Copyright

by

Ren Wu

2008

The Dissertation Committee for Ren Wu

certifies that this is the approved version of the following dissertation:

Multiple-Antenna Wireless Communications:

Detection and Estimation with Smart Antennas,

and Space-Time Code Design Considerations

Committee:

Ioannis Psaromiligkos, Supervisor

Milica Popovich

Jan Bajcsy

Multiple-Antenna Wireless Communications:

Detection and Estimation with Smart Antennas,

and Space-Time Code Design Considerations

by

Ren Wu, M.Eng.

Dissertation

Presented to the Faculty of Engineering of

McGill University

in Partial Fulfillment of the Requirements

for the Degree of

Doctor of Philosophy

McGill University

December 2008

To my family.

Acknowledgments

This research work would not have been possible without the help/support

of many people. In the first place the author wishes to express his gratitude to his

supervisor, Prof. Dr. Ioannis Psaromiligkos who was abundantly helpful and offered

invaluable assistance, support and guidance. By guiding the author through the ini-

tial research work, Prof. Dr. Ioannis Psaromiligkos insured that the author was

well positioned from the early stage. Prof. Dr. Ioannis Psaromiligkos has provided

endless encouragement and support, instructions and guidance, scientific insights and

inspiration, and academic and research knowledge and experience to the author in

every aspect of his PhD study. Prof. Dr. Ioannis Psaromiligkos has given valuable

instructions and supervision in the scientific and engineering area to explore and on

the direction to go to pursue the scientific truth. Along the way, all of the advice, sug-

gestions and discussions turned out to be extremely helpful in clarifying uncertainty

and removing ambiguousness.

Deepest gratitude are also due to the members of the supervisory committee,

Prof. Dr. Milica Popovich and Dr. Jan Bajcsy without whose knowledge and as-

sistance this study would not have been successful. The author has attended both

professors’ classes and has learnt essential knowledge that has been beneficiary to

his study. Lessons given by Prof. Dr. Milica Popovich provided the author with

solid background in the area of antenna systems. Lessons taught by Prof. Dr. Jan

Bajcsy equipped the author with fundamental and advanced wireless digital commu-

nications theory. In addition, they kindled author’s interest in research in wireless

v

digital communications.

The author cannot end without thanking his family, whose constant encour-

agement and dedication inspired him and supported him all along the way through

his study. He is grateful to his wife Yiying Zuo, and his parents Hanrong Wu and

Jianfen Li.

Ren Wu

McGill University

December 2008

vi

Multiple-Antenna Wireless Communications:

Detection and Estimation with Smart Antennas,

and Space-Time Code Design Considerations

Publication No.

Ren Wu, Ph.D.

McGill University, 2008

Supervisor: Ioannis Psaromiligkos

The main theme of this thesis is wireless communications using multiple anten-

nas. The thesis consists of four topics on smart antenna technology, its applications

to direct sequence code division multiple access (DS/CDMA) communications, and

multiple-input multiple-output wireless communications. The first problem under

consideration is the joint estimation of direction-of-arrival (DoA), propagation de-

lay, and complex channel gain for antenna-array DS/CDMA communications over

frequency selective multipath channels. We propose a subspace based MUSIC-type

estimation algorithm which utilizes the spatial smoothing preprocessing technique.

The proposed algorithm essentially breaks the multipath induced coherency within

the received signals and recovers the full signal subspace spanned by the dominant

vii

signal paths of all users. This allows for the use of MUSIC-type DoA and delay

estimators for individual paths of a particular user. We then describe a new crite-

rion for detecting the number of signals impinging on a uniform linear array (ULA),

which exploits eigenvector information of the sample array covariance matrix and

makes explicit use of the peak information of the MUSIC spectrum. In the third part

we present an iterative weight matrix approximation (IWMA) algorithm. IWMA

computes an approximation to the optimum weight matrix used by weighted spatial

smoothing (WSS) to completely decorrelate input sources and generate a diagonal

source covariance matrix. A useful observation regarding IWMA is that the gen-

erated matrix is suitable as a basis for subspace-type DoA estimation. In the last

part we discuss two deterministic measures for designing linear processing space-time

block codes (a.k.a. linear dispersion codes). The first measure is obtained by ap-

plying Jensen’s Inequality to the mutual information criterion for linear dispersion

codes. We show that there is a tractable relationship between this measure and the

mutual information criterion. The second measure is a natural extension of the con-

ditions required for complex linear processing orthogonal designs. The relationship of

the second measure to the total-squared-correlation (TSC) is revealed. The connec-

tion and difference between the set of conditions and the LDC mutual information is

illustrated via the first and the second measures we obtained.

viii

Systemes de Communications Sans Fil

Multi-Antennes: Detection et Evaluation avec les

Antennes Intelligentes, et Considerations de

Conception de Code D’espace-temps

Publication No.

Ren Wu, Ph.D.

Universite McGill, 2008

Superviseur: Ioannis Psaromiligkos

Les communications sans fil utilisant des antennes multiples constituent le theme

principal de cette these qui traite de quatre sujets concernant la technologie des an-

tennes intelligentes, son application aux communications a sequence directe CDMA,

et des communications a entrees multiples et sorties multiples (MIMO). Le premier

sujet est celui de l’estimation simultanee des directions d’arrivee, du retard de propa-

gation et du gain complexe du canal pour des communications CDMA sur des canaux

multi-trajets a frequences selectives. Nous proposons un algorithme d’estimation

de type MUSIC a base de sous-espaces, utilisant un pretraitement par lissage spa-

tial. L’algorithme propose brise essentiellement la coherence induite par les trajets

ix

multiples pour recouvrer entierement le sous-espace du signal cree par les signaux

dominants de tous les utilisateurs. Ceci permet l’utilisation d’estimateurs MUSIC

des directions d’arrivee et du retard pour les signaux d’un utilisateur donne. Nous

decrivons un nouveau critere pour detecter le nombre de signaux captes sur un reseau

d’antennes uniforme et lineaire qui exploite le vecteur propre de la matrice de co-

variance des echantillons et qui utilise l’information des pics de spectre de MUSIC.

Troisiemement, nous presentons un algorithme iteratif d’approximation de la matrice

des poids (IWMA) qui calcule une approximation de la matrice des poids optimale

utilisee pour le lissage spatial (WSS), afin de completement decorreler les sources

d’entrees. Avec IWMA la matrice qui est generee peut etre utilisee comme base

de l’estimation de type sous-espace des directions d’arrivee. Nous discutons finale-

ment de deux mesures deterministiques pour concevoir des codes dispersion lineaires.

La premiere mesure s’obtient par l’application de l’Inegalite de Jensen au critere

d’information mutuelle pour codes de dispersion lineaire. Nous montrons qu’il y a

une relation tractable entre cette premiere mesure et le critere d’information mutuelle.

La deuxieme mesure est une extension naturelle des conditions requises pour le traite-

ment lineaire complexe d’un design orthogonal. Nous mettons a jour la relation de

la deuxieme mesure avec la correlation quadratique totale (TSC). Ces deux mesures

illustrent le lien et la difference entre le jeu de conditions et l’information mutuelle

des codes de dispersion lineaires.

x

Contents

Acknowledgments v

Abstract vii

Abrege ix

List of Figures xv

Chapter 1 Introduction 1

1.1 Wireless Digital Communications in Spatial and Temporal Dimensions 2

1.2 Multiple-Antenna Wireless Communications . . . . . . . . . . . . . . 4

1.3 Smart Antenna Based Techniques . . . . . . . . . . . . . . . . . . . . 5

1.4 The Parameter Estimation Problem in Antenna Array Signal Processing 5

1.4.1 The Problem of Number of Signals Detection . . . . . . . . . 6

1.4.2 Estimation of Directions-of-Arrival . . . . . . . . . . . . . . . 7

1.4.3 Joint DoA and Delay Estimation in Array CDMA Communi-

cations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

1.5 Multiple-Input Multiple-Output Wireless Communications . . . . . . 14

1.6 Objective of the Thesis and Summary of the Contributions . . . . . . 16

1.6.1 Objective of the Thesis . . . . . . . . . . . . . . . . . . . . . . 16

1.6.2 Summary of the Contributions . . . . . . . . . . . . . . . . . . 17

1.7 Organization of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . 18

xi

Chapter 2 Background 22

2.1 DS/CDMA Wireless Communications . . . . . . . . . . . . . . . . . . 23

2.2 Antenna Array - ULA . . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.3 Multiple-Input Multiple-Output and Space-Time Block Coding . . . . 35

2.4 Notes on the Notations Used In The Thesis . . . . . . . . . . . . . . 41

2.5 Systems, Models, Signals and Assumptions . . . . . . . . . . . . . . . 42

Chapter 3 Spatial-Smoothing Based MUSIC-Type Joint DoA and Time-

Delay Estimation 43

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

3.2 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

3.3 Spatial Smoothing Based Joint Direction of Arrival and Delay Estimation 47

3.4 Channel Estimation And Removal of Timing Ambiguity . . . . . . . 52

3.5 SS Based Joint DoA-Delay Estimation Using Chip-Shifted Estimates

of the Covariance Matrix . . . . . . . . . . . . . . . . . . . . . . . . . 54

3.5.1 Joint DoA and Delay Estimation Using Chip-Shifted Covari-

ance Matrix Estimates . . . . . . . . . . . . . . . . . . . . . . 54

3.5.2 Joint DoA and Delay Estimation Using the Non-Shifted Co-

variance Matrix Estimate . . . . . . . . . . . . . . . . . . . . 58

3.6 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

3.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

Chapter 4 The MUSIC MDL Criterion 70

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

4.2 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

4.3 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

4.3.1 Optimal MDL-based signal enumeration criterion . . . . . . . 73

4.3.2 Suboptimal MDL-based criterion . . . . . . . . . . . . . . . . 74

4.4 Detection Criterion Exploiting Peaks in the MUSIC Spectrum . . . . 75

xii

4.4.1 Observations and Remarks . . . . . . . . . . . . . . . . . . . . 81

4.5 Simulation and Performance Evaluation . . . . . . . . . . . . . . . . . 84

4.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

4.7 Appendix I - Proof of Theorem 3 . . . . . . . . . . . . . . . . . . . . 84

4.8 Appendix II - Proof of Theorem 4 . . . . . . . . . . . . . . . . . . . . 85

Chapter 5 Weighted Spatial Smoothing Based Iterative Weight Matrix

Approximation 89

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

5.2 System Model and Background . . . . . . . . . . . . . . . . . . . . . 92

5.2.1 Weighted Spatial Smoothing . . . . . . . . . . . . . . . . . . . 94

5.3 Proposed Iterative Weight Matrix Approximation Algorithm . . . . . 96

5.4 Theoretical Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

5.5 DoA Estimation Using Wn . . . . . . . . . . . . . . . . . . . . . . . . 107

5.6 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

5.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

Chapter 6 On Two Deterministic Measures for Linear Processing Space-

Time Block Codes 113

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

6.2 System Model and the Linear Processing ST Coding Scheme . . . . . 118

6.2.1 The Equivalent MIMO Channel . . . . . . . . . . . . . . . . . 120

6.3 Jensen’s Inequality and Relaxation of LDC Mutual Information . . . 122

6.4 The GTSC and TSA Metrics . . . . . . . . . . . . . . . . . . . . . . . 126

6.5 Lower Bounds for the GTSC Metric . . . . . . . . . . . . . . . . . . . 128

6.5.1 Bound That is Analogous to Welch’s Bound . . . . . . . . . . 128

6.5.2 Lower Bounds for Other Cases of r and n . . . . . . . . . . . 135

6.5.3 Lower Bounds - Further Results . . . . . . . . . . . . . . . . . 141

6.6 Computer Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . 143

xiii

6.6.1 Jensen’s Relaxation of LDC-MI . . . . . . . . . . . . . . . . . 143

6.6.2 Examples of LP-STBCs: Constellation Rotation and Product

Distance Gain . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

6.6.3 GTSC-TSA Metric and the LDC Mutual Information Criterion 152

6.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

Chapter 7 Conclusion, Discussion, and Future Work 155

7.1 Spatial Smoothing Based JADE-MUSIC . . . . . . . . . . . . . . . . 155

7.2 The MUSIC-MDL Criterion . . . . . . . . . . . . . . . . . . . . . . . 156

7.3 The IWMA Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . 156

7.4 Two Deterministic Design Criteria for LP-STBC . . . . . . . . . . . . 157

7.5 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158

Bibliography 160

xiv

List of Figures

2.1 DS/CDMA transmitters and a DS/CDMA receiver with antenna array 23

2.2 Direct-sequence spread spectrum transmission . . . . . . . . . . . . . 24

2.3 Multipath fading channel . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.4 Uniform Linear Array . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

2.5 ULA system model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

2.6 Conventional spatial smoothing . . . . . . . . . . . . . . . . . . . . . 32

2.7 Multiple-Input Multiple-Output system . . . . . . . . . . . . . . . . . 35

3.1 Spatial-smoothing-based MUSIC spectrum vs. DoA for six possible

values of the delay of user 0 . . . . . . . . . . . . . . . . . . . . . . . 60

3.2 MSE of joint DoA and delay estimator vs. SNR of user 0 . . . . . . . 61

3.3 MSE of joint DoA and delay estimator vs. number of samples . . . . 62

3.4 MSE of channel estimator vs. SNR of user 0 . . . . . . . . . . . . . . 63

3.5 Probability of ambiguity resolution vs. SNR of user 0 . . . . . . . . . 64

3.6 Spectrum of the proposed spatial-smoothing-based MUSIC algorithm

(using chip-shifted covariance matrices). . . . . . . . . . . . . . . . . 66

3.7 Spectrum of the proposed spatial-smoothing-based MUSIC algorithm

(using a non-time-shifted matrix). . . . . . . . . . . . . . . . . . . . . 67

3.8 Probability of acquisition vs. SNR of user 0 . . . . . . . . . . . . . . 68

3.9 MSE of joint DoA and delay estimator vs. SNR of user 0 . . . . . . . 69

xv

4.1 Number of signals detection: MDLMUSIC vs. MDL; 10-element ULA, 4

equal-power sources; in terms of SNR (dB); number of samples is 1500;

averaged over 400 Monte-Carlo runs. . . . . . . . . . . . . . . . . . . 82

4.2 Number of signals detection: MDLMUSIC vs. MDL; 10-element ULA,

8 equal-power sources; in terms of number of samples; averaged over

400 Monte-Carlo runs. . . . . . . . . . . . . . . . . . . . . . . . . . . 83

5.1 Operations of the IWMA algorithm: averaged squared error of Wi

versus number of iterations. . . . . . . . . . . . . . . . . . . . . . . . 109

5.2 Performance of DoA estimation using IWMA-generated weight matrix:

MSE of DoA estimates versus system SNR in dB. . . . . . . . . . . . 110

5.3 Operations of the IWMA algorithm: averaged squared error of Wi

versus system SNR in dB. . . . . . . . . . . . . . . . . . . . . . . . . 111

5.4 Performance of DoA estimation using IWMA-generated weight matrix:

MSE of DoA estimates versus system SNR in dB. . . . . . . . . . . . 112

6.1 The analogy of the Row Column Equivalence . . . . . . . . . . . . . . 131

6.2 Graphical representation of {Ai}’s for [3,2,2] . . . . . . . . . . . . . . 134

6.3 LDC-MI vs. 12T

log det(I2r + M ρNZ) for 3I2O: SNR=20dB; r = 3;

produced from 105 randomly generated LPM’s. . . . . . . . . . . . . 144

6.4 A STBC obtained by 2 × [4, 4, 4] design of ψ = 43.639 and optimally

rotated QAM. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

6.5 A LP-STBC obtained by the two-step design procedure: firstly obtain

a set of LPM designs with ψ ≤ 48; then use QAM signalling and

maximizes with respect to the product distance gain. . . . . . . . . . 149

6.6 A STBC obtained by 2 × [4, 4, 4] design of ψ = 68 and QAM constel-

lation; versus QOSTBC (QAM). . . . . . . . . . . . . . . . . . . . . . 150

6.7 LDC-MI vs. GTSC metric for 2I2O: SNR=20dB; r = 3; produced

from 105 randomly generated LPM’s. . . . . . . . . . . . . . . . . . . 151

xvi

6.8 LDC-MI vs. GTSC metric for 3I2O: SNR=20dB; r = 3; produced

from 105 randomly generated LPM’s. . . . . . . . . . . . . . . . . . . 153

xvii

Glossary

AIC Akaike Information Criterion

BER Bit Error Rate

CDMA Code Division Multiple Access

CML Conditional Maximum Likelihood

CMLE Conditional Maximum Likelihood Esti-

mate/Estimator

CRB Cramer Rao Bound

CSI Channel State Information

CSS Conventional Spatial Smoothing

DEML DEcoupled Maximum Likelihood

DoA Direction of Arrival

DS/CDMA Direct Sequence/Code Division Multiple Access

EDGE Enhanced Data Rates for GSM Evolution

ESPRIT Estimation of Signal Parameters via Rotational In-

variance Techniques

ETSI European Telecommunications Standards Institute

EVD EigenValue Decomposition

xviii

FDMA Frequency Division Multiple Access

FER Frame Error Rate

GIS Geographic Information System

GPRS General Packet Radio Service

GPS Global Positioning System

GSM Global System for Mobile communications

GTSC Generalized TSC

IQML Iterative Quadratic Maximum Likelihood

IWMA Iterative Weight Matrix Approximation

JADE Joint Angle and Delay Estimation

LBS Location-Based Service

LDC Linear Dispersion Code

LDC-MI Linear Dispersion Code-Mutual Information

LP-STBC Linear Processing STBC

LPM Linear Processing Matrix

LS Least Squares

LSML Large Sample Maximum Likelihood

LTE Long Term Evolution

MAI Multiuser Access Interference

MAN Metropolitan Area Network

MDL Minimum Description Length

xix

MIMO Multiple-Input Multiple-Output

MLE Maximum Likelihood Estimate/Estimator

MODE Method Of Direction Estimation

MUSIC MUltiple SIgnal Classification

MVDR Minimum Variance Distortionless Response

OFDM Orthogonal frequency Division Multiplexing

OSMDL Order Statistics MDL

OSTBC Orthogonal STBC

PN Pseudo Noise

QAM Quadrature Amplitude Modulation

QOSTBC Quasi-Orthogonal STBC

RHS Right-Hand Side

SISO Single Input Single Output

SNR Signal to Noise Ratio

SS Spatial Smoothing

STBC Space-Time Block Code

SVD Single Value Decomposition

TDMA Time Division Multiple Access

TLS Total Least Squares

TSA Total Squared Amicability

TSC Total Squared Correlation

xx

ULA Uniform Linear Array

UMTS Universal Mobile Telecommunications System

V-BLAST Vertical Bell Laboratories Layered Space-Time

WiMAX Worldwide Interoperability for Microwave Access

WSF Weighted Subspace Fitting

WSS Weighted Spatial Smoothing

xxi

Chapter 1

Introduction

1

1.1 Wireless Digital Communications in Spatial and

Temporal Dimensions

Since the last decade of the 20th century, mobile telecommunications and wire-

less personal communications have experienced an unprecedented and globally spread-

ing boom. This is particularly due to the success of GSM (Global System for Mobile

communications) [1] digital cellular phone system, which has been widely deployed

around the world. GSM is a system based on the TDMA (Time Division Multiple

Access) and FDMA (Frequency Division Multiple Access) technologies. In addition

to TDMA and FDMA, the technology of CDMA (Code Division Multiple Access) [2]

has been standardized and widely deployed1. Recent developments in digital wireless

communication technology are further driven by new upper-layer user applications.

For example, the incorporation of GPS (Global Positioning System) devices in cellular

phones created GIS (Geographic Information System) related services and applica-

tions (e.g., location-based services (LBS)) which further drive and push the limits of

wireless communications.

Currently, wide-range wireless personal communications are mainly narrow-

band voice communications and low-to-medium rate data communications. Broad-

band wireless communications, on the other hand, are expected to become increas-

ingly popular in the near future. Smart phones and laptops/notebooks (possibly with

reduced dimensions) that are enabled by broadband wireless communication technolo-

gies are envisioned to grow in popularity. Customers could benefit from these devices

and technologies and obtain substantially-improved access to the future mobile Inter-

net. One important broadband MAN-scale (Metropolitan Area Network) mobile data

communication system is the WiMAX (Worldwide Interoperability for Microwave

Access) system [3]. Important features of WiMAX include the OFDM (Orthogonal

frequency Division Multiplexing) wideband modulation technology and the multiple-

1CDMA first appeared as a proposed standard in the early 90s.

2

antenna enabled MIMO (Multiple-Input Multiple-Output) communication scheme.

OFDM is a physical layer communication technique that is suitable for broadband

wireless transmission and reception in hostile multipath frequency-selective environ-

ments. The multipath problem is tackled conventionally by time-domain equalization

at the receiver which is computationally demanding. In an OFDM system, to combat

multipath fading effect, the strategy is to transmit in parallel several low-rate data

streams across the spectrum available where each stream experiences a flat-fading

channel. On the other hand, the MIMO technology is capable of increasing a commu-

nication system’s capacity multi-fold by means of the spatial dimension of communica-

tion that becomes available via deploying multiple antennas at both the transmitter

and receiver. With MIMO systems, a high spectral efficiency can be achieved by

exploiting the new spatial channels existing within the systems. The MIMO and

OFDM technologies underlie not only WiMAX but also the 802.11n WLAN (Wire-

less Local Area Network) wireless system. Furthermore, the 3GPP (3rd Generation

Partnership Project) LTE (Long Term Evolution) standardization body also advo-

cates OFDM and MIMO as key technologies for the physical layer. 3GPP LTE is

a strong competitor to WiMAX and consists of a comprehensive set of technologies,

systems, protocols and specifications. Its goal is to combine the most recent state-

of-the-art developments in wireless digital communications with ETSI’s (European

Telecommunications Standards Institute) successful engineering practices in the field

of cellular communications. These practices in the past resulted in the introduction

of GSM, GPRS (General Packet Radio Service), EDGE (Enhanced Data Rates for

GSM Evolution) and UMTS (Universal Mobile Telecommunications System) wireless

systems and standards.

This thesis presents research work in the area of multiple-antenna wireless

communications. In the thesis two scenarios of multiple-antenna communications are

considered: in the first scenario only the receiver employs multiple antennas; while in

the second both the transmitter and receiver employ multiple antennas.

3

1.2 Multiple-Antenna Wireless Communications

The use of multiple antennas (at either the receive side or both the transmit

and receive sides) in wireless point-to-point communications adds an extra spatial

dimension to the existing time and frequency dimensions. With the addition of the

new dimension, new transmission and reception methods and techniques can be used

to improve systems’ reliability and spectral efficiency compared to the original single-

antenna systems. Several representative techniques have been developed for this

purpose, such as [4], [5]:

1. Spatial multiplexing that improves spectral efficiency and increases data transfer

rate of wireless systems;

2. SDMA (Space-Division Multiple Access) that provides a new type of multiple

access technology;

3. Transmit beamforming that takes advantage of partial channel knowledge at

the transmitter side;

4. Interference-nulling achieved through array signal processing techniques such as

receiver-side beamforming;

5. Spatial transmit and receive diversity can be achieved via space-time coding

and diversity combining at receiver, respectively.

The above list provides only a glimpse of the possible benefits that multiple

antenna systems can bring to contemporary wireless communications. But generally

speaking, we can divide the current multiple-antenna techniques into two categories:

smart antenna based techniques and MIMO based techniques.

4

1.3 Smart Antenna Based Techniques

The smart antenna technology consists of signal processing techniques and

methods that use multiple antennas to improve the systems’ performance and relia-

bility over conventional single antenna systems. As an example, antenna array can

be used as a measure to mitigate fading. With multiple antennas at the receiver,

and assuming that there is sufficient separation between the antennas, the perfor-

mance of the receiver countering the fading effect (i.e., the system reliability) can

be substantially improved by combining independent copies of the transmitted signal

obtained from multiple antennas. This is a spatial diversity technique. On the other

hand, the array signal processing technique of beamforming works on received signals

that are correlated with each other. It maximizes the Signal-to-Noise Ratio (SNR)

of the intended signal while achieving interference-nulling (cancelation) of unwanted

signals. In the next section we discuss several topics within the research area of smart

antenna.

1.4 The Parameter Estimation Problem in Antenna

Array Signal Processing

When using an antenna array at the receiver side, several key parameters of

the input signals must first be estimated before subsequent processing of the signals

and symbol detection can be performed. Of these parameters of interest, the number

of signals and the directions-of-arrival (DoAs) are two of the most important ones.

This section provides an overview of these two parameter estimation problems. The

major focus is on Uniform Linear Arrays (ULAs), the antenna elements of which lie

along one dimension in space and are uniformly spaced.

5

1.4.1 The Problem of Number of Signals Detection

In general, detection of the number of signals (for this parameter, “detection”

is more often used instead of “estimation” due to some historical reasons, although

both terms can be used) is done via computing various statistics using the eigenvalues

of the sample covariance matrix of the antenna array. The traditional approach to the

number of signals estimation problem proposed and discussed by Wax and Kailath [8]

is based on information theoretical criteria for model order selection, namely the MDL

(Minimum Description Length) principle [6] and the Akaike Information Criterion

(AIC) [7]. In this method, the eigenvalues of the sample covariance matrix of the

antenna array are obtained and sorted, and then the multiplicity of the smallest

eigenvalues is identified. Within the algorithm a penalty term is introduced which is

formulated according to the AIC or MDL criteria. The resultant test is a function of

the number of signals k (the hypothesis), the minimization of which is the estimate

of the number of input signals.

It has been shown that (e.g., [9]) the estimator obtained via the AIC criterion

is not a consistent estimator2 and that it tends to overestimate the true value of

the parameter asymptotically with increasing number of samples, while the estima-

tor obtained via the MDL principle provides a consistent estimate. Furthermore, it

can be shown that the detection that is based on the MDL criterion has moderate

performance when the SNR of the system is low.

The MDL estimator proposed in [8] is an unstructured estimator, i.e., it does

not take into consideration that the system is a ULA and the signal model is parame-

terized by the directions-of-arrival. In [10] the author described an estimator that has

the DoAs parameterized and at the same time utilizes the MDL criterion. However,

the algorithm is based upon the maximum likelihood estimates of the DoAs meaning

it needs to perform a multi-dimensional search which is computationally demanding.

2An estimator is said to be consistent if the estimate converges in probability to the true valuewhen the number of samples increases to infinity.

6

In [11], the authors proposed the use of order statistics3 to improve the per-

formance of the MDL detection criterion for a finite number of snapshots. The paper

pointed out the fact that when only a finite number of snapshots are available at the

receiver side, the one-to-one relationship between the sorted eigenvalue estimates and

the actual eigenvalues (sorted also in the same order) is not valid with high proba-

bility. As a solution to this problem the asymptotic distributions (when number of

samples increases to infinity) of the eigenvalue estimates are utilized to compute the

order statistics. The computed order statistics are then used to obtain a different

version of the MDL criterion called OSMDL (Order Statistics MDL).

A different solution to the problem is proposed in [12], where the authors

consider the upper thresholds for the estimates of the eigenvalues, which, together

with their counterparts, the lower thresholds, define regions such that the probabil-

ity of the eigenvalues falling outside of them are equal to some predefined values.

These thresholds are adaptively predicted using the asymptotic distributions of the

estimated eigenvalues and are used to perform hypothesis testing for the number of

input signals. It can be shown that the performance of that method is superior to

that of MDL in the low SNR regime. Yet another source enumeration solution is

described in [13], based on testing the equality of all pairwise eigenvalues. The tech-

nique utilizes the bootstrap method to remove the dependence of the algorithm on

the Gaussian assumptions of its input signals. This makes the algorithm more robust

to deviations of the underlying model from the Gaussian assumptions, but it also

implies higher computational complexity.

1.4.2 Estimation of Directions-of-Arrival

In array signal processing, the estimate of the directions-of-arrival of the wave-

fronts impinging on a ULA was a research topic of considerable interest as the knowl-

3Consider n random variables X1, X2, . . . , Xn, that if rearranged in descending order producethe sequence X(1), X(2), . . . , X(k), . . . , X(n), the random variable X(k) is the kth order statistic of theoriginal n random variables.

7

edge of the DoAs is essential for subsequent signal processing and for such practical

applications as passive source localization. DoA is usually represented by the electri-

cal angle, which is defined with respect to the actual incidence angle. The electrical

angle φ of an impinging wave is φ = 2πdλ

sin θ, where d is the antenna spacing, λ is

the wavelength, and θ is the actual angle of incidence. When the input signals have

incidence angles (measured with respect to the straight line that is perpendicular to

the line of ULA) within the range of [−π2, π

2] and when the distance between array

elements is constrained to be less than one half of the wavelength of the input sig-

nal, there is a one-to-one correspondence between the electrical angle and the actual

incidence angle.

In this thesis, especially the first three parts of it, it is assumed that the

inputs are narrowband signals, the reason being that the ULA behaves differently to

the frequency components of a broadband signal. Firstly, if it is not a narrowband

signal, spatial aliasing could happen to the high-frequency components (while not

to the low-frequency components). Secondly, if the inputs are narrowband signals,

i.e., the bandwidth of individual signals is such that its product with the maximum

travel time of the wavefronts across the antenna array is much smaller than 1, the

difference between the signal’s arrival times at the different antenna elements would

have a negligible effect on the phase of the signal. Thus the only phase shifts that

are significant between the antenna elements are those caused by the incidence angles

of the input signals, which is desired as we can derive the incidence angles (i.e., the

DoAs) from the phase shifts.

There exist many direction-of-arrival (DoA) estimation methods within the

literature. Among them are Capon’s method [14], MUSIC [15], Min-Norm [22] [23],

stochastic maximum likelihood [17], [18], conditional maximum likelihood [19], WSF

[26], MODE [27], Root-MUSIC [16], ESPRIT [28] and IQML (Iterative Quadratic

Maximum Likelihood) [29] methods, etc.

Capon’s algorithm [14] is also known as the minimum variance distortionless

8

response (MVDR) spectral estimator. In the algorithm, the DoAs are estimated by

the locations of the peaks of the MVDR power spectrum that represents the output

power of the MVDR filter as a function of the steering direction. The filter is ob-

tained by minimizing its output power subject to the constraint that the response to

a given steering direction is constrained to be unity. In the MUltiple SIgnal Classi-

fication (MUSIC) algorithm, the vector of the received signals is viewed as a linear

combination of signal vectors embedded in additive Gaussian noise. In MUSIC, the

so-called noise subspace consists of the set of eigenvectors of the covariance matrix

of the array output that have the M −D smallest eigenvalue, where M is the num-

ber of antenna elements, D is the number of input signals and D < M . When the

estimated array covariance matrix is ideal, it can be shown that the noise subspace

is orthogonal to the steering vectors of the input signals. The DoA estimates can

be obtained as the directions that yield steering vectors orthogonal to the noise sub-

space. For noisy observations, the noise subspace of the estimated covariance matrix

of the array vector is first identified, and then the MUSIC spectrum is obtained by

plotting the magnitudes of the projections of the vectors upon this noise subspace as

a function of the steering direction. The DoA estimates are obtained by identifying

the points where the magnitude of the projection achieves local minima. A more

detailed description of the algorithm is given in subsection 2.2. In [20] and [21], the

authors discussed the connection between the MVDR and MUSIC algorithms. It was

shown that the MVDR algorithm utilizes the covariance matrix raised to the power

of −1 and decreasing this power coefficient to −n (n > 1) and in the limit that n

goes to infinity the MVDR estimator is transformed into the MUSIC estimator.

