Channel Estimation and Signal-Space Diversity for Vector

Channel Estimation and Signal-Space

Diversity for Vector-Valued Transmissions

DISSERTATION

zur Erlangung des akademischen Grades eines

Doktor–Ingenieurs

(Dr.–Ing.)

der Fakultat fur Ingenieurwissenschaften und Informatikder Universitat Ulm

von

SHAWKI ABDELFATTAH AHMED SAAD ABDELKADER

AUS KAIRO, AGYPTEN

1. Gutachter: Prof. Dr.-Ing. Jurgen Lindner2. Gutachter: Prof. Dr.-Ing. Hans-Jorg PfleidererAmtierender Dekan: Prof. Dr. rer. nat. Helmuth Partsch

Ulm, 31. Januar 2007

Acknowledgements

Praise to God, the most gracious and the most merciful. Without His blessing andguidance, my accomplishments would have never been possible.

I would like to thank all who helped me during the course of this work. My words willfail to express my deepest heartfelt thanks to my supervisor, Prof. Dr.-Ing. Jurgen Lindner,for giving me the opportunity to carry out this thesis and for his guidance, support, motivationand encouragement1). Without his continuous support and patience this work would not havebeen possible. I thank him also for his concern and assistance even with other things in mylife.I also would like to take this opportunity to extend my thanks to all my colleagues workingin the Institute for their truthful and sincere concerns about my study and myself and for thefavorable working environment. I would like to particularly acknowledge the help of Dr. rer.nat. Werner Teich, and I would like to express my special thank to Doris Y. Yacoub, ChristianPietsch, Ivan Perisa and Markus Dangl for their friendly support. A special thank is due to myparents who gave me always support during this work. Finally, and most important, I shouldmention that I probably would never have finished this thesis at all without the persuasion ofmy wife Heba.

1)The disseration report was concluded within my research period in the Institute of Information TechnologyUniversity of Ulm

V

Contents

1 Introduction and Motivation 1

2 Fundamentals 5

2.1 Conventions of notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.2 MIMO channel identification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.2.1 Rayleigh fading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.2.2 MIMO channel model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.2.3 Linear algorithms for MIMO channel identification . . . . . . . . . . . . 10

2.2.4 General space model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.3 MMSE for MIMO channel estimation . . . . . . . . . . . . . . . . . . . . . . . . 11

2.3.1 Approximate MMSE estimator . . . . . . . . . . . . . . . . . . . . . . . 13

2.4 Detection based on the estimated channel . . . . . . . . . . . . . . . . . . . . . 13

2.5 Signal-space diversity and fading effects . . . . . . . . . . . . . . . . . . . . . . . 15

2.5.1 Signal-space diversity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.5.2 Lattice constellations and optimum design . . . . . . . . . . . . . . . . . 17

2.5.3 Multidimensional constellations design . . . . . . . . . . . . . . . . . . . 18

2.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3 MIMO Channel Estimation in the Presence of Errors 21

3.1 Channel capacity with imperfect channel knowledge . . . . . . . . . . . . . . . . 22

3.1.1 Lower bound of the capacity with erroneous estimated channel . . . . . . 22

3.2 The influence of channel estimation on optimum and suboptimum receiver . . . 24

VII

Contents

3.2.1 Modelling channel estimation errors . . . . . . . . . . . . . . . . . . . . . 26

3.2.2 Cramer-Rao bound as a quality measure for channel estimation . . . . . 27

3.3 Diversity combining with channel estimation errors . . . . . . . . . . . . . . . . 28

3.3.1 Optimal combining and its error performance . . . . . . . . . . . . . . . 28

3.3.2 Suboptimal combining and its error performance . . . . . . . . . . . . . . 31

3.4 MIMO with signal-space diversity in the presence of channel errors . . . . . . . 32

3.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

4 Adopting Signal-Space Diversity to Combat Channel Effects 39

4.1 From Rayleigh fading to Gaussian channel . . . . . . . . . . . . . . . . . . . . . 40

4.1.1 High diversity order rotated constellation techniques . . . . . . . . . . . 41

4.1.2 Unequal values of the rotation matrix element weights . . . . . . . . . . 44

4.1.3 Upper bounds of signal-space diversity . . . . . . . . . . . . . . . . . . . 44

4.2 The construction of the rotation matrix . . . . . . . . . . . . . . . . . . . . . . . 45

4.2.1 Hadamard and Fourier spreading transforms . . . . . . . . . . . . . . . . 45

4.2.2 Rotation using rotated spreading transforms . . . . . . . . . . . . . . . . 46

4.3 Signal-space diversity - DAST . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

4.3.1 DAST codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

4.3.2 Maximum achievable rate of the rotated constellation . . . . . . . . . . . 48

4.3.3 Constructing DAST codes . . . . . . . . . . . . . . . . . . . . . . . . . . 49

4.3.4 DAST codes and combining technique effect . . . . . . . . . . . . . . . . 51

4.4 The similarity between DAST and MC-CDM . . . . . . . . . . . . . . . . . . . . 53

4.5 Simulation results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

4.5.1 Half diversity-order of signal space . . . . . . . . . . . . . . . . . . . . . 60

4.5.2 Full diversity-order of signal space . . . . . . . . . . . . . . . . . . . . . . 62

4.5.3 Signal-space with space-time diversity techniques . . . . . . . . . . . . . 63

4.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

VIII

Contents

5 Recursive Algorithms for MIMO Channel Estimation 75

5.1 Adjustable time-varying channel model . . . . . . . . . . . . . . . . . . . . . . . 76

5.1.1 Clarke’s channel model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

5.1.2 Pop&Beaulieu’s channel model . . . . . . . . . . . . . . . . . . . . . . . 78

5.1.3 Modified-Pop&Beaulieu’s channel model . . . . . . . . . . . . . . . . . . 79

5.1.4 Establishing Rayleigh time-varying channel model . . . . . . . . . . . . . 80

5.2 Channel estimation and Alamouti Scheme . . . . . . . . . . . . . . . . . . . . . 80

5.2.1 Alamouti’s space-time block coding scheme . . . . . . . . . . . . . . . . . 83

5.2.2 STBCs from complex notations to a linear (real) transformation . . . . . 85

5.3 LS and linear-MMSE channel estimation . . . . . . . . . . . . . . . . . . . . . . 88

5.3.1 Constructing the correlation matrices . . . . . . . . . . . . . . . . . . . . 89

5.4 LMS and normalized-LMS algorithms for MIMO-channel estimation . . . . . . . 90

5.4.1 LMS channel estimator . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

5.4.2 Normalized-LMS for channel estimation . . . . . . . . . . . . . . . . . . 93

5.5 Channel estimation using RLS and adaptive-λ-RLS algorithms . . . . . . . . . . 94

5.5.1 RLS for channel estimation . . . . . . . . . . . . . . . . . . . . . . . . . 94

5.5.2 Adaptive forgetting factor for RLS algorithm . . . . . . . . . . . . . . . . 95

5.6 Simulation results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

5.6.1 Channel estimation and Alamouti scheme . . . . . . . . . . . . . . . . . 96

5.6.2 Channel estimation for DAST codes . . . . . . . . . . . . . . . . . . . . . 102

5.6.3 DAST in the presence of channel estimation errors . . . . . . . . . . . . . 105

5.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

6 Summary, Conclusion, and Suggestions for Future Work 113

6.1 Summary and conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

6.2 Suggestions for future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

A List of Abbreviations 117

Bibliography 119

IX

List of Figures

2.1 Rotated constellation in two dimensional with BPSK modulation is an exampleof the signal-space diversity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.2 General Block diagram of vector-valued transmission system using signal-spacediversity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.1 Transmission vector blocks generally contain pilot and data symbols slots . . . . 26

3.2 Theoretical capacities for MIMO (8×2) system with different average percentagesof channel estimation errors (Equation (3.16)). . . . . . . . . . . . . . . . . . . . 36

4.1 Transmission model on symbol basis for MIMO system using general coding ma-trix containing signal-space and space-time matrices. The matched filter matrixin general maximizes form the SNR at the receiving end. . . . . . . . . . . . . . 54

4.2 Rotated constellations of BPSK of different diversity orders. Half diversity orderis represented by Boutros 1,2, and 3 in addition to full diversity order whichrepresented by Bury, Damen and UEW. . . . . . . . . . . . . . . . . . . . . . . 59

4.3 BPSK and QPSK are rotated (spread) by using Hadamard matrix. The diversityorder is 8 and ”Diversity8” represents the diversity upper bound. . . . . . . . . 60

4.4 Different half diversity order (Boutros1, 2, and 3) rotation angles. The rotationmatrix is the the rotated Hadamard matrix with diversity 8 and data constella-tions are BPSK. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

4.5 Using only real values of the rotated Hadamard matrix. The rotation anglesare Boutros 1 , 2, and 3 (half diversity order) the constellations are BPSK withdiversity 8. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

4.6 QPSK constellations are rotated with rotated Hadamard matrix (diversity order8). The rotation angles design based on half diversity order. . . . . . . . . . . . 62

4.7 Real values of the rotated matrix are used for QPSK. The diversity order is 8with Boutros 1, 2, and 3 (half diversity rotation angles). . . . . . . . . . . . . . 63

X

List of Figures

4.8 BPSK data constellations with half diversity (Boutros 1, 2, and 3) comparedwith full diversity order design of Bury and Damen. . . . . . . . . . . . . . . . . 64

4.9 Rotated constellations of BPSK of different half and full diversity orders (onlyreal values of rotation matrices). . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

4.10 QPSK data constellations are rotated with half and full diversity order withdiversity 8. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

4.11 Only real values of the rotation matrix are used for QPSK with diversity 8 (halfand full diversity order design). . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

4.12 Comparison of different matrices (half and full) using rotated Hadamard androtated Fourier matrices for BPSK with diversity order 8. . . . . . . . . . . . . . 66

4.13 Real valued of different rotation matrices (half and full diversity order). The ro-tation is by using rotated Hadamard and rotated Fourier for QPSK with diversityorder 8. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

4.14 Rotated Hadamard matrix by using rotation angles of Damen (full diversityorder). The data constellations are BPSK and QPSK with diversity order 8. . . 67

4.15 UEW rotation matrix is compared with equal weighting matrix elements of halfor full diversity order. The constellations are BPSK of diversity 8. . . . . . . . . 67

4.16 Rotated BPSK with different half and full diversity orders (only real values ofrotation matrices) by using equal and unequal weights of the rotation matrixelements. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

4.17 Different half and full diversity orders by using equal and unequal weights of therotation matrix elements For QPSK. . . . . . . . . . . . . . . . . . . . . . . . . 68

4.18 Half and full diversity orders (only real values of rotation matrices) are used forQPSK. The rotation matrices based on equal and unequal weights design. . . . . 69

4.19 DAST code for 8 × 1 MIMO system. The rotation matrix is rotated Hadamardmatrix or rotated Fourier matrix. The constellations are BPSK and the detectiontechnique is ML. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

4.20 Real values of the rotation matrices are used for DAST code for MIMO 8 × 1.The rotation matrix is rotated Hadamard or rotated Fourier matrix. The dataconstellations are BPSK and ML is the detection technique. . . . . . . . . . . . 71

4.21 Half and full diversity of the rotation matrix are used for DAST code for MIMO8× 1. The rotation is by using rotated Hadamard or Fourier matrices for BPSKwith ML detection. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

4.22 DAST code and MIMO 8 × 1 for QPSK with ML detection by using differentrotation matrices. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

4.23 Real values of the rotation matrices are used for DAST codes and MIMO 8 × 1for QPSK with ML detection. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

XI

List of Figures

5.1 A Multipath propagation consists of 50 waves arriving from random directions [1]. 77

5.2 The cross-correlation function (CCF) between real and imaginary componentsof channel realizations, for fdTs = 0.001 and M = 12, calculated according to [2]. 81

5.3 The autocorrelation function (ACF) of real and imaginary components of channelrealizations, for fdTs = 0.001 and M = 12, calculated according to [2]. . . . . . . 81

5.4 The amplitude distribution of channel realizations, for fdTs = 0.001 and M = 12,calculated according to [2]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

5.5 The phase distribution of channel realizations, for fdTs = 0.001 and M = 12,calculated according to [2]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

5.6 General Block Diagram of Transmission System with channel estimation. Thechannel estimation could be estimated by using pilot symbols (training mode)or detected symbols (tracking mode) . . . . . . . . . . . . . . . . . . . . . . . . 93

5.7 ML is used for channel estimation of the Alamouti scheme. Four received anten-nas are used with QPSK with channel time variance of 100 symbols. The channelestimation is updated only at the training mode for each block as indicated by”initial channel estimate”. ”Updating channel estimate” means the channel isinitialized at the beginning of each transmission block and there is a continuesupdating by using the detected data. . . . . . . . . . . . . . . . . . . . . . . . . 97

5.8 BER of the Alamouti scheme using ML, where ML is used for, training or track-ing. To calculate the bounds, the transmitted vectors are assumed to be perfectlyknown to the receiver. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

5.9 Using different algorithms for training and ML technique is for tracking thechannel estimation. As an example, LMS-ML is LMS algorithm in training thechannel estimation and ML for tracking its estimates. . . . . . . . . . . . . . . . 99

5.10 Channel estimation for Alamouti scheme by using different algorithms for train-ing (ML, LMS, NLMS, and RLS) and LMS for tracking the estimates. . . . . . . 100

5.11 RLS algorithm is for tracking the channel estimates of the Alamouti schemeusing different algorithms for training. . . . . . . . . . . . . . . . . . . . . . . . 100

5.12 Different algorithms for training and Tracking are used for channel estimation.Four receive antennas are used for Doppler frequency fdTs = 0.001. Differentparameters have different value to compare their effect on the channel estimation.101

5.13 Training and tracking channel estimation are done using different algorithms.Different values of algorithms parameters are used for Alamouti scheme withfour receive antennas and Doppler frequency of (fdTs = 0.01). . . . . . . . . . . 102

5.14 BER of the Alamouti scheme using different algorithms for training and trackingof the channel estimation. The normalized Doppler frequency is (fdTs = 0.1)with four receive antennas. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

5.15 Different smoothing length are used with MMSE for channel estimation. TheSTC is DAST with 4 transmit and 2 receive antennas. . . . . . . . . . . . . . . 104

XII

List of Figures

5.16 MMSE, LS, and LMS are used for channel estimation for DAST with 4 transmitand 2 receive antennas. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

5.17 LMS algorithm is used for tracking channel estimates with DAST and QPSKdata constellations. Different algorithms are used for initializing LMS channelestimator for a channel having a 0.001 normalized Doppler frequency. . . . . . . 105

5.18 NLMS algorithm is used for tracking channel estimates of DAST with QPSKdata constellations. Different algorithms of training mode are used for a channelhaving a 0.001 normalized Doppler frequency. . . . . . . . . . . . . . . . . . . . 106

5.19 DAST with QPSK using RLS algorithm for tracking the channel estimation.Different algorithms are used for the training mode for a channel having a 0.001normalized Doppler frequency. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

5.20 MIMO system with 8 × 2 uses DAST with BPSK data constellations. BERcurves represent different average percentages of channel estimation errors. . . . 107

5.21 DAST is used for different MIMO systems with 4× 1, 4× 2 and 4× 4 (transmitantennas × receive antennas). The Data constellations are BPSK with MRC orEGC as a combining technique at the receiver. . . . . . . . . . . . . . . . . . . . 108

5.22 MIMO systems 8 × 1, 8 × 2 and 8 × 4 using DAST. The data constellations areBPSK with MRC or EGC to combine at the received signal. . . . . . . . . . . . 108

5.23 MIMO system 2× 1 using DAST code with QPSK. The channel is estimated byusing LMS with MRC or EGC. . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

5.24 MIMO system 2 × 2 with DAST code and LMS for channel estimation. . . . . . 110



XIII

List of Figures

XIV

Chapter 1Introduction and Motivation

One important part in our daily life is mobile communications. Now, the physicaldistances among people are eliminated. Because the prices of mobile communication servicesand handsets decrease, more and more users are joining the wireless communication society.The main problems of wireless communication are the limited resources of bandwidth andtransmit power. Therefore, the aim of researchers is to develop transmission schemes that usethe resources as efficiently as possible. This means that the bandwidth and power efficienciesshould be as high as possible. The higher the bandwidth efficiency is (number of bits that canbe transmitted per second and Hertz), the better the frequency resource is used. On the otherhand, the power efficiency is a measure of the power needed for a certain bit error rate. Thechallenge to reach this goal is the mitigation of interference and distortions that are introducedon the mobile radio link. Such distortions are path loss, shadowing, Doppler shift, intersymbolinterference due to multipath propagation, multiple-access interference and noise.

To combat the degradation effects of the communication channel, which increase the probabil-ity of error at the receiver, researchers have turned to the spatial dimension. As a matter offact, the spatial dimension has a two fold advantage: it can combat the effect of fading andshadowing and it also increases the channel capacity. The channel capacity determines theminimum required signal to noise ratio (SNR) to achieve error-free transmission at a givenbandwidth efficiency. There is no general rule on how to reach the theoretical capacity in

1

1 Introduction and Motivation

practice. Therefore, much research has been undertaken to find signal construction methodswhich can approach this bound as close as possible while employing the signal space and/orthe spatial dimension. In this thesis, we concentrate on the problem of signal diversitydesign, channel estimation and channel estimation error effects on the communication systemperformance. In real situations, channel estimation is constrained by some factors like thelimited training length and the efficiency of the estimation technique. Because of these factors,channel estimation errors result. In order to have a reliable transmission, the parametersdescribing the channel are required at the receiver. One common way to measure theseparameters is to send a known training sequence just before the transmission of unknown data.The training symbols enable estimation of the channel characteristics at moderate cost.

The most important aspects of this thesis are as follows:

• Effects of channel estimation errors on MIMO transmission

• Different design criteria of signal-space diversity

• Construction of the rotation matrices by using Hadamard or Fourier transform

• Applying signal-space concept for MIMO transmissions

• Comparison of different models for time-varying channels to be used for MIMO transmis-sions

• Channel estimation methods for MIMO transmissions

We employ a vector transmission model that is suitable to include MIMO channels.The outline of this thesis is as follows:

• Chapter 2 gives the relevant definitions, fundamentals and detailed assumptions whichwill be used throughout the whole thesis.

• Chapter 3 studies the effect of channel estimation errors with the assumption of theminimum mean square error (MMSE) criterion. We consider the effect of the channelestimation errors on the theoretical channel capacity. Furthermore, the influence of chan-nel estimation errors on the bit error rate (BER) performance is presented. Moreover, westudy the optimum combining for perfect and imperfect channel knowledge.

• Chapter 4 introduces the signal space diversity as a concept to combat the fading ef-fects on the channel. Furthermore, the design criteria for different diversity orders arepresented. The combination of signal-space and MIMO systems are presented with differ-ent combining techniques. The diagonal algebraic space-time (DAST) code constructionand its performance are also discussed.

2

• Chapter 5 describes of MIMO-channel estimation based on regularly inserted pilot sym-bols. As it has been explained in [3], the optimal channel estimation may be separatedinto a ML estimator and consecutive Wiener filter. Here, we make some modificationsof this method by using the maximum likelihood (ML) and MMSE channel estimationto perform near-optimal. The channel estimation was done for a MIMO system usingdifferent space-time codes (Alamouti and DAST) and employing different channel esti-mation algorithms. Most of them depend on the MMSE estimation technique or one of itsmodifications. Beside MMSE we use least-squares (LS), ML, least mean square (LMS),normalized-LMS, recursive mean square (RLS), and and adaptive forgetting factor λ forRLS. In addition, we consider also the possibility of using different algorithms for trainingand tracking the channel.

3

Chapter 2Fundamentals

The aim of this chapter is to give an introduction to some concepts required for under-standing this work. First, we introduce the multiple-input multiple-output (MIMO) channel toimprove the communication system performance. We include a brief overview over the mainconventions of mathematical notation in this thesis. However, specific definitions needed onlyonce or not very often will be introduced where it is necessarily. Channel state information playsan important role in the whole MIMO transmission system performance. So, its identificationwill be the main focus of this work. The minimum mean square error (MMSE) techniques areemployed to estimate the MIMO channel in concatenation with the optimal detection to get thenear-optimal receiver. Finally, signal-space diversity concept will be introduced which combatthe Rayleigh fading channel effects.

2.1 Conventions of notations

The following conventions are used through this thesis. Perfect time and frequency syn-chronization is assumed. Variables are normalized and no units are required. For mathematicalnotation, vectors have a single underline, whereas matrices have a double underline. Generally,if not stated otherwise, all quantities are assumed to be complex. Further conventions are asfollows:

5

2 Fundamentals

Symbols:j Imaginary unit. j =

√−1.

R Field of real numbers.C Field of complex numbers.0nt

Zero matrix of dimension nt × nt.

Int

Identity matrix where the dimension follows from the context.

Operators, vectors and matrices:(.)∗ Complex conjugate.(.)T Transpose.(.)H Hermitian (complex conjugate transpose).<{x} Real part of x. <{x} = 1

2(x∗ + x).

={x} Imaginary part of x. ={x} = j 12

(x∗ − x).φ Usually represents a correlation matrix.

sgn(x) Sign function of the real number x, where sgn(x) =

1 if x > 00 if x = 0

−1 if x < 0

Si(x) The sinc function, where Si(x) =

{

1 if x = 0sin(x)x

if x 6= 0‖.‖ The Frobenius norm, sometimes also called the Euclidean norm.

A matrix norm of an nr × nt matrix defined as the squareroot of the sum of the absolute squares of its elements, ‖A‖ =√∑nr

i=1

∑nt

i=1 | Aij |2 =√

trace(

AAH)

λ Eigenvalues vector of the square matrix φ with dimension nt.

rank {φ} Rank of the matrix φ.

trace{φ} Trace of the square matrix φ. trace{φ} =∑nt

i=1 λi, where λipresents the eigenvalues of the matrix φ.

det{φ} Determinant of the square matrix φ. det{φ} =∏nt

i=1 λi.

diag{x} Constructs a diagonal matrix of vector x.

Gaussian tail function Q(x) Q(x) = 1π

∫ π/2

0exp

(

− x2

2 sin2 u

)

d u.

a⊗ b Kronecker product. The Kronecker product of two matrices a ∈C{m×n} and b ∈ C{p×q} is the mp× nq matrix.

a⊗ b =

a11b a12b · · · a1nba21b a22b · · · a2nb

......

......

am1b am2b · · · amnb

vec(h) All columns of matrix h are stacked in a single column vec(h) =[

hT1 · · ·hTnt

]T.

6

2.2 MIMO channel identification


For future wireless communication systems, it is necessary to have high transmissionrate, spectral efficiency, power efficiency, and reliable transmission. Due to the multipatheffect (unlike Gaussian channels), wireless channels suffer from attenuation. Multiple copiesof the transmitted signal arrive at the receiver with a slight time shift and lead to a severeattenuation. According to this severe attenuation, it is difficult for the receiver to determinethe transmitted signal. Employing diversity techniques provides less attenuated replicas ofthe transmitted signals at the receiver. Using multiple antennas at both the transmitter andreceiver in multipath-rich wireless scenario setups to constitute MIMO (multiple-input multiple-output) antenna systems. This way, we achieve high data rate and combat the attenuation effectdue to multipath without increasing the total transmitted power or bandwidth. In addition,MIMO channel systems provides significant increase of channel capacity [4, 5, 6, 7]. If thechannel impulse response is perfectly known at the receiver, the capacity has been shown togrow linearly with number of antennas. Most MIMO detection schemes assume perfect channelknowledge to be at the receiver. In most wireless systems, the channel estimation task is doneby the aid of pilots that are known to the receiver. In addition to the reduced spectral efficiencyby using pilots, the system performance depends on the quality of the channel estimation.

2.2.1 Rayleigh fading

If a very short impulse is transmitted over a time-varying multipath channel, the receivedsignal appears as a train of impulses, which is due to the multipath progagation. The time-variations effect is caused in many applications in practice by moving the transmitter and/orreceiver. The Rayleigh fading channel is used to model the stochastic time-variant transmissionchannels if some assumption are met. A typical example is given if there is a transmission be-tween a base station and a car which moves through a city. For Rayleigh fading one imprortantassumption is, that there is an infinite number of multipath components with zero delay. Inpractice this means that the bandwidth of the transmitted signal must be small compared tothe inverse of the maximum delay observed for the multipath propagation. If the transmittedsignal in equivalent lowpass domain is sT (t), the received signal in case of a Rayleigh fadingchannel with no additive noise is given by

yT (t) = sT (t) · nT (t) . (2.1)

nT (t) is a sample function of a complex-valued Gaussian process with real and imaginary are realGaussian process, statistically independent and equal distributed with zero mean. The outputsignal of the Rayleigh fading channel can also be calculated with time–variant convolution:

yT (t) ==1

2

∫ ∞

−∞hT (τ, t) · s(t− τ)dτ . (2.2)

hT (τ, t) is the time-variant impulse response:

hT (τ, t) = δT (τ) · nT (t) , (2.3)

7

2 Fundamentals

δT (τ) is the Dirac delta impulse in the equivalent lowpas domain, i.e. δT (τ) = 2Si(2πfgτ) withfg being the cut–off frequency of the equivalent lowpass signal. δT (τ) means that the Rayleighfading channel is an ideal lowpass channel. The stochastic characteristics of the channel vari-ations are with respect to t. In this thesis, only the dependence on t will be considered, andsamples of hT (τ, t) in t–direction are taken according to the the Doppler bandwidth Bd. Thefollowin abbreviation will be used

h[k] = hT (τ, k · ∆t); ∆t =1

2Bd. (2.4)

Bd is the Doppler bandwidth of the Rayleigh channel.

