Thesis Adaptive Antennas For CDMA WIRELESS Networks

ADAPTIVE ANTENNAS FOR CDMA WIRELESSNETWORKS

A DISSERTATION

SUBMITTED TO THE DEPARTMENT OF ELECTRICAL ENGINEERING

AND THE COMMITTEE ON GRADUATE STUDIES

OF STANFORD UNIVERSITY

IN PARTIAL FULFILLMENT OF THE REQUIREMENTS

FOR THE DEGREE OF

DOCTOR OF PHILOSOPHY

By

Ayman F. Naguib

August 1996

c Copyright by Ayman F. Naguib 1996

All Rights Reserved

ii

I certify that I have read this thesis and that in my opinion it

is fully adequate, in scope and in quality, as a dissertation for

the degree of Doctor of Philosophy.

Arogyaswami Paulraj(Principal Adviser)




Thomas Kailath




Donald Cox

Approved for the University Committee on Graduate Stud-

ies:

iii

Abstract

Wireless cellular communication is witnessing a rapid growth in markets, technology, and

range of services. A major current thrust for cellular communication systems is improved

economics through enhanced coverage early in the life cycle of a network and high spectrum

efficiency later in the life cycle. An attractive approach for economical, spectrally efficient,

and high quality digital cellular and personal communication services (PCS) is the use of

spread spectrum modulation with code division multiple access (CDMA) technology. Yet

another very promising dimension for improving performance of all types of cellular net-

works is the use of antenna arrays at the base station. This thesis explores techniques for

integrating these two dimensions.

We study the use of multiple antennas at the base station and the associated advanced

signal processing in CDMA wireless networks. Our focus is on the mobile to base or re-

verse link. We begin with a space-time cellular channel model for spread spectrum net-

works. We then describe the appropriate signal and interference models. We then propose

a ”Beamformer-RAKE” receiver structure for exploiting multiple antennas at the base sta-

tion. An estimator for the beamforming weight vector in the presence of angle and Doppler

spreads is derived using a code filtering approach and its tracking performance is estab-

lished.

The above Beamformer-RAKE structure is then applied to a specific M-ary orthogonal

modulation and noncoherent RAKE combining receiver used in an existing interim CDMA

standard. We propose an overall antenna array base station receiver architecture for this

iv

system and study the system performance in terms of BER and power control loop perfor-

mance. The effect of angle spread, number of antennas, multiple resolved paths and Doppler

spread are studied.

v

Acknowledgments

First and foremost, I am thankful to God, the most gracious most merciful for helping

me finish this dissertation. It is my belief in him that helped me persevere at times

when it seemed impossible to go on.

Many individuals have profoundly influenced me during my graduate studies at Stan-

ford, and it is a pleasure to acknowledge their guidance and support. I would like to begin

by expressing sincere thanks to my advisor, Professor Arogyaswami Paulraj, for his sup-

port for the past three years. Professor Paulraj has also provided me with generous finan-

cial support, both in terms of research assistantships and support for attendance at numerous

technical meetings.

I wish to express my appreciation to members of my orals and reading committees: Pro-

fessors Thomas Kailath, John Cioffi, Donald Cox, and Leonard Tyler for taking the time to

read and critique my thesis. Professor Cox was especially helpful in this regard. A special

thanks to Professor Kailath not only for recommending me to Professor Paulraj but also for

his continuous support that I felt throughout my studies at Stanford. I truly appreciate the

support of Professor Abbas El Gamal, during my first two years at Stanford and for his en-

couragement and for being a sincere friend when I really needed one. Our group’s Secretary

Christine Linke was instrumental in getting things done. I am grateful to her for being such

a good listener on those many occasions when I just needed to talk.

My association with Professor Paulraj’s and Professor Kailath’s groups has been also a

source of invaluable experience and friendship for me. I would particularly like to thank

vi

(in alphabetical order): Dr. Hamid Aghajan, Suhas Diggavi, Khalid El Awady, Alper Er-

dogan, Derek Gerlach, Bijit Halder, Babak Hassibi, Babak Khalaj, Jen-Wei Liang, Tushar

Moorti, Boon Ng, Gregory Raleigh, Sumeet Sandhu, Dr. Ali Sayed, Shilpa Talwar, and Dr.

