMO channel matrix. In that way, the method is transparent, meaning there is no need to transmit SI, which enables the application of this technique without standardisation. The resulting PAPR reduction will, however, be less than in the SI case since the peak minimisation is done jointly for the full packet. Hence, the application seems primarily interesting for transmissions with short packets.

6.4 PAPR reduction by spatial shifting

6.4.6

215

Numerical results

The performance of the proposed PAPR reduction methods was tested by the use of Monte Carlo simulations, the results of which are reported in this section. The parameters of the simulations are Nc = 64 carriers, number of TX branches Nt ∈ {2,4} and QPSK modulation is used. The performance is given in terms of the complementary cumulative distribution function (CCDF) of the PAPR, which indicates the probability that the PAPR is larger than PAPR0 . When clipping occurs at power level PAPR0 the CCDF values indicate the probability of clipping. In all ﬁgures the reference curve, i.e., the PAPR without SS or PS, is indicated by P = 1. Note that PAPR0 is given on a linear scale in the ﬁgures. Figure 6.11(a) shows PAPR results for the SS method with SI, as resulting from the optimisation in (6.81). The results are given for 2 and 4 TX branches and for 2 and 4 PTSs. The pseudo-random (PR) carrier grouping scheme is applied. It can be concluded from this ﬁgure that at a clipping probability of 10−3 and for 2 TX branches, 0.9 dB and 2.2 dB PAPR reduction is achieved for P = 2 and P = 4, respectively. For 4 TX branches the reduction is 1.4 dB and 2.6 dB, respectively. As a reference, Fig. 6.11(b) depicts the results for a system that applies PS, where the same phase shifts are applied for the corresponding PTSs on the diﬀerent TX branches, as in (6.85), and bp ∈ [−1, 1]. In this way, the additional complexity due to the PS method is low. It can be concluded from the results that at a clipping probability of 10−3 and for 2 TX branches, 1.2 dB and 2.6 dB PAPR reduction is achieved for P = 2 and P = 4, respectively. For 4 TX branches, the reduction is 0.8 dB and 1.9 dB, respectively. It is interesting to note that a better performance is achieved for Nt = 2 than for Nt = 4, which can be attributed to the fact that common phase shifts are applied to the PTSs on all TX branches. The results for the combined SS/PS method are depicted in Fig. 6.11(c), where the same phase shifts are applied for the corresponding PTSs on the diﬀerent TX branches, as in (6.85), and bp ∈ [−1, 1]. It can be concluded from the results that at a clipping probability of 10−3 and for 2 TX branches, 1.9 dB and 3.6 dB PAPR reduction is achieved for P = 2 and P = 4, respectively. For 4 TX branches, the reduction is 1.7 dB and 3.2 dB, respectively. Clearly, improved performance is achieved compared to solely applying SS or PS. We note that a slightly better performance is achieved for Nt = 2 than for Nt = 4, as for PS, due to the fact that common phase shifts are applied on all TX branches. The inﬂuence of the diﬀerent subcarrier grouping schemes, as presented in Section 6.4.4, is evaluated in Fig. 6.12. Figure 6.12(a) depicts the CCDF of the PAPR resulting from the SS method for a system with 2 TX branches, applying 2 and 4 PTSs. Although from [155] the results

216

6 Nonlinearities 100 Nt=2, Nt=2, Nt=4, Nt=4,

P =2 P =4 P =2 P =4

P (PAPR > PAPR0)

P (PAPR > PAPR0)

100

10−1

P=1

10−2

10−3

2

4

6 8 PAPR0

10

12

Nt=2, P =2 Nt=2, P =4 Nt=4, P =2 Nt=4, P =4

10−1

10−2

P =1

10−3 2

4

(a) SS method

6

8 PAPR0

10

12

(b) PS method

P (PAPR > PAPR0)

100 Nt=2, P =2 Nt=2, P =4 Nt=4, P =2 Nt=4, P =4

10−1

P =1

10−2

10−3

2

4

6 8 PAPR0

10

12

(c) Combined SS/PS method

Figure 6.11.

CCDF of the PAPR for the PR subcarrier grouping scheme.

were anticipated to diﬀer for the various grouping schemes, it can be concluded that the performance of SS does not depend on the selected grouping. For the combined SS/PS scheme, the results of which are in Fig. 6.12(b), the PAPR performance does depend on the selected subcarrier grouping. The SInt scheme performs worst and the PR grouping yields the best performance. This conﬁrms the ﬁndings of the authors of [155] for SISO OFDM. Finally, Fig. 6.13 presents the PAPR results for the transparent SS and SS/PS method with P = 2 and P = 4 for a system with 2 TX branches. It is noted that the PAPR for the transparent method is calculated

217

6.4 PAPR reduction by spatial shifting 100

100

10−1

P=1

P =2

10−2

PR SInt NC P (PAPR > PAPR0)

P (PAPR > PAPR0)

PR SInt NC 10−1

P =2 10−2

P =1

P =4

P =4 10−3

2

4

6

8

10

12

10−3

2

4

PAPR0

(a) SS method

Figure 6.12. Nt = 2.

6 8 PAPR0

10

12

(b) combined SS/PS method

CCDF of the PAPR for the diﬀerent subcarrier grouping schemes, for

for the entire 10 OFDM symbol packet. As reference the curves for the PAPR without SS or PS (“P = 1”) and for the SI-based SS/PS method with P = 4 (“SI-based”) are depicted. It can be concluded from the results that at a clipping probability of 10−3 and for the SS method, 0.9 dB and 1.8 dB PAPR reduction is achieved for P = 2 and P = 4, respectively. For the SS/PS method the reduction is 1.4 dB and 2.7 dB, respectively. This enforces the conclusion that, although its PAPR performance is worse than that of the SI-based method, the PAPR reduction achieved by the transparent method is still signiﬁcant.

6.4.7

Summary and discussion

A technique named spatial shifting (SS) was introduced to reduce the peak-to-average power ratio (PAPR) of the time-domain signals in multiple-antenna OFDM systems. For that purpose, the extra degree of freedom provided by the MIMO system is used to reshuﬄe groups of subcarriers such that the resulting TX signals attain a low PAPR. The numerical results in this section support the conclusion that signiﬁcant gain in PAPR reduction can be achieved by combining SS with phase shifting (PS) of the subcarrier groups. Moreover, this section proposes the application of the SS and the combined SS/PS method in a transparant mode, where no extra overhead is introduced by the transmission of side-information. It can, furthermore, be concluded from the presented results that the performance of SS, diﬀerently from PS, does not depend on the chosen

218

6 Nonlinearities 100

P (PAPR > PAPR0)

SS, P = 2 SS/PS, P = 2 SS, P = 4 SS/PS, P = 4 10−1

P=1 10−2

SI-based 10−3 2

4

6

8

10

12

14

PAPR0

Figure 6.13. CCDF of the PAPR for the transparant SS and combined SS/PS method for the PR subcarrier grouping scheme, for Nt = 2 and K = 10. PAPR is calculated for the entire 10 OFDM symbol packet.

subcarrier grouping. Therefore, the grouping scheme can be chosen such to minimise the computational complexity of the implementation of SS. It is noted that the PAPR reduction approaches were proposed and evaluated for discrete-time signals. Therefore, the presented results will only be directly applicable to continuous-time signals when ideal bandpass ﬁltering of the signals can be assumed. When this is not the case, the achieved PAPR reduction might be lower and sub-optimal [157, 162]. To achieve eﬀective PAPR reduction for these cases, the presented algorithms could be modiﬁed to apply oversampling in the algorithm, i.e., by (p) zero-padding of the signal sm,nt and by increasing the size of the DFT matrix in (6.77). The reduction of the computational complexity of the proposed methods is an interesting subject for further research in this ﬁeld. To that end, for example, sub-optimal techniques can be investigated as previously proposed for conventional OFDM in, e.g., [156]. Another subject of further research is PAPR reduction in MIMO OFDM systems, that already exploit the extra degree of freedom by transmit processing, like, e.g., transmit beamforming systems. The performance of this kind of systems does, diﬀerently from space division multiplexing [19], depend on the speciﬁc ordering of the TX streams. Therefore, the straightforward application of SS for this kind of systems is not recommended.

6.5 RX-based correction of nonlinearities

6.5

219

RX-based correction of nonlinearities

In the approach introduced in Section 6.4, the impact of nonlinearities on system performance is minimised by reducing the PAPR of the signals input to the nonlinearities. In the approach proposed in this section, which can be applied in combination with that of Section 6.4, nonlinear distortion of the signals to be transmitted is not prevented. Instead, the distortion of the signals is corrected for in the baseband part of the RX. We consider the typically occurring scenario where the most severe nonlinearities are experienced in the PAs, located in the TX chain. In this scenario, the approach highlighted above will have the advantage that a lower input power backoﬀ to the PAs can be applied and that, consequently, the eﬃciency of the PAs will be seriously improved. In contrast, several promising approaches have been proposed in the last few years that digitally compensate for the TX nonlinearities in the TX baseband, i.e., digital predistortion [141, 155]. The major disadvantage of these approaches is, however, that extra hardware and processing are required at the TX side of the transmission. In a downlink scenario, where the base station acts as TX, this makes sense since the cost of the additionally required hardware and the extra consumed power are relatively marginal. In the uplink scenario, where the mobile station acts as TX, this is evidently not the case. It is for this scenario that RX-based compensation, as proposed in this section, is promising, since no extra hardware is required and since most of the required processing is carried out in the base station. In the here proposed RX-based compensation of TX-induced nonlinearities, we can only compensate for the in-band inﬂuences. The second problem caused by TX nonlinearities is the spectral regrowth in neighbouring bands, which can cause a system not to meet the spectral mask imposed by regulations/standards. This limits the amount of nonlinear distortion that can be allowed by the system. Considerable nonlinear distortion can be allowed in-band, however, without substantial loss in achievable rate [35, 149]. To overcome this problem, the signal applied as input to the nonlinear device, i.e., the PA, can ﬁrst be clipped in the baseband part of the system to reduce the PAPR. Since the clipping is done in baseband it can be followed by frequency-domain ﬁltering to remove the out-of-band power. The in-band experienced nonlinearity is then a combination of the baseband clipping response and the PA nonlinearity, while the generated out-of-band distortion is limited. The outline of this section is as follows. First Section 6.5.1 describes the combined baseband clipping and ﬁltering approach. Subsequently, an estimation approach for the linear and nonlinear MIMO channel is proposed in Section 6.5.2. In the compensation approaches of Sections 6.5.3

220

6 Nonlinearities

and 6.5.4, the resulting estimates are used to apply postdistortion and distortion noise removal, respectively. The performance of the diﬀerent algorithms is evaluated using a simulation study, the results of which are reported in Section 6.5.5. Finally, Section 6.5.6 summarises the nonlinearity compensation approach presented in this section.

6.5.1

Baseband amplitude clipping

With the aim of limiting the out-of-band spectral regrowth, when the signals are fed through the nonlinearities, we consider the application of amplitude clipping in the TX digital baseband. This baseband clipping is a straightforward and low-complexity approach to reduce the PAPR, and its amplitude response is equal to that of the clipping ampliﬁer presented in (6.5). The inﬂuence of amplitude clipping, here referred to as hard clipping, on the time-domain signal is illustrated in Fig. 6.14. The ﬁgure depicts the clipping level and the amplitude of the original signal as a function of the signal sample index. Also, the amplitude of the clipped signal is depicted, which is equal to that of the original signal below the clipping level and is equal to the clipping level for larger input amplitude values. To achieve the same reduction in peak power for the continuous-time signal, the clipping is best applied to an oversampled signal [157, 162]. The disadvantage of hard clipping is, however, that spectral regrowth is generated outside the signal bandwidth. This can be explained by the sharp edges generated in the time-domain signal by the hard clipping.

2.5

Clipping level Original signal Hard clipped signal Peak windowing

amplitude

2

1.5

1

0.5

0

0

10

20 30 sample index

40

50

Figure 6.14. Inﬂuence of baseband amplitude clipping and peak windowing on a sampled time-domain OFDM signal.

6.5 RX-based correction of nonlinearities

221

This disadvantage can easily be overcome by the use of appropriate ﬁltering following the clipping. An implementation of such an approach was proposed in [163], where the ﬁltering is carried out in the frequency domain. The c-times oversampled hard clipped signal is transformed to the frequency domain using a c-times oversampled DFT. The subcarriers not carrying data are put to zero and the signal is transformed back to the time domain. The IDFT processing will introduce a small amount of spectral regrowth, which can be further removed by applying the clipping and frequency-domain ﬁltering in an iterative procedure. The complexity of the frequency-domain ﬁltering might, however, be too high since it requires additional oversampled DFT and IDFT processing. A computationally less complex approach was proposed in [150] and [151], which applies peak windowing instead of hard clipping. Here the samples preceding and following a peak are also reduced in amplitude using a window that determines the attenuation proﬁle. Different windowing approaches can be chosen, e.g., based on Gaussian, sinc or exponential windows. The width of the window determines how much the out-of-band radiation is suppressed, i.e., the wider the window, the more it is suppressed. A wider window, however, also yields more in-band distortion. The application of a sinc-window for peak windowing is illustrated in Fig. 6.14, from which it is clear that the sharp edges are successfully removed, compared to the hard clipping. It is, however, also concluded from the ﬁgure that the signal has changed more from the original signal than for hard clipping, i.e., more in-band distortion is introduced. Since our main interest lies in the in-band caused distortion, we will in the following no longer regard these clipping techniques, under the assumption that we will meet the imposed spectral masks with them. We note, however, that when one wants to reduce the nonlinearity-caused out-of-band spectral regrowth, the above presented clipping methods can be applied and regarded in-band as an additional nonlinear transfer, which can also be corrected by using, e.g., the method presented in Section 6.5.4.

6.5.2

Estimation of linear and nonlinear channel

For the nonlinearity correction methods proposed in Sections 6.5.3 and 6.5.4 and the MIMO detection algorithms, we require estimates of the experienced (nonlinear MIMO) transfer. A straightforward method would be to estimate the Nr Nt elements of the nonlinear MIMO transfer matrix, i.e., the combination of the nonlinearities and the MIMO wireless channel. This, however, does not use the knowledge that only Nt distinct nonlinearities occur, i.e., in the TX PAs. To better exploit this

222

6 Nonlinearities Preamble L train TX1 TX2

NL train p

DATA 1

p

DATA 2

Figure 6.15. Frame format for a 2 TX MIMO system enabling estimation of both the TX nonlinearities and the MIMO multipath channel.

characteristic, and the fact that the nonlinear and linear MIMO channel inﬂuence the signal sequentially, we propose the separation of the estimation of the nonlinear transfers and the linear MIMO channel. To that end, we introduce a frame format as depicted schematically in Fig. 6.15 for a two TX system. In this frame format the actual data transmission is preceded by a preamble consisting of known training data. The training data consists of a part used for the estimation of the linear MIMO channel, indicated by “L train”, and a part used for the estimation of the TX-caused nonlinearities, indicated by “NL train”. The estimation of the linear MIMO channel, based on the presented preamble structure, is presented in Section 6.5.2.1. The estimation of the nonlinearities and their inverses is treated in Sections 6.5.2.2 and 6.5.2.3, respectively.

6.5.2.1 Linear MIMO channel estimation For the estimation of the MIMO multipath channel we propose the application of a constant modulus sequence. For eﬃcient estimation, the application of the (constant modulus) Frank-Zadoﬀ-Chu (FZC) based training, as introduced in Section 3.2.2.6, is suggested. The Nr Nc ×1 received signal vector during the training for this FZC-based preamble can be expressed as ˘ xp = (FΥ ⊗ INr )Gg((Θ ⊗ INt )up ),

(6.89)

where up denotes the Nt Np ×1 (time-domain) training symbol without cyclic preﬁx (CP) and g(·) denotes the Nt -dimensional nonlinear function, i.e., it applies the nonlinear function gnt (·) to the part of the vector ˘ denotes the Nr Nc ×Nt Nc macorresponding to the nt th TX branch. G trix, modelling the MIMO channel impulse response matrix, where we ˇ = (Υ ⊗ IN )G(Θ ˘ recall that G ⊗ INt ) forms a block-circulant matrix. t Since we regard memoryless nonlinearities and since the training signal up has constant modulus, we can rewrite (6.89) to ˇ ˇ xp = (F ⊗ INr )Gg(u p ) = (F ⊗ INr )G(INc ⊗ η)up ,

(6.90)

6.5 RX-based correction of nonlinearities

223

where we deﬁned η = diag{η1 , . . . , ηNt } as an Nt -dimensional diagonal matrix. The nonlinear distorted signal vector corresponding to the nt th TX branch is given by gnt (up,nt ) =

gnt (|up,nt |) up,nt = ηnt up,nt , |up,nt |

(6.91)

where up,nt denotes the Nc ×1 subvector of up for the nt th TX branch and |a| denotes the vector of absolute values of the entries of a. In the last step of (6.91), the constant modulus property of the training vector up,nt is exploited. The complex parameter ηnt thus models the amplitude change and phase shift of the constant modulus training sequence, induced by the AM-AM and AM-PM response of the nonlinearity of the nt th branch, respectively. We note that for a nonlinearity only causing AM-AM distortion, ηnt is a real parameter. When, subsequently, MIMO channel estimation is applied using the received signal vector xp in a similar way as done Section 3.4, it is readily found that the estimate of the Nr Nc ×Nt Nc channel matrix is given by ˜ = H(IN ⊗ η), H c

(6.92)

where we have, for now, assumed a noiseless system. We conclude from ˜ corresponding to the nt th TX diﬀer by (6.92) that the columns of H the complex factor ηnt from the actual channel response H. Although these complex factors are deﬁned by (6.91), they can not be calculated without knowledge of the nonlinearities. Hence, for now they are indistinguishable from the actual MIMO channel.

6.5.2.2 Nonlinearity estimation The second part of the preamble, as depicted in Fig. 6.15, is used for the estimation of the nonlinearities. The sequence used as training must exhibit a high PAPR to enable accurate estimation of the nonlinear responses. Since the nonlinearities are estimated in the RX, i.e., after transmission through the frequency-selective fading channel, the signal must, moreover, exhibit some level of whiteness to avoid the total training sequence to coincide with a channel fade. To meet these requirements, i.e., high PAPR and whiteness, a complex white Gaussian signal is chosen. The sequence is generated in a pseudo-random manner and the variance of the training is chosen to be equal to that of the signals in the data part of the transmission. The same training is transmitted simultaneously from the diﬀerent TX branches, yielding an overhead that is independent of the number of TX branches. The time-domain version of the Nc ×1 training vector is denoted by p = [p(1), . . . , p(Nc )]T , to which a CP is added before transmission.

224

6 Nonlinearities

Since the training symbol p is transmitted simultaneously on all TX branches, the resulting RX signal vector (after DFT processing) is found by x = H(FΥ ⊗ INt )g(Θp ⊗ 1Nt ) = H(F ⊗ INt )g(p ⊗ 1Nt ) = Hqd ,

(6.93)

where qd is the frequency-domain version of the nonlinear distorted training symbol and 1Nt denotes the Nt -dimensional all-ones column vector. The estimate of qd can be found using the estimated MIMO channel in (6.92), yielding ˜ † x = (IN ⊗ η)† H† Hqd ˜d = H q c −1 = (INc ⊗ η )qd .