2.2.2 MIMO channel model

In general form, assume for block k (i.e., certain block at certain time instant k)

y[k] = x h[k] + n[k]. (2.5)

where h[k] is the M × 1 channel coefficient vector to be estimated, x is an N × M matrixcontaining the known training data, y is the received vector with dimension N × 1 induced bythe training data and n is the N × 1 vector of the noise with zero mean and unit variance.Through the following, some assumption are considered

• N ≥M and x has a full-rank.

• n[k] and h[k] have zero mean and mutually independent and statistically characterizedby

E{h[k]hH [k]} = φhh

[k]δ[k − k], (2.6)

E{n[k]nH [k]} = σ2n[k]δ[k − k]I

N. (2.7)

where k is another time instant and k 6= k. φhh

is the autocorrelation matrix of channel and

σ2n is the noise variance.

General assumptions are considered through the channel estimation:

1. The symbol matrix x contains the hard decision of the (detected) symbols. The space-timeinterleaved sub-streams are independently mapped into the symbol matrix to constitutex.

2. Iterative channel estimation is performed using the detected pilot symbol located in apreamble and the transmitted symbols as well.

3. At the beginning of each data block, an initial channel state information (CSI) is estimatedusing only known pilot symbols. At the initialization phase of the channel estimation,the LS (least squares) estimation technique is applied since there is no need to know theknowledge of the noise power as will be explained later in this chapter.

8


Some common assumptions and considerations are considered before studying the channel es-timation of the MIMO channel. Again, if nt and nr are the number of transmit and receiveantennas, respectively.

y[k] =[

y1[k], · · · , ynr[k]]T

is the received or output vector of the channel at instant k with the dimension nr × 1,

n[k] =[

n1[k] · · · , nnr[k]]T

is the channel noise vector with dimension nr × 1, and

x[k] =[

x1[k], · · · , xnt[k]]T

is the channel input vector with dimension nt×1 and h is a matrix with dimension nr×nt whichrepresents the CSI matrix of linear MIMO system. Our assumption is generally the MIMOcommunication system has uncorrelated transmit and receive antennas and the transmissionsystem is an independent block transmission system. Also, the noise vector n consists ofcomponents which are stationary and independent of the channel input sequence blocks x. Boththe channel input x and channel noise n are white with zero-mean. As a general assumption,different antenna inputs may have different symbol powers, but it is assumed here that allsymbols have zero mean and identical variance σ2

x. h is generally the composite channel impulseresponse, which includes transmitter and receiver filter as well as the physical channel response.Without loss of generality, the channel is approximated by FIR of only one tap for each path.According to the last assumptions, there are a maximum of ntnr unknown parameters thatidentify the channel matrix h. At this point the whole MIMO system is represented by

y[k] =

h[k] 0 · · · 00 h[k] 0...

.... . .

...0 0 · · · h[k]

x[k] + n[k]

= diag(h[k])x[k] + n[k],

(2.8)

where the covariance matrices of the channel input vectors φxx

, noise vectors φnn

and channel

output vectors φyy

are

φxx

= E{x[k]xH [k]} = σ2xInt

φnn

= E{n[k]nH [k]} = σ2nInr

φyy

= E{y[k]yH [k]}

= σ2xdiag(h)(diag(h))H + σ2

nInr.

(2.9)

σ2x and σ2

n, are the input and noise variances respectively.

9

2 Fundamentals

2.2.3 Linear algorithms for MIMO channel identification

For a noiseless MIMO system, the system can be presented by

y[k] = h[k] x[k]. (2.10)

If h[k] is a full rank matrix, then there exists an nt × nr FIR estimator filter h this reduces toa ZF filter for a single tap channel such that

hH

[k] h[k] = Int

, (2.11)

that is,

x[k] = hH

[k]y[k]. (2.12)

Equation (2.11) implies that y[k] is also full rank for all time instants greater than zero. Thistechnique of channel estimation called “linear prediction algorithm“.

Then, for the noiseless channels, if h is a full rank matrix, then (h)−1 is calculated andgenerally, x[k − 1] = (h)−1y[k − 1]. Several different approaches of linear prediction algorithmextension have been presented and discussed in [8].

2.2.4 General space model

Under the assumption of uncorrelated channel paths for sufficiently largely spaced an-tennas an nr × nt MIMO channel can be separated into nr multiple-input single-output MISOsubchannels. The aim is separate channel estimation for each receive antenna. So, it is conve-nient to use the notation of the channel vector

h[k] , vec{hH [k]}=[

h1[k], h2[k], · · · , hnr[k]]

,(2.13)

where h[k] in Equation (2.13) is a vector of ntnr elements that is obtained by stacking allcolumns of the Hermitian transpose of the channel h. So, the wireless channel can be modelledas a stochastic first-order AR (auto-recursive) process of the form

hi[k] = φshi[k − 1] + ∆hi[k] ∈ C(nt×1), (2.14)

where i = 1, 2, · · · , nr and ∆h[k] is the deriving noise vector. The state transition matrix φs

models the spatio-temporal correlation of the channel as

φs

= BI ∈ R(nt×nr), (2.15)

where B = J0(2πfDTs) ≤ 1 where J0(.) is the 0-th-order Bessel function of the first kind, fDand Ts are the Doppler frequency and the reciprocal of the bandwidth (i.e. symbol period)respectively.

10

2.3 MMSE for MIMO channel estimation

2.3 MMSE for MIMO channel estimation

One of the widely used (suboptimum) parameter estimation techniques is the MMSEtechnique. MMSE technique is adopted for the channel matrix coefficients estimation h. Fora single user, the binary symbols are xi[k] ∈ {−1, 1}; i = 1, 2, · · · , nt. Assume there areindependent data block streams and no channel memory. Assume also the channel fadingsare constant within one data block and change block by block. As a matter of fact, the lastassumption is reasonable if the block duration is not longer than the channel coherence time.There is also no interblock interference (IBI) because we assume a guard interval between datablocks. Each data block contains a preamble of pilot symbols which are well known to thereceiver. The received signal at receiver l, where l = 1, 2, · · · , nnr

in the symbol interval k isgiven by

yl[k] =

nt∑

i=1

hlixi[k] + nl[k], (2.16)

Since there is no IBI, we may stack received elements of all receiver antennas in one vector yof size nr × 1

y =(

I ⊗ x)

h+ n = xh+ n (2.17)

where y = [y1, y2, · · · , ynr]T , h =

[

hT1 , hT2 , · · · , hTnr

]T, hl = [hl1, hl2, · · · , hlnt

]T , n =

[n1, n2, · · · , nnr]T and I is nr × nr identity matrix. This way, x is an nr × nrnt block diag-

onal data matrix

x = I ⊗ x

where

x =[

x1, x2, · · · , xnt

]T(2.18)

where ⊗ and nt are Kronecker product and the data block length, respectively.

This signal model is suitable for channel estimation since nrnt unknown channel parametersare collected in the vector h. Here, in our system model, we need to estimate nrnt channelcoefficients hji[k] based on the nt observations collected in the received vector y. As a matterof fact, the optimum solution to this estimation problem is obtained by maximizing the jointposteriori probability density function (pdf) of h and x

(

h, x)

= arg maxh,x

{

f(

h, x|y)}

. (2.19)

For any general data model, if the observed data and the unknown channel are jointly Gaussiandistributed, the resulting MMSE channel estimate is a linear function of the data [9]. Due tothe shape of the pdf, f(y), the exact MMSE estimator is a complicated task. So, we constrainthe channel estimator to be a linear function of y.

At a certain time instant, assuming the channel estimation to be a linear function of y, i.e.,

hLMMSE

= φ yT (2.20)

11

2 Fundamentals

where, hLMMSE

is the channel estimate matrix using the least minimum mean square error(LMMSE) technique and the matrix φ satisfies the Wiener-Hopf equation [10], where

φyyφH = φ

yh. (2.21)

φyy

and φyh

are the autocorrelation matrix of the channel output and the covariance matrix of

the channel output and the channel impulse response matrix, respectively. Remembering thath and the noise vector n have zero mean and the elements of h are i.i.d. then:

φyy

= Ex{

Eh{

En{yyH}}}

= Ex{xxH} + σ2nI

φyh

= Ex{

Eh{

En{yhH}}}

= Ex{x} = x,(2.22)

where the index of the expectation operator refers to the random variable that the expectationis taken of. x indicates with the constellation alphabets. x are the exact (hard) values forknown pilot symbols or the detected data symbols. For simplicity we omit the notation indexof the expectations, then the linear MMSE (LMMSE) channel estimator is represented by

hLMMSE

= φHyhφ−1

yyy

= x(

E{xxH} + σ2nI)−1

y.(2.23)

There is an independent estimator for each antenna l = 1, · · · , nnrwhich are decoupled

hlLMMSE = x(

E{x xH} + σ2nI)−1

yl (2.24)

and

x = diag (x)

x = diag (x).(2.25)

To simplify Equation (2.24), E{x xH} will be divided into two parts as follows

E{xxH} = {x xH + ∆x}, (2.26)

where matrix elements of ∆x are the all off-diagonal elements of matrix E{x xH}, then

hlLMMSE = x(

x xH + ∆x + σ2nI)−1

yl. (2.27)

According to [11], the last vector expression will be

hlLMMSE = (xHε−1x+ I)−1xHε−1yl, (2.28)

where ε = (∆x + σ2nI) represents the error of the MMSE estimator. If the transmitted data

are known to the receiver (as training or pilot sequences), then the estimation error, ε = σ2nI

is only due to the noise, in this case the LMMSE can be only pure MMSE. The MMSE at thetraining mode will be

hlLMMSE = (xHx+ I)−1xHyl. (2.29)

The previous arguments mean that at the training phase the exact MMSE is used for channelestimation for given pilot symbols. In data phase, the variance of unknown data symbols areincluded together with the noise of the matrix ε, which introduces the channel estimation error.

12

2.4 Detection based on the estimated channel

2.3.1 Approximate MMSE estimator

According to the result of the previous section, we replaced the vector x with its estimatedvector x. The assumption is applied that the channel estimation error equals zero. Then theestimator will be

hl =(

xH x+ σ2nI)−1

xHyl. (2.30)

This approximation leads to an acceptable performance with a small error if the number ofreceive and transmit antennas is small. The influence of the assumption of zero channel es-timation error will lead to degradations in the performance for high number of transmit andreceive antennas [11]. Both exact or simplified channel estimator using LMMSE assume in mostcases independent, complex Gaussian channel coefficients and well known noise power at thereceiver. If the prior statistical knowledge of the channel and noise are not assumed in advance,the least-squares (LS) estimation approach can be used. If x is known, then LS estimation ofthe channel will be

hlLS|x =(

xHx)−1

xHyl. (2.31)

Also, if the data vector x is unknown, then the approximate LS can be used by replacing thevector x with its estimate

hjl =(

xH x)−1

xHyl, (2.32)

where the vector x represents the APP estimates of x.

2.4 Detection based on the estimated channel

Assume x to be the generally complex-valued transmitted symbol. Here for the sake ofsimplicity, it is BPSK, where x ∈ {√Es,−

√Es}. Es is the energy per symbol. x is drawn from

an i.i.d. source. The fading channel is modelled by a correlated Rayleigh fading model wherethe channel state is

h = [h1, h2, · · · , hL]H . (2.33)

L is the number of channel paths and h is a proper 1) complex Gaussian random vector h ∼CN(0, φ

hh) and the channel noise vector is n ∼ CN(0, φ

nn) . n has also zero mean and proper

complex white Gaussian noise with covariance matrix φnn

.

φnn

= E{nnH} = N0I (2.34)

I is identity matrix with dimension L× L. The received random variable vector y will be

y = hx + n, (2.35)

where y = [y1, y2, · · · , yL]H . Using the combiner with weighting vector w,

w = [w1, w2, · · · , wL]H , (2.36)

1)The complex variance is sufficient to determine the probability density of a complex Gaussian random variablewith zero mean, this characteristic is known as proper [12]

13

2 Fundamentals

the combined decision random variable y will be

y = wHy = wH (hx + n) . (2.37)

If the channel state information h is known at the receiver side, then MRC can be applied tomaximize SNR, thus minimize BER. However, in the real situation the channel is not perfectlyknown at the receiver and only imperfect channel information is presented through a pilot (ortraining) scheme.

For general purposes we assume the imperfect channel information vector is h (it is h ifthe channel is estimated by MMSE), where

h =[

h1, h2, · · · , hL]H

, (2.38)

which is also assumed to be proper complex Gaussian with zero mean and covariance matrixφhh

, where

φhh

= E{hhH}, (2.39)

furthermore,

φhh

= E{hhH} (2.40)

is the crosscorrelation matrix of the channel observation. h and the channel state h are bothjointly proper. Although, the imperfect channel is the only one available at the receiver side,perfect channel statistics are also known. Such assumption is reasonable since the channelstatistics vary much more slowly than the channel state itself and thus can be obtained byusing enough training sequences to initialize the channel estimation. This training is repeatedoften in a continuous way to achieve this initialization.

If h is given as the channel observation, the MMSE channel estimation of h is

h = E{h|h}= φ

hhφ−1

hhh,

(2.41)

where h has zero mean and variance matrix φhh

.

φhh

= φhhφ−1

hhφHhh. (2.42)

So, the estimated channel (assuming the channel estimation task is done by using MMSE) willbe the combination of linear estimator h and the channel estimation error ∆h

h = h+ ∆h

where h (it is possible to be h) and ∆h are orthogonal and ∆h ∼ CN (0, φ∆h∆h

),

φ∆h∆h

= φhh

− φhhφ−1

hhφHhh

= φhh

− φhh. (2.43)

14

2.5 Signal-space diversity and fading effects

As a matter of fact, h is not available at the receiver, the MAP detection rule uses the conditionon the channel vector h and combiner output y

x = arg maxx

p(

x|y, h)

. (2.44)

Assume x and h are given and y is complex Gaussian distributed with mean and variance whichare respectively

E{

y|x, h}

= wH hx

VAR{y|x, h} = wH(

φ∆h∆h

Es +N0I)

w,(2.45)

and the conditional probability function pdf of the combined received vector is

p(y|x, h) =1

π(

VAR{y|x, h}) exp

(

−|y − E{y|x, h}|2VAR{y|x, h}

)

, (2.46)

then if the channel observation is independent of the transmitted symbols, the decision rule issimply the minimum distance decision rule

x = arg maxx

(x|y, h)

= arg minx

|y − wH hx|2.(2.47)

With BPSK modulation, the decision rule reduces to a threshold test

|y − wH hx0|2x=x0

≶x=x1

|y − wH hx1|2 (2.48)

or

<{

yH(

wH h)} x=x0

≶x=x1

0. (2.49)


The diversity advantage due to spreading the constellation points in the signal-space iscalled “signal-space diversity“. This diversity advantage is used for combat the fading effects ofchannels and thus approaching Gaussian channel performance as the diversity order increases.The modulation diversity techniques are commonly regarded as a good means for transmissionwith high spectral efficiency without any extension of the bandwidth or power, only with someincrease of the complexity. It depends mainly on finding new signal constellations which havemore advantages over the original constellation to overcome the channel effects by employingone more degree of freedom which are offered by the signal-space dimension. Restated theproblem in terms of finding good signal constellations for Rayleigh fading channels. That is,

15

2 Fundamentals

it is a problem of designing lattice sphere packings [13]. As a matter of fact, the linearity andhighly symmetrical structure of lattices give the chance to simplify the decoding task. Actually,the problem is still the same with more details, how to construct signal constellations withminimum average energy for a desired error rate given a certain spectral efficiency. The aim nowis to increase the diversity order in the signal-space dimension by using the multidimensionalmodulation diversity schemes.

2.5.1 Signal-space diversity

The main task to achieve this type of diversity is to make a certain rotation to the originalsignal constellation in such a way that any two points are distinguishable in the maximumnumber of distinct components. For example, through the M-PSK data constellation, if a deepfade hitting only one of the components of the transmitted vector, we can see that two pointmay collapse together due to the effect of noise. An important feature we may find (whichshould be considered in the design) in the rotation matrix is that the rotated signal has thesame performance of the non-rotated one, if they pass through the same Gaussian channel.As a matter of fact, the signal constellations which were designed in advance for the Gaussianchannels are not suitable when used over Rayleigh fading channels due to its limited diversityorder. On the other hand, the signal constellations which is matched to the Rayleigh fadingchannels, leads to a bad performance in the Gaussian channels due to the low packing densityof the lattices [13].

As an example, we consider the original constellation in two dimensional with BPSK modulationin Figure 2.1, on left side. We assumed that we have independent fadings in the two signal-space dimension. If a fade affects only one of components of the transmitted signal vector, thenas an example received points of symbol 1 and 4, 2 and 3 are close together and it is difficultfor the receiver to distinguish which symbols, 1 or 4, 2 or 3 were transmitted. Specially, incase of a fully deep fade, received points of symbol 1 and 4, 2 and 3 collapse together becausethe imaginary coordinate components of points 1 and 4, 2 and 3 are not distinguished. Now,considering the rotated constellation in Figure 2.1, on right side we can see that the receivedpoints of symbols 1 and 4, 2 and 3 do not collapse together even if fully deep fading becausewith rotated constellation all coordinate components of 4 points are distinguished. Obviously,it is easier now for the receiver to recognize the received symbols. There is a possibility ofsome signal constellations to be suitable for both channels (Gaussian and Rayleigh) and maybe also used for the Ricean channels see section 4.1. Boutros and Viterbo stated in [14] that inorder to reduce the error probability of multidimensional signal sets by employing signal-spacediversity, there are four dominant factors:

1. Minimizing the average energy per constellation point.

2. Maximizing the diversity order.

3. Maximizing the minimum-product distance between the transmitted and received set ofsignal constellations. The definition of the product distance is presented in section 2.5.2in Equation (2.56)

16


1 2

34

1

2

3

4

Rotated constellationOriginal constellation

Figure 2.1: Rotated constellation in two dimensional with BPSK modulation is an example ofthe signal-space diversity.

4. Minimizing the total number of points at the minimum-product distance (i.e. minimizingthe product kissing number). The minimum kissing number of a packing is the minimumnumber of balls touching a given one. Note that for a lattice packing this is simply thenumber of balls touching any given one, since every ball touches the same number ofothers. The problem of existence and construction of lattice packings with high kiss-ing numbers received a considerable amount of attention, and there are several knownconstructions that show that the kissing number of a lattice packing of balls [15].

2.5.2 Lattice constellations and optimum design

The optimality measure of constellation design is the error probability of such con-stellations. Due to the symmetrical shape of lattices, the error probability is written asPe(Ξ) = Pe(Ξ|x), where the lattice is Ξ and any transmitted points vector x will be x ∈ Ξ. Theunion bound gives the upper bound to the point error probability [13]

Pe (S ≤ Pe(Ξ)) ≤∑

y 6=xP(

x→ y)

, (2.50)

where S is carved from the lattice Ξ, y and is the received vector. P (x→ y) gives the pairwiseerror probability that the received vector is closer to y than to x. For AWGN, the pairwiseerror probability will be

Pe(Ξ) ≤ τc2

erfc

(

dmin/2√2N0

)

, (2.51)

where τc is the kissing number and dmin is the minimum Euclidean distance of the lattice. Theerror probability per point of a cubic constellation as a function of SNR can be upper bounded

17

2 Fundamentals

by [13]

Pe(S) ≤ τc2

erfc

(

√

3κe2κe+1

EbN0

γ(Ξ)

)

(2.52)

where κe is the spectral efficiency, κe = 2mnE

, m is the number of bits per symbol and nE is thenumber of dimensions of the Euclidean space, γ(Ξ) is the fundamental gain of Ξ and Eb is theenergy per bit.

γ(Ξ) =d2

min

vol(Ξ)2/nE

, (2.53)

where γ(Zn) = 1 (Zn is the n-dimensional integer lattice), so γ(Ξ) is the asymptotic gain of Ξover Zn. For spherical constellations the total gain should take into account the shape gain.vol(Ξ) is the fundamental volume of lattice Xi, some examples of its calculation are presentedin [16]. For the Rayleigh fading channel, the approximate of the pairwise error probability is

P (x→ y) ≤ 1

2

nE∏

i=1

1

1 + (xi−yi)2

8N0

. (2.54)

For high values of SNR

P (x→ y) ≤ 1

2

∏

xi 6=yi

1(xi−yi)2

8N0

=1

2

1

(κe

8Eb

N0)ldlmin(x, y)2

, (2.55)

where dlmin(x, y)2 is the normalized l-distance of x from y when these two points differ in lcomponents;

dlmin(x, y)2 =

∏

xi 6=yi(xi − yi)

2

(Eb/nE)l. (2.56)

Equation (2.56) explains the reason of using the property of maximizing the minimum productdistance in the design of the rotation matrix. Of course, the reason is minimizing the pair wiseerror probability.

2.5.3 Multidimensional constellations design

Constructing multidimensional lattice constellations with high diversity needs a math-ematical tool which can be found in algebraic number theory. A lot of work was done usingalgebraic number theory to reach the optimization which may fulfill the four items that werementioned in section 2.5.1, see for example [13], [14], [16], [17], and many others. The aim isto get a new constellation which can overcome the effects of Rayleigh fading channel. Let x bea rotated vector from the rotated constellations, where

x = ms (2.57)

s is the original constellation vector (their points belong in general belonging to M-PSK con-stellations). m is the rotation matrix, which may be used to overcome the effects of fadingchannel. Figure 2.2 gives a schematic diagram which represents the block diagram of such a

18

2.6 Summary

VED+MOD

Bit-to-symbols Signal-spacematrix

Multipath/MIMOchannel

Vectordetection

Modification to have nearGaussian channel performance

PSfrag replacements

a s x

n

y x → a

m h

Figure 2.2: General Block diagram of vector-valued transmission system using signal-spacediversity.

system with rotated constellation. The channel is characterized as an independent Rayleighfading channel. Also assume perfect channel state information and phase recovery at the re-ceiver without intersymbol interference, i.e., the channel is assumed memoryless. The aim is toapply a mathematical tool to construct lattice multidimensional constellations with maximum(high) diversity order (gain).

2.6 Summary

After introducing the notations used in this thesis and the MIMO channel and how it canbe identified. An interesting method to use the linear algorithms to identify the MIMO channeland accordingly one can establish a general space model. The general MMSE technique wasused to estimate the channel and the optimal detection rule is also introduced. The signal-space diversity was given as a tool to combat the Rayleigh fading channel effects. The followingchapters extend our previous discussion and address some details about dealing with MIMOchannel effects from the transmitter and receiver sides.

19

Chapter 3MIMO Channel Estimation in the Presence of

Errors

Because of the importance and the large number of channel parameters to be estimatedin MIMO systems, the channel estimation errors have an important effect on the systemperformance. For system analysis and design, there is a theoretical and practical relevance toderive a lower bound depending on the channel estimation errors [18]. In [19], the author isdealing with the SISO (single-input single-output) system assuming perfect and imperfect CSIat the receiver (assuming there is no feedback link between receiver and transmitter). Thecontrol or prediction of the many variables of wireless communication channels are hard tobe achieved in most cases. When the receiver moves and faces many obstacles, the receivedsignal suffers of shadowing and multipath. The channel capacity might vanish in the presenceof imperfect channel knowledge at the receiver. The assumption of known signal-to-noise ratiocan lead to tight upper and lower bound of the capacity which are independent of the chan-nel distribution [19]. In [20], authors extend the work of the SISO case in [19] to the MIMO case.

This chapter deals with the MIMO channel estimation errors which are required for un-derstanding the effect of the estimation errors from the capacity bound, error modelling,optimum and sub-optimum combining point of views. First, we introduce the capacity withimperfect channel and its lower bound in the presence of estimation errors. Then we deal withestimation errors if the optimum or sub-optimum receiver is used. Subsequently, the MMSE

21

3 MIMO Channel Estimation in the Presence of Errors

and the approximate MMSE for channel estimation are introduced. Afterward, we investigatethe effect of channel estimation error performance with optimal and sub-optimal combiningtechniques. Finally, we introduce the signal space-diversity to reduce the estimation errorseffect.

3.1 Channel capacity with imperfect channel knowledge

The main advantage of using MIMO systems is the capacity gain by exploiting thespatial dimension [4, 5, 6, 7]. But this capacity could be affected if there are some errors in thechannel estimation. Consider a MIMO system with nt transmit and nr receive antennas. Thediscrete-time transmission system is modeled as

y = h x + n. (3.1)

Assuming both h and n to be quasi-stationary (through a certain block length which equalsat least the number of transmitting antennas nt), and the entries of each are i.i.d and have zero-mean. The channel and noise variances are normalized such that the entries of h and n haveunit variance (normalized power). As a matter of fact, the suitable scaling of the transmittedpower does not change the mutual information (or capacity) of the channel. It is also assumedthat both the fading process of the channel and the noise processes are uncorrelated through thetime instances. Through the described system above, if the fading process is perfectly knownto the receiver, the mutual information between the input vector x and the output vector y ofthe channel is given by

I(x; y) = E{(log2 | I + hHhφxx

|}, (3.2)

where φxx

= E{xxH} is the input covariance matrix and E{.} is representing the expectation

operator.