The Min-Norm algorithm [22] [23] belongs to the category of subspace-based

DoA estimators (same as the MUSIC algorithm) and more specifically to the category

of weighted MUSIC algorithms. It uses a single vector in the noise subspace, which

is a linear combination of the noise subspace vectors. The localization is done via

projection of steering vector of scanning electrical angle on this single vector. In

9

deriving the estimator, the goal is to find a single vector4 v in the noise subspace which

has v1, the first element of v, equal unity and the Euclidean norm minimized. This is

equivalent to require that the M −D zeros of the polynomial D(z) =∑M

i=1 viz−(i−1)

that do not belong to the DoAs of the input signals be inside the unit circle and that

these zeros be uniformly distributed in area where the other D zeros that correspond

to the input DoAs do not exist. The minimum norm argument and properties of the

algorithm are given by [23].

The DoA estimation problem can also be solved by using maximum likelihood

techniques. Two maximum likelihood estimators (MLEs) exist within the literature

that differ on the assumptions about the underlying data. The first one assumes

that the input data are deterministic but unknown, and they are formulated as nui-

sance parameters; the resultant estimator is called the conditional MLE (CMLE)

[19]. In the second treatment it is assumed that the input data are samples from

some probability distribution (e.g., from the Gaussian distribution) and the resultant

estimator is known as the stochastic MLE [17] [18]. In [25] the relationship between

the MUSIC algorithm and the CML (Conditional Maximum Likelihood) estimator

was established and elaborated on. It was shown that MUSIC achieves an estimation

accuracy equal to that of conditional maximum likelihood estimator for large sample

size as long as the source signals are uncorrelated.

Weighted Subspace Fitting (WSF) was developed in [26] and it was shown

that its performance approached the Cramer Rao Bound (CRB5) asymptotically with

increasing sample size. However, the computational requirements of WSF are high

as it relies upon multidimensional search. It is observed that another high-definition

DoA estimation algorithm, the Method Of Direction Estimation (MODE) [27], is

equivalent to WSF if the impinging signals are not coherent.

In Root-MUSIC, the requirement that the array manifold be orthogonal to

4Normally we use bold lower case letters to denote column vectors and bold upper case lettersto denote matrices.

5For any unbiased estimator of a deterministic parameter, the CRB is a lower bound to thevariance of these estimators.

10

the noise subspace is expressed via a polynomial representation and the DoAs are

estimated by first finding the roots of the polynomial and then selecting the D roots

(D again is the number of signals) that are closest to the unit circle. The error that

occurs in the radial part of the estimates would not contribute to the error of the

estimates of the DoAs as only the phases are important. In this sense root-MUSIC is

expected to perform better than MUSIC. However, MUSIC is more general in that it

can be used for a large variety of antenna shapes and geometries while the application

of root-MUSIC is limited to ULA.

Estimation of Signal Parameters via Rotational Invariance Techniques (ES-

PRIT) [28] is another important direction-of-arrival estimation algorithm. ESPRIT

essentially requires that a translational invariance [28] exists within the antenna array,

a condition that is satisfied by ULAs. In ESPRIT the rotational invariance [28] that

corresponds to the translational invariance is exploited. The main idea is as follows.

Mathematically, two overlapping subarrays of length M − 1 (the first array is formed

by antennas 1 to M − 1 and the second one by elements 2 to M) differ by a transla-

tional displacement, and the second subarray’s manifold can be written as a product

of the diagonal matrix of DoAs and the first subarray’s manifold. Correspondingly,

the partitions of the eigenvectors that correspond to the subarray manifolds are re-

lated by a linear transformation described by a transformation matrix. It can be

shown that this transformation matrix has the same eigenvalues (which are actually

the DoAs) as the translational matrix between the two subarray manifolds. This fact

can be exploited to obtain the DoA estimates. In ESPRIT, this linear transformation

is estimated using either least squares (LS) or total least squares (TLS) methods. An

improvement to ESPRIT is the unitary ESPRIT algorithm [24]. In Unitary ESPRIT

the computational load is reduced by real-valued computations which is obtained by

mapping the original centro-Hermitian matrices (a matrix A is centro-Hermitian if

JAJ = A, where J is a matrix which has only ones on the anti-diagonal while other

elements are zeros) of ESPRIT to real-valued matrices. The estimation accuracy is

11

improved by exploiting the unit magnitude property of the phasors representing the

DoAs. Further its performance is enhanced through forward/backward averaging of

the data, a processing inherent in the operation of the algorithm.

Finally, iterative quadratic maximum likelihood (IQML) [29] is a computa-

tional algorithm for the minimization of the deterministic maximum likelihood esti-

mator. While the maximum likelihood estimator provides an analytic answer to the

direction-finding problem, the IQML algorithm gives an efficient computational im-

plementation. It is based upon the polynomial parameterization of the deterministic

ML estimator, after which an iterative procedure can be established.

1.4.3 Joint DoA and Delay Estimation in Array CDMA Com-

munications

In recent years, multiple antennas were successfully applied to DS-CDMA com-

munications. For DS-CDMA communications employing antenna arrays, we similarly

need to obtain full knowledge of the channel parameters before the detection of the

transmitted symbols from each user can be performed. Of these parameters we are

particularly interested in the direction-of-arrival and the time delay of individual

CDMA users (or individual multipath components of each user) as well as the chan-

nel fading coefficients.

One solution to the joint DoA, delay and channel estimation problem was

proposed in [31] which is based upon the work described in [32] and a variant of

it (presented in [33]). This method requires the transmission of training sequences

and is valid for additive noise of unknown covariance matrix. It does not provide an

explicit formula for the estimation of the direction-of-arrival of individual multipath

component. Instead, it computes a composite channel impulse response utilizing the

DEML (DEcoupled Maximum Likelihood) [32], [33] estimator. It then directly uses

the obtained channel impulse response for detecting the symbols of a particular user.

The idea is similar to the notion of received effective signature waveform.

12

Two other solutions to the joint estimation problem are JADE-MUSIC (Joint

Angle and Delay Estimation) [34] and JADE-ESPRIT [35]. Both are blind estima-

tion procedures and belong to the category of subspace based estimation algorithms.

JADE-MUSIC extends the MUSIC technique to the joint angle and delay estimation

problem. In JADE-ESPRIT the composite channel matrix containing the unknown

DoAs, time-delays and channel coefficients needs to be estimated first. Then by tak-

ing the Fourier transform of the rows of the channel matrix estimate the ESPRIT

algorithm can be applied to the joint angle and delay estimation problem, and a

closed-form solution is therefore obtained.

Another maximum-likelihood-based estimation algorithm was proposed in [36],

where the received signals are transformed to the frequency domain and the distinct

time delays are converted to phasors. The algorithm requires the transmission of

a data sequence that is known a priori at the receiver side, i.e., it is a supervised

approach. Further, it does not directly estimate the DoAs of the individual multipath

components of a single source. Instead, the effective spatial signature vectors, which

are linear combinations of all coherent paths from one specific source, are estimated.

Further, the formulation of the system model exchanges the positions of the matrix of

transmitted data symbols and the matrix of spatial signature vectors, which makes the

resultant model analogous to a standard array DoA estimation model. ML estimators

can then be formulated. One advantage of this technique is that it has the ability to

estimate more unknowns than the number of antenna elements of the system.

In [37] the authors extend the code acquisition process in time domain to

two-dimensional angular and time domain, where the continuous angular domain

is divided into discrete bins and a search of the optimum inside these bins can be

performed.

In [38], TST-MUSIC (Time-Space-Time) was proposed as a solution to the joint

DoA and time-delay estimation problem. In the TST-MUSIC approach, the received

signals are arranged into two-dimensional data (the space and time dimension). The

13

TST-MUSIC algorithm has a tree structure and consists of two temporal T-MUSIC

algorithms and one spatial S-MUSIC algorithm. First the T-MUSIC algorithm in

the temporal dimension is performed to obtain initial estimates of the parameters

and then temporal filters are constructed using these estimates. S-MUSIC is then

performed on the filter outputs to obtain estimates of the spatial parameters and

the estimates are used to construct beamformers filtering the signals in the spatial

dimension. Note that in this stage the originally hard-to-differentiate temporal pa-

rameters, such as time delays that are close to each other, can be separated by their

spatial parameters. Finally, T-MUSIC is performed again to accurately recover the

time-delay parameters. The grouping of the time-delays and the incident angles can

be done automatically. In this algorithm the number of antenna elements can be less

than the total number of input signals.

1.5 Multiple-Input Multiple-Output Wireless Com-

munications

The second fundamentally different view point of multiple-antenna systems is

the concept of Multiple-Input Multiple-Output (MIMO). MIMO studies systems with

multiple antennas at both the transmitting and receiving sides. In MIMO systems

the received signals from different antenna elements are usually uncorrelated with

each other. This is in general not the case for an antenna-array system. Also the

geometry of the array of antennas is not as important for a MIMO system as it is for

an antenna-array systems. Another important difference between the techniques for

MIMO systems and those for antenna arrays is that a large part of research effort for

the former is put on the design of the transmission schemes.

In the late 90s, several seminal works laid out the foundations for the concept

of MIMO. In [39] (see also [40] and [41]) the channel capacity of the MIMO system

was discussed. It was shown that the MIMO structure, with rich scattering between

14

transmitting and receiving pairs, contains orthogonal spatial channels the number of

which is linear with respect to the smallest of the numbers of transmit and receive

antennas. When perfect channel knowledge is available at both the transmitter and

the receiver side, the MIMO structure enables new spatial dimension and the ad-

ditional degrees of freedom can be exploited to obtain a spectral efficiency that is

integer multiple that of a single input single output (SISO) system.

Significant research efforts have been devoted to developing MIMO-related the-

ories and many important research results emerged after the discovery of the potential

capacity gain. Various MIMO configurations and channel models were under investi-

gation and techniques and algorithms that can be used to exploit the system capacities

for these systems were proposed. Generally speaking, the MIMO system’s capacity is

affected by the variation of the channel coefficients and the cross-correlations between

the transmitting and receiving antennas. Recent research also shows that feedback

from the receiver, even if it is partial, can dramatically improve the data rate of

MIMO systems [42].

The technique of water filling can be used to allocate more power (within the

constraint of fixed total power) to favorable channels in order to achieve system capac-

ity. As noted before, the MIMO channel’s capacity can be affected by many factors.

For example, the “keyhole” phenomenon could possibly exist [100]. Mathematically,

the keyhole effect can be expressed as the cascading of one MISO and one SIMO

Raleigh channels, and in this case the rank of the overall MIMO channel matrix is re-

duced to only 1, meaning that only one SISO channel can be possibly used to transfer

data. The phenomenon could be observed when there is a vertical array at the base

station on top of a building, with the mobile beneath the building, and transmission

takes place via diffraction over the rooftop.

Transmit beamforming is a powerful technique for wireless transmissions in

MIMO channels [43], [44]. It was shown that when the channel matrix contains

certain structure due to the existence of distinct multipath propagations, transmit

15

beamforming can be used to minimize the error probability. Note that in [44] the

receiver is assumed to have only one antenna. The combined effect of transmit beam-

forming with space-time block codes was investigated in [43] and [45]. The transmit

beamforming techniques developed in [45] using the maximized averaged SNR crite-

rion can cause the diversity that is available by using a STBC to disappear with only

a single channel being favored; while this is not the case in [43].

Alternatively, transmit beamforming transmission can be derived using the

capacity criterion, as in [46], [47] and [44], where the optimization problem of max-

imizing the mutual information between the inputs and the outputs with transmit

beamforming is considered.

1.6 Objective of the Thesis and Summary of the

Contributions

1.6.1 Objective of the Thesis

To summarize the previous sections, the design of a multiple-antenna system

is an important engineering practice and multiple-antenna systems have attracted

enormous research interest ever since its inception. The adoption of multiple anten-

nas provides substantial capacity and performance improvement to current single-

antenna communication systems. The MIMO technology is envisioned to become

popular within the near future. These observations have suggested that the field of

multiple-antenna wireless communication deserves much more research effort, which

further motivates this PhD study into this area. In conducting this PhD research, the

objective and goal are to investigate present-day problems in multiple-antenna wire-

less communications and propose practical solutions and alternatives; and to expand

the theory and knowledge in this area through original work and fresh perspectives.

16

1.6.2 Summary of the Contributions

The contributions of the thesis are summarized below:

• We propose a subspace-based MUSIC-type joint DoA-delay estimation algo-

rithm which utilizes the spatial smoothing preprocessing technique. The pro-

posed technique addresses the problem of joint estimation of direction-of-arrival

(DoA), propagation delay, and complex channel fading coefficients of individ-

ual multipath components of a particular user for antenna-array DS/CDMA

communications over frequency selective multipath channels and provides an

efficient and attractive alternative.

• A new criterion for detecting the number of signals impinging on a uniform linear

array (ULA) is described. The criterion is unique in that it makes explicit use

of the peak information of the MUSIC spectrum. It exhibits a systematic way

of utilizing the eigenvector information of the array covariance matrix in the

number of signals detection problem.

• In the next we describe the iterative weight matrix approximation (IWMA) al-

gorithm, which is a novel smart antenna technology used to obtain an approxi-

mation to the optimum weight matrix for performing weighted spatial smooth-

ing (WSS) to obtain diagonal signal covariance matrix. We provide detailed

analysis of the algorithm’s behavior for the case when the estimate of the array

covariance matrix is ideal, which shows that the approximation matrix obtained

not only contains the DoAs information but also its structure is optimized such

that it is suited as a basis upon which subspace-based DoA estimation can be

performed. We illustrate this DoA estimation strategy by computer simula-

tions. The improvement in performance also suggests the effectiveness of the

IWMA algorithm.

• The last contribution is a study of two deterministic design measures for linear

processing space-time block codes (LP-STBC’s) which consist of sets of linear

17

processing matrices (LPM’s) and scalar signal constellations. The measures are

deterministic in the sense that their computations do not involve any statistical

operators and are defined solely with respect to the set of LPM’s. The first mea-

sure is obtained by applying Jensen’s Inequality to the linear dispersion code

mutual information [112] and it has a tractable relationship to the LDC mu-

tual information. The measure has the advantage of simplifying the LP-STBC

design as it separates the design of LPM’s from the statistical properties of the

channel. The second measure is a natural extension and generalization of the

conditions required for the amicable orthogonal design, which lays a theoretical

foundation for orthogonal space-time block codes. We studied its properties

and investigated lower bounds for it in the case of real design. We illustrate the

analogy between a derived lower bound to that of the total-squared-correlation

(TSC) for designing CDMA sequence set. We also illustrate the connection and

difference between the conditions for amicable orthogonal design and the LDC

mutual information via the first and the second measures we obtained.

1.7 Organization of the Thesis

In what follows we provide brief descriptions of the remaining chapters of the

thesis.

The thesis presents several research works in the field of multiple-antenna wire-

less communication. Before these works are discussed in the individual chapters, a

background chapter is included which provides necessary background information on

the main topics of the thesis.

The main contents of this thesis are arranged in four chapters. First, we con-

sider the problem of joint estimation of direction-of-arrival, propagation delay, and

complex channel fading coefficients for individual multipath components of a partic-

ular user in the context of antenna-array DS/CDMA communications over frequency

18

selective multipath channels. We propose a subspace-based MUSIC-type joint DoA-

delay estimation algorithm which utilizes the spatial smoothing preprocessing tech-

nique. The proposed algorithm essentially breaks the multipath induced coherency

within the received signals and recovers the full signal subspace spanned by all dom-

inant signal paths of all users. This allows for the use of MUSIC-type joint DoA

and time-delay estimators for the individual path components of the user of interest.

Based on the angle and timing information, we then estimate the multipath fad-

ing coefficients. We also consider two variants of the spatial-smoothing based joint

DoA-delay MUSIC technique, which are based upon chip-shifted estimates of the

space-time autocorrelation matrix. Another feature of the second type of algorithms

is that they utilize space-time received vectors that span only a single information

symbol period and exhibit superior performance when the data record size available

for parameter estimation is limited. This work is described in Chapter 3.

In Chapter 4 we describe a new criterion for detecting the number of signals

impinging on a uniform linear array (ULA). The criterion makes explicit use of the

peak information of the MUSIC spectrum. Specifically, we consider two maximum

likelihood estimates (MLEs) of the noise variance, that is, the MLE which is derived

from the unstructured eigenvalue decomposition (EVD) based parameterization and

the MLE that is obtained using structured DoA parameterization. Based upon a

large-sample formulation of the difference between these two MLEs, and by applying

the minimum description length (MDL) principle, we obtain the proposed criterion.

We prove that the proposed criterion provides a consistent estimate of the number

of signals and demonstrate that it has a better performance at low SNR for equal-

power sources when compared with the original MDL-based signal number detection

criterion.

In Chapter 5 we describe the iterative weight matrix approximation (IWMA)

algorithm. We consider the design of weighted spatial smoothing (WSS) which was

proposed as a technique to obtain a diagonal source covariance matrix for array sig-

19

nal processing. A diagonal source covariance matrix is a desired feature for subspace-

based direction-of-arrival (DoA) estimation algorithms as the cross-correlation among

the input signals can markedly impair the performance of these estimators. However,

the optimal weight matrix for such a purpose requires explicit knowledge of the DoAs.

In this thesis, we present an iterative weight matrix approximation (IWMA) algorithm

which is capable of obtaining an approximation to optimal weight matrix in an itera-

tive fashion. The algorithm is applicable when the input covariance matrix is positive

definite. The algorithm starts from a scaled identity matrix as an initial guess and

carries out a series of weighted spatial smoothing operations. After each WSS the

algorithm computes a new weight matrix, which is to be used for the next iteration.

The algorithm is based on the use of an effective correlation matrix6, which is nat-

urally brought about by the operations performed in each iteration and on the fact

that for a positive definite Hermitian matrix, the set of eigenvalues of its Hadamard

product with a correlation matrix is majorized by its own set of eigenvalues. While

WSS that is based on IWMA is an effective method to decorrelate highly correlated

signals, it is also interesting to note that with the IWMA algorithm the approximate

matrix generated can form the basis for subspace-type DoA estimation. Simulation

results illustrate the effectiveness of this estimation strategy which also suggests the

effectiveness of the IWMA algorithm.

The last topic of the thesis is the study of two deterministic design measures

for linear processing space-time block codes (LP-STBC’s) which consist of sets of

linear processing matrices (LPM’s) and scalar signal constellations. The measures

are deterministic in the sense that their computations do not involve any statistical

operators and are defined solely with respect to the set of LPM’s. The first measure

is obtained by applying Jensen’s Inequality to the mutual information criterion for

linear dispersion codes [112] denoted by CLD. The expectation operator is moved into

the log det() operator following Jensen’s rule. By assuming channel coefficients that

6A correlation matrix is defined as a square matrix with diagonal of all 1’s and off-diagonalelements less than or equal to 1 in magnitudes [134].

20

are independent and identically distributed (i.i.d.) Gaussian we compute the expec-

tations after which we obtain a deterministic design measure. We shall show that

there is a tractable relationship between this measure and CLD and will show that the

design of LP-STBC using this relationship can be simplified. The second measure is a

natural extension of the conditions required for complex linear processing orthogonal

design or amicable orthogonal design. For the LPM’s of a LP-STBC, we associate

with them two measures of non-orthogonality: total-squared-skew-symmetry and

total-squared-amicability (TSA). The relationship of total-squared-skew-symmetry

to total-squared-correlation (TSC) is revealed. TSC measures the non-orthogonality

(cross-correlation) properties of a vector set, and is commonly used in the design of se-

quence sets for Code Division Multiple Access (CDMA) systems. It can be shown that

total-squared-skew-symmetry is a generalization of total-squared-correlation (GTSC).

For GTSC a lower bound analogous to Welch’s lower bound for TSC exists, which

establishes itself upon the Hurwitz-Radon numbers and the Hurwitz-Radon family

of matrices. By computer simulations, we shall establish that the second measure

is less revealing than the first one for the performance of the final codes. However,

the lower bound derived can still be a good indicator of the performance of real de-

sign of size 3× 3. Comparing the two deterministic measures reveals to some extend

the difference and similarities between CLD and the criterion for amicable orthogonal

design.

Finally, the conclusions are given in the last chapter of the thesis together with

a discussion of the future work.

21

Chapter 2

Background

22

Figure 2.1: DS/CDMA transmitters and a DS/CDMA receiver with antenna array

The intent of this chapter is to provide necessary background information for

the major topics of this thesis, including brief descriptions of code division multiple

access systems, multipath fading, block fading, synchronization, channel estimation,

uniform linear array, MUSIC, spatial smoothing, multiple-input multiple-output sys-

tems and space-time block codes.

2.1 DS/CDMA Wireless Communications

In the first part of this PhD work (which is given in Chapter 3), our focus

is the design of signal processing techniques for DS/CDMA (Direct Sequence/Code

Division Multiple Access) wireless communications over frequency selective multipath

channel. The receiver is equipped with a uniform linear array (ULA), as shown on the

right side of Fig. 2.1. More details about the ULA structure will be given at a later

section of this chapter. Antenna array is a vital component of contemporary digital

wireless communication systems and the adoption of antenna array in a DS/CDMA

system can dramatically improve its performance. For example, a CDMA system’s

23

Figure 2.2: Direct-sequence spread spectrum transmission

performance is limited by the MAI (Multiuser Access Interference) within the system.

By utilizing antenna array at receiver side, the system’s resistance to MAI can be

enhanced through appropriate signal processing algorithms.

Fig. 2.1 shows a DS/CDMA wireless communications system. DS/CDMA is

the underlying physical layer transmission technology of several major third-generation

wireless standards like CDMA2000 and UMTS. DS/CDMA uses the direct-sequence

spread spectrum transmission technique, in which user’s data sequence is modulated

on to a code sequence, e.g., a PN (Pseudo Noise) code sequence. In DS/CDMA,

each user is assigned a different code sequence and at the same time all users share

a common communication media. Fig. 2.2 is a diagram showing the generation of a

DS/CDMA signal. The generated signal is given by:

s(t) =∑

i

N−1∑

l=0

b[i]d[l]PTc(t− iT − lTc), (2.1)

where b[i] ∈ {−1, +1} is the ith information symbol (bit) of a CDMA user; T

is the symbol duration; Tc is the chip duration; N = TTc

is the bandwidth ex-

pansion factor or system processing gain; d4= ( d[0] d[1] . . . d[N − 1] )T , with

d[l] ∈ {−1/√

N, +1/√

N}, is the unique code (signature) vector assigned to the user;

and PTc(t) is the chip pulse waveform.

24

The generated DS/CDMA signal goes through a frequency-selective multipath

Rayleigh fading channel (see Fig. 2.1), which is typically modeled as a tapped delay

line:

y(t) = s(t) ? h(t), (2.2)

where “?” denotes convolution of two functions; and

h(t) =L−1∑n=0

αnδ(t− τn). (2.3)

The quantities αn and τn denote the amplitude and delay of the nth tap respectively.

Fig. 2.3 illustrates a typical multipath fading channel with two path components that

can be modeled as a tapped delay line with two taps using (2.3). The amplitude αn

is a complex Gaussian random variable and its magnitude is Rayleigh distributed.

The delay τn is assumed to have a value that is an integer multiple of the chip period

Tc. The frequency-selective fading channel is static for some period such that at the

receiver meaningful estimates of signal and channel parameters can be obtained and

used for detection.

The mobile DS/CDMA users/devices are assumed to be stationary or moving

slowly. Stationary or slowly moving users/devices typically experience slow fading.

On the other hand, fast fading should be expected when there is a relative quick mo-

tion between the CDMA user and the ULA-receiver, which usually causes noticeable

amount of Doppler spread/shift.

At the receiver, the continuous-time received signal is chip-matched-filtered

and the output is sampled (at chip-rate) to produce discrete-time received sinal. We

collect N outputs (N is the processing gain of a DS/CDMA system, i.e., the length

of the code signature sequence) into a column vector:

y[i] =(y[iN ] y[iN + 1] . . . y[iN + N − 1]

)T

. (2.4)

25

Figure 2.3: Multipath fading channel

Suppose that there is one user transmitting signals. The vector y[i] can be written

as follows [48]:

y[i] = D

b[i− 1]

b[i]

+ n[i], (2.5)

where n[i] is a column vector that consists of the additive Gaussian noise. D4= [dldu]

consists of the two effective signature vectors for the user, where

dl =

0 d[N − 1] . . . d[0]

0 0 . . . d[1]

0 0. . .

......

...... d[N − 1]

0 0 0 0

·

h0

h1

...

hN−2

hN−1

, (2.6)

26

and

du =

d[0] 0 . . . 0

d[1] d[0] . . . 0...

.... . .

...

d[N − 2] d[N − 3]... 0

d[N − 1] d[N − 2] . . . d[0]

·

h0

h1

...

hN−2

hN−1

. (2.7)

We define

h4=

h0

h1

...

hN−2

hN−1

(2.8)

as a column vector that consists of the channel fading coefficients. Note that some

entries of h might be zeros. For example, suppose that there are two propagation

paths in the channel with delays 0 ≤ τ1 = 2 ≤ N − 1 and 0 ≤ τ2 = 4 ≤ N − 1,

respectively, then only the two elements h2 and h4 are non-zero and denote the two

complex channel coefficients α1 and α2, respectively.

The estimation of the channel vector h can be carried out by using subspace

based methods [48]. (A typical subspace based method is the MUSIC algorithm. See

Section 2.2 for a detailed description of the algorithm.) This is done via projections

of dl and du onto the noise subspace found from the sample covariance matrix of

y[i] and solving for h (note that h is included in both dl and du respectively) that

minimizes the Euclidean norm of the projections.

Denote the ideal covariance matrix of y[i] by Ryy. The noise subspace is the

span of the eigenvectors of the N − 2 smallest eigenvalues of Ryy. Here 2 (in N − 2)

is specific to our example of one user and two effective signature vectors dl and

du. From the theory for subspace based methods, the projections of dl and du on

the noise subspace will be zero. In reality, instead of the ideal Ryy, sample-average

27

estimation of the covariance matrix is used. Denote the sample covariance matrix

by Ryy (normally we use “(·)” to denote the estimate of a parameter, a vector, or a

matrix, etc.). The noise subspace is known from the eigenvectors that correspond to

the N − 2 smallest eigenvalues of Ryy. Suppose that the noise subspace is given by

Vn , we then have:

h = arg minh‖(dl)HVn‖2 + ‖(du)HVn‖2. (2.9)

After the estimate of h is obtained the synchronization problem is readily solved.

This is done as follows [48]:

• Given h, find the least-square fit to hi, denoted as {αi}, for i = 0, 2, . . . , N − 1;

• Select the maximum of {αi} and the corresponding i gives the delay in number

of chips;

• Subtract the found path from h and repeat the above process, until known

number of paths have been found.

2.2 Antenna Array - ULA

The usage of antenna arrays keeps growing, especially for the uplink of a mobile

wireless communication system. In this thesis, the first three topics focus on the

smart antenna technology and more specifically on ULA arrays. ULA requires that

the antenna elements lie along a straight line in three-dimensional space and the inter-

element spacing be fixed as shown in Fig. 2.4. The electrical angle φ of an impinging

wave onto a ULA is given by φ = 2πdλ

sin θ, where d is the antenna spacing, λ is the

wavelength, and θ is the actual angle of incidence.

Fig. 2.5 shows the system diagram that is used in the first three parts of

this thesis. In Fig. 2.5, y(t) = Ax(t) + n(t), where x(t) is a D × 1 column vector

consisting of the input signals. Matrix A has dimension M × D and is the array

28

Figure 2.4: Uniform Linear Array

Figure 2.5: ULA system model

29

manifold of the ULA. A is parameterized by the directions-of-arrival (DoAs) of the

input signals and more specifically the corresponding electrical angles, i.e., we have

A4= [a(φ1),a(φ2), . . . , a(φD)]. The ith column vector a(φi) is known as the steering

vector of the ith input signal. The M × 1 column vector n consists of spatially and

temporally uncorrelated zero-mean Gaussian random variables.

In Chapters 3, 4 and 5, MUSIC or MUSIC-type algorithms play an essential

role in the problems under consideration. In what follows we briefly describe the

MUSIC algorithm. We start from the properties of the array covariance matrix for

ULA, when the additive noise is spatially white. The exact covariance matrix of the

received signal vector y(t) is given by:

Ryy = ARxxAH + σ2I, (2.10)

where Rxx4= E{x(t)x(t)H} is the covariance matrix of the input signal vector x(t)

and σ2 is variance of the additive Gaussian noise. The eigenvalue decomposition of

Ryy is given by:

Ryy =M∑i=1

λivivHi = V ΛV H , (2.11)

where

Λ =

λ1 ∅

λ2

. . .

∅ λM

(2.12)

consists of the eigenvalues of Ryy (i.e., λ1, λ2, . . . , λM) that are sorted in descend-

ing order and V = [v1,v2, . . . , vM ] consists of the corresponding eigenvectors. The

eigenvalues satisfy the defining conditions

det(Ryy − λiI) = 0, i = 1, 2, . . . , M. (2.13)

30

Substitute ARxxAH + σ2I for Ryy in the above equation and we have:

det[ARxxAH − (λi − σ2)I] = 0, i = 1, 2, . . . , M. (2.14)

This says that the eigenvalues of ARxxAH are λi−σ2, i = 1, 2, . . . , M . Suppose that

M > D and A has full column rank. Suppose that Rxx has full rank and is positive

definite. We have that ARxxAH has rank D and the smallest M −D eigenvalues are

zeros, meaning that:

λi − σ2 = 0, i = D + 1, D + 2, . . . ,M, (2.15)

that is,

λi = σ2, i = D + 1, 2, . . . , M. (2.16)

Further, Ryy and its eigenvalues and eigenvectors satisfy the following defining con-

ditions:

(Ryy − λiI)vi = 0, i = 1, 2, . . . , M. (2.17)

For i = D + 1, D + 2, . . . , M , we then have:

ARxxAHvi = 0, i = D + 1, D + 2, . . . ,M, (2.18)

which implies that the array manifold and the steering vectors are orthogonal to the

set of eigenvectors that correspond to the M − D smallest eigenvalues σ2. The set

Vn4= [vD+1,vD+2, . . . , vM ] is known as the noise subspace.