(6.94)

If we then deﬁne the time-domain nonlinear distorted training signal (after removal of the CP) as rd = g(p ⊗ 1Nt ),

(6.95)

then its estimate is found using (6.94) by ˜rd = (F−1 ⊗ INt )˜ qd −1 = (F ⊗ INt )(INc ⊗ η −1 )(F ⊗ INt )g(p ⊗ 1Nt ) = (INc ⊗ η −1 )rd .

(6.96)

The estimated signal corresponding to the nt th TX branch can then be written as ˜rd,nt = ηn−1 r = ηn−1 g (p). t d,nt t nt

(6.97)

We conclude from (6.97) that the MIMO processing can successfully separate the signals originating from the diﬀerent TX branches and that the result enables us to estimate the diﬀerent nonlinear responses. We note that the factor ηn−1 will be estimated and corrected for using the t nonlinearity estimation and nonlinearity correction algorithms, respectively. Now that the TX signals are separated, we ﬁrst consider the leastsquares estimation (LSE) of the parameters of the polynomial model for the nonlinearity. In Section 6.2.1.4, it was shown that the other nonlinearity models can be mapped to the polynomial model, making this a viable approach for the diﬀerent models of nonlinearity. Using

225

6.5 RX-based correction of nonlinearities

the polynomial model of (6.9), the estimate expressed in (6.97) can be rewritten for the nth sample as p(n) r˜d,nt (n) = ηn−1 t

N −1

βm+1 |p(n)|m .

(6.98)

m=0

Subsequently, we deﬁne ⎡ p(1) p(1)|p(1)| ⎢ p(2) p(2)|p(2)| ⎢ Φ=⎢ .. .. ⎣ . .

··· ··· .. .

p(Nc ) p(Nc )|p(Nc )| · · ·

p(1)|p(1)|N −1 p(2)|p(2)|N −1 .. .

p(Nc )|p(Nc )|N −1

⎤ ⎥ ⎥ ⎥, ⎦

(6.99)

and β = [β1 , β2 , . . . , βN ]T .

(6.100)

Here Φ is given by diag{p} times the Vandermonde matrix [46] of the amplitudes of the training vector values and β contains the parameters of the N th order polynomial model. Using these expressions, we can rewrite (6.98) in matrix notation as ˜rd,nt = ηn−1 Φβ = Φβ , t

(6.101)

β. where β = ηn−1 t The LSE of the parameters of the polynomial nonlinearity model for the nt th TX branch is then found by β˜ = Φ† ˜rd,nt ,

(6.102)

where we recall that Φ† = (ΦH Φ)−1 ΦH denotes the pseudo-inverse of Φ. The complexity of this solution can be reduced considerably by precalculating Φ† , which is possible since it is based on the known training vector p. When only odd orders occur in the nonlinearity, as for instance in (6.11), the basis matrix Φ can be simpliﬁed by ommiting the even orders. For the model of (6.11), the basis Φ reduces to ⎡ ⎤ p(1) p(1)|p(1)|2 p(1)|p(1)|4 ⎢ p(2) p(2)|p(2)|2 p(2)|p(2)|4 ⎥ ⎢ ⎥ Φ=⎢ (6.103) ⎥, .. .. .. ⎣ ⎦ . . . p(Nc ) p(Nc )|p(Nc )|2 p(Nc )|p(Nc )|4 and β is given by β = [1, β3 , β5 ]T ,

(6.104)

226

6 Nonlinearities

where β3 and β5 are real parameters. Finally, for this nonlinearity the estimates of all parameters can be found from (6.102) as −1 η˜nt = β˜ (1) , β˜3 = η˜nt β˜ (2), β˜5 = η˜n β˜ (3),

(6.105) (6.106) (6.107)

t

where β˜ (n) denotes the nth element of the estimated vector β˜ . The estimate of the MIMO channel in (6.92) can be corrected for the constant amplitude error by multiplying with (6.105) for the diﬀerent TX branches. The disadvantage of the polynomial basis as chosen above is that the matrix ΦH Φ is often ill-conditioned for complex Gaussian signals. The inversion of the matrix will thus incur numerical errors [164]. It was shown by the authors of [164] that this problem does not occur for orthogonal polynomials. Diﬀerent orthogonal polynomial bases can be applied for this estimation, but here we derive a basis using GramSchmidt orthonormalisation. Although this basis can be derived for the general problem of (6.101), we will here regard the speciﬁc problem where the nonlinearity can be modelled by (6.11), i.e., a 5th order AMAM nonlinearity. Since we only require the odd orders, the orthogonal polynomials are derived using the set of basis functions {A, A3 , A5 }, where A(n) = |p(n)| denotes the amplitude of the elements of the training vector. The derivation is deferred to Appendix E, but the result is given here. The qth polynomial basis is denoted by ψq (A) and the total set of orthonormal basis polynomials is given by 1 ψ1 (A) = √ A, µ2

(6.108)

ψ2 (A) = α1 A3 + α2 A, 1 α3 α4 ψ3 (A) = √ A5 + √ A3 + √ A, α5 α5 α5

(6.109) (6.110)

where we have deﬁned µq = E[Aq (N )] = E[|p(N )|q ] =

Nc 1 |p(n)|q , Nc

(6.111)

n=1

where the expectation is calculated over N , which is discrete uniform distributed over {1, . . . , Nc }. The other parameters in (6.108) to (6.110)

227

6.5 RX-based correction of nonlinearities

or deﬁned as :

µ2 , µ2 µ6 − µ24 ; µ24 , α2 = − µ22 µ6 − µ2 µ24

α1 =

(6.112) (6.113)

α3 = −(µ8 α1 − µ6 α2 )α1 ,

(6.114) µ6 (6.115) α4 = (µ8 α1 − µ6 α2 )α2 − , µ2 α5 = µ10 + α32 µ6 + α42 µ2 + 2α3 µ8 + 2α4 µ6 + 2α3 α4 µ4 . (6.116) For this orthonormal polynomial basis, Φ is given by 6 Φ = diag

7 p(1) p(2) p(Nc ) , ,..., |p(1)| |p(2)| |p(Nc )| ⎡ ψ1 (|p(1)|) ψ2 (|p(1)|) ⎢ ψ1 (|p(2)|) ψ2 (|p(2)|) ⎢ ·⎢ .. .. ⎣ . .

ψ3 (|p(1)|) ψ3 (|p(2)|) .. .

⎤ ⎥ ⎥ ⎥ . (6.117) ⎦

ψ1 (|p(Nc )|) ψ2 (|p(Nc )|) ψ3 (|p(Nc )|) The parameters of the nonlinearity can, similarly as for the other basis, be estimated using (6.102).

6.5.2.3 Inverse nonlinearity estimation Instead of estimating the nonlinearity, as presented in Section 6.5.2.2, we can also directly estimate the inverse of the nonlinearity using the preamble structure presented in Fig. 6.15. This will show to be useful for the postdistortion procedure introduced in Section 6.5.3, since no inverse of the estimated nonlinearity has to be calculated. For the approach presented in Section 6.5.4, on the other hand, it might not be beneﬁcial, since we require an estimate of the nonlinearity there. The polynomial estimation of the inverse nonlinearity is very similar to the estimation of the nonlinearity as presented in Section 6.5.2.2. That is why we ﬁrst rewrite (6.97) as p = nt (ηnt ˜rd,nt ) = ˆnt (˜rd,nt ),

(6.118)

where ˆnt (a) = nt (a · ηnt ) and nt (·) denotes the inverse nonlinear function of gnt (·), i.e., by deﬁnition, nt (gnt (x)) = x.

(6.119)

228

6 Nonlinearities

Here we make the assumption that the experienced nonlinearity gnt (·) is invertible, which will be met when the experienced nonlinearity is smooth. Under the assumption that nt (or equivalently ˆnt ) can be modelled by a polynomial model of order M , (6.118) can be rewritten for the nth sample as M −1 p(n) = r˜d,nt (n) χm+1 |˜ rd,nt (n)|m . (6.120) m=0

We note that (6.120), with ﬁnite M , is an approximation for certain polynomial nonlinearities of ﬁnite order. It can be shown, however, that the equality in (6.120) is valid for the TWTA and SSA models of Sections 6.2.1.2 and 6.2.1.3, given that M is chosen suﬃciently large. In matrix notation (6.120) is given by p = Φnt χ,

(6.121)

χ = [χ1 , χ2 , . . . , χM ]T ,

(6.122)

where we have deﬁned

and

⎡

⎢ ⎢ Φnt = ⎢ ⎣

r˜d,nt (1) r˜d,nt (2) .. .

r˜d,nt (1)|˜ rd,nt (1)| r˜d,nt (2)|˜ rd,nt (2)| .. .

··· ··· .. .

rd,nt (Nc )| · · · r˜d,nt (Nc ) r˜d,nt (Nc )|˜

r˜d,nt (1)|˜ rd,nt (1)|M −1 r˜d,nt (2)|˜ rd,nt (2)|M −1 .. .

⎤ ⎥ ⎥ ⎥. ⎦

r˜d,nt (Nc )|˜ rd,nt (Nc )|M −1 (6.123)

The LSE of the parameters of the polynomial model in (6.120) are then given for the nt th TX branch as ˜ nt = Φ†nt p. χ

(6.124)

We note that, as for the estimation of the nonlinearity, diﬀerent bases can be applied, which might have a better performance for the speciﬁc experienced nonlinearity. The construction of Φ and χ, then change similarly to Φ and β in Section 6.5.2.2. The problem, however, is that Φnt depends on ˜rd here. Therefore, the parameters of the orthonormal basis derived in Appendix E have to be calculated for every realisation separately. Alternatively, a basis could be derived based on the distribution of rd , as in [164]. Note that additionally, in contrast to the method presented in Section 6.5.2.2, the inverse of (6.123) can not be precalculated, since it contains the estimates of the nonlinearly distorted TX signals. Furthermore,

229

6.5 RX-based correction of nonlinearities 3

Output amplitude

2.5

2

1.5

1

Linear response NL response Inv. NL response Est. NL response Est. inv. response

0.5

0

0

0.5

1

1.5

2

2.5

3

Input amplitude Figure 6.16. Example of input-output amplitude transfer of the actual and estimated nonlinearity (NL) and its inverse (inv.). The nonlinearities are modelled using the SSA model with p=1 and A0 =3.

it has to be calculated separately for every TX branch. Consequently, the complexity of the presented estimation is considerably higher than that of the estimation of the nonlinearity. The application of the estimated inverse of the nonlinearity will, however, result in a much lower complexity for the postdistortion implementation, as will be highlighted in Section 6.5.3. An example of the estimates of the nonlinearities and their inverses are given in Fig. 6.16 for a scenario with an SNR of 20 dB. It depicts the nonlinear input-output amplitude response, which is here modelled using the SSA model of (6.8) with p=1 and A0 =3. As a reference the linear response is given by the dashed line. The estimated nonlinear response using the method proposed in Section 6.5.2.2, is depicted in asterisks and matches the original nonlinear curve well for low and moderate input amplitudes. The estimate diverges from the actual nonlinear transfer for high input amplitudes, since higher orders in the nonlinearity become dominant there. These orders are not included in the model used for the estimation. Furthermore, the number of observations at high input amplitudes is small, due to the speciﬁc construction of the training sequence, which yield low accuracy for the estimates for high input amplitude values. This error, however, is not very signiﬁcant

230

6 Nonlinearities

when the estimate is used for, e.g., postdistortion, since also the data signal exhibits a low probability of high amplitudes. The ﬁgure also depicts the estimated inverse nonlinear response using the approach of Section 6.5.2.3, which reveals a similar error for high input amplitudes. For a more thorough evaluation of the performance of the presented estimation algorithms, the reader is referred to Section 6.5.5.

6.5.3

Postdistortion

In this section we propose a method for RX-based correction of the inband caused distortion due to TX nonlinearities. The approach is based on postdistortion, for which the required RX processing is schematically summarised in Fig. 6.17. In this algorithm we apply the estimates of the linear MIMO channel and the (inverse of the) nonlinearities as found from the methods proposed in Sections 6.5.2 and 6.5.2.3, respectively. First the nonlinear distorted TX signal vector is recovered applying MIMO estimation, using the estimate of the MIMO channel of ˜ . We can either ﬁrst correct this channel estimate for (6.92), yielding u η as found using the method of Section 6.5.2.2, or correct for the η in the following postdistortion. A ﬁrst approach is postdistortion based on the estimates of the inverse nonlinearities as found from the method proposed in Section 6.5.2.3. Using the expression for the inverse nonlinearity in (6.120) it is easily found that the postdistorted signal corresponding to the nt th TX stream

y Nr .. . 1

˜ H ···

x .. . CP .. Removal, . S/P

DFT

Nr .. . 1 .. 0 .

MIMO proc.

.. Detector

. .. .

..

Nt

. 1 0 . ..

..

. .. .

IDFT

P/S

1

.. DFT

..

Nt

. 1 0. ..

S/P

..

Nc−1 ˜s

Nt

Nc−1

Nc−1

Nt .. . 1

˜ u

˜s d

˜ u

.

Nt .

Postdistortion

1

··· ˜ β˜ or χ

Figure 6.17. Block diagram of the RX processing in a MIMO OFDM system applying postdistortion.

231

6.5 RX-based correction of nonlinearities

is given by u ˜nt (n)

=u ˜nt (n)

N −1

χ ˜nt ,m+1 |˜ unt (n)|m .

(6.125)

m=0

A second approach is based on the estimates of the nonlinearities, as found from the methods presented in Section 6.5.2.2. To retrieve the data symbols, we implicitly calculate the inverse of the polynomial by applying Lagrange interpolation on the time-domain signals. The Lagrange method is chosen due to its relatively low computational complexity. The AM-AM postdistorted signal is for this method given by u ˜nt (n) =

M u ˜nt (n) a(m) |˜ unt (n)| m=1

M < l=1,m=l

|˜ unt (n)| − bnt (l) , bnt (m) − bnt (l)

(6.126)

where a(m) is the M -dimensional basis used for the Lagrange interpolation, which is chosen in a tradeoﬀ between the quality of ﬁt and smoothness of the output signal. The points used for the interpolation are found using the estimated nonlinear function g˜A and given by bnt (m) = g˜A,nt (a(m)) .

(6.127)

Here another degree of freedom to increase the quality of ﬁt is the basis used for a(m). In the study presented here, a linear and exponential basis were considered, deﬁned by amax alin (m) = (m − 1) M −1

aexp (m) = amax

max )−1 exp( (m−1)a M −1

exp(amax )−1

for

m = 1, . . . , M, (6.128)

for

m = 1, . . . , M, (6.129)

respectively. The maximum amplitude level amax can be chosen such that it maximises the performance of the interpolation.

6.5.4

Iterative distortion removal

An alternative approach for the correction of the distortion caused by nonlinearities in SISO OFDM systems was proposed in [165] and [35] and is here referred to as iterative distortion removal (IDR). It applies an iterative decision-directed approach, which uses that, due to Bussgang’s theorem, the nonlinear distorted signal can be rewritten as a scaled version of the original signal plus a distortion term. The impact of Bussgang’s theorem on the signal model was shown in (6.21). For the following it is useful to rewrite, using Bussgang’s theorem, the distorted

232

6 Nonlinearities y

Nr .. . 1

x ..

CP Removal, S/P

. .. .

DFT

Nr .. . 1 .. 0 .

Nc−1

˜ H ··· MIMO proc.

˜sd ..

˜s

Nt

. 1 0 . ..

Nc−1

..

IDR

. .. .

Nt .. Detector . 1

···

β˜

Figure 6.18. Block diagram of the RX processing in a MIMO OFDM system applying iterative distortion removal (IDR).

TX vector as function of the non-distorted frequency-domain vector: sd,nt = FΥgnt (unt ) = FΥ(αnt unt + dnt ) = αnt snt + ent ,

(6.130)

where we recall that F denotes the Fourier matrix, Υ denotes the removal of the CP and gnt (·) denotes the nonlinearity experienced by the nt th Ns ×1 time-domain TX vector unt . The scaling factor αnt was deﬁned by (6.24) and dnt and ent denote the time-domain and frequency-domain distortion vector, respectively. Finally, the nt th Nc ×1 non-distorted frequency-domain TX vector is denoted by snt . The RX structure for a system applying IDR is schematically depicted in Fig. 6.18. First conventional OFDM and MIMO processing, ˜ is applied to separate the diﬀerent using the estimated linear channel H, TX streams. The resulting signal vector is given by ˜sd . Subsequently, processing is applied for every estimated TX stream separately in the frequency domain. The iterative detection process is summarised in the block diagram of Fig. 6.19. ˜nt In this processing ﬁrst an estimate of the distortion noise vector e is subtracted from the signal ˜sd,nt . In the ﬁrst iteration this estimate is equal to 0. Subsequently, the vector corresponding to the nt th branch is scaled by 1/α ˜ nt . An estimate of αnt is found by substituting the parameters of the polynomial model βnt , found from the estimation algorithm of Section 6.5.2.2, into (6.27). The resulting signal is input to the hard slicing device D(·), the output of which can be expressed by . / ˜nt αnt snt + ent − e ˜snt = D . (6.131) α ˜ nt For perfect knowledge of both αnt and ent and no noise, (6.131) equals snt and the distortion is perfectly removed. Since in the ﬁrst iteration

233

6.5 RX-based correction of nonlinearities α ˜ n- 1 t

˜sd,n

t

˜sn t

+

˜sn t

Slicing D(·)

-

IDFT α ˜nt -

˜n e

t

DFT

Figure 6.19. stream.

+

g˜n (·) t

Block diagram of the iterative distortion removal for the nt th TX

˜nt = 0 and noise does occur, errors will appear in ˜snt and additional e iterations are required. The estimate of the transmitted signal is therefore used to ﬁnd the estimate of the distortion noise used in the next iteration, which is given by ˜nt = FNc g˜nt (F−1 (6.132) e snt ) − α snt , ˜ nt F−1 Nc ˜ Nc ˜ where g˜nt (·) denotes the estimate of the nonlinearity found using the algorithm of Section 6.5.2.2 and FNc denotes the Nc -dimensional Fourier ˜nt , and thus also matrix. By applying several iterations the estimate of e ˜snt , improves, as was shown for SISO OFDM systems experiencing hard clipping in [35, 165]. The baseband clipping, as treated in Section 6.5.1, can also be considered as a nonlinear function and modelled, e.g., using (6.5). Therefore, the experienced nonlinear function gnt (·) can also be regarded as a superposition of the clipping and the PA nonlinearities. Obviously, the IDR method can, when the transfer is known, jointly correct for the combined nonlinearity.