Mats Viberg. The stimulating discussions that we had during Tuesday and Friday afternoon

group meetings helped me shape and redefine my understanding of the field, and had a di-

rect influence on this thesis. Suhas Diggavi, Babak Khalaj, Gregory Raleigh, and Dr. Mats

Viberg were especially helpful in this regard. I have also enjoyed the friendship of many

others within the Information Systems Lab. In particular, I would like to thank Barry An-

drews, Michael Grant, Paul Dankoski, and Vincent K. Jones. I have very fond memories of

our lively discussions and the good times we shared.

Outside the Lab, I have also cherished the company of several friends throughout my

stay at Stanford. In particular, I would like to thank Hisham Abdelhamid, Naofal Al Dhahir,

Walid Azzam, Jalel Azaiez, Khalid El Awady, Yaser Haddarah, Youssef Ismail, Sadok Ka-

llel, Wael Lutfi, Aladdin Nassar, and Mazhar Islam Raja. In addition, I shared unforgettable

hours of fun, intellect, and spiritual uplift (not to forget the Friday dinners!) with many

members (too numerous to mention) of the Islamic Society at Stanford. Our Friday meeting

was always an event to look forward to after a long week.

I am indebted to my mother and my late father for teaching me the importance of hard

work and perseverance and for instilling in me the confidence that I could succeed at what-

ever I chose to do. Finally, I would like to dedicate this work to my wife Fatma and to my

three kids Ahmed, Sumayyah, and Omar. I want to thank Fatma for her patience, love, her

desire that I succeed, for putting up with me, and for the many sacrifices she has had to

make over the past seven years. I also wish to thank Ahmed, Sumayyah, and little Omar

for their smiles, their reckless enjoyment of life, and for always reminding me that the most

important work that I will ever do is within the walls of my own home!.

vii

Contents

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv

Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi

List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xii

List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii

1 Introduction 1

1.1 Cellular Systems and Standards . . . . . . . . . . . . . . . . . . . . 3

1.1.1 European Global System for Mobile (GSM) . . . . . . . . . . 5

1.1.2 North American TDMA Digital Cellular (IS-54) . . . . . . . . 7

1.1.3 North American CDMA Digital Cellular (IS-95) . . . . . . . . 8

1.2 Why Antenna Arrays? . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.3 Thesis Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.4 Thesis Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2 Description and Modeling of Wireless Channels 13