The above property is utilized by the MUSIC algorithm to estimate the directions-

of-arrival of the input signals, as detailed in what follows. Eq. (2.18) holds true only

for ideal Ryy. When the array covariance matrix is estimated from noisy observation

data, MUSIC performs DoA search by identifying the peaks of the following spatial

31

Figure 2.6: Conventional spatial smoothing

spectrum:

P(φ)4=

1

aH(φ)VnV Hn a(φ)

, φ ∈ [−π, π] , (2.19)

where Vn = [vD+1, vD+2, . . . , vM ] consists of the M − D eigenvectors of Ryy, the

estimated array covariance matrix. In practice, Ryy can be obtained by sample-

averaging over N observed vectors as follows:

Ryy4=

1

N

N∑i=1

y(ti)y(ti)H . (2.20)

The MUSIC algorithm requires that the signal covariance matrix Rxx has full

rank. When there are coherent signals, the signal covariance matrix will be rank

deficient and the algorithm is no longer applicable. Spatial smoothing ([75] [76]) is

a technique designed to overcome this difficulty. It is a preprocessing technique ma-

32

nipulating the estimated array covariance matrix such that the corresponding signal

covariance matrix can have full rank.

In spatial smoothing, the M -element ULA is divided into Q overlapping sub-

arrays of P elements each. Variables Q and P satisfy the condition Q = M − P + 1.

The qth subarray, q = 1, 2, . . . , Q, is formed by the q, (q + 1), . . . , (q + P − 1)th

elements of the ULA. Let us denote by yq(t) the received signal and nq(t) the ad-

ditive noise over the qth subarray, and by Aq the submatrix of A formed by the

q, (q + 1), . . . , (q + P − 1)th rows of A. Then we have:

yq(t) = Aqx(t) + nq(t), q = 1, 2, . . . , Q. (2.21)

To perform spatial smoothing, we compute the weighted sum of the Q auto-correlation

matrices E{yq(t)y

Hq (t)

}, for q = 1, 2, . . . , Q (we use “ ~(·)” to denote vectors or ma-

trices after spatial smoothing preprocessing):

~Ryy4=

1

Q

Q∑q=1

E{yq(t)y

Hq (t)

}. (2.22)

The matrix ~Ryy of (2.22) can be rewritten as:

~Ryy = ~A ~Rxx~AH + σ2IP , (2.23)

where ~A4= [~a(φ1), ~a(φ2), . . . , ~a(φD)] is the subarray manifold with ~a(φk)

4= [1, e−jφk ,

. . . , e−j(P−1)φk ]T , k = 1, 2, . . . , D. The D×D matrix ~Rxx is the signal autocorrelation

matrix after spatial smoothing and is given by

~Rxx4=

1

Q

Q∑q=1

ΦqRxxΦ−q =

1

QRxx ◦ (BHB). (2.24)

In (2.24), Φ4= diag

([e−jφ1 , e−jφ2 , . . . , e−jφD ]T

), B

4= [b(φ1)b(φ2) . . . b(φD)] with b(φk)

4=

33

[1, ejφk , . . . , ej(Q−1)φk ]T for k = 1, 2, . . . , D, and “◦” denotes the matrix Hadamard

product. It can be shown that ~Rxx has full rank, as required by MUSIC. In Fig. 2.6

the spatial smoothing preprocessing is illustrated using a simple diagram.

In practice, Ryy is used as the basis to perform spatial smoothing:

~Ryy =

1

Q

Q∑q=1

FqRyyFTq , (2.25)

where Fq =[0P×(q−1) IP×P 0P×(M−P−q+1)

], q = 1, 2, . . . , Q are matrices used to

obtain the submatrices of Ryy.

After we obtain~Ryy, the MUSIC estimation is then performed as follows.

The noise subspace spanned by the eigenvectors associated with the P −D smallest

eigenvalues of~Ryy is first identified. If

~Vn is the matrix whose columns are formed

by the eigenvectors spanning the noise subspace, then we can estimate the DoAs

φk, k = 1, 2, . . . , D, from the locations of the D largest peaks of the spectrum P(φ)

which is defined as follows:

P(φ)4=

∥∥∥ ~V H

n · ~a(φ)∥∥∥−2

, φ ∈ [−π, π] , (2.26)

where ~a(φ)4= [1, e−jφ, . . . , e−j(P−1)φ]T . To obtain the locations of the D largest peaks

a search on the real line would need to be performed.

Having described the MUSIC algorithm and the spatial smoothing preprocess-

ing technique, in the next we briefly discuss the reception at ULA of transmitted

signal passing through frequency selective multipath fading channel h(t) (see Eq.

(2.2)). The received baseband signal at ULA is given by:

y(t) = s(t) ? h(t), (2.27)

34

Figure 2.7: Multiple-Input Multiple-Output system

where

h(t) =L−1∑n=0

a(φn)αnδ(t− τn), (2.28)

and a(φn) is the steering vector for the nth multipath from (electrical) angle φn.

In the next section we give a brief description of multiple-input multiple-output

systems and space-time block coding.

2.3 Multiple-Input Multiple-Output and Space-Time

Block Coding

The last part of the thesis presents research work on the MIMO wireless com-

munication technology. A typical MIMO system is shown in Fig. 2.7. The received

signal matrix is given by:

Y =

√ρ

MSH + N , (2.29)

where S is a T × M matrix consisting of the symbols to be transmitted. Here M

denotes the number of transmitting antennas and T is the number of symbol periods.

The elements of matrix S are such arranged that (i, j)th element of it denotes the

transmission of a symbol at the ith symbol period from the jth transmitting antenna.

35

The channel matrix H is of dimension M × N . The (i, j)th element of it denotes

the channel coefficient between the ith transmitting antenna and the jth receiving

antenna. Variable N is the number of receiving antennas. The received matrix Y

is of dimension T × N and consists of the received signals from N antennas over T

symbol periods. The normalization√

ρM

ensures a receiver side signal-to-noise ratio

per antenna equal to ρ. Rewrite (2.29) as follows:

Y T =

√ρ

MHT ST + NT , (2.30)

and suppose that the single value decomposition (SVD) of HT is given by:

HT = UΣV H , (2.31)

where “(·)T ” denotes matrix transposition and “(·)H” denotes the Hermitian trans-

pose of a matrix, respectively. We then have:

UHY T =

√ρ

M·Σ · V HST + UHNT . (2.32)

As Σ is a diagonal matrix, we can further rewrite (2.32) as follows:

[UHY T ]i =

√ρ

Mσi[V

HST ]i + [UHNT ]i, i = 1, 2, . . . , min{M, N}, (2.33)

where σi is the ith diagonal entry of Σ and “[·]i” denotes the ith row of a matrix. From

Eq. (2.33) we see that when there is a perfect knowledge of H at both the transmitter

and receiver, the MIMO channel is decomposed into min{M, N} independent scalar

channels, which implies the potential capacity gain that is available by using MIMO

instead of SISO.

STBC (Space-Time Block Coding) is an important class of coding techniques

for a MIMO system, which is also the focus of the last part of this work (Chapter 6).

In STBC, a block of input symbols is transformed into a code matrix C, the ith row

36

of which represents the antenna outputs at ith symbol period. An example of STBC

is the Alamouti code [49]. Suppose that symbols s1 and s2 are to be transmitted

via two transmitting antennas, Alamouti scheme constructs the following space-time

code word:

C =

s1 s2

−s∗2 s∗1

. (2.34)

The first row of C denotes that at time interval t1, s1 will be transmitted from the

first antenna and s2 will be transmitted from the second antenna. The second row of

C denotes that at time interval t2, −s∗2 will be transmitted from the first antenna and

s∗1 will be transmitted from the second antenna. The importance of such a coding

scheme lies in the simplicity of its decoding and the fact it explores the full transmit

diversity of the two transmitting antenna system.

Suppose that at receiver there is one receiving antenna, i.e., we have a 2 × 1

MIMO system. Suppose that H is given by

H =

h1

h2

. (2.35)

Let S of Eq. (2.29) be equal to C. We then have:

y1

y2

=

√ρ

M

s1 s2

−s∗2 s∗1

h1

h2

+

n1

n2

, (2.36)

which can be rewritten as:

y1

y∗2

=

√ρ

M

h1 h2

h∗2 −h∗1

s1

s2

+

n1

n2

. (2.37)

37

Note that

h1 h2

h∗2 −h∗1

is orthogonal (so is

s1 s2

−s∗2 s∗1

), i.e.,

h1 h2

h∗2 −h∗1

H h1 h2

h∗2 −h∗1

=

|h1|2 + |h2|2 0

0 |h1|2 + |h2|2

. (2.38)

Thus we can left multiply Eq. (2.37) by

h1 h2

h∗2 −h∗1

H

and this gives:

h∗1y1 + h2y

∗2

h∗2y1 − h1y∗2

=

√ρ

M

|h1|2 + |h2|2 0

0 |h1|2 + |h2|2

s1

s2

+

h∗1n1 + h2n2

h∗2n1 − h1n2

, (2.39)

which shows that the detections of s1 and s2 are decoupled (as manifested by the di-

agonal matrix

|h1|2 + |h2|2 0

0 |h1|2 + |h2|2

in Eq. (2.39)); and that the full transmit

diversity of “2I1O” channel is obtained (as manifested by the term |h1|2 + |h2|2 in Eq.

(2.39)), without altering the whiteness of the noise vector. Further, the Alamouti

scheme obtains full-rate transmission, i.e., it transmits two symbols over two symbol

periods.

Unfortunately, the coding scheme given in Eq. (2.34) can not be extended to

different number of transmission antennas other than two as explained in [104], in

which the Alamouti’s scheme is shown to be a special case of the orthogonal space-

time block codes (OSTBC) from the theory of orthogonal designs [113]. By using

the theory of orthogonal designs it can be proved that it is not possible to construct

similar codes to the Alamouti code for spatial dimensions other than two. Instead

of OSTBC, quasi-orthogonal space-time block codes can be constructed [131], where

38

the orthgonality is relaxed:

C =

s1 s2 s3 s4

s∗2 −s∗1 s∗4 −s∗3

s3 −s4 −s1 s2

s∗4 s∗3 −s∗2 −s∗1

. (2.40)

These codes are full-rate codes but they do not achieve the full transmit diversity

that is available.

In [112] the authors introduced linear dispersion codes (LDCs), together with

a criterion for designing the codes, which is to maximize the mutual information

obtainable by the constructed codes. The linear dispersion codes with the mutual

information design criterion, is an important class of space-time coding schemes for

wireless MIMO communications.

Suppose that r information symbols si, i = 1, 2, . . . , r are to be transmitted.

The linear dispersion code for {si} is given by [112]:

C =r∑

i=1

siAi + j

r∑i=1

siBi. (2.41)

Here j :=√−1 (we use “:=” and “

4=” interchangeably to denote the notion of “by

definition”), while si := Re {si} and si := Im{si} are the real and imaginary parts

of si, respectively. The matrices Ai and Bi ∈ RT×M are real-valued linear dispersion

matrices of size T×M . The choices of the dispersion matricesAi and Bi, i = 1, 2, . . . , r

decide the final form of the code, under the power constraint that E{tr(βC(βC)∗)} =

MT with a normalizing factor β. It can be seen from the definition given in Eq.

(2.41) that linear dispersion codes include orthogonal space-time block codes as their

special cases.

Linear dispersion codes are an important class of codes due to the fact that on

one hand the linear coding structure simplifies the code design process, on the other

39

hand a large part of the MIMO’s potential can be achieved via such a coding scheme.

Further, the decoding (not necessarily the optimal decoding) can be done linearly.

The design of a linear dispersion code centers around the design of the dispersion

matrices. The mutual information criterion described in [112] is used by the authors

to search for the dispersion matrices via numerical optimization. In what follows we

introduce the mutual information criterion.

As will be seen from the account given in Chapter 6, the linear dispersion

matrices {Ai} and {Bi}, i = 1, 2, . . . , r, transform the original MIMO channel H into

an equivalent MIMO channel represented by matrix H ∈ R2NT×2r. The equivalent

MIMO channel H is a function of the linear dispersion matrices Ai and Bi for i =

1, 2, . . . , r and the original H . The inputs to the equivalent MIMO channelH are now

the real and imaginary parts of si for i = 1, 2, . . . , r, while the outputs being the real

and imaginary parts of the (i, j)th entry of the received matrix Y , for i = 1, 2, . . . , T

and j = 1, 2, . . . , M . The linear dispersion code mutual information is defined as the

following quantity with respect to H:

CLD :=1

2TEH

{log det

(I2NT +

ρ

MHHT

)}

=1

2TEH

{log det

(I2r +

ρ

MHTH

)}. (2.42)

The selection of Ai and Bi (i = 1, 2, . . . , r) by maximizing the above quantity

was demonstrated to produce linear dispersion codes that have outstanding perfor-

mance in terms of probability of bit error. The main difficulty of this design process

is in that it requires a search for the dispersion matrices via numerical optimization

algorithms.

In the following we review the two important criteria for designing space-time

codes, i.e., the rank and determinant criteria [99] [102].

Suppose that C is transmitted and in the receiver E 6= C is detected. Under the

assumption of quasi-static Rayleigh fading, the Chernoff bound [50] of the pairwise

40

probability of error is given by:

P{C → E|H} ≤ det[IM + ρA(C, E)]−N , (2.43)

where

A(C, E) = B(C, E)B∗(C, E), (2.44)

and

B(C, E)4= (C − E)T . (2.45)

The rank criterion requires that the minimum rank of A(C, E) over all distinct code-

word pairs {C, E} to be at least r in order to achieve a diversity gain of rN . Suppose

that the non-zero eigenvalues of A(C, E) are λ1, λ2, . . . , λr. The determinant criterion

requires that the minimum of (∏r

i=1 λi)1/r over all distinct codeword pairs {C, E} be

maximized in order to maximize the coding gain.

In Chapter 6, the MIMO channel that is considered is mainly flat-fading

Rayleigh model. The fading is quasi-static with coherence time T channel uses during

which the fades are supposed to be constant, though they may change from one block

of time T to the other.

2.4 Notes on the Notations Used In The Thesis

In the thesis, we let either “(·)∗” or “(·)H” denote the Hermitian transpose of

a matrix and either “4=” or “:=” denote a new definition, i.e., “

4=” or “:=” means

“by definition”. We let either “conj(X)” or “X” denote the element-wise complex

conjugation of the matrix X.

The thesis is divided into four parts and a few notations have different mean-

ings/definitions in different parts. We identify here those notations. We note that

there is no ambiguity in understanding these notations within their contexts. M is

used to denote the number of receiving antennas in Chapters 3, 4 and 5 and is used

41

to denote the number of transmission antennas in Chapter 6. N is used to denote

DS/CDMA system gain in Chapter 3 and is used to denote the total number of sam-

ples in Chapters 4 and 5. N is used to denote the number of receiving antenna in

Chapter 6. T is used to denote symbol period in Chapter 3 and is used to denote the

number of channel uses in Chapter 6.

2.5 Systems, Models, Signals and Assumptions

Different parts of the thesis have different systems, models, signals and as-

sumptions, which are listed in Table 2.1.

Table 2.1: Systems, models, and signals considered in the thesis

Chapter Model

3 Multiuser DS/CDMAUniform Linear Array

Mobile users are assumed to be stationary or slow-motionedNo Doppler effect

Frequency-selective multipathRayleigh fading channel

Parameters φk,n, κk,n and αk,n

No angular spreadStatic fading

4 Uniform Linear ArrayD narrowband sources

No multipath propagationStatic fading or no fading

5 Uniform Linear ArrayD narrowband sources

No multipath propagationStatic fading or no fading

6 MIMO systemFlat Rayleigh fading

Block fadingQuasi-static fading

42

Chapter 3

Spatial-Smoothing Based

MUSIC-Type Joint DoA and

Time-Delay Estimation

43

3.1 Introduction

In this chapter, we consider the problem of joint estimation of direction-

of-arrival (DoA), propagation delay, and complex channel gain for antenna-array

DS/CDMA communications over frequency selective multipath channels. Accurate

timing and direction-of-arrival estimation is a prerequisite for successful demodulation

of the received data stream. Subspace-based joint direction-of-arrival (DoA) and de-

lay estimation algorithms for antenna-array DS/CDMA systems have been considered

in [51] and [52]. However, extension of these algorithms to realistic multipath fading

channels is difficult due to the degeneration of the signal subspace caused by multipath

propagation. While the approach given in [53], based on the DEcoupled Maximum

Likelihood (DEML) [54], provides a possible solution, it is not a blind scheme and

requires the transmission of a training symbol sequence. The approach reported in

[55] resorts to separate spatial- and temporal- processing of the 2-D space-time data.

Maximum likelihood type algorithms based on e.g., Deterministic Maximum Likeli-

hood [56] or Weighted Subspace Fitting [57] perform multidimensional searches and

are computationally intensive.

To overcome this difficulty, in this work, we propose a blind MUSIC-type

estimation algorithm that utilizes the spatial smoothing preprocessing technique [58].

By properly transforming the received covariance matrix, the algorithm essentially

breaks the coherency within the received signals and recovers the full signal subspace

spanned by all dominant multipaths of all users while keeping the additive noise white

and Gaussian. MUSIC-type joint DoA and delay estimation can then be carried out

for individual paths. Based on obtained angle and timing estimates, we proceed to

evaluate the multipath fading coefficients. The algorithm requires only N parallel

exhaustive searches in the real line to find out the DoA and time delay, where N is

the system processing gain, and exhibits reduced computational complexity compared

to maximum likelihood based algorithms that rely on multidimensional searches in

space of at least 2KL dimensions where K is the number of users and L is the number

44

of multipaths (assumed to be the same for all users).

Our treatment takes into account large multipath delays (greater than a symbol

period). As we explain later, there is an intrinsic ambiguity associated with the

channel synchronization for this case. We show how this ambiguity can be resolved

during channel coefficient estimation.

Two variants of the proposed spatial-smoothing based MUSIC-type joint DoA-

delay estimation scheme are further studied, which utilize ST received vectors that

span a single information symbol period i.e., they are of length NP and their co-

variance matrix has dimensions NP ×NP (P is the length of the antenna subarray

after spatial smoothing). This is motivated by the findings in [60], [61] where it was

shown, in the context of minimum-variance-distortionless-response (MVDR)-based

synchronization, that the dimension of the matrix can severely impact the accuracy

of sample-average covariance matrix estimates: the larger the dimension, the higher

the number of data vector samples that are required to achieve a given accuracy.

This chapter is organized as follows: in Section 3.2 we define the system model;

in Section 3.3 we describe the spatial-smoothing based joint DoA and delay estima-

tion algorithm. The ambiguity caused by long delay spread is dealt with in Section

3.4. In Section 3.5 we describe the two variants of the proposed estimation scheme.

We present an estimation algorithm that uses a sequence of chip-shifted covariance

matrices and a lower complexity variant that utilizes a sequence of chip-shifted ST

signature vectors. The simulation results are presented in Section 3.6.

3.2 System Model

We consider the uplink of a K-user asynchronous direct sequence code division

multiple access (DS/CDMA) system. The continuous time baseband transmitted

45

signal of the kth user, k = 0, 1, . . . , K − 1, can be expressed as follows:

sk(t)4=

+∞∑i=−∞

N−1∑

l=0

bk[i]√

Ekdk[l]PTc(t− iT − lTc). (3.1)

In (3.1), bk[i] ∈ {−1, +1} is the ith information symbol (bit) of the kth user; T is the

symbol duration; Ek is the transmitted energy of the kth user; N is the system pro-

cessing gain; dk4= ( dk[0] dk[1] . . . dk[N − 1] )T , with dk[l] ∈ {−1/

√N, +1/

√N},

is the unique code (signature) vector assigned to the kth user; Tc is the chip duration;

and PTc(t) is the chip pulse waveform.

The transmitted signal propagates through a multipath fading channel with L

paths (for simplicity in presentation we assume that the number of paths is the same

for all users). The receiver consists of a uniform linear antenna array (ULA) of M

antenna elements which are spaced half-the-wavelength apart. At the mth antenna

element, m = 0, 1, . . . , M − 1, the received baseband signal is given by

ym(t) =K−1∑

k=0

L−1∑n=0

αk,nsk(t− τk,n)e−jmφk,n + ηm(t), (3.2)

where φk,n is the electrical angle1 of the impinging signal corresponding to the nth

path of the kth user, ηm(t) is AWGN, and αk,n is the channel fading coefficient of

the nth path of the kth user, modeled as a complex deterministic unknown. The

delay variable τk,n denotes comprehensively the transmission delay and the multipath

propagation delay of the nth path of the kth user. We assume that τk,n = κk,nTc for

some κk,n ∈ {0, 1, . . . , 2N − 1} i.e., we consider large delay spreads that can be up to

(2N − 1)Tc in value.

After chip matched filtering and chip-rate sampling we obtain M discrete time

1The electrical angle φ of an imping wave is φ = 2πdλ sin θ, where d is the antenna spacing, λ is

the wavelength, and θ is the actual angle of incidence.

46

received signals ym[l], m = 0, 1, . . . , M − 1:

ym[l] =K−1∑

k=0

L−1∑n=0

√Ekαk,nbk[b l − κk,n

Nc] · dk[b(l − κk,n)%Nc] · e−jmφk,n + ηm[l], (3.3)

where “%” denotes the modulo operator, “b·c” denotes the floor operator, and ηm[l]

is zero-mean spatially and temporally uncorrelated Gaussian random variable with

variance σ2.

The problem we consider here is the estimation of the direction of arrival φ0,n,

time delay τ0,n (equivalently κ0,n) and complex channel gain α0,n of the nth multipath

component, n = 0, 1, . . . , L−1, of the user of interest e.g., user 0. The only quantities

assumed known are the system processing gain N , the signature vector d0, the number

of active users K and the number of paths L.

3.3 Spatial Smoothing Based Joint Direction of

Arrival and Delay Estimation

The M -antenna array is divided into Q overlapping subarrays of P elements

each, i.e. Q = M − P + 1. The qth antenna subarray, q = 0, 1, · · · , Q− 1, is formed

by the q, q + 1, · · · , (q + P − 1)th elements of the original array. Let yq[i] be the 2N -

long space-time received vector of the qth subarray over the ith information symbol

period, given by

yq[i]4= (yq[iN −N ], . . . , yq+P−1[iN −N ], . . . ,

yq[iN + N − 1], . . . , yq+P−1[iN + N − 1])T

=K−1∑

k=0

S(q)k Hk

bk[i− 2]

bk[i− 1]

bk[i]

bk[i + 1]

+ n, (3.4)

47

where Hk is defined as

Hk4=

hk ∅∅∅

hk

hk

∅∅∅ hk

, k = 0, 1, . . . , K − 1, (3.5)

in which hk4=√

Ek · [αk,0, . . . , αk,L−1]T is the complex channel gain vector for the kth

user that incorporates the transmitted energy Ek; n4= (ηq[iN −N ], . . . , ηq+P−1[iN −

N ], . . . , ηq[iN + N − 1], . . . , ηq+P−1[iN + N − 1])T is a 0-mean Gaussian noise vector

with covariance matrix σ2I; and S(q)k is given by

S(q)k

4= Sk

Ψk ∅∅∅

Ψk

Ψk

∅∅∅ Ψk

q

= SkJqk , q = 0, 1, . . . , Q− 1. (3.6)

We note that one observation period of 2N chips would span four transmitted bits,

therefore yq[i] in (3.4) will contain up to four bits bk[i−2], bk[i−1], bk[i] and bk[i+1].

In (3.6), Ψk is defined as

Ψk4= diag

([e−jφk,0 , e−jφk,1 , . . . , e−jφk,L−1 ]T

), (3.7)

while Sk is the space-time signature matrix of the kth user, k = 0, 1, . . . , K − 1,

defined as

Sk4=

Sl

k Smk Su

k 0

0 Slk Sm

k Slk

. (3.8)

The matrices Suk , Sm

k and Slk in (3.8), each of dimension NP × L, are the upper,

48

middle and lower part, respectively, of the following matrix of the kth user i.e.,

Suk

Smk

Slk

= (sk,0 sk,1 . . . sk,L−1), (3.9)

where sk,n is the space-time signature vector of the nth path of the kth user, given

by

sk,n = ( [0 . . . 0]︸ ︷︷ ︸κk,n

dk[0] . . . dk[N − 1]︸ ︷︷ ︸N

[0 . . . 0]︸ ︷︷ ︸2N−κk,n

)T ⊗ ak,n. (3.10)

In (3.10), ak,n = [1, e−jφk,n , . . . , e−j(P−1)φk,n ]T is the steering vector associated with

the nth path of the kth user, and ⊗ denotes the Kronecker product.

The proposed method is based on the eigenvalue decomposition (EVD) of the

spatially-smoothed received vector autocorrelation matrix

Ryy4=

1

Q

Q−1∑q=0

E{yq[i]y

Hq [i]

}

=1

Q

K−1∑

k=0

Sk

[Q−1∑q=0

J qkHk(J

qkHk)

H

]SH

k + σ2I. (3.11)

Theorem 1 deals with the rank of the following matrices

Q−1∑q=0

J qkHk(J

qkHk)

H , k = 0, 1, . . . , K − 1 (3.12)

and shows that spatial smoothing preprocessing when applied to the space-time co-

variance matrix can restore the full signal subspace spanned by all signal paths from

all users.

Theorem 1 The matrices∑Q−1

q=0 J qkHk(J

qkHk)

H , k = 0, 1, . . . , K−1, are of full rank

if and only if Q ≥ L and the DoAs of the multipath signals of the kth user are distinct.

49

Proof: Let us define the vectors ϕk and matrices Fk for k = 0, 1, . . . , K − 1, by

ϕk4= [e−jφk,0 , e−jφk,1 , . . . , e−jφk,L−1 ]T (3.13)

and

Fk4=

ϕk ∅∅∅

ϕk

ϕk

∅∅∅ ϕk

, (3.14)

respectively. Then

Ryy =1

Q

K−1∑

k=0

Sk

[(Q−1∑q=0

F •qk (F •q

k )H

)• (HkH

Hk )

]SH

k + σ2I, (3.15)

where “•” denotes the Hadamard (element-wise) product and power and

F •0k =

1 ∅∅∅

1

1

∅∅∅ 1

, (3.16)

where “1” denotes a L× 1 column vector of all ones. Furthermore we have

Q−1∑q=0

F •qk (F •q

k )H =

∑q ϕ•q

k (ϕ•qk )H ∅∅∅

∑q ϕ•q

k (ϕ•qk )H

∑q ϕ•q

k (ϕ•qk )H

∅∅∅∑

q ϕ•qk (ϕ•q

k )H

(3.17)

which is positive definite if and only if Q ≥ L and the directions-of-arrival of the

50

signal paths from user k are distinct. This conclusion is obtained by observing that∑Q−1

q=0 ϕ•qk (ϕ•q

k )H = BHB where B = (bk,0 bk,1 . . . bk,L−1) and bk,n4= [1, ejφk,n ,

. . . , ej(Q−1)φk,n ]T . Finally,

[Q−1∑q=0

ϕ•qk (ϕ•q

k )H

]• (hkh

Hk ) =

diag(hk)

[Q−1∑q=0

ϕ•qk (ϕ•q

k )H

]diag(hH

k ) (3.18)

which implies that∑Q−1

q=0 J qkHk(J

qkHk)

H , k = 0, 1, . . . , K − 1, are of full rank.

Applying the EVD on Ryy we obtain

Ryy = V ΛV H , (3.19)

where the matrix V consists of the eigenvectors of Ryy and Λ is the (diagonal) eigen-

value matrix. It is well known that the noise subspace is spanned by the eigenvec-

tors associated with the 2NP − 3KL smallest eigenvalues. From the above theorem

and (3.19) we conclude that, if Vn is the matrix whose columns are formed by the

eigenvectors spanning the noise subspace then the space-time signature matrices are

orthogonal to Vn i.e.,

V Hn Sk = 0, k = 0, 1, . . . , K − 1. (3.20)

Therefore, we obtain the estimates φ0,n and κ0,n of the direction-of-arrival φ0,n and

the delay κ0,n, n = 0, 1, . . . , L − 1, of all paths of user 0 from the locations of the L

largest peaks of the spectrum P(φ, κ) defined as

P(φ, κ)4=

∥∥V Hn · s(φ, κ)

∥∥−2, φ ∈ [−π, π] , k = 0, 1, . . . , K − 1, (3.21)

51

where

s(φ, κ)4= ([0 . . . 0]︸ ︷︷ ︸

κ

d0[0] . . . d0[N − 1]︸ ︷︷ ︸N

[0 . . . 0]︸ ︷︷ ︸N−κ

)T ⊗

([1, e−jφ, . . . , e−j(P−1)φ]T ). (3.22)

To obtain the locations of the L largest peaks N parallel searches on the real line

need to be performed.

In practice, Ryy is not known. The most commonly used method for estimating

it is through sample-average over H received vectors. The resulting estimate is given

by

Ryy =1

QH

Q−1∑q=0

H−1∑i=0

yq[i]yq[i]H . (3.23)

3.4 Channel Estimation And Removal of Timing

Ambiguity

For large delay spreads (i.e. NTc ≤ τ0,n ≤ (2N −1)Tc) for some path n, timing

estimation is subject to the following ambiguity: the estimated κ0,n will be within the

range [0, N − 1], while the correct estimate would be either κ0,n or κ0,n + N . In the

following, we show how this ambiguity can be resolved during the channel estimation

stage.

Suppose that the angle and timing estimates (see section 3.3) are given by

φ0,n and κ0,n for the nth path of user 0. Define y[i]4= (y0[iN − N ], . . . , yM−1[iN −

N ], . . . , y0[iN + N − 1], . . . , yM−1[iN + N − 1])T and the autocorrelation matrix

R′yy

4= E

{y[i]yH [i]

}. (3.24)

52

Construct the following space-time signature matrices with φ0,n and κ0,n

Sk(κ0,0 + δ0,0, . . . , κ0,L−1 + δ0,L−1, φ0,0, . . . , φ0,L−1)

4=

Sl

k Smk Su

k 0

0 Slk Sm

k Slk

, (3.25)

where Suk , Sm

k and Slk are similarly defined as in (3.8), (3.9) and (3.10), with the size

of the steering vector being M .

In (3.25) δ0,n, n = 0, 1, . . . , L − 1, can take two values 0 or N corresponding

to the two possible values for the actual delay κ0,n or κ0,n + N . We find δ0,n, n =

0, 1, . . . , L − 1, by minimizing the smallest eigenvalue of G (see below) over all 2L

values of the vector δ04= [δ0,0, δ0,1, . . . , δ0,L−1]

T . The associated eigenvector is the

estimate of the channel coefficients h0. In other words, the channel estimate is

arg min‖h0‖=1,δ0

hH0 G (δ0,0, δ0,1, . . . , δ0,L−1) h0. (3.26)

The matrix G (δ0,0, δ0,1, . . . , δ0,L−1) in (3.26) is defined by

G4=

Sl

k

0

H

(R′yy)

−1

Sl

k

0

+

Sm

k

Slk

H

(R′yy)

−1

Sm

k

Slk

+

Su

k

Smk

H

(R′yy)

−1

Su

k

Smk

+

0

Slk

H

(R′yy)

−1

0

Slk

,

(3.27)

where Suk , Sl

k and Smk as functions of δ0,0, δ0,1, . . . , δ0,L−1 are given by (3.25). This

technique is based on the minimum-variance-distortionless response (MVDR) channel

estimation method. The reason behind this choice, as opposed to the use of MUSIC-

type methods, is the unavailability of the rank of the signal subspace of Ryy. In

contrast to subspace algorithm, MVDR-type algorithms do not require knowledge of

the rank of the signal subspace.