6.5.5

Simulation results

The performance of the algorithms proposed in this section is evaluated here by means of Monte Carlo simulations for diﬀerent ZF-based MIMO OFDM conﬁgurations. The simulated systems apply Nc = 64 subcarriers, which all carry 16-QAM or 64-QAM data. A packet-based system is simulated, the format of which was schematically depicted in Fig. 6.15. The preamble introduced in this section is used for the estimation of both the linear MIMO wireless channel and the TX nonlinearities. For the estimation of the nonlinearities and inverse nonlinearities the polynomial basis of (6.103) is used. Throughout this section all TX

234

6 Nonlinearities

branches experience the same nonlinearity, but they are separately estimated. The length of the training symbols is equal to the length of an OFDM symbol. One symbol is used for the estimation of the MIMO channel and one for the estimation of the nonlinearities. For all simulations the channel is modelled as a spatially uncorrelated Rayleigh faded MIMO channel with a rms delay spread of 50 ns. MSE in nonlinearity estimation We will ﬁrst evaluate the performance of the proposed nonlinearity and inverse nonlinearity estimators, as proposed in Section 6.5.2.2 and 6.5.2.3, respectively. For the performance of the linear channel estimator proposed in Section 6.5.2.1 the reader is referred to Section 3.4.2. A straightforward method to evaluate the estimation performance would be the calculation of the mean squared error (MSE) of the estimated polynomial parameters. Since the input signals, however, exhibit a limited amplitude range, diﬀerent polynomial functions might result in a similar transfer for this low amplitude range, but still result in a high MSE, making this evaluation not useful. A second method could be to calculate the MSE in the (estimated) resulting nonlinear transfer for uniformly spaced input values over the range of the input signal, see Fig. 6.16 for an illustration. This, however, does not take into account that the envelope of the input data signal is approximately Rayleigh distributed. Consequently, errors in the large input amplitude region are over weighted, making also this measure unsuitable. Hence, as a better performance measure we regard the MSE in the estimated signals using the estimated nonlinearities and inverse nonlinearities. Therefore, we compare three MSEs in the following, which are deﬁned as MSEofthenonlineardistortedsignal: MSE1 = E |˜ ud,nt (n)−unt (n)|2 , ˜ † x for a system that does experience the ˜ d = (FΥ ⊗ INr )H where u nonlinearities. MSE of the postdistorted inverse nonlin signal using2 the estimated earities: MSE2 = E |˜ ˜nt (n) is found using unt (n) − unt (n)| , where u (6.125). MSE of the distorted signal using the estimated nonlinearities: ˜ †x ˜ lin = (FΥ ⊗ INr )H MSE3 = E |˜ gnt (˜ unt ,lin (n)) − ud,nt (n)|2 , where u for a system without nonlinearities. These MSE results are depicted in Fig. 6.20 for a 16-QAM 2×4 system, where the TX nonlinearities are modelled using the TWTA AM-AM model of (6.6). The results are given for χA = 1 and κA ∈ {0.05, 0.1, 0.2}.

235

6.5 RX-based correction of nonlinearities 100

MSE

10−1

10−2

10−3

κA = 0.05 κA = 0.1 κA = 0.2 linear

10−4 0

20 40 60 Average SNR per RX antenna (dB)

80

Figure 6.20. MSE in a 2×4 system experiencing TWTA nonlinearities with χA = 1 and κA ∈ {0.05, 0.1, 0.2}. MSE is given for the nonlinear distorted signal, MSE1 , (no lines), for postdistortion using the estimated inverse nonlinearities, MSE2 , (dashed lines) and for distortion using the estimated nonlinearities, MSE3 , (solid lines).

In these simulations, and those for Fig. 6.21, the linear channel was assumed to be perfectly known. So only estimation of the 3rd and 5th order of the nonlinearity was required. Note that here, and throughout this section, the SNR is deﬁned as the per RX branch experienced SNR. We conclude from the results for the nonlinear system without compensation (in markers without lines) in Fig. 6.20 that MSE1 increases with increasing κA . The MSE for the linear system is given as reference (with triangles), for which the error is caused by the additive RX noise. Using these curves we can conclude that for low SNR values the MSE is dominated by the AWGN, while the ﬂooring at high SNR values is caused by the nonlinearities. The results for the nonlinearity estimation (in solid lines) show that MSE3 decreases for decreasing κA , i.e., for decreasing inﬂuence of nonlinearities. This can be explained by the fact that the 5th order polynomial model better ﬁts the experienced nonlinearity when κA gets smaller, something that also can be concluded from (6.14). Also, it is partly caused by the fact that the impact of the nonlinearity is smaller with smaller κA . For the estimation of the inverse nonlinearities (in dashed lines), i.e., MSE2 , we can draw the same conclusions.

236

6 Nonlinearities 100

MSE

10−1

3rd order

10−2

5th order

10−3

10−4

linear nonlinear, MSE1 inverse est, MSE2 NL est, MSE3 0

20

40

60

80

Average SNR per RX antenna (dB)

Figure 6.21. MSE for a 1×4 (solid lines) and a 2×4 (dashed lines) system experiencing SSA nonlinearities with p=1 and A0 =3. The MSE results are depicted for the reference system (“linear”), a nonlinear distorted system MSE1 , postdistored signals using the estimated inverse nonlinearities MSE2 and nonlinear distorted signals using the estimated nonlinearities MSE3 .

The impact of the polynomial order and the MIMO conﬁguration is analysed in Fig. 6.21. The ﬁgure depicts results for a 1×4 and a 2×4 16-QAM system experiencing nonlinearities modelled by the SSA model of (6.8) with p=1 and A0 =3. It can be concluded from Fig. 6.21 that for low SNR values MSE1 , MSE2 and MSE3 are governed by the AWGN and for high SNR by the errors in the (inverse) nonlinearity estimates. Therefore, the MSE for low SNR depends on the MIMO conﬁguration, while the ﬂooring at high SNR is independent of the MIMO conﬁguration. The ﬁgure also compares the performance of estimation using a 3rd and 5th order basis matrix Φ. We conclude that for both the nonlinearities and inverse nonlinearities estimation, the algorithms based on the 5th order model perform better, since it better approximates the experienced SSA model. We note that in both cases the ﬁrst order is equal to 1, since the MIMO wireless channel is assumed to be perfectly known. The inﬂuence of imperfect wireless MIMO channel knowledge is evaluated in Fig. 6.22 for a 16-QAM 2×4 system. It compares the MSE performance for a system with perfect channel knowledge (in dashed

237

6.5 RX-based correction of nonlinearities 100

MSE

10−1

10−2

10−3

linear nonlinear, MSE1 inverse est, MSE2 NL est, MSE3

10−4 0

20 40 60 Average SNR per RX antenna (dB)

80

Figure 6.22. MSE for a 2×4 system experiencing SSA nonlinearities with p=1 and A0 =3 with perfect (dashed lines) and estimated (solid lines) MIMO wireless channel knowledge.

lines) with that of a system using an estimated MIMO channel (in solid lines), both using the approach of Section 6.5.2.1. We conclude from Fig. 6.22 that the degradation in MSE for low SNR is equal to that of the reference curve (linear). Interestingly the MSE performance of the uncorrected nonlinearity, i.e., MSE1 , improves for imperfect linear MIMO channel knowledge for high SNR values, compared to the case of perfect channel knowledge. This is explained by the fact that the estimated channel is scaled by a factor ηnt , which is generally smaller than 1, with respect to the actual channel. When we recall (6.37), it can be concluded that the nonlinear distorted signal on the nt th TX branch can be written as the αnt -scaled version of the original signal plus a distortion noise signal. Generally also αnt is smaller than 1. In this way, the ηnt -scaled channel estimate partly corrects for the αnt -scaling of the signal, which yields a (slightly) increased MSE performance for high SNR values. The MSE performance for the estimated inverse nonlinearities, MSE2 , is almost equal to the perfect channel knowledge case, while a small MSE degradation is observed for MSE3 . This can be explained by the fact that the ﬁrst order now also has to be estimated due to unknown scaling of the linear MIMO channel estimate.

238

6 Nonlinearities

MSE in postdistortion Subsequently, we study the MSE performance of the postdistortion approach introduced in Section 6.5.3. We ﬁrst consider the MSE in the estimate of the time-domain TX signal after postdistortion for a 16QAM 2×4 system. The MSE of postdistortion (PD) with the estimated inverse nonlinearity, as described by (6.125) and deﬁned above as MSE1 , and with the estimated nonlinearity using Langrange interpolation, as described by (6.126), are regarded. It is now useful to deﬁne the MSE for the latter: MSE of the postdistorted signal using the estimated nonlinearities: MSE4 = E |˜ ˜nt (n) are unt (n) − unt (n)|2 , where the elements of u found using the Lagrange interpolation in (6.126). For the Lagrange interpolation based postdistortion (denoted by “LG PD” in Fig. 6.23) we regard both the linear and the exponential basis, with diﬀerent vales of amax . For all simulations the order of the Lagrange interpolation M was chosen to be 5. The nonlinearity is again modelled using the SSA model with p=1 and A0 =3. Both the linear MIMO channel and the nonlinearities are estimated using the method proposed in this section. A large SNR range is simulated to reveal the ﬂooring behaviour. We conclude from the results in Fig. 6.23 that severe MSE ﬂooring occurs due to the nonlinearity, when no postdistortion is applied. A large improvement in MSE is clearly achieved when postdistortion using the estimated inverse nonlinearities (MSE1 , denoted by “PD using est inverse”) is applied: a degradation compared to the linear system occurs only at high SNR values. We conclude that this approach performs superior to the Lagrange based interpolation method for all the simulated cases. For the Lagrange-based interpolation, MSE4 reaches its minimum for amax = 2. For larger amax the performance decreases again, both at high and low SNR. It can be concluded that the performance of the linear and exponential basis are similar for the considered cases. The performance at low SNR decreases for increasing amax . This is explained by the fact that the points spanning the basis are spaced far apart, which results in a non-smooth curve for low accurate estimates of the nonlinearities. Judging from Fig. 6.23, there must exist an optimum value of amax which maximises the postdistortion performance, i.e., minimises MSE4 . Therefore, the MSE as a function of amax was investigated for a 2×4 system, the result of which is depicted in Fig. 6.24. The top plot investigates the ﬂooring for high SNR values and is simulated for an SNR of 200 dB. The lower plot investigates the performance degradation for low SNR

239

6.5 RX-based correction of nonlinearities 100

amax= 0.5

MSE

10−1

amax= 4

10−2 amax= 1 linear nonlinear PD using est inverse LG PD - lin basis LG PD - exp basis

10−3

10

amax= 2

−4

0

20

40

60

80

Average SNR per RX antenna (dB)

Figure 6.23. MSE in postdistortion for a 2×4 system experiencing SSA nonlinearities with p=1 and A0 =3 and an unknown MIMO wireless channel. Lagrange-based postdistortion (“LG PD”) results are depicted for amax = 4 (solid lines), amax = 2 (dashed lines), amax = 1 (dotted lines) and amax = 0.5 (dash-dot lines).

values and is simulated for SNR = 10 dB. The results for postdistortion using the estimated inverse nonlinearities are given as reference. Figure 6.24 shows that for high SNR values the MSE decreases to a minimum as a function of amax , and then increases again. This is explained by the fact that low amax values result in inaccurate modelling for the postdistortion of high amplitude values, while high values of amax result in non-smooth curves. The minimum MSE for the exponential and linear basis is found for amax values of 1.7 and 1.8, respectively. For low SNR, judging from the lower ﬁgure, it can be concluded that below amax = 1.7 no severe degradation occurs. It can, thus, be concluded that for both bases, and this antenna conﬁguration, amax = 1.7 is a good choice. We note that we derived one value of amax to be applied over the whole range of occurring SNR values. A more optimal, although also more diﬃcult, solution could be found when we would allow amax to vary as function of the SNR. BER performance of systems applying PD and IDR Although the presented MSE results provide insight in the performance of the algorithms, they do not directly translate to system performance of a system applying them. Therefore bit-error rate (BER) simulations were carried out, the results of which are depicted in Fig. 6.25. Results

240

6 Nonlinearities

MSE

10−1 10−2 10−3

10−4

0

MSE

102

0.5

1

1.5 2 amax

SNR = 200 dB 2.5 3 3.5

LG PD - lin basis LG PD - exp basis PD using est inverse

101 100 10−1

0

0.5

1

1.5 2 amax

SNR = 10 dB 2.5 3

3.5

Figure 6.24. MSE in the postdistortion-based TX signal estimates as a function of the maximum basis values amax for a 2×4 system experiencing SSA nonlinearities with p = 1 and A0 = 3 for the diﬀerent algorithms of Section 6.5.3.

are depicted for a 2×4 system applying 64-QAM modulation and no coding. Simulations were carried out for a system experiencing nonlinearities modelled using the SSA model with p = 1. For Fig. 6.25(a) and 6.25(b) the clipping levels A0 were 3 and 2, respectively. For the correction methods, the MIMO linear channel and the nonlinearities were estimated using the preamble and algorithms presented in this section. The BER results of a system not experiencing nonlinearities (“linear”) and experiencing nonlinearities, but not applying correction (“nonlinear”) are given as reference in the ﬁgures. The Lagrange-based PD results are given for the linear basis with amax = 1.7. The results for a system applying IDR are given for the initial step (“it=0”), i.e., where the estimated distortion noise is taken to be an all-zeros vector, and the two following iterations. When we ﬁrst consider Fig. 6.25(a), we can conclude that severe BER degradation occurs due to the introduced nonlinearities; ﬂooring occurs at a level of 7·10−3 . PD using either the estimated inverse nonlinearities or the estimated nonlinearities and Lagrange interpolation can successfully reduce the inﬂuence of the nonlinearities. The resulting BER curves are shifted 0.5 dB compared to that of a system not experiencing nonlinearities. It is noted that the Lagrange-based postdistortion performs a little worse than the other postdistortion method for low SNR values. The IDR method already improves the BER performance in the initial detection of the data considerably due to the α ˜ n−1 scaling of the symbols. t In the next iteration the BER improves further since an estimate of the

241

6.5 RX-based correction of nonlinearities 100

10−1

10−1

10−2

10−2 BER

BER

100

10−3

10−3 Linear Nonlinear PD inverse LG Interpol. IDR, it=0 IDR, it=1 IDR, it=2

10−4

10−5

0

10 20 30 40 50 Average SNR per RX antenna (dB)

(a) SSA, p = 1, A0 = 3

10−4

10−5 0

Linear Nonlinear PD inverse LG Interpol. IDR, it=0 IDR, it=1 10 20 30 40 50 Average SNR per RX antenna (dB)

(b) SSA, p = 1, A0 = 2

Figure 6.25. BER results for a 2×4 system applying diﬀerent RX-based nonlinearity correction methods.

distortion noise term is subtracted from the data. At a BER of 10−4 , a degradation of 2 dB compared to the ideal performance is achieved. When more iterations are applied, the performance does not improve further considerably. When we, subsequently, regard the results for a clipping level of A0 = 2 in Fig. 6.25(b), it can be concluded that these nonlinearities degrade the BER performance even more. This can be understood by the fact that higher probability of clipping and more distortion occur, as can be observed from the AM-AM transfer depicted in Fig. 6.2. Again, the performance of both postdistortion methods is similar. The achieved degradation at a BER of 10−4 is now 1 dB. This can be explained by the increased inﬂuence of clipping, which can not be corrected by the postdistortion. Also the performance of the IDR algorithms is worse than that of Fig. 6.25(a), which can be explained by the higher BER in the initial step, which yields a worse estimate of the distortion noise and consequently results in less suppression of the inﬂuence of the nonlinearities. The results are only shown for the ﬁrst iteration, since no further improvement was found after this iteration.

242

6.5.6

6 Nonlinearities

Summary

The possibility to correct the in-band inﬂuence of nonlinearities was illustrated in this section. To this end, an approach for the correction of the inﬂuence of TX-caused nonlinearities in a MIMO OFDM system was introduced in this section. The method consists of an estimation procedure for the linear MIMO wireless channel response and the TX nonlinearities followed by compensation. The estimation is enabled by a preamble consisting of a constant modulus MIMO training part and a sequence with high peak power for nonlinearity estimation. Both the least-squares estimation of the nonlinearities and their inverses was introduced. For correction two algorithms were presented, based on these estimates: postdistortion and iterative distortion removal. A numerical performance evaluation revealed that both correction methods can be successfully applied to signiﬁcantly reduce the inﬂuence of nonlinearities.

6.6

Conclusions

The inﬂuence, estimation and compensation of nonlinearities in multiple-antenna OFDM systems were treated in this chapter. First diﬀerent models for nonlinearities experienced in wireless communication systems were introduced and their similarity was shown. Then, the inﬂuence of these nonlinearities on the signal model of an OFDM system was derived. It is concluded that the main eﬀect is a scaling of the constellation points and the generation of an additive distortion term. Also, the inﬂuence of nonlinearities on the probability of erroneous detection of transmitted M -QAM symbols in MIMO OFDM systems was derived in this chapter. To that end, the inﬂuence of TX and RX nonlinearities was studied separately. Analytical expressions were derived for the symbol-error rate (SER) of nonlinear distorted MIMO OFDM systems. Results were found for both AWGN and Rayleigh faded MIMO channels. Compact expressions were derived for these SERs, which were shown, using simulation results, to accurately predict the performance of such a system. To decrease the impact of nonlinearities, a technique for peak-toaverage power ratio (PAPR) reduction of the time-domain signals in multiple-antenna OFDM systems was introduced. This technique is named spatial shifting (SS), and uses the extra degree of freedom provided by MIMO to reshuﬄe groups of subcarriers such that the resulting TX signals attain a low PAPR. It was concluded from simulation results that a signiﬁcant PAPR reduction can be achieved using SS. Also, more PAPR reduction was achieved by combining SS with phase shifting (PS)

6.6 Conclusions

243

of the subcarrier groups. Moreover, this chapter proposed the application of the SS and the combined SS/PS method in a transparent mode, which introduces no overhead since it does not require the transmission of side-information. Finally, this chapter also presented an approach for the RX-based correction of TX-caused nonlinearities in MIMO OFDM systems. In this method the experienced nonlinearities are estimated at the RX, which is enabled by a new, eﬃcient preamble design. Two correction methods were presented, one based on postdistortion and one based on a decision-directed removal of the nonlinear distortion noise. Numerical simulation results reveal that both methods can successfully be applied to signiﬁcantly reduce the inﬂuence of nonlinearities in MIMO OFDM systems.

Chapter 7 A GENERALISED ERROR MODEL

7.1

Introduction

The previous chapters have treated the inﬂuence of diﬀerent radiofrequency (RF) impairments in multiple-antenna OFDM systems, independently. In this chapter a ﬁrst step is made towards the modelling of the composite severeness and impact of diﬀerent RF impairments. Such a generalised impairment model has the major advantage that not all eﬀects have to be investigated separately, but that the combined impact of all impairments can be captured. This combined inﬂuence of all impairments is also the measure that is of interest to system designers. One commonly used measure for the aggregate severity of imperfections in system design is the error vector magnitude (EVM) [166], which basically measures the root mean square (rms) error in the estimated symbols. In order to unambiguously relate this measure to the ﬁnal system performance measure, i.e., the error rate of the detection, the distribution of the error term resulting from all implementation deﬁciencies has to be known. Generally, though, it is assumed to be a zero-mean complex Gaussian process. It was shown in the previous chapters, however, that when the impairments induced by the RF part of the system prevail, this will not be a good approximation, making a performance estimation using EVM far from accurate. Furthermore, the traditional EVM measure does not allow to take the nonstationarity of the impairments into account, which will be present when systems apply power-save modes and switch between transmitter/receiver (TX/RX) modes. In this chapter we, therefore, introduce an error model that next to an additive part also has an amplitude scaling and phase rotational part. To separate the inﬂuence of TX and RX impairments, which will be diﬀerent

246

7 A generalised error model

when the system experiences a fading channel, the model incorporates impairments at both sides of the wireless channel. To deal with the nonstationary behaviour of the RF-impairments, a two-step model is introduced where the parameters for the error model are diﬀerent in the transient and stable phase of the reception. Finally, expressions are derived to map this error model to the probability of erroneous symbol detection for a zero-forcing (ZF) based MIMO OFDM system. The outline for the rest of this chapter is as follows. First Section 7.2 introduces the proposed error model and reviews the inﬂuence of the different impairments treated in the previous chapters and their mapping to the model. The performance of a MIMO OFDM system experiencing the error model is, subsequently, derived in Section 7.3. Section 7.4 compares these analytical ﬁndings with results from Monte Carlo simulations. Furthermore, the prediction is compared with that of the EVMbased model. Finally, conclusions are drawn in Section 7.5.