2.1 Fundamentals of Radio Propagation . . . . . . . . . . . . . . . . . . 14

2.1.1 Propagation Path Loss . . . . . . . . . . . . . . . . . . . . . 16

2.1.2 Multipath Fast Fading . . . . . . . . . . . . . . . . . . . . . 19

2.1.3 Log-Normal Slow Fading . . . . . . . . . . . . . . . . . . . 25

2.2 Vector Multipath Channels . . . . . . . . . . . . . . . . . . . . . . . 27

2.2.1 Array Response Vector . . . . . . . . . . . . . . . . . . . . . 27

viii

2.2.2 Vector Channel Modeling . . . . . . . . . . . . . . . . . . . 31

2.2.3 Path Amplitudes and Power-Delay Profile . . . . . . . . . . . 41

2.3 Cellular Channels . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

2.3.1 Macro Cells . . . . . . . . . . . . . . . . . . . . . . . . . . 43

2.3.2 Micro Cells . . . . . . . . . . . . . . . . . . . . . . . . . . 44

2.3.3 Pico or Indoor Cells . . . . . . . . . . . . . . . . . . . . . . 44

2.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

3 Adaptive Beamforming with Antenna Arrays 46

3.1 Adaptive Beamforming Techniques . . . . . . . . . . . . . . . . . . . 48

3.1.1 Direction-Finding Based Beamforming . . . . . . . . . . . . . 49

3.1.2 Beamforming Based on Training-Signals . . . . . . . . . . . . 54

3.1.3 Signal-Structure-Based Beamforming . . . . . . . . . . . . . 55

3.2 Adaptive Beamforming for Wireless CDMA . . . . . . . . . . . . . . 57

3.2.1 CDMA Signal Models . . . . . . . . . . . . . . . . . . . . . 58

3.2.2 Code-Filtering Approach for Adaptive Beamforming . . . . . . 60

3.3 CDMA Beamforming with Multipath . . . . . . . . . . . . . . . . . . 65

3.3.1 Space-Time Matched Filter . . . . . . . . . . . . . . . . . . . 67

3.3.2 Beamformer-RAKE Receiver Structure . . . . . . . . . . . . . 69

3.3.3 A Simulation Example . . . . . . . . . . . . . . . . . . . . . 72

3.4 ”Beamformer-RAKE” Receiver Examples . . . . . . . . . . . . . . . 75

3.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

4 Beamforming for Time-Variant Channels 83

4.1 Recursive Adaptive Beamforming . . . . . . . . . . . . . . . . . . . 84

4.1.1 Recursive Estimation of the Channel Vector . . . . . . . . . . 85

4.1.2 Time-Update of Covariance Estimates . . . . . . . . . . . . . 87

4.1.3 Algorithm Summary . . . . . . . . . . . . . . . . . . . . . . 92

ix

4.2 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

4.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

5 Overview of the IS-95 CDMA Standard 100

5.1 CDMA Forward Link . . . . . . . . . . . . . . . . . . . . . . . . . 102

5.2 CDMA Reverse Link . . . . . . . . . . . . . . . . . . . . . . . . . . 104

5.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

6 CDMA Base Station Receiver with Antenna Arrays 108

6.1 Received Signal Vector Model . . . . . . . . . . . . . . . . . . . . . 110

6.2 Receiver Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

6.3 Signal Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

6.3.1 Noise Analysis . . . . . . . . . . . . . . . . . . . . . . . . . 118

6.3.2 Self and Multiple Access Interference Analysis . . . . . . . . . 119

6.3.3 Decision Statistics . . . . . . . . . . . . . . . . . . . . . . . 122

6.4 Probability of Error Analysis . . . . . . . . . . . . . . . . . . . . . . 123

6.4.1 Low Doppler Frequency . . . . . . . . . . . . . . . . . . . . 124

6.4.2 High Doppler Frequency . . . . . . . . . . . . . . . . . . . . 126

6.5 Numerical and Simulation Results . . . . . . . . . . . . . . . . . . . 129

6.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

7 Performance of Power Control in CDMA 138

7.1 CDMA Reverse Link Open Loop Power Control . . . . . . . . . . . . 140

7.2 CDMA Reverse Link Closed Loop Power Control . . . . . . . . . . . 141

7.3 Closed Loop Power Control Model . . . . . . . . . . . . . . . . . . . 142

7.4 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

7.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

x

8 Conclusions 154

8.1 Thesis Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

8.2 Future Directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

A Multipath Fading Correlation 157

A.1 Fading Correlation for Scalar Channels . . . . . . . . . . . . . . . . . 157

A.2 Fading Correlation for Vector Channels . . . . . . . . . . . . . . . . . 158

B Probability Distributions 162

Bibliography 165

xi

List of Tables

3.1 Optimum weight vector for SINR, ML, and MMSE performance measures 54

3.2 Estimated multipath parameters . . . . . . . . . . . . . . . . . . . . . . 73

4.1 Power recursion for estimating principal eigenvector . . . . . . . . . . . 86

4.2 Inverse square root time-update . . . . . . . . . . . . . . . . . . . . . . 89

4.3 Beamforming algorithm summary . . . . . . . . . . . . . . . . . . . . . 91

4.4 Floating point operations count . . . . . . . . . . . . . . . . . . . . . . 93

6.1 Percent reduction in capacity at Pb = 10−2 and Pb = 10−3 for high Doppler