53

3.5 SS Based Joint DoA-Delay Estimation Using

Chip-Shifted Estimates of the Covariance Ma-

trix

One of the characteristics of the algorithm given in Section 3.3 is that it em-

ploys space-time (ST) received vectors that span a time interval equal to twice the

information symbol period i.e., their length is equal to 2NP , where P is the length

of the antenna subarray and N the CDMA processing gain. As a result, the algo-

rithm requires the estimation of a 2NP × 2NP received vector covariance matrix.

However, in the context of minimum-variance-distortionless-response (MVDR)-based

synchronization, it was recently shown ([60], [61]) that the dimension of the matrix

can severely impact the accuracy of sample-average covariance matrix estimates: the

larger the dimension the higher the number of data vector samples that are required

to achieve a given accuracy. Therefore, motivated by the findings in [60], [61] we

consider two variants of the proposed scheme for the joint estimation of the DoA and

the timing of a user in a DS/CDMA system. The proposed algorithms utilize ST

received vectors that span a single information symbol period i.e., they are of length

NP and their covariance matrix has dimensions NP ×NP . Simulation studies reveal

that this reduction results in improved performance in short data record situations.

The treatment given in the following subsections will not include the case of

large delay spread.

3.5.1 Joint DoA and Delay Estimation Using Chip-Shifted

Covariance Matrix Estimates

The M -antenna array is divided into Q overlapping subarrays of P elements

each i.e. Q = M − P + 1. The qth antenna subarray, q = 0, 1, . . . , Q − 1, is formed

by the q, (q + 1), . . . , (q + P − 1)th elements of the original array. Let y(κ)q [i] be the

54

space-time received vector of the qth subarray over the ith information symbol period,

that is observed with a delay of κ chips, κ = 0, 1, . . . , N −1. The reason for collecting

κ-chip shifted ST observations will become apparent later. The vector y(κ)q [i] is given

by

y(κ)q [i]

4= (yq[iN + κ], . . . , yq+P−1[iN + κ], . . . ,

yq[iN + N − 1 + κ], . . . , yq+P−1[iN + N − 1 + κ])T

=K−1∑

k=0

S(q,κ)k

hk ∅∅∅

∅∅∅ hk

bk[i− 1]

bk[i]

+ n(κ), (3.28)

where n(κ) 4= (ηq[iN +κ], . . . , ηq+P−1[iN +κ], . . . , ηq[iN +N −1+κ], . . . , ηq+P−1[iN +

N − 1 + κ])T is 0-mean Gaussian noise vector with covariance matrix σ2I; hk is the

channel gain vector; and S(q,κ)k is a NP × 2L matrix given by

S(q,κ)k

4= S

(κ)k

Ψk ∅∅∅

∅∅∅ Ψk

q

= S(κ)k Kq

k , q = 0, 1, . . . , Q− 1. (3.29)

In (3.29), Ψk is as defined in (3.7), while S(κ)k is the space-time signature matrix of

the kth user, k = 0, 1, . . . , K − 1, defined as

S(κ)k

4=

(l(κ)k,0 . . . l

(κ)k,L−1 u

(κ)k,0 . . . u

(κ)k,L−1

). (3.30)

The vectors u(κ)k,n and l

(κ)k,n in (3.30) are the upper and lower halves, respectively, of the

space-time signature vector s(κ)k,n for the nth path of the kth user that is given by

s(κ)k,n

4= ([0 . . . 0]︸ ︷︷ ︸

κk,n+κ′

dk[0] . . . dk[N − 1]︸ ︷︷ ︸N

[0 . . . 0]︸ ︷︷ ︸N−κk,n−κ′

)T ⊗ ak,n. (3.31)

55

R(κ)yy =

1

Q

(S

(κ)0 . . . S

(κ)K−1

)

∑Q−1q=0 Kq

0L0(Kq0L0)

H ∅∅∅. . .

∅∅∅∑Q−1

q=0 LqK−1KK−1(L

qK−1KK−1)

H

(S(κ)0 )H

...

(S(κ)K−1)

H

+σ2I

(3.32)

In (3.31), κ′ is defined as follows:

κ′ =

−κ if κ ≤ κk,n

N − κ if κ ≥ κk,n

. (3.33)

The two estimation algorithms that we will describe are based on the eigenvalue

decomposition (EVD) of the spatially-smoothed received autocorrelation matrix

R(κ)yy

4=

1

Q

Q−1∑q=0

E{y(κ)q [i]y(κ)

q [i]H}

=1

Q

K−1∑

k=0

S(κ)k [

Q−1∑q=0

KqkLk(K

qkLk)

H ](S(κ)k )H + σ2I, (3.34)

where Lk is defined as:

Lk4=

hk ∅∅∅

∅∅∅ hk

, k = 0, 1, . . . , K − 1. (3.35)

Alternatively, (3.34) can be written in matrix form as shown in (3.32) at the top of

this page.

The following theorem deals with the rank of the matrices

Q−1∑q=0

KqkLk(K

qkLk)

H , k = 0, 1, . . . , K − 1. (3.36)

Theorem 2 The matrices∑Q−1

q=0 KqkLk(K

qkLk)

H , k = 0, 1, . . . , K−1, are of full rank

56

if and only if Q ≥ L and the DoAs of the multipath signals of the kth user are distinct.

Proof: The proof is similar to that for Theorem 1 and is omitted here.

The first one of the two estimation algorithms is based on the use of a sequence

of chip-shifted estimates of the ST covariance matrix R(κ)yy , κ = 0, 1, . . . , N − 1. The

second one uses the non-shifted estimate R(0)yy .

Applying eigenvalue decomposition on R(κ)yy , κ = 0, 1, . . . , N − 1 we obtain

R(κ)yy = V (κ)Λ(κ)(V (κ))H , (3.37)

where the matrix V (κ) contains in its columns the eigenvectors of R(κ)yy and Λ(κ) is

the (diagonal) eigenvalue matrix. If V(κ)

n is the matrix whose columns are formed by

the eigenvectors spanning the noise subspace then the space-time signature vectors

are orthogonal to V(κ)

n i.e.

[V (κ)n ]Hu

(κ)k,n = 0 and [V (κ)

n ]Hl(κ)k,n = 0,

for k = 0, 1, . . . , K − 1, n = 0, 1, . . . , L− 1. (3.38)

In the special case when κ is equal to κk,n, the lower half (l(κ)k,n) of the space time signa-

ture vector becomes zero while the upper half u(κ)k,n becomes equal to (dk[0] . . . dk[N −

1])T ⊗ ak,n. This simplifies expression (3.38) to

(V (κ)n )Hu

(κ)k,n = 0, when κ = κk,n. (3.39)

Therefore, we may estimate the direction-of-arrival φ0,n and the delay κ0,n,

n = 0, 1, . . . , L−1, of all paths of user 0 as the locations of the peaks of the spectrum

P(φ, κ) defined as follows:

P(φ, κ)4= ‖(V (κ)

n )Hs(φ)‖−2, (3.40)

57

where s(φ) is the space-time signature vector defined as

s(φ)4= (d0[0] . . . d0[N − 1])T ⊗ [1, e−jφ, . . . , e−j(P−1)φ]T . (3.41)

In practice, R(κ)yy is not known and as described earlier is estimated as follows:

R(κ)yy =

1

QH

Q−1∑q=0

H−1∑i=0

y(κ)q [i]y(κ)

q [i]H . (3.42)

3.5.2 Joint DoA and Delay Estimation Using the Non-Shifted

Covariance Matrix Estimate

The aforementioned algorithm requires the EVD of N chip-shifted matrices

R(κ)yy , κ = 0, 1, . . . , N−1, a task that computationally can be complex. In the following

subsection we describe an algorithm of lower complexity that requires the EVD of a

single estimated covariance matrix.

Applying EVD on R(0)yy we obtain

R(0)yy = V (0)Λ(0)(V (0))H , (3.43)

where the matrix V (0) consists of the eigenvectors of R(0)yy and Λ(0) is the (diagonal)

eigenvalue matrix. Let V(0)

n be the matrix whose columns consist of the noise eigen-

vectors (that in turn are associated with the smallest eigenvalues). The space-time

signature vectors are orthogonal to V(0)

n , i.e.

[V (0)n ]Hu

(0)k,n = 0 and [V (0)

n ]Hl(0)k,n = 0,

for k = 0, 1, . . . , K − 1, n = 0, 1, . . . , L− 1. (3.44)

Therefore, we may estimate the direction-of-arrival φ0,n and the delay κ0,n,

n = 0, 1, . . . , L−1, of all paths of user 0 as the locations of the peaks of the spectrum

58

P(φ, κ) defined as follows:

P(φ, κ)4=

1

‖[V (0)n ]Hs(L)(φ, κ)‖2 + ‖[V (0)

n ]Hs(U)(φ, κ)‖2, (3.45)

where s(U)(φ, n) and s(L)(φ, n) are the upper and lower half, respectively, of the

space-time signature vector,

s′(φ, κ)4= [[0 . . . 0]︸ ︷︷ ︸

κ

d0[0] . . . d0[N − 1]︸ ︷︷ ︸N

[0 . . . 0]︸ ︷︷ ︸N−κ

]T

⊗ ([1, e−jφ, . . . , e−j(P−1)φ]T ). (3.46)

In practice, R(0)yy is not known and is estimated through sample-averaging over H

received vector samples (as in (3.42)).

3.6 Simulation Results

In the first simulation we consider a 4-user system with a processing gain

equal to 35. The number of paths for all users is 3. The receiver ULA consists of 9

sensors while the subarray size is 5. The first figure shows the spectrum P(φ, κ) for the

proposed algorithm (see eq. (3.21)). It is plotted against φ and, for simplicity, we show

six curves corresponding to the delays k = 0, 1, . . . , 5. The actual angles of arrival are

shown by the vertical lines (φ0,0 = 0.7250, φ0,1 = −2.6583 and φ0,2 = 1.6916) while the

actual delays are κ0,0 = 3, κ0,1 = 4 and κ0,2 = 5. We see that the three largest peaks

are located at the actual DoAs, and correspond to the correct delays. In Fig. 3.2

we plot the MSE (squared error between the estimates [φ0,0, κ0,0, . . . , φ0,L−1, κ0,L−1]T

and the true values [φ0,0, κ0,0, . . . , φ0,L−1, κ0,L−1]T averaged over 250 runs) for the

conventional and the proposed MUSIC-based algorithm as a function of the SNR of

user 0 with the other users’ SNRs fixed at 16, 10 and 15 dB respectively. In Fig.

3.3 we give the MSE performance in terms of number of receiving samples. In Fig.

59

−pi −pi/2 0 pi/2 pi−10

−8

−6

−4

−2

0

2

4

6

8

10

Electrical Angle

Spa

tial−

Sm

ooth

ing−

Bas

ed M

US

IC S

pect

rum

(dB

)

0,00,1 0,21,0 1,11,2 2,0 2,12,23,0 3,13,2

k0=0

k0=1

k0=2

k0=3

k0=4

k0=5

Figure 3.1: Spatial-smoothing-based MUSIC spectrum vs. DoA for six possible valuesof the delay of user 0

3.4 the MSE of the channel estimation with respect to the SNR of user 0 is given

assuming correct timing estimation, while in Fig. 3.5 we set κ0,1 = 39 and we plot

the probability of ambiguity resolution of the proposed approach against the SNR of

user 0. The superiority of the proposed spatial smoothing technique is evident.

In the second simulation we consider a 4-user system with a processing gain

N = 35. The users’ signal-to-noise ratios (SNRs) are 10, 13, 15 and 16dB, respectively.

The number of paths for each channel is L = 3 (assumed to be the same for all users).

The receiver is a ULA of M = 9 sensors while the subarray size is P = 5.

In Fig. 3.6 and 3.7 we plot the spectrum P(φ, κ) for the algorithms presented

in Section 3.5 as a function of the DoA φ. Each curve corresponds to a different

60

5 10 15 20 250

1

2

3

4

5

6

7

SNR of User 0 (dB)

MS

E o

f Joi

nt D

oA a

nd D

elay

Est

imat

or

Without Spatial SmoothingWith Spatial Smoothing

Figure 3.2: MSE of joint DoA and delay estimator vs. SNR of user 0

61

600 700 800 900 1000 1100 1200 13001.22

1.24

1.26

1.28

1.3

1.32

1.34

1.36

1.38

1.4

1.42x 10

−3

Number of Samples

MS

E o

f Joi

nt D

oA a

nd D

elay

Est

imat

or

Without Spatial SmoothingWith Spatial Smoothing

Figure 3.3: MSE of joint DoA and delay estimator vs. number of samples

62

5 10 15 20 250

0.01

0.02

0.03

0.04

0.05

0.06

SNR of User 0 (dB)

MS

E o

f Cha

nnel

Est

imat

or

Figure 3.4: MSE of channel estimator vs. SNR of user 0

63

5 10 15 20 250.75

0.8

0.85

0.9

0.95

1

SNR of User 0 (dB)

Pro

babi

lity

of A

mbi

guity

Res

olvi

ng

Figure 3.5: Probability of ambiguity resolution vs. SNR of user 0

64

value of κ (for clarity of presentation we assume the maximum possible path delay

is 5). The actual delays are κ0,0 = 3, κ0,1 = 4, κ0,2 = 5. Each of the vertical dotted

lines denotes the electrical angle of the nth path of the kth user and is marked by

the pair k, n. The corresponding values are φ0,0 = 0.7205 rad, φ0,1 = 1.6916 rad

and φ0,2 = −2.6583 rad, respectively. In Figs. 3.6 and 3.7 the data record size is

H = 570. We see that the chip shifted matrices based algorithm exhibits taller and

sharper peaks compared to its lower complexity counterpart.

In Fig. 3.8 we compare the algorithm given in Section 3.3 (that utilizes receive

vectors of length 2NP ) with that given in Section 3.5 (that utilizes receive vectors of

length 1NP ). The data record size is H = 570. We plot the probability of acquisition

for each path of user 0, against the SNR of user 0. This is the probability that the

estimated delay is equal to the actual delay (in number of chips). In Fig. 3.9 we plot

the total mean squared error (MSE) of the joint DoA and delay estimators defined

as |κk,n − κk,n|2 + |φk,n − φk,n|2. The MSE was evaluated over 40 Monte-Carlo runs.

The superiority of the proposed algorithms is apparent.

3.7 Conclusion

We described a blind MUSIC-type DoA and delay estimation algorithm that is

suitable for antenna-array-based DS/CDMA system in multipath environments. The

proposed algorithm is based on the spatial smoothing preprocessing technique and

requires only N one-dimensional parameter searches to estimate the timings (within

a described ambiguity factor) and DoAs. These estimates are then used for channel

gain estimation and timing ambiguity resolution. We further study two variants of

the proposed estimation schemes which are suited for short-data-record scenarios.

65

−pi −pi/2 0 pi/2 pi

−5

0

5

10

15

Electrical Angle

MU

SIC

Spe

ctru

m (

dB)

0,00,1 0,21,0 1,11,2 2,0 2,12,23,0 3,13,2

k0=0k0=1k0=2k0=3k0=4k0=5

Figure 3.6: Spectrum of the proposed spatial-smoothing-based MUSIC algorithm(using chip-shifted covariance matrices).

66

−pi −pi/2 0 pi/2 pi

−5

0

5

10

15

20

Electrical Angle

MU

SIC

Spe

ctru

m (

dB)

0,00,1 0,21,0 1,11,2 2,0 2,12,23,0 3,13,2

k0=0k0=1k0=2k0=3k0=4k0=5

Figure 3.7: Spectrum of the proposed spatial-smoothing-based MUSIC algorithm(using a non-time-shifted matrix).

67

−20 −15 −10 −5 0 5 100

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

SNR of User 0 (dB)

Pro

babi

lity

of A

cqui

sitio

n

MUSIC With Spatial Smoothing (1NP): path 0MUSIC With Spatial Smoothing (1NP): path 1MUSIC With Spatial Smoothing (1NP): path 2MUSIC With Spatial Smoothing (2NP): path 0MUSIC With Spatial Smoothing (2NP): path 1MUSIC With Spatial Smoothing (2NP): path 2

Figure 3.8: Probability of acquisition vs. SNR of user 0

68

−20 −15 −10 −5 0 5 1010

−3

10−2

10−1

100

101

102

103

SNR of User 0 (dB)

MS

E o

f Joi

nt D

oA a

nd D

elay

Est

imat

or

Conventional MUSIC Using Chip−Shifted RMUSIC With Spatial Smoothing Using Chip−Shifted RMUSIC With Spatial Smoothing (1NP)MUSIC With Spatial Smoothing (2NP)

Figure 3.9: MSE of joint DoA and delay estimator vs. SNR of user 0

69

Chapter 4

The MUSIC MDL Criterion

70

4.1 Introduction

In array signal processing, the classical approach to signal enumeration is the

method proposed by M. Wax and T. Kailath [62]. It is based on the information the-

oretical minimum description length (MDL) criterion and is simple and computation-

ally efficient. The most recent research results regarding the MDL criterion include

for example [63], where the authors present a detailed analysis of the MDL criterion

and of the situation of overmodeling, based on their findings regarding the conditions

for the MDL criterion in analytic form. In addition to the MDL detection strategy,

there exist many other techniques and solutions to the problem of source enumera-

tion, which exploit specific features and properties of a particular signal model. For

example, the authors in [64] consider closely spaced sources and identify the asymp-

totic covariance matrix and its eigenvectors under this assumption. In their method,

the estimated eigenvectors are tested against the derived asymptotic pattern. In [65]

the authors suggest a detection method which takes into account the measured array

manifold and uses the eigenvectors of the estimated covariance matrix. Finally in

[66], the author designs a subspace based detection method specifically for CDMA

communications. While these techniques constitute better alternatives that utilize

the eigenvectors of the sample covariance matrix (in contrast, the MDL criterion uti-

lizes only the eigenvalues), they are specific to their respective system models and

they do not provide further insight into how we can exploit the extra information

provided by the eigenvectors.

In this work, we approach the problem of number of signals detection from the

direction of joint order detection and direction-of-arrival estimation. More specifically,

for each possible value of the number of signals k, the proposed solution constructs

first the (M − k)-dimensional (M is the total number of array elements) testing

subspaces, for k = 1, 2, . . . , M−1, and generates the corresponding MUSIC spectra. It

then utilizes the DoA estimates and the corresponding values of the MUSIC spectrum,

together with the M − k smallest eigenvalues of the sample covariance matrix, to

71

compute an MDL-type metric. This metric uses not only the eigenvalues but also the

eigenvectors of the sample covariance matrix, and jointly exploits the model order

information contained within. This results in better performance in low SNRs. In

addition, this method is based on a generic ULA system and, provided that certain

minor assumptions (discussed in later sections) are satisfied, it can be applied to a

wide variety of signal structures.

The rest of the chapter is structured as follows: in Sections 4.2 and 4.3 we

define the array system and give a brief description of the MDL criterion, respectively;

in Section 4.4 we detail the model order detecting criterion and its properties of

consistency. The simulation results are presented in Section 4.5, while Section 4.6

concludes the chapter.

4.2 System Model

We consider D signals x1(t), x2(t), . . . , xD(t) each impinging with an electrical

angle φk ∈ [−π, π] on an M -element uniform linear array (ULA). Let y(t) be the

vector consisting of the M received signals from the ULA at time t. Sampling y(t)

at instants ti (i = 1, 2, . . . , N) we obtain the discrete-time received vector sequence

y(ti) given by

y(ti) = Ax(ti) + n(ti), i = 1, 2, . . . , N. (4.1)

In (4.1), x(ti)4= [x1(ti), x2(ti), . . . , xD(ti)]

T is the vector containing the D transmitted

signals and A4= [a(φ1), a(φ2), . . . , a(φD)] is the array manifold whose kth column

a(φk)4= [1, e−jφk , . . . , e−j(M−1)φk ]T , k = 1, 2, . . . , D, is the steering vector associated

with the input xk(t). In general, the angles φk, k = 1, 2, . . . , D, are distinct, which

implies that the matrix A is of full column rank. We assume that the transmitted

vectors x(ti), i = 1, 2, . . . , N , are random, identically distributed, with a covariance

matrix given by Rxx4= E{x(t)x(t)H}, where E{·} is the expectation operator. Fi-

nally, n(ti) is the additive Gaussian noise assumed to be zero-mean, spatially and

72

temporally white, with covariance matrix σ2IM , where IM is the M × M identity

matrix.

The problem we consider in this chapter is the estimation of the number of

signals D (1 ≤ D < M) and the DoAs φk, k = 1, 2, . . . , D, given N observed vector

samples y(ti), i = 1, 2, . . . , N .

4.3 Background

Given N observations that are generated from some distribution, the informa-

tion theoretical MDL [67] principle is a criterion for model selection among the family

of competing models (probability densities) {f(·|ξ), ξ ∈ Ξ}, parameterized with pa-

rameter vector ξ, that best describes the data. It seeks the minimum encoding length

of the N observations, which is formulated in [67] and is given by

− ln f(y(t1), . . . , y(tN)|ξ)

+ 12m ln N. (4.2)

In (4.2) the first term is the minus log-likelihood of the observations y(t1),y(t2), . . . , y(tN)

with ξ being the maximum likelihood estimate of the vector of parameter based on

the same observations. The second term is a penalty term that is given in terms

of the number of data N and the number of independent parameters m within the

vector ξ.

4.3.1 Optimal MDL-based signal enumeration criterion

The optimal signal enumeration criterion in the minimum description length

(MDL) sense can be obtained as follows. Let Ryy4= 1

N

∑Ni=1 y(ti)y

H(ti) be the sample

covariance matrix of the N observations. Assuming that there are k sources in the

73

system, the MLE of the DoA vector ψ(k) 4= [φ1, φ2, . . . , φk] is given by [70]

ψ(k) = arg maxψ(k)

{−N ln det[PA[ψ(k)]RyyPA[ψ(k)] +

1M−k

tr(P⊥

A[ψ(k)]Ryy

) ·P⊥A[ψ(k)]

]}.

(4.3)

Here PA[ψ(k)]4= A[ψ(k)][AH [ψ(k)]A[ψ(k)]]−1AH [ψ(k)] is the projection matrix onto

A[ψ(k)]. A[ψ(k)] is the array manifold with parameter vector ψ(k) i.e., A[ψ(k)]4=

[a(φ1), a(φ2), . . . , a(φk)]. Also, P⊥A[ψ(k)]

= I − PA[ψ(k)]. Removing the “arg max{·}”operator and the minus sign, and adding a penalty term, we apply the MDL principle

to (4.3), which gives:

MDLopt(k)4= N ln det

[PA[ψ(k)]RyyPA[ψ(k)] +

1M−k

tr(P⊥

A[ψ(k)]Ryy

) · P⊥A[ψ(k)]

]

+ 12[k(k + 1) + 1] ln N. (4.4)

In (4.4) ψ(k) = [φ1, φ2, . . . , φk] consists of the k ML DoA estimates obtained from

(4.3). The counting of the freely adjusted parameters and the formulation of the

penalty term follow that of [71]. The number of signals is then given by arg mink

MDLopt(k). This estimate is optimal in the MDL sense, but it exhibits prohibitively

high computational complexity since for each k it performs a multi-dimensional search

for the evaluation of the MLE ψ(k) of the DoAs. For this reason, the most commonly

used signal enumeration method is the one proposed in [62] and is summarized next.

4.3.2 Suboptimal MDL-based criterion

The criterion described here is based on a generic parameterization by the

model’s eigensystem first discussed by T. W. Anderson [68]. Let λ1 ≥ λ2 ≥ . . . ≥ λM

be the eigenvalues of Ryy in descending order. For each k ∈ {0, 1, . . . , M − 1},calculate [62]

MDL(k)4= (M − k)N ln

[1

M−k

∑Mi=k+1 λi

(∏Mi=k+1 λi

) 1M−k

]+

1

2[k(2M − k) + 1] ln N. (4.5)

74

The number of signals is then given by arg mink MDL(k). As dictated by the MDL

principle [67] the first term is the minus log-likelihood function with parameters (the

eigenvalues and the eigenvectors) substituted by their ML estimators. The second

term of (4.5) is the penalty term which is directly proportional to the number of

freely adjusted parameters of the model, k(2M − k) + 1 [62].

In the next section we consider appropriate simplifications/approximations of

the RHS (right-hand side) of (4.4) by which a good tradeoff between accuracy and

computation complexity is achieved.

4.4 Detection Criterion Exploiting Peaks in the

MUSIC Spectrum

In this section we describe the proposed technique. Eq. (4.4), as a function of

ψ(k), is equivalent to [71]

L1(k, ψ(k))4= N ln

[det Rs[ψ

(k)] ·det(

1M−k

tr Rn[ψ(k)] ·I)]+ 1

2[k(k+1)+1] ln N (4.6)

where Rs[ψ(k)] is a k × k matrix and Rn[ψ(k)] is a (M − k) × (M − k) matrix such

that

U [ψ(k)]

Rs[ψ

(k)] ∅

∅ ∅

U [ψ(k)]H = PA[ψ(k)]RyyPA[ψ(k)], (4.7)

and

U [ψ(k)]

∅ ∅

∅ Rn[ψ(k)]

U [ψ(k)]H = P⊥

A[ψ(k)]RyyP⊥A[ψ(k)], (4.8)

respectively, with U [ψ(k)] being the unitary matrix suitable for this coordinate trans-

formation [71]. We note that

tr Rn[ψ(k)] = tr P⊥A[ψ(k)]RyyP

⊥A[ψ(k)] = tr P⊥

A[ψ(k)]Ryy, (4.9)

75

which is by itself related to the conditional1 MLE of ψ(k) [69]:

ψ(k) = arg maxψ(k)

[−MN ln(

1M

tr P⊥A[ψ(k)]

Ryy

)]. (4.10)

Regarding tr P⊥A[ψ(k)]

Ryy we have the following theorem. The proof is included

in the Appendix I at the end of this chapter.

Theorem 3 Let the eigenvalue decomposition (EVD) of Ryy be

Ryy = V (k)s Λ(k)

s [V (k)s ]H + V (M−k)

n Λ(M−k)n [V (M−k)

n ]H , (4.11)

where Λ(k)s denotes the k largest eigenvalues and V (k)

s consists of the corresponding

eigenvectors while Λ(M−k)n and V (M−k)

n denote respectively the M−k smallest eigenval-

ues and their eigenvectors. Then, under the assumption that [V (k)s ]HA is nonsingular,

we have

tr P⊥A[ψ(k)]

Ryy ≤ tr Λ(M−k)n + tr[A[ψ(k)]HV (M−k)

n [V (M−k)n ]HA[ψ(k)]][K], (4.12)

where ψ(k) 4= [φ1, φ2, . . . , φk] and K is defined by

K4= [[V (k)

s ]HA[ψ(k)]]−1[Λ(k)s − λMI][A[ψ(k)]HV (k)

s ]−1. (4.13)

For k = D, K is asymptotically equal to Rxx and we have by [72]

tr P⊥A[ψ(D)]

Ryy ' tr Λ(M−D)n + tr[A[ψ(D)]HV (M−D)

n [V (M−D)n ]HA[ψ(D)]][Rxx]. (4.14)

¤

We shall show that the approximation of tr P⊥A[ψ(k)]

Ryy by RHS of (4.12) does not

affect the detection of L1(k, φ1, φ2, . . . , φk) in the large sample limit. As we know an

1In the conditional model, the transmitted signal vectors x(ti) i = 1, 2, . . . , N are assumed to bedeterministic unknown.

76

MLE by itself can not detect the model order. The addition of the MDL penalty

term transforms the (monotonically decreasing) curve of MLE-versus-k so that the

resultant shape is convex with a minimum at k = D. The two observations given by

Theorem 3 suggest that the approximation of tr P⊥A[ψ(k)]

Ryy by the RHS of (4.12) can

be utilized: it approximately holds true when k = D, as manifested by Eq. (4.14); for

k 6= D, the approximation would uniformly contribute to bending the curve upward.

In the case of diagonal Rxx, (4.14) can be further simplified

tr P⊥A[ψ(D)]

Ryy ' tr Λ(M−D)n +

∑Di=1 r′i · a(φi)

HV (M−D)n [V (M−D)

n ]Ha(φi), (4.15)

where r′i denotes the ith diagonal entry in Rxx and a(φi)HV (M−D)

n [V (M−D)n ]Ha(φi) is

the inverse of P(φi), the MUSIC spectrum point. When k 6= D and supposing K is

diagonal with its ith diagonal element denoted by r′′i , the inequality equation (4.12)

may be similarly rewritten as follows

tr Λ(M−k)n + tr[A[ψ(k)]HV (M−k)

n [V (M−k)n ]HA[ψ(k)]][K] ≤

tr Λ(M−k)n +

∑ki=1 r′′i · a(φi)

HV (M−k)n [V (M−k)

n ]Ha(φi). (4.16)

Using K for both k = D and k 6= D, and denoting consistently K’s diagonal entry

by ri, we have the following substitute for tr P⊥A[ψ(k)]

Ryy:

tr Λ(M−k)n +

∑ki=1 ri · a(φi)

HV (M−k)n [V (M−k)

n ]Ha(φi). (4.17)

Substituting (4.17) into (4.6) we obtain the following cost function

L2(k, φ1, φ2, . . . , φk)4= (M − k)N ln

[1

M−k

∑Mi=k+1 λi + 1

M−k

∑ki=1 ri · a(φi)

HV (M−k)n

· [V (M−k)n ]Ha(φi)

]+ N ln

[det Rs[ψ

(k)]]+ 1

2[k(k + 1) + 1] ln N.