7.2

Error model

In this chapter we will consider the same MIMO OFDM system and use the same notation as applied throughout this book. In this chapter, however, we consider the addition of two blocks modelling the RF impairments in the TX and RX, i.e., εTX and εRX , respectively. The baseband model for the system is depicted in Fig. 7.1.

s Nt .. . 1 .. 0 .

u

..

. .. .

IDFT

Add CP P/S

..

.

εTX

Nt .. . 1

Nr .. .1

G

εRX

Nc − 1 Nt ..

MIMO

.1 .. 0 .

detector

..

. .. .

Nc − 1 ˜s

DFT

..

CP Removal, S/P y

.

Channel Estimation

. ···

Nc − 1 x

..

Figure 7.1. ments.

Nr .. . 1 .. 0 .

..

v

.

Baseband model for a MIMO OFDM system with TX and RX impair-

247

7.2 Error model

We recall that the Nt Nc ×1 frequency-domain symbol vector s is transformed into the time-domain Nt Ns ×1 transmit vector u by the OFDMprocessing, where u was deﬁned in (2.42). The signal then passes though three blocks modelling the TX impairments (εTX ), the propagation channel (G) and RX impairments (εRX ), respectively. At the RX, the signal experiences AWGN, denoted by v, yielding the Nr Ns ×1 time-domain RX vector y. This vector is used to estimate the MIMO channel response and, after OFDM processing, construct the Nr Nc ×1 frequency-domain RX vector x. Subsequently, the channel estimate and x are used to retrieve the estimate of the transmitted data ˜s.

7.2.1

TX and RX impairment model

The inﬂuence of the proposed error model is schematically represented in Fig. 7.2. The ﬁgure depicts the non-impaired complex symbol s, given by the white circle, and the impaired equivalents s˘, which lie in a noise cloud around the mean E(˘ s), given by the black circle. We note that there thus exist a bias, since E(˘ s) = s. To allow for more freedom in modelling than inherently assumed by the classical EVM model, this error model introduces, next to the additive error term, a scaling and phase rotation of the mean, i.e., E(˘ s) = κejϕ s = χs. When we, subsequently, rewrite the inﬂuence of the impairment blocks εTX and εRX on the frequency-domain MIMO OFDM signal vectors, we ﬁnd that the inﬂuence of the error model on the kth subcarrier of the mth symbol of the signal vector a is given by ˘m (k) = χx,m (k)am (k) + ηx,m (k), a

(7.1)

where x∈{t, r} for the TX and RX impairment model, respectively. Here χx,m (k) is a diagonal matrix modelling the scaling and phase shift and

Figure 7.2.

Schematic representation of the proposed error model.

248

7 A generalised error model

ηx,m (k) models the additive error vector with zero-mean entries. To be able to distinguish between the inﬂuence of TX and RX impairments, the error model is implemented at both sides of the communication link. Then, using (7.1) and (2.45), we can deﬁne the resulting frequencydomain RX signal vector as x = χr H˘s + ηr + n = χr H (χt s + ηt ) + ηr + n,

(7.2)

where we omitted the symbol index m to increase readability. The additive TX and RX impairment vectors are given by ηt and ηr , respectively. The amplitude and rotational error are modelled in the Nc Nt ×Nc Nt diagonal matrix χt and the Nc Nr ×Nc Nr diagonal matrix χr for the TX and RX, respectively. Note that we here separately modelled the commonly modelled additive white Gaussian noise (AWGN) source, denoted by n, which enables the separation of the inﬂuence of the AWGN and the other impairments. For simplicity, we will assume, in the remainder of this chapter, that all carriers for one TX/RX branch experience the same phase and amplitude error, i.e., χt = (INc ⊗ χt ) with χr = (INc ⊗ χr ) with

χt = diag{χt,1 , . . . , χt,Nt }, χr = diag{χr,1 , . . . , χr,Nr }.

(7.3) (7.4)

Under these assumptions, (7.2) can be rewritten to x = (INc ⊗ χr )H((INc ⊗ χt )s + ηt ) + ηr + n.

(7.5)

The inﬂuence of the error model in (7.5) on the reception of a 16-QAM modulation is illustrated in Fig. 7.3. Here the results are shown for a 1×1 system experiencing an AWGN channel and χt,1 χr,1 = 0.8ejπ/18 . The original 16-QAM modulation is given by white circles, while the means of the resulting symbols are given by black markers. The received symbols, including the additive errors, are given by the scatter plots.

7.2.2

Mapping of RF impairments

In this subsection we demonstrate the mapping of the inﬂuence of the impairments treated in Chapters 4 to 6 to the error model proposed in Section 7.2.1. Phase Noise We ﬁrst consider phase noise (PN), which was in Chapter 4, see (4.23), shown to results in a common rotation for all detected carriers on all branches, i.e., the common phase error (CPE). Furthermore, an additive term occurs due to leakage of the diﬀerent carriers into each other,

249

7.2 Error model 1.5 1

Q

0.5 0 −0.5 −1 −1.5 −1.5

−1

−0.5

0 I

0.5

1

1.5

Figure 7.3. Inﬂuence of the error model on the reception of 16-QAM symbols. The original 16-QAM modulation is given by white circles, while the means of the resulting symbols are given by black markers. The received symbols, including the additive errors, are given by the scatter plots.

i.e., the inter-carrier interference (ICI). We showed that for systems experiencing fading channels the structure and inﬂuence of the ICI depend on whether it is induced by TX or RX PN. The CPE can now be modelled in the phase error ϕ of the proposed error model and the ICI in the additive terms ηt and ηr . IQ imbalance From Chapter 5 it can be concluded that IQ imbalance in a (MIMO) OFDM system results in leakage from the mirror subcarrier signal into, and a phase rotation and attenuation of, the signal on the desired subcarrier, see (5.28) and (5.35). Obviously, the ﬁrst eﬀect can be modelled by the additive error terms, while the amplitude and phase error are well represented by χt and χr . Nonlinearities For nonlinearities it was concluded in Chapter 6 that an OFDM signal experiencing an AM-AM nonlinearity results in scaling by α of the desired signal plus an additional (independent) noise term, see (6.37) and (6.64). When also AM-PM nonlinearities occur, α is a complex parameter and, hence, an additional phase error occurs. Again the phase and amplitude error can be modelled in our error model by χt and χr and the distortion noise can be modelled in the additive terms ηt and ηr . Multiple impairments From the above we conclude that the inﬂuence of the regarded impairments can all be modelled as aﬃne linear functions, which means that, by deﬁnition, their superposition is again aﬃne linear. The combined

250 Nr ..

7 A generalised error model

.1

εRX

Nr .. .

1

≡

Figure 7.4.

Nr .. . 1

Nr NL

..

.

PN

..

.

IQ imb.

..

. 1

Superposition of the RX impairments.

inﬂuence of the studied impairments can thus also be modelled by the proposed error model. This is illustrated for the RX impairments in Fig. 7.4, where the RX error model εRX in Fig. 7.1 is replaced by the superposition of a nonlinearity, a PN source and IQ imbalance. It is then easily veriﬁed using the derivations in the previous chapters, that the impaired received signal vector for the kth subcarrier of the mth symbol can be written as xm (k) = χr,m (k)H(k)sm (k) + ηr,m (k),

(7.6)

where the amplitude scaling and phase rotation are modelled by χr,m (k), which is given by the Nr ×Nr diagonal matrix χr,m (k) = γ0,m K1 α, and ηr,m (k) is the Nr ×1 additive impairment vector given by ηr,m (k) = K1 γ0,m em (k) + ξm (k) ∗ ∗ αH∗ (−k)s∗m (−k) + e∗m (−k) + ξm +K2 γ0,m (−k) .

(7.7)

(7.8)

We recall that γ0,m and ξm denote the phase noise caused CPE and ICI, respectively. The nonlinearity parameters are α and em , which model the amplitude/phase distortion and the nonlinear distortion noise, respectively. The RX IQ imbalance parameters are denoted by K1 and K2 . Our conclusion that the error model can also be applied for the superposition of multiple non-idealities is, thus, conﬁrmed by (7.6) to (7.8).

7.2.3

Nonstationarity of the impairments

In this subsection we propose a model to deal with the nonstationarity of RF impairments. It is, therefore, important to understand how this time-dependent behaviour originates, which, therefore, is illustrated in this subsection. Note that many wireless systems, like, e.g., systems based on the IEEE 802.11 standard, operate in packet-based transmissions. This is governed by the higher OSI layers, which are based on the

7.2 Error model

251

TCP/IP protocol. This means that a user terminal (UT), in a downlink scenario, at one point might receive a long burst of packets, but also that at another moment it might not receive any packet for long time intervals. Keeping the UT fully functional during this idle time is very power-consuming. Therefore, to design an energy eﬃcient implementation for an UT, it is important to incorporate some kind of power-save mode. In this mode the UT periodically senses the medium to determine whether it detects a packet, but is not fully functional to receive the packet. When the UT identiﬁes the start of a packet, it switches to the fully functional reception mode. During the power-save mode, many power consuming parts in the RF-subsystem are switched oﬀ. Since the RF-subsystem includes several ﬁlters, it will clearly take a certain time period for the system to settle in a stable mode when it is switched on, making the experienced imperfections nonstationary. Furthermore, in an IEEE 802.11 network setup, an acknowledgement for a packet has to be transmitted/received in several µs after reception/transmission of the previous packet. That is why the device has to switch between RX/TX mode very rapidly. The allowed time is often referred to as TX-RX turn around time. Again, here it would take some time period for the RF-subsystem to settle after the switching. The above mentioned nonstationarity of the imperfections results in an interval with severe RF-impairments and a period where they are less of an issue. This is the motivation for modelling the impairments in the ﬁrst phase of the reception with other settings than in the second phase. However, for simplicity and applicability, we apply the same model in both phases, as proposed above, and only change the model parameters. As an example of this nonstationarity, we regard the frequency synthesis (FS) of the RF or intermediate frequency stages in the radio system. This frequency synthesis is generally based on a phase-locked loop (PLL). It is well known that the locking time of a PLL, when it switches from one frequency to another, depends on the frequency jump and the required accuracy. The settling time grows with increasing jump and accuracy [167, pp. 1-39]. MIMO OFDM systems require stringent phase noise speciﬁcations as was shown in Chapter 4, but the settling time to achieve a high phase noise accuracy in one step would be too high and part of the preamble would not be correctly received. This is why often a two-step approach is chosen [167]. In this method a ﬁrst stage is applied where the accuracy requirement is low and the locking is fast. Using these settings, the ﬁrst part of the packet is detected. Then, to increase accuracy and to meet the PN speciﬁcations, the frequency accuracy is increased by changing the loop-ﬁlter settings of the PLL in the second stage. Since

252

7 A generalised error model

the frequency jump is much smaller now, the settling time is manageable. As a result, the PN is worse in the reception of the ﬁrst part of the packet, mapping to our two-step model as proposed above.

7.2.4

Transmission structure

Here we shortly review the packet structure for a MIMO OFDM system applying coherent detection. It was shown in Chapter 2 that the transmission consists of a preamble phase and a data phase. We recall that the preamble part is used in the receiver to set the automatic gain control, to perform time- and frequency synchronisation and to acquire estimates of the MIMO channel matrices for the diﬀerent subcarriers. In the data phase the transmitted data is estimated from the received signal, as was described in Section 2.3.1. It is clear that there is a parallel with the two periods in the RF impairments, which we have identiﬁed in Section 7.2.3. Therefore, the preamble phase is now deﬁned to be the part where the RF-impairments are severe and the data phase is the second part, where the inﬂuence of imperfections is less.

7.3

Performance evaluation

In a next step, it is important to understand the link between the introduced error model and the ﬁnal system performance measure symbolerror rate (SER). Therefore, we will illustrate the mapping to system performance by the derivation of the probability of error for a MIMO OFDM system experiencing the proposed error model. Although the model proposed above is more general applicable, we will in the remainder of this chapter, to make the performance evaluation more tangible, make the following assumptions: We assume that the elements of ηt and ηr are zero-mean complex Gaussian distributed and that their covariance matrices are given by σt2 I and σr2 I, respectively. Additionally, we assume that the diﬀerent additive impairments are independent. We assume that the rotational and scaling errors are equal on all TX and RX branches. We assume that the inﬂuence of the impairments experienced during the preamble only aﬀect the channel estimates. The phase and amplitude errors are assumed to be constant during the preamble period. Under these assumptions (7.5) simpliﬁes to x = χt χr H(s + ηt ) + ηr + n,

(7.9)

7.3 Performance evaluation

253

where the elements of ηt are zero-mean complex Gaussian distributed with a variance that is κ2t = |χt |2 times smaller than that of the elements of ηt .

7.3.1

Preamble phase

First we regard the inﬂuence in the preamble phase, where the Nr Nc ×Nt received signal block is given by Xp = χt,p χr,p H(Sp + Et,p ) + Er,p + N,

(7.10)

where the subscript “p” refers to the preamble period. Furthermore, the Nt Nc ×Nt preamble block is given by Sp = [sp,1 , . . . , sp,Nt ] and Xp , Et,p , Er,p and N have the same structure as Sp , and Et is the block version of ηt . The estimate of the MIMO channel matrix is now found by leastsquares estimation (LSE), i.e., by multiplication with the pseudo-inverse of the preamble block Sp and is given by ˜ = Xp S† = χp H + (χp HEt,p + Er,p + N) S† H p p = χp H + E,

(7.11)

where χp = χt,p χr,p and E are the multiplicative and additive error in the channel estimate, respectively. To have a better estimate of the channel, multiple preamble blocks can be used to average over and improve the channel estimate. When P training blocks are used, and the element of E are i.i.d., this yields a P -times smaller variance of the elements of E.

7.3.2

Data phase

During the data phase the received signal x is given by (7.9). We estimate the transmitted signal from our received signal by multiplying it with the pseudo-inverse of the estimated channel matrix in (7.11). When we deﬁne χ = χr χr , the estimates of the transmitted signal are by ˜ †x ˜s = H

† = H(χp I + H† E) (χH(s + ηt ) + ηr + n) −1 † † † = χ−1 p (I + χp H E) (χ(s + ηt ) + H (ηr + n)) −1 † † ≈ χ−1 p (I − χp H E)(χ(s + ηt ) + H (ηr + n)) χ s + δ, = χp

(7.12)

where we apply an approximation of the pseudo-inverse using the ﬁrst two terms of its Taylor expansion. Here the additive error in the

254

7 A generalised error model

estimation of s is given by χ χ † χ † † −2 † † δ= η + χ−1 p H (ηr + n) − 2 H Es − 2 H Eηt − χp H EH (ηr + n). χp t χp χp (7.13) It is then found that δ is a zero-mean Gaussian variable and that for small additive errors, its covariance matrix Ω is well approximated by Ω ≈ σk2 I + σ2 (HH H)−1 , where κ2 σk2 = 2 κp σ2

1 = 2 κp

4 4

2 σt,p σt2 + κ2t P κ2t,p

(7.14)

5 ,

2 κ2 σr,p P + κ2 2 σn + σr2 + P P

(7.15) 5 .

(7.16)

We note that we left out terms which included the multiplication of two additive error terms in the approximation for (7.14). Considering (7.12), it can be concluded that the estimated symbols are given by a scaled and rotated version of the transmitted symbols plus an additive error term, as was already schematically depicted in Fig. 7.2.

7.3.3

Probability of error

To calculate the probability of error, we ﬁrst consider the additive error terms. This results in an eﬀective signal-to-noise ratio (SNR) for the nt th branch and the kth subcarrier, aﬀected by a given channel, imperfect channel knowledge and TX and RX non-idealities, which is given by σs2 σk2 + σ2 [(HH (k)H(k))−1 ]nt nt −1 Nt [(HH (k)H(k))−1 ]nt nt 1 ℘ k ℘λ = + = . ℘k ℘ ℘k + ℘λ

℘ =

(7.17)

Here [A]mm denotes the mth diagonal element of matrix A and ℘ . (7.18) ℘λ = H Nt [(H (k)H(k))−1 ]nt nt Furthermore, the expressions for the transmitter and receiver SNR in (7.17) are deﬁned by ℘k = σs2 /σk2 , ℘ = Nt σs2 /σ2 ,

(7.19) (7.20)

255

7.3 Performance evaluation

respectively. When the channel matrix H has i.i.d. complex Gaussian elements, often referred to as a Rayleigh faded channel, ℘λ in (7.17) is chi-square distributed with 2R = 2(Nr − Nt + 1) degrees of freedom. The probability density function of ℘λ is denoted by p℘λ (ρ) and given by (6.44). We note that ℘k is deterministic. The SER for the nt th branch and kth subcarrier of an uncoded system can then be found by ∞ Pe = Pe,M -QAM (ρ; ℘k , χp , χ)p℘λ (ρ)dρ, (7.21) 0

where Pe,M -QAM (ρ; ℘k , χp , χ) denotes the SER expression for an M -QAM system, experiencing the proposed error model with parameters ℘k , χp and χ. The average SER is now found by averaging Pe over the diﬀerent subcarriers, TX branches and symbols. To illustrate the inﬂuence of the rotation and scaling of the con stellation by χ = χ/χp = κ ejϕ , as described in (7.12), we derive Pe,M -QAM (℘, χ ) for the 4-QAM modulation. For this purpose, the inﬂuence of χ on two of the 4-QAM symbols, s1 and s2 , is schematically depicted in Fig. 7.5. In the ﬁgure the original constellation points are depicted by white dots, while the impaired symbols, s1 and s2 , are given by black dots. In the calculation we will, as in the previous chapters, use an approach where we separate the detection of the I and Q part of the signal, i.e., the real and imaginary part of the signal, yielding the detection of two parallel 2-PAM signals. Here we will exploit, similarly as in Chapters 5 and 6, the independence of the real and imaginary part of the estimated signal for a given χ and SNR ℘. Q

s2

s1

q1

s1

q 1, q 2

s2

q2 ϕ ϕ I

i2 i2

0

i1

i1

Figure 7.5. Inﬂuence of the multiplicative error on the reception of the 4-QAM symbols s1 and s2 .