frequency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

xii

List of Figures

1.1 If I had a cellular telephone . . . . . . . . . . . . . . . . . . . . . . . 2

1.2 Overview of a cellular telephone network . . . . . . . . . . . . . . . . 4

1.3 GSM frame and time-slot structure . . . . . . . . . . . . . . . . . . . 5

1.4 IS-54 frame and time-slot structure . . . . . . . . . . . . . . . . . . . 7

2.1 Illustration of the wireless propagation environment. . . . . . . . . . . 15

2.2 Direct and indirect paths on a flat-terrain environment. . . . . . . . . . 16

2.3 Magnitude of a Rayleigh fading channel: fd = 80 Hz and Ts = 104.2µs. 23

2.4 Fast fading amplitude correlation vs. delay (in wavelengths traveled). . 24

2.5 A Wireless communication system employing antenna arrays. . . . . . 28

2.6 A one-dimensional array manifold. . . . . . . . . . . . . . . . . . . . 29

2.7 The uniform linear array scenario. . . . . . . . . . . . . . . . . . . . 31

2.8 An illustration of the vector channel. . . . . . . . . . . . . . . . . . . 32

2.9 Model geometry. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

2.10 Spatial envelope correlation vs. antenna spacing: mean AOA θ = 0◦ . . 36

2.11 Spatial envelope correlation vs. antenna spacing: mean AOA θ = 30◦ . 37

2.12 Space-Time Fading: fd = 50 Hz, � = 0◦, and Ts = 208.3µs. . . . . . . 38

2.13 Space-Time Fading: fd = 50 Hz, � = 3◦, and Ts = 208.3µs. . . . . . . 39

2.14 Space-Time Fading: fd = 50 Hz, � = 40◦, and Ts = 208.3µs. . . . . . 40

3.1 Block diagram of an adaptive array system . . . . . . . . . . . . . . . 47

3.2 Constant modulus property restoral beamformer . . . . . . . . . . . . 56

xiii

3.3 Mobile transmitter block diagram . . . . . . . . . . . . . . . . . . . . 59

3.4 CDMA receiver-beamformer block diagram . . . . . . . . . . . . . . 63

3.5 RAKE receiver block diagram . . . . . . . . . . . . . . . . . . . . . 65

3.6 Multipath CDMA RAKE receiver . . . . . . . . . . . . . . . . . . . 66

3.7 Space-Time Matched Filter . . . . . . . . . . . . . . . . . . . . . . . 69

3.8 Beamformer-RAKE receiver structure . . . . . . . . . . . . . . . . . 70

3.9 Estimated multipath profile . . . . . . . . . . . . . . . . . . . . . . . 74

3.10 Balanced DQPSK Beamformer-RAKE receiver with incoherent combining 75

3.11 Pe for balanced DQPSK with incoherent combining and � = 0◦ . . . . 77

3.12 Pe for balanced DQPSK with incoherent combining and large � . . . . 78

3.13 Balanced QPSK Beamformer-RAKE receiver with coherent combining . 79

3.14 Pe for balanced QPSK with coherent combining and � = 0◦ . . . . . . 81

3.15 Pe for balanced QPSK with coherent combining and large � . . . . . . 82

4.1 Mismatch loss as a function of fd and � for µ = 0.98 . . . . . . . . . 94

4.2 Mismatch loss as a function of fd and � for µ = 0.95 . . . . . . . . . 95

4.3 Mismatch loss as a function of fd and µ for � = 5◦ . . . . . . . . . . 96

4.4 Optimum forgetting factor µ as a function of fd and � . . . . . . . . . 97

4.5 Mismatch loss as a function of fd and � with optimum forgetting factor 98

5.1 CDMA IS-95 forward link waveform generation . . . . . . . . . . . . 102

5.2 CDMA IS-95 reverse link waveform generation . . . . . . . . . . . . 105

5.3 CDMA Base station receiver: IS-95 approach . . . . . . . . . . . . . 106

6.1 CDMA Base station receiver: beamforming approach . . . . . . . . . 109

6.2 Base station receiver block diagram . . . . . . . . . . . . . . . . . . 113

6.3 Correlators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

6.4 Optimum Beamforming and Incoherent RAKE . . . . . . . . . . . . . 117

6.5 Simulation scenario . . . . . . . . . . . . . . . . . . . . . . . . . . 129

6.6 I-channel: first antenna interference distribution . . . . . . . . . . . . 130

xiv

6.7 Q-channel: first antenna interference distribution . . . . . . . . . . . . 131

6.8 Pb for fd = 5 Hz and closed loop power control. . . . . . . . . . . . . 132

6.9 PDF of z(n)1 for n = h at high fd. . . . . . . . . . . . . . . . . . . . . 133

6.10 Pb for high fd and � = 0◦, and power control. . . . . . . . . . . . . . 134

6.11 Pb for high fd and � = 3◦, and power control. . . . . . . . . . . . . . 135

6.12 Pb for high fd and � = 60◦, and power control. . . . . . . . . . . . . 136

7.1 Feedback Power Control Model . . . . . . . . . . . . . . . . . . . . 142

7.2 Power Control Algorithm . . . . . . . . . . . . . . . . . . . . . . . . 145

7.3 Power-controlled received signal vs. simulated Rayleigh fading: fd = 5

Hz, K = 5, L = 4. . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

7.4 Power-controlled received signal vs. simulated Rayleigh fading: fd =100 Hz, K = 5, L = 4. . . . . . . . . . . . . . . . . . . . . . . . . . 147

7.5 Power control error vs. number of paths: K = 5. . . . . . . . . . . . . 148

7.6 Power control error vs. power step size: L = 2,K = 5. . . . . . . . . . 149