Assume the receiver performs MMSE estimation of the channel, which means

h = h+ ∆h, (3.3)

where h is the estimated channel matrix and ∆h is its estimation error matrix. By this repre-

sentation assumption h and ∆h are uncorrelated, and the entries of ∆h are zero-mean complexstationary Gaussian with variance

σ2∆h = MMSE = E{h2

ij} − E{h2ij} (3.4)

The variance σ2∆h is the measure quality of the channel estimation, which may be known to the

transmitter and/or the receiver.

3.1.1 Lower bound of the capacity with erroneous estimated channel

The capacity C of the channel is the maximum of the mutual information I(x; y | h),

given that the channel estimation h is known. To calculate the lower bound of the mutual

22

3.1 Channel capacity with imperfect channel knowledge

information, it will be expanded to the differential entropy

I(x; y | h) = h(x | h) − h(x | y, h), (3.5)

given φxx

and choosing (x | h) to be Gaussian 1) [20]. The first term of RHS in Equation (3.5),

the differential entropy will be E{log2(πeφxx

)} and the second term will be upper bounded by

the entropy of a Gaussian random variable whose variance is equal to the mean square error ofthe linear MMSE estimate of x given y and h. So, the lower bound of the mutual informationwill be

Ilower(x; y | h) = E{log2 | I + hH

(I + φ∆hx

)−1hφxx

|}. (3.6)

Because of the independence of entries of the channel matrix h, the channel estimation errors∆hij are also independent and identically distributed, which means that

E{∆hij∆hmn} = σ2∆hδi−m,j−n (3.7)

andφ

∆hx= σ2

∆hPI, (3.8)

where P is the total transmitted power and then the definition of the mutual informationbecomes 2)

Ilower(x; y | h) = E{log2 | I +1

1 + σ2∆hP

hHh φ

xx|}. (3.9)

From Equations (3.2) and (3.9), the channel estimation error results play the same role asSNR loss factor of η = (1 − σ2

∆h)/(1 + σ2∆hP ). Aiming to reach an upper bound of the mutual

information, we will write the mutual information in an alternative way

I(x; y | h) = h(y | h) − h(y | x, h). (3.10)

Then the upper bound will be reached if

h(x | h) ≤ E{log2 | πe(hφxxhH

+ (1 + σ2∆hP )I) |} (3.11)

where the Gaussian distribution maximizes the entropy over all distributions with the samecovariance. Since the channel error elements of the matrix ∆h are Gaussian distributed (by

assumption), then h(y | x, h) is also complex Gaussian with mean hx, and variance (φ∆hx

+ I)

andh(y | x, h) = Ex{log2 | πe(1 + σ2

∆h ‖ x ‖2)I |} (3.12)

and the upper bound of the mutual information will be

Iupper(x; y | h) = Ilower(x; y | h) + nrEx

{

log2

(

σ2∆hP + 1

σ2∆h ‖ x ‖2 +1

)}

(3.13)

From the definition expressions of the lower and upper bounds of mutual information in Equa-tions (3.9) and (3.13), it is important to say that, the difference between these two bounds is

1)If the channel is not perfectly known, the distribution could be considered as Gaussian.2)The whole derivations could be found in [19].

23


usually small for a Gaussian inputs unless nr � nt. In other words, the two bounds are approx-imately the same for the Gaussian mutual information. Taking the limit of the second termfor high SNR in Equation (3.13), it approaches (nr/nt)log2

√e ≈ 0.71(nr/nt) for Gaussian input.

In Rayleigh fading channels, the optimal correlation matrix φxx

that maximizes the mu-

tual information (capacity) is spatially white, φxx

= (P/nt)I (i.e, temporal power adaptation

does not increase the capacity [20]); it means that P = p, where p is the average power. Thechannel matrix can be represented by single value decomposition,

h = u d vH , (3.14)

where u and v are unitary matrices and d = diag(P1, · · · , Pnt) is a diagonal matrix and its

elements are determined from the eigenvalues

Λ = hHh (3.15)

λi is the (i, i)th element of Λ.Therefore, the capacity lower bound is given by

Clower =nt∑

i=1

E

{

log2

(

1 +p/nt

1 + σ2∆hp

λi

)}

(3.16)

As a conclusion at this point, despite the channel estimation error, the mutual informationincreases linearly with the smaller number of transmit and receive antennas, but it is limitedby the estimation errors in the high SNR [20].

3.2 The influence of channel estimation on optimum and

suboptimum receiver

In the MIMO wireless channels, the transmitted signal could suffer from multiple accessinterference (MAI)( if multiuser access is employed), multiple propagation and additive noise.Iterative receiver algorithms mitigate these signal impairments, while offering a good tradeoffbetween performance and complexity. The correspondence between the CDMA and MIMOsystems may be useful to mitigate the effect of MAI and also ISI [21], [11] by using the sametechniques for both systems according to this correspondence. The direct correspondencebetween CDMA and MIMO systems can be easily established: a CDMA with Ns users ischaracterized by an N × Ns spreading matrix, whose columns are N-chips long spreadingsequences for each user. A MIMO system, with Ns = nt transmit antennas and N = nrreceive antennas, is characterized by an nr × nt channel matrix whose entries are channelimpulse responses for every transmit-receive antenna pair (for a flat-fading channel the entriesare scalar coefficients while in the frequency-selective channel, the entries are vectors of thechannel coefficients). While the decoding and detection for the two systems are obviouslyvery similar, channel estimation in wireless systems in MIMO and in CDMA is considerably

24

3.2 The influence of channel estimation on optimum and suboptimum receiver

different task [21], [11]. The optimum receiver performs multiuser detection and decoding ina combined way. In fact, the channel estimation process will be included in parallel with thedetection and decoding process, where both have the information required for the estimatoroperation. Conversely, the optimum receiver suffers from large complexity (interpretably,together with number of users and block/codeword length). The commonly used suboptimummethodology is the separation of the detection and decoding procedure 3).

We assume that, the channel state information (CSI) is quasi-static and the transmitted datapower is equally distributed through the transmitted antennas. For block transmission, everyblock contains nt symbols. Now, x = [x1, x2, · · · , xnt

] represents the data block which is sentthrough channel. s is the spreading matrix, which represents how x will be spread through thetransmit antennas with dimensions nt × nt. Additionally,

x = s diag(x), (3.17)

where x represents the spread transmitted data

x =

x11 · · · x1nt

.... . .

...xnt1 · · · xntnt

∈ Cnt×nt, (3.18)

and the received matrix will be

y = hx + n ∈ Cnr×nt, (3.19)

where the channel matrix is represented by

h =

h11 · · · h1nt

.... . .

...hnr1 · · · hnrnt

∈ Cnr×nt (3.20)

and the noise matrix is

n =

n11 · · · n1nt

.... . .

...nnr1 · · · nnrnt

(3.21)

So, at each time instant, the received vector will be

y = h x+ n = x h+ n. (3.22)

x = xT ⊗ Inr

where ⊗ is the Kronecker product symbol; x = [x1, · · · , xnt]T and h = vec(h) =

[

hT1 , · · · , hTnt

]Tis the vector in which the columns of h are stacked in a big column with dimen-

sion ntnr and the same for n = vec(n) =[

nT1 , · · · , nTnt

]T.

3)The capacity loss due to the separation between the detection and decoding could be found in [22]. Whenthe channel is assumed to be unknown to the receiver, doing the optimum receiving could be a hard task.

25


Pilots Data

Transmission symbols block

Pilotsymbols

Data symbols

PSfrag replacements xp,i xd,i

xi

Figure 3.1: Transmission vector blocks generally contain pilot and data symbols slots

In the training based channel estimation, the transmitted power is divided between the dataand pilot symbols, then the transmitted symbol block xi is composed of data and pilots partsas explained in Figure 3.1 as:

xi = xd,i + xp,i, (3.23)

where xd,i are either zero or i.i.d data symbols with zero mean and variance σ2d and xd,i are

pilot symbols with power σ2p. Whatever the pilot design, the channel input/output relationship

is represented by the modified version of Equation (5.6) as

y = hdxd + h

pxp + n = (x

d+ x

p)h + n (3.24)

3.2.1 Modelling channel estimation errors

In the presence of the channel estimation error, the channel matrix is represented (if thechannel estimation is done by using MMSE technique as explained in section 3.1) by

h = h+ ∆h,

then the Equation (5.6) will be

y = h x + n

n = ∆h x+ n.(3.25)

Equation (3.25) is based on the following assumptions [23]:

• The elements hij of the channel matrix h are complex Gaussian i.i.d. and the coefficients∆hij of the channel estimation error matrix ∆h are also complex Gaussian.

• Unbiased channel estimator, i.e., the channel estimation error has a zero mean.

• There is no correlation between channel estimation error ∆h and the data vector x.

According to the last assumptions the covariance matrix of n is φn, where

φn

= E{

∆h x xH∆hH}

+ σ2nI, (3.26)

26

3.2 The influence of channel estimation on optimum and suboptimum receiver

the coefficients of this matrix are given by

φn(i,j) = E{

nin∗j

}

=

nt∑

k=1

nt∑

l=1

E{

∆hikxkx∗l ∆h

∗jl

}

+ E {ninj} . (3.27)

Because there is no correlation between channel estimation error and data symbols then

E{

∆hikxkx∗l ∆h

∗jl

}

= E {xkx∗l }E{

∆hik∆h∗jl

}

(3.28)

The assumption of i.i.d data symbols leads to E {xkx∗l } = σ2d +σ2

p or, E {xkx∗l } = σ2x. This way,

φn(i,j) = σ2x

nt∑

k=1

E{

∆hik∆h∗jk

}

; i 6= j

= σ2x

nt∑

k=1

E{

| ∆hik |2}

+ σ2n; j = i,

(3.29)

in generalφn

= σ2xE{

∆h∆hH}

+ σ2nI. (3.30)

3.2.2 Cramer-Rao bound as a quality measure for channel estimation

The Cramer-Rao bound (CRB) is applied to specify the theoretical and the necessaryconditions of performance limits for some information signal processing. CRB is the lowerbound on the mean square error in sense of an unbiased MSE estimation of a certain parameter.Assuming the parameter now, hi is the estimate of the channel vector parameter hi, then

E{

(ˆhi − hi)(ˆhi − hi)

T}

,≥ CRBi (3.31)

where i is the vector number indication and the bar over the vectors represents the real valueindication 4) ∀ i = 1 · · ·nr and CRBi is the CRB for the vector i. CRBi is relative to thechannel estimation vector which can be calculated by the inversion of the Fisher informationmatrix; more insight details are found in [24], where

nr∑

i=1

E[

∆h∆hH]

≥nr∑

i=1

CRBi, (3.32)

this way, the effective noise has a lower bound of the covariance matrix φn,

φn

= σ2x

nr∑

i=1

CRBi + σ2nI. (3.33)

In [24], CRB is calculated for assessing the performance of the channel estimators, if there issome priori information about the transmitted sequence by the receiver side. Both calculationsof CRB for the deterministic and random channels are also considered.

4)Converting the complex vectors or matrices to the real value representation could be found in section 5.2.2.

27


3.3 Diversity combining with channel estimation errors

The use of the combining techniques have been employed at the receiver to combat theeffect of channel fading by collecting the received signals with a certain combining technique.Moreover, many researches are considered and assumed perfect channel knowledge for usingthe combining techniques at the receiver side. Maximum ratio combining MRC is the optimumcombining technique to minimize the bit error rate (BER) if the channel state informationis perfectly known to the receiver. Unfortunately, in real life the channel is not perfectlyknown to the receiver and this estimation errors may cause some degradation of the BERperformance. The optimal performance is obtained, if the detection technique is optimal, basedon the maximum a posteriori (MAP) detection technique with the assumption of perfectlyknown channel at the receiver (which is not the case in most real situations). Consideringhybrid system that combines space-time coding (STC) with signal-space diversity 5). In reality,the signal-space diversity introduces the interleaving without any loss of power or bandwidthefficiency but in the traditional interleaving there is an additional complexity and delay due tothe interleaving procedure. In addition, if the number of transmitting antennas is nt = 8, thenit is enough to achieve close to the performance of a perfectly interleaved system [25].

3.3.1 Optimal combining and its error performance

The measure of the system performance is the average BER (P∆) over additive noise n,channel state h, channel observation h and the transmitted symbol x, is calculated as

P∆ =∑

x

Pr(x)

∫ ∫ ∫

Pr(∆h|n, h, h, x) p(n, h, h) dn dh dh, (3.34)

where

Pr(∆h|h, x) =

∫ ∫

Pr(∆h|n, h, h, x) p(n, h|h, x) dn dh (3.35)

and Pr(.) represents the average probability. According to the assumption that the channel his independent of the symbols x, we get

P∆ =∑

x

Pr(x)

∫

Pr(

∆h|h, x)

p(h) dh. (3.36)

In case of perfect channel knowledge:This means that h = h = h and Equation (3.36) will be

P∆ =∑

x

Pr(x)

∫

Pr (∆h|h, x) p(h) dh. (3.37)

5)Signal-space diversity is discussed in some details in chapter 4.

28


Equation (3.37) indicates that the error is only because of the additive Gaussian noise. Giventhe perfect channel knowledge, the MAP detection rule shows that the conditional error prob-ability Pr (∆h|h, x1) is

Pr (∆h|h, x1) = Pr(

<{yH(

wH h)

} ≤ 0|h, s1

)

= Q

(√

2|wHh|2Es|wHw|N0

)

.(3.38)

Due to the symmetry in BPSK

Pr (∆h|h, x0) = Pr (∆h|h, x1) (3.39)

This way, the conditional error probability is minimized when w = h and after averaging overx, we get

minwPr (∆h|h) = Q

√

2|h|2EsN0

(3.40)

Equation (3.40) is exactly the MRC, which is optimal when the channel is perfectly known atthe receiver and the noise n is white. At the combiner output the average BER will be

P∆ =

∫

Q

(

√

2SNRh

)

p(h) dh (3.41)

where

SNRh =|h|2N0

Es (3.42)

SNRh is the output SNR of the combiner for BPSK and the channel is perfectly known.

In case of MRC (as optimal combiner) with imperfect channel knowledge:Now, the channel state information statistics (φ

hh, φ

hhand φ

hh) are known at the receiver

and the channel state is not perfectly known. MMSE for estimating the channel can be doneas h = φ

hhφ−1

hhh. In general the channel is decomposed as h = h + ∆h where ∆h has zero

mean, variance φ∆h∆h

and is independent of h. By applying the MAP, the conditional error

probability will be

Pr(∆h|h, x1) = Pr(

<{yH(

wH h)

} ≤ 0|h, x1

)

= Q

√

√

√

√

2|wH h|2Es(wH(φ

∆h∆hEs +N0I)w)

.(3.43)

For BPSK, Pr(∆h|h, x0) = Pr(∆h|h, x1) = Pr(∆h|h, x) and the required w is to minimizePr(∆h|h, x) which becomes

w = (φ∆h∆h

Es +N0I)−1h, (3.44)

29


averaging over all x,

minwPr(∆h|h) = Q

(

√

2 hH

(φ∆h∆h

Es +N0I)−1hEs

)

(3.45)

It is clear that the channel estimation error ∆h is unknown, but the estimation of the channelby using MMSE is known and the estimation error ∆h is independent of h and also h. This way,the estimation error ∆h can be added to the Gaussian noise vector n and the total noise willbe ∆h + n which has zero mean and variance, φ

nn= φ

∆h∆hEs +N0I. With the fact that, the

elements of ∆h are mostly not white. So to get the suitable combiner, a whitening operationshould be done for the both noise terms ∆h+n. The output SNR (for BPSK) of the combinerwill be

SNR0 = hH(

φ∆h∆h

Es +N0I)−1

hEs

= hHφ−1

nnhEs

(3.46)

The average error probability will be

P∆ =

∫

Q(√

2SNR0) p(h) dh. (3.47)

As soon as Pr(∆h|h) is determined, the error performance P∆ is evaluated by averaging theconditional error over all possible channel observation as in Equation (3.47). Using Equation(3.46) to obtain the distribution of SNR0, Equation (3.47) becomes

P∆ =

∫

Q

(

√

2 SNR0

)

p(SNR0) d SNR0, (3.48)

where the Gaussian tail function is defined as

Q(x) =1

π

∫ π/2

0

exp

(

− x2

2 sin2 u

)

d u. (3.49)

When the combiner output SNR0 is in a quadratic form as it can be found here, P∆ can becalculated in another way as

P∆ =

∫

h

1

π

∫ π/2

0

hHφ−1

nnhEs

sin2 u

.1

πNdet(φhh

).exp

(

−hHφ−1

hhh)

du dh

=1

π

∫ π/2

0

∫

h

1

πNdet(φhh

)exp

(

hH

(

φ−1

nnEs

sin2 u+ φ−1

hh

)

h

)

dh du

(3.50)

Let

Λ = φhhφ−1

nnEs (3.51)

30


where {λi}Ni represents the set of eigenvalues of Λ, then after integration with respect to the

h, we get

P∆ =1

π

∫ π/2

0

(

det

(

Λ

sin2 u+ I

))−1

du

=1

π

∫ π/2

0

(

N∏

i=1

(

λisin2 u

+ 1

)−1)

du

=1

π

∫ π/2

0

(

N∏

i=1

Ni∏

k=1

βi,k

(

λi

sin2 u+ 1

)k)

du,

(3.52)

where N is the number of distinct eigenvalues, Ni is the eigenvalue’s multiplicity and βi,k isthe i− th residue associated with the k− th power in the partial fraction expansion (the wholederivation is found in [26]). As a conclusion, P∆ will be as follows

P∆ =1

π

∫ π/2

0

(

λ

sin2 u+ 1

)−Ndu

=N−1∏

i=1

(

N − 1 + i

i

)

(

1 − f(λ))i (

f(λ)N)

(3.53)

where, f(λ) = 12

(

1 −√

λ1+λ

)

and all the eigenvalues are the same and equal to λ. In the case

of different eigenvalues

P∆ =N∑

i=1

βi1

π

∫ π/2

0

(

λisin2 u

+ 1

)−Ndu

=

N∑

i=1

βi f(λi)

(3.54)

and

βi =∏

i6=k

(

1 − λi

λk

)−1

(3.55)

f(λi) = 12

(

1 −√

λi

1+λi

)

. (3.56)

Note that: The error performance P∆ in Equation (3.41) under perfect channel channelknowledge is in the same form as in Equation (3.52); the only difference is that {λi}Ni=1 shouldbe replaced by the eigenvalues {λi}Ni=1 of

Λ = φhhN−1

0 Es. (3.57)

3.3.2 Suboptimal combining and its error performance

The scenario of using MRC combining technique with h as the true channel state is notoptimum any more due to the channel estimation errors. This way, it is hard to implement

31


the optimum combining technique. An example of sub-optimal combining is to treat h as if itis the true channel h and to be used for MRC combining. In this case, the combining weightvector is set to be w = h, and the real part of the decision variable y = wH (hx + n) will beused for the threshold test

y = <{

hH

(hx+ n)} x=x0

≶x=x1

0. (3.58)

The sub-optimum BER will be

P∆ =∑

x

Pr(x)

∫

Pr(

∆h|h, x)

p(h)dh (3.59)

where

Pr(

∆h|h, x)

= Pr(

y ≥ 0|h, x = −√

Es

)

= Q

(√

2(

<{hH h})2

EshH(

φ∆h∆h

Es +N0I)

h

)

.(3.60)

Then for the effective SNR, SNReff (BPSK data constellation) it holds

SNReff =

(

<{hH h})2

Es

hHφnnh

≤ |hH h|2EshHφnnh

≤| (√

φnn

)−1h |2 Es = hHφ−1

nnhEs = SNR0 (3.61)

Equation (3.61) shows that the suboptimal BEP is lower bounded by the optimal performanceor the minimum BEP [26].

3.4 MIMO with signal-space diversity in the presence of

channel errors

Space-time codes combined with simple linear operations at the receiver, can achieve fulldiversity gain. Unfortunately, full-rate space-time block codes do not exist for general complexsignal constellation (except for two transmit antennas as in Alamouti’s scheme [27]). Moreover,decoding space-time block codes requires good channel estimation. In the presence of channelmismatching and the resulting loss of orthogonality at the receiver among transmitted antennasignals, ISI could be introduced and the performance is degraded. For nt transmit antennas, thedata stream is split into nt copies, each is rotated and fed to the transmit antennas. Basically,the transmitted data can be spread using a specific STBC to be sent over multiple transmitantennas. The data are split into frames and repeated with time to send through transmittingantennas via space to the receiving antennas. The symbols will be repeated depending onthe used space-time code and the code rate. If the code rate is one it means the number oftime slot will equal the transmit antennas nt, and this will be the assumption here. Now,the idea is to rotate the data constellation through the signal space direction and sending it

32

3.4 MIMO with signal-space diversity in the presence of channel errors

through space-time directions. Assuming the number of receiving antennas is nr = 1 and theextension to multiple receive antennas is straightforward. The input vector of the rotation partis x = [x(0), x(1), · · · , x(L− 1)] and the output will be x = [x(0), x(1), · · · , x(L− 1)],

xT = mxT , (3.62)

where L is the frame size and m is the rotation matrix of L×L complex valued elements 6) andit is well known to the receiver. A Rayleigh quasi-static fading channel is considered in whichthe fading remains constant during the whole frame L and varies frame by frame. It can bedescribed by the independent fading vector

h = [h1, h2, · · · , hnt]

= [|h1|ejθ1, |h2|ejθ2, · · · , |hnt|ejθnt ],

(3.63)

where |hi| is a Rayleigh distributed random variable with unit variance, θi is uniformly dis-tributed, where 0 ≤ θ ≤ π and hi is the fading gain from the i − th transmit to the receiveantenna.The received data is a noisy superposition of the nt transmitted signal. For the aim ofillustration assume that we have only one receiving antenna and the transmitted data streamswill be in sub-stream with length of the number of transmitting antennas nt and each element ofthis sub-stream will be multiplied by a general complex scaler so, x = [x(1), x(2), · · · , x(nt)] andthe output of the rotation part will be x = [x(1), x(2), · · · , x(nt)]. Generally, x(i) = m(i)x(i);m(i) is a complex scaler value. In this case, assuming there is no ISI and the total receivedpower is Es. The complex matched filter output for the symbol x(i) with the all copies is

y(i) =

√

Esnt

nt∑

k=1

mk(i)|hk|ejθkx(i) + n(i) (3.64)

∀i = 1, 2, · · · , nt, n(i) is a complex Gaussian random variable with zero mean and varianceN0

2in each dimension. For PSK (or QAM) mk(i) = ejφn(i), where φn(i) is a discrete variable ;

0 ≤ φn ≤ π, then Equation (3.64) will be

y(i) =

√

Esnt

nt∑

k=1

|hk|ej(θk+φk(i))x(i) + n(i)

=√

Esα(i)x(i) + n(i),

(3.65)

where α(i) =√

1nt

∑nt

k=1 |hk|ejψk(i) and ψk(i) = θk + φk(i) is the effective fading over i− th

symbol and it has zero-mean, complex Gaussian random variable with variance 0.5 in eachdimension. Although, the quasi-static fading is assumed for each channel path, the effectivefading is non-static, thus a diversity gain can be achieved.

Perfect channel performance:With perfect CSI at the receiver, the optimum detection performs as

x = arg min{x(i)}

| y(i) − α(i)x(i) |2 . (3.66)

6)More details of the design and characteristics of such matrices can be found in chapter 4

33


The pairwise error probability of transmitting x and decoding in favor of x given CSI and mcan be upper bounded by

P(

x→ x|m, {|h|}, {θj})

≤ 1

2e

−SNR·A

4 (3.67)

Where SNR = Es

N0and A =

∑

i∈τ D(i)|α(i)|2; τ is the set of time indices such that x(i) 6= x(i)

and D(i) = |x(i) − x(i)|2. The size of τ is the effective length and will be represented by din the sequence.

For two transmit antennas, let m be a matrix contains all the rotation angles as

m =

(

φ1(1) φ1(2), · · · , φ1(L)φ2(1) φ2(2), · · · , φ2(L)

)

(3.68)

and D =∑

i∈τ D(i). According to [25], Equation (3.67) will be

P(

x→ x|m)

=1

2

1

(D · SNR/4 + 1)2

∞∑

k=1

(

rc · SNR

D · SNR + 4

)2k

, (3.69)

where r2c =

(∑

i∈τ D(i) cosφ(i))2

+(∑

i∈τ D(i) sinφ(i))2

. By taking the expectation of Equation(3.69) with respect to m, we get

P(

x→ x|m)

≤ 1

2

1

(D · SNR/4 + 1)2

∞∑

k=1

(

SNR

D · SNR + 4

)2k

Em{r2kc } (3.70)

Em{.} is the expectation with respect to m. In [25], the calculation of P (x→ x) for theAlamouti’s scheme is presented for using random phase rotation of the original constellation.