256

7 A generalised error model

For the real part of the signal, we can conclude from Fig. 7.5 that √ i1 = R{s1 } = |s1 | cos(π/4) = |s1 |/ 2 = d, (7.22) √ (7.23) i2 = R{s2 } = |s2 | cos(3π/4) = −|s2 |/ 2 = −d, i1 = R{s1 } = R{χ s1 } = κ |s1 | cos(π/4 + ϕ ) √ (7.24) = κ d 2 cos(π/4 + ϕ ), i2 = R{s2 } = R{χ s2 } = κ |s2 | cos(3π/4 + ϕ ) √ (7.25) = κ d 2 cos(3π/4 + ϕ ), similarly we ﬁnd for the not depicted points √ i3 = R{s3 } = κ d 2 cos(5π/4 + ϕ), √ i4 = R{s4 } = κ d 2 cos(7π/4 + ϕ).

(7.26) (7.27)

The probability of an erroneous detection of the I-signal is now given by Pe,I 2-PAM =

=

1 P(i1 + δI < 0) + P(i2 + δI > 0) 4

+P(i3 + δI > 0) + P(i4 + δI < 0)

√ 1 P(δI > κ d 2 cos(π/4 + ϕ )) 2 √ +P(δI > κ d 2 sin(π/4 + ϕ )) . (7.28)

Since the error term δI = R{δ} is Gaussian distributed, (7.28) can be rewritten as & 1% Pe,I 2-PAM,Es (℘) = Q(c1 2℘) + Q(c2 2℘) , (7.29) 2 √ √ where √ ℘ = Es /N0 , c1 = 2κ cos(π/4 + ϕ ), c2 = 2κ sin(π/4 + ϕ ) and d = 2Es . It is easily veriﬁed from the symmetry of the 4-QAM constellation that the SER for the Q-signal Pe,Q2-PAM (℘) is equal to (7.29). Hence, due to the independence of the estimated I- and Q-signal, the SER for the distorted 4-QAM signal is given by Pe,4-QAM (℘) = 1 − (1 − Pe,I 2-PAM,Es /2 (℘))(1 − Pe,Q2-PAM,Es /2 (℘)) = 1 − (1 − Pe,I 2-PAM,Es /2 (℘))2 = 2Pe,I 2-PAM,Es /2 (℘) − (Pe,I 2-PAM,Es /2 (℘))2 √ √ Q(c1 ℘)Q(c2 ℘) √ √ = Q(c1 ℘) + Q(c2 ℘) − 2 √ √ Q2 (c1 ℘) + Q2 (c2 ℘) . (7.30) − 4

7.4 Numerical results

For high SNR values, (7.30) can be approximated by √ √ Pe,4-QAM (℘) ≈ Q(c1 ℘) + Q(c2 ℘).

257

(7.31)

We note that for the case that there is only AWGN, i.e., no other impairments, c1 = c2 = 1 and (7.30) reduces to the well-known SER expression √ √ for 4-QAM, i.e., 2Q( ℘) − Q2 ( ℘). Similarly to the derivation in Chapters 5 and 6, we can consider a case with only TX imperfections and one with only RX imperfections. For the ﬁrst case the SER is found by substituting ℘ = ℘k into (7.30) or the high SNR approximation of (7.31). For the RX impairment case, we substitute (7.31) into (7.21) and apply the ﬁndings of Section 5.3.2.2. We then ﬁnd that the SER expression for a MIMO system applying 4-QAM and experiencing a Rayleigh faded channel and the proposed error model is given by / . √ (1 − ℘ c3 c1 ) 1 1 3 −c21 ℘ Pe ≈ ,R + ; ; 2 F1 2 2 2 2 2 / . √ (1 − ℘ c3 c2 ) 1 1 3 −c22 ℘ ,R + ; ; , (7.32) + 2 F1 2 2 2 2 2 where we assumed moderate 9 phase1 error such that c1 > 0 and c2 > 0. (R− 2 )! and 2 F1 denotes the hypergeometHere we have deﬁned c3 = π2 (R−1)! ric function. For R = 1 we can simplify (7.32) in a similar way as in (5.80).

7.4

Numerical results

In this section results from the SER expressions derived in the previous section are compared with results from Monte Carlo simulations. As a test case, a MIMO extension of the IEEE 802.11a standard was studied. The applied parameters are: modulation is 4-QAM, bandwidth is 20 MHz, number of subcarriers Nc = 64 and no coding is applied. The channel was modelled to have i.i.d. complex Gaussian elements. We now assume that the additive impairments during the preamble 2 = qσ 2 and period are q times worse than in the data phase, i.e., σt,p t 2 = qσ 2 . We depict the SER as a function of the RX SNR for a σr,p r system with only AWGN receiver noise, i.e., Nt σs2 /σn2 , to clearly reveal the inﬂuence of the impairments. Only one training block is used for the estimation of the channel matrices, thus P =1. Figure 7.6 shows the results for a 2×2 and 2×4 system impaired by additive white Gaussian RX noise n, an amplitude error κ and a phase error ϕ . The system does not experience additive RX and TX impairments, i.e., σr = σt = 0, additionally κp = 1. The ﬁgure compares the

258

7 A generalised error model 100

100

κ =1 κ =0.6

κ =1 κ =0.6 10−1

SER

SER

10−1

10−2

10−3

10−3

10−4

10−2

0

10 20 30 40 SNR per RX antenna (dB)

(a) 2×2

50

10−4

0

10

20

30

40

50

SNR per RX antenna (dB)

(b) 2×4

Figure 7.6. SER performance of a 2×2 and 2×4 system experiencing the multiplicative part of the error model. Theoretical results are depicted by lines and the simulations results by markers. Results are depicted for amplitude errors κ = 1 and κ = 0.6 and phase errors ϕ = 0 (solid lines), ϕ = π/10 (dashed lines) and ϕ = π/5 (dotted lines).

results from simulations with the analytical results found by combining (7.21) and (7.30). It is obvious from Fig. 7.6 that there is a close agreement between the theoretical and simulation results for both MIMO conﬁgurations. It can, furthermore, be concluded that for high SNR values both the phase and amplitude error yield an SNR shift of the SER curves and no ﬂooring occurs. This can be understood from the high SNR SER approximation in (7.31), which shows an SNR scaling that depends on the amplitude and phase errors. In Fig. 7.7 results are depicted for the same MIMO conﬁgurations, for systems experiencing both additive and multiplicative TX and RX impairments. The imperfections during the preamble phase are twice as high as during the data phase, i.e., q = 2. Again there is close agreement between the analytical results and the simulation results, which proves the applicability of the theoretical results for performance prediction. We conclude that for the additive errors ﬂooring occurs in the SER curves. The ﬂooring level depends on the location of the additive impairment, i.e., in the TX or RX. Here the ﬂooring, additionally, depends

259

7.4 Numerical results κ κ κ κ

100

=1, ϕ =0, σt =0.1, σr =0 =1, ϕ =0, σt =0, σr =0.1 =0.6, ϕ =π/10, σt =0.1, σr =0 =0.6, ϕ =π/10, σt =0, σr =0.1

10−1

SER

10−2

10−3

10−4

10−5

0

10

20

30

40

50

60

70

Average SNR per RX antenna (dB)

Figure 7.7. SER performance of a 2×2 (dashed lines) and 2×4 (solid lines) system for diﬀerent additive and multiplicative TX/RX impairments with q = 2. Theoretical results are depicted by lines and the simulations results by markers. 100 10−1

SER

10−2

10−3 10−4 κ=1, ϕ =0 κ=0.6, ϕ =0 κ=1, ϕ =π /10 κ=0.6, ϕ =π /10

10−5 10−6 0

10 20 30 Average SNR per RX antenna (dB)

40

Figure 7.8. Predicted SER performance of a 2×4 system experiencing the proposed error model. Results are given for the derived analytical expression in (7.21) (solid lines) and the EVM approach (dashed lines).

260

7 A generalised error model

on the experienced multiplicative errors. As for phase noise and nonlinearities in Fig. 4.13 and Fig. 6.8, respectively, we can conclude that TX additive impairments result in a SER ﬂoor for high SNR values that does not depend on the MIMO conﬁguration, while it does for additive RX impairments. Finally, we want to prove our hypothesis that the predictability of the system performance based using EVM is low. Therefore, Fig. 7.8 depicts analytical SER prediction results for a system experiencing the proposed error model. It compares the analytical results of Section 7.3.3 with results that are found using the EVM-approach. For the latter, the MSE of the symbols is calculated and used to calculate the eﬀective SNR. The EVM is found as the square root of the MSE in the estimated symbols for a system not experiencing a channel, i.e., where the channel matrix can be expressed by the identity matrix. For this case, the MSE for the nt th TX branch is found using (7.12) and given by snt − snt |2 = E |χnt − 1|2 + |δnt |2 EVM2nt = E |˜ = κ2 + 1 − 2κ cos(ϕ) + σk2 + σ2 .

(7.33)

The SER for the nt th TX branch is then found by substituting the eﬀective SNR ℘0 = ℘eﬀ = σs2 /EVM2nt into (7.21) for a non-impaired system, i.e., with χp = χ = 1 and ℘k = ∞. In the simulations for Fig. 7.8 we used κp =1 and σt = σr = 0, i.e., the system only experiences multiplicative imperfections and RX AWGN. The results for the EVM-based SER evaluation are given in dashed lines and the results for the proposed error model are given in solid lines. It can be concluded from Fig. 7.8 that the EVM-based approach achieves the same results as derived for the error model in Section 7.3.3 for a system only experiencing additive errors. This is because the EVM2 is equal to the variance of the detection error for this case, i.e., the effective SNR is suﬃcient to correctly model the inﬂuence of the additive impairments. When multiplicative errors occur, however, the EVMbased approach largely overestimates the inﬂuence of the imperfections. Flooring occurs for the estimated curves, while this is not the case for the actual SER which results in an SNR shift. This can be explained by the fact that EVM2 or MSE no longer equals the second moment of the error, since the multiplicative error results in a bias in the symbol estimate. This supports the conclusion that a performance evaluation merely based on the EVM-measure is unsuitable for systems experiencing RF impairments. Alternatively, one could consider the here proposed error model as an extension to the EVM-model, which is preferred above the conventional model when multiplicative errors occur.

7.5 Conclusions

7.5

261

Conclusions

In this chapter we made a ﬁrst step in coming to a measure for the aggregate severeness of RF impairments. To this end, we introduced an approach for modelling the combined inﬂuence of diﬀerent radio frontend impairments in wireless systems. The model incorporates an additive and a multiplicative error term, which are modelled for both transmitter and receiver. The multiplicative errors result in a phase shift and amplitude scaling of the received signal constellation. The additive errors result in a noise cloud around the detected symbols. Furthermore, this method models the inﬂuence of the impairments in two steps: a ﬁrst period with severe imperfections and a second period with less inﬂuence of the non-idealities. This chapter demonstrated that the diﬀerent non-idealities treated in the previous chapters of this book, i.e., phase noise, IQ imbalance and nonlinearities, and moreover their combined inﬂuence, properly map onto the proposed error model. Furthermore, it was illustrated, both analytically and numerically, that the proposed error model can be mapped to the performance of the system. To this end, the study considered the symbol-error rate for a ZF-based multiple-antenna OFDM system. Additionally, the generally applied system performance measure error vector magnitude (EVM) is shown to be unsuitable for probability of error evaluation in systems experiencing RF impairments, since it leads to overestimation of the imperfection performance impact. The traditional EVM requirement, as applied in many standards, therefore leads to overdesign for this kind of systems and as such to unnecessarily high manufacturing costs. Interesting extensions to the here proposed model might lie in the statistical characterisation of the additive and multiplicative term. Note that here only deterministic multiplicative errors where considered, but that it might be useful to considered a range of realisations. Such an extension would also enable the design of a performance characterisation and measurement method for wireless systems with RF imperfections.

Chapter 8 DISCUSSION AND CONCLUSIONS

8.1

Summary and conclusions

The potential of multiple-input multiple-output orthogonal frequency division multiplexing (MIMO OFDM) as the basis for next-generation high data rate broadband wireless systems has been reported in many contributions over the last few years. A ﬁrst reason is the high spectral eﬃciency, which is achieved by both techniques: OFDM places the subcarriers very close together and MIMO applies multiple spatial branches to transmit multiple parallel data streams. A second reason can be found in the possibility to operate MIMO OFDM systems in dispersive multipath environments. It is shown that these multiple-antenna OFDM systems even beneﬁt from the dispersiveness of the multipath channel. Many challenges, however, remain when regarding the eﬃcient implementation of this kind of systems, due to their extremely high sensitivity to synchronisation errors and RF front-end imperfections. This book studies, and proposes tools to remove, several of the major hurdles on the path to successful application of these promising multiple-antenna and multicarrier techniques. In the remainder of this section, we will summarise and discuss the conclusions of this book grouped by the main subject areas: synchronisation for MIMO OFDM-based wireless systems, the veriﬁcation of the MIMO OFDM concept by means of a test system, baseband equivalent RF impairment models, applicable in the MIMO OFDM context, the performance impact of RF imperfections in multiple antenna OFDM,

264

8 Discussion and conclusions

mitigation algorithms/approaches for the inﬂuence of front-end impairments, a generalised error model mapping the impact of the key performance limiting non-idealities.

8.1.1

Synchronisation for MIMO OFDM systems

Synchronisation is an essential task for any digital communication system and a prerequisite for reliable reception of transmitted data. Although blind synchronisation algorithm are being proposed, in practical systems the synchronisation tasks are generally enabled by inserting known training data in the transmission. For packet-based transmissions this is often placed in front of the actual data packet and it is referred to as the preamble. In Chapter 2 diﬀerent preamble structures were proposed for a MIMO OFDM system. These preambles are designed such that they provide enough information for accurate synchronisation, but at the same time impose only a limited overhead. In the proposed designs, backward compatibility with other, non-MIMO systems was taken as an extra design constraint. With more and more wireless systems appearing in frequency bands that are currently already occupied by other systems, this will become an increasingly important design constraint. Based on these preamble structures, methods for frequency synchronisation, symbol timing and MIMO channel estimation were presented in Chapter 3, to provide the reader with insights in the typically required synchronisation processing in the implementation of a MIMO OFDM system. The introduced frequency synchronisation algorithm, which is based on correlation between repeated preamble symbols, reveals a linear and cubical increase in performance with the number of receiver antennas and the preamble length, respectively. It was additionally concluded that the algorithm eﬀectively exploits the spatial diversity in fading channels. The applicability was proven by simulations revealing that a system applying the introduced synchronisation only yielded a minor performance degradation compared to a perfectly synchronised system. Estimation of the MIMO channel matrix was evaluated for the diﬀerent preamble structures, which revealed that for low timing oﬀsets and rms delay spreads eﬃcient structures can be applied which do not increase the overhead compared to a single-input single-output (SISO) system. For longer channels, however, less eﬃcient structures are required to achieve good performance. Combinations of the proposed preamble structures then become promising to balance performance with overhead level.

8.1 Summary and conclusions

265

In symbol timing for MIMO OFDM, the special case of systems applying cyclic preﬁxes that are short compared to the channel length was regarded. Diﬀerent approaches to symbol timing were presented and evaluated. From this evaluation it was concluded that the algorithm designed to minimise the inter-symbol interference (ISI) performed best for accurate channel knowledge and double-cluster channels. For single-cluster channels and less reliable channel knowledge, the reduced-complexity algorithm, however, outperformed the ISI-minimisation algorithm.

8.1.2

Multiple-antenna OFDM proof of concept

With the aim of proving the applicability of the MIMO OFDM concept in a realistic propagation environment, an implementation of a wireless LAN system based on this concept was developed, as detailed in Chapter 3. The system has 3 transmitter (TX) and 3 receiver (RX) branches and operates in the 5 GHz band. The transmission scheme was based on a MIMO extension of the IEEE 802.11a standard. For the baseband implementation of the receiver processing the synchronisation approaches presented in Chapter 3 formed the basis. Indoor measurements in a typical oﬃce environment revealed that the achieved performance of the test system is slightly worse than as was expected from the idealised simulations. For a given range the implementation of the 3×3 multiple-antenna OFDM system achieved a throughput that is on average 2 times higher than that of its 1×1 counterpart. The discrepancy between the empirical and theoretical results can likely be attributed to correlations between the received signals, caused by either the propagation channel or mutual coupling between the transceiver branches. Another probable explanation may lie in the impact of front-end impairments on system performance, which were not included in these (idealised) system evaluations.

8.1.3

Behavioural RF impairment modelling

To elucidate the performance degradation observed in the system implementation presented in Chapter 3 and as the basis of a theoretical sensitivity analysis of MIMO OFDM systems to RF front-end impairments, a set of baseband equivalent models for the most important analogue front-end non-idealities was presented in Chapters 4 to 6. In this approach, not the actual RF front-ends were modelled, but the inﬂuence of their imperfections on the baseband signals. To provide insightful and meaningful results, models were chosen with a minimum number of, essential, parameters. A major advantage of these behavioural models is that they generally do not depend on the speciﬁc front-end design, but

266

8 Discussion and conclusions

do have a direct link with the resulting performance measures of these designs. As such, a wide range of implementations can be evaluated using the same models. All models were applied for both the TX and RX of the system, to be able to distinguish between their impact. For phase noise a free-running carrier model was selected in Chapter 4, characterised by the −3 dB bandwidth of the oscillator. A common oscillator for the diﬀerent MIMO branches was assumed, yielding the same phase noise on the diﬀerent MIMO branches. IQ imbalance was modelled in Chapter 5 using an asymmetric mismatch model, where the I-branch was modelled as perfect and a phase and an amplitude error were modelled in the Q-branch. Additionally, a model for ﬁlter mismatch, resulting in frequency selective IQ imbalance, was introduced in this work. For the nonlinearities various existing models were reviewed and their similarity was shown in Chapter 6. It was shown how these models map to commonly used measures for RF system characterisation. The models for IQ imbalance and nonlinearities were implemented independently for the diﬀerent MIMO branches, since these impairments can vary for the diﬀerent front-ends due to e.g. process spread.