(4.18)

The first term within the RHS of (4.18) contains in its logarithm an approximation

77

to the MLE of the noise variance, which is given by [70]

σ2 = 1M−k

tr P⊥A[ψ(k)]

Ryy. (4.19)

This approximation consists of the summation of two terms: the average of the

M − k eigenvalues, 1M−k

∑Mi=k+1 λi, and a weighted sum of the k inverses of the

MUSIC spatial spectrum heights, 1M−k

∑ki=1 ri · a(φi)

HV (M−k)n [V (M−k)

n ]Ha(φi). The

term ln[det Rs[ψ

(k)]]

in (4.18) has the following interpretation. Observe that

tr Ryy = tr PA[ψ(k)]RyyPA[ψ(k)] + tr P⊥A[ψ(k)]

RyyP⊥A[ψ(k)]

= tr Rs[ψ(k)] + tr Rn[ψ(k)],

(4.20)

which is a constant for the N observations. As tr Rn[ψ(k)] on the RHS of Eq. (4.20)

is replaced by

tr Λ(M−k)n +

∑ki=1 ri · a(φi)

HV (M−k)n [V (M−k)

n ]Ha(φi), (4.21)

tr Rs[ψ(k)] is replaced by

tr Λ(k)s −∑k

i=1 ri · a(φi)HV (M−k)

n [V (M−k)n ]Ha(φi). (4.22)

This says that on one hand the noise estimate σ2 = 1M−k

tr Λ(M−k)n is increased by an

amount equal to

1M−k

∑ki=1 ri · a(φi)

HV (M−k)n [V (M−k)

n ]Ha(φi), (4.23)

while on the other hand the sum of the signal eigenvalues tr Λ(k)s is decreased by

the same amount. Thus det Rs[ψ(k)], which is the product of the non-zero eigen-

values of PA[ψ(k)]RyyPA[ψ(k)], is then given by det Λ(k)s − δ(φ1, φ2, . . . , φk), where

δ(φ1, φ2, . . . , φk) is the amount of modification obtained by the knowledge of (the max-

imum likelihood estimates of) the directions-of-arrival. Approximating ln[det Rs[ψ

(k)]]

78

by ln det Λ(k)s yields the following cost function

L3(k, φ1, φ2, . . . , φk)4= (M − k)N ln

[1

M−k

∑Mi=k+1 λi + 1

M−k

∑ki=1 ri · a(φi)

HV (M−k)n

· [V (M−k)n ]Ha(φi)

]+ N ln det Λ(k)

s +1

2[k(k + 1) + 1] ln N. (4.24)

Technically, it is feasible to use L3(k) as a source enumerator. However, there

are several difficulties involved. First, we need a way to associate K’s diagonal entries

with the MUSIC peaks2. Second, as it turns out in practice, for L3(k) to work properly

in the high signal-to-noise ratio regime, the number of data samples N should be large.

In what follows, we consider replacing the ri’s with a single small-valued coefficient

r.

The first observation regarding the coefficient r is that its value should not be

1N

, as in this case the criterion (4.24) would degenerate to

(M − k)N ln[

1M−k

∑Mi=k+1 λi

]+ N ln det Λ(k)

s + 12[k(k + 1) + 1] ln N, (4.25)

which is obviously not an efficient estimator when compared term by term to the

original MDL criterion (cf. (4.5)).

In [73], the asymptotic mean and variance of the MUSIC null spectrum 1P(φ)

is

given by (for φ around the true direction-of-arrival)

(M −D)[

σ2

N

∑Di=1

λi

(λi−σ2)2a(φ)Hviv

Hi a(φ)

]and

(M −D)[

σ2

N

∑Di=1

λi

(λi−σ2)2a(φ)Hviv

Hi a(φ)

]2, (4.26)

respectively, where vi is the ith eigenvector of Ryy that corresponds to λi. When the

signal-to-noise ratio is low, the quantities λi

(λi−σ2)2, i = 1, 2, . . . , D, (see (4.26)) increase

2One possible way of doing this is to sort both the diagonal entries and the MUSIC peaks by theirmagnitudes and then relate them, i.e., the largest diagonal entry will weight the highest MUSICpeak, the second largest one weights the second highest one, and so on. We do not consider thisapproach here.

79

in their values as the denominator is small. Suppose SNR is such that λi

(λi−σ2)2are

O(N), then the entries of the signal covariance matrix Rxx = [[V (D)s ]HA]−1

[Λ(D)

s −σ2I][AHV (D)

s ]−1 will be of magnitude O( 1√N

). This corresponds to a single r = 1√N

.

We note that in this case r 1P(φ)

still has a Gaussian asymptotic behavior with mean

and variance given by

(M −D) 1√N

[σ2

N

∑Di=1

λi

(λi−σ2)2a(φ)Hviv

Hi a(φ)

]and

(M −D) 1N

[σ2

N

∑Di=1

λi

(λi−σ2)2a(φ)Hviv

Hi a(φ)

]2, (4.27)

respectively. Since the EVD-parameterization-based MLE of the noise variance σ2 =

1M−D

tr Λ(M−D)n is asymptotically Gaussian with mean σ2 and variance σ4

N(M−D), it

should be expected that its correction term (due to the DoA-parameterization) is

on the scale of 1√N

or less. These observations suggest a choice of the coefficient

r = 1√N

. Actual simulation shows the validity of this choice for a reasonable range of

N . Setting r equal to 1√N

, we obtain

MDLMUSIC(k)4= (M − k)N ln

[1

M−k

∑Mi=k+1 λi + 1

M−k

∑ki=1

1√N· a(φi)

HV (M−k)n

· [V (M−k)n ]Ha(φi)

]+ N ln det Λ(k)

s +1

2[k(k + 1) + 1] ln N, (4.28)

which is our proposed signal enumeration criterion. Evaluation of the value of the k

can be performed efficiently using Algorithm 1.

In Algorithm 1, vi is the eigenvector of the ith eigenvalue λi, i = 1, 2, . . . ,M .

The number of signals is determined by the k that minimizes MDLMUSIC(k).

The consistency property of MDLMUSIC(k) is summarized in the following the-

orem.

Theorem 4 The estimator, as given by (4.28), is asymptotically consistent. ¤

The proof is given in the Appendix II.

80

Algorithm 1 MUSIC MDL Number of Signals Detection

for k = 1 to M − 1 doV (M−k)

n ← (vk+1 vk+2 . . . vM

)for φ = −π to π doP(φ) = 1

aH(φ)V(M−k)

n [V(M−k)

n ]Ha(φ)

end forif number of peaks ≥ k then

Obtain the k largest maxima: P(φi), i = 1, 2, . . . , kCompute MDLMUSIC(k) using λk+1, λk+2,. . ., λM and 1

P(φ1), 1

P(φ2), . . . , 1

P(φk)

end ifend forD = arg mink MDLMUSIC(k)

4.4.1 Observations and Remarks

The derivation of MDL(k) is based on a generic eigenvalue decomposition

(EVD) model first discussed by T. W. Anderson [68]. For a particular system, such

as that in (4.1), a more specific parameterization that yields a better detector is pos-

sible, but usually at a much higher computation cost. By adopting the large sample

approximation, we are able to reduce the computations required while maintaining

sufficient accuracy.

Compared to the original MDL criterion, (4.28) improves in such a way that

the MUSIC maxima are included within the computation of the noise variance (while

counting differently the number of independent parameters). This is easily seen by

inspecting the conditional MLE (Eq. (4.19)), its large sample approximation (Eq.

(4.15)), and the MLE that is discussed in [68]

σ2 = 1M−D

∑Mi=D+1 λi = 1

M−Dtr Λ(M−D)

n . (4.29)

We see that the estimator (4.19) is specific to array signal processing, and that its

calculation involves not only the M − D eigenvalues but also the estimates of the

DoAs, as manifested by the second term on the RHS of Eq. (4.15).

With the estimator in (4.28), we make explicit usage of the MUSIC spectra

81

−25 −24 −23 −22 −21 −20 −19 −18 −17 −16 −150

0.5

1

1.5MUSIC MDL Number of Signals Detecion

SNR (dB)

Pro

babi

lity

of D

etec

tion

MDL PrincipleMUSIC MDL (r=1/√N)

Figure 4.1: Number of signals detection: MDLMUSIC vs. MDL; 10-element ULA, 4equal-power sources; in terms of SNR (dB); number of samples is 1500; averaged over400 Monte-Carlo runs.

peaks. As mentioned before, several other source enumerators [64], [65], [66] are

distinct from the MDL method in that they exploit the eigenvectors of the received

autocorrelation matrix. Criterion (4.28) presents another example illustrating this

point.

Another feature of the MUSIC MDL estimator is that it eliminates spurious

hypotheses because of the fact that the number of MUSIC peaks within the testing

spectra is insufficient. This is an observation drawn from the consistency proof in

Appendix II.

82

1000 2000 3000 4000 5000 6000 7000 8000 9000 100000

0.5

1

1.5MUSIC MDL Number of Signals Detecion

Number of Samples

Pro

babi

lity

of D

etec

tion

MDL PrincipleMUSIC MDL (r=1/√N)

Figure 4.2: Number of signals detection: MDLMUSIC vs. MDL; 10-element ULA, 8equal-power sources; in terms of number of samples; averaged over 400 Monte-Carloruns.

83

4.5 Simulation and Performance Evaluation

We simulate a 10-element ULA and consider two sets of simulations. In the first

simulation there are 4 source signals with equal power. Their directions-of-arrival are

evenly distributed within the range [−π, π]. In Fig. 4.1 we plot out the probability of

correct detection versus the system’s signal-to-noise ratio, which is defined as tr Rxx

Dσ2 .

The probability of correct detection is calculated by averaging over 400 Monte-Carlo

runs. The number of input samples is N = 1500. The second simulation is an 8-

source scenario and we evaluate the probability of detection against the number of

input samples N . We fixed the input SNRs of all sources at −24 dB. The results are

shown in Fig. 4.2. In each of the above two plots we compare the new technique with

the original MDL. The performance improvement is apparent.

4.6 Conclusion

It is shown that we can utilize the MUSIC maxima information to improve

the detection of the number of signals based on the MDL principle. This information

is carefully incorporated and the computation burden is controlled. The resultant

estimator is computationally efficient while achieving perceptible performance im-

provement.

4.7 Appendix I - Proof of Theorem 3

In this section we provide proof for Theorem 3. We start from

tr PA[ψ(k)]Ryy = tr[[V (k)s ]HA(AHA)−1AHV (k)

s Λ(k)s ]

+ tr[[V (M−k)n ]HA(AHA)−1AHV (M−k)

n Λ(M−k)n ], (4.30)

84

where on the RHS we omit the dependence of A on ψ(k). With some manipulations,

we obtain

tr PA[ψ(k)]Ryy = tr Λ(k)s + tr[[V (k)

s ]HA(AHA)−1AHV (k)s − I]Λ(k)

s

− tr[[V (k)s ]HA(AHA)−1AHV (k)

s − I]Λ(M−k)n

≥ tr Λ(k)s + tr[[V (k)

s ]HA(AHA)−1AHV (k)s − I][Λ(k)

s − λMI], (4.31)

where the inequality is a result of Λ(M−k)n º λMI and “º” is the symbol of positive

semidefiniteness.

Assuming that [V (k)s ]HA is nonsingular, i.e., the k largest peaks correspond to

the signal subspace, we have [72]:

[V (k)s ]HA(AHA)−1AHV (k)

s − I =

[I + [AHV (k)

s ]−1AHV (M−k)n [V (M−k)

n ]HA[[V (k)s ]HA]−1

]−1 − I. (4.32)

Now by noting that for a positive semidefinite matrix Γ

(I + Γ)−1 − I º −Γ, (4.33)

we have

tr PA[ψ(k)]Ryy ≥ tr Λ(k)s − tr[AHV (M−k)

n [V (M−k)n ]HA][K], (4.34)

where K4= [[V (k)

s ]HA]−1[Λ(k)

s − λMI][AHV (k)s ]−1. for k = D, the proof is given by

[72].

4.8 Appendix II - Proof of Theorem 4

In this section, following the methodology given in [62], and by using the

consistency result of [74], we prove the consistency of the estimator (4.28) (Theorem

85

4). In the proof we will use parameter r with no substitution. First, we see that

MDLMUSIC(k)−MDLMUSIC(D) = ∆MLE(k) + ∆Penalty(k) ={

(M − k)N ln[

1M−k

∑Mi=k+1 λi + 1

M−k

∑ki=1 r · aH(φi)V

(M−k)n [V (M−k)

n ]Ha(φi)]−

(M −D)N ln[

1M−D

∑Mi=D+1 λi + 1

M−D

∑Di=1 r · aH(φi)V

(M−D)n [V (M−D)

n ]Ha(φi)]+

N ln det Λs(ψ(k))−N ln det Λs(ψ

(D))}

+{

12[k(k + 1) + 1] ln N − 1

2[D(D + 1) + 1] ln N

}. (4.35)

For the terms on the RHS of Eq. (4.35), let us denote the expression enclosed in the

first pair of curly brackets as ∆MLE(k) and that of the second pair as ∆Penalty(k). In the

following we consider two cases according to the values of k and D: k > D and k < D.

We shall show that in both cases MDLMUSIC(k) > MDLMUSIC(D) asymptotically.

For k > D, except for an additional correction term and a different count of

the degrees of freedom, the other parts of (4.35) are the same as the original MDL.

Thus we only need to show that the additional correction term obtained from testing

MUSIC spectra increases the probability that MDLMUSIC(k) > MDLMUSIC(D) (when

compared to the original MDL).

Partitioning V (M−D)n into two submatrices

V (M−D)n

4=

[V

(k−D)n V (M−k)

n

], (4.36)

we have

aH(φi)V(M−k)

n [V (M−k)n ]Ha(φi) ≤ aH(φi)V

(M−D)n [V (M−D)

n ]Ha(φi), (4.37)

which says that for k > D, there are at least D peaks (of the k peaks) that can be

derived from the MUSIC spectrum using V (M−D)n . Asymptotically these D peaks are

distinct and the corresponding a(φi), i = 1, 2, . . . , D span the true signal subspace.

86

For the remaining k − D peaks we have two possibilities. They are either

completely indistinguishable from the previous D peaks, or some of them are spurious

and the corresponding a(φi)’s fall within the true noise subspace, in which case we

would have some peaks whose null spectrum aH(φi)V(M−k)

n [V (M−k)n ]Ha(φi) are non-

decreasing. For this second case, the probability of being positive is increased.

For the first case (when the k −D peaks are not separable from the other D

peaks), we obtain by Taylor series expansion

(M − k)N ln[

1M−k

∑Mi=k+1 λi + 1

M−k

∑ki=1 r · aH(φi)V

(M−k)n [V (M−k)

n ]Ha(φi)]'

(M − k)N ln[

1M−k

∑Mi=k+1 λi

]+

(M−k)N ·r·O( 1N

)1

M−k

PMi=k+1 λi

, (4.38)

and

(M −D)N ln[

1M−D

∑Mi=D+1 λi + 1

M−D

∑Di=1 r · aH(φi)V

(M−k)n [V (M−k)

n ]Ha(φi)]'

(M −D)N ln[

1M−D

∑Mi=D+1 λi

]+

(M−D)N ·r·O( 1N

)1

M−D

PMi=D+1 λi

. (4.39)

Thus the effect of the correction terms is negligible. In reality this corresponds to a

situation where for a given k there are not enough MUSIC peaks.

Since the correction term improves the detecting probability, then by the con-

sistency of the original MDL for k > D, we have that MDLMUSIC(k) > MDLMUSIC(D)

is asymptotically positive with probability one.

For k < D, let us define

∆eigenvalues(k)4= 1

D−k

∑Di=k+1 λi + 1

D−k

∑ki=1 r · aH(φi)V

(M−k)n [V (M−k)

n ]Ha(φi)

− 1D−k

∑Di=1 r · aH(φi)V

(M−D)n [V (M−D)

n ]Ha(φi). (4.40)

And we have

[1

M−k

∑Mi=k+1 λi + 1

M−k

∑ki=1 r · aH(φi)V

(M−k)n [V (M−k)

n ]Ha(φi)]M−k

=

87

[M−DM−k

1M−D

∑Mi=D+1 λi + D−k

M−k·∆eigenvalues(k)+

M−DM−k

1M−D

∑Di=1 r · aH(φi)V

(M−D)n [V (M−D)

n ]Ha(φi)]M−k

. (4.41)

Using the generalized arithmetic-geometric means inequality, (4.41) is larger than or

equal to

[∆eigenvalues(k)

]D−k·[

1M−D

∑Mi=D+1 λi+

1M−D

∑Di=1 r · aH(φi)V

(M−D)n [V (M−D)

n ]Ha(φi)]M−D

.

(4.42)

Rewrite exp[

1N

∆MLE(k)]

and we have

exp[

1N

∆MLE(k)]

=

[∆eigenvalues(k)

]D−k

QDi=k+1 λi

· [ 1M−k

∑Mi=k+1 λi

+ 1M−k

∑ki=1 aH(φi)V

(M−k)n [V (M−k)

n ]Ha(φi)]M−k

/{[

∆eigenvalues(k)]D−k·[ 1

M−D

∑Mi=D+1 λi

+ 1M−D

∑Di=1 r · aH(φi)V

(M−D)n [V (M−D)

n ]Ha(φi)]M−D}

. (4.43)

In the large sample limit, V (M−k)n approaches V (M−k)

n and includes some of the dimen-

sions from the signal subspace. Therefore the expression∑k

i=1 aH(φi)V(M−k)

n [V (M−k)n ]Ha(φi)−

∑Di=1 aH(φi)V

(M−D)n [V (M−D)

n ]Ha(φi) is either pos-

itive or O( 1N

). Besides, the eigenvalues λi, i = k + 1, k + 2, . . . , D are not asymptot-

ically equal. Thus, as N is approaching the limit, ∆eigenvalues(k) > 1D−k

∑Di=k+1 λi >

∏Di=k+1 λi. Here the second inequality is the (basic) arithmetic mean-geometric mean

(AM-GM) inequality.

Summing up, 1N

∆MLE(k) is asymptotically positive with probability one. Com-

bining this with the fact that 1N

∆Penalty(k) approaches zero with increasing N , we

conclude that for k < D, MDLMUSIC(D)−MDLMUSIC(k) is positive with probability

one in the large sample limit.

88

Chapter 5

Weighted Spatial Smoothing Based

Iterative Weight Matrix

Approximation

89

5.1 Introduction

In this chapter we describe the weighted spatial smoothing (WSS) based it-

erative weight matrix approximation (IWMA) algorithm. As we have mentioned, in

wireless communication systems employing antenna-array at the receivers, the signal

parameter directions-of-arrival (DoAs) can be estimated using subspace-based tech-

niques and algorithms. In realistic communication environments, correlated sources

and propagation paths often exist, due to deliberate jamming or unavoidable multi-

path propagation. This situation can severely affect the accuracy of subspace-based

estimators. As a remedy to the situation, the technique of spatial smoothing and its

variations have been extensively studied. Spatial smoothing was originally proposed

in [75] and its concept was further analyzed and formulated in [76]. In [79] an im-

provement to the technique was discussed based on a strategy to take advantage of

the cross correlations between the subarrays’ outputs. In [80], backed by theoretical

analysis of the robustness of eigenvectors to noise interference (which determines the

estimator’s performance), it was suggested that squaring the array covariance ma-

trix can improve the performance of forward or forward/backward spatial smoothing.

In [81] correlated and uncorrelated signals are separately estimated. MUSIC is first

applied to the array covariance matrix to have the DoA estimates of the uncorre-

lated signals. The obtained DoAs are used to construct a covariance matrix which

is subtracted from the original covariance matrix. Spatial smoothing is then applied

iteratively, starting from 3-element array, and 2 overlapping subarrays and increasing

iteratively to 4 and 3, respectively, and so on. This process should continue until the

MUSIC peaks emerge. It was shown that the computational complexity required is

much lower than that of standard methods. In [82], the author designed a method

which utilizes not only the sum of the forward and backward spatially smoothed co-

variance matrices, but also the difference between the two matrices. It was shown

that this method benefits from generating the basis for the signal subspace from the

columns of the sample covariance matrix. Methods handling coherent signals include

90

also [83, 84, 85], where the spatial filtering method and enhancements to it are con-

sidered. In [85] it was shown that while the scheme’s requirement on the antenna

aperture is the same as corresponding forward/backward spatial smoothing meth-

ods, the estimation performance is improved. More traditionally, DoA estimation for

coherent cases can be obtained by maximum likelihood algorithms such as Determin-

istic Maximum Likelihood [77] or Weighted Subspace Fitting [78]. They both perform

multidimensional searches and are computationally intensive.

In this work we consider cases in which the incoming signals are highly-

correlated and their cross-correlation coefficients are not of value one, but can come

very close to it. We present a directions-of-arrival estimation scheme, the key step of

which is a weighted spatial smoothing (WSS) based iterative weight matrix approx-

imation (IWMA) algorithm. WSS [87] is a generalization of the spatial smoothing

technique. The latter was originally proposed in [75] and the concept was further

analyzed and formulated in [76]. In WSS the spatially-smoothed covariance matrix is

obtained as a weighted sum of the cross-covariance matrices between the ith and the

jth subarrays (the ith subarray is formed by the ith, i+1th, . . ., i+P − 1th antenna

elements), i = 1, 2, . . . , Q and j = 1, 2, . . . , Q, where Q is the total number of subar-

rays. The (i, j)th covariance matrix is weighted by a coefficient wij. Accordingly, a

weight matrix for a WSS processing is defined as a matrix consisting of the weights

wij’s i.e., W4= [wij]. WSS includes conventional spatial smoothing (CSS) ([75] [76])

as a special case, in which case the weight matrix is given by W = 1QI. The weights

of a WSS pre-processing scheme are determined by different design criteria [87][88].

In this work we choose wij’s such that the smoothed source covariance matrix after

the application of WSS, ~Rxx (denoted using an over-vector ~(·)), is diagonal. Diagonal

~Rxx is a desired feature for subspace-based estimation algorithms: a generic MU-

SIC estimator is a Large Sample Maximum Likelihood (LSML) estimator if and only

if the signal covariance matrix is diagonal [86]. The optimum weight matrix Wopt

that produces diagonal ~Rxx can be obtained analytically [87]. But its computation

91

requires explicit knowledge of the DoAs, which are actually unknowns that need to

be estimated. In [87] it is suggested that the coarse DoA estimates obtained from

Capon’s spectrum are used to calculate an approximation to Wopt.

IWMA is a procedure that is capable of approximating Wopt in an iterative

fashion and without any prior knowledge of the DoAs. It is applicable as long as

the input covariance matrix is positive definite. The algorithm starts initially with

a weight matrix W0 = 1QIQ and carries out a series of weighted spatial smoothing

steps with respect to the estimated noise-free array covariance matrix. Within each

iteration it performs WSS using the weight matrix that is obtained from the previous

iteration step. We shall show that for noise-free array covariance matrix the above

procedure is convergent with a finite number of iterations.

Suppose that the algorithm stops at the nth iteration where we obtain the

weight matrix Wn. Two features of the ideal Wn make it a suitable basis upon which

subspace-based DoA estimation can be performed. First, Wn is parameterized by

the directions-of-arrival of the input signals. Second, its structure is optimized as

a result of the IWMA algorithm (this point will become clear in a later section).

We will illustrate the operations of the IWMA algorithm and the performance of

IWMA-generated weight matrix based estimation strategy.

The remainder of the chapter is organized as follows. In Section 5.2 we present

the system model and briefly describe the weighted spatial smoothing preprocessing

technique. In Section 5.3 we present the proposed IWMA algorithm. Theoretical

analysis of the algorithm is provided in Section 5.4. Finally, computer simulations

are presented and discussed in Section 5.6.

5.2 System Model and Background

The system model is similar to that of the previous chapter. We consider D

signals x1(t), x2(t), . . . , xD(t) (D is assumed to be known a priori) each impinging

92

with an electrical angle φk ∈ [−π, π] on an M -element uniform linear array (ULA).

Let y(t) denote the vector consisting of the M received signals from the ULA at time

t. Sampling y(t) at time instants ti (i = 1, 2, . . .) we obtain a discrete-time received

vector sequence y[i] = y(ti) that is given by:

y[i] = Ax[i] + n[i], i = 1, 2, . . . , N, (5.1)

where x[i]4= [x1(ti), x2(ti), . . . , xD(ti)]

T is the vector containing the D transmitted

signals and A4= [a(φ1), . . . , a(φD)] is the array manifold, the kth column of which is

the steering vector a(φk)4= [1, e−jφk , . . . , e−j(M−1)φk ]T associated with the kth input

xk(t), k = 1, 2, . . . , D. In general, the angles φk, k = 1, 2, . . . , D are distinct, which

implies that the matrix A is of full column rank. We assume that the transmitted

vectors x[i], i = 1, 2, . . . , N , are random, identically distributed, with a covariance

matrix given by Rxx4= E{x[i]x[i]H} where E{·} denotes the expectation operator.

Finally, in (5.1), n[i] = n(ti) is additive Gaussian noise assumed to be zero-mean,

spatially and temporally white, with covariance matrix σ2IM , where IM is the M×M

identity matrix.

The problem we consider in this work is the estimation of the direction-of-

arrival (DoA) φk of the kth input signal xk(t). Traditionally, DoA estimation is done

via subspace-type algorithms [89]. However, these estimators can perform poorly

in situations when the signals xk(t) are highly correlated. In these cases the signal

covariance matrix Rxx is no longer diagonal and can even be ill-conditioned which

can noticeably limit the effectiveness of subspace based DoA estimation algorithms.

In this chapter, we consider exactly such cases where the signal cross-correlation

coefficients can be very close to unity.

In the rest of this chapter, we use “conj(X)” to denote the element-wise com-

plex conjugation of the matrix X. To simplify the notation, we will also use X to

denote the same operation. Also, for a Hermitian matrix X of size D we will denote

its eigenvalues (sorted in descending order) as λl(X), l = 1, 2, . . . , D, its eigenvalue

93

vector as λ(X)4= [λ1(X), λ2(X), . . . , λD(X)]T , and its maximum and minimum

eigenvalues as λmax(X)4= λ1(X) and λmin(X)

4= λD(X), respectively.

5.2.1 Weighted Spatial Smoothing

In WSS, the M -element ULA is divided into Q = M − P + 1 overlapping

subarrays of P elements each. The qth subarray, q = 1, 2, . . . , Q, is formed by the

q, (q + 1), . . . , (q + P − 1)th elements of the ULA. Let us denote by yq[i] (nq[i]) the

received signal (noise) over the qth subarray at the ith time instant, i = 1, 2, . . . , N ,

and by Aq the submatrix of A formed by the q, (q + 1), . . . , (q + P − 1)th rows of A.

Then

yq[i] = Aqx[i] + nq[i], q = 1, 2, . . . , Q. (5.2)

To perform WSS, we first compute the weighted sum of the Q2 cross-correlation

matrices between yp[i] and yq[i], for p, q = 1, 2, . . . , Q:

~Ryy4=

Q∑p=1

Q∑q=1

wpqE{yp[i]y

Hq [i]

}=

Q∑p=1

Q∑q=1

wpqFpRyyFTq , (5.3)

where wpq denotes the (p, q)th weight for the (p, q)th cross-correlation matrix and Fi =[0P×(i−1) IP×P 0P×(M−P−i+1)

], i = 1, 2, . . . , Q. It is easily seen that conventional

spatial smoothing (CSS) [75] [76] is a special case of WSS obtained using wpq = δ(p−q)

where δ(·) is the Kronecker delta. The matrix ~Ryy of (5.3) can be decomposed into

two terms:

~Ryy = ~A ~Rxx~AH + ~Rnn, (5.4)

where ~Rnn4=

∑Qp=1

∑Qq=1 wpqE

{np[i]n

Hq [i]

}denotes the spatially-smoothed noise co-

variance matrix and ~A4= [~a(φ1), ~a(φ2), . . . , ~a(φD)] is the subarray manifold with

~a(φk)4= [1, e−jφk , . . . , e−j(P−1)φk ]T , k = 1, 2, . . . , D. Finally, ~Rxx is the signal autocor-

94

relation matrix after WSS and is given by [87]

~Rxx4=

Q∑p=1

Q∑q=1

wpqΦpRxxΦ

−q = Rxx ◦ (BHWB). (5.5)

In (5.5), W = [wpq], p, q = 1, 2, . . . , Q is the Q×Q weight matrix, Φ4= diag([e−jφ1 , e−jφ2 ,

. . . , e−jφD ]T ), B4= [b(φ1)b(φ2) . . . b(φD)] with b(φk)

4= [1, ejφk , . . . , ej(Q−1)φk ]T for

k = 1, 2, . . . , D, and “◦” denotes the matrix Hadamard product. The weight matrix

W is chosen such that the signal autocorrelation matrix after applying WSS, ~Rxx, is

diagonal. It can be shown [87] that the matrix W that results in a diagonal covari-

ance matrix is given by Wopt = (BBH)†, where “(·)†” denotes the Moore-Penrose

matrix inverse.

After WSS the subspace-based DoA estimation is then performed as follows.

The noise subspace spanned by the eigenvectors associated with the P −D smallest

eigenvalues of the spatially smoothed covariance matrix is first identified. If~Vn is

the matrix whose columns are formed by those eigenvectors of the spatially smoothed

estimated covariance matrix that span the noise subspace, then we can estimate the

DoAs φk, k = 1, 2, . . . , D, of all sources from the locations of the D largest peaks of

the spectrum P(φ) defined as

P(φ)4=

∥∥∥ ~V H

n · ~a(φ)∥∥∥−2

, φ ∈ [−π, π] , (5.6)

where ~a(φ)4= [1, e−jφ, . . . , e−j(P−1)φ]T . To obtain the locations of the D largest peaks

a search on the real line would need to be performed.

Clearly, the inherent difficulty associated with the calculation of Wopt is that

it depends on the unknown φk, k = 1, 2, . . . , D. In the next section we propose the

iterative weight matrix approximation (IWMA) algorithm that can approximate the

optimum weight matrix Wopt in an iterative fashion without any knowledge of the

DoAs φk.

95

5.3 Proposed Iterative Weight Matrix Approxima-

tion Algorithm

The proposed algorithm starts with an initial weight matrix W0 set equal to

1QIQ and applies a series of weighted spatial smoothing operations on the “de-noised”

array covariance matrix Rdn4= Ryy − σ2I where Ryy is the autocorrelation matrix

of the spatial observation vector i.e., Ryy4= E{y[i]y[i]H}. Each WSS operation uses

a weight matrix obtained from the previous iteration step. This iterative procedure

is formally described in Algorithm 2. We shall show in the next section that in the

ideal and noise-free case, the weight matrix Wi iteratively approaches a matrix of the

form (B ·D ·BH)†, where D is a diagonal matrix.

Algorithm 2 Iterative Weight Matrix Approximation

Input: Rdn = Ryy − σ2IInitialization:W0 ⇐ 1

QIQ and F ⇐ [

IQ×Q 0(P−Q)×Q

]n ⇐ number of iterations, n is a positive even numberMain Loop:for i = 1 to n do

Perform WSS (see (5.3)) on Rdn using Wi−1

Obtain ~Rdn,i = ~A[Rxx ◦ (BHWi−1B)] ~AH

Wi ⇐ [conj(F ~Rdn,iFT )]†

end forOutput: Wn

In practice, the exact covariance matrix Ryy is not known and is estimated by

sample-averaging over N observed vectors as follows:

Ryy4=

1

N

N∑i=1

y[i]y[i]H . (5.7)

The “de-noised” version of the array covariance matrix can then be estimated by first

applying an eigenvalue decomposition on Ryy. More specifically, let Λs be the diag-

onal matrix formed by the signal eigenvalues λ1(Ryy), λ2(Ryy), . . . , λD(Ryy) and Vs

96

the matrix containing in its columns the corresponding signal eigenvectors. Similarly,

let Λn be the diagonal matrix formed by the noise eigenvalues

λD+1(Ryy), λD+2(Ryy), . . . , λM(Ryy) and Vs the matrix containing in its columns the

corresponding noise eigenvectors. Then, σ2 and Rdn can be estimated by σ2 =

1M−D

tr Λn and Rdn4= Vs(Λs − σ2ID)V H

s , respectively. The procedure is described

by Algorithm 3 below.