Imperfect channel performance:In this case the channel fadings will be

h = h+ ∆h,

where ∆h has the same statistical properties as h but with variance σ2∆, then

α(i) =1√nt

nt∑

k=1

hkejφk(i)

= α(i) + ∆α(i),

(3.71)

where α(i) = 1√nt

∑nt

k=1 hkejφk(i) and ∆α(i) = 1√

nt

∑nt

k=1 ∆hkejφk(i). This way, the channel

estimation error with rotation introduces additional noise to the received signal, which is acomplex, zero-mean, Gaussian variable with variance σ2Es, given symbol energy Es. Thisextra noise is correlated in time and using this erroneous channel estimation will lead to asub-optimum receiver and the conditional pairwise error probability will be

P (x→ x|{α(i)}) = P

(

∑

i∈τ<{α∗(i) (x∗(i) − x∗(i)) y(i)} < 0|{x(i)}{h(i)}

)

. (3.72)

34

3.4 MIMO with signal-space diversity in the presence of channel errors

Let

V =∑

i∈τ<{α∗(i) (x∗(i) − x∗(i)) y(i)}

= <{

∑

i∈τ

(

|α(i)|2x(i) (x∗(i) − x∗(i)))

+∑

i∈τ(α∗(i)∆α(i)x(i) (x∗(i) − x∗(i)))

}

+∑

i∈τ<{(α∗(i) (x∗(i) − x∗(i))n(i))} ,

(3.73)

where V is a real Gaussian random variable condition on {α(i)}, {x(i)} and {x(i)}. For BPSK;V has a mean of

µv = 2√

Es∑

i∈τ|α(i)|2 (3.74)

and the variance is bounded by

σ2v =

(

|∑

i∈τα(i)e−jφ1(i)|2σ2Es + |

∑

i∈τα(i)e−jφ2(i)|2σ2Es + 2

∑

i∈τ|α(i)|2N0

)

≤ 2d · σ2∑

i∈τ|α|2Es +

∑

i∈τ|α|2N0

= 2Es

(

d · σ2 +1

SNR

)

∑

i∈τ|α(i)|2.

(3.75)

From Equations (3.74) and (3.75), Equation (3.72) can be bounded by

P (x→ x|{α(i)}) ≤ 1

2exp(−

∑

i∈τ |α(i)|2d · σ2 + 1/SNR

) (3.76)

In Equation (3.76), α(i) is a function of h1 and h2 for high SNR, assuming the pdf of hj tobe the same as that of hj for all j = 1, 2. Taking the expectation with respect to the channelestimates {α(i)}, the pairwise error probability is upper bounded by

P(

x→ x|{h(i)})

≤ 1

2

1

( ˜SNR + 1)2

∞∑

k=1

(

˜SNR/d

4 · ˜SNR + 4

)2k

Em{r2kc } (3.77)

where˜SNR =

1

σ2 + 1/(d · SNR). (3.78)

For the Alamouti’s scheme with random rotation angles of the data constellation, the pairwiseerror probability for general MPSK or QAM could be found in [25]. When the free distance dis large, the channel estimation error dominates the error probability performance. For highernt the performance gain is also achieved when the signal constellation is rotated. However, thegain becomes less significant as nt increases. The diversity order achieved by the constellationrotation is upper bounded or can be maximized by the min (nt, d), i.e., when nt increases, the

35


−4 −2 0 2 4 6 8 10 120

1

2

3

4

5

6

7

8

9

SNR [dB]

Cap

acity

(bp

s/H

z)

Perfectly known channel10% channel estimation error30% channel estimation error

Figure 3.2: Theoretical capacities for MIMO (8× 2) system with different average percentagesof channel estimation errors (Equation (3.16)).

effective fading becomes progressively independent in time and the correlation coefficient ofeffective fading amplitude at distinct time is inversely proportional to the number of antennasnt, the proof of this can be found in [25]. It is important here to state that, the extra noisedue to the channel estimation error for the Alamouti scheme with rotation has smaller variancecompared with the traditional Alamouti without rotation, the proof can be found also in [25].

3.5 Summary

Modelling the error of the channel estimation as a Gaussian distributed error isa way of studying the effect of channel estimation error on the channel capacity. Thisassumption is principally based on using MMSE or any of its modification versions forchannel estimation. Figure 3.2 presents the effects of some accuracies, assuming percentagesof channel estimation error 10% and 30%, where the channel capacity is calculated forMIMO system with two receive and EIGHT transmit antennas. Figure 3.2 shows that theMIMO channel capacity is dramatically decreased if we have only 30% channel estimation error.

In this chapter, the presence of the channel estimation error is studied to show how it candecrease the whole MIMO system performance. The channel capacity is a convenient measurefor describing the system performance in terms of channel estimation error point of view. Theeffect of channel estimation error is included for calculating the theoretical bound of the channel

36

3.5 Summary

capacity. The optimum receiver based on the perfect channel knowledge is converted to sub-optimum one in presence of channel estimation error. In MIMO communication systems, thecombining techniques are mainly used for maximizing the SNR based on the channel knowledge.The error performance of the sub-optimal combining could be the solution for the imperfectchannel knowledge. In addition, in this chapter we studied the modulation diversity or signal-space diversity with imperfect channels. Signal-space diversity can be employed to combat thechannel estimation error if we have large number of transmit antennas and effective length 7).Chapter 4 will introduce signal-space diversity aspects in more details.

7)The effective length will be studied further in chapter 4 as product distance.

37

Chapter 4Adopting Signal-Space Diversity to Combat

Channel Effects

The rotation of M-PSK constellation was employed and proved to increase the diversityorder of the Ralyleigh fading channel [14] [28] [13] [16]. The first statement about a solutionemploying the rotation of the original constellation to improve the performance of the Rayleighfading channel was by G. R. Lang in [29]. He described the problem as the construction ofn-dimensional lattice. However, that was only the formulation of the problem without solution.The main idea is spreading the information contained in each component over and through sev-eral components of the constellation points. Algebraic number theory was employed in [14] toproduce rotated constellations in real or complex space with half or full diversity order throughsignal space dimension without any increase of the power or bandwidth. In [28] the authorshave also constructed lattice constellation with high diversity order to be used in the space timecoding after using the Hadamard transform. As a common principle, maximizing the diversityis one of the best ways to reduce the error probability through the Rayleigh fading channel.The problem could not be so easy if the desired aim was to search for a multidimensional rota-tion which can offer a diversity order up to L, where L represents the maximum diversity order.

Therefore, in this section we compare different signal space diversities (half or full diversity

39

4 Adopting Signal-Space Diversity to Combat Channel Effects

order) from the design criteria. Furthermore, we introduce the different techniques of designingmultidimensional constellations to have near Gaussian channel performance. We also proposeunequal values of the rotation values of the rotation matrix elements weights. The use of rotatedHadamard and Fourier transforms will be compared. To gain both benefits of signal and spacediversities, the DAST codes will be studied from the construction and combining techniquespoint of view.

4.1 From Rayleigh fading to Gaussian channel

The received vector is y, where

y = diag(h)x + n, (4.1)

and diag(h) is a matrix containing the vector of the channel coefficients h in its main diagonal.The vectors x and n are representing the transmit and channel noise respectively. To definethe pairwise error probability P (x1 → x2), which is the probability of the received pointy to be closer to x2 than x1, assuming x1 is transmitted. Assuming the ML detector willbe used to minimize the metric m(x1|y, h) =

∑li=1 | yi − hixi |2, where l is the number of

channel paths and hi is the channel fading coefficient of path i. The detector will select x2 ifm(x1|y, h) ≤ m(x2, x1, h) and the conditional pairwise error probability could be representedas

P (x1 → x2|h) = P (

l∑

i=1

|yi − hix2|2 ≤l∑

i=1

|yi − hix1|2) = P (χ ≥ ζ), (4.2)

where χ =∑l

i=1 hi(x1 − x2)ni is a Gaussian random variable and ζ = 12

∑li=1 h

2i (x1 − x2)2 is a

constant. χ has a zero mean and a variance σ2χ = 2N0ζ. Then the conditional pairwise error

probability can be defined as P (x1 → x2|h) = Q(ζ/σχ) or

P (x1 → x2|h) = Q

√

∑li=1 hi(x1 − x2)2

4N0

, (4.3)

where Q is the Gaussian tail function which is defined as in Equation (3.49). By taking theaverage over the fading coefficients hi, the pairwise error probability P (x1 → x2) could berepresented by

P (x1 → x2) =

∫

P (x1 → x2|h)f(h)dh, (4.4)

where f(h) is the probability density function pdf of the fading coefficients. There exists atleast one Hamming distance between constellations x and y with the diversity order L. Assumeas an example |x1−x2| = 1 for the first L components and equals zero for the n−L components,so

P (x1 → x2|h) = Q

√

∑Li=1 h

2i

4N0

(4.5)

40


Now, for the Gaussian channel as a special case

P (x1 → x2) = Q

(

√

L

4N0

)

= Q

(

dE(x1, x2)

2σ

)

(4.6)

The squared Euclidean distance between x1 and x2 is d2E(x1, x2) = L and the noise variance

σ2n = N0. If L goes to infinity then E[

∑Li=1 h

2i ] = L. This way, the Rayleigh fading channel

has the same effects as the Gaussian one if L is large enough. Therefore the fadings have slighteffect at the presence of large L. In other words, when L goes to infinity the pdf of fΥ(x2)tends to be a Dirac impulse δ(x2), where

Υ =

∑Li=1 (h2

i − 1)

L

more details of this could be found in [14]. According to [14], P (x1 → x2) will be defined as afunction of SNR where SNR = L

N0

P (x1 → x2) =

(

1 − SNR

2

)L

.L−1∑

k=0

(

L+ k − 1

k

)(

1 + SNR

2

)k

(4.7)

Where SNR is the average of SNR which defined as

SNR =

√

SNR8L

1 + SNR8L

(4.8)

Boutros and et al. [14] showed that the fading effect is reduced when diversity is larger thanor equal to 8. Also in [25], it was proved that if the number of transmit antennas nt = 8 withrotations of block length equals nt, it is enough to reach near the Gaussian channel performance.

4.1.1 High diversity order rotated constellation techniques

The problem of designing a constellation with high diversity order is a problem of findinga rotated multidimensional cubic lattice Zn with high diversity. Assuming that Ξn,L is ann-dimensional lattice with diversity L. The generator or rotation matrix m is the generatormatrix of the lattice Zn which transforms all the integer component vectors into a set of vectorswith predefined diversity. In general it is possible to apply a coding technique to the rotatedlattice for the sake of coding gain. The design techniques for rotated constellations could besummarized as follows:

• Zn lattices construction from known rotated integral lattice:If we have two latices, we may say that they are equivalent if they are equal up to arotation and scaling factor. So, the equivalent generator matrices m

1and m

2are related

asm

2= Km β m1

R (4.9)

41


Where Km is a scaling factor, R is a unitary rotation matrix (det(R) = ±1) and β is a

lattice basis transformation matrix, where β is also a unitary matrix (det(β) = ±1). Any

non-rotating matrix could be defined with Ξn,1

, since it has diversity L = 1 and with

Ξn,n/2

the corresponding rotated lattice with diversity L = n2. Ξ

n,1and Ξ

n,n/2are defined

by the generator matrices of m1

and m2

respectively. After determining Km factor andmatrix β, we can obtain the desired matrix R. Based on some assumptions which found

in [14], the rotation matrix R will be

R = m2m−1

1. (4.10)

• Construction of Zn,n/2 lattices algebraically:The aim here now is constructing a family of orthogonal matrices with diversity L = n

2for

n = 2e13e2, e1, e2 = 0, 1, 2, · · · . We need to construct the complex lattice Ξ of dimension

n/2 through the ring of integers VE = Z[j](θ), where θ = e2πj/N is an N-th root of unity.E is an algebraic extension of V [j] = {a + jb|a, b ∈ V } of degree Φ(N)/2; Φ(.) is theEuler function. Then, the generator matrix of the complex lattice will be

m =

1 1 · · · 1θ1 θ2 · · · θ(n/2)...

......

...

θ(n/2)−11 θ

(n/2)−12 · · · θ

(n/2)−1(n/2)

=

m1

m2...

m(n/2),

(4.11)

where the lattice vectors mi, i = 1, 2, · · · , n/2, represent the rows of m. It is possibleto have the real lattice of dimension n by replacing each complex entry with real rep-resentation [14] [30]. Selecting the roots of θi where i = 1, 2, · · · , n/2, because of theorthogonality of the matrix m, the orthogonality among the complex vectors also meansorthogonality between the real vectors. By finding the solution which leads to the rootsof θ, we may obtain n/2 distinct values of θi with

φi = 2ψ + π

n+

4π(i− 1)

n, (4.12)

where θi = ejφi; i = 1, 2, · · · , n/2 and ψ = ±π3,±π

2,±2π

3. The summary is given in Table

4.1

• Construction of Zn,n lattices algebraically:Using the real algebraic number field V (2 cos(2π/N)), the constellation has full diversityL = n and some of these lattices maximize the product distance of the constellation.The problem is to choose the rotation matrices not only for achieving the maximumdiversity order L, but also for the maximization of the minimum product distance. Thisway, it would be better to make the constraints of the rotation lattices as simple aspossible. The optimization of maximizing the minimum product distance does not dependexperimentally on the size of the constellation. In two-dimensional case, it is possible tohave an optimum solution of the maximum of minimum product distance. On the otherhand, the higher the dimensions (for L ≥ 8), the best rotation matrices could be achieved

42


Model name ψ φi

Boutros1 −π3

4π3n

+ 4π(i−1)n

Boutros2 −π2

πn

+ 4π(i−1)n

Boutros3 − 2π3

2π3n

+ 4π(i−1)n

Table 4.1: Different values of rotation angle

by using computer simulations. We reach the best but not the optimum because of thelot of iterations which limit the accuracy of the search [14]. Algebraic construction of Zn,nlattices aims at choosing a family of number fields. These number fields are arbitrary andnot all of them can maximize the product distance of the constellation. Rotation as a toolin the sense of the signal space diversity can be optimum if we have maximum diversityorder and maximum product distance at the same time. The quasi-optimal rotation withthe best values (maximum) of the minimum product distance are reported in [14], [28].These values of the rotation matrix components could be achieved iteratively by choosingthe maximum of the minimum product distances. This method works good for diversityorder L ≤ 8, for L > 8, the method looses the efficiency due to the excess of parameters inthe higher dimensions diversity order. For the diversity order L ≤ 8, the rotation angleswill be as follows

θi = ejπ

4L.(4i−1)(2i−1) ∀ i = 1, 2, · · · , L (4.13)

In the same way, Bury and et al [31] suggested other angles of rotation as follows

θi = ej2π

Cm. i−1

L and i = 1, 2, · · · , L (4.14)

Where Cm is a constant depending on the transmit symbol alphabet. For example Cm = 2for BPSK and Cm = M for M-QAM.

Some remarks of this part could be addressed here:

• The use of modulation or signal space diversity is an efficient way to combat the effectsof the Rayleigh fading channel, which does not need any additional power or bandwidth.

• The diversity order L and the minimum product distance are not the only importantdesign parameters, but could be considered as the most dominant parameters.

• The constellation design, which takes into account the product kissing number as anotherparameter which must be considered, is an open problem.

43


4.1.2 Unequal values of the rotation matrix element weights

In the above techniques of the rotation matrix design there is an assumption of equalweight of all elements in the rotation matrix. In [29] the author assumed equally weight elementsof the rotation matrix, but he showed out, that the possibility of employing unequal weightingelements of the rotation matrix (which could be not the optimum case).

Rayleigh fading channel model is the most common model of communication system. So itis possible to deal with this model only. To design a rotation matrix for the Rayleigh fadingchannel, we might use a rotation matrix with unequal weight elements (UEW). Equation (4.1.2)gives an example of such matrices

u2

=1√5

(

1 −22 1

)

(4.15)

and

u4

=1√17

(

u2

−22u2

22u2

u2

)

(4.16)

In general form

uN

=1

√

1 + (2(N/2))2

(

uN/2

−2N/2uN/2

2N/2uN/2

uN/2

)

(4.17)

This way, the spreading is done only by using scaling in concatenation with Hadamard trans-form.

4.1.3 Upper bounds of signal-space diversity

The upper bound of the bit error probability BEP could be considered as a measure ofthe maximum gain which can be achieved by using the spreading transform. Bury and et al.in [32] introduced the way of calculating the upper bound of BEP and [33] [34] also gave theexpression of maximum BEP as a function of L diversity order

Pb =

(

1 − SNR

2

)L L−1∑

l=1

(

L− 1 + l

l

)(

1 − SNR

2

)l

, (4.18)

where SNR is a constant dependent on SNR and represents the average SNR;

SNR =

√

SNR

1 + SNR(4.19)

In [35], analysis of memoryless channel could be found with perfect channel estimation from thepoint of view of the upper and lower bound of the achievable pair wise error probability PER ofthe rotated constellation. It showed that by increasing the diversity through the space directionwhile keeping the Euclidean distance constant improves the diversity gain. Of course, the codedtransmission could have improved performance if the signal space diversity was employed.

44

4.2 The construction of the rotation matrix

4.2 The construction of the rotation matrix

The aim of the rotated constellation is to overcome the effects of Rayleigh fading channeland get closer to the Gaussian channel as the diversity order L increases. This rotation matrixdoes not add redundancy to the original information bits but only one-to-one rotation mapping.It means that there is no increase in the power or bandwidth and the gain is now achievedthrough the rotation with some increase of the demodulation complexity. A block rotationcould be achieved by the matrix m, which may be done in general after three function steps:

1. Mapping the data bit information. E.g. for M-PSK, data will be represented in general asa gray mapping. Other mapping technique are also possible. Actually, the mapping couldbe a key parameter if we have channel coding, but here no channel coding is assumed.

2. The rotation should satisfy the four factors [14], which were stated in section 2.5.1.

3. Guarantee that there are uncorrelated independent diversity paths. Using a very longlength interleaver, which can guarantee that we have independent channels (equal thenumber of diversity order(L)) paths. If the length of the rotated vector is long enoughnear Gaussian channel performance can be achieved [25].

4.2.1 Hadamard and Fourier spreading transforms

Two most famous spreading matrices are Hadamard and Fourier matrices. Both matricesare recursively generated, the Walsh-Hadamard is

wh(1) = 1√2

[

1 11 −1

]

(4.20)

, wh(n) = wh(n−1) ⊗ wh(1),

where ⊗ represents the Kronecker product and the matrix size is Ns = 2n which equals the blocklength. Furthermore, extending constructions of Hadamard transforms are possible for manysizes of multiples of four. The effective implementations of such multiplications of Hadamardmatrix is given by the fast Hadamard transform, whose complexity is of order O(Ns log2Ns)[32].

The other spreading matrix is Fourier matrix f . For block size Ns its elements are defined as

fi,k = 1√N.exp

(

−j2π (ik)N

)

, 0 < i, k ≤ Ns (4.21)

Different efficient implementations of this matrix exist by using fast Fourier transform (FFT)where Ns is power of two and the complexity order is O(Ns log2Ns) [32].

In general, the spreading transform matrices should satisfy the followingproperties:

45


• Invertability and square shape matrix which introduce a suitable way for bandwidthefficiency.

• Equal spreading gain which is an important condition for maximum diversity gain.

• Orthogonality, which means all Euclidean distances between vector pairs remain the samebefore and after spreading. This condition is very important to insure the spreadingmatrix affects only the Rayleigh fading channels and not the Gaussian channel.

• Low complexity implementation, and accordingly the Hadamard matrix is the much pop-ular choice for spread spectrum transmission systems.

The complexity comparison between Hadamard and Fourier matrices implementation could befound in [31].

4.2.2 Rotation using rotated spreading transforms

The idea here is to spread the alphabet with size M through the L paths if the spreadingblock length is L = Ns, i.e. we have a maximum diversity order. This way, we have ML distinctpoints at each path. It means that each point in the multidimensional signal constellation hasenergy distributed all over paths. This technique leads to the possibility of detection even ifone of paths is fade out. Actually, the rotation (spreading) of the BPSK as an example dependson the m if it is complex or real matrix. The complex matrix gives one more degree of freedomby spreading through the complex plain [32]. The use of Hadamard (or the rotated version ofHadamard) matrix for the spreading has an advantage of maximizing the spreading gain of therotated constellation [28]. At the same time, using the Hadamard matrix is useful for reducingthe effects of the peak-to-mean envelope power ratio (PMEPR), which results from sending theinformation bits through the diagonal of m and zero elsewhere [28]. The rotated transform ingeneral is defined by column rotation of the original transform by a certain designed angle

whrot

= wh.diag(θ), (4.22)

where

θ =[

θ1 θ2 · · · θL]T, (4.23)

wh and whrot

are Hadamard matrices before and after rotation respectively and θ is a vectorwhich contains the rotation angle for each column of the original matrix wh. In [14], it wasshown that the pairwise error probability of the Rayleigh fading channel could reach to thepairwise error probability of the Gaussian channel if the diversity order L is very large. Inother words, the fading effect is reduced when diversity order L is greater than or equals to 8.Thus,, the diversity order L plays a big role in the gain which may be offered by rotation. Itis better at this point to focus on finding the rotation angle which could increase the diversitygain assuming that the diversity order L is maximum, and also maximize the minimum productdistance between any two constellation points.

46

4.3 Signal-space diversity - DAST


While employing multiple transmit and receive antennas in a communication systemincreases the capacity or the maximum bit rate, signal space diversity provides a method tocombat the channel degradation effects. The combination of multiple antennas at both sides(transmitting and receiving) and signal space diversity is presented in many research papers. In[16], the authors established a simple rotation between high diversity multidimensional rotationsobtained from complex cyclotomic fields and discrete Fourier transform. This new rotationexhibits a good diversity distribution and can be combined to QAM constellations to combatthe channel fading. In [36], the authors use also algebraic number theoretic tools to designlinear space time-constellation rotation block codes. With arbitrary number of nt and nr, thisSTC (space-time codes) design achieves 1 symbol/sec rate and achieves maximum diversitygains ntnr over quasi-static fading channel. In [37] the authors introduced a way to constructa rotation with full diversity order for STBC. This STC achieves the simultaneously gooddiversity and multiplexing performance. In the diagonal algebraic space-time (DAST) blockcodes [28], [38], one gains the advantages of both techniques at the same time.

The problem with orthogonal full diversity space-time block coding is the non-existence of thefull rate orthogonal-STBC for general QAM in the case of more than two transmit antennas.[14] introduced the multidimensional rotated QAM modulation scheme to be used in singleantenna Rayleigh fading channels with QAM modulation. This scheme is extended later tothe space time coding in [28]. The rotation in [28] is based on a rotated matrix which is carvedfrom the real part of the cyclotomic number field, which has more advantages (relativelygood values of the minimum product distance) than any other complex rotation matrix evenwith having both full rate and also full diversity. All schemes which employ unitary (ororthogonal) rotation matrices, guarantee no performance loss in non-fading AWGN channels.In [39] a unitary rotation matrices achieve full diversity for QAM with any number of transmitantennas. The Alamouti space time code with constellation rotation is approximately 1.5 and2 dB superior to the space time with only space constellation code at BER equals to 10−6 forQPSK and 16-QAM, respectively. More results about the rotation of the Alamouti schemecould be found in [39] [40].

In [41], the authors presented a new approach of designing and constructing a modula-tion diversity complex lattice for the Rayleigh fading channel and also to be used in the spacetime coding. The used criteria is the minimum product distance with minimizing the averageenergy of the signal constellation through a new lattice family. This structure is general andcould be employed for any dimension.

4.3.1 DAST codes

Damen in [28] combined the rotated Hadamard matrix transform and MIMO systemsto achieve full rate transmit diversity over quasi-static or fast fading channels. The aim ofthis combination is to combat the Rayleigh fading effects in one side and increase the capacityof the system in the other side. DAST codes have their diversity gain and coding gain for all

47


constellations whether real or complex.Assuming, we already have a system with nt transmittedantennas and nr received antennas with independent paths of the quasi-static channels betweenthe transmitter and the receiver. The transmission is done assuming block transmission oflength L and white Gaussian noise. The channel is constant through at least the L block lengthand changing block by block. h is the channel matrix of dimension nr × nt, with hi,j denotingthe fading coefficient between transmit antenna j and receive antenna i. The space block isrepresented with information symbol vector a = (a1, a2, · · · , ad) for d > 0. After transformationthrough the matrix m, the rotated data vector will be x = m a of dimensions nt×L containingthe elements xjt, j = 1, 2, · · · , nt and t = 1, 2, · · · , L. It means that majt is sent over transmitantenna j at time t. If the symbol period is normalized, then the normalized rate of the codex is d/L symbol/sec. For nr receive antennas and L time periods, the received signal will be

y = h x+ n (4.24)

Where n is the complex noise vector of dimension nr with independent Gaussian distributedrandom variables entries having variance σ2

n per real dimension. To minimize the pair errorprobability PEP, the rank criterion and determinant criterion are employed [28]:

• The rank criterion is defined as the minimum rank ”rank” of m(a − e), where e is theerror vector of the code word a. This rank is taken over all codewords pairs.

• The determinant criterion is defined as the minimum of the geometric mean of the nonzeroeigenvalues of m(x− e) which equal (

∏rankj=1 λj)

1/rank and are taken over all distinct code-words pairs.