8.1.4

Performance impact of front-end imperfections

Phase noise (PN) was shown, in Chapter 4, to yield a rotation of all carriers, named common phase error (CPE), and an additive error term due to leakage between the subcarriers, named inter-carrier interference (ICI). In contrast to commonly assumed in the literature, it was shown and proven in this work that this ICI term can not be accurately modelled as an additive Gaussian noise source. The derived ICI limit distribution was shown to exhibit thicker tails than the normal distribution. Because of this, the commonly applied Gaussian approach leads to a severe underestimation of the error rate performance, while estimates using the derived limit distribution generate accurate predictions. Further, the diﬀerence between the inﬂuence of TX and RX-caused PN on the error in signal detection was derived analytically. It was shown that for TX PN the power of this error term is equal for all MIMO conﬁgurations. For the case of per subcarrier independent Rayleigh faded channels, the RX PN is concluded to have less impact than TX PN when the number of RX branches is larger than twice the number of TX branches. IQ imbalance was studied for the popular direct-conversion architecture. It was show in Chapter 5 that IQ mismatch in a MIMO OFDM transceiver yields both leakage between a pair of DC-mirrored subcarriers, and a scaling and rotation of the signal component. Compact

8.1 Summary and conclusions

267

expressions were derived for the symbol-error rate (SER) of MIMO OFDM systems applying M -QAM modulation and experiencing either TX or RX IQ imbalance. These analytical expressions were shown, using simulation results, to accurately predict the performance of such systems over a wide range of impairment parameters. Overall it can be concluded that in fading channels the performance impact of RX IQ imbalance is more severe than that of TX IQ imbalance, since the latter does not results in SER performance ﬂoors for moderate values of mismatch. Nonlinearities were shown, in Chapter 6, to result in a scaling of the transmitted MIMO OFDM signals plus an additive distortion term. The impact hereof on the SER of M -QAM based systems was derived analytically. Again, the inﬂuence of TX and RX nonlinearities was studied separately, to reveal the diﬀerence between their impact. Compact expressions for the SER were found for both AWGN and Rayleigh faded MIMO channels. These results were shown to accurately predict the performance of nonlinear impaired multiple-antenna OFDM systems using numerical simulations. The outcomes support the conclusion that TX nonlinearities cause ﬂooring, for high SNR values, at a level that is independent of the MIMO conﬁguration. For RX nonlinearities, however, the performance in this high SNR region does depend on the MIMO conﬁguration. The analytical performance results derived in this book constitute a powerful tool for system designers to derive the tolerable levels of a speciﬁc RF impairment. This level might be the level to be achieved by front-end design, or by the combination of the front-end and the mitigation algorithm design. Although the results were derived for multiple antenna OFDM, they can also be applied for conventional single-antenna OFDM systems, which can be seen as a special case of MIMO OFDM.

8.1.5

Mitigation approaches for RF impairments

To illustrate the possibilities to digitally mitigate the inﬂuence of phase noise, a compensation approach for the CPE term in MIMO OFDM system experiencing channel correlation was proposed in Chapter 4. For the estimation of the CPE a maximum-likelihood estimator (MLE) was proposed together with an iterative Gauss-Newton-based implementation. To come to a practical implementation, a sub-optimal approach with reduced computational complexity was studied, based on a least-squares estimator (LSE). Results from a performance evaluation showed that for uncorrelated MIMO channels the LSE and MLE perform similarly, but that for spatially correlated MIMO channels the MLE obtains an improved mean squared error performance. From a bit-error rate study it was concluded that the diﬀerences between the

268

8 Discussion and conclusions

system performance obtained with LSE and MLE are only marginal and that, therefore, the LSE-based compensation approach is well applicable for CPE correction in a multiple-antenna OFDM system. For the compensation of the ICI an iterative decision-directed approach was studied. In this approach the ICI signal component is estimated using initial estimates of the data symbols. A new detection is made on the received signal, after the estimated ICI is subtracted from it. It is illustrated how the performance can be improved by only considering the dominant contributions to the ICI. A numerical performance analysis focussing on the BER revealed that indeed a considerable performance improvement can be achieved using this compensation approach. Due to the decision-directed nature of the compensation method, however, the full potential of the approach is not leveraged. Hence, methods to further improve the performance of the compensation algorithm remain a topic for future research. Data-aided estimation and mitigation approaches for frequency-independent (FI) IQ imbalance were treated in Chapter 5. Diﬀerent algorithms were designed for the cases of TX, RX and combined TX/RX FI IQ imbalance. All methods are based on a, for this purpose designed, preamble design. The algorithms exploit the stability of the IQ imbalance parameters over frequency and time by separating their estimation from that of the MIMO wireless channel matrix, since the wireless channel response varies over frequency and time. Additionally, the decision-directed compensation of frequency-selective IQ imbalance, based on adaptive MIMO ﬁltering, was studied. The ﬁlter is initialised using the estimates from the data-aided approach. Eﬀectively this approach applies MIMO detection for a pair of DC-mirrored subcarriers simultaneously. Hence, the MIMO IQ imbalance problem is transformed into an equivalent 2Nt ×2Nr MIMO problem. Since this approach does not allow for separation of the IQ imbalance and wireless channel response, however, it is application is less beneﬁcial for FI IQ imbalance. From a numerical performance study we conclude that both compensation approaches provide a considerable improvement in performance compared to a system not applying compensation for the inﬂuence of IQ mismatch. Two methods for the reduction of the impact of nonlinearities in a MIMO OFDM system were illustrated in Chapter 6, i.e., a transmitterand a receiver-based approach. In the ﬁrst approach, the peak-toaverage power ratio (PAPR) of the transmit vector is reduced by a novel technique named spatial shifting. The PAPR reduction is achieved by reshuﬄing groups of subcarriers between the TX branches and selecting the realisation with the lowest overall PAPR. As such the algorithm

8.1 Summary and conclusions

269

exploits the extra degree of freedom provided by MIMO. To detect the symbols at the receiver additional information about the selected permutation has to be conveyed to the receiver. A method to minimise or even remove the overhead introduced by the transmission of this information was also treated. In a second approach, compensation for TX-caused nonlinearities is applied in the RX. For this approach the out-of-band spectral regrowth caused by the nonlinearity is assumed to be manageable or to be treated in another way. In the proposed method the experienced nonlinearities and MIMO propagation channel are estimated simultaneously in the RX, which is enabled by an eﬃcient, for this purpose designed, preamble. Two postdistortion methods are introduced, which apply these estimates; one based on Lagrange interpolation and one based on a decision-directed iterative technique. Simulation results revealed that both methods can successfully be applied to signiﬁcantly reduce the inﬂuence of nonlinearities in multiple-antenna OFDM systems. The choice for a TX- or RX-based approach for the reduction of the inﬂuence of nonlinearities has to be based on the available processing power in both TX and RX. One could, of course, also apply a combination of the two methods. Although the computational complexity of the kind of mitigation approaches presented in this book would previously have been prohibitive for implementation, the constant increase in functional density of digital IC-technology, according to Moore’s law [168], makes their implementation currently very feasible. This is especially true since this functional scaling has much less impact on the size and cost of the analogue frontend. Therefore, the mitigation techniques are very promising to provide system designers with the possibility to solve certain design challenges in either the analogue or digital part of the transceiver.

8.1.6

A generalised error model

A model for the combined inﬂuence of RF impairments was proposed in Chapter 7, which has the major advantage that the diﬀerent impairments do not have to be modelled separately, but that their joint impact can be considered. The model consists of a multiplicative and an additive error term, both occurring in TX and RX. The multiplicative errors result in a phase shift and amplitude scaling of the signal constellation, while the additive errors result in a noise cloud around the detected symbols. Furthermore, the model includes a two-step concept to incorporate the nonstationarity of the non-idealities, where the parameters for the error model are diﬀerent in the transient and stable phase of signal reception.

270

8 Discussion and conclusions

It was demonstrated that the inﬂuence of the impairments treated in Chapters 4 to 6, i.e., phase noise, IQ imbalance and nonlinearities, as well their combined inﬂuence, properly map to the proposed model. Furthermore, it was shown, both analytically and numerically, how the proposed error model maps to SER for a multiple-antenna OFDM system. Additionally, the generally applied system performance measure error vector magnitude (EVM) was proven to be unsuitable for probability of error evaluation in systems experiencing RF impairments, since it leads to a considerable overestimation of the imperfection performance impact. The traditional EVM requirement, applied in many standards, therefore leads to overdesign for this kind of systems and as such to unnecessarily high manufacturing costs.

8.2

Scope of future research

The inﬂuence and digital compensation of RF imperfections is still an emerging research ﬁeld. Although the number of journal and conference publications on this subject has been increasing over the last years, this is one of the ﬁrst books on this subject. Hence, it is not possible for this book to give a complete view of this research area, since new contributions are constantly appearing. This also means that many interesting research challenges remain in this ﬁeld. Therefore, this section will address relevant research topics, which could lead to extensions or broadening of the results achieved in previous literature and in this book. The organisation of this section is as follows. First Section 8.2.1 suggests promising directions for the evaluation of the impact of RF frontend non-idealities. Then possible extension to the mitigation approaches are discussed Section 8.2.2. Finally, Section 8.2.3 discusses the application of, and the therefore required extensions to, the results of this book to emerging high-speed wireless systems in the millimetre-wave (mmwave) band.

8.2.1

Inﬂuence of RF impairments

The sensitivity analyses presented in this book focussed on uncoded multiple-antenna OFDM systems, where MIMO processing is applied in the RX baseband. Although the results are also applicable to the more conventional SISO, SIMO and MISO OFDM systems, extension of the work towards other system designs and other impairments are also of interest. To that end, a study into the inﬂuence of RF impairments in MIMO systems applying pre-coding, where the MIMO processing is applied in

8.2 Scope of future research

271

the TX instead of the RX, would be of interest. Furthermore, a study into the MIMO architecture named space division multiple access [39], where a base station with multiple antennas is combined with single antenna users, would be of beneﬁt. Clearly other impairment conditions would exist in such a system since, e.g., all users have their own independent RF oscillator. Furthermore, the work in this book mainly focussed on linear MIMO processing, while a performance evaluation of systems applying other MIMO detection strategies, e.g., maximum-likelihood detection, would be beneﬁcial. Since OFDM-based systems typically apply channel coding and interleaving, it would be interesting to study the impact of the impairments on such systems analytically. Similarly, the increased interest in multicarrier code division multiple access (MC-CDMA) justiﬁes a careful study of the impact of impairments in these OFDM-related systems. The performance evaluations for the generalised error model of Chapter 7 can serve as a basis to derive pragmatic system requirements based on system performance. To this end, extensions to the model lie in the statistical characterisation of the additive and multiplicative term for diﬀerent impairments and combinations of impairments. For this purpose, it would be useful to experimentally derive typical model parameters for several implemented test systems. Such an extension would also enable the design of a performance characterisation and measurement method for wireless systems with RF imperfections. The system requirements derived based on the model can then replace the currently applied EVM-based requirements in new standards for OFDM-based wireless communication systems. It this way, the allowed impairment levels can more accurately be speciﬁed, which will prevent overdesign and unnecessarily high manufacturing costs. Although we treated three of the major front-end impairments in this work, in speciﬁc system designs other RF impairments like, e.g., DC oﬀset, 1/f noise, nonlinearities with memory, limited word length due to the DAC/ADC and front-end cross-talk might prevail. It might, therefore, be of importance to study the performance impact of other RF impairments. Moreover, in this book the performance analyses were focussed on the diﬀerent impairments separately to clearly reveal their impact. Interesting extension might lie, however, in analyses of the joint impact of several impairments.

8.2.2

Mitigation approaches

Several mitigation approaches for the inﬂuence of individual RF impairments were presented in this book, mainly for application in the baseband part of the system during data transmission. Some techniques

272

8 Discussion and conclusions

are based on detected data, i.e., decision-directed, other use speciﬁc training structures, i.e., data-aided. For the decision-directed approaches it might be interesting to use, instead of the here applied hard slicing outputs, either the soft outputs [169] of the symbol detector(s) or the hard/soft outputs of the decoder(s). Using the decoder outputs, the reliability of detected symbols can be increased, while the soft information can be used as an estimate of the reliability of the decisions. Furthermore, for certain problems the application of turbo processing [170, 171] might be beneﬁcial. In such a turbo mitigation approach, one could, similarly to turbo synchronisation [172], improve the performance of the impairment parameter estimates by iterative processing. A next step could also include the joint estimation and compensation of multiple impairments, which might improve performance when not only a single impairment dominates system performance. The generalised model, as proposed in Chapter 7, can be used as a starting point for such a compensation approach. For the data-aided approaches to be viable, it is essential that industry standards for new wireless communication systems use the ﬁndings in this ﬁeld to deﬁne preamble structures that allow for receiver-based compensation of the inﬂuence of RF imperfections. The currently applied preambles are typically designed to enable frequency synchronisation and wireless channel estimation. It was shown in this work that by small changes and additions to these training structures, also the estimation of the parameters of the RF imperfections can be enabled. A performance and complexity comparison between baseband mitigation, as proposed in this work, and approaches applying calibration within the front-end can be used to reveal the best location to estimate and compensate for the diﬀerent impairments. When one, however, wants to achieve optimal overall designs, co-design of the baseband and RF part will be essential. In such an approach the traditional design boundary located between the analogue front-end and baseband part is removed and functions are performed in the domain yielding the best overall system performance. As such, one could imagine that a part of the front-end could be adaptive and would be adjusted based on measurements in the baseband with the aim to improve the overall system performance. Clearly, many research challenges remain concerning this co-design approach.

8.2.3

Towards mm-wave systems

There is an increasing interest in the application of the mm-wave band for high-speed wireless communication purposes [21–23], since several

8.2 Scope of future research

273

GHz of bandwidth is available around 60 GHz for license-free use. Since current low-cost CMOS-based IC technology is enabling operation at these high frequencies, it is likely that next-generation high-speed wireless systems will exploit this frequency band. Since process spread is much larger at these frequencies, however, dealing with RF impairments will be a key design challenge in this kind of systems. Therefore, the work presented in this book will be useful in the design of eﬃcient mmwave systems. Speciﬁc aspects of communications at these frequencies, however, should be taken into account when applying the presented results to this new frequency band. One of these main characteristics concerns the mm-wave propagation channel, which reveals lower rms delay spreads and Ricean fading [173]. It is anticipated that the diﬀerence between the inﬂuence of TX and RX impairments will decrease for these Ricean fading conditions, but more thorough investigation is needed to substantiate these surmises. Furthermore, the low delay spreads make us believe that non-OFDM system designs might also be of interest for application in this band. A promising example could be the application of single-carrier (SC) transmissions with frequency-domain equalisation. Therefore, a study into the performance tradeoﬀs between OFDM- and SC-based systems, considering both the propagation channel and RF impairment characteristics, would be interesting.

Glossary

General Notation X x x(p) (X)l,m (x)l XT XH X∗ X−1 X† |X| X det{X} tr{X} IN FN 0 1 X⊗Y diag{x} x∗ x ˜ E{x} E{x|y} var{x} R{x} I{x} Γ(·)

matrix vector vector x during the pth iteration (l, m)th element of matrix X lth element of vector x transpose of matrix X conjugate transpose of matrix X conjugate of matrix X inverse of matrix X pseudo-inverse of matrix X, given by (XH X)−1 XH matrix of absolute values of the entries of matrix X L2 norm of matrix X determinant of matrix X trace of matrix X N × N identity matrix N × N Fourier matrix all zeros matrix all ones matrix Kronecker or direct matrix product of matrices X and Y diagonal matrix containing the elements of vector x on its diagonal complex conjugate of x estimate of x expected value of x conditional expected value of x given y variance of x real part of x imaginary part of x Gamma function

276 Q(·) δ(·) P(x) ◦ x x Cn {·} N (µ, Ω) CN (µ, Ω)

Glossary

Gaussian error integral or Q-function delta function probability of occurrence of x convolution element-wise multiplication smallest integer larger than x largest integer smaller than x cyclicly shifting over n samples normal distribution with mean of µ and variance Ω complex normal distribution with mean of µ and variance Ω

Acronyms and abbreviations 1G 2G 3G 3G-LTE A/D ADC AGC AF AM-AM AM-PM AP AWGN BEP BER BFWA BO BPF BPSK BWC CCDF CCER CFO CIR CP CPE CRLB CSI DAB DAC

ﬁrst generation second generation third generation long term evolution of 3G analogue-to-digital analogue-to-digital converter automatic gain control adaptive ﬁlter amplitude-to-amplitude amplitude-to-phase access point additive white Gaussian noise bit-error probability bit-error rate broadband ﬁxed wireless access backoﬀ band-pass ﬁlter binary phase shift keying backward compatibility complementary cumulative distribution function CIR-to-CIR-error ratio carrier frequency oﬀset channel impulse response cyclic preﬁx common phase error Cram´er-Rao lowerbound channel state information digital audio broadcasting digital-to-analogue converter

277

Glossary

D/A dB dBi DC DFT DVB-H DVB-T ECDF EVM FDM FI FPGA FS GI GSM HDTV HSDPA HSUPA I IBO IC ICI IDFT IEEE IF i.i.d. IP ISI LG LL LMS LNA LO LPF LSE LT MAC Mb MB-OFDM MIMO MISO MLD

digital-to-analogue decibel decibel relative to isotropic direct current discrete Fourier transform digital video broadcasting - handheld digital video broadcasting - terrestrial empirical cumulative distribution function error vector magnitude frequency division multiplexing frequency independent ﬁeld programmable gate array frequency selective guard interval global standard for mobile communications high-deﬁnition television (HDTV) high-speed downlink packet access high-speed uplink packet access in-phase input backoﬀ integrated circuit inter-carrier interference inverse discrete Fourier transform institute of electrical and electronics engineers intermediate frequency independent and identically distributed intercepting point inter-symbol interference Lagrange log-likelihood least-mean-squares low-noise ampliﬁer local oscillator low-pass ﬁlter least-squares estimation long training medium access control megabit multiband OFDM multiple-input multiple-output multiple-input single-output maximum-likelihood detection

278 MLE MMSE MNC MRC MSE NDR NL NLOS OFDM OSI PA PAC PAM PAPR PD pdf PDP PER PHY PLL PN PS PSD PSK PTS Q QAM QPSK RF RLS rms RX SC SDM SER SI SIC SIR SIMO SISO SM SNR

Glossary

maximum-likelihood estimation minimum mean squared error maximum normalised correlation maximum ratio combining mean squared error nonlinear distortion removal nonlinear non line of sight orthogonal frequency division multiplexing open system interconnection power ampliﬁer per-antenna coding pulse-amplitude modulation peak-to-average power ratio postdistortion probability density function power delay proﬁle packet-error rate physical layer phase-locked loop phase noise phase shifting power spectral density phase shift keying partial transmit sequence quadrature quadrature amplitude modulation quadrature phase shift keying radio frequency recursive least-squares root mean square receiver / received single carrier space division multiplexing symbol-error rate side information successive interference cancellation signal-to-ISI ratio single-input multiple-output single-input single-output subcarrier multiplexed signal-to-noise ratio

279

Glossary

SO SS SSA ST STA STC TM TO TWTA TX UMTS UT UWB V-BLAST WLAN ZF

subcarrier orthogonal spatial shifting solid-state ampliﬁer short training station space-time coding time multiplexed time orthogonal travelling wave tube ampliﬁer transmitter / transmitted universal mobile telecommunications system user terminal ultra wideband vertical Bell labs space time wireless local area network zero-forcing

Symbols fc fs H(k) G(n) sm sm (k) um um (n) xm xm (k) ym ym (n) vm nm Θ Υ Nt Nr Nc Ng Ns Ts ω

carrier frequency sample frequency Nr ×Nt MIMO channel matrix for the kth subcarrier Nr ×Nt matrix of the nth tap of the channel impulse response Nt Nc ×1 TX frequency-domain vector for the mth symbol Nt ×1 subvector of sm for the kth subcarrier Nt Ns ×1 TX time-domain vector for the mth symbol Nt ×1 subvector of um for the nth sample Nr Nc ×1 RX frequency-domain vector for the mth symbol Nr ×1 subvector of xm for the kth subcarrier Nr Ns ×1 RX time-domain vector for the mth symbol Nr ×1 subvector of ym for the nth sample Nr Ns ×1 time-domain RX noise vector for the mth symbol Nr Nc ×1 frequency-domain equivalent of vm addition of the GI removal of the GI number of transmit antennas number of receive antennas number of OFDM subcarriers number of samples in the GI length of the total OFDM symbol, Nc + Ng sample time, 1/fs angular frequency

Appendix A MSE in CFO estimation

This appendix derives the mean squared error (MSE) in the estimation of the normalised carrier frequency oﬀset (CFO) for the MIMO OFDM estimator proposed in Section 3.3.2. It was shown in (3.21) that the estimation error could be approximated for high SNR as

Nc

n+Np −1

I vH (m)t(m) + tH (m)v(m + Np )e−jθ

m=n

ε(n) ≈ εˆ(n) =

(A.1)

.

n+Np −1

2πNp

t(m)2

m=n

First we ﬁnd that the mean value of εˆ is given by E[ˆ ε(n)] = 0,

(A.2)

since the elements of v are zero-mean complex normally distributed and independent of the elements of t that are zero-mean complex discrete uniformly distributed. The variance of the error term is then given by var (ˆ ε(n)) = E[ˆ ε2 (n)]

⎡4

⎢ ⎢ Nc2 ⎢ E = ⎢ 4π 2 Np2 ⎢ ⎣

52 ⎤

n+Np −1

−jθ

I v (m)t(m) + t (m)v(m + Np )e H

m=n

H

4

52

n+Np −1

t(m)2

⎥ ⎥ ⎥ ⎥. (A.3) ⎥ ⎦

m=n

When we now use that t(n)2 is constant and given by Nr σt2 , we can split the nominator and denominator. The denominator of (A.3) can be rewritten as ⎛

m+Np −1

⎝

n=m

⎞2 2⎠

t(n)

⎛ =⎝

Nr

m+Np −1

nr =1

n=m

=

. 2

⎞2 2⎠

|tnr (n)|

/ Nr Np σt4 + Nr Np σt4 2 / + Nr Np σt4 = Nr2 Np2 σt4 . (A.4) .