Algorithm 3 Iterative Weight Matrix Approximation

Input: Ryy, estimate of Ryy

Initialization:Perform eigenvalue decomposition (EVD) on Ryy = VsΛsV

Hs + VnΛnV H

n

Obtain σ2 = 1M−D

tr Λn and Rdn4= Vs(Λs − σ2ID)V H

s

W0 ⇐ 1QIQ and F ⇐ [

IQ×Q 0(P−Q)×Q

]n ⇐ number of iterations, n is a positive even numberMain Loop:for i = 1 to n do

Perform WSS (see (5.3)) on Rdn using Wi−1 to obtain~Rdn,i

Wi ⇐ [conj(F~Rdn,iF

T )]†

end forOutput: Wn

5.4 Theoretical Analysis

In this section we study the behavior of Algorithm 2. We start with an ob-

servation about its iterative process: assuming that at the ith iteration the weight

matrix is given by Wi = (BDBH)†, where D is a D×D diagonal matrix, the entries

of which are strictly larger than zero, we have Wi+2 = (BDBH)† = Wi. The matrix

Wi+1 at the next iteration is given by:

Wi+1 = [conj(F ~RdnFT )]† = [conj(F ( ~A ~Rxx

~AH)F T )]† = (B conj( ~Rxx)BH)†

= (B conj(Rxx ◦ (BHWiB))BH)† = (BDRD†BH)† = (BH)†DD†RB†, (5.8)

97

where DR is a diagonal matrix whose main diagonal is equal to that of the signal

covariance matrix Rxx. Similarly, at the (i + 2)th iteration the matrix Wi+2 is

Wi+2 = (B conj(Rxx ◦ (BH(BH)†DD†RB†B))BH)†

= (BDRD†RDBH)† = (BDBH)† = Wi. (5.9)

Eqs. (5.8) and (5.9) reveal that if at the ith iteration Wi is equal to the optimum

weight matrix1 Wopt = (BBH)†, or to some matrix of the form α(BBH)† for α > 0,

then the algorithm will enter a loop where its output will alternate between two

values, one of which is equal to Wopt. Motivated by this observation, we will treat

the IWMA algorithm’s iteration steps in pairs, starting at an even iteration step.

The behavior of the proposed IWMA algorithm will be investigated by studying

how the source correlation matrix adapts after every two consecutive iterations. Let

~Rxx,i (i = 1, 2, 3, . . . , n) (again n is a positive even number) denote the spatially

smoothed source covariance matrix implicit in the ith iteration i.e., a D ×D matrix

such that ~Rdn,i = ~A ~Rxx,i~AH . We will now show that two consecutive iterations of

the IWMA algorithm modify the ith source covariance matrix ~Rxx,i to the (i + 2)th

source covariance matrix ~Rxx,i+2 as follows:

~Rxx,i+2 = Rxx ◦ [Rxx ◦ ( ~Rxx,i)†]†, i = 0, 2, . . . , n, (5.10)

where ~Rxx,0 is defined as ~Rxx,04= ( 1

QB

HB)†. Indeed, for the first four iteration steps

of the IWMA algorithm we have

~Rxx,1 = Rxx ◦ (1

QBHB) (5.11)

~Rxx,2 = Rxx ◦ (BH(B(Rxx ◦ (1

QB

HB))BH)†B) = Rxx ◦ (Rxx ◦ (

1

QB

HB))†.

(5.12)

1We note that Wopt = (BDBH)† for D = I.

98

~Rxx,3 = Rxx ◦ (conj( ~Rxx,2))† (5.13)

~Rxx,4 = Rxx ◦ (Rxx ◦ ( ~Rxx,2)†)†. (5.14)

Proceeding by induction on i and assuming that the updating formula is true for

some even-numbered integer i and any even numbers less than i, we have

~Rxx,i+1 = Rxx ◦ (conj( ~Rxx,i))† (5.15)

~Rxx,i+2 = Rxx ◦ (Rxx ◦ ( ~Rxx,i)†)†, (5.16)

which prove the updating formula as given in (5.10). The updating procedure de-

pends on the two matrices ~Rxx,0 and Rxx both of which are positive definite2. As

the Hadamard product of Hermitian matrices maintains positive definiteness [76],

the above iteration process is non-degenerate and ~Rxx,i is always positive definite

and Hermitian. Thus the pseudo-inverse operation in (5.10) can be replaced by the

ordinary matrix inverse i.e., ~Rxx,i+2 = Rxx ◦ [Rxx ◦ ( ~Rxx,i)−1]−1, i = 0, 2, . . . , n.

The next theorem establishes the fact that two consecutive iterations of the

IWMA algorithm result in a reduction of the Frobenius norm of the source covariance

matrix.

Theorem 5 For i = 0, 2, . . . , n, we have

∥∥∥ ~Rxx,i+2

∥∥∥2

F≤

∥∥∥ ~Rxx,i

∥∥∥2

F, (5.17)

where ‖ · ‖F denotes the Frobenius matrix norm.

Proof: Letting Σi4= ( ~Rxx,i)

−1 we need to show that

∥∥Rxx ◦ (Rxx ◦Σi)−1

∥∥2

F≤

∥∥Σ−1i

∥∥2

F. (5.18)

2The matrix ~Rxx,0 = 1QB

HB is positive definite as long as Q ≥ D and the DoAs of the input

signals are distinct. Rxx is assumed to be positive definite throughout this text.

99

We first note that

~Rxx,i+2 = Rxx ◦ (Rxx ◦Σi)−1 = CR ◦ (CR ◦Σi)

−1, (5.19)

where CR4= URRxxUR, CR

4= URRxxUR and UR

4= diag

([r11

−1/2, r22−1/2, . . . , rDD

−1/2]T

)

with rii denoting the ith diagonal entry of Rxx. CR is a correlation matrix and so is

CR; therefore we have

∥∥Rxx ◦ (Rxx ◦Σi)−1

∥∥2

F=

∥∥CR ◦ (CR ◦Σi)−1

∥∥2

F≤

∥∥(CR ◦Σi)−1

∥∥2

F. (5.20)

Thus it is sufficient to show that

∥∥(CR ◦Σi)−1

∥∥2

F≤

∥∥Σ−1i

∥∥2

F. (5.21)

We have the following properties regarding the Hadamard product of a Hermitian

matrix and a correlation matrix [92, Theorem 5.5.11]: the set of eigenvalues of CR◦Σi,

λ(CR ◦Σi) is majorized3 by the set of eigenvalues of Σi, λ(Σi). Combined with the

fact that the function f(λl) = λ−2l is convex, we have [90, p. 64]

D∑

l=1

[λl(CR ◦Σi)]−2 ≤

D∑

l=1

[λl(Σi)]−2, (5.22)

which is exactly the relationship (5.21). Together with (5.20) this proves the relation-

ship (5.18). It is easily seen that the equality in (5.18) holds if and only if (CR◦Σi)−1

is diagonal which, in turn, is diagonal if and only if Σi is diagonal.

Moreover, two consecutive iterations of the IWMA algorithm lead to a reduc-

3For real vectors a and b of the same size n, a is said to majorize b if (i)∑n

i=1 ai =∑n

i=1 bi and(ii)

∑ki=1 a[i] ≥

∑ki=1 b[i], k = 1, 2, . . . , n− 1, where a[i] denotes the ith largest element of a and b[i]

denotes the ith largest element of b. If condition (i) is relaxed to∑n

i=1 ai ≥∑n

i=1 bi then a is saidto weakly submajorize b. Intuitively, majorization is a partial order over vectors of real numbersand the statement “a majorizes b” means that the components of a are more spread out than thoseof b.

100

tion of the eigenvalue range of the source signal correlation matrix. This is formally

stated by the following theorem.

Theorem 6 The range of the eigenvalues of ~Rxx,i+2 is confined within that of ~Rxx,i

i.e.,

λmin( ~Rxx,i) ≤ λmin( ~Rxx,i+2) ≤ λmax( ~Rxx,i+2) ≤ λmax( ~Rxx,i). (5.23)

Proof: It is equivalent to show that

λmin(Σ−1i ) ≤ λmin[CR ◦ (CR ◦Σi)

−1] ≤ λmax[CR ◦ (CR ◦Σi)−1] ≤ λmax(Σ

−1i ). (5.24)

The set of eigenvalues of Σi, λ(Σi), majorizes the set of eigenvalues of CR ◦ Σi,

λ(CR ◦Σi). Therefore,

λmin(Σi) ≤ λmin(CR ◦Σi) ≤ . . . ≤ λmax(CR ◦Σi) ≤ λmax(Σi). (5.25)

This in turn gives

λ−1max(Σi) ≤ [λmax(CR ◦Σi)]

−1 ≤ . . . ≤ [λmin(CR ◦Σi)]−1 ≤ λ−1

min(Σi). (5.26)

We then have

λmin(Σ−1i ) ≤ λmin[(CR ◦Σi)

−1] ≤ λmin[CR ◦ (CR ◦Σi)−1]

≤ . . . ≤ λmax[CR ◦ (CR ◦Σi)−1] ≤ λmax[(CR ◦Σi)

−1] ≤ λmax(Σ−1i ) (5.27)

which completes the proof.

A consequence of Theorem 6 is that the eigenvalues of ~Rxx,i+2 are always kept

within the range [λ−1max(

1QB

HB), λ−1

min(1QB

HB)] which is the range of the eigenvalues

of Σ0 = ( ~Rxx,0)−1.

Theorem 7 The eigenvalue vector of ~Rxx,i+2, λ( ~Rxx,i+2) is weakly submajorized by

101

the eigenvalue vector of ~Rxx,i, λ( ~Rxx,i). Moreover,

λ( ~Rxx,i+2) = Qiλ( ~Rxx,i), (5.28)

where Qi is a doubly substochastic matrix i.e., a square matrix entries of which are

nonnegative and the row sum and column sum of it is less than or equal to one.

Proof: It suffices to show that

λ( ~Rxx,i+2) = λ(Rxx ◦ (Rxx ◦Σi)

−1)

= λ(CR ◦ (CR ◦Σi)

−1)

is weakly submajorized

by λ(Σ−1i ) and that λ

(CR ◦ (CR ◦Σi)

−1)

= Qλ(Σ−1i ), where Q is a doubly sub-

stochastic matrix. We first see that the vector of eigenvalues λ(CR ◦Σi) is majorized

by that of Σi, λ(Σi). Since f(λl) = λ−1l is a convex function for positive eigen-

values λl, l = 1, 2, . . . , D, λ[(CR ◦ Σi)−1] is weakly submajorized by λ(Σ−1

i ) [90,

p. 115]. Next, since λ[CR ◦ (CR ◦ Σi)−1] is majorized by λ[(CR ◦ Σi)

−1], we con-

clude that λ[CR ◦ (CR ◦Σi)−1] is submajorized by λ(Σ−1

i ). Finally, since each value

of λ[CR ◦ (CR ◦ Σi)−1] and λ(Σ−1

i ) is nonnegative, we then have that [90, p. 27]

λ[CR ◦ (CR ◦Σi)−1] = Qiλ(Σ−1

i ).

The update from λ( ~Rxx,i) to λ( ~Rxx,i+2) as given by (5.28), combined with

(5.23), shows that the eigenvalue range of ~Rxx,i+2 is squeezed by the algorithm with

a best effort. This is because each eigenvalue is a weighted sum of the elements

of λ( ~Rxx,i) and the weights are given by the pth row of Qi. As Qi is a doubly

substochastic matrix, the weights are non-negative and their sum is less than or

equal to one. Furthermore, the eigenvalues can not be squeezed below λmin( ~Rxx,i), as

seen from (5.23).

To summarize, the working principles of the proposed algorithm as described

by Theorems 5-7 are:

1. The Frobenius matrix norm of ~Rxx,i+2 is a decreasing function with increasing

i.

2. With increasing i, the eigenvalue range of ~Rxx,i+2, [λmin( ~Rxx,i+2), λmax( ~Rxx,i+2)],

102

is kept within the eigenvalue range two iterations before, i.e.

Eigenvalue range of ~Rxx,i︷ ︸︸ ︷

λmin( ~Rxx,i) ≤Eigenvalue range of ~Rxx,i+2︷ ︸︸ ︷

λmin( ~Rxx,i+2) ≤ λmax( ~Rxx,i+2) ≤ λmax( ~Rxx,i) . (5.29)

3. Furthermore, the eigenvalue range is reduced by the algorithm on a best effort

basis, as prescribed by the second and the third theorems. Note that the range

may or may not be reduced. A strict decrease of the maximum eigenvalue

happens when at the i + 2th iteration the elements of the first column of Qi

are less than 1. If Rxx or ~Rxx,i+2 is diagonal, then the eigenvalue range is not

reduced. Whether or not the range is reduced is dependent on Qi and eventually

on the inputs to the algorithm and the current conditions for the iteration. This

point is further elaborated on in the next theorem.

Theorem 8 Let ~Rxx,i+2 be as defined before (Eq. (5.10)). ~Rxx,i+2 approaches a

diagonal matrix D of the same size with finite (and even) number of IWMA iterations.

The entries of the main diagonal D are positive and their range is reduced by the

algorithm in a best effort.

Proof: We proceed by considering the two possible cases when the algorithm

is at the (i + 2)th iteration: (i) the matrix (CR ◦ Σi)−1 is diagonal; (ii) the matrix

(CR ◦Σi)−1 is not diagonal.

We shall use the following term to quantify how close (CR ◦ Σi)−1 is to a

diagonal matrix:D∑

p,q=1p6=q

|[(CR ◦Σi)−1]pq|2(1− |[CR]pq|2). (5.30)

In (5.30), [X]pq is used to denote the (p, q)th element of the matrix X. As noted

before, CR is a correlation matrix and it is positive definite (CR has the same positive

definiteness as the signal covariance matrix Rxx). A correlation matrix has its main

103

diagonal elements equal to 1 and its off-diagonal elements less than or equal to 1.

Since the input signals are correlated but not coherent by assumption, none of the

off-diagonal entries of Rxx is equal to one. When (CR ◦Σi)−1 is a diagonal matrix,

(5.30) is equal to 0. Otherwise (5.30) is larger than 0 and the magnitude of its value

corresponds to the deviation of (CR ◦ Σi)−1 from a diagonal matrix. When (5.30)

approaches 0, (CR ◦ Σi)−1 approaches a diagonal matrix. Likewise, the converse is

true.

Consider the first case i.e., at the (i + 2)th iteration the matrix (CR ◦ Σi)−1

is diagonal, which is either because CR is diagonal (equivalently because Rxx is

diagonal) or Σi is diagonal (equivalently ~Rxx,i is diagonal), we know from Eq. (5.19)

that ~Rxx,i+2 is diagonal, which proves the first part of the theorem.

Suppose now that the matrix (CR ◦ Σi)−1 is not diagonal at the (i + 2)th

iteration. We then have

‖ ~Rxx,i‖2F−‖ ~Rxx,i+2‖2

F =∥∥Σ−1

i

∥∥2

F− ‖CR ◦ (CR ◦Σi)

−1‖2F

≥‖(CR ◦Σi)−1‖2

F − ‖CR ◦ (CR ◦Σi)−1‖2

F

=D∑

p,q=1p6=q

|[(CR ◦Σi)−1]pq|2(1− |[CR]pq|2) > 0, (5.31)

where the second line follows from the inequality (5.21). From (5.31) we have that the

magnitude of the Frobenius norm ‖ ~Rxx,i+2‖2F is decreasing, the amount of which is

larger than or equal to (5.30), which quantifies how close (CR ◦Σi)−1 is to a diagonal

matrix. If the amount of decreasing is approaching zero, we see that (CR ◦ Σi)−1

is approaching a diagonal matrix. And this proves the first part of the theorem.

Otherwise we continue our discussion as below.

As long as (CR ◦Σi)−1 deviates from a diagonal matrix for i = 0, 2, 4, . . ., we

104

have

∑i=0,2,4,...

(‖ ~Rxx,i‖2

F − ‖ ~Rxx,i+2‖2F

)≥

∑i=0,2,4,...

D∑p,q=1p6=q

|[(CR ◦Σi)−1]pq|2(1− |[CR]pq|2),

(5.32)

and

‖ ~Rxx,i+2‖2F = ‖ ~Rxx,0‖2

F −∑

i=0,2,4,...

(‖ ~Rxx,i‖2

F − ‖ ~Rxx,i+2‖2F

)

≤ ‖ ~Rxx,0‖2F −

∑i=0,2,4,...

D∑p,q=1p6=q

|[(CR ◦Σi)−1]pq|2(1− |[CR]pq|2). (5.33)

Note that ‖ ~Rxx,i+2‖2F can not be decreased indefinitely towards 0, which can be

explained by viewing the vector of the eigenvalues of the smoothed covariance matrix

~Rxx,i+2 at the (i+2)th iteration, i.e. λi+2 = [λ1( ~Rxx,i+2), . . . , λD( ~Rxx,i+2)]T , as a point

within the D-dimensional real space RD. λi+2 lies on the surface of a hypersphere

centered at the origin [0, 0, . . . , 0]T with radius ‖ ~Rxx,i+2‖F = [∑D

l=1 λ2l (

~Rxx,i+2)]−1/2.

The decrease of ‖ ~Rxx,i+2‖2F with increasing i corresponds to the shrinking of the

hypersphere, on top of which λi+2 lies. At the same time, λi+2 lies within an imag-

inary hypercube, each side of which runs from [0, . . . , 0, λmin( ~Rxx,i+2), 0, . . . , 0]T to

[0, . . . , 0, λmax( ~Rxx,i+2), 0, . . . , 0]T . According to Theorem 6 the hypercube may or

may not shrink with increasing i, but is always contained within the previous hyper-

cube, as suggested by the inequality (5.23). As λi+2 is confined within the hypercube,

this prevents the ‖ ~Rxx,i+2‖2F from decreasing indefinitely. Furthermore, the shrinking

of the hypersphere combined with the shrinking of the hypercube defines the behavior

of the iteration algorithm.

With increasing i, the value of ‖ ~Rxx,i+2‖2F decreases as (5.33), and we will

have ‖ ~Rxx,i+2‖2F = λ2

max(~Rxx,i+2) +

∑D−1l=1 λ2

min(~Rxx,i+2), beyond which point further

iterations will strictly decrease the maximum eigenvalue. For example, with ad-

105

ditional j (j even and j > 0) iterations we will have ‖ ~Rxx,i+2+j‖2F < ‖ ~Rxx,i+2‖2

F

and λmin( ~Rxx,i+2) ≤ λmin( ~Rxx,i+2+j) ≤ λmax( ~Rxx,i+2+j) ≤ λmax( ~Rxx,i+2), which im-

plies that λmax( ~Rxx,i+2+j) < λmax( ~Rxx,i+2). Eventually we will have ‖ ~Rxx,i+2+j‖2F =

∑Dl=1 λ2

min(~Rxx,i+2+j) for some j. At this time the hypershere and the hypercube

will have only one intersection point, which is λi+2+j and the eigenvalues are seen to

be squeezed towards some real number α which is within the predetermined range

[λ−1max(

1QB

HB), λ−1

min(1QB

HB)]. The quantity ‖ ~Rxx,i+2+j‖2

F can no longer be decreased

with increasing j (and the hypersphere will not shrink). From Theorem 5 we see that

this means that the equality of (5.18) holds and ~Rxx,i+2+j is αI. This completes the

proof of the first part of the claim.

As the eigenvalue range is decreased by the algorithm in a best effort, we see

that the range of the final diagonal matrix D’s entries is decreased by the algorithm

in a best effort.

With the above, we see that the IWMA algorithm is convergent to the two

alternating stages (5.8) and (5.9) and will stabilize.

Some remarks are now in order.

• Although the algorithm in a best effort reduces the range of the entries of D, it

is possible that ~Rxx,i+2 will converge to some diagonal matrix D instead of αI

(which is the case when the algorithm converges before the hypersphere and the

hypercube intersects at only one point) as this is dependent on the inputs to the

algorithm over which the algorithm has no control. Extensive simulations have

shown that for the assumptions of interest within this text, D is only slightly

deviated from an ideal diagonal matrix αI.

• The dependence of IWMA upon the correlation matrix CR (Eq. (5.20)) im-

plies that the algorithm is suited to highly correlated input signals, the cross-

correlation coefficients of which are very close to the value one; and it is not

applicable in cases where coherent signals and the cross-correlation coefficients

among them are all one.

106

5.5 DoA Estimation Using Wn

With IWMA we have Wn = (BDBH)† for sufficiently large n. Wn is well

formatted and is a suitable basis upon which DoA estimation can be directly carried

out. This is because D is a diagonal matrix such that the spread of its diagonal

elements is minimized by the algorithm in a best effort (again this is suggested by

Theorem 6 and 7). DoA estimation is carried out by identifying the Q × (Q − D)

dimensional noise subspace that is associated with Wn, and using the noise subspace

to perform MUSIC-type DoA estimation.

5.6 Simulations

In the first simulation (shown in Fig. 5.1) we demonstrate the operation of the

proposed algorithm by plotting the approximation error of Wopt by Wi, as a function

of the number of iterations for different values of the array snapshot sample size N .

The approximation error is defined as minc e(c) where e(c)4= ‖Wi− cWopt‖2

F . It can

be shown that the value of c that minimizes e(c) is given by

c =

∑Qp,q=1 [Wi]

∗pq[Wopt]pq +

∑Qp,q=1 [Wopt]

∗pq[Wi]pq

2∑Q

p,q=1 [Wopt]∗pq[Wopt]pq

. (5.34)

In this study, we consider D = 2 signals impinging at electrical angles −0.1098

and 1.4654, respectively, on a 5-element ULA that is divided into Q = 5 − 4 +

1 = 2 sub-arrays of size P = 4. The system’s signal-to-noise-ratio (SNR), defined

as tr Rxx/(Dσ2), is set at 5dB. The two sources have equal power and the cross-

correlation between them is 0.9. Each curve shown in Fig. 5.1 represents averages over

20 Monte-Carlo runs. The curves show that when the number of samples is sufficient

the error of weight matrix estimations is decreased by the number of iterations. Note

that since the ideal Wi may not necessarily converge to a scaled version of Wopt, the

floors of the curves within the figure are anticipated.

107

In our next simulation, we evaluate the performance of the DoA estimator

which is based on the generated weight matrix by the IWMA algorithm as described

at the end of the previous section. We consider a 5-element ULA and D = 2 incident

source signals with electrical angles π/7 and π/4, respectively. The subarray size is

P = 3 which implies that Q = 5 − 2 + 1 = 3. The two sources are assumed to have

equal power and correlation 0.999. In Fig. 5.2 we plot the total MSE∑D

k=1 |φk−φk|2

of the two DoA estimates versus the input SNR. The number of observation snapshots

is N = 900 while the number of iterations of the IWMA algorithm is 300. In this

figure we include for comparison purposes the performance of MUSIC, CSS-based

MUSIC (for which the dimension of the subarray size is also P = 3), LMUSIC [93]

and SSMUSIC [94]. All curves shown are averages over 800 Monte-Carlo runs. The

performance improvement is apparent. In Fig. 5.3 we also plot the squared error

of the generated weight matrix versus the system’s SNR. The error has the same

definition as that of the first simulation.

In the third simulation we compare the performance of an IWMA-based DoA

estimator against that of a CSS-based MUSIC estimator for the case of M = 13

antennas. The subarray size is P = 7 (therefore, Q = 7). We consider the scenarios

of D = 2, 3, 4 or 5 incoming sources. In each case, the electrical angles are given by

the first D elements of the vector [0.4488, 0.6283, 0.8976, 1.5708, 2.6180]. As in Fig.

5.2 the sources are assumed to have equal powers and the cross correlation coefficients

between them are all 0.9999. The number of observed snapshots is N = 1500 while

the number of iterations of the IWMA algorithm is 300. In Fig. 5.4 we plot the

total MSEs of the DoA estimates versus SNR. All results shown are averages over 500

Monte-Carlo runs.

108

5.7 Conclusion

We described an iterative weight matrix approximation (IWMA) procedure,

which is able to efficiently generate an approximating weight matrix that can be

used in performing weighted spatial smoothing (WSS). The procedure can effectively

decorrelate the incoming signals received by an antenna array. By theoretical analysis

and computer simulations, we study the performance of the algorithm. We finally

note that the application area of IWMA and the estimation methods based on it is

different from that of conventional spatial smoothing and weighted spatial smoothing.

2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 4410

−5

10−4

10−3

10−2

10−1

ith Iteration

Sqa

ured

Err

or o

f W

Number of samples: N=100Number of samples: N=1000Number of samples: N=10000Number of samples: N=100000

Figure 5.1: Operations of the IWMA algorithm: averaged squared error of Wi versusnumber of iterations.

109

−10 −7 −4 −1 2 5 8 11 14 17 20 23 26 29 32 35 38 41 44 47 50

10−3

10−2

10−1

100

SNR (dB)

MS

E o

f DoA

Est

imat

ion

MUSICSSMUSICLMUSICDoA Estimation UsingIWMA Weight MatrixCSS Based MUSIC

Figure 5.2: Performance of DoA estimation using IWMA-generated weight matrix:MSE of DoA estimates versus system SNR in dB.

110

−10 −7 −4 −1 2 5 8 11 14 17 20 23 26 29 32 35 38 41 44 47 500.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

SNR (dB)

Squ

ared

Err

or o

f W

Figure 5.3: Operations of the IWMA algorithm: averaged squared error of Wi versussystem SNR in dB.

111

−10 −7 −4 −1 2 5 8 11 14 17 20 23 26 29 32 35 38 41 44 47 50 53

10−3

10−2

10−1

SNR (dB)

MS

E o

f DoA

Est

imat

ion

2 sources: DoA Estimation Using IWMA Weight Matrix2 sources: CSS Based MUSIC3 sources: DoA Estimation Using IWMA Weight Matrix3 sources: CSS Based MUSIC4 sources: DoA Estimation Using IWMA Weight Matrix4 sources: CSS Based MUSIC5 sources: DoA Estimation Using IWMA Weight Matrix5 sources: CSS Based MUSIC

Figure 5.4: Performance of DoA estimation using IWMA-generated weight matrix:MSE of DoA estimates versus system SNR in dB.

112

Chapter 6

On Two Deterministic Measures

for Linear Processing Space-Time

Block Codes

113

6.1 Introduction

In this chapter we consider deterministic measures for linear processing space-

time block codes. As we know, Multiple Input and Multiple Output (MIMO) is a

recent and promising technology for wireless communications, the main idea behind

which is the recognition and full exploitation of the spatial degrees of freedom within

the systems using multi-transmit multi-receive antenna arrays. In the 90s, the ca-

pacity gain of multiple-transmitting-multiple-receiving-antenna systems was revealed

[95], [96] and [97]. This led to an enormous research interest in exploiting MIMO’s

inherent degrees of freedom.

There are two fundamental viewpoints of MIMO’s functionalities and its us-

age. The first one exploits the possible capacity increase and is represented by the

V-BLAST (Vertical Bell Laboratories Layered Space Time) transmission scheme [100]

and the transmit beamforming technique [101]. Transmit beamforming utilizes chan-

nel state information (CSI) at the transmitter side. For this purpose, research has

been focused on analyzing various MIMO channels, their corresponding capacities and

efficient transmission strategies. On the other hand, a completely different MIMO

strategy is to improve the reliability of transmission and achieve full transmit di-

versity, a representative method of which is the orthogonal space-time block coding

(OSTBC) proposed by [103] and [104], which assumes no knowledge of CSI at the

transmitter side.

Since the works on [103] and [104], many space-time block coding schemes have

been proposed. B. Hassibi and B. M. Hochwald in [112] proposed the linear disper-

sion codes (LDCs) and a design criterion which maximizes the mutual information

between the (multiple) input symbols and the (multiple) output signals. In [117] the

authors proposed unitary space-time constellations, a technique similar to that of

[118]. Su and Xia in [129] considered several quasi-orthogonal schemes and studied

optimal rotations of the scalar signal constellations which can be used to obtain full

transmit diversity and maximize product distance gain. Constellation rotation was

114

also considered by N. Sharma and C. Papadias in a well-known work [128]. In [115]

the authors designed a space-time code specifically for systems with two transmitting

antennas which achieves both diversity and multiplexing gain. In [119] the authors

proposed a design criterion which seeks to minimize a quasi-orthogonality measure

of the space-time codes. The criterion is claimed to be equivalent to LDC’s mutual

information. In [121] space-time codes are designed using the mathematical tool of

division algebra (a division algebra has multiplicative inverses) where the rank crite-

rion (correspondingly the transmit diversity [104]) is of major concern and this rank

condition is maintained by the properties of division algebra.

In general, an STBC design involves spatial- and temporal- dimensions and can

be based on diverse performance and reliability criteria. We may divide the current

approaches to this problem into two main categories:

(i) Designing the overall space-time constellations. This includes, for example,

ST block codes based on division algebra, unitary STBC and minimal quasi-

orthogonality STBC.

(ii) Employing a linear processing (LP) structure ([104], [112]) and separately de-

signing the linear processing matrices (LPM’s) [104] (this definition is similar to

that of the so-called dispersion matrices in [112], for a formal definition within

this context, please refer to Eq. (6.2)) and the scalar signal constellations of

the individual transmitted symbols [104], [112] (see Eq. (6.2)). This cate-

gory includes, for example, OSTBC, quasi-orthogonal STBC (QOSTBC) [104],

QOSTBC with constellation rotation [128] and the linear dispersion codes.

We identify in the following several important considerations when designing

a space-time coding scheme:

(i) Diversity. It requires that the minimum rank of all differences of the ST code-

words be maximized. (The rank- and the determinant- criterion are given and

discussed in [99] and [102]).

115

(ii) Coding gain (product distance gain). Maximizing the product distance gain is

a more desirable characteristic as it is directly related to the frame-error-rate

performance of the ST code.

(iii) Linear decoding. Linear/fast decoder is an important design consideration for

ease of implementation and real-time decoding of space-time block codes.

(iv) Adaptability to scalar signal constellations. The consideration of this is specific

to the linear processing schemes, which separate the design of the set of linear

processing matrices from that of the transmitted (complex) signal constella-

tions. A space-time block code from orthogonal design, for example, achieves a

complete decoupling of the design into two separate procedures. On the other

hand, an STBC design with respect to the overall space-time constellation would

not have such problems.

In this work we study linear processing space-time block coding (LP-STBC)

and consider features that characterize the set of linear processing matrices (LPM’s)

associated with an LP-STBC. We discuss deterministic measures defined for the

LPM’s and investigate the relations of these measures to the final performance of

the STBC’s and to other design criteria. These measures are deterministic in their

nature and involve no statistical operators (such as expectations) and are solely con-

cerned with the LPM’s. Within this theme we study and discuss next two such

measures for LP-STBC’s.

The first one is a design criterion obtained by Jensen’s relaxation of the LDC

mutual information. In general the LPM set should be combined with the transmit-

ting constellations in order to evaluate meaningful performance criteria. For exam-

ple, QOSTBC’s FER (Frame Error Rate) performance can be substantially improved

through constellation rotations. Fortunately, the two can be effectively decoupled

using the LDC mutual information criterion [112]. The derivation of the criterion

is based upon the observation that the set of linear processing matrices effectively

116

reshapes the original MIMO channel. The criterion requires only the statistical dis-

tribution of the channel coefficients.