4.3.2 Maximum achievable rate of the rotated constellation

For the constellation V ; V ⊂ C and C is the field of complex numbers, if the size of V is2b elements and the diversity gain is ntnr, then the upper bound of the transmission rate will beb bits/second/Hz. In the STBCs, the code matrix representing the constellation is V ntL. Usingrotated constellations of dimension d = nt, the original constellation V will be transformed intoa new constellation V1, where

V1.=

(

x, x =

nt∑

i=1

mijai j = 1, 2, · · · , d = nt

)

(4.25)

In Equation (4.25), ai ∈ V , and mji are the elements of the j-th row of the rotation matrix m

which has a dimension d and the size of V1 equals 2db. It means that the maximum achievablerate after rotation will be db bit/second/Hz. At this point it is important to say that the DASTand Alamouti space time codes have the same full diversity with the only difference being thenonlinear processing for DAST and the linear processing for Alamouti.

48


4.3.3 Constructing DAST codes

The main part of the DAST is the rotation matrix. The rotation is based on the factthat the constellation V which has dimension d. Consider the vector x ∈ V and its componentsare x1, x2, · · · , xd, which are different from other vectors of V . Let us assume that the vector xis affected by independent fadings to be h1x1, h2x2, · · · , hdxd. If the receiver can recover thevector x unless all the components fall in deep fading, this property is called full signal-spacediversity of the constellation V . Now, the measure of the signal-space diversity could be theminimum product distance 1) for all elements of V , which is defined as

dmin(d).= min

dis=x1−x2,x1 6=x2∈V

d∏

j=1

|disj| (4.26)

The optimal design criterion of the rotation matrix according to [28] is the maximization of theminimum product distance. Due to the complexity of reaching the optimum rotation matrixdesign values, the rotation matrices of dimensions L > 4 could be classified to be quasi-optimumdesign.Assuming the rotation of a vector s, which is in general the complex information source vectorand will be represented by

x = ms,

where m is the optimal or suboptimal rotation matrix designed to maximize the modulationdiversity order (or gain),

s = [s1, s2, · · · , snt]T

and the complex rotated source vector is

x = [x1, x2, · · · , xnt]T

The source signal is divided into vectors x of length nt and delivered to nt transmit antennasfor nt time slots. The transmit signal matrix x achieving space-time diversity is as follows

x =

x11 · · · xnt

1...

. . ....

x1nt

· · · xnt

nt

= wh diag(x) (4.27)

where wh is the Hadamard matrix of nt dimension, and the channel is assumed to be a quasi-static fading channel for at least during nt time slots. After the channel, the received matrixwill be

y = hx + n

= h wh diag(x) + n(4.28)

1)The minimum product distance comes from the calculations of the pair error probability as could be foundin section 3.4 or in [25]

49


and

y =

y11 · · · ynt

1...

. . ....

y1nt

· · · ynt

nt

, (4.29)

h =

h11 · · · y1nt

.... . .

...hnrnt

· · · ynrnt

, (4.30)

where n is nr×nt complex matrix contains independent Gaussian distributed random variablesof variance 0.5 per real dimension. Based on the quasi-static channel model, the received signalmatrix y will be at the input of the receiver. To do the task of the decoding for the DAST

code, at first the received matrix y1

will be rearranged to a vector y as follows

y1

= vec(yT ) =[

y11, · · · , ynt

1 , · · · , ynr, · · · , ynt

nr

]T(4.31)

y1

= h x + n1 (4.32)

and

h =

h11 0 · · · 0...

. . .. . .

...

0 0 · · · hntnt

......

......

hnr1 0 · · · 0...

. . .. . .

...

0 0 · · · hnrnt

(4.33)

wh h =

h11 · · · hntnt

.... . .

...

hnr1 · · · hnrnt

(4.34)

At this point, we can state that h plays the same role as a new channel impulse responsematrix. This assumption is considered as a hidden assumption in the calculation of the BERin section 3.4. To maximize the SNR and also diversity gain, the received signal y

1should be

combined at the receiver (we use MRC in the optimum case, which looks like the matched filterin the optimal case, if the channel state information is well known at the receiving side). Themaximization of the SNR, could be achieved if the received signal y

1is matched to the matrix

h

y2

= hHy

1

= hHhx + h

Hn1

= diag(

nr∑

i=1

|hi1|2, · · · ,nr∑

i=1

|hint|2)x+ n2

(4.35)

50


The correlation of the noise after matching will be

E{n2nH2 } = E{hHn1n

H1 h} = σ2

nhHh (4.36)

Now, there is a need of a whitening filter (hHh)

1

2 and the input of the vector detector is

x = (hHh)

1

2 y2

= (hHh)

1

2x+ (hHh)

1

2n2

= diag

√

√

√

√

nr∑

i=1

|hi1|2, · · · ,

√

√

√

√

nr∑

i=1

|hint|2

x + n3

= diag(c)x + n3,

(4.37)

where c =

(

√

∑nr

i=1 |hi1|2, · · · ,√

∑nr

i=1 |hint|2)

, since the Hadamard transform matrix is orthog-

onal, we can also say that the matrix which contains cij, where i = 1, · · · , nr and j = 1, · · · , ntis also orthogonal because its elements are identically distributed complex Gaussian variableswith variance nr

2per real dimension.

4.3.4 DAST codes and combining technique effect

Assumingˆh is the resulting estimate of h which may contain some estimation errors,

then the combining (matching) at the receiver will beˆhH

.

y2

=ˆhH

y1

= ˆhH

hx+ ˆhH

n1

(4.38)

andˆhH

h = diag(nr∑

i=1

ˆh∗i1hi1, · · · ,nr∑

i=1

ˆhinthint

) (4.39)

Using whitening process of the noise by (ˆhH ˆh)1/2 and the complex input vector of the detector

will bexMRC = diag(c)x + nMRC, (4.40)

where

diag(c) =

(

ˆhH ˆh

)− 1

2 ˆhH

h

= diag

∑nr

i=1ˆhi1hi1

√

∑nr

i=1 |ˆhi1|2

, · · · ,∑nr

i=1ˆhint

hint√

∑nr

i=1 |ˆhint

|2

(4.41)

51


According to Equation (4.41), if there is a channel estimation error, then perfect combinationand detection also will not be reached.

The maximum ratio combining in the optimum case means the matched filter problem. On theother hand, the equal gain combining means only the phase matching of the received signal.

So, if the received signal vector y1

is matched to the phase arg(hH

) instead of hH

then

y2

= arg (hH

)y1

= arg (hH

)hx+ arg (hH

)n1

= diag(

nr∑

i=1

|hi1| · · ·nr∑

i=1

|hint|)x+ n2

(4.42)

and there will be no need now for a whitening filter because

E{n2nH2 } = E{arg (h

H)n1n

H1 arg(h)} = nrσ

2nI (4.43)

In this case, the input of the detector will be

xEGC = arg (hH

)h x+ arg(hH

)n1

= diag(nr∑

i=1

|h1i|, · · · ,nr∑

i=1

|h1nt|)x + nEGC

= diag(c1)x + nEGC

(4.44)

where c1 = (∑nr

i=1 |h1i|, · · · ,∑nr

i=1 |h1nt|).

In the following we compare the two combining techniques regarding to the SNR:

• For MRC, the matrix hH

will be used for maximizing the combination gain or as a matchedfilter

SNR1 =E{∑nr

i=1 |hij|2}2σ2

n

=nr

2σ2n

. (4.45)

• For the EGC, the matrix arg(hH

) will be used for maximizing the gain

SNR2 =E{(

∑nr

i=1 |hij|)2}2nrσ2

n

=E{∑nr

i=1 |hij|2} + 2E{∑i≤i1≤nr,i1≤i2≤nr|hi1j||hi2j|}

2nrσ2n

,

(4.46)

where hij =∑nt

j=1 hijwhjj and the average or expectation E is calculated over time and for onetransmit antenna. From the comparison of the SNR, we can conclude that:

• SNR2 = SNR1 if nr = 1.

52

4.4 The similarity between DAST and MC-CDM

• The Hadamard transformation has orthogonal vectors. If the real and imaginary com-ponents, <{hik} and ={hik} of the channel coefficients hik are independently Gaussian-distributed (as could be found here) with zero mean and variance of 0.5

nt, i.e., E{|hik|2} =

1/nt then hij should be the same.

SNR1 =nr

2σ2n

(4.47)

2E{∑i≤i1≤nr,i1≤i2≤nr|hi1j||hi2j|}

2nrσ2n

≈ nr(nr − 1)(Γ(3/2))2

2nrσ2n

(4.48)

andSNR2

SNR1≈ nr + nr(nr − 1)(Γ(3/2))2

nr(4.49)

Where Γ(.) is Gamma function. Equation (4.49) gives the indication that the ratio of the twoSNR’s are the same if nr = 1 and equals π/4 at large number of receiving antennas nr

2).It means that for a high number of receiving antennas, the difference between the SNR forEGC and MRC is only one dB with total complexity saving of O(nrn

3t ) by using EGC. The

comparison from the complexity point of view between MRC and EGC could be found in [42].


Digital transmission schemes with multiple transmit/receive antennas and the resultingtransmission over MIMO channels have drawn much attention in recent years. MIMO channelshave channel capacities which increase linearly with the minimum number of transmit/receiveantennas [4, 5, 6, 7]. MIMO channels have also the potential for transmit and/or receivediversity, which can still be exploited in cases where no spatial multiplexing gain is given. Inthis section, we discuss this topic based on the space-time-coding scheme, which is shown to bemathematically equivalent to multicarrier code division multiplexing (MC-CDM).

Figure 4.1 shows the basic model of the transmission we assume here. This vector transmissionmodel is on symbol basis, k is the discrete symbol time. The sequence s(k) of transmit symbolvectors has components si(k), i = 1, 2, ...,M . They are in general complex-valued, e.g. xi(k) ∈{±1 ± j} in case of QPSK. The mapping from source symbols (bits) to be transmitted to thesi(k) is not shown in the figure. With respect to conventional channel coding, we assume anuncoded transmission and only transmit diversity or space-time-coding (STC) is considered,especially ST block coding. s(k) is fed to an STC matrix u, which leads to the new vectorsequence

x(k) = u s(k). (4.50)

x(k) is the input of the block channel which gives y(k) at its output:

y(k) = hblock

x(k) + n(k). (4.51)

2)Γ(3/2) =√

π

2 , where Γ(1/2) =√

π

53


+

VED

Symbol vector channel

Block channel

Vector detection

Transmitsymbolvector

Space-time-coding matrix MIMO

blockchannelmatrix

PSfrag replacementss

hblock

w

ux

n

xx

Figure 4.1: Transmission model on symbol basis for MIMO system using general coding ma-trix containing signal-space and space-time matrices. The matched filter matrix in generalmaximizes form the SNR at the receiving end.

In general, the sequence of noise vectors n(k) has correlations in signal component and time-kdirection. After ST matched filtering (diversity combining in time and space) with matrix wand vector detection (VED), the detected vector sequence x(k) results. With a properly chosenw, one obtains sufficient statistics at the input of VED with respect to maximum likelihood(ML) or maximum a posteriori (MAP).The STC matrix u has two functions:

• Spreading s(k) in signal space:The components si(k) are spread over the axes of the s(k) signal space. If properly done,this produces signal space diversity, which is important because components can be fadedout by the channel.

• Spreading in space:Space means here transmit antennas. This part spreads the symbols to be transmittedproperly in spatial direction and in general it uses more than one symbol interval (calledtime slots in this context). In each time slot, different spatial spreading may be applied.

As a consequence, u can be represented as a product of two matrices:

u = uSTuSSD

. (4.52)

uSSD

is the signal space diversity matrix. Its size is M × M with M being the number ofcomponents of s(k). The ST matrix u

SThas size Lnt x M , with nt being the number of

transmitted antennas and L the number of time slots used for the ST block code. Therefore,x(k) has Lnt components. The ST block code has rate M

L, i.e. the ratio between the number

of symbols in s(k) at the input and the number of time slots used to transmit x(k) for fixed k.In general, the code rate is reflected through the ratio between columns and rows of u. h

block

54


is a block-diagonal matrix, where each block h is the discrete-time MIMO channel matrix onsymbol basis for one time slot. For L = 4 time slots, we get

hblock

= diag(h, h, h, h) (4.53)

=

h 0 0 00 h 0 00 0 h 00 0 0 h

; L = 4

with 0 being the zero matrix. h has size nr x nt with nr and nt being the number of received andtransmitted antennas respectively. From this, we get the size of h

blockas Lnr x Lnt. Because

all four time slots in the example have the same h, the physical channel has to be time-invariantover those four time slots. Furthermore, h is memoryless in this example, otherwise we wouldhave a sequence of matrices and h

blockwould not be block diagonal matrix anymore. For the

linear ST block codes considered here, an optimum receiver consists of two parts, see Figure4.1. The space-time MF matrix w is followed by vector detection (VED). To get an overall MLreceiver, w must be matched to the STC matrix u used at the transmit side and the MIMOblock channel h

blocki.e.

w = uHhHblock

= uHSSD

uHSThHblock

, (4.54)

and VED must be a ML vector detection algorithm. While the first two matrices from theright mean matched filtering in space and time, uH

SSDmeans despreading in signal space. If we

denote the vector sequence after spreading at the transmit side with x(k) = uSSD

x(k), we getfor its counterpart at the receiving side

x(k) = φhhx(k) + n1(k) (4.55)

φhh

= uHSThHblock

hblock

uST

n1(k) = uHSThHblock

n(k).

φhh

is a channel (autocorrelation) matrix and n1(k) is a sequence of colored noise vectors with

correlation matrix N0 φhh

, where N0 is the power density of the continuous-time AWGN vector

process on the channel. φhh

depends on the special STC used for transmission. If we include

the SSD spreading and despreading we get

x(k) = uHSSD

φhhuSSD

x(k) + uHSSD

n1(k) (4.56)

= φx(k) + n(k).

As expected, there is only one single matrix φ which maps x(k) to x(k). All individual matrices

at the transmit side – i.e. uSSD

and uST

– should be such that φ is as close as possible to an

55


identity matrix multiplied by a constant. The size of φ is M x M , which in this context is

independent of the number of transmitted and received antennas. But the influence of thereceive antennas and the resulting effect of received diversity can be expressed directly. It canbe shown that φ can be decomposed into a sum:

φ =nr∑

l=1

φ(l). (4.57)

φ(l) in this sum is the individual contribution from received antenna l. In general, VED includes

equalization and conventional channel decoding (which is not present here). Equalization heremeans coping with the effects produced by the off-diagonal elements of φ. With respect to

practical applications, many suboptimum VED techniques were used and discussed. Some ofthose techniques come very close to ML performance for many MIMO channels, see e.g. [30].

This short description of the transmission model assumes that there is no interblock interferencei.e. the vectors x(k) are received independently (with respect to k). For MIMO channels withmemory (or MIMO channels with multipath propagation) we get vector transmissions withinterblock interference. More about those general cases and more details on vector-valuedtransmission can be found in [21]. DAST code could be constructed as explained before insection 4.3.3, where the spreading matrix u

SSDis optimized with respect to signal space diversity

(or modulation diversity) which again results in maximum transmit diversity. The number oftime slots equals the number of components in x(k) i.e. L = M . u

STis an M0 x M matrix with

columns of a Walsh Hadamard matrix (WH matrix) on the main diagonal. The wh matrix hassize nt x nt. For M = 4, we get e.g.

uST

= diag(wh1, wh2, wh3, wh4) (4.58)

whi is the ith column of the wh matrix. The number of rows in uST

is therefore (M0 = nt) ×(M = nt) x 4. If x(k) = u

SSDs(k) is the vector sequence at the transmit side after spreading,

then the matrix uST

has the following effect: In time slot i the ith component of x(k) istransmitted over all transmitted antennas in parallel. The nt components of whi serve asweighting factors for the corresponding antenna.

The vector sequence y(k) at the output of the block channel can be calculated for the examplewith M = 4 as follows:

y(k) = h diag(wh1, wh2, wh3, wh4) x(k) + n(k)

=

hwh1 0 0 0

0 hwh2 0 00 0 hwh3 00 0 0 hwh4

x(k) + n(k), (4.59)

with h being the memoryless nr × nt MIMO channel matrix and n(k) the sequence of WGNvectors. Equation (4.59) shows an important characteristic of the DAST scheme: There is nocrosstalk between the components of x(k) because the column vectors h whi are staggered in

56


the matrix in vertical direction without overlap. For a 2 x 2 MIMO channel we get

y(k) =

h11 + h12 0h21 + h22 0

0 h11 − h12

0 h21 − h22

x(k) + n(k). (4.60)

The value hij belongs to the path from transmitted antenna j to received antenna i. The vectorsequence x(k) at the input of despreading (uH

SSD) is calculated from y

0(k) as follows

x(k) = uHSThHblock

y0(k). (4.61)

This means MRC is applied. The DAST channel matrix φhh

– which transforms x(k) into x(k),

see (4.55) – can be calculated as

φhh

= uHSThHblock

hblock

uST

. (4.62)

The 2 x 2 MIMO example from above yields explicitly

φhh

=

[

φ11 00 φ22

]

(4.63)

φ11 = |h11 + h12|2 + |h21 + h22|2

φ22 = |h11 − h12|2 + |h21 − h22|2

As a consequence of the diagonal matrix discussed before, see Equation (4.59), φhh

is also a

diagonal matrix. As explained before, the noise vector term in x(k) represents colored noisewith correlation matrix N0 φ

hh. So – depending on the VED – a whitening matrix might be

necessary.

DAST and MC-CDM:The mathematical description of DAST has much in common with multicarrier code divisionmultiplexing (MC-CDM). MC-CDM is based on orthogonal frequency division multiplexing(OFDM) with spreading over the OFDM subcarriers at the transmit side and despreadingafter the OFDM transmission at the receive side. For a SISO transmission, the overall modelon symbol basis can be expressed by

x(k) = φhhx(k) + n1(k) (4.64)

φhh

= φOFDM

= f hHOFDM

hOFDM

f−1

n1(k) = f hHOFDM

n(k).

This is a special case of Equation (4.55), f is the Fourier matrix. Spreading and despreading

– which turns OFDM to MC-CDM – is the same as for DAST. The matrix hOFDM

is a cyclicmatrix with shifted versions of the inverse-time SISO channel impulse response on its rows(counted in symbol intervals). Because h

OFDMis cyclic, φ

OFDMbecomes a diagonal matrix.

57


The main diagonal entries in φOFDM

are the squared absolute values of the SISO channel

transfer function at discrete frequency i (or OFDM subchannel i).

For DAST, the equivalence with MC-CDM becomes obvious. For MC-CDM the inverse Fouriermatrix f−1 = fH is taken instead of the wh matrix. This can be explained in a concise way

with the help of the 2 x 2 MIMO example from before. h11 +h12 in Equations (4.60) and (4.63)is the WH transform of the path vector [h11, h12] at l = 0 and h11 − h12 the WH transformof [h11, h12] at l = 1, respectively. l numbers the values in WH domain and [h11, h12] belongsto receiving antenna one. If we replace the WH transform in DAST by the inverse Fouriertransform, we get the transfer function values of the path vectors. For one receiving antenna,the equivalence of SISO OFDM and Fourier matrix DAST is given, if we interpret [h11, h12] asimpulse response of the SISO OFDM channel. Additional receiving antennas play the samerole in both systems if we extend the SISO OFDM to SIMO OFDM with the same numberof receiving antennas as in DAST. The corresponding φ

hhmatrices must be added according

to Equation (4.57), see also [21].

As a result, we can say that DAST with Fourier matrix is equivalent to SIMO MC-CDM, iffor all i the DAST path vectors [hi1, hi2, ..., hint

] to receiving antenna i are the same as theSIMO impulse responses to receiving antenna i in case of SIMO MC-CDM. Of course, for manytransmit antennas in case of DAST, we need a ”rich scattering environment” for SIMO MC-CDM to get the equivalence. The performance of DAST with Fourier matrix is statisticallyequivalent, so the conclusion is expected to hold also for the original DAST. A consequenceof those comparisons is that spreading matrices u

SSDcan be discussed for both transmission

schemes in the same way. More about good spreading matrices can be found in [31], [32]. Thebasic advantage of spreading in time for Rayleigh fading channels was already shown in [43].For MIMO OFDM better schemes can be taken, see MC-CAFS [44], [45], [46], which spreadsin frequency and space.

To concentrate on the effects caused by spatial and temporal matched filtering (4.61), we takea sphere detector [47] for VED because of its ML or near ML performance. Optimal MRC willbe compared with EGC on the basis of the symbol error rate (SER). For channel estimationwe take an LMS (orthogonal-LMS) algorithm because of its simplicity [48]. Some simulationresults about this part could be found in section 5.6.3

4.5 Simulation results

Based on section 4.1.1, where the rotation of Hadamard matrix will be done throughrotated vectors of the Hadamard matrix as in section 4.2.1. As a matter of fact, the designof these rotation angles could be based on half or full diversity order. Full diversity orderlike Bury and Damen in section 4.1.1, in addition to UEW in section 4.1.2, and also the halfdiversity order will be like Boutros 1, 2, and 3 in section 4.1.1. Figure 4.2 shows the rotatedconstellations of full and half diversity order techniques with BPSK data constellations andEIGHT diversity order. Figure 4.3 shows the spreading or rotation using rotated Hadamardmatrix for both BPSK and QPSK with 8 diversity order. ”Diversity8” in Figure 4.3 represents

58


0

0

Real

Imag.

Damen

0

0

RealIm

ag.

Bury

0

0

Real

Imag.

Boutros1

0

0

Real

Imag.

Boutros2

0

0

Real

Imag.

Boutros3

0

0

Real

Imag.

UEW

Figure 4.2: Rotated constellations of BPSK of different diversity orders. Half diversity order isrepresented by Boutros 1,2, and 3 in addition to full diversity order which represented by Bury,Damen and UEW.

59


0 2 4 6 8 10 12

10−4

10−3

10−2

10−1

Eb/N

0[dB]

BE

R

unspread

QPSK

BPSK

Diversity8

AWGN

Figure 4.3: BPSK and QPSK are rotated (spread) by using Hadamard matrix. The diversityorder is 8 and ”Diversity8” represents the diversity upper bound.

the theoretical upper bound of spreading diversity gain, which we haver calculated as in section4.1.3.

4.5.1 Half diversity-order of signal space

According to the rotation angles of Boutros [14], which were stated also in section4.1.1, where half diversity order for general QAM rotated constellation can be achieved byusing rotated Hadamard matrix. The results of Figure 4.4 are shown for the complex rotationmatrix and Figure 4.5 for only real values of rotation matrix with the same assumption of thetransmission as in the complex case and using BPSK.Using half diversity order rotation angles (as in [14] and section 4.1.1) with QPSK data

constellations with diversity order EIGHT, the results are presented in Figures 4.6 and 4.7 forcomplex and real of rotation matrices respectively.

Figures 4.5 and 4.7 show that the real values of the rotation matrix for the three angles ofBoutros [14] do not have the same effect on the Rayleigh fading channels. Although the spreadof the constellations through the complex plain in Figure 4.2 appear to be the same, only onof the Boutros angles in case of using the real values of the rotation matrices has the ability toimprove the performance at high SNR as can be seen in Figures 4.5 and 4.7(even with BPSKor QPSK). As a matter of fact, we use the real value of the rotation matrix to study the effectof spreading only in one dimension.

60


0 5 10 15 2010

−5

10−4

10−3

10−2

10−1

100

Eb/N

0 [dB]

BE

R

Unspread

Boutros1

Boutros2

Boutros3

Diversity−8 bound

Figure 4.4: Different half diversity order (Boutros1, 2, and 3) rotation angles. The rotationmatrix is the the rotated Hadamard matrix with diversity 8 and data constellations are BPSK.

0 5 10 15 20

10−4

10−3

10−2

10−1

100

Es/N

0 [dB]

BE

R

Unspread

Boutros1

Boutros2

Boutros3

Diversity−8 bound

Figure 4.5: Using only real values of the rotated Hadamard matrix. The rotation angles areBoutros 1 , 2, and 3 (half diversity order) the constellations are BPSK with diversity 8.

61


0 5 10 15 2010

−5

10−4

10−3

10−2

10−1

Eb/N

0 [dB]

BE

R

Unspread

Boutros1

Boutros2

Boutros3

Diversity−8 bound

Figure 4.6: QPSK constellations are rotated with rotated Hadamard matrix (diversity order8). The rotation angles design based on half diversity order.

From Figures 4.4 and 4.6 it is clear that the half diversity order which is offered by usingthe designed angles of Boutros (two of it) could be used with BPSK constellations and theperformance is very close to the diversity upper bound. In contrast is the performance ofBoutros angles with QPSK and the real valued rotation matrices with both BPSK and QPSK.

4.5.2 Full diversity-order of signal space

In this part of simulation results, we introduce the full diversity rotation using rotatedHadamrad matrix. We also compare between different rotation matrices (half and full diversity).It is important to state here that the design of the rotation angles is based on half or fulldiversity order but the rotation matrix is the rotated Hadamard matrix according half or fulldiversity design. Figures 4.8 and 4.9 show half and full diversity order for rotated Hadamradmatrices with BPSK data constellation with diversity EIGHT. Figures 4.10 and 4.11 show halfand full (only real values of rotation matrices) for rotated Hadamrad matrices for QPSK dataconstellation with diversity EIGHT. Figure 4.8 shows that Bury, Boutros1 and Damen have thesame performance with BPSK constellations and diversity EIGHT, while Boutros 2 (as a halfdiversity) has a slight difference in performance. Figure 4.9 shows that Damen rotations anglesoutperforms all others and even with the used real valued rotation matrices, the differencebetween the Damen rotation angles is about 2.0 dB at 10−5 BER from the upper bound of thediversity EIGHT. It is possible to state the same comments in Figures 4.8 and 4.9 for Figures

62


0 5 10 15 20

10−4

10−3

10−2

10−1

100

Eb/N

0[dB]

BE

R

Unspread

Boutros1

Boutros2

Boutros3

Diversity−8 bound

Figure 4.7: Real values of the rotated matrix are used for QPSK. The diversity order is 8 withBoutros 1, 2, and 3 (half diversity rotation angles).