=2

(Nr Np )! 2!(Nr Np − 1)!

282

Appendix A: MSE in CFO estimation

The expected value of in the nominator of (A.3) can be rewritten as n+N p −1 % & 2 E I vH (m)t(m) + tH (m)v(m + Np )e−jθ m=n

Nr = E nr =1

n+Np −1

sin(αnr (m))|vnr (m)||tnr (m)|

m=n

2 + sin(βnr (m))|tnr (m)||vnr (m + Np )|

Nr = E nr =1

n+Np −1

sin2 (αnr (m))|vnr (m)|2 |tnr (m)|2

m=n

+ sin2 (βnr (m))|tnr (m)|2 |vnr (m + Np )|2

= Nr Np σt2 σv2 ,

(A.5)

where αnr (m) = ∠{vn∗ r (m)tnr (m)} and βnr (m) = ∠{t∗nr (m)vnr (m + Np )e−jθ }. When we now substitute (A.4) and (A.5) into (A.3) we ﬁnd that the variance of the error in the estimation of the normalised CFO δ is given by var (ε(n)) ≈

Nc2 (2π)2 N

3

r Np ℘

,

(A.6)

Here ℘ = σt2 /σv2 = Nt σu2 /σv2 = P/σv2 denotes the SNR per receive antenna and P is the total transmit power. The variances of the elements of u, t and v are given by σu2 , σt2 and σv2 , respectively. Since the estimator is unbiased, the variance of the estimator derived in (A.6) equals the MSE.

Appendix B MSE in channel estimation

B.1

MSE in linear channel interpolation

In this section we derive the MSE in the estimation of the channel elements for the SM preamble, which are found through linear interpolation as deﬁned in (3.29). When we omit the MIMO channel element index for readability, the error in channel estimation for carrier k = nt − 1 + aNt + i (for i ∈ {1, . . . , Nt − 1} and a ∈ {0, . . . , Nc /Nt − 2}) is given by [76] ε(k) =

(Nt − i)(H(k + Nt − i) + ε1 ) + i(H(k − i) + ε2 ) − H(k), Nt

(B.1)

where the error in the estimation of H(k + Nt − 1) and H(k − i) are deﬁned as ε1 and ε2 , respectively. From (B.1) the MSE can be derived, which yields (Nt − i)2 2 i2 σε1 + 2 σε22 + b1 R(0) + b2 [R(Nt ) + R(−Nt )] 2 Nt Nt +b3 [R(i) + R(−i)] + b4 [R(Nt − i) + R(i − Nt )], (B.2)

E[|ε(k)|2 ] =

where the channel frequency correlation is deﬁned by R(a) = E [H(k)H ∗ (k + a)]

(B.3)

and where (Nt − i) i + 2, Nt2 Nt Nt − i = − , Nt

b1 = 1 + b3

(Nt − i)i , Nt2 i b4 = − . Nt

b2 =

When we now use that σε21 and σε22 are found from (3.28) and given by σn2 /(2Nt σs2 ), the average MSE over all interpolated elements is given by Nt −1 1 E[|ε(nt − 1 + aNt + i)|2 ] E[|ε| ] = Nt − 1 i=1 2

=

5Nt − 1 Nt + 1 2Nt − 1 σn2 + R(0) + R{R(Nt )} + η, 6Nt2 σs2 3Nt 3Nt

(B.4)

284

Appendix B: MSE in channel estimation

where we used that R(a) = R∗ (−a) and η =

Nt −1 −1 (Nt − i)[R(i) + R(−i)] + i[R(Nt − i) + R(i − Nt )] Nt (Nt − 1) i=1

= −

Nt −1 2 (Nt − i)[R(i) + R(−i)] Nt (Nt − 1) i=1

Nt −1 4 (Nt − i)R{R(i)}. = − Nt (Nt − 1) i=1

(B.5)

The frequency correlation of the channel R(a) in (B.4) can now be derived from our channel model, as deﬁned in Section 2.2. Here we exploit that the channel exhibits an exponentially decaying power delay proﬁle, as deﬁned in (2.18). Furthermore, it is assumed that the average channel attenuation equals 0 dB, i.e. Pr = 1. The resulting correlation factor is then found to be given by ' #L−1 L−1 ∗ j 2π(k+a)l −j 2πkl ∗ N N c c g(l)e g (l )e R(a) = E [H(k)H (k + a)] = E l =0

l=0

=

L−1

j 2πal N

e

c

E |g(l)|2 =

l=0

L−1

j 2πal N

Pd (l)e

c

l=0

L−1 −lT /σ j 2πal 1 e s τ e Nc −l Ts /στ l =0 e l=0

= L−1 − σ s τ LT

=

e

Ts

Ts

ej2πa − 1 e στ − 1

e στ − e

j2πa L

− σ s τ LT

e

−1

.

(B.6)

It is easily veriﬁed from (B.6) that R(0) = 1. Now, for a given normalised rms delay spread στ /Ts , we can calculate the average MSE on the carriers where interpolation is applied, which is given by EIP = E[|ε|2 ] =

B.2

Nt −1 2Nt − 1 σn2 5Nt − 1 Nt + 1 4 Nt − i + + R{R(N )}− R{R(i)}. t 6Nt2 σs2 3Nt 3Nt Nt i=1 Nt − 1 (B.7)

MSE in linear channel extrapolation

In this section we derive the MSE in the estimation of the channel elements for the SM preamble, which are found through linear extrapolation as deﬁned by (3.32). When we again omit the MIMO channel element index for readability, the error in channel estimation for carrier k = Nc − Nt + nt − 1 + i is given by ε(k) =

−i[H(k − Nt − i) + ε1 ] + (Nt + i)[H(k − i) + ε2 ] − H(k). Nt

(B.8)

When we deﬁne the error in the estimation of H(k − Nt − i) and H(k − i) to be denoted by ε1 and ε2 , respectively, we can write resulting MSE as E[|ε(k)|2 ] =

i2 2 (Nt + i)2 2 σ + σε2 + b1 R(0) + b2 [R(Nt ) + R(−Nt )] 2 ε1 Nt Nt2 +b3 [R(i) + R(−i)] + b4 [R(Nt + i) + R(−Nt − 1)], (B.9)

285

B.2 MSE in linear channel extrapolation

where R(a) was deﬁned in (B.3) and for the exponentially decaying PDP given by (B.6). The other parameters in (B.9) are deﬁned as (Nt + i)2 i2 + 2, 2 Nt Nt Nt + i = − , Nt

b1 = 1 +

b2 = −

b3

b4 =

(Nt + i)i , Nt2

i . Nt

The average MSE in channel estimation on the extrapolated carriers is then found to be given by EEP

Nt −1 1 = E[|ε| ] = E[|ε(Nc − Nt + nt − 1 + i)|2 ] Nt − 1 i=1 2

= where =

11Nt − 1 5Nt − 1 8Nt − 1 σn2 + −2 R{R(Nt )} − , 6Nt2 σs2 3Nt 3Nt Nt −1 i Nt + i 2 R{R(Nt + i)} − R{R(i)}. Nt − 1 i=1 Nt Nt

(B.10)

(B.11)

Appendix C Measurement setup

This appendix gives an extended description of the test system used for the throughput evaluation presented in Section 3.6.1. As described in Section 3.6.1, both the TX and RX platform are built in a PC with three boards, where every board represents a TX or RX branch, resulting in a 3×3 MIMO system, as shown in Fig. C.1. Every board consists of a baseband part, intermediate frequency (IF) part and radio frequency (RF) front-end based on a 5.x GHz GaAs radio chip developed within Agere Systems. Figure C.1 shows the inside of the RX platform and indicates these three parts. The test system operates in the 5.x GHz ISM band, and is capable of transmitting broadband signals with a bandwidth up to 20 MHz. For ease of veriﬁcation of the test system, the reception at the RX platform is trigged by a cable from the TX. For the measurements the TX and RX platform are both positioned on a trolley, as illustrated in Fig. C.2. The antenna array is mounted at the front of the trolley.

C.1

Baseband

The baseband processing is built around two ﬁeld programmable gate arrays (FPGAs), i.e., Xilinx Virtex 800, per board. Although the FPGAs are in the current

Figure C.1. Receiver equipment: a PC with three boards containing the baseband, IF and RF parts.

288

Appendix C: Measurement setup

Figure C.2.

Illustration of the measurement setup.

RS 485 MAC c

MAC c

MAC c low - IF in

ADC Xilinx Virtex 800

Xilinx Virtex 800

Xilinx Virtex 800

I out Q out

Dual DAC

to IF and RF board

ISA connector Figure C.3.

Schematic representation of the used baseband board.

setup mainly used as memory banks, they can be used to implement a real-time baseband solution. The baseband boards are capable of transmission and reception, while the IF and RF part are only set up to transmit or receive for the TX and RX platform, respectively. Consequently only the respective parts of the baseband boards are used.

289

C.2 IF stages

Figure C.4.

GUI of the MATLAB software for the 3×3 system.

The board provides connections to a medium access control (MAC) board by the MAC connectors, as denoted by “MAC c” in Fig. C.3. When this is implemented a complete point-to-point link can be implemented. The RS 485 connector shown in Fig. C.3 is used as a triggering interface between the TX and RX, as explained before. To exploit the ﬂexibility provided by the design of the test system, the baseband processing of both TX and RX was implemented in MATLAB. In the TX part of the software complex data was generated, where diﬀerent packet formats, modulations and coding rates could be selected. This digital data was subsequently loaded in the memory banks of the TX boards. When transmitted through the wireless channel the RX data was captured from the RX board and loaded into the MATLAB interface again. There synchronisation, channel estimation, detection and decoding were applied. The diﬀerent detection algorithms described in Section 2.3.1 were implemented here. BER measurements were possible by comparing the transmitted and received data. Figure C.4 shows the graphical user interface (GUI) developed with the MATLAB software. It shows a successful, i.e. error-less, 108 Mbit/s transmission, with 3 TX antennas and 16-QAM modulation.

C.2

IF stages

The TX IF stage is schematically illustrated in Fig. C.5. Here the baseband signal (I and Q) is fed through a low-pass ﬁlter (LPF), mixed up to IF, i.e. 1489 MHz. The IF signal subsequently fed to the RF front-end which up-converts the signal to RF, i.e., 5.15 GHz. The IF part of the RX, as illustrated in Fig. C.6, has the output signal of the RF front-end at 1489 MHz as input. This signal is properly scaled to exploit the full range of the ADC. The bandpass ﬁltered signal is then down-converted to low-IF centred

290

Appendix C: Measurement setup

Figure C.5.

Block diagram of the baseband-to-IF stage of a transmitter branch.

Figure C.6.

Block diagram of the IF-to-low-IF stage of a receiver branch.

around 15 MHz. This signal is fed to the baseband board, where it is sampled and digitally down-converted.

C.3

Antennas

The MBA-5 (Miniature Broadband Antenna) wideband antenna of ASCOM is used for the measurements. This antenna is designed primarily for wireless LAN applications in the 5 GHz band. The speciﬁcations of the antenna can be found in [85].

Appendix D Proof of Theorem 4.1

In this appendix we will provide the proof for Theorem 4.1. We ﬁrst reduce the proof of Theorem 4.1 to four key convergence statements (D.11), (D.18), (D.24) and (D.25). After this, we prove these four statements separately. Proof of Theorem 4.1 subject to (D.11), (D.18), (D.24) and (D.25). We P recall that XNc converges in probability to X, and write XNc −→ X, when, for every > 0, (D.1) lim P(|XNc − X| > ) = 0. Nc →∞

We will make frequent use of the fact that if XNc and YNc converge in probability to X and Y , then also XNc + YNc converges to X + Y in probability. We will start by rewriting the sum N c −1

ej

n

i=0

ε (i) j 2πln N

e

c

(D.2)

.

n=0

For this, we will use partial summation, which states that for any two sequences of m numbers {a(n)}m n=1 and {b(n)}n=1 , we have m−1

a(n)[b(n + 1) − b(n)] = a(m)b(m) − a(0)b(0) −

n=0

m−1

[a(n + 1) − a(n)]b(n + 1). (D.3)

n=0

We apply this to a(n) = ej

n

i=0

ε (i)

j 2πln N

b(n + 1) − b(n) = e

,

so that b(n) =

n−1

j 2πli N

e

c

c

,

.

(D.4)

(D.5)

i=0

For this choice, we can compute that b(0) = b(Nc ) = 0, and j 2πln N

b(n) =

e

c

j 2πl N

e

c

−1 −1

.

(D.6)

Therefore, we arrive at N c −1

ej

n

i=0

ε (i) j 2πln e Nc

n=0

=

N c −1

1 j 2πl N

e

c

−1

n=0

2πl(n+1) n j Nc ejε (n+1) − 1 ej i=0 ε (i) 1 − e .

(D.7)

292

Appendix D: Proof of Theorem 4.1

Now, ε (n + 1) is small, since it has variance σε2 =

σ2 . Nc

Therefore, we can expand

ejε (n+1) − 1 ≈ jε (n + 1)

(D.8)

to arrive at N c −1

ej

n

i=0

εi j 2πln e Nc

≈

j 2πl N

e

n=0

N c −1

j c

−1

εn+1 ej

n

i=0

εi

j

1−e

2πl(n+1) Nc

(D.9)

.

n=0

Using (D.9), we deﬁne an approximation of ξ given by ξ =

N c −1

j

l=1

N c −1

jsl Nc [e

2πl Nc

− 1]

εn+1 ej

n

i=0

εi

j

1−e

2πl(n+1) Nc

(D.10)

.

n=0

Indeed, we will prove that ηNc , which is deﬁned to be the diﬀerence between ξ and ξ , converges to zero in probability, i.e., ηNc = ξ − ξ −→ 0, P

(D.11)

in probability. Therefore, to prove the claim, it now suﬃces to prove that ξ −→ ζ. j 2πln j 2πl When we use the periodicity of e Nc and e Nc , we can more conveniently rewrite this as P

ξ ≈

N c −1

js(l) j

0<|l|≤Nc /2

Nc [e

2πl Nc

− 1]

ε (n + 1)ej

n

i=0

ε (i)

j

1−e

2πl(n+1) Nc

,

(D.12)

n=0

where, for l < 0, we deﬁne s(l) = s(Nc − l). Clearly, {s(l)}0<|l|≤Nc /2 now is an i.i.d. sequence of random variables with the same distribution as s(1). Here we note that we have, for Nc even, counted a term too many. However, for this term, we have that l = N2c , so that the diﬀerence is equal to, with l = N2c , N c −1

is(l) j

Nc [e

2πl Nc

− 1]

ε (n + 1)ej

n

i=0

ε (i)

j

1−e

2πl(n+1) Nc

n=0

=

Nc −1 n P js(l) ε (n + 1)ej i=0 ε (i) (−1)n − 1 −→ 0, 2Nc n=0

(D.13)

so that this change is negligible. From now on, we will for convenience deﬁne ξ =

0<|l|≤Nc /2

N c −1

js(l) j 2πl N

Nc [e

c

− 1]

ε (n + 1)ej

n

i=0

ε (i)

j

1−e

2πl(n+1) Nc

.

(D.14)

n=0

When Nc → ∞, we see that for every l ﬁxed, j 2πl N

Nc [e

c

− 1] → 2πlj,

(D.15)

so that it is natural to assume that ξ ≈

c −1 n s(l) N 2πl(n+1) j Nc . ε (n + 1)ej i=0 ε (i) 1 − e 2πl n=0

l∈Z:l =0

(D.16)

293

Appendix D: Proof of Theorem 4.1 To make this precise, we deﬁne S(t) =

s(l) 1 − ej2πlt . 2πl

(D.17)

l∈Z:l =0

This random sum is a well-deﬁned random variable, and, in particular, has ﬁnite second moment. We will show that 1 ejσB(t) S(t)dB(t), (D.18) ζ=σ 0

and we will also prove that ξ −→ ζ, by using (D.18). First, with P

SNc (t) =

0<|l|≤Nc /2

so that

ξ =

N c −1

1 − ej2πlt ,

js(l)

j 2πl N

Nc [1 − e n

j

ε (n + 1)e

i=0

ε (i)

c

(D.19)

] .

SNc

n=0

n+1 Nc

/ (D.20)

.

P

We will prove that ξ −→ ζ in two steps, namely, with δNc and κNc given by δNc =

N c −1

ε (n + 1)ej

n

i=0

ε (i)

. SNc

n=0

and κNc =

N c −1

ε (n + 1)ej

n

i=0

n+1 Nc

ε (i)

/

. S

n=0

. −S

n+1 Nc

n+1 Nc

/ (D.21)

/ − ζ,

(D.22)

we have that ξ = ζ + ηNc + δNc + κNc .

(D.23)

Therefore, it suﬃces to prove (D.11) and (D.18) and P

(D.24)

P

(D.25)

δNc −→ 0, as well as κNc −→ 0.

Together, these claims prove Theorem 4.1. Equations (D.11), (D.18), (D.24) and (D.25) will be proved below. The above completes the proof of Theorem 4.1 subject to (D.11), (D.18), (D.24) and (D.25). Subsequently, we prove the technical results (D.11), (D.18), (D.24) and (D.25). We make frequent use of the fact that, for a sequence of (complex) random variables YNc , P the convergence YNc −→ 0 follows if E[|YNc |2 ] → 0. This follows from the Chebychev inequality, as well as lim P(|YNc | > ) ≤ lim E[|YNc |2 ] = 0.