7.7 Power control error vs. loop delay: L = 2,K = 5. . . . . . . . . . . . 150

7.8 Power control error vs. forward link error rate: L = 2, K = 5. . . . . . 151

7.9 Power control error vs. angle spread: L = 4, and K = 5. . . . . . . . . 152

7.10 Power control error vs. angle spread: L = 4, and K = 9. . . . . . . . . 153

xv

Chapter 1

Introduction

The realization of wireless communications providing high-speed and high-quality informa-

tion exchange between two portable terminals that might be located anywhere in the world

is the new communications challenge for the next decade. The great popularity of cordless

phones, cellular phones, radio paging, and other emerging portable communication tech-

nologies demonstrates a great demand for such services. For example, from 1990 to 1994,

the number of cellular telephone users in the US has grown from 5.1 million to 23.3 million

subscribers, and by the year 2000 it is projected that the number will nearly double to 46.9

million subscribers.

What has emerged from the world wide research and development activity in this area

is the need for new technology advances to meet such an explosive growth. These include:

• New techniques to improve the quality and spectral efficiency of communication over

wireless channels.

• Better techniques for sharing the limited spectrum to accommodate different wireless

services.

• New signal processing techniques to implement various functions of the cellular sys-

tem.

1

Chapter 1. Introduction 2

Figure 1.1: If I had a cellular telephone ... (Herman c Jim Unger [1]. Reprinted with per-mission of Universal Press Syndicate. All rights reserved)

The physical limitations of the wireless channel present a fundamental technical chal-

lenge for reliable communications. The channel is susceptible to time-variant noise, inter-

ference, and multipath. Moreover, radio spectrum is now a limited resource, and even with

the recent increase in the spectrum allocation for wireless services, it will be stretched out

to its capacity limit to accommodate various current and emerging wireless services.

Also, limitation in the power and size of the communication and computing device is

another major design consideration. Most personal communications and wireless services

devices are meant to be carried in a briefcase or a pocket and, therefore, must be small and

lightweight, which translates to a low power requirement, since small batteries must be used.


However, many of the signal processing techniques used for efficient spectral utilization

demand significant processing power, precluding the use of low power devices. Continuing

advances in VLSI and integrated circuit technology for low power applications will provide

a partial solution to this problem. However, placing most of the signal processing burden

on fixed location sites with large power resources has and will continue to be the common

trend in wireless systems design.

Cellular communication systems involve two radio links: the reverse link from the mo-

bile to the base station, and the forward link from the base station to the mobile. In this the-

sis, we study the use of multiple antennas at the base station and the associated advanced

signal processing for CDMA wireless systems and the effect of such processing on system

performance measures such as capacity, range, and mobile transmit power. In this work we

will focus on the reverse (mobile to base) link.

1.1 Cellular Systems and Standards

In order to accommodate the demand for wireless communication services, efficient use

of limited available frequency spectrum is essential. Cellular systems exploit the power fall-

off with distance of a transmitted signal to reuse a communication channel (where a com-

munication channel can be either a frequency band, a time slot, or a unique code) at another

spatially separated location. The coverage area is divided into smaller regions or cells, each

containing a subset of the mobile users of the cellular system. In each cell, only one user is

assigned to a particular communication channel. Operation within a cell is controlled by a

base station, which is responsible for serving calls to and from users located in their respec-

tive cells. The base stations are connected to the mobile telephone switching office (MTSO)

that serves as a controller to a group of base stations and as an interface between the mobile

users and the fixed public switching telephone network (PSTN). When a mobile user crosses


µT1/ Wave

Base Station

Mobile

MTSO

PSTN

Co

Figure 1.2: Overview of a cellular telephone network

the boundary between two cells, its communication channel is switched, or handed off, to

the base station in the new cell. The shape of each cell is determined by the power footprint

of the transmitting base station. An illustration of a cellular telephone network is shown in

Figure 1.2.