4.10 and 4.11. The only difference is that the Damen rotation angles is about 3.5 dB at 10−5

BER from the upper bound of the diversity EIGHT.

Figures 4.12 and 4.13 also show some comparisons between the rotated Hadadmard and therotated Fourier matrices using Bury [31] and Damen [28] rotation angles for both BPSK andQPSK respectively. Figure 4.12 shows the same performance by using rotated Hadamard matrix(using Bury or Damen rotation angles) and the rotated Fourier matrix (using Bury rotationangles). It also shows that the Fourier matrix performs better than the Hadamard one withoutrotation for both. Figure 4.13 show some different performance of the rotated Hadamard andFourier matrices specially at high SNR.

Figure 4.14 shows more about the difference between Figures 4.12 and 4.13, if the rotationangles of Damen [28] are used and the diversity order is 8.Now, the rotation matrix which using UEW (unequal weighting values, see section 4.1.2) willbe added. Figures 4.15 and 4.16 are for BPSK complex and real rotation matrices respectively.The same could be shown in Figures 4.17 and 4.18 for QPSK data constellations.

4.5.3 Signal-space with space-time diversity techniques

The combination of MIMO systems (which employee space-time diversity) with signal-space diversity is a well known to have the advantages of using both techniques at the sametime.To introduce more rotation matrices with DAST codes [28], Figure 4.19 gives the compar-isons between Hadamrd and Fourier for rotation, Damen (Hadamard with Damen [28] rotation

63


0 2 4 6 8 10 12 14 16 1810

−5

10−4

10−3

10−2

10−1

100

Eb/N

0 [dB]

BE

R

Unspread

Bury

Damen

Boutros1

Boutros2

Boutros3

Diversity−8 bound

Figure 4.8: BPSK data constellations with half diversity (Boutros 1, 2, and 3) compared withfull diversity order design of Bury and Damen.

0 5 10 15 20

10−4

10−3

10−2

10−1

100

Eb/N

0 [dB]

BE

R

Unspread

Bury

Damen

Boutros1

Boutros2

Boutros3

Diversity−8 bound

Figure 4.9: Rotated constellations of BPSK of different half and full diversity orders (only realvalues of rotation matrices).

64


0 5 10 15 2010

−5

10−4

10−3

10−2

10−1

Eb/N

0 [dB]

BE

R

Unspread

Bury

Damen

Boutros1

Boutros2

Boutros3

Diversity−8 bound

Figure 4.10: QPSK data constellations are rotated with half and full diversity order withdiversity 8.

0 5 10 15 20

10−4

10−3

10−2

10−1

100

Eb/N

0[dB]

BE

R

Unspread

Bury

Damen

Boutros1

Boutros2

Boutros3

Diversity−8 bound

Figure 4.11: Only real values of the rotation matrix are used for QPSK with diversity 8 (halfand full diversity order design).

65


0 2 4 6 8 10 12

10−4

10−3

10−2

10−1

Eb/N

0[dB]

BE

R

unspreadHadamardDamen−realFourierrot.Hadamard (Bury)rot.Fourier (Bury)Damen−complexDiversity8AWGN

Figure 4.12: Comparison of different matrices (half and full) using rotated Hadamard androtated Fourier matrices for BPSK with diversity order 8.

0 2 4 6 8 10 12

10−4

10−3

10−2

10−1

Eb/N

0[dB]

BE

R

unspreadHadamardDamen−realFourierrot.Hadamard (Bury)rot.Fourier (Bury)Damen−complexDiversity8AWGN

Figure 4.13: Real valued of different rotation matrices (half and full diversity order). Therotation is by using rotated Hadamard and rotated Fourier for QPSK with diversity order 8.

66


0 2 4 6 8 10 12

10−4

10−3

10−2

10−1

Eb/N

0[dB]

BE

R

unspread

QPSK

BPSK

Diversity8

AWGN

Figure 4.14: Rotated Hadamard matrix by using rotation angles of Damen (full diversity order).The data constellations are BPSK and QPSK with diversity order 8.

0 2 4 6 8 10 12 14 16 1810

−5

10−4

10−3

10−2

10−1

100

Eb/N

0 [dB]

BE

R

Unspread

Bury

UEW

Damen

Boutros1

Boutros2

Boutros3

Diversity−8 bound

UEW bound

Figure 4.15: UEW rotation matrix is compared with equal weighting matrix elements of halfor full diversity order. The constellations are BPSK of diversity 8.

67


0 5 10 15 20

10−4

10−3

10−2

10−1

100

Es/N

0 [dB]

BE

R

Unspread

Bury

UEW

Damen

Boutros1

Boutros2

Boutros3

Diversity−8 bound

Figure 4.16: Rotated BPSK with different half and full diversity orders (only real values ofrotation matrices) by using equal and unequal weights of the rotation matrix elements.

0 5 10 15 2010

−5

10−4

10−3

10−2

10−1

Eb/N

0 [dB]

BE

R

Unspread

Bury

UEW

Damen

Boutros1

Boutros2

Boutros3

Diversity−8 bound

Figure 4.17: Different half and full diversity orders by using equal and unequal weights of therotation matrix elements For QPSK.

68


0 5 10 15 20

10−4

10−3

10−2

10−1

100

Eb/N

0[dB]

BE

R

Unspread

Bury

UEW

Damen

Boutros1

Boutros2

Boutros3

Diversity−8 bound

Figure 4.18: Half and full diversity orders (only real values of rotation matrices) are used forQPSK. The rotation matrices based on equal and unequal weights design.

angles), rotated Hadamrd with Bury [31] rotation angles and Fourier with Bury rotation matrixalso. Figure 4.19 is for DAST code, MIMO 8 × 1 and BPSK constellations. Figure 4.20 showsthe same rotation as Figure 4.19 but with only real part of matrices considered.

Is there any possibility for using half diversity rotation matrices in DAST, Figure 4.21 answersthis question for BPSK using the same rotations as in Figure 4.19. It is clear in Figure 4.21 thatthe rotation angles of Boutros 3 are the worst. In addition, the rotation angles of Boutros1 andBoutros 2 have somehow bad performance at high SNR when compared with rotation matricesof Bury and Damen. For QPSK constellation through DAST, Figures 4.22 and 4.23 presentcomplex and real rotation matrices respectively for full diversity order design based matrices(Bury and Damen). The results of Figures 4.22 and 4.23 are for MIMO 8 × 1.

We may now conclude that the performance of half and full diversity orders with BPSK areapproximately the same. The difference could be recognized if the used spreading (rotation)matrix is real valued matrix which could be an indication of how bad the channel is. For thehigher data constellation levels, for example QBSK or higher, the full diversity order gives thebest performance and some of the half diversity orders improve the performance within limit.It is worth also to state that there is no recognized difference if the rotation is done by usingrotated Hadamard or rotated Fourier matrices at low SNR but there are slight differences athigh SNR.

69


0 2 4 6 8 10 1210

−5

10−4

10−3

10−2

10−1

100

Es/N

0[dB]

BE

R

Damen−complex rotation

AWGN

Fourier spreading matrix

Fourier and Bury rotation

Hadamard spreading matrix

Hadamard−Bury for rotation

Figure 4.19: DAST code for 8 × 1 MIMO system. The rotation matrix is rotated Hadamardmatrix or rotated Fourier matrix. The constellations are BPSK and the detection technique isML.

4.6 Summary

The most important contributions of this chapter were to use different signal-space di-versity techniques to combat the channel fading effects. These techniques aim to force thechannel performance to the Gaussian channel. The design of the rotation matrix is dependenton the maximum diversity order, if it is half or full diversity order. The design criteria areintroduced and compared and the upper bound is presented. Also, constructing the space-timecode in concatenation with the signal space diversity by using the rotated Hadamard or Fourierspreading transform is studied. The similarity between the DAST and MC-STBC is introducedso that aspects can be applied to MIMO systems. In the simulation results, we can concludethat using real values of the rotation matrices are very effective to study the differences amongthe different rotation matrices. It means, the rotation matrix performance is better if its realvalues play a reasonable role in spreading. In other words, the spreading of real constellationslike BPSK can be achieved with only real values of a good rotation matrix. The rotated BPSKconstellation has a special feature of a slight performance differences either with half or fulldiversity design.

70

0 2 4 6 8 10 12

10−4

10−3

10−2

10−1

100

Es/N

0[dB]

SE

R

Damen − realAWGNFourier−Bury realHadamard−Bury realHadamard spreading

Figure 4.20: Real values of the rotation matrices are used for DAST code for MIMO 8 × 1.The rotation matrix is rotated Hadamard or rotated Fourier matrix. The data constellationsare BPSK and ML is the detection technique.


0 2 4 6 8 10 1210

−5

10−4

10−3

10−2

10−1

100

Es/N

0[dB]

SE

R

Boutros1

AWGN

Boutros2

Boutros3

Fourier

Damen

Fourier−Bury

Hadamard

Hadamard−Bury

Figure 4.21: Half and full diversity of the rotation matrix are used for DAST code for MIMO8 × 1. The rotation is by using rotated Hadamard or Fourier matrices for BPSK with MLdetection.

0 2 4 6 8 10 1210

−4

10−3

10−2

10−1

100

Es/N

0[dB]

SE

R

Damen − Complex

AWGN

Fourier

Fourier−Bury

Hadamard−Bury

Hadamard

Figure 4.22: DAST code and MIMO 8 × 1 for QPSK with ML detection by using differentrotation matrices.

72

4.6 Summary

0 2 4 6 8 10 12

10−4

10−3

10−2

10−1

100

Es/N

0[dB]

SE

R

Damen − real

AWGN

Fourier+Bury real

Hadamard−Bury−real

Hadamard

Figure 4.23: Real values of the rotation matrices are used for DAST codes and MIMO 8 × 1for QPSK with ML detection.

73


74

Chapter 5Recursive Algorithms for MIMO Channel

Estimation

The communication channel is the path which connects both the receiver and transmit-ter. Through this path some distortions occur to the transmitted signal. These distortionscould lead to a false detection of the original transmitted signal at the receiver. To study whatmight happen to the transmitted signal, there is an enormous need to model the communica-tion channel. A lot of research work has been done in the area of modelling the communicationchannel. One of the most benefits of the communication channel modelling is the representationof how the communication channel changes between the transmitter and receiver with time.Actually, modelling the time-variant channel is to estimate it according to a certain modelassuming this model reflects a realistic communication channel by using some adjustable pa-rameters. In addition, as a result of the multipath behavior of the communication channel, thequality of the received signal is reduced, which could lead to a detection error at the receiverside. To overcome the multipath effects of the channel, there is a need to employ the multi-path effects itself if possible. The space-time coding (STC) has this special ability to spreadthe transmitted data through time (or frequency) and space, thus employing the space as onemore degree of freedom in addition to the time (or frequency) direction. Accordingly, to modelan effective transmission of information, the possibility of time-varying or frequency-selectivechannels should be considered. Depending on whether the channel is time-varying and how

75

5 Recursive Algorithms for MIMO Channel Estimation

fast it changes, there are three general type of channels:

1. Fast time-varying, where the channel changes within one symbol.

2. Medium time-varying, where the channel changes every few durations (e.g., after tens) ofsymbols.

3. Slow time-varying, where the channel changes very slowly (e.g., after hundreds of sym-bols).

Most wireless channels could be considered as slowly time-varying channels since the relativespeed of motion between the transmitter and receiver is much slower than the speed of wavepropagation. As a result, the performance of such a system will be thoroughly anticipated asquasi-static time-varying channels, where the channel remains nearly constant within one blockbut varies block by block, assuming independent block transmission. The block transmissionoffers more flexibility in designing optimal transceivers compared to its serial counterpart.However, the design of filter bank transceivers to optimize the average performance whencommunicating over random channels is still a problem deserving further exploration. One ofthe solutions to this problem is a linear precoding (LP) and proper interleaving.

As it was shown in the pervious chapters, the transmission of information could be donethrough MIMO communication systems by using space-time codes with or without signalspace diversity. The problem which could be seen in a realistic communication system withchannel estimation is doing the estimation with minimum estimation errors. In this chapter,we introduce some different channel estimation techniques for MIMO systems with differentscenarios. The outline of this chapter is as follows: First, we introduce a general adjustablechannel model which is derived through some modification of the original Clarkes’ model [49].This modification of Clarke’s model was done in two steps, first the Pop&Beaulieu’s and thenthe modified-Pop&Beaulieu’s model 1). Then introducing the Alamouti’s scheme and how canchannel estimation be done by using the duality between the data and channel estimationand introducing also a mechanism for updating the channel estimation. Subsequently, weuse the conversion from the complex notation to real one to estimate the channel. The restof this chapter is arranged as follows. After introducing the linear recursive algorithms forchannel estimation, MMSE and LS channel estimation methods will be introduced. Finally,the channel estimation using LMS, Normalized-LMS, RLS, and RLS with adaptive forgettingvector (λ) will be presented.

5.1 Adjustable time-varying channel model

Time varying linear channels can be characterized in time and/or frequency. Thosecharacterizations are specialized by considering three classes of channels:

1)Pop&Beaulieu and modified-Pop&Beaulieu are referred to two channel models and the names are given byus.

76


Figure 5.1: A Multipath propagation consists of 50 waves arriving from random directions [1].

• The wide-sense stationary (WSS) channels.

• The uncorrelated scattering (US) channels.

• The wide-sense stationary uncorrelated scattering (WSSUS) channels.

Owing to the limitation of the time duration or bandwidth, all real-life channels can be classifiedas randomly time-variant linear channels with finite degrees of freedom when dealing with chan-nels which have the same characteristics as the wide-band filters (time variant). For channelswhere inputs and outputs are narrow-band, it is possible to replace these filters with equiva-lent narrow-band filters leading to input-output relationship that seems to be time invariant[50]. This hypothesis encourages dealing with a block transmission not a serial transmission.Figure 5.1 gives an example of a fading pattern for a single carrier with frequency 2 GHz. Themultipath propagation environment consists of 50 reflections modelling incoming waves fromrandom directions. An area of 0.75m × 0.75m is shown. Moving the transceiver just a fewcentimeters can induce a drop of the power by over 30dB [1].

5.1.1 Clarke’s channel model

Establishing a discrete-time, single-input single-output (SISO) and multiple-inputmultiple-output (MIMO) channel models was a motivation for many researchers. Finding amodel which is statistically precise and computationally useful to portray the continuous-timeMIMO Rayleigh fading channel is a challenge task. These channels may have spatial cor-relation, temporal correlation or time dispersive (or frequency selectiveness). To reflect the

77


realistic fading scenario, the discrete-time model will presented. This model will reflect theeffects of transmit filter, physical multipath channel fading and receive filter into receiver’ssampling-period spaced stochastic channel model.

In [50], the time varying channels are classified into three kinds as was stated above insection 5.1. We concluded that the WSSUS channels have the characteristics of WSS andUS channels. The duality between the time and frequency to describe the time varyingchannels could also be found in section 5.1. Jakes Rayleigh fading channel model based on thesum-of-sinusoids model to generate a correlated Rayleigh envelope is presented in [51]. Jakessimulator model produces a fading output signal that is not WSS when averaged across thesamples of fading channels [52]. Consider the frequency nonselective fading process of Clarkesreference model, h(t) is given by

h(t) =

√

2

N

N∑

i=1

ej(wdt cos(αi+φi))

= hc(t) + jhs(t),

(5.1)

where

hc(t) =

√

2

N

N∑

i=1

cos(wdt cos(αi + φi))

hs(t) =

√

2

N

N∑

i=1

sin(wdt cos(αi + φi)).

(5.2)

N , αi and φi in Equations (5.1) and (5.2) are the number of paths, angle of incoming wave andinitial phase associated with propagation path i, wd is the maximum angular doppler frequencyrespectively. The central limit theorem justifies that the quadrature components of h(t) canbe approximated as Gaussian random processes when N is large. Assuming that αi and φi aremutually independent and uniformly distributed on [−π, π] [49].

5.1.2 Pop&Beaulieu’s channel model

Pop&Beaulieu [52] developed a Rayleigh fading simulator. This model, based on Clarkes’model [49] and its low-pass fading process, is given by

h(t) = hc(t) + jhs(t) (5.3)

hc(t) =

√

1

N

N∑

i=1

cos(wdt cos2πi

N+ φi) (5.4)

hs(t) =

√

1

N

N∑

i=1

sin(wdt cos2πi

N+ φi), (5.5)

where wd is the maximum angular doppler frequency, φi’s are mutually independent and uni-formly distributed over all [−π, π]. This assumed that there is a normalization constant used

78


to make h(t) values have unit power. In [53], Xiao and Zheng presented some second-orderstatistics of this model

φhchc(τ) =

√

1

2N

N∑

i=1

cos(wdτ cos2πi

N) (5.6)

φhshs(τ) =

√

1

2N

N∑

i=1

cos(wdτ cos2πi

N) (5.7)

φhchs(τ) =

√

1

2N

N∑

i=1

sin(wdτ cos2πi

N) (5.8)

φhshc(τ) =

√

1

2N

N∑

i=1

sin(wdτ cos2πi

N) (5.9)

φhh(τ) = 2φhchc(τ) + j2φhchs

(τ). (5.10)

φhchc(τ) and φhshs

(τ) are the autocorrelations of the quadrature components, φhshc(τ) and

φhchs(τ) are the crosscorrelations of the quadrature components and φhh(τ) is the autocorrela-

tion of the complex envelope respectively.

Xiao and Zheng in [53] concluded some remarks for this model:

• The statistical properties of Pop&Beaulieu’s model tend to the desired ones of Clarke’smodel [49], as N → ∞, but when N is finite then the statistics are different from Clarke’sones.

• When N is finite and odd, the imaginary part of φhh(τ) can be significantly different fromzero (the desired statistical property for Clarke’s model [49]), which implies that the realand imaginary components of this model are statistically correlated in this case.

5.1.3 Modified-Pop&Beaulieu’s channel model

Due to the shortcoming of Pop&Beaulieu’s model, Xiao and Zheng [2] proposed a modifiedsimulation model as follows,

h(t) = hc(t) + jhs(t) (5.11)

hc(t) =

√

1

N

N∑

i=1

cos(wdt cosαi + ϑi) (5.12)

hs(t) =

√

1

N

N∑

i=1

sin(wdt cosαi + ϑi). (5.13)

assuming that αi = 2πi+θi

Nfor i = 1, 2 , · · · , N , and ϑi with θi to be statistically independent

and uniformly distributed over [−π, π] for all i. It is clear that the only difference between themodified model [2] and Pop&Beaulieu’s model [52] is the random variables θi which are added

79


to the angle of arrival.

In the modified Pop&Beaulieu’s model, the autocorrelations of the real or the imagi-nary parts are constant, equal and do not depend on the number of the sinusoids N andthe crosscorrelation of quadrature components are zeroes. Furthermore, the autocorrelationfunction of the squared envelope approaches the desired one as N approaches infinity.However, good approximation has been observed when N is as small as 8 [2]. The modifiedPop&Beaulieu’s simulation model can be directly used to generate uncorrelated paths forfrequency selective Rayleigh channels and diversity combining techniques. So, hk(t) is givenfor the path number k by

hk(t) = hc(k, t) + jhs(k, t)

hc(k, t) =

√

1

N

N∑

i=1

cos

(

wdt cos(2πi+ θi,k

N) + ϑi,k

)

hs(k, t) =

√

1

N

N∑

i=1

sin

(

wdt cos(2πi+ θi,k

N) + ϑi,k

)

.

(5.14)

θi,k and ϑi,k are mutually independent and uniformly distributed over [−π, π] for all i and fur-thermore hk(t) and hl(t) are uncorrelated for all k 6= l. This is due to the mutual independenceof θi,k , ϑi,k, and ϑi,l when k 6= l.

5.1.4 Establishing Rayleigh time-varying channel model

For a realistic simulation, there is a need to have a realistic time-varying channel withadjustable normalized maximum Doppler frequencies. We have here some statistical propertiesof the channel simulators that could be used for the simulations. The development of thismodel is presented in the above sections. Figures 5.2 and 5.3 give the cross-correlation betweenthe real and imaginary parts and the auto-correlation of the generated time-varying channelusing the modified version of Pop&Beaulieu channel model with fdT = 0.001 and the numberof fading paths is 12 (M = 12) respectively. Figures 5.4 and 5.5 give the amplitude and phasedistribution of the generated time varying channel using the same model of the modified −Pop&Beaulieu with the same parameters as in Figures 5.2 and 5.3.

According to Figures 5.2, 5.3, 5.4 and 5.5 the modified-Pop&Beaulieu could be used to generateindependent time-varying channel fadings with certain maximum Doppler frequency. Thesegenerated channel fadings, which are uncorrelated and could represent the real time varyingchannel, will be used through the simulation as a time varying channel model.

5.2 Channel estimation and Alamouti Scheme

The well known ”space-time codes” express the two-dimensional signals utilized in mul-tiple transmitted and received antennas MIMO systems. In the coherent case, where CSI is

80


Figure 5.2: The cross-correlation function (CCF) between real and imaginary components ofchannel realizations, for fdTs = 0.001 and M = 12, calculated according to [2].

Figure 5.3: The autocorrelation function (ACF) of real and imaginary components of channelrealizations, for fdTs = 0.001 and M = 12, calculated according to [2].

81


Figure 5.4: The amplitude distribution of channel realizations, for fdTs = 0.001 and M = 12,calculated according to [2].

Figure 5.5: The phase distribution of channel realizations, for fdTs = 0.001 and M = 12,calculated according to [2].

82


obtainable a priori at the receiver side, Tarokh et al. [54] derived the design condition for fulldiversity space-time codes in quasi-static fading channels. The work in [55] presented new trellisand graphical space-time codes with enhanced coding advantages, additional block space-timecodes and developments to more realistic frequency-selective and time-selective channels. Inthe face of this contemporary progress, the design of full-diversity, high-rate and low-complexityspace-time codes for quasi-static channels stays an open subject. As an example, the complex-ity of maximum-likelihood (ML) decoding for full diversity space-time trellis codes increasesexponentially with the transmission rate and number of data constellation levels. In contrast,the layered space-time architectures achieve high transmission rates with low complexity re-ceivers at the sacrifice of compact diversity advantages [4]. Last but not least, the orthogonalspace-time codes [56] and the Diagonal Algebraic Space-Time (DAST) codes in [28] accomplishfull diversity and permit support of a low-complexity receiver, but cause a considerable lossin the transmission rates (capacity), as a general rule. Only a small number of examples ofspace-time codes are known in the literature to accomplish full rate and full diversity with apolynomial complexity receiver [38]. It is common to use multiple antennas and employ STCat the transmitter and form some kind of combining technique at the receiver to enhance thereceived signal quality.

5.2.1 Alamouti’s space-time block coding scheme

S. Alamouti in [27] proposed a transmit diversity technique or a special STC to improvethe signal quality at the receiver. Alamouti constrained the number of the transmittedantennas to two for general complex data constellations. The scheme may be generalizedto nr receive antennas. It requires no bandwidth expansion and the redundancy spreads thesignal over the space domain not over the time or frequency domain. There are some trendsto extend Alamouti scheme to more than two transmit antennas to improve the diversity gainby employing more transmitting antennas, but at the expense of loss of the code orthogonalityand rate (the loss of full rate advantage) [57].

Alamouti scheme is suited for all applications where the system capacity is limited fad-ing at the remote units. It works by dividing the original symbol stream into two separatesymbol streams, which are transmitted using two transmit antennas. Assuming there is onlyone receiving antenna, the received signal y is

y[k] = x[k]h1[k] + x[k + 1]h2[k] + n[k] (5.15)

y[k + 1] = −x∗[k + 1]h1[k + 1] + x∗[k]h2[k + 1] + n[k + 1], (5.16)

where h1[k] and h2[k] are the channel taps h1 and h2 at instant k, x[k] and n[k] is the transmittedsymbol and additive white noise (AWGN) respectively at instant k, and ∗ denotes complexconjugate. The most important property of Alamouti scheme is the orthogonality of transmittedsymbol blocks. A general representation can be easily used for the Alamouti scheme. For theabove described multiple-input single-output (MISO) system, the transmission model in vectornotation is

y = h x + n. (5.17)

83


The vector y =[

y[k] y∗[k + 1]]T

contains the received signal, the vector x =

[ x[k] x[k + 1] ]T represents the transmitted information, n = [ n[k] n∗[k + 1] ]T representsthe additive white Gaussian noise and the channel matrix h is

h =

(

h1[k] h2[k]h∗2[k + 1] −h∗1[k + 1]

)

, (5.18)

where h1[k] = h1[k + 1] and h2[k] = h2[k + 1]. To detect the transmitted symbols, we computethe hard estimate of the vector

y = [ y[k] y[k + 1] ]T , (5.19)

where(

y[k]y[k + 1]

)

= hH(

y[k]y[k + 1]

)

. (5.20)

The independent estimation of the transmitted symbol x[k] and x[k + 1] is possible due tothe fact that the outputs y[k] and y[k+1] are decoupled. This is because of the orthogonalityproperty of the matrix h, where hhH is a diagonal matrix. Assuming flat fading channels,

the symbols [ x[k] x[k + 1] ] are estimated from the vector [ y[k] y[k + 1] ] using MLdetector. It is clear from the previous discussion that the orthogonality condition of h is thekey property that must be preserved at both the transmitter and receiver in order to achieveerror free decoupling. So, good estimates of the channel should be available at the receiver.