Nc →∞

Nc →∞

(D.26)

For convenience, we will prove the statements in the order (D.24), (D.11), (D.18) and (D.25).

294

Appendix D: Proof of Theorem 4.1

Proof of (D.24). We bound the second moment of the summands with |l| > K, for any K, by ⎡ $ $ E ⎣$

js(l)

K≤|l|≤Nc /2

≤

N c −1

N [e

j 2πl N

c

− 1]

σs2

ε (n + 1)ej

n=0

j 2πl N

Nc2 |e

c

i=0

ε

(i)

2πl(n+1) Nc

j

1−e

⎤ $$2 $ ⎦

' c −1 $ $ N j n ε (i) $2 i=0 $ , E ε (n + 1)e #

1

K≤|l|≤Nc /2

n

− 1|2

(D.27)

n=0

where we use the independence of s(l) for diﬀerent l, and we write σs2 = E[|s(l)|2 ]. We write out ' # N −1 c $ $ j n ε (i) $2 i=0 $ ε (n + 1)e E n=0

=

N c −1 N c −1

n1 n2 E ε (n1 + 1)ε (n2 + 1)ej i=0 ε (i) e−j i=0 ε (i) .

(D.28)

n1 =0 n2 =0

When n1 < n2 , we have that ε (n2 + 1) is independent of ε (n1 + 1)ej

n1

i=0

ε (j) −j

e

n2

i=0

ε (i)

(D.29)

,

so that the expected value equals zero. Therefore, we arrive at ' N −1 c −1 c n $2 $ N 2 j ε (i) $ = E $ ε (n + 1)e i=0 E ε (n + 1) = σ 2 , #

n=0

(D.30)

n=0

so that we obtain ⎡ $ $ E ⎣$

K≤|l|≤Nc /2

N c −1

js(l) j 2πl N

Nc [e

− 1]

c

j

ε (n + 1)e

i=0 ε (i)

n

j

1−e

2πl(n+1) Nc

⎤ $$2 $ ⎦

n=0

≤ σs2 σ 2

K≤|l|≤Nc /2

1 j 2πl N

Nc2 |e

c

− 1|2

.

(D.31)

It is not hard to see that when |l| ≥ Nc /4, we have that j 2πl N

|e

c

$ /$ . $ 2πl $$ ≥ 1. − 1| ≥ $$1 − cos Nc $

Furthermore, using the fact that | sin(x)| ≥ 2|x| for all |x| ≤ π |l| ≤ N4c $ . /$ $ 4|l| 2πl $$ j 2πl ≥ |e Nc − 1| ≥ $$sin . Nc $ Nc We conclude that j 2πl N

|e

c

6 − 1| ≥ min

7 |l| ,1 . Nc

(D.32) π , 2

we obtain that for (D.33)

(D.34)

295

Appendix D: Proof of Theorem 4.1

As a consequence, we obtain that ⎡ ⎤ N c −1 $ n 2πl(n+1) $2 is(l) $ $ j j ε (i) Nc 1−e ε (n + 1)e i=0 E ⎣$ $ ⎦ j 2πl Nc − 1] n=0 K≤|l|≤Nc /2 N [e 1 ≤ σs2 σ 2 , (D.35) min{|l|2 , Nc2 } K≤|l|≤Nc /2

so that, uniformly in Nc , the variance of the summands with |l| > K is bounded by a constant times K −1 . Therefore, when K = KNc tends to inﬁnity, the variance of this term tends to zero, so that this term tends to zero in probability by (D.26). An identical computation shows that ⎤ ⎡ N c −1 $ n 2πl(n+1) $2 s(l) $ $ j j ε (i) Nc 1−e ε (n + 1)e i=0 (D.36) E ⎣$ $ ⎦ → 0, 2πl n=0 KNc ≤|l|<∞

which implies that the contribution due to |l| ≥ KNc in ζ converge to 0 in probability. We are left to investigate the contribution due to 0 < |l| ≤ KNc , since we know that δNc − δN c −→ 0, P

(D.37)

where

δNc =

# j

s(l) N [e

1<|l|≤KNc

j 2πl N

c

1 − 2πl − 1]

' N −1 c

ε (n + 1)ej

n

i=0

ε (i)

j

1−e

2πl(n+1) Nc

.

n=0

(D.38) Therefore, it suﬃces to prove that δN c −→ 0. P

(D.39)

For this, we note that δN c =

j 2πl 2πjl − Nc [e Nc − 1]

s(l)

j 2πl N

Nc [e

1<|l|≤KNc

·

N c −1

c

ε (n + 1)ej

− 1](2πl) n

i=0

ε (i)

j

1−e

2πl(n+1) Nc

.

(D.40)

n=0

Again we have similarly to (D.31) that

2 2 2 E[|δN c | ] = σs σ

$ $2 $ $ j 2πl $2πjl − Nc [e Nc − 1]$

1<|l|≤KNc

j 2πl N

Nc2 |e

c

− 1|2 (2πl)2

.

(D.41)

A Taylor expansion yields that j 2πl N

|2πjl − Nc [e

c

− 1]| ≤ C

l2 . Nc

(D.42)

296

Appendix D: Proof of Theorem 4.1

Therefore, also using (D.33), we arrive at

2 2 2 E[|δN c | ] ≤ Cσs σ

1 cKNc ≤ . Nc Nc

1<|l|≤KNc

This converges to zero for every KNc = o(Nc ), for example for KNc = proves the convergence in (D.24).

(D.43) √

Nc . This

Proof of (D.11). Recall that ηNc =

1 Nc

N c −1

s(l)

0<|l|≤Nc /2

j 2πl N

e

c

−1

2πl(n+1) n j Nc ejε (n+1) −1−jε (n+1) ej i=0 ε (i) 1−e .

n=0

(D.44) We show that E[|ηNc |]2 converges to zero, which implies that ηNc converges to zero in probability by (D.26). To prove that the second moment of ηNc converges to zero, we use the computations in the proof of (D.24) to obtain that 2 2 E[ηN c ] ≤ σs

1 j 2πl N

Nc |e c − 1|2 ' # N −1 c $ jε (n+1) j n ε (i) $2 i=0 $ $ . e ·E − 1 − jε (n + 1) e

0<|l|≤Nc /2

2

(D.45)

n=0

Similarly to (D.30), we obtain that # N −1 ' c $ jε (n+1) n $2 E $ e − 1 − jε (n + 1) ej i=0 ε (i) $ n=0

=

N c −1

$ $2 E $ejε (n+1) − 1 − jε (n + 1) $

n=0

≤

N c −1 n=0

3σ 4 4 E ε (n + 1) = , Nc

4 where we use that E ε (n + 1) = 2 E[ηN c] ≤

3σ 4 σs2 Nc

3σ 4 . Nc2

(D.46)

We arrive at

0<|l|≤Nc /2

1 j 2πl N

Nc |e 2

c

−

1|2

≤

c , Nc

(D.47)

using a similar argument as in (D.34) with K = 1, where c is a constant. This proves that the approximation in (D.11) is valid, and even shows that the error in the 1 approximation is, with high probability, smaller than N − 2 +δ and any δ > 0. Proof of (D.18). This again follows from a second moment calculation. We can interchange the order of summation when we have a ﬁnite sum, but for an inﬁnite sum that is not so clear. However, by a similar computation as in the proof of (D.24), we see that 1 P ejσB(t) (S(t) − S≤KNc (t))dB(t) −→ 0, (D.48) σ 0

297

Appendix D: Proof of Theorem 4.1 where

S≤KNc (t) =

0<|l|≤KNc

s(l) 1 − ej2πlt . 2πl

Also, it is not hard to see that, again using a second moment computation, σs(l) 1 jσB(t) P e [1 − ej2πlt ]dB(t) −→ 0. 2πl 0

(D.49)

(D.50)

|l|>KNc

Noting that

1

ζ=σ

ejσB(t) S(t)dB(t) +

0

|l|>KNc

σs(l) 2πl

−σ 0

1

1

ejσB(t) [1 − ej2πlt ]dB(t)

0

ejσB(t) (S(t) − S≤KNc (t))dB(t)

(D.51)

then completes the proof.

i Proof of (D.25). We use that ε (i) = σ B( i+1 ) − B( ) , where {B(t)}t≥0 is a Nc Nc standard Brownian motion. Then, N c −1 n=0

ε (n + 1)ej

n

i=0

ε (i)

. S

n+1 Nc

/

/ . / . / n+2 n+1 n+1 B −B Nc Nc Nc n=0 / . / . / . N c −1 n+1 n n jσB( Nn ) c S =σ B −B . (D.52) e Nc Nc Nc n=0

=σ

N c −1

jσB( n+1 ) N

e

c

.

S

The above is the usual approximation to the stochastic integral 1 σ ejσB(t) S(t)dB(t).

(D.53)

0

Since for every t, the integrand ejσB(t) S(t) has a ﬁnite second moment, and since the process t → ejσB(t) S(t) is predictable, the sum in (D.51) converges in probability to the integral in (D.53). See, e.g., [104, Section 13.8] for convergence results of stochastic integrals.

Appendix E Orthonormal polynomial basis

We construct an orthonormal polynomial basis for the pseudo-random Gaussian preamble using Gram-Schmidt, based on the set of real values {x, x3 , x5 }, and where we apply a weighting function equal to 1. The nth polynomial in the basis is denoted by πn (x), the normalised version of which is given by ψn (x). The ﬁrst polynomial is given by π1 (x) = x, 2

(E.1) 2

E[|π1 (x)| ] = E[|x| ] = µ2 , π1 (x) x = √ , ψ1 (x) = µ2 E [|π1 (x)|2 ]

(E.2) (E.3)

where we deﬁne µn = E[xn ].

(E.4)

Subsequently, the second polynomial basis is found by π2 (x) = x3 − x3 , ψ1 (x)ψ1 (x) µ4 x, = x3 − µ2 1

x3 , ψ1 (x) = √ E[x4 ]. µ2

(E.5) (E.6)

The normalised version of (E.5) is given by π2 (x) , ψ2 (x) = E[|π2 (x)|2 ] #. /2 ' µ4 µ4 µ2 2 3 = E x6 − 2 x4 + 42 x2 E[|π2 (x)| ] = E x − x µ2 µ2 µ2 = µ6 −

(E.7)

µ24 , µ2

(E.8)

ψ2 (x) = α1 x3 + α2 x,

(E.9)

thus where we have deﬁned the coeﬃcients : α1 = α2

µ2 , µ2 µ6 − µ24 ; µ24 = − . 2 µ2 µ6 − µ2 µ24

(E.10) (E.11)

300

Appendix E: Orthonormal polynomial basis

Finally, the third polynomial basis is found by π3 (x) = x5 − x5 , ψ2 (x)ψ2 (x) − x5 , ψ1 (x)ψ1 (x),

x5 , ψ2 (x)ψ2 (x) = E α1 x8 + α2 x6 (α1 x3 + α2 x), 6 x x

x5 , ψ1 (x)ψ1 (x) = E √ √ . µ2 µ2

(E.12) (E.13) (E.14)

This yields π3 (x) = x5 + α3 x3 + α4 x,

(E.15)

where we have deﬁned the coeﬃcients α3 = −(µ8 α1 − µ6 α2 )α1 , α4

µ6 = (µ8 α1 − µ6 α2 )α2 − . µ2

(E.16) (E.17)

The normalised version of (E.20) is given by π3 (x) ψ3 (x) = , E[|π3 (x)|2 ] E[|π3 (x)|2 ] = α5 = E (x5 + α3 x3 + α4 x)2 = µ10 + α32 µ6 + α42 µ2 + 2α3 µ8 + 2α4 µ6 + 2α3 α4 µ4 ,

(E.18)

(E.19)

which ﬁnally yields ψ3 (x) =

x5 + α3 x3 + α4 x . √ α5

(E.20)

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Index

3G-LTE, 2 60 GHz, see mm-wave systems adaptive ﬁltering, 178 aﬃne linear impairment inﬂuence, 249 analytical performance analysis, 91, 150, 195, 252 antenna speciﬁcations, 290 AWGN channel, 45 backward compatibility, 34 baseband, 7 equivalent impairment models, 8 implementation, 74, 287 Brownian motion, 85, 112 Bussgang’s theorem, 192 carrier frequency oﬀset, 40 channel estimation, 52, 253 algorithm, 52 extrapolation, 284 interpolation, 283 nonlinearities, 222 numerical results, 58 performance, 52, 283 channel modelling, 11 correlation, 17 mean propagation delay, 13 multipath propagation, 12 power delay proﬁle, 13, 17 Rayleigh fading, 68 rms delay spread, 14 stochastic, 16 cyclic preﬁx, 22 decision-directed compensation, 130, 178 digital video broadcasting (DVB), 2 direct-conversion, 141 error vecor magnitude (EVM), 245, 260

frame format, 32, 252 IEEE 802.11a, 32 MIMO OFDM, 34 free-running oscillator, 85 frequency synchronisation, 39, 264 algorithm, 42 imperfect, 40 numerical results, 47 performance analysis, 44, 281 generalised error model, 245, 269 impairment mapping, 248 IQ imbalance, 249 multiple impairments, 249 nonlinearities, 249 phase noise, 248 nonstationarity, 250 probability of error, 254 system model, 246 system performance, 252 data phase, 253 numerical results, 257 preamble phase, 253 TX and RX impairments, 247 Gram-Schmidt orthonormalisation, 226, 299 guard interval, 22 homodyne, see direct-conversion IEEE 802.11a, 3, 32 impairments, 6, 249 inter-carrier interference (ICI) carrier frequency oﬀset, 40 compensation, 130 distribution, 91 large Nc , 95 small σ, 99 tail probabilities, 102

316 error term, 108 RX phase noise, 110 TX phase noise, 109 estimation, 131 phase noise, 89 properties, 95 system performance, 104, 108 inter-symbol interference (ISI) OFDM, 22 timing synchronisation, 63 IQ imbalance, 139, 143, 249 estimation, 163 performance, 172 preamble, 164 RX IQ imbalance, 167 TX IQ imbalance, 166 TX/RX IQ imbalance, 168, 170 frequency ﬂat, 143, 163 frequency-selective, 148, 178 mitigation, 163, 178 performance, 172, 180 modelling, 140 frequency ﬂat, 143 RX ﬁlter imbalance, 148 receiver, 144 signal model, 145 system performance, 150 numerical results, 159 RX IQ imbalance, 157 TX IQ imbalance, 152 transmitter, 144 IQ mismatch, see IQ imbalance

Index travelling wave tube ampliﬁer, 189 RX-based correction, 219 inverse nonlinearity estimation, 227 iterative distortion removal, 231 least-squares estimation, 224 MIMO channel estimation, 222 nonlinearity estimation, 223 orthonormal basis, 226, 299 performance, 233 postdistortion, 230 signal model, 192 system performance, 195 numerical results, 203 RX nonlinearities, 201 TX nonlinearities, 196 OFDM, 3, 22 oscillator model, 85 power spectral density, 86

measurements system setup, 287 test system, 76 MIMO, 4, 19 MIMO OFDM, 26, 265 backward compatibility, 34 baseband implementation, 74, 287 frame format, 34 front-end, 289 measurement results, 76 system performance, 73, 79 test system, 73, 287 mm-wave systems, 7, 272 multicarrier, see OFDM

PAPR reduction baseband clipping, 220 spatial shifting, 208 performance analysis, 91, 150, 195, 252 phase noise, 83, 248 Brownian motion, see Brownian motion CPE, 89 correction, 117 Cram´er-Rao lower bound, 122 estimation performance, 124 maximum-likelihood estimator, 117, 119 sub-optimal estimator, 121 ICI, see inter-carrier interference (ICI) modelling, 84 system model, 88 preamble, 32, 252 Frank-Zadoﬀ-Chu codes, 38 frequency synchronisation, 43 IEEE 802.11a, 32 IQ imbalance, 164 MIMO OFDM, 34 nonlinearities, 221 subcarrier-multiplexed, 37 subcarrier-orthogonal, 38 time-multiplexed, 36 time-orthogonal, 36

nonlinearities, 185 intermodulation, 190 modelling, 187 ideal clipping ampliﬁer, 188 polynomial model, 190 solid-state ampliﬁer, 189

Rayleigh fading, 17, 68 RF impairments, 6, 265, 270 generalised error model, 245 IQ imbalance, 139 nonlinearities, 185 phase noise, 83

Lagrange interpolation, 231

317

Index rms delay spread, see channel modelling, rms delay spread spatial shifting, 208, 210 algorithm, 210 combined with phase shifting, 211 performance, 215 signal model, 209 subcarrier grouping schemes, 213 transparency, 213 symbol timing, see timing synchronisation synchronisation frequency, see frequency synchronisation timing, see timing synchronisation system model, 18 carrier frequency oﬀset, 40 generalised error model, 246 MIMO, 19 MIMO OFDM, 26 nonlinearities, 187 OFDM, 22 phase noise, 88, 91

timing synchronisation, 63 system performance, 73 generalised error model, 252 IQ imbalance, 150 nonlinearities, 195 phase noise, 94, 104, 113 RF impairments, 266 timing synchronisation, 62, 264 algorithms, 66 dominant path detection, 66 reduced complexity algorithm, 68 SIR optimisation, 67 performance, 68 ultra wideband (UWB), 3 Vandermonde matrix, 225 WiMAX, 2 wireless communications, 1 zero-IF, see direct-conversion

About the Author

Tim Schenk was born in Heerle, The Netherlands, on June 28, 1978. After ﬁnishing the “Gymnasium β” (secondary school) at the Norbertuscollege in Roosendaal in 1996, he started studying electrical engineering at the Eindhoven University of Technology (TU/e) in Eindhoven, The Netherlands. In 2001 he spend 4 months, as part of the M.Sc. programme, at the Communications Research Centre in Ottawa, Canada, performing research on radio propagation in microcellular environments. The research for his M.Sc. thesis was performed in the wireless systems research department of Agere Systems in Nieuwegein, The Netherlands, and was concerned with synchronisation for wireless LAN systems based on multiple-antenna techniques. He received the M.Sc. degree in 2002. In August 2002, he started working towards the Ph.D. degree within the radiocommunications chair at the TU/e, in cooperation with, and sponsored by, Agere Systems. A large part of the Ph.D. research was performed within the Dutch cooperative research project B4 Broadband Radio@hand. The ﬁrst two years of the research were carried out in the wireless systems research department of Agere Systems in Nieuwegein. In the framework of his Ph.D. project, Tim was a visiting researcher at CEA-Leti in Grenoble, France, from January to March 2006. He received the Ph.D. degree in Nov. 2006. In November 2003, dr. Schenk received a best paper award at the IEEE SCVT2003 symposium. He was also awarded the 2006 Veder Prize by the Dutch Scientiﬁc Radio Fund Veder for his contributions in the ﬁeld of system optimisation for multiple-antenna systems. Since October 2006, he has been a senior research scientist in the connectivity systems and networks department at Philips Research in Eindhoven, The Netherlands. His research interests are in the ﬁeld of wireless communications, wireless control and signal processing.