To obtain the first measure, we proceed by applying Jensen’s Inequality to the

mutual information measure of the linear dispersion code, denoted by CLD, and then

evaluate the statistical expectations. This gives a design criterion that is deterministic

as it does not involve statistical operators and is solely concerned with the LPM

set. We shall show using computer simulations that the criterion maintains a close

relationship with the LDC mutual information, and it has the advantage of simplifying

the LP-STBC design as it separates the design of LPM’s from the statistical properties

of the channel.

The second criterion that will be discussed consists of two metrics that measure

the non-orthogonality of the LPM’s and are obtained by generalizing of the concept

of orthogonal STBC [104]. OSTBC and QOSTBC signaling schemes are special cases

of the designs obtained by minimizing these metrics. The first non-orthogonality

metric, termed total-squared-skew-symmetry, is defined with respect to {Ai} (see

Eq. (6.2)), the set of linear processing matrices that is assigned to the real parts of

the transmitting constellations. Similarly, we have non-orthogonality that is defined

with respect to {Bi} (Eq. (6.2)), which are the LPM’s associated with the imaginary

parts. The second metric, termed total-squared-amicability (TSA), involves both sets

of matrices (Ai’s and Bi’s).

We establish that the total-squared-skew-symmetry is a generalized total-squared-

correlation (GTSC). TSC [122], [123] is a measure of non-orthogonality and is com-

monly used in the design of the sequence set for Code Division Multiple Access

(CDMA) systems. We give a lower bound for GTSC that is analogous to that of TSC

i.e., the Welch’s bound. We then discuss the relationship between minimizing GTSC

and the orthogonal Procrustes rotation problem [126] together with the closed form

solution to the problem. Furthermore, the lower bounds for GTSC are established

using the Hurwitz-Radon family of matrices.

117

By computer simulations we give several examples of LP-STBC’s using this

measure. We also demonstrate the relationship of this measure to CLD. We estab-

lish that this measure is less revealing than the first one of the performance of the

final codes. However, a lower bound that is derived here can still indicate well the

performance limit of some code designs.

The rest of the chapter is organized as follows. In Section 6.2 we give the system

model. In Section 6.3 we study Jesen’s relaxation of the LDC mutual information. In

Sections 6.4 and 6.5 we study the GTSC-TSA design criterion and the lower bounds

for the GTSC metric are given in Section 6.5. In Section 6.6 we describe computer

simulations and give several examples of LP-STBC’s. We conclude the work in 6.7.

6.2 System Model and the Linear Processing ST

Coding Scheme

The system we consider is a narrow-band wireless communication system con-

sisting of M transmitting antennas and N receiving antennas. The MIMO channel is

assumed to be flat-fading. The fading is quasi-static with coherence time T channel

uses during which the fades are supposed to be constant, though they may change

from one block of time T to the other. The received signal matrix containing the

received signal vectors during T channel uses is given as

Y =

√ρ

MSH + N . (6.1)

Here Y is of dimension T×N and H of dimension M×N is the channel matrix whose

(i, j)th element [H ]ij denotes the fading coefficient between the ith transmit antenna

and the jth reception antenna. The channel coefficients [H ]ij, i = 1, 2, . . . , M and

j = 1, 2, . . . , N are assumed to be i.i.d. zero mean unit variance circularly symmetric

complex Gaussian random variables i.e., [Hij] ∝ CN (0, 1), E{vec(H)vec(H)∗} =

118

IMN , where “(·)∗” denotes the Hermitian transpose of a matrix and “E(·)” is the

expectation operator. Also S represents the T×M space-time transmit code word, the

entries of which depend on the ST transmission scheme adopted. Finally, N denotes

the matrix of additive Gaussian noise independent of the received signals. It consists

of entries that are CN (0, 1) distributed and is spatially and temporally white. In (6.1)

the baseband transmission S is power constrained such that E{vec(S)∗vec(S)} =

E{tr(SS∗)} ≤ MT . The normalization√

ρM

ensures a receiver side signal-to-noise

ratio per antenna equal to ρ.

The design of an ST transmission scheme can be viewed as constructing a

linear or non-linear mapping from a block of r constellation points (input symbols)

{s1, s2, . . . , sr} to a space-time code word matrix C of dimension T×M , whose (i, j)th

element is transmitted during the ith channel use and at the jth antenna. The linear

dispersion code, or linear processing space-time block code, is given by [112]

C =r∑

i=1

siAi + j

r∑i=1

siBi. (6.2)

Here j :=√−1, while si := Re {si} and si := Im{si} are the real and imaginary part

of si, respectively. The matrices Ai and Bi ∈ Rn×s are real-valued linear processing

(dispersion) matrices of size n× s (with n ≥ s). In this work we consider only LPM’s

that are real and orthogonal i.e., Ai’s and Bi’s satisfy the following conditions:

ATi Ai = AiAT

i = Is, i = 1, 2, . . . , r, (6.3)

and

BTi Bi = BiBT

i = Is, i = 1, 2, . . . , r, (6.4)

where “(·)T ” denotes matrix transposition. Henceforth the dimension of the identity

matrix will be omitted if it can be inferred from the context. In general r, s and n

are parameters that we can choose. The parameter r denotes the number of input

119

information symbols to be transmitted; s is equivalent to the number of transmission

antennas M and n is equivalent to the block length T , both of which are defined in

the system model given by (6.1). From (6.3) and (6.4) and assuming i.i.d. CN (0, 1)

inputs si’s, we see that the power constraint can be satisfied by a normalizing factor

β such that E{tr(βC(βC)∗)} = sn.

6.2.1 The Equivalent MIMO Channel

In this subsection an equivalent MIMO channel model ([112]) to the model of

(6.1) is given. Neglecting the noise matrix N we can rewrite (6.1) as follows:

Y = Re {Y }+ jIm{Y }

=

√ρ

M(

r∑i=1

siAi + j

r∑i=1

siBi)× (H + jH)

=

√ρ

M

r∑i=1

(siAiH − siBiH)

+ j

√ρ

M

r∑i=1

(siAiH + siBiH). (6.5)

In (6.5), the operators (·) and (·) denote the Re {·} and Im{·}, respectively. We

will interchangeably use these two sets of notations. Expressing Re {Y }, Im{Y },Re {H} and Im{H} in vector form:

Re {Y } =(

y1 y2 · · · yN

),

Im{Y } =(

y1 y2 · · · yN

),

Re {H} =(

h1 h2 · · · hN

), and

Im{H} =(

h1 h2 · · · hN

), (6.6)

120

we have:

y1

...

yN

y1

...

yN

=

√ρ

M

A1h1 −B1h1 · · · Arh1 −Brh1

......

. . ....

...

A1hN −B1hN · · · ArhN −BrhN

A1h1 +B1h1 · · · Arh1 +Brh1

......

. . ....

...

A1hN +B1hN · · · ArhN +BrhN

︸ ︷︷ ︸H

s1

s1

s2

s2

...

sr

sr

. (6.7)

The equivalent MIMO channel H ∈ R2NT×2r is the left multiplying matrix that is

on the right hand side (RHS) of Eq. (6.7), and is a function of the original channel

H and the set of linear processing matrices {Ai,Bi, i = 1, 2, . . . , r}. After (6.7)’s

transformation, suboptimal linear decoders can be employed for fast recovery of the

si’s.

The elements of HTH, [HTH]κι, κ = 1, 2, . . . , 2NT and ι = 1, 2, . . . , 2NT , and

their counterparts across the main diagonal (denoted by ↔), [HTH]ικ, are given by:

[HTH]κι =N∑

n=1

hTnAT

i Ajhn +N∑

n=1

hTnAT

i Ajhn ↔

[HTH]ικ =N∑

n=1

hTnAT

j Aihn +N∑

n=1

hTnAT

j Aihn, (6.8)

κ = 2i− 1,ι = 2j − 1, i = 1, 2, . . . , r and j = 1, 2, . . . , r,

[HTH]κι =N∑

n=1

hTnBT

i Bjhn +N∑

n=1

hTnBT

i Bjhn ↔

[HTH]ικ =N∑

n=1

hTnBT

j Bihn +N∑

n=1

hTnBT

j Bihn, (6.9)

κ = 2i,ι = 2j, i = 1, 2, . . . , r and j = 1, 2, . . . , r,

121

[HTH]κι = −N∑

n=1

hTnAT

i Bjhn +N∑

n=1

hTnAT

i Bjhn ↔

[HTH]ικ = −N∑

n=1

hTnBT

j Aihn +N∑

n=1

hTnBT

j Aihn, (6.10)

κ = 2i− 1, ι = 2j, i = 1, 2, . . . , r and j = 1, 2, . . . , r,

and

[HTH]κι = −N∑

n=1

hTnBT

i Ajhn +N∑

n=1

hTnBT

i Ajhn ↔

[HTH]ικ = −N∑

n=1

hTnAT

j Bihn +N∑

n=1

hTnAT

j Bihn, (6.11)

κ = 2i, ι =2j − 1, i = 1, 2, . . . , r and j = 1, 2, . . . , r.

In the next section we discuss a relaxation of the LDC mutual information CLD by

using Jensen’s Inequality.

6.3 Jensen’s Inequality and Relaxation of LDC Mu-

tual Information

As seen from (6.7), the set of linear processing matrices transforms the original

MIMO channel H into an equivalent channel represented by the matrix H. The LDC

mutual information is defined as

CLD :=1

2TEH

{log det

(I2NT +

ρ

MHHT

)}

=1

2TEH

{log det

(I2r +

ρ

MHTH

)}. (6.12)

122

By Jensen’s inequality and the concavity of the log det() function, we have

CLD ≤ 1

2Tlog det

(I2r +

ρ

MEH{HTH}

), (6.13)

where the elements of the product matrixHTH are given by Eqs. (6.8) through (6.11),

respectively. To compute the expectation in the RHS of (6.13), we have to evaluate

three types of quantities existing within HTH. The first type are those quantities

given by (6.8) i.e., {[HTH]κι, κ = 2i−1, ι = 2j−1, i = 1, 2, . . . , r and j = 1, 2, . . . , r}.We note that HTH = HTH+(HTH)T

2, which implies that they can be expressed as

follows:

[HTH]κι =N∑

n=1

hTn

ATi Aj +AT

j Ai

2hn +

N∑n=1

hTn

ATi Aj +AT

j Ai

2hn

=

h1

h2

...

hN

h1

h2

...

hN

T

ATi Aj+AT

j Ai

2∅ . . . ∅

∅ ATi Aj+AT

j Ai

2. . . ∅

......

. . ....

∅ ∅ . . .AT

i Aj+ATj Ai

2

︸ ︷︷ ︸G

h1

h2

...

hN

h1

h2

...

hN

,

κ = 2i− 1, ι = 2j − 1, i = 1, 2, . . . , r and j = 1, 2, . . . , r.

(6.14)

Assuming that H ’s entries are independent and identically distributed (i.i.d.) zero-

mean, variance-one, circularly-symmetric and complex Gaussian random variables,

the characteristic function of (6.14) are given by:

ϕ(jυ) =

∫. . .

∫dh1 . . . dhMdh1 . . . dhNπ−MN

123

· exp[−[hT

1 . . . hTN hT

1 . . . hTN ](I2MN − jυG) · [hT

1 . . . hTN h1 . . . hT

N ]T]

= det(I2MN − jυG)−12 . (6.15)

In (6.15), G denotes the square block-diagonal matrix on the RHS of (6.14), with

diagonal blocks given byAT

i Aj+ATj Ai

2. The last step follows from the general multi-

variate Gaussian integral [112]. If we define K := I − jυG, then the derivative of

ϕ(jυ) with respect to υ is given by:

dϕ(jυ)

dυ= −1

2det(I − jυG)−

32 · d det(I − jυG)

dυ

= −1

2det(I − jυG)−

32 · tr

[(∂ det K

∂K

)T∂(I − jυG)

∂υ

]

= −1

2det(I − jυG)−

32 · tr

[[det K

(K−1

)T]T

(−j)G

]

=1

2j det(I − jυG)−

12 tr[(I − jυG)−1G]. (6.16)

Using (6.16), we can show that the expectations of the quantities given in (6.14),

which are given by:

E{[HTH]κι} = −jdϕ(jυ)

dυ

∣∣∣υ=0

=1

2tr G = N tr

AiATj +AjAT

i

2,

κ = 2i− 1, ι = 2j − 1, i = 1, 2, . . . , r and j = 1, 2, . . . , r. (6.17)

The second type of quantities are those given by (6.9). The computation of the ex-

pected values of these quantities is similar to that of the first type, the only difference

being that these quantities are defined with respect to Bi’s. Thus we have:

E{[HTH]κι = N trBT

i Bj + BTj Bi

2, κ = 2i, ι = 2j, i = 1, 2, . . . , r and j = 1, 2, . . . , r.

(6.18)

124

Finally, the third type of quantities are those given by (6.10) and (6.11). Because the

two are similar, we will only consider one of them e.g., those given by (6.10).

[HTH]κι =N∑

n=1

hTn

BTj Ai −AT

i Bj

2hn +

N∑n=1

hTn

ATi Bj − BT

j Ai

2hn

=[hT

1 hT2 . . . hT

N hT1 hT

2 . . . hTN

]

·

∅ ∅ ∅ BTj Ai−AT

i Bj

2. . . ∅

∅ ∅ ∅ .... . .

...

∅ ∅ ∅ ∅ . . .BT

j Ai−ATi Bj

2

ATi Bj−BT

j Ai

2. . . ∅ ∅ ∅ ∅

.... . .

... ∅ ∅ ∅

∅ . . .AT

i Bj−BTj Ai

2∅ ∅ ∅

h1

h2

...

hN

h1

h2

...

hN

,

κ = 2i− 1, ι = 2j, i = 1, 2, . . . , r and j = 1, 2, . . . , r. (6.19)

The expectations can be computed similarly to the computations of the first type and

we have

E{[HTH]κι} = −1

2tr G′ = 0, κ = 2i− 1, ι = 2j, i = 1, 2, . . . , r and j = 1, 2, . . . , r,

(6.20)

where G′ denotes the square matrix within the RHS of Eq. (6.19). Similarly, we

have:

E{[HTH]κι} = 0, κ = 2i, ι = 2j − 1, i = 1, 2, . . . , r and j = 1, 2, . . . , r. (6.21)

Having obtained the three types of expectations, we can now define a criterion

for choosingAi and Bi, i = 1, 2, . . . , r, which is to search forAis and Bis that maximize

125

the following quantity, derived from CLD by applying Jensen’s Inequality:

% :=1

2Tlog det

(I2r + N

ρ

MZ

), (6.22)

where the matrix Z is defined as follows:

trAT

1 A1+AT1 A1

2∅ tr

AT1 A2+AT

2 A1

2∅ . . .

∅ trBT

1 B1+BT1 B1

2∅ tr

BT1 B2+BT

2 B1

2. . .

trAT

2 A1+AT1 A2

2∅ tr

AT2 A2+AT

2 A2

2∅ . . .

∅ trBT

2 B1+BT1 B2

2∅ tr

BT2 B2+BT

2 B2

2. . .

......

......

. . .

, (6.23)

which has as its elements trAT

i Aj+ATj Ai

2and tr

BTi Bj+BT

j Bi

2, 1 ≤ i, j ≤ r, and zeros

elsewhere.

By computer simulations, we see that there exists an almost monotonic rela-

tionship between Jensen’s relaxation and CLD (see Section 6.6) and their relation is

tractable. Besides, % is a deterministic criterion and, as it does not contain any statis-

tic operators, its computation is fairly easy. Thus we anticipate that its application

in LP-STBC design would greatly reduce the search space of the problem.

6.4 The GTSC and TSA Metrics

In what follows, we discuss another deterministic design criterion by which

we extract an important feature of the linear processing ST codes. In this criterion,

the goal is to find a set of orthogonal matrices Ai and Bi ∈ Rn×s that minimize the

following measure (“‖ · ‖F ” denotes the matrix Frobenius norm):

ψ :=r∑

i=1

r∑j=1

∥∥∥ATi Aj +AT

j Ai

2

∥∥∥2

F+

r∑i=1

r∑j=1

∥∥∥BTi Bj + BT

j Bi

2

∥∥∥2

F

126

+r∑

i=1

r∑j=1

∥∥∥ATi Bj − BT

j Ai

2

∥∥∥2

F. (6.24)

Let us denote the three summing terms of (6.24) as (the notations ψGTSC(B), ψGTSC(B)

and ψTSA will be used later):

ψGTSC(A) :=r∑

i=1

r∑j=1

∥∥∥ATi Aj +AT

j Ai

2

∥∥∥2

F, (6.25)

ψGTSC(B) :=r∑

i=1

r∑j=1

∥∥∥BTi Bj + BT

j Bi

2

∥∥∥2

F, (6.26)

and

ψTSA :=r∑

i=1

r∑j=1

∥∥∥ATi Bj − BT

j Ai

2

∥∥∥2

F. (6.27)

We note that ‖ATi Aj+AT

j Ai

2‖2

F = tr(AT

i Aj+ATj Ai

2

)2

. The measure ψ, together with the

orthogonality requirements (6.3) and (6.4), is a natural generalization of the set of

conditions of the following problem [113]:

Find Ai and Bi ∈ Rn×s such that they satisfy the following system of equations

(cf. (6.28)):

ATi Ai = Is, i = 1, 2, . . . , r

BTi Bi = Is, i = 1, 2, . . . , r

ATi Aj = −AT

j Ai, 1 ≤ i 6= j ≤ r

BTi Bj = −BT

j Bi, 1 ≤ i 6= j ≤ r

ATi Bj = BT

j Ai, 1 ≤ i, j ≤ r. (6.28)

The set of matrices {Ai,Bi, i = 1, 2, . . . , r} satisfying these equations yield the so

127

called amicable orthogonal design [113], which can be viewed as a generalization

of the real orthogonal design [104] [113]. In [104] it is also called complex linear

processing orthogonal design. As can be easily seen, matrices satisfying (6.28) would

yield a minimum measure ψ = 2rs. Thus ψ-minimizing designs include orthogonal

designs (amicable orthogonal designs and real orthogonal designs) as special cases.

In the next section we study the lower bounds for ψGTSC. Our main focus

would be the real quasi-orthogonal design using square matrices (s = n and Ai ∈Rn×n, i = 1, 2, . . . , r).

6.5 Lower Bounds for the GTSC Metric

6.5.1 Bound That is Analogous to Welch’s Bound

There exists an analogy between total-squared-skew-symmetry and total-squared-

correlation (TSC) [123], which is a design criterion for the signature sequence set

of a synchronous Direct Sequence Code Division Multiple Access (DS-CDMA) sys-

tem. To illustrate this let us consider a K-user DS/CDMA system and let us define

L := {s1, s2, . . . , sK} as the signature set, where the signatures si are column vectors

of length equal to the system processing gain L. The TSC is defined as a measure of

the correlations between the signatures si’s:

TSC(L) :=K∑

i=1

K∑j=1

|sTi sj|2 = ‖LT L‖2

F = ‖LLT‖2F = tr

(LT L

)2, (6.29)

where each si is assumed to be normalized i.e., sTi si = 1. Thus minimization of TSC

would yield sequence sets that have the least cross correlation sum. If we denote the

128

transmitted information bit of the ith user as bi, then we may write (cf. (6.31))

(b1s1 + . . . + bKsK)T (b1s1 + . . . + bKsK) =K∑

i=1

b2i ‖si‖2 +

K−1∑i=1

K∑j>i

bibj(sTi sj + sT

j si).

(6.30)

Therefore we may alternatively interpret the TSC-minimization criterion as one whose

objective is to suppress the magnitudes of the pairing sums sTi sj + sT

j si. Complete

suppressing sTi sj + sT

j si = 0 is equivalent to requiring that sTi sj = −sT

j si which,

since sTj si = sT

i sj, can be rewritten as sTi sj = −sT

i sj. This in turn is equivalent to

requiring that sTi sj = 0. This observation is the motivation behind the TSC criterion

that seeks to minimize the magnitudes of (sTi sj)

2.

This alternative viewpoint helps illustrate the analogy between TSC and the

measure ψ. Indeed, in linear processing STBC we write

(s1A1 + · · ·+ srAr)T (s1A1 + · · ·+ srAr)

=r∑

i=1

s2iAT

i Ai +r−1∑i=1

r∑j>i

sisj

(ATi Aj +AT

j Ai

)

=(r∑

i=1

s2i )I +

r−1∑i=1

r∑j>i

sisj

(ATi Aj +AT

j Ai

). (6.31)

While the linear processing orthogonal design seeks to eliminate the second term in

(6.31), it can be generalized as a minimization problem with respect to the metric:

ψ =r∑

i=1

r∑j=1

∥∥∥ATi Aj +AT

j Ai

2

∥∥∥2

F, (6.32)

where the symmetric part of the product matrix ATi Aj (

ATi Aj+AT

j Ai

2) is measured using

the matrix Frobenius norm.

Comparing (6.30) to (6.31) we can establish a one-to-one correspondence be-

tween the DS-CDMA signature sequences si’s and the space-time linear processing

129

matrices Ai’s. Thus the total-squared-skew-symmetry of LP-STBC can be viewed as

a generalization of TSC for DS-CDMA. In the sequel we refer to the former as GTSC.

In what follows, we will establish a lower bound for GTSC similar to that of TSC

(the Welch’s bound [124], [125]).

First, we have the following theorem that describes properties regarding the set

of linear processing matrices {Ai}, which is analogous to the row column equivalence

Lemma given in [125].

Theorem 9 Let ψ[r,s,n] denote the measure ψ with respect to the r n × s matrices.

Let also ψ[s,r,n] denote ψ with respect to the s n× r matrices. We have the following:

(i) ψ[r,s,n] = ψ[s,r,n];

(ii) ψ[r,n,s] = ψ[n,r,s];

(iii) When s = n and ATi Ai = AiAT

i , ψ[r,s,n] = ψ[s,r,n] = ψ[r,n,s] = ψ[n,r,s].

¤

Proof: We only need to provide the proof of the first claim. The proofs of the

other claims follow immediately.

Suppose that the matrices Ai’s are stacked along the z-axis (illustrated in Fig.

6.1), and let Dκ’s denote the matrices that are “sliced” along the x-axis. Let aκ[i]

stand for the κth column of Ai, and di[κ] stand for the ith column of Dκ. It can be

seen that di[κ] = aκ[i]. Thus we have:

ψ[r,s,n] =r∑

i=1

r∑j=1

∥∥∥ATi Aj +AT

j Ai

2

∥∥∥2

F

=r∑

i=1

r∑j=1

s∑κ=1

s∑ι=1

∣∣∣aTκ [i]aι[j] + aT

κ [j]aι[i]

2

∣∣∣2

=s∑

κ=1

s∑ι=1

r∑i=1

r∑j=1

∣∣∣dTi [κ]dj[ι] + dT

j [κ]di[ι]

2

∣∣∣2

=ψ[s,r,n]. (6.33)

130

Figure 6.1: The analogy of the Row Column Equivalence

The following properties will be used in proving the lower bound.

Theorem 10 Given r matrices Ai ∈ Rn×n that are orthogonal i.e., ATi Ai = AiAT

i =

I, i = 1, 2, . . . , r, we have the following inequality

n∑κ=1

√ψ′κκ ≥

r∑i=1

√ψii = r

√n, (6.34)

where ψ′κκ = ‖DTκDκ+DT

κDκ

2‖2

F and ψii = ‖ATi Ai+AT

i Ai

2‖2

F . ¤

Proof: We consider two cases: (i) r ≥ n and (ii) r ≤ n.

(i) For r ≥ n, we have

n∑κ=1

√ψ′κκ :=

n∑κ=1

∥∥DTκDκ

∥∥F

=n∑

κ=1

(r∑

i=1

r∑j=1

∣∣dTi [κ]dj[κ]

∣∣2) 1

2

131

≥n∑

κ=1

(r2

n

) 12

= nr√n

= r√

n

=r∑

i=1

∥∥ATi Ai

∥∥F, (6.35)

where the inequality∑r

i=1

∑rj=1

∣∣dTi [κ]dj[κ]

∣∣2 ≥ r2

ncomes from the Welch’s

bound.

(ii) For r ≤ n, we have

n∑κ=1

√ψ′κκ :=

n∑κ=1

∥∥DTκD

∥∥F

=n∑

κ=1

(r∑

i=1

r∑j=1

∣∣dTi [κ]dj[κ]

∣∣2) 1

2

≥n∑

κ=1

(r∑

i=1

∣∣dTi [κ]di[κ]

∣∣2) 1

2

= n√

r ≥ r√

n. (6.36)

With the above two properties, we can now prove a lower bound of ψ[r,n,n] for

r ≥ n, which is similar to the Welch’s Bound. Indeed,

ψ[r,n,n] = ψ[n,r,n] =n∑

κ=1

n∑ι=1

∥∥∥DTκDι +DT

ι Dκ

2

∥∥∥2

F

≥n∑

κ=1

∥∥∥DTκDκ +DT

κDκ

2

∥∥∥2

F=

n∑κ=1

∥∥DTκDκ

∥∥2

F

≥ 1

n

(n∑

κ=1

∥∥DTκDκ

∥∥F

)2

≥ 1

n

(r√

n)2

. (6.37)

132

The second inequality in (6.37):

n∑κ=1

∥∥DTκDκ

∥∥2

F≥ 1

n

(n∑

κ=1

∥∥DTκDκ

∥∥F

)2

, (6.38)

is a form of Cauchy-Schwarz inequality (also see [125]). It becomes an equality if and

only if∥∥DT

1D1

∥∥F

=∥∥DT

2D2

∥∥F

= . . . =∥∥DT

nDn

∥∥F. (6.39)

This lower bound is achievable by an LPM design of size [r, 2, 2] for r ≥ 2 i.e.,

ψ[r,2,2] ≥ r2 for r ≥ 2. For example, when r = 2, there exists an orthogonal design

which gives ψ[2,2,2] = 4, with∥∥AT

i Ai

∥∥2

F= 2. For r = 3, the minimum ψ[3,2,2] = 9 is

approximately achieved by the following set of matrices:

A1 =

0.2764 −0.9610

0.9610 0.2764

, A2 =

−0.9684 0.2493

−0.2493 −0.9684

and A3 =

−0.6923 −0.7216

0.7216 −0.6923

,

(6.40)

where ψ = 9.0004 and is given by:

Ψ =

2.0000 0.5147 0.5042

0.5147 2.0000 0.4814

0.5042 0.4814 2.0000

, (6.41)

where for a better presentation we use matrix Ψ which is defined by Ψ := [ψij] and

ψij :=∥∥AT

i Aj+ATj Ai

2

∥∥2

F. Graphic representation of {Ai} is given in Fig. 6.2 for a visual

understanding of the matrices obtained from the minimization, where the two arrows

of the same line-type denote the two column vectors, respectively, of the matrices Ai

for i = 1, 2, 3.

This bound is also applicable to LPM design of size [5, 4, 4]. In this case we

have ψ[5,4,4] ≥ 25. By numerical optimization with respect to LPM over the integer

field (and essentially over {−1, 0, 1}), we can obtain a minimum of ψ = 28.

133

0.2

0.4

0.6

0.8

1

30

210

60

240

90

270

120

300

150

330

180 0

Figure 6.2: Graphical representation of {Ai}’s for [3,2,2]

134

6.5.2 Lower Bounds for Other Cases of r and n

The bound given in the previous section is valid for [r, n, n] and r ≥ n. In this

section we take a look at lower bounds for several other cases of r and n. We first

have the following properties regarding a general lower bound.

Theorem 11 Given r n × n matrices Ai ∈ Rn×n such that ATi Ai = AiAT

i = I, i =

1, 2, . . . , r, we have the following inequality:

(i) ψ[r,n,n] ≥ rn, when 1 ≤ r ≤ ρ(n);

(ii)

ψ[r,n,n] ≥ rn + r(r − 1)ψ[ρ(n)+1,n,n] − (ρ(n) + 1)n

ρ(n)(ρ(n) + 1), (6.42)

when r ≥ ρ(n) + 1.

In the above ρ(n) is the Hurwitz-Radon number which is given by ρ(n) = 8c + 2d for

n = 2ab where b is odd and a = 4c + d for 0 ≤ d ≤ 3; a, b, c, d, n ∈ Z+. ¤

Case (i) is trivial as the lower bound can be achieved by the Hurwitz-Radon

family of matrices. Case (ii) states that for r ≥ ρ(n)+1, ψ[r,n,n] can be lower bounded

by the minimal ψ[ρ(n)+1,n,n]. The proof can be deduced by contradiction: suppose we

can find a design ψ[r,n,n] which has a smaller value than that given in (6.42). We are

going to have ψ[ρ(n)+1,n,n] < ψ[ρ(n)+1,n,n] which can not be true.

Thus, in general, the procedure to evaluate a lower bound will be to find

the minimum of ψ[ρ(n)+1,n,n] using optimization algorithms. The numerical solution

obtained provides a lower bound for ψ[r,n,n] for r ≥ n.

In the following section we will study several cases where the minimum of

ψ[n+1,n,n] can be evaluated analytically.

The first case is when n is odd i.e., n4= 1 (mod 2). In this case ρ(n) is 1

and ψ[r,n,n] can be lower-bounded by the minimum skew-symmetry incurred by any

ρ(n) + 1 = 2 linear processing matrices. We have the following theorem about the

minimum skew-symmetry between any two LPM’s and the lower bound on ψ[r,n,n]:

135

Theorem 12 Given n × n matrices Ai that satisfy (6.3) and n4= 1 (mod 2), we

have: (i)∥∥AT

i Aj+ATj Ai

2

∥∥2

F≥ 1 for any 1 ≤ i 6= j ≤ r; (ii) ψ[r,n,n] ≥ rn + r(r − 1). ¤

Proof: We first observe that

∥∥∥ATi Aj +AT

j Ai

2

∥∥∥2

F=

∥∥∥I +(AT

i Aj

)2

2

∥∥∥2

F, (6.43)

since the Frobenius norm is invariant under unitary multiplication. The determinant

of(AT

i Aj

)2is given by

det(AT

i Aj

)2=

(detAT

i Aj

)2(6.44)

and is positive. Thus the orthogonal matrix(AT

i Aj

)2is special orthogonal1. An

orthogonal matrix can only have eigenvalues from the set {1,−1, ejφ, e−jφ} for some

φ ([126]), and since {−1,−1} and {ejφ, e−jφ} must occur in pairs because(AT

i Aj

)2

is special orthogonal, there must exist a real eigenvalue λ = 1 for n4= 1 (mod 2).

Therefore, ∥∥∥ATi Aj +AT

j Ai

2

∥∥∥2

F≥ 1. (6.45)

By applying Theorem 11, we then have ψ[r,n,n] ≥ rn + r(r − 1).

The equality of (6.45) can be attained by explicit construction of the following or-

thogonal matrices:

A1 =

1 0T

0 V1

and A2 =

1 0T

0 V2

, (6.46)

where V1 and V2 ∈ R(n−1)×(n−1) are two matrices from the Hurwitz-Radon family of

matrices of order n− 1. Note that ρ(n− 1) ≥ 2 for n− 14= 0 (mod 2) ([113]), so it

is always possible to have two such matrices.