STBC represented by the Alamouti scheme is a special transmit diversity technique, whichcan be used to improve the signal quality at the receiver by using simple processing. Theintroduced scheme by Alamouti [27] is based on well known channel at the receiver in orderto decode the transmitted symbols. Actually not only the channels estimates are required atthe receiver but there is also the need to track the channel variations assuming there is a timevarying channel. Since high bandwidth efficiency is one of the most important motivationsfor MIMO systems, it would be rather hindering than useful to use long training sequencesto obtain reliable channel estimates. Consequently, channel estimation should better, at leastin some extent be derived from transmitted data. In addition, if data symbols are working totrack the channel estimates, time variance of the channel could additionally be managed tosome extent without the necessity to transmit any new training signals.

By means of a complex valued notation, it is impractical to include STBCs in a lineartransmission model, for the reason that the complex conjugate of the transmit symbols cannotbe articulated as a linear transformation of the original symbols. Accordingly, handing overa linear real valued transmission model could solve this problem. Actually, STBCs maybe considered as an extraordinary form of spreading that uses the space as a direction ofspreading in addition to the time direction. The real valued transmission model and the specialstructure of Alamouti’s scheme offer a simple approach of updating the channel estimatesthrough the transmitted data. This can be done by using the mathematical duality betweenthe transmitted data and the channel coefficients. Indeed, it may possibly be shown thatindependent estimates of the channel impulse response are simply obtained by means of asimple linear transformation. In accordance with the physical channel, the aim is to find a

84


strategy that could be used, through the new estimates, to update the previously availablechannel estimates.

5.2.2 STBCs from complex notations to a linear (real) transformation

A MIMO system has nt transmitted antennas and nr received antennas, with channelimpulse response h where

h =

h11 · · · h1nt

.... . .

...hnr1 · · · hnrnt

.

The vector x contains Ls symbols of l subsequent time slots and s contains the transmit signalsfor all l time slots stacked on top of each other, which gives l Ls entries in total, then

s = υlJLs

lx, (5.21)

where JLs

lconsists of l identity matrices of size Ls put on top of each other with purpose

of stacking the vector x of size l Ls. The actual mapping and reordering is done by υl. It is

important to state that υlis always a block diagonal matrix structure and each block carries out

the mapping of a particular time slot. Moreover, the combination of υlJLs

lcould be considered

as a spreading matrix and also could be seen as a single matrix that maps between x and s.Now, we can introduce a modified version of the channel impulse response matrix

hl=

h · · · 0...

. . ....

0 · · · h

. (5.22)

The original channel impulse response matrix h is repeated on the main diagonal l-times to

construct the matrix hl

in Equation (5.22). With the exception of JLs

la subscript always

indicates a block diagonal matrix whose blocks may differ from each other if the channel variesblock by block. The same applies in the case of υ

l. The received vector y using h

lis determined

as followsy = h

ls+ n, (5.23)

y contains the received signal from l subsequent time slots and n is a vector of l nr samples of

white Gaussian noise. By using matched filter at the receiver hHl

and the decoding (despread-ing) matrix finally leads to a simple matrix transmission model, which still provides sufficientstatistics at the output of the matched filter

x = JLs

l

HυHlhHly

= JLs

l

HυHlhHlhlυlJLs

lx + JLs

l

HυHlhHln

= JLs

l

HυHlhHlhlυlJLs

lx + n

= φspread

x + n.

(5.24)

85


φspread

determines also the correlation of n except for a constant factor. Using the real-valued

representation, the system model becomes

¯x = J2Ls

l

TυTlhT

lhlυlJ2Ls

lx+ J2Ls

l

TυTlhT

ln

= J2Ls

l

TυTlhT

lhlυlJ2Ls

lx+ ¯n

= φspread

x+ ¯n,

(5.25)

where . is the real valued representation of .. It is clear from Equation (5.25) that the matchedfilter now simply becomes the transpose instead of the Hermitian transpose of the channelmatrix [30]. Generally, a complex valued counterpart does not necessary exist for some υ

l.

STBCs may be incorporated into the real valued linear transmission model that was describedabove. The Alamouti’s scheme [27] or any other STBC could be represented in a similar way.Considering the bases of the Alamouti’s scheme, the symbols x1 and x2 are transmitted at thesame time through two transmitting antennas during l = 2 time slots assuming there is nointerference between the two symbols after the matched filter. At the first time slot x1 and x2

are transmitted from antennas 1 and 2 respectively, at the second time slot −x∗2 and x∗1 are

transmitted from antennas 1 and 2 respectively. It means that the two symbols are transmittedthrough two time slots, i.e,

J2Ls

l= J4

2=

1 0 0 00 1 0 00 0 1 00 0 0 11 0 0 00 1 0 00 0 1 00 0 0 1

. (5.26)

The actual reordering of the symbols, with the complex conjugate operation is represented byυl

where

υ2

=

1 0 0 0 0 0 0 00 1 0 0 0 0 0 00 0 1 0 0 0 0 00 0 0 1 0 0 0 00 0 0 0 0 −1 0 00 0 0 0 1 0 0 00 0 0 0 0 0 0 10 0 0 0 0 0 −1 0

. (5.27)

Generally for any type of STBCs, υl

is a block diagonal matrix with l blocks where each blockmaps the symbols to the transmit antenna for a certain time slot. According to Equation(5.27), the υ

2does the mapping of the Alamouti scheme. It could be sufficient for υ

2and J4

2to be described as a single spreading/transformation matrix, but it is useful to expand theminto two matrices to employ them in channel estimation. Now, at the receiver the transmission

86


matrix can be represented as

φA

= J4

2

TυT

2hT

2h

2υ

2J4

2

=nr∑

k=1

(h∗k1hk1 + h∗k2hk2)I4.

(5.28)

A in Equation (5.28) is an indication of Alamouti, I4

is an identity matrix of size 4 and φA

as

represented before is thus a scaled identity matrix. This means there is no crosstalk betweensymbols and provides full diversity, and the scaling factor is the sum of the squared absolutevalues of all entries of h. Actually, there is a duality between the channel impulse responsecoefficients and the transmitted data, if we are going to use real-valued representation of thecomplex transmission matrix calculations. How we can use this duality to estimate the channelimpulse response coefficients will be explained in the following. Assuming the received vectory is in real representation form as

y =

<{y1}={y1}<{y2}={y2}

, (5.29)

the superscript indicates the time index. Equation (5.29) is based on the transmission of thereal valued vector

x =

(

<{x}={x}

)

. (5.30)

since the aim is to estimate the channel impulse response coefficients (it’s a system not a signal),so there is a need to rearrange the signal x as a real valued system x as follows

x =

(

<{x} −={x}={x} <{x}

)

. (5.31)

The real value extension of x in Equation (5.31) seems to be like a system and the secondcolumn of x is the complement of the first column. This way, y is reconstructed in the real-valued notation as

y =

<{y1} −={y1}={y1} <{y1}<{y2} ={y2}={y2} −<{y2}

. (5.32)

Also, the last definition of x is modified to the following

x2

=

(

x 00 x

)

. (5.33)

y is extended be

g2

=

<{y1} −={y1} 0 0={y1} <{y1} 0 0

0 0 <{y2} ={y2}0 0 ={y2} −<{y2}

, (5.34)

87


where superscript 2, gives the indication of using two time slots for constructing the matrices.By using the above real-matrices representation, there is a possibility now for obtaining inde-pendently distributed maximum likelihood (ML) estimates for the channel matrix by using alinear transformation

¯h =

1

cJ2nr

2

Tg

2xT

2υT

2J4

2

=1

cJ2nr

2

T(h

2υ

2x

2xT

2υT

2+ n

2xT

2υT

2)J4

2

= h+1

(x∗1x1 + x∗2x2)J2nr

2

T(n

2xT

2υT

2)J4

2.

(5.35)

Generally, the whole problem described above aims to enable independent usage of each timeslot assuming correct decisions are taken for the received symbols. We can get independentestimates of the channel impulse response matrix by using the orthogonality of MIMO channelpaths because of the special code structure of the Alamouti scheme. As an example of someupdating procedures of the channel matrix by using old estimates, one can use a scaled valueof the present estimate which can be added to the old estimate as explained in [58]

¯h

2i+4=

1

2i+ 2(2i · ¯

h2i

+ 2 · ¯h

2i+2), (5.36)

i = 1, 2, . . . . Here is assumed that the channel impulse response is updated every two timeslots and the channel impulse response at instants zero and two are initialized [59]. Actually,the above Equation (5.36) of updating might be not effective in some time-varying channelmodels. So there is a need for another mechanism to update the channel impulse response.For example using moving average with different averaging length according to the channel-variation model (some simulation results for the previous two updating techniques are presentedin section 5.6.1).

5.3 LS and linear-MMSE channel estimation

The least squares (LS) channel estimator vector hLS[k] is defined as

hLS[k] = x y[k] = h[k] + x n[k], (5.37)

where x = (xH [k]x[k])−1xH [k] and the part of the training data is already contained in x[k] (xis defined in Equation (2.5)), which implies that the LS estimator does not require any priorstatistical knowledge. To have such long term channel properties (prior knowledge), the wellknown linear-MMSE channel estimator is employed

h[k] = φhh

[k]xH [k](x[k]φhh

[k]xH [k] + σ2[k]I)−1y[k] (5.38)

= x(I − σ2[k]φ−1

yy[k])y[k]. (5.39)

88

5.3 LS and linear-MMSE channel estimation

For stationary environment φyy

[k] = φyy

and φnn

[k] = φnn

, so the channel correlation matrices

will be replaced by the respective sample matrices

φyy

[k] =1

k

k∑

i=1

y[i]yH [i] (5.40)

φLS

hh[k] =

1

k

k∑

i=1

hLS[i]hH

LS[i], (5.41)

where φLS

hh[k] = xφ

yy[k]xH and hLS is calculated using Equation (5.38).

5.3.1 Constructing the correlation matrices

The sample correlations φLS

hhand φ

yydo not take into account the correlation structure

φyy

[k] = x[k]φhh

[k]xH [k] + σ2n[k]I (5.42)

In case of nonstationary channel:It means that, φ

hh[k], φ

yy[k], and σ2

n[k] depend on the block time index k. Then the channel

correlation φhh

[k] and the noise variance σ2n[k] are considered as parameters to be estimated

by the LS approach from observed receive correlation matrix φyy

[k]. Consider Equation (5.42)

where the noise variance σ2n[k] (or SNR) is given

(φhh

[k], σ2n[k]) = arg min

φhh

[k],σ2n[k]

‖φyy

[k] − xφhh

[k]xH [k] + σ2n[k]I‖2, (5.43)

and (‖.‖) is the Frobenius norm. As a matter of fact, the sample correlation of y[k] could becalculated by a scaled value of the stationary correlation matrix of y as

φyy

[k] = (1 − µ2)

k∑

i=1

µk−12 y[i]yH [i], (5.44)

where µ2 is the forgetting factor 0 ≤ µ2 ≤ 1, and the correlation of the channel estimation is

φhh

[k] = x φyy

[k]xH − σ2n[k](xH [k]x[k])−1

= x(φyy

[k] − σ2n[k]I)xH .

(5.45)

In Equation (5.45), (xH [k]x[k])−1 = x xH . Assuming z = x[k]x and z⊥ = I − z, the noisevariance estimate will be

σ2n = 1nr − nttrac{z⊥φ

yy[k]z⊥}

= 1 − µ2nr − nt

k∑

i=1

µk−12 ‖z⊥y[i]‖2.

(5.46)

89


This way, z⊥y[k] is interpreted as estimate of the noise, and the structured correlation estimate

of φyy

[k] is φs

yy[k] and calculated by

φs

yy[k] = x[k]φ

s

h[k]xH [k] + σ2

n[k]I = zφy[k]z + σ2

n[k]z⊥. (5.47)

In case of stationary channel:For the stationary environments, the correlations φ

yy[k], φ

hh[k] and σ2

n[k] are independent of

k and the forgetting factor µ2 can be used to calculate the sample correlation φyy

[k] but the

noise variance will be [11]

σ2n[k] =

1

nr − nt

1

k

k∑

i=1

‖z⊥y[i]‖2. (5.48)

5.4 LMS and normalized-LMS algorithms for

MIMO-channel estimation

Assume the MIMO channel paths are Rayleigh fading for all time steps. Generally, thereceived vector y ∈ C(nr×1) at instant k will be

y[k] = h[k]x[k] + n[k]. (5.49)

The channel matrix is h ∈ C(nr×nt) and x ∈ C(nt×1) is the symbol vector simultaneously

transmitted by nt transmit antennas. n ∈ C(nr×1) is the complex additive white Gaussian noisevector with zero mean and covariance matrix φ

nn

φnn

= σ2nInr

(5.50)

where Inr

is an identity matrix with dimensions nr × nr and σ2n is the noise variance.

5.4.1 LMS channel estimator

The optimal estimation can only be accomplished when the input statistical infor-mation matches ’a-priori ’ information for which the estimator is designed. For nearly allcommunication systems, this ’a-priori ’ knowledge is either only partially or not known to thedesigners. Two opportunity approaches are possible. The first one optimizes the estimator tothe ’medium’ case, which is a representative of an average estimation status and average signalstatistics. This trade-off visibly results in a poor estimation performance, because most of theestimation conditions diverge from this ’medium’ case. Otherwise, an adaptive algorithm canbe built-in to adjust the estimator parameters dynamically along with the input statisticalinformation. In this approach, ’a priori ’ knowledge of the input is no longer essential in theestimation process, but in its place, a costly recursive algorithm is considered necessary to

90

5.4 LMS and normalized-LMS algorithms for MIMO-channel estimation

perform the self-adjustment of the estimator’s parameters. The complexity of this kind ofalgorithm is based on the choice of the optimization method used in the recursive algorithm.

Two solutions are commonly used in recursive algorithms based on the stochastic gradientand least squares estimation approaches. Their representatives of the recursive algorithms arethe LMS and RLS (Recursive Least Square) algorithms respectively. Commonly, the LMSalgorithm is more widely employed than the RLS algorithm because of its low computationalcomplexity and robust performance. On the other hand, the RLS algorithm has a fasterconvergence rate than the LMS algorithm. Despite the fact that latest modifications to bothLMS and RLS algorithms have enhanced their performance, it is still complicated to decide forthe most suitable algorithm for any certain application. For instant, in real-time applications,the usage of the RLS-based algorithm is now no longer an obstruction. For this reason, thefast RLS algorithm was introduced, whose computational complexity is comparable to theLMS algorithm [60].

A range of the LMS-based solutions was also developed around the same time aimingto enhance adaptive estimation performance. These contain a variable step-size LMS algo-rithm for improving the convergence and steady-state performance, and the delayed LMSalgorithm which permits the pipelining of the LMS algorithm to enhance its throughput rateperformance [61]. Both the improved LMS and RLS algorithms are well-suited in terms ofcomputational complexity and numerical stability. The optimum selection of the algorithmcurrently comes down to a fine balance of convergence, steady-state performance (SNRimprovement), and tracking capability where a stable convergence performance is essential forthe identification and tracking the channel estimation.

Using weight control technique with the LMS algorithm is a way to improve its perfor-mance. The criterion employed in the LMS algorithm is the minimization of the mean squareerror (MSE) between the reference signal and the estimator output signal. The weights canbe controlled according to the steepest descent algorithm. Generally, the solution in thesense of optimum stationary linear estimation problem is the Wiener filter, which dependson correlation matrix of estimator input and cross-correlation between estimator input anddesired output in the training mode (or the output after decision in the tracking mode). Thisfinds minimum of mean-squared error cost function. Alternative procedure to find a solutionis based on the method of steepest descent which summarized as follows [10]:

1. Start with initial estimate of estimator weights.

2. Compute gradient vector of cost function (which represents the mean square error (MSE)of the estimated values) at current weight estimate.

3. Adjust weight vector in direction opposite to gradient vector to obtain next estimate.

4. Repeat steps 2 and 3 until convergence to the solution of estimator weight (steady statewith slight differences between estimates).

91


A simple recursive weight update rule for the channel estimator (one path-single tap) is givenby:

h[k + 1] = h(k) − ∆∇J [k] (5.51)

h[k + 1] and h[k] are the new and old estimates respectively, ∆ is a convergence step-size and∇J [k] is the gradient of the MSE cost function J [k] of the estimation at instant k. In thevectors form, channel estimation using LMS algorithm is calculated

h[k + 1] = h(k) − ∆∇J [k], (5.52)

where h[k + 1] and h[k] are the new and old estimate vectors respectively, ∆ is a convergencestep-size and ∇J [k] is the gradient vector of the MSE cost function J(k). Steepest-descent is ageneral optimization technique that only requires the knowledge of gradient of cost function asshown in Equations (5.51) and (5.52). This method is simple, but sometimes slow to convergeor has high error variance. This tradeoff depends on the value of the convergence step size ∆.

The LMS Algorithm is a practical method for calculating the gradient ∇J(k) which re-quires prior knowledge of ensemble-averaged correlation and crosscorrelation matrices. Inpractice, correlation, crosscorrelation and hence gradient have to be estimated. Theseestimations come from data standard estimators, where one can replace ensemble averageswith time averages. It means that this version uses a single sample to form instantaneousestimates without averaging. The algorithm which uses weight update with instantaneousgradient estimate is known as least-mean-square (LMS) algorithm [10].

LMS is a stochastic gradient algorithm (as opposed to deterministic gradient of steepest-descent) where the estimated gradient generally points in random direction (such as ”gropingin the dark”); LMS performs steepest descent on instantaneous squared error. Instantaneousestimates have large variances, but are eventually smoothed by the recursive nature of theLMS, so there is no need to measure signal correlations directly, no matrix inversion required,and computational and storage requirements are not large. As a result, LMS algorithm issimple to be implemented and adapts a linear FIR filter for channel estimation. Therefore, it’srobust and could be suitable for tracking slowly time-varying environments. LMS algorithmscan be used in the MIMO channel estimation as follows. The received signal in Equation(5.49) at the i-th antenna can be written as

yi[k] = xH [k] hi[k] + ni[k] ∈ C(1×1) (5.53)

where i = 1, 2, · · · , nr. The error signal at time k for receive antenna i can be calculated by

ei[k] = yi[k] − hH

i x[k], (5.54)

where x[k] denotes the hard decision of the output of the space-time detector as indicated inFigure 5.6. The channel state vectors hi are updated according to the LMS solution as

hi[k + 1] = hi[k] − ∆ xi[k]e∗i [k] (5.55)

92

5.4 LMS and normalized-LMS algorithms for MIMO-channel estimation

PilotsymbolsModulation concatenated with

signal and/or space diversity

Detection&Channel

estimation

+

+ VED

VectorDetection

+-

MOD

Multipath/MIMOchannel

PSfrag replacements

a x

n

h

x → ah

y

Figure 5.6: General Block Diagram of Transmission System with channel estimation. Thechannel estimation could be estimated by using pilot symbols (training mode) or detectedsymbols (tracking mode)

where ∆ is the convergence step size parameter and e∗i is the complex conjugate of the error eiat instant k.

Using ∆ as an adaptation or convergence step-size depends on the SNR and time variationrate of the channel. Also, it depends on the number of taps of the estimator to be estimatedin one iteration [10] (here we have assumed a single tap estimator for each path between atransmit and receive antenna). It is clear from the Equation (5.55) that the only factor thatcontrols the convergence of the channel estimator is the convergence step size ∆. Due to theeasy implementation of the LMS algorithm, it operates as a basis for the derivation of a numberof further algorithms like the normalized-LMS.

5.4.2 Normalized-LMS for channel estimation

As it could be recognized in Equation (5.55), the updating of the channel estimation isdirectly proportional to the vector of the estimated data x. So, if the vector values x are large,the LMS algorithm experiences “a gradient noise amplification“ problem. To deal with thisproblem, the normalized-LMS algorithm could be used. Normalized-LMS could be consideredto be as the solution to a constrained minimization problem 2). The whole derivations of thenormalized-LMS can be found in [10]. As a summary of the normalized-LMS, the adaptive stepsize ∆ in Equation (5.55) will be

∆1 =∆

∑ ‖x‖ , (5.56)

2)It was shown that the normalized-LMS algorithm is exactly the minimum-norm solution to a linear least-squares problem involving a single error equation [10].

93


where∑ ‖x‖ is the normalized value of the detected vector power and 0 < ∆ < 1. Through

this way of normalization, it is possible to control the change in the estimated channel vectorfrom one iteration to the next without changing its direction. In addition to what is stated asthe advantages of the normalized-LMS, the single value decomposition provides the connectionlink between the form of least-squares estimation and the LMS theory [10].

5.5 Channel estimation using RLS and adaptive-λ-RLS

algorithms

Considering the faster convergence rate of the channel estimator, additional sophisticatedadaptation algorithms such as the Recursive Least Square (RLS) algorithm can be used. TheRLS algorithm has more rapid convergence rate and less significant missadjustment than theLMS and even less than the normalized-LMS (NLMS) algorithms. Consequently, the algorithmis able to enhance the bit error rate (BER) performance. However, improvement is achieved atthe price of expanding computational complexity and storage requirements.

5.5.1 RLS for channel estimation

The performance index to be minimized by RLS algorithm is defined as an exponentiallywindowed sum of the squared error ε[k]

ε[k] =

Ls∑

i=1

λL−i|en[i]|2 (5.57)

where Ls is the window length and λ is the forgetting factor; λ ≤ 1. The general RLS methodcan be found in [10] and how it can be employed for MIMO channel estimation is summarizedas follows:

• Initialize the channel estimator;

h[0] = 0 and P [0] = δ−1Int

(5.58)

0 is a zero elements vector, δ is a small positive constant for high SNR and large positiveconstant for low SNR.

• for each time instant k = 1, 2, · · · and receive antenna i compute:

I.π[k] = P [k − 1]x[k]

II.

K[k] =π[k]

λ+ xHπ[k]

ei[k] = yi[k] − hH

[k − 1]x[k]

hi[k] = hi[k − 1] −K[k]e∗i [k]

94

5.5 Channel estimation using RLS and adaptive-λ-RLS algorithms

III.

P [k] = λ−1P [k − 1] − λ−1K[k]xH [k]P [k − 1],

where

x = [x[k], x[k − 1], · · · , x[k − nt + 1]]T ,

and

hi[k] = [hi1[k], hi2[k], · · · , hint[k]]T .

Concerning the RLS algorithm, the computation is on track with known initial condi-tions, and employs the information contained in new data samples to update the previousestimates. The RLS algorithm is designed to minimize the exponentially weighted leastsquares cost function, where λ is a positive constant forgetting factor close to but lessthan one and the inverse of (1 − λ) can be used for determining the memory of the al-gorithm. When λ equals to one, we have the ordinary method of least squares and itsmemory is infinite. Incorporating a forgetting factor guarantees that the data input inthe distant past are ”forgotten”, so that the algorithm can track changes in the systemand maneuver in a time-variant environment.

5.5.2 Adaptive forgetting factor for RLS algorithm

RLS with a constant forgetting factor λ is often used to update coefficients of channelestimator, but sometimes using a constant forgetting factor may not yield a good performancein time-varying environments. The adaptive forgetting factor for RLS is obtained and basedon the updated estimator error measurements. To track slow statistical variations of the fadingchannel status, the forgetting factor gives a larger weight to more recent data in order to copewith the channel variations or to follow its variation.

A good steady state performance could be achieved for the RLS algorithm if the envi-ronment is stationary. In the case of the non-stationary environment, the used forgetting factorλ has a finite memory aiming to track the variation of the channel fading status. If λ = 1 alldata are weighted equally and the memory of RLS algorithm is infinite which is perfect tosuppress the estimation noise effect. On the other hand, if λ is a small value then the algorithmhas a short memory which is better to track the channel dynamics. If RLS algorithm is usedin channel estimation, there is an assumption that the channel is time invariant during sametime. In order to overcome the channel estimation error, trial and error method can be usedto search for ”a suitable” forgetting factor λ , which may not be appropriate. Furthermore,the optimal value of λ depends on the channel variation rate and the noise component valuein the received signal. Due to the effects of input noise and channel variations, there may beerrors in the estimated received signal. If the received signal has a low SNR value, there willbe a big effect of the noise in the channel estimation. As a result, it will be better to use amechanism of updating the forgetting factor which can be represented as follows

E {e[k]e∗[k]} = σ2nA[k], (5.59)

95


where e[k] are uncorrelated noise sequences, σ2n is the noise variance resulting from the input

additive noise which can be calculated, and A[k] is an unbiased error calculated and dependsonly on the error of the received signal

A[k] =1

σ2n

λ[k]A[k − 1] + e[k]e∗[k]

γ[k](5.60)

γ[k] = 1 + λ[k − 1]γ[k − 1] (5.61)

with initial values λ[0] = 1, A[0] = 0 and γ[0] = 0. Because of the effects of the input noiseand channel variations, there may exist errors in the A[k] estimation. As a result of this error,λ[k] may be larger than 1, so in practice λ[k] must be controlled to be not more than ONE toguarantee that 0 < λ[k] < 1 and this can be done by

λ[k] = λ[k + 1] + µ1 sgn(λ[k] − λ[k − 1]), (5.62)

where µ1 is a step-size proportional to the channel fading, variation rate and inversely propor-tional to SNR value of the received signal [62]. sgn(.) denotes the sign function

sgn(u) =

1 if u > 00 if u = 0

−1 if u < 0(5.63)


This chapter part is divided into three main parts covering the channel estimation pointof view. First, the channel estimation for MIMO if the used STC is Alamouti scheme. Second,the channel estimation for MIMO if the used STC is DAST. The last part is the effect ofthe channel estimation error with different combining techniques and how this effect could beminimized by using the suboptimum combining techniques.