In the US, cellular telephone service is provided mostly by the AMPS (Advanced Mobile

Phone Service) system [2-4] which evolved from extensive research at Bell Laboratories in

the 1970’s. In this system analog voice signals are frequency modulated onto carriers in the

800 MHz band (824-849 for the downlink and 869-894 for uplink). To prevent significant

adjacent-channel interference, each user channel is allotted a 30 kHz bandwidth. Only a

certain set of the carrier frequencies is available in a given cell, and neighboring cells must

use different sets of carriers. This method of frequency reuse results in reduced cochannel

interference, but the resulting system capacity is only one call per 210 kHz [4]. The increas-

ing demand for cellular telephone service has pushed this technology to its capacity limits


1 2 3 4 5 6 7 8

Frame = 4.6155 ms

Data

26F

1

F

1Train

26

Data

26

T3

Guard8.25

T3

Slot = 577 sµ

T: Tail bitsF: FlagTrain: Equalizer training sequenceGuard: Guard time interval

Figure 1.3: GSM frame and time-slot structure

and resulted in proposals for a ”second generation” of cellular systems, employing digital

modulation techniques and alternative multiple access techniques. A brief description of

several of these techniques follows.

1.1.1 European Global System for Mobile (GSM)

The Pan-European standard for digital cellular telephony, called Global System for Mobile

Communications (GSM) has two objectives: pan European roaming , which offers compat-

ibility through the European continent, and interaction with the integrated service digital

network (ISDN). The first commercial GSM system, called D2, was implemented in Ger-

many in 1992.


The bands allocated to GSM are 890-915 MHz for downlink and 935-960 MHz for up-

link. The band in either direction is divided into 124 frequency channels, each with car-

riers spaced 200 kHz apart. Each cell site in a GSM system has a fixed number of fre-

quency channels (two way) ranging from only one to usually not more than 15. In GSM,

an FDMA/TDMA (frequency division multiple access/time division multiple access) radio

channel structure is used. Each FDMA frequency channel supports a multiple user signal

format. Each 200 kHz FDMA channel uses an aggregate bit rate of 270.833 kbits/sec, car-

ried over the radio interface using GMSK (Gaussian-filtered minimum-shift keying) mod-

ulation with a bandwidth-time product of 0.3. The use of a transmitted bit rate as high as

270 kbits/sec requires the implementation of adaptive equalization techniques to deal with

channel multipath, and GSM specifications require that equipment be built to accommodate

RMS delay spread up to 16 µs. The GSM standard provides for the use of slow frequency-

hopping as a means of reducing other user interference, though no GSM systems currently

implements this technique.

The 270-kbits/sec data stream in each FDMA channel is divided into 8 fixed-assignment

TDMA channels or time slots termed logical channels. Each slot is 577 µs, which cor-

responds to the transmission time for about 156.25 bits, though only 148 bits are actually

transmitted in each slot. The remaining time, 8.25 bits time duration or about 30.5 µs, is

guard time in which no signal is transmitted to prevent overlapping of signal bursts arriv-

ing at a base station from different mobile terminals. Figure 1.3 shows the GSM frame and

time-slot structure. The logical channels are organized into a hierarchical frame structure

that provides each mobile terminal with a two-way traffic channel and a separate two-way

control channel. The numbering of time slots is offset between two directions on the down-

link to prevent a mobile terminal from transmitting and receiving at the same time. More

details on the GSM frame hierarchy and network issues can be found in [4-7].