This problem of finding ATi Aj of minimum skew-symmetry can be cast as an

orthogonal Procrustes problem [126]. Let F ∈ Rm×n and G ∈ Rm×n, m ≥ n, denote

1An orthogonal matrix X is said to be special orthogonal if detX = 1 [126].

136

two sets of n points within an m-dimensional space. In the Procrustes problem, the

goal is to seek an orthogonal rotation matrix Q ∈ Rn×n such that the following is

minimized:∥∥F − GQ

∥∥2

F. (6.47)

When m = n, the Procrustes rotation matrix Q can be expressed analytically:

Qmin = UV T , (6.48)

where U and V are obtained from the singular value decomposition (SVD) of the

inner product of G and F as follows:

GTF = USV T . (6.49)

The following theorem describes the relations between the problem of finding

ATi Aj of minimum skew-symmetry and orthogonal Procrustes problem:

Theorem 13 If we choose F and G such that F is a skew-symmetric matrix of rank

n − 1 for n4= 1 (mod 2) or n for n

4= 0 (mod 2), and G = I, then minimization of

(6.47), rewritten as follows:∥∥F −Q

∥∥2

F, (6.50)

is equivalent to minimization of the skew-symmetry measure

∥∥Qmin +QTmin

∥∥2

F. (6.51)

¤

Noting that Qmin = UV T (cf. (6.48)), we see that the solution to the Procrustes

problem yields two orthogonal matrices U and V that minimize the measure of

skew-symmetry given by ∥∥∥UV T + V T U

2

∥∥∥2

F. (6.52)

137

This is of the exact same form as∥∥AT

i Aj+ATj Ai

2

∥∥2

F. This suggests that for the cases

of n4= 1 (mod 2), the problem can be solved analytically by the solution to the

Procrustes rotation problem as outlined above.

Before giving the proof for Theorem 13, we need first the following theorem:

Theorem 14 For given skew-symmetric matrices M and N ∈ Rn×n and for a uni-

tary matrix P ∈ Cn×n we have:

minP∗P=I

trMP∗NP =n∑

i=1

λiσi, (6.53)

where λi and σi are the eigenvalues of M and N , respectively, sorted in the order of

decreasing imaginary parts e.g., {λ1 = 2j, λ2 = j, λ3 = 0j, λ3 = −j, λ4 = −2j}. Note

that the eigenvalues of skew-symmetric matrix are either pure imaginary or zero, and

they appear in conjugate pairs. ¤

Proof: (This proof follows that of [127].) Suppose that the eigenvalue decom-

position of M and N are given by M = UΛU ∗ and N = V ΣV ∗. We write

trMP∗NP = tr UΛU ∗P∗V ΣV ∗P= trΛ (U ∗P∗V )Σ (V ∗PU ) . (6.54)

Letting W := U ∗P∗V we then have

trΛ (U ∗P∗V )Σ (V ∗PU ) =n∑

i=1

λiW∗ΣW

=n∑

i=1

n∑j=1

∣∣[W ]ij∣∣2λiσj

=n∑

i=1

n∑j=1

[uvT ◦ F ]ij, (6.55)

where u := diag(U ) and v := diag(V ) are column vectors consisting of the eigen-

values of M and N , respectively, arranged in the order described previously. The

138

operator “◦” denotes the element-wise matrix Hadamard product. The matrix F :=

[|[W ]ij|2] is doubly stochastic as each of its columns and rows add up to one i.e.,

n∑i=1

∣∣[W ]ij∣∣2 = 1, for j = 1, 2, . . . , n (6.56)

andn∑

j=1

∣∣[W ]ij∣∣2 = 1, for i = 1, 2, . . . , n. (6.57)

This is due to the fact that W is unitary. We will now show that F = In is a

minimizer of (6.55).

Suppose we have k ∈ {1, 2, . . . , n} such that

∣∣[W ]kk

∣∣2 < 1,∣∣[W ]ii

∣∣2 = 1, for i < k,∣∣[W ]ij

∣∣2 = 0, for 1 ≤ i 6= j < k. (6.58)

Then by the properties of doubly stochastic matrices, we have that for some k < p ≤ n

and k < q ≤ n:

∣∣[W ]kq

∣∣2 > 0,∣∣[W ]pk

∣∣2 > 0, and∣∣[W ]pq

∣∣2 < 1. (6.59)

Given some 0 < ε < 1, we can construct a new doubly stochastic matrix F ′, which

has the following entries updated (denoted by →) from F :

∣∣[W ]kk

∣∣2 →∣∣[W ]kk

∣∣2 + ε,∣∣[W ]kq

∣∣2 →∣∣[W ]kq

∣∣2 − ε,∣∣[W ]pk

∣∣2 →∣∣[W ]pk

∣∣2 − ε,∣∣[W ]pq

∣∣2 →∣∣[W ]pq

∣∣2 + ε. (6.60)

139

We then have:

n∑i=1

n∑j=1

[uvT ◦ F ′]ij −n∑

i=1

n∑j=1

[uvT ◦ F ]ij

=n∑

i=1

n∑j=1

[uvT ◦ F ′ − uvT ◦ F ]ij

=n∑

i=1

n∑j=1

[uvT ◦ (F ′ − F )]ij

= ελkσk − ελkσq − ελpσk + ελpσq

= ε (λk − λp) (σk − σq) ≤ 0. (6.61)

The last inequality comes from the fact that the eigenvalues are so arranged that

λk − λp = aj and σk − σq = bj (k < p ≤ n and k < q ≤ n) for some nonnegative

a, b ∈ R. This updating can be performed for every∣∣[W ]ii

∣∣2 < 1 and in the end we

would have a Fopt = I where∣∣[W ]ii

∣∣2 = 1, i = 1, 2, . . . , n. This is achievable since W

is unitary and the set of unitary matrices is closed under multiplication.

In the next we give the proof for Theorem 13.

Proof: We will first show that:

minQTQ=In

∥∥F −Q∥∥2

F= min

QTQ=In

trFQ−QT

2. (6.62)

We first see that:

∥∥F −Q∥∥2

F=

∥∥∥F − Q+QT

2− Q−QT

2

∥∥∥2

F

=∥∥∥Q+QT

2

∥∥∥2

F+

∥∥∥F − Q−QT

2

∥∥∥2

F− 2 tr

Q+QT

2

(F − Q−QT

2

)

=∥∥∥Q+QT

2

∥∥∥2

F+

∥∥∥F − Q−QT

2

∥∥∥2

F, (6.63)

where in the last line the first term contains the symmetric part of F − Q and

the second term contains the skew-symmetric part. The last step follows from the

140

fact that for any symmetric matrix A and skew-symmetric matrix B of the same

dimension, tr AB = 0. We then have:

∥∥∥Q+QT

2

∥∥∥2

F+

∥∥∥F − Q−QT

2

∥∥∥2

F

=∥∥∥Q+QT

2

∥∥∥2

F+

∥∥F∥∥2

F+

∥∥∥Q−QT

2

∥∥∥2

F− 2 trFT Q−QT

2

=∥∥∥Q+QT

2

∥∥∥2

F+

∥∥F∥∥2

F+

∥∥∥Q−QT

2

∥∥∥2

F+ 2 trFQ−Q

T

2. (6.64)

Since∥∥Q+QT

2

∥∥2

F+

∥∥Q−QT

2

∥∥2

Fis constant for any orthogonal matrix Q, the minimiza-

tion of∥∥F −Q∥∥2

Fwith respect to Q is equivalent to the minimization of trF Q−QT

2.

Regarding the minimum of trF Q−QT

2we have the following. From Theorem 14 we

see that for skew-symmetric F and Q−QT

2, trF Q−QT

2has a lower bound

∑ni=1 λiσi,

where {λi} and {σi} are the eigenvalues of F and Q−QT

2respectively, sorted by de-

creasing imaginary parts. As F is fixed, we anticipate that the minimization of

trF Q−QT

2with respect to Q would yield {σi} adjusted according to {λi}. Note

that F must have the largest possible rank i.e., n − 1 for n4= 1 (mod 2) and n for

n4= 0 (mod 2), otherwise the minimization would not be meaningful e.g., if F is an

all-zero matrix. Furthermore, the minimization will maximize the total sum of the

magnitudes of σi’s, that is,∥∥Q−QT

2

∥∥2

F=

∑ni=1 |σi|2 will be maximized. Now since

∥∥Q−QT

2

∥∥2

F+

∥∥Q+QT

2

∥∥2

F=

∥∥Q∥∥2

F= n, this is equivalent to requiring that

∥∥Q+QT

2

∥∥2

Fbe

minimized, as claimed.

6.5.3 Lower Bounds - Further Results

Minimization of the skew-symmetry measure would require the minimization

of ∥∥∥I +Q2

2

∥∥∥2

F, (6.65)

where Q is orthogonal and Q ∈ Rn×n. Note that since Q is orthogonal, Q2 is also

orthogonal. Further, det (Q2) = 1. Thus Q2 belongs to the so-called special or-

141

thogonal group2 so(n). As discussed before, the eigenvalues of Q2, assumed to be

λi, i = 1, 2 . . . , n, are within the set {1,−1, ejφ, e−jφ}. In addition, except for n4= 1

(mod 2), where there exists one eigenvalue λi = 1, for all other cases, the eigenval-

ues must exist in pairs. So, there are either two eigenvalues such that λi = λj = 1

(i 6= j), two eigenvalues such that λi = λj = −1 or tow eigenvalues such that λi = ejφ

and λj = e−jφ (i 6= j). This eigenvalue pattern is closely related to our analysis of

the lower bounds for the skew-symmetry measure ψ. For example, a study of the

Hurwitz-Radon family of matrices would reveal that for two matrices Ai and Aj from

this family, the eigenvalues of ATi Aj would all be −1, such that

∥∥ I+Q2

2

∥∥2

F= 0. With

this in mind, we are able to derive the lower bound for the case of n4= 2 (mod 4).

Theorem 15 For n4= 2 (mod 4), we have:

ψ[r,n,n] ≥ rn + r(r − 1)12

(3√

2)2 − 3× 2

3× 2. (6.66)

¤

Proof: Given n4= 2 (mod 4), we let r = ρ(n) + 1 = 3. Now since there do not

exist three orthogonal matrices such that Ψ is given by

Ψ =

n 0 0

0 n 0

0 0 n

, (6.67)

we can not find three matrices A1, A2 and A3 such that each pair ATi Aj is skew-

symmetric. This means that the eigenvalues of each(AT

i Aj

)2(i 6= j) would not

possibly be all −1. So there must exist one(AT

i Aj

)2such that two of its eigenvalues,

λk and λl, are λk = λl = 1, or λk = ejφ and λl = e−jφ. For n4= 2 (mod 4), we have

ρ(n) = 2, thus there are at least two distinct pairs(AT

i1Aj1

)2(i1 6= j1) and

(ATi2Aj2

)2

2The special orthogonal matrices of size n× n form a special orthogonal group.

142

(i2 6= j2), each of which will have eigenvalues that is of the pattern as described

above. Thus a lower bound can be given by explicit constructing the following three

matrices:

A1 =

U1 0T

0 V1

, A2 =

U2 0T

0 V2

and A3 =

U3 0T

0 V3

, (6.68)

where 0 denotes all-zero matrix of size (n−2)×2; Ui, for i = 1, 2, 3, are 2×2 orthogonal

matrices, and Vi, i = 1, 2, 3, are orthogonal matrices of dimension n−24= 0 (mod 4).

Note that since ρ(n− 2) ≥ 4, for A1, A2 and A3, we can find Vi, i = 1, 2, 3 from the

Hurwitz-Radon family of order n − 2. We find U1, U2 and U3 such that the set of

Ui’s has the minimum skew-symmetry measure. The minimum ψ of the set of Ui’s is

given by the generalized Welch’s bound 1n

(r√

n)2

(cf. Eq. (6.37)). Thus we have the

following lower bound for n4= 2 (mod 4) from Theorem 11:

ψ[r,n,n] ≥ rn + r(r − 1)12

(3√

2)2 − 3× 2

3× 2. (6.69)

6.6 Computer Simulations

6.6.1 Jensen’s Relaxation of LDC-MI

In this subsection, we study the relationship between the LDC mutual infor-

mation and its Jensen’s relaxation %. We consider an LPM design of size 2× [3, 3, 3]

(“2×” within 2×[3, 3, 3] denotes a complex design). The number of receiving antennas

is 2. The system SNR is set to ρ = 20dB. The MIMO channel H consists of entries

that are independent and identically distributed (i.i.d.) zero-mean, variance-one,

circularly-symmetric and complex Gaussian random variables. The linear processing

matrices are generated randomly. After the raw data is collected, the final data points

143

5.5 6 6.5 7 7.5 85.4

5.6

5.8

6

6.2

6.4

6.6

6.8

7

7.2

7.4

LDC Mutual Information vs. 1/(2T)logdet(I+Nρ/MZ)(2×[3,3,3]: 3 complex symbols, 3x3 LPM’s; N=2; SNR=20dB)

1/(2T)logdet(I+Nρ/MZ)

LDC

−M

I

Figure 6.3: LDC-MI vs. 12T

log det(I2r + M ρNZ) for 3I2O: SNR=20dB; r = 3; pro-

duced from 105 randomly generated LPM’s.

are produced in the following way: for each fixed interval within the total range of

%, we use the largest LDC-MI value obtainable by the generated LPM’s. The result

is plotted in Fig. 6.3. We observe that the curve of LDC-MI versus % is almost

monotonic. Thus there is a tractable connection between the two measures (at least

for the LPM design under investigation).

6.6.2 Examples of LP-STBCs: Constellation Rotation and

Product Distance Gain

In this subsection we examine several examples of LPM designs based on com-

puter simulations.

By numerical optimization, we can show that for Ai,Bi ∈ Z4×4, i = 1, 2, . . . , 4,

144

the minimum of GTSC-TSA is given by ψmin = 48. We can see that each LPM set

associated with the quasi-orthogonal STBC’s as described in [130] and [131] exhibits

a GTSC-TSA equal to ψmin.

For an LPM design over R, there exist sets {Ai} and {Bi} (i = 1, 2, . . . , 4) that

yield ψ of smaller value. For example, the GTSC-TSA value of the following linear

processing matrices is ψ = 43.639.

A1 =

−0.0000 0.9005 −0.0371 0.4332

−0.9105 0.0961 0.3651 −0.1686

−0.3081 −0.4028 −0.2871 0.8127

0.2758 −0.1326 0.8848 0.3514

, (6.70)

A2 =

−0.0000 0.1306 0.5138 −0.8479

−0.0149 −0.1251 −0.8397 −0.5282

−0.1010 0.9787 −0.1730 0.0460

−0.9948 −0.0975 0.0302 0.0033

, (6.71)

A3 =

−0.1773 0.3907 0.8544 0.2930

−0.2390 −0.1000 0.3098 −0.9148

−0.9474 0.0653 −0.2768 0.1466

−0.1176 −0.9127 0.3120 0.2361

, (6.72)

A4 =

−0.4678 −0.5800 0.5099 0.4299

0.6680 −0.3823 0.5061 −0.3891

−0.5461 −0.2118 −0.0605 −0.8082

0.1914 −0.6875 −0.6929 0.1027

, (6.73)

145

and

B1 =

−0.0000 −0.2127 −0.8874 0.4091

0.5022 −0.4101 −0.2355 −0.7240

0.8579 0.3437 0.0889 0.3715

0.1088 −0.8176 0.3863 0.4129

, (6.74)

B2 =

−0.2275 −0.8650 −0.0725 0.4414

0.9724 −0.1919 0.0286 0.1299

−0.0218 0.4637 −0.1289 0.8763

−0.0476 0.0026 0.9886 0.1429

, (6.75)

B3 =

−0.0376 −0.6019 0.0389 −0.7967

0.0501 −0.0960 0.9870 0.1184

−0.1601 −0.7794 −0.1384 0.5896

−0.9851 0.1448 0.0712 −0.0594

, (6.76)

B4 =

−0.5155 −0.1848 0.7256 0.4166

0.2485 0.0303 −0.3369 0.9077

−0.7461 0.5184 −0.4166 0.0323

0.3404 0.8344 0.4318 0.0392

. (6.77)

With {Ai} and {Bi} given as above and by employing a rotated QAM con-

stellation for s4 (rotated by ej2/9π) we obtain the first example of LP-STBC. Its

FER-vs-SNR (Frame Error Rate, the rate that there is an error in T channel use)

curve is plotted against the performance of the QOSTBC (QAM) scheme in Fig.

6.4. In the figure “J-P-F scheme” (short for Jafarkhani-Papadias-Foschini) denotes

QO-STBC’s, and “2×” within 2× [4, 4, 4] denotes a complex design.

The above design involves an independent determination of the Ai’s and Bi’s

with respect to the GTSC-TSA metric. The design procedure is:

(i) choose a superset of LPM sets that have a certain range of ψGTSC +ψTSA values;

146

5 10 15 2010

−6

10−5

10−4

10−3

10−2

10−1

100

A LP−STBC Design with ψ≈43 and Optimally Rotated 4QAM

SNR (dB)

Pro

babi

lity

of F

ram

e E

rror

J−P−F Scheme2x[4,4,4] LP−STBC,ψ≈43,Rotated 4QAM

Figure 6.4: A STBC obtained by 2 × [4, 4, 4] design of ψ = 43.639 and optimallyrotated QAM.

147

(ii) design suitable scalar signal constellations and optimize the performance of the

final code over the LPM superset.

In practice there are several performance criteria we can adopt for the second step:

diversity (rank), product distance gain and obtainable channel capacity, etc. The

following is another example of the above procedure. Focusing on LPM designs with

ψ ≤ 48, which is the GTSC-TSA value of QOSTBC’s LPM set, and assuming that the

input symbols use QAM signalling, we specifically search for LPM’s that maximize

the product distance gain [102] of the final code. The {Ai} and {Bi} are given as

below: The FER of the final code is plotted in Fig. 6.5. We observe from simulations

that for many candidate LPM sets, their combinations with QAM signaling give full

transmit diversity and that it is the product distance gain that is more decisive in

the final choice.

The last example has its complete space-time constellation given as follows

C =

s1 s2 s3 s4

−s2 −s1 −s∗4 s∗3

−s3 s∗4 −s1 −s∗2

−s4 −s∗3 s∗2 −s1

. (6.78)

Its LPM set has a non-orthogonality measure of ψGTSC + ψTSA = 68, which is larger

than that of QOSTBC. Using QAM signalling, we compare its performance to that

of QOSTBC (QAM). The FER-vs-SNR curves of the two schemes are given in Fig.

6.6.

As can be seen from the above examples the procedure is less tractable than

designing LP-STBC by the first measure/criterion. This is also illustrated by the fol-

lowing subsection showing the relationship between GTSC-TSA and the LDC mutual

information criterion. However, we still find that the GTSC metric be an indicator

of the performance of the final code in some real designs.

148

5 10 15 2010

−5

10−4

10−3

10−2

10−1

100

A LP−STBC Design with ψ<48 and Maximized wrt Product Distance Gain

SNR (dB)

Pro

babi

lity

of F

ram

e E

rror

J−P−F Scheme2x[4,4,4] LP−STBC,ψ<48,4QAM

Figure 6.5: A LP-STBC obtained by the two-step design procedure: firstly obtaina set of LPM designs with ψ ≤ 48; then use QAM signalling and maximizes withrespect to the product distance gain.

149

5 10 15 2010

−5

10−4

10−3

10−2

10−1

100

A LP−STBC Design with ψ=68

SNR (dB)

Pro

babi

lity

of F

ram

e E

rror

J−P−F Scheme2x[4,4,4] LP−STBC,ψ=68,4QAM

Figure 6.6: A STBC obtained by 2× [4, 4, 4] design of ψ = 68 and QAM constellation;versus QOSTBC (QAM).

150

9 10 11 12 13 14 15 16 17 185

5.5

6

6.5

7

7.5

8

8.5

9

9.5

10

LDC Mutual Information vs. GTSC Metric([3,2,2]: 3 real symbols, 2x2 LPM’s; N=2; SNR=20dB)

ψGTSC

LDC

−M

I

Figure 6.7: LDC-MI vs. GTSC metric for 2I2O: SNR=20dB; r = 3; produced from105 randomly generated LPM’s.

151

6.6.3 GTSC-TSA Metric and the LDC Mutual Information

Criterion

In this subsection we study the relationship between the GTSC-TSA metric

and the LDC mutual information. In Fig. 6.7, we plot out the LDC mutual infor-

mation of randomly generated LPM sets of size [3, 2, 2] versus their GTSC measure.

The [3, 2, 2] LPM design corresponds to a 2I2O system with 3 input symbols (r = 3).

The SNR in dB used to calculate the LDC-MI is 20. The plot of Fig. 6.7 is obtained

as follows: for small increase of the ψGTSC, the set of LPM’s that maximizes the

LDC-MI measure is determined and the corresponding LDC-MI value is plotted. The

data are collected with a step size 0.22 for ψGTSC. We see that in the plot there exists

an abrupt jump of LDC-MI in the neighborhood of ψGTSC = 14. LDC-MI actually

achieves its maximum around ψGTSC = 14, which suggests that by focusing on LPM’s

with ψGTSC about the value of 14, we are able to obtain good LPM sets in terms of

LDC-MI.

In Fig. 6.8 we plot the curve of LDC-MI-versus-GTSC for a 3I2O system and

LPM design of size [3, 3, 3]. The SNR is also 20dB. In this case, the LDC-MI measure

is roughly monotonic with respect to the GTSC metric. By focusing on LPM’s with

smaller GTSC we obtain good STBC’s in terms of LDC-MI. The lower bounds given

in the theoretical analysis help us understand this region of GTSC. Since GTSC is a

deterministic metric and involves no statistical operator the design of LPM’s in this

case can be made easier by using GTSC.

We note that the matrix Z (cf. (6.23)) in some way shows the difference and

connection between the GTSC-TSA measure and the LDC mutual information. As a

natural extension of the complex linear processing orthogonal design a measurement

of the symmetry of ATi Bj is included, i.e. the TSA term

∑ri=1

∑rj=1

∥∥∥ATi Bj−BT

j Ai

2

∥∥∥2

Fis

calculated, while in (6.22), we find there are no similar term(s).

152

14 15 16 17 18 19 20 21 22 23 24 25 26 27 286.5

6.6

6.7

6.8

6.9

7

7.1

7.2

7.3

LDC Mutual Information vs. GTSC Metric([3,3,3]: 3 real symbols, 3x3 LPM’s; N=2; SNR=20dB)

ψGTSC

LDC

−M

I

Figure 6.8: LDC-MI vs. GTSC metric for 3I2O: SNR=20dB; r = 3; produced from105 randomly generated LPM’s.

153

6.7 Conclusion

We investigated two deterministic measures for the design of linear processing

space-time block codes (LP-STBC’s). The measures are deterministic in the sense

that their computations do not involve any statistical operators and are defined solely

with respect to the set of LPM’s. The first measure is obtained by applying Jensen’s

Inequality to the mutual information criterion for linear dispersion codes. The expec-

tation operator is moved into the log det() operator following Jensen’s rule. By as-

suming channel coefficients that are i.i.d. Gaussian, we compute the expectations and

this gives a deterministic design measure. We show that there is a tractable relation-

ship between this measure and CLD and show that the design of LP-STBC using this

relationship can be simplified. The second measure is a natural extension of the condi-

tions required for complex linear processing orthogonal design or amicable orthogonal

design. For the LPM’s of an LP-STBC, we associate with them two measures of non-

orthogonality: total-squared-skew-symmetry and total-squared-amicability (TSA).

The relationship of total-squared-skew-symmetry to total-squared-correlation (TSC)

is revealed. TSC measures the non-orthogonality (cross-correlation) properties of a

vector set, and is commonly used in the design of sequence sets for Code Division Mul-

tiple Access (CDMA) systems. It can be shown that total-squared-skew-symmetry

is a generalization of total-squared-correlation (GTSC). For GTSC a lower bound

analogous to Welch’s lower bound for TSC exists, which establishes itself upon the

Hurwitz-Radon numbers and the Hurwitz-Radon family of matrices. By computer

simulations, we observe that the second measure is less tractable than the first one.

However, the lower bound derived can still be a good indicator of the performance of

real designs of size 3× 3. Comparing the two deterministic measures reveals to some

extend the differences and connections between CLD and the criterion for amicable

orthogonal design.

154

Chapter 7

Conclusion, Discussion, and Future

Work

In this thesis we have presented the results of several research works, conducted

within a unified theme of multiple-antenna technology.

7.1 Spatial Smoothing Based JADE-MUSIC

In Chapter 3 we considered the problem of joint estimation of direction-of-

arrival (DoA), propagation delay, and complex channel gain for antenna-array DS/CDMA

communications over frequency selective multipath channels and proposed a subspace

based MUSIC-type estimation algorithm which utilizes the spatial smoothing prepro-

cessing technique. The proposed algorithm essentially breaks the multipath induced

coherency within the received signals and recovers the full signal subspace spanned

by all dominant signal paths of all users. This allows for the use of MUSIC-type DoA

and delay estimators for the individual paths of the user of interest. Based on the

angle and timing information, we then estimated the multipath fading coefficients.

Simulation results illustrated the effectiveness of this approach. We further consid-

ered two variants of the proposed spatial smoothing based MUSIC-type estimation

155

scheme. The proposed algorithms utilize space-time received vectors that span only a

single information symbol period and exhibited superior performance when the data

record size available for parameter estimation is limited.

7.2 The MUSIC-MDL Criterion

In Chapter 4 we described a new criterion for detecting the number of sig-

nals impinging on uniform linear array (ULA). Our criterion made explicit use of the

peak information of the MUSIC spectrum. We considered two maximum likelihood

estimates (MLEs) of the noise variance, σ2: the MLE that is derived from the eigen-

value decomposition (EVD) parameterization and the MLE that is based upon the

direction-of-arrival (DoA) parameterization. Based upon a large-sample formulation

of the difference between these two MLEs, and by applying the minimum description

length (MDL) principle, we obtained the proposed criterion. For each hypothesis of

k sources, in addition to computing σ2 using the M − k smallest eigenvalues of the

sample covariance matrix, the new criterion applies an additional correction term

calculated from the k largest peaks of the MUSIC spectrum, which is generated from

the testing noise subspace of dimension M−k. We proved that the proposed criterion

provides a consistent estimate of the number of signals and demonstrate that it has

a better performance at low SNR for equal-power sources when compared with the

original MDL-based signal number detection criterion [62].

7.3 The IWMA Algorithm

In Chapter 5 we presented an iterative weight matrix approximation (IWMA)

algorithm which is capable of obtaining an approximate to the optimal weight matrix

Wopt in an iterative fashion for performing weighted spatial smoothing to obtain

diagonal source covariance matrix for array signal processing. WSS was proposed

156

in [87] as a technique to obtain diagonal source covariance matrix for array signal

processing. Diagonal source covariance matrix is a desired feature for subspace-based

direction-of-arrival (DoA) estimation algorithms, as the cross-correlations among the

input signals can markedly deteriorate the performance of these estimators. But

the optimum weight matrix [87] for such a purpose requires explicit knowledge of

the DoAs. The algorithm is applicable when the input covariance matrix is positive

definite. The algorithm starts from a scaled identity matrix as an initial guess of

Wopt. It then carries out a series of weighted spatial smoothing iteratively. Each

WSS is performed using a weight matrix obtained from previous iteration. After

each WSS the algorithm computes a new weight matrix, which is to be used for

the next iteration. The principle of the algorithm is in its utilization of an effective

correlation matrix, which is naturally brought about by the operations performed in

each iteration, and the fact that for a positive definite Hermitian matrix, the set of

eigenvalues of its Hadamard product with a correlation matrix is majorized by the set

of eigenvalues of itself. WSS based on IWMA can be shown to be an effective method

to decorrelate highly correlated signals. Besides this, a useful observation regarding

the IWMA algorithm is that the approximate matrix it generates is suited as a basis

for subspace-type DoA estimation. Simulation results illustrated the effectiveness of

this estimation strategy, which also suggests the effectiveness of the IWMA algorithm.

Some consideration of the future work for this research will be given in a later section.

7.4 Two Deterministic Design Criteria for LP-STBC

In Chapter 6 we discussed two deterministic measures for designing linear pro-

cessing space-time block codes (LP-STBC’s). The measures are deterministic in the

sense that their computations do not involve any statistical operators and are defined

solely with respect to the set of LPM’s. The first measure is obtained by applying

Jensen’s Inequality to the mutual information criterion for linear dispersion codes

157

[112] (CLD). The expectation operator is moved into the log det() operator following

Jensen’s rule. By assuming channel coefficients that are independent and identically

distributed (i.i.d.) Gaussian, we computed the expectations and this gives a deter-

ministic design measure. We showed that there is a tractable relationship between

this measure and CLD and showed that the design of LP-STBC using this relation-

ship can be simplified. The second measure is a natural extension of the conditions

required for complex linear processing orthogonal design or amicable orthogonal de-

sign. For the LPM’s of a LP-STBC, we associated with them two metrics of non-

orthogonality: total-squared-skew-symmetry and total-squared-amicability (TSA).

The relationship of total-squared-skew-symmetry to total-squared-correlation (TSC)

was revealed. TSC measures the non-orthogonality (cross-correlation) properties of a

vector set, and is commonly used in the design of sequence sets for Code Division Mul-

tiple Access (CDMA) systems. It can be shown that total-squared-skew-symmetry

is a generalization of total-squared-correlation (GTSC). For GTSC a lower bound

analogous to Welch’s lower bound for TSC exists, which establishes itself upon the

Hurwitz-Radon numbers and the Hurwitz-Radon family of matrices. By computer

simulations, we established that the second measure is less tractable than the first

one. However, the lower bound derived can still be a good indicator of the perfor-

mance of real design of size 3×3. Comparing the two deterministic measures to some

extend revealed the differences and connections between CLD and the criterion for

amicable orthogonal design.

7.5 Future Work

In this section we describe from our understanding some possible future work

for the study of the IWMA algorithm.

Within the thesis, the initial weight matrix for the IWMA algorithm is chosen

to be I. The different choices for the initial weight matrix can be further investigated

158

in order to have a better understanding of how these choices will have an impact on

the behavior of the algorithm. A generalization of the currently developed theory to

include also generic initial weight matrices can be carried out.

Another important aspect of the IWMA algorithm is its statistical character-

istics. Further studies in this direction can be carried out in order for us to have a

better understanding of the algorithm’s performance in different sample sizes. With

solid mathematical formulations for the algorithm’s statistical properties, we will have

a thorough understanding of the advantages and shortcomings of the algorithm, and

with this understanding we may come up with improvements to the algorithm and

new algorithms/schemes that perform better.

The computational complexity of an algorithm is an important characteristic

and future efforts are needed in order to completely understand the computational

requirements of the IWMA algorithm.

Finally, we are also very interested in knowing what other kinds of applications

this algorithm can have. The algorithm is quite self-contained and complete by itself,

and because of this it might have other usages in other areas different from array

signal processing.

159

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