5.6.1 Channel estimation and Alamouti scheme

Using ML for channel estimation

Based on the orthogonality which is recognized between the data blocks of the Alamoutischeme which could lead in the same time to the orthogonality between channel paths. Assum-ing the received detected data is correct (by using training or pilots sequences in the trainingmode) and the ML detection technique is used in the training and tracking modes. This way,the ML technique could be applied to estimate the channel as stated in section 5.2. A movingaverage will be used to induce a certain smoothing to the channel estimates as explained byEquation (5.36) or by using moving average which may have a length depending on the channelvariation model.

Figure 5.7 shows the BER for method described in section 5.2.2 using ML and Alamouti scheme(the duality between data detection and channel state information) for channel estimation. The

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Figure 5.7: ML is used for channel estimation of the Alamouti scheme. Four received antennasare used with QPSK with channel time variance of 100 symbols. The channel estimation isupdated only at the training mode for each block as indicated by ”initial channel estimate”.”Updating channel estimate” means the channel is initialized at the beginning of each trans-mission block and there is a continues updating by using the detected data.

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Figure 5.8: BER of the Alamouti scheme using ML, where ML is used for, training or tracking.To calculate the bounds, the transmitted vectors are assumed to be perfectly known to thereceiver.

training is 4% of each block of length 100 symbols assuming the channel is constant through100 symbol duration length. The weighting (or the smoothing procedures) factor could be asstated before in Equation (5.36) and the data constellation is QPSK with 4 receiving antennas.Figure 5.7 indicates that there is a very slight difference between the performance of the MLand the perfectly known channel. It introduces also the effect of the updating procedure usingthe scaled moving average in Equation (5.36). This updating procedure gives 1dB improvementover the initial estimates (i.e, estimating the channel only at the training mode by pilots) atBER= 10−4.

ML with LMS, NLMS and RLS for channel estimation

Figure 5.8 gives some comparisons assuming the channel to be constant during the blocklength of 100 symbols and 4% training with ML channel estimation for training mode (i.e.,a new realization of the channel each 100 symbol block length) and ML or LMS algorithmfor tracking. Figure 5.8 shows the BER using LMS algorithm for the channel estimation andindicates that the LMS estimation hits the performance of the upper bound at high SNR if theML is used in the training phase (where the ML estimation makes the initialization of the LMSestimator).To calculate the upper bound, the transmitted vectors are assumed to be perfectlyknown to the receiver. Figure 5.8 is also indicated that the ML (using Alamouti scheme) fortraining and tracking without using a smoothing procedure for estimates will fail to be close tothe upper bound of the performance.

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Figure 5.9: Using different algorithms for training and ML technique is for tracking the channelestimation. As an example, LMS-ML is LMS algorithm in training the channel estimation andML for tracking its estimates.

Figure 5.9 gives different tracking techniques (LMS, NLMS and RLS) using ML for trainingwith time-varying channel fdTs = 0.001 and Alamouti scheme with two receiving antennas andQPSK constellation. It can be recognized from Figure 5.9 that it will be hard for the ML totrack the channel estimation if the used algorithm in the training mode is not ML. Figures 5.10and 5.11 introduce more channel estimation techniques comparisons using different algorithms(ML, RLS and Normalized-LMS (NLMS)). For training and tracking the channel estimationis done by LMS and RLS respectively. The channel is time-varying with fdTs = 0.001 usingAlamouti with 2 receiving antennas. From Figures 5.10, we conclude that the ML and RLScan be used as training algorithms for LMS. In the RLS channel estimation, it is possible touse ML or RLS for training as shown in Figure 5.11.

Effects of algorithms parameters with channel estimation

We are trying now to improve the performance of algorithms by modifying some pa-rameters of these algorithms, which can improve the performance of the estimation. In LMS,the step size ∆ can be time varying, where at the beginning of the sub-blocks inside the blocktransmission the step size will have a big value and after that it will be reduced gradually in alinear way. This way, LMS can convergence faster at the beginning and after that the reduced

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Figure 5.10: Channel estimation for Alamouti scheme by using different algorithms for training(ML, LMS, NLMS, and RLS) and LMS for tracking the estimates.

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Figure 5.11: RLS algorithm is for tracking the channel estimates of the Alamouti scheme usingdifferent algorithms for training.

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Figure 5.12: Different algorithms for training and Tracking are used for channel estimation.Four receive antennas are used for Doppler frequency fdTs = 0.001. Different parameters havedifferent value to compare their effect on the channel estimation.

step size will follow the variation of the channel in a smooth way. Similarly, the forgetting factorλ in the RLS algorithm could be also time-variable. Another parameter of the channel estima-tion which can be considered is the sliding average window to smooth the channel estimationvalues. The length of this sliding window is inversely proportional with the speed of the channeltime variations. Figures 5.12, 5.13 and 5.14 present the comparisons of traditional LMS, RLS,LMS with variable step-size and RLS with variable forgetting factor for different time-varyingchannels: slow (fdTs = 0.001), medium (fdTs = 0.01) and fast (fdTs = 0.1). Figure 5.13 showsthe assumed noise variance σ2

n in Equation (5.60). If there is a range information given for LMSor RLS channel estimation in the legend of Figures 5.12, 5.13 and 5.14, this means that the ∆for LMS or λ for RLS is linearly time varying through the whole block of channel estimationbeing the largest at the beginning and acquiring the lowest value at the end of the block.

For the time varying step-size ∆ of the LMS-channel estimator, its value ranges from asmall (slow convergence with low error variance) to a high (fast convergence with high errorvariance). The same applies for the linear variable forgetting factor λ. If λ0 or µ1 is presented,it means that the adaptive forgetting factor is used for channel estimator as in section5.5.2. The smoothing length of the channel estimator is also indicated in Figures 5.12, 5.13and 5.14, which is inversely proportional to the value of the normalized Doppler frequency fdTs.

According to the results in Figures 5.12, 5.13 and 5.14, we conclude that the smoothing windowlength is the most dominant factor in the ML estimation technique where it should be wide for

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Figure 5.13: Training and tracking channel estimation are done using different algorithms. Dif-ferent values of algorithms parameters are used for Alamouti scheme with four receive antennasand Doppler frequency of (fdTs = 0.01).

slowly time varying channels (fdTs = 0.001) and is thus inversely proportional with the channelvariation rate. In the LMS and RLS algorithms, ∆ and λ are the main control factors and thesmoothing window length plays a small role in the channel estimation (as it was stated before insections 5.4 and 5.5, there is a hidden smoothing procedure in the LMS and RLS algorithms). Asa matter of fact, the difference between the different algorithms performances can be recognizedfor medium and fast time-varying channels. For medium time-varying channels in Figure 5.13,RLS is the best algorithm for tracking the channel variations, but with the suitable forgettingfactor λ, the performance of the adaptive techniques can be further enhanced at high SNR. InFigure 5.14, the ML is the best algorithm that can be used for the channel estimation.

5.6.2 Channel estimation for DAST codes

In this section, the DAST codes reflect the use of signal space diversity and MIMO systemat the same time. In the following, the channel estimation was done by using 4% training at thebeginning of each 100 symbol block length. The training was done by using a Walsh–Hadamardmatrix which contains orthogonal vectors of BPSK alphabets. After the training mode, thetracking depends on the detected symbols. The variance time of channel equals two symbols.

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Figure 5.14: BER of the Alamouti scheme using different algorithms for training and trackingof the channel estimation. The normalized Doppler frequency is (fdTs = 0.1) with four receiveantennas.

Channel estimation using MMSE and LS algorithms for DAST

Averaging window length is an important factor when using MMSE for channel estimation.Figure 5.15 shows the effect of the smoothing length of the MMSE channel estimator. Where thedata constellations are BPSK, the rotated constellations were achieved by rotated-Hadamardfrom Bury [31], and the channel time variance equals the time of 100 block symbol lengthand 4% training. We conclude now that the smoothing window length is an important factorwith MMSE channel estimator which should be chosen based on the knowledge of the channelvariation rate.Figure 5.16 compares between LMS, LS and MMSE for channel estimation assuming the channelis static through 100 symbol and using 4% training at the beginning of each block. For MIMO-DAST the smoothing length is indicated in the figure. The data constellations in Figure 5.16,like in Figure 5.15, are BPSK, the MIMO-DAST system has 2 receiving and 4 transmittingantennas. The performance of MMSE, LS and ML-LMS channel estimation techniques can beadjusted to have approximately the same performance by adjusting the smoothing or averaginglength and also by using a suitable step size ∆ for LMS algorithm.

Using LMS, NLMS, and RLS for channel estimation with DAST

Figures 5.17, 5.18 and 5.19 represent the simulation results using LMS, NLMS and RLS fortracking the channel estimation assuming the training was done using different algorithms. ML

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Figure 5.15: Different smoothing length are used with MMSE for channel estimation. The STCis DAST with 4 transmit and 2 receive antennas.

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Figure 5.16: MMSE, LS, and LMS are used for channel estimation for DAST with 4 transmitand 2 receive antennas.

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Figure 5.17: LMS algorithm is used for tracking channel estimates with DAST and QPSKdata constellations. Different algorithms are used for initializing LMS channel estimator for achannel having a 0.001 normalized Doppler frequency.

represents the using of Walsh-Hadamard vectors and ML for channel estimation. ML is alsoused for initializing channel estimation if another algorithms is used in the training mode. Thenumber of transmitted antennas is nt = 2 and the number of received antennas is nr = 4.Figures 5.17, 5.18 and 5.19 have common feature in that none of the combination of algorithmscould achieve the channel estimation task. In addition, all Figures show an error-floor forSNR > 10 dB and the reason of this is some channel estimation errors which have a somehowsteady state performance at high SNR (look like additive noise). The difference between thecombined algorithms (if they are working probably) is recognized at high SNR.

5.6.3 DAST in the presence of channel estimation errors

Figure 5.20 shows the effect of channel estimation errors on the Bit Error Rate (BER)for the DAST STC with two receive and EIGHT transmit antennas. With only 30% channelestimation error, the BER can increase to a critical level. In the following, we are going to dealwith channel estimation error by studying the effect of channel estimation (with errors) andthe combining techniques.

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Figure 5.18: NLMS algorithm is used for tracking channel estimates of DAST with QPSK dataconstellations. Different algorithms of training mode are used for a channel having a 0.001normalized Doppler frequency.

Different combining techniques for DAST with imperfect channel knowledge

As was shown in section 5.6.2, there are some estimation errors which can be recognized bythe appearance of an error-flow at high SNR. In DAST codes, we use some sort of combiningtechniques at the receiving side assuming we have perfect channel knowledge at the receivingside which is not the case if the channel is to be estimated. Figure 5.21 and 5.22 presentdifferent combining techniques (maximum ratio combining (MRC) and equal gain combining(EGC)) with different number of receive antennas (1, 2, and 4) using 4 and 8 transmit antennasrespectively.

For the optimum receiver, the MIMO channel must be perfectly known at the receiving side,see equation (4.61), and the question now arises, what happens if this knowledge is not perfect,because the estimation is based on a noisy input signal. Since estimation errors occur, it is notstraightforward to say that algorithms which are optimum for perfect channel knowledge willremain optimum. Simpler algorithms may possibly lead to better overall performance as willbe explained later in this chapter. We discuss this topic with the example of diagonal algebraicspace time (DAST) codes [28]. They promise full transmit and receive diversity (with rate 1)and can be taken as a candidate for a robust transmission over MIMO channels.

To concentrate on the effects caused by spatial and temporal matched filtering as in Equation(4.61), we take a sphere detector [47] for VED because of its ML or near ML performance.Optimal MRC will be compared with EGC on the basis of the symbol error rate (SER). For

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Figure 5.19: DAST with QPSK using RLS algorithm for tracking the channel estimation.Different algorithms are used for the training mode for a channel having a 0.001 normalizedDoppler frequency.

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Figure 5.20: MIMO system with 8× 2 uses DAST with BPSK data constellations. BER curvesrepresent different average percentages of channel estimation errors.

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Figure 5.21: DAST is used for different MIMO systems with 4 × 1, 4 × 2 and 4 × 4 (transmitantennas × receive antennas). The Data constellations are BPSK with MRC or EGC as acombining technique at the receiver.

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Figure 5.22: MIMO systems 8 × 1, 8 × 2 and 8 × 4 using DAST. The data constellations areBPSK with MRC or EGC to combine at the received signal.

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channel estimation, we take an LMS (orthogonal-LMS) algorithm [10] because of its simplicity.

Remark

In genereral, if the channel does not assume to be perfectly known, the optimal detector forOFDM and MC-CDM is a detector that may be decomposed into a training-based LMMSEchannel estimator followed by a decision stage which searches for the minimum of a generalMahalanobis distance, see [63], [64]. This means that, irrespective of the signal symbolalphabet (PSK, QAM, etc.), no optimality is lost by adopting a common LMMSE channelestimator – which estimates both amplitude and phase. The subsequent detector appliesthe vector detection rule as given in [64], but generally this results in receiver complexityincreasing exponentially with the number of OFDM data-subcarriers. Because of the equiv-alence between OFDM/MC-CDM and DAST, we conclude that this remark is also relevanthere. More details about the equivalence between DAST and MC-CDM are found in section 4.4.

Figure 4.1 shows the basic model of the transmission that we assume here. This vectortransmission model is on symbol basis, k is the discrete symbol time. For the linear ST blockcodes considered here, an optimum receiver consists of two parts: The space-time MF matrixw followed by vector detection (VED).

The transmission was organized in blocks of 100 symbols, where the first 4 symbols (+ ++−) were training symbols used to initialize the LMS algorithm. The remaining 96 datasymbols were transmited with QPSK and the detected data symbols were taken to track thechannel variations with the LMS algorithm. The blocks were transmitted independent of eachother. For all paths of the MIMO channels considered here, independent Rayleigh fadingwas generated according to the modified-Pop&Beaulieu channel model 5.1.3. The normalizedmaximum Doppler frequency was 0.001. The time-variant behavior generated with this modelwas taken only in two symbol intervals, i.e. over subblocks of two symbols it was time-invariant.For performance comparison of different schemes, the symbol error rate (SER) was measured.As explained before, we look for two alternatives related to the first part of signal processingof the received vector y – see (4.61): MRC and EGC.

Figure 5.23 shows simulation results for a 2 x 1 MIMO channel. For a SER of 2 · 10−3, thedifference between a perfectly known channel and the estimated channel is about 6 dB. Thisreduces to 3 dB if EGC is taken instead of MRC. Similar results are shown in Figure 5.24 toFigure 5.26 for 2 x 2, 2 x 4 and 4 x 4 MIMO. EGC is better than MRC if channel estimationerrors are present. The opposite is true in case of ideal channel knowledge.

The simulation results with diagonal algebraic space-time (DAST) block codes Figures 5.24,5.25 and 5.26 show that in case of channel estimation errors equal gain combining (EGC) isbetter than the maximum ratio combining (MRC), despite of the fact that EGC is suboptimumfor perfect channel knowledge. Additionally, the signal processing complexity is lower for EGCthan for MRC. It must be noted that the estimation algorithm (LMS) has an influence on theseconclusions.

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Figure 5.23: MIMO system 2 × 1 using DAST code with QPSK. The channel is estimated byusing LMS with MRC or EGC.

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Figure 5.24: MIMO system 2 × 2 with DAST code and LMS for channel estimation.

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5.7 Summary

This chapter includes modelling the communication Rayleigh channel through an ad-justable channel variation model. We have introduced three channel models. The first one isthe Clarke’s model and the other two are modified version of it to obtain more realistic models.This models which introduced were useful to model a time varying channel with adjustableDoppler frequencies in the simulations. The Alamouti scheme has its own characteristic withorthogonal data transmission blocks, which can be converted to orthogonality between channelpaths. This way, we employed ML for channel estimation with Alamouti scheme. In addition,we introduced and compared different algorithms (ML, MMSE, LS, LMS, NLMS RLS andadaptive forgetting factor-RLS) to be used in the channel estimation of Alamouti and DASTcodes. We found that the soothing length is a key factor of the channel estimation in MIMOsystem (here we used Alamouti and DAST codes). The combination of two algorithm for MIMOchannel estimation is also introduced and we concluded the suitable combinations. The channelestimation error is studied on performance when we use different combining techniques (MRCand EGC) for DAST codes. We concluded that the performance of sub-optimum combining byusing EGC can be better the the optimum one of MRC for imperfect channel estimation.

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Chapter 6Summary, Conclusion, and Suggestions for

Future Work

In this chapter we summarize the basic ideas and conclusions of the thesis. We deal withthe problem of efficient digital communication over wireless channels when the transmittedsignal suffers from Rayleigh fading. We mean by “efficient“ that the power efficiency is as highas possible, i.e. the bit error rate is as low as possible for a certain SNR, and the bandwidthefficiency (bit/sec/Hz) is as high as possible at the same time. These are the main motivationsbehind the concept of the signal-space diversity. Within this framework we focus on channelestimation and influence of signal-space diversity. The considerations include the investigationof channel estimation and introducing the idea of using two algorithms for channel estimationto deal with the channel estimation error. We also include several suggestions to extend thework presented in this thesis.

6.1 Summary and conclusions

Channel estimation plays an important role especially if the number of parameters tobe estimated is big as it is usually given for MIMO systems. Due to the bandwidth efficiencyrequirements, transmit power restrictions, and the general time-varying nature of the channel,

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6 Summary, Conclusion, and Suggestions for Future Work

training signal energy and time are always limited in practice. This leads to imperfect channelestimation which impairs the accuracy of detecting the transmit data and in turn degradesthe overall system performance. With Chapter 2 devoted to the fundamentals, we gavein introduction to linear MIMO channel identification, MMSE for channel estimation andmodelling the estimation error based on it as an estimation technique. First in Chapter 3, weconsidered the effects of channel estimation error in a multiple-input multiple-output Rayleighchannel environment. This study was then extended to a different aspect in Chapter 4, wherewe investigated the signal-space concept to combat the channel effect at the transmit side.In this chapter we assumed perfect channel knowledge of channel fading coefficients at thereceiver side. In Chapters 3 and 5, we assumed at the receiving side there is a need to estimatethe channel. This channel estimation is studied from two points of view. The first one is thechannel estimation error effect in Chapter 3 as stated above and the second is the channelestimation techniques and algorithms in Chapter 5. The transmitted signal throughout thethesis was modulated using BPSK and QPSK, although the concepts we introduced can beextended to other modulation schemes.

We considered only slow fading on the channel where “slow“ is with respect to thesymbol duration of the digital transmission. The fading process is modelled as a Gaussianrandom variable. A new realization for the random variable is generated for a block of transmitsignals. This model is used in the thesis, since it is a good approximation for many realscenarios which employ signal-space or space-time.

The influence of imperfect channel knowledge and how it can be modelled are the keyfactors for understanding the subject. The transmitter and receiver design can be developedto match or overcome the bad effects of the channel or imperfect channel knowledge.

• Optimal MMSE estimation. Throughout this thesis the error model of the channelestimation was based on using the minimum mean square error (MMSE) criterion forchannel estimation. On one hand, using maximum likelihood (ML) estimation and usingMMSE for smoothing the estimates leads to near the optimum performance. On theother hand, the channel estimation error is modelled according to the MMSE channelestimation as an additive Gaussian noise. This modelling of the channel estimation isused to study the channel estimation effect on capacity and performance aspects.

• Imperfect MIMO-channel effects. We studied how the channel capacity will beaffected with the imperfect channel knowledge, considering the channel estimation isdone by using MMSE estimation. The optimal receiver can be affected, if the channelis known imperfectly. We further studied the optimal detection rule for perfect channelknowledge and how it becomes sub-optimal in the presence of channel estimation error.For this reason, we introduced signal–space diversity and combining techniques forMIMO channels taking into account imperfect channel estimates. We found that theoptimal combining techniques lose its optimality with imperfect channel knowledge.According to this result, we suggested suboptimal combing techniques to overcome theimperfect channel effects.

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6.2 Suggestions for future work

• Combined signal and spatial diversity to combat fading. We combine signal-space and space time diversity to combat channel fading. We dealt with signal-space(modulation) diversity from many point of views. First, the design techniques ofmultidimensional constellation have the ability to combat the effect of Rayleigh fadingchannels. In addition, the concatenation of signal-space diversity with MIMO system isemployed to construct diagonal algebraic space-time (DAST) codes. We also introducedsome rotation matrices to be used in DAST codes. We concluded that for a BPSK dataconstellation there is a possibility to use different rotation matrics with either half orfull diverity design with neglegable differences in performance. For half diversity orderwe found that some good rotation matrices could have the same performance for highersignal constellation. we concluded also the equivalence between DAST and MC-CDM.

• Channel estimation. Through the whole thesis, there is an assumption to have atraining phase to estimate the channel and a tracking phase in which the informationsymbols are transmitted. During the tracking phase the channel is estimated with the aidof the detected symbols. Moreover, two channel modes were used, a block fading channelmodel and continuously time-varying channel. We also gave a comparison between threemodels of generating Rayleigh fading channels and suggested one of them to be used inthe simulation with time varying MIMO system. Some estimation techniques are studiedfor the MIMO systems using the Alamouti scheme (as a space-time code (STBC)) andthe DAST scheme (as STBC using signal space diversity). These estimation algorithmscontain maximum likelihood (ML), minimum-mean square error (MMSE), least-squares(LS), least-mean square (LMS), normalized-LMS (NLMS), recursive least squares (RLS)and RLS with adaptive step-size. At the same time some signal-space diversities (fulland half diversity) are investigated. In order not to complicate the investigation someassumptions are made throughout this thesis. This means that, the channel is Rayleigh(memoryless) with quasi-static channel for certain block lengths. As a conclusion wecan state that the channel estimation can be done for MIMO systems by using manyalgorithms from MMSE with near the optimal performance. The only condition is to usea good initialization for the algorithm with a suitable smoothing mechanism.

6.2 Suggestions for future work

The presented work in this thesis can be extended and continued in different directions. Somesuggestions for future work can be given:

• In this thesis, we have assumed an uncoded transmission. Using a coded transmission is awide area of further research. In this case, we can use joint vector detection and decodingwith channel estimation.

• The analysis in this thesis is mainly discussed for BPSK and QPSK, although it is possibleto extend to higher signal constellations which can be used for future research direction.

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6 Summary, Conclusion, and Suggestions for Future Work

• Throughout this thesis, a Rayleigh fading channel model is assumed. Other channelmodels can also be applied in the case of imperfect channel estimation with some changesin the notations. Nevertheless, the appropriate design of the receiver in this case, includinga decision-directed improvement of channel estimates, is a wide area for further research.

• Channel estimation in this thesis is mainly based on the MMSE and/or ML criterion.Other criteria could be used and modelling the channel estimation error in this case is anopen area.

• In designing STC with channel estimation and iterative detection and/or decoding, thesystem complexity is always an issue faced by the system designers. There is a tradeoffamong various parameters, and one usually has to find moderate complexity to obtain agood balance. Therefore, the idea of reducing computational complexity of detection andchannel estimation can be explored further.

• We used only the optimal detection techniques ML or MAP. These techniques can notbe used for higher order signal constellations due to the higher complexity. Hence, thesub-optimum detection techniques like sphere decoding, recurrent neural networks anditerative algorithms can be considered.

116

Appendix AList of Abbreviations

AWGN Additive white Gaussian noise.BER Bit error rate.BEP Bit error probability.BPSK Binary phase-shift keying.CDMA Code division multiple access.CSI Channel state information.CRB Cramer-Rao bound.DAST Diagonal algebraic space time.EGC Equal gain combining.FIR Finite impulse response.IBI Interblock interference.i.i.d Independent and identically distributed.ISI Intersymbol interference.LMS Least mean square.LMMSE Linear minimum mean square error.LS Least squares.MAI Multiple access interference.MAP Maximum a posteriori.MC-CAFS Multicarrier-cyclic antenna frequency spread.MC-CDM Multi-carrier CDM.

117

A List of Abbreviations

SER Symbol error rate.ML Maximum likelihood.MMSE Minimum mean square error.MPSK M-phase shift keying.MQAM M-quadrature amplitude modulation.MRC Maximum ratio combining.MSE Mean square error.NLMS Normalized-LMS.PMEPR Peal-to-mean envelope power ratio.OFDM Orthogonal frequency division multiplexing.QAM Quadrature amplitude modulation.QPSK quadrature PSK.RLS Recursive least squares.SIMO Single-input multiple-output.SNR Signal-to-noise ratio.STBC Space-time block code.STC Space-time code.UEW Unequal weights.VED Vector detection.WH Walsh-Hadamard transform.WSSUS Wide-sense stationary uncorrelated scattering.

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