1 2 3 4 5 6

Frame = 40 ms

G6

R6

Data16

Synch28

Data122

CCH12

DVCC12

Data122

Mobile to Base

Slot = 6.67 ms

Synch28

CCH12

Data130

DVCC12

Data130

RSVD12

Base to Mobile

G: Guard timeR: Ramp timeSynch: Synchronization

DVCC: Digital verification color codeRSVD: Reserved for future useCCH: control channel

Figure 1.4: IS-54 frame and time-slot structure

1.1.2 North American TDMA Digital Cellular (IS-54)

The North American TDMA Digital Cellular standard (IS-54) has been developed by

the digital cellular standards subcommittee as a replacement for the existing analog AMPS

system. As with GSM, the IS-54 radio channel structure is a combination of FDMA and

TDMA, with user traffic and control channels built upon the logical channels provided by

TDMA time slots. The designated frequency channels in IS-54 are the same as those in


AMPS, with carriers spaced 30 kHz apart. This was done so as to allow conversion of in-

dividual analog channels to digital operation. In the initial phase of TDMA implementa-

tion, the mobile phones are dual-mode devices, capable of operating on either AMPS analog

channels or TDMA digital channels, and cellular operators are required to continue support-

ing AMPS users as digital service is introduced.

The IS-54 standard specifies the radio channel modulation as π/4-shift DQPSK (dif-

ferentially encoded phase shift keying), to be implemented with square-root raised cosine

filtering and rolloff parameter 0.35. The principal advantage of this modulation scheme is

bandwidth efficiency. The channel transmission rate is 48.6 kbits/sec; and with a channel

spacing of 30 kHz, this yields channel utilization of 1.62 bits/sec/Hz, a 20% improvement

over GSM. The principal disadvantage of a linear modulation scheme is a lower efficiency

of linear transmitter power amplifiers as compared to nonlinear amplifiers that can be used

with constant envelope modulation. This is reflected in the size and weight of the hand set.

In IS-54, each 30-kHz digital channel has a transmission rate of 48.6 kbits/sec. The 48.6-

kbits/sec stream is divided into 6 TDMA channels of 8.1 kbits/sec each. The IS-54 time slot

and frame format, as shown in Figure 1.4, is much simpler than that of the GSM standard.

The 40 ms frame is composed of six 6.67-ms time slots corresponding to 324 bits each.

Although needed in only a few specific places for 48.8 kbits/sec transmission rate, a delay

spread equalizer is required in the IS-54 standard.

1.1.3 North American CDMA Digital Cellular (IS-95)

As an alternative to IS-54’s TDMA standard for digital cellular, a new digital cellular system

based on code-division multiple access (CDMA) technology was proposed [8]. In contrast

to the IS-54 standard, which uses the same set of carrier frequencies, spaced 30 kHz apart, as

are used by the analog AMPS system, the CDMA system uses spread spectrum signals with

1.2288 MHz spreading bandwidth, a frequency span equivalent to 41 AMPS channels. (The


forward and reverse links actually use separate carrier frequencies, spaced 45 MHz apart.)

Clearly, this design does not lend itself to selective channel replacement of analog AMPS,

unlike the TDMA IS-54 system. Instead, large blocks of channels will be replaced at one

time by the CDMA system [5]. Therefore, the IS-95 standard specifies a dual-mode design

of the mobile station so that the CDMA digital cellular units can be still used in areas with

existing analog cellular coverage.

As one of the goals of this thesis is to apply advanced array signal processing techniques

derived therein to CDMA systems, a full description of the IS-95 CDMA system will be

given later in Chapter 5.

1.2 Why Antenna Arrays?

As mentioned earlier, the goal of wireless communication systems is to provide a wide va-

riety of wireless services such as voice, data, facsimile, and electronic mail. The current

design trend for those systems is to deploy a large number of small cells, each served by a

base station using a conventional antenna system. While this might provide a solution for

the coverage problem, it has a number of significant drawbacks. A large number of small

cells will result in increased number of base station equipment, increased networking and

coordination requirements, increased handoff, and reduced trunking efficiency.

Optimum combining (or beamforming) and adaptive signal processing with antenna ar-

rays at the base station can be used to improve system coverage, base station capacity, and

link quality for moderate to large-sized cells. With antenna arrays, the base station can re-

duce or suppress the interference due to cochannel users (either from adjacent cells or same

cell), and through spatial diversity and beamforming gain, it provides improved desired sig-

nal level. This improvement that antenna arrays bring can be traded off in a number of ways.


For example, when a network is first deployed, capacity is not an issue (especially in ru-

ral areas). As the system matures, more and more users will request wireless services and

the need for more capacity develops. In this later stage, the gain due to spatial processing

with antenna arrays can facilitate a denser use of the available bandwidth and, therefore,

an increase in system capacity. In the early stages of system deployment, service providers

strive to minimize the initial deployment cost by covering a given service area with the least

number of base stations. The gain provided by spatial processing with antenna arrays can

be used to help meet this coverage goal. This gain can also be used to improve the signal

quality itself, which leads to low outage probability, better voice quality, and the ability to

support higher data rates for wireless data services. Other possible benefits of spatial pro-

cessing with antenna arrays include the ability to reduce the loss in performance when low

power terminals co-exist with high power terminals.

1.3 Thesis Overview

In this introductory chapter, we have attempted to lay the background for the subject mate-

rial of this thesis. The first half of Chapter 2 is an overview of radio channel modeling that

describes some problems associated with a wireless radio channel. The models described

therein are for scalar channels. To support the core subject of this thesis, which is the use of

adaptive antenna arrays at the base station, in the second half of Chapter 2 we develop a sta-

tistical vector channel model based on the physical propagation environment and its statis-

tical properties. In Chapter 3, we start by reviewing previous techniques and the conditions

under which they are applicable for optimum beamforming. We then articulate the reasons

why those techniques are not applicable for wireless CDMA systems. We then introduce

our space-time approach for estimating the channel vector and for optimum beamforming,

and construct a space-time receiver model which we call a ”Beamformer-RAKE” to exploit


both the temporal and spatial structure of the received multipath signal to maximize perfor-

mance. In Chapter 4, we develop a recursive algorithm for estimating the vector channel and

the optimum beamforming weights. We then study its tracking performance under realistic

channel conditions.

Chapters 5 and 6 describe an application of the above space-time processing techniques

to the current CDMA standard IS-95. Chapter 5 gives a brief description of the proposed

IS-95 system, with more emphasis on describing the reverse link modem. In Chapter 6, we

present an overall base station receiver structure with antenna arrays and describe its build-

ing blocks. We then study the performance of the proposed receiver in terms of the uncoded

bit error rate (BER) as a function of loading (number of users) under different channel con-

ditions. In our analysis, we assume perfect channel estimates and do not include tracking

errors. As we mentioned earlier, closed-loop power control is a key element of the current

implementation of the IS-95 system. In Chapter 7, we describe a method for closed-loop

control. We also present a simulation study to characterize the performance of closed-loop

power control under different operating scenarios. Chapter 8 contains concluding remarks

and a summary of the thesis. In Appendix A, we derive the fading correlation for both scalar

and vector multipath channels. In Appendix B, we derive the probability distributions used

in the analysis in Chapter 6.

Several of the main ideas in this thesis have already been published by the author in the

course of his research. The work on the vector channel model has been partially published

in [9]. The idea of space-time processing and ”Beamformer-RAKE” is presented in [10, 11].

The performance enhancement in CDMA systems due to the use of antenna arrays has been

published in [12-14]. The recursive algorithm for estimating and tracking the channel vector

is presented in [15, 16]. The proposed base station receiver architecture and its performance

analysis are presented in [17-19]. Results on the power control performance are presented

in [20].


1.4 Thesis Contributions

We have made several contributions in this thesis research to the body of communication

systems engineering. These are

• The development of a statistical vector channel model based on the physical propa-

gation environment for wireless channels when multiple antennas are used at the base

station (this work is a joint contribution with Gregory Raleigh and Suhas Diggavi);

• A new space-time processing technique for estimating the vector channel and for op-

timum beamforming in CDMA systems;

• The derivation of the space-time matched filter receiver for vector multipath signals

over AWGN channels;

• The construction of a ”Beamformer-RAKE” receiver model for exploiting both spa-

tial and temporal structure in the received CDMA multipath signal to maximize sys-

tem performance;

• A recursive algorithm for estimating the vector channel and a quantification of its

tracking performance under different channel conditions;

• An antenna array-based base station receiver architecture for the current CDMA wire-

less standard based on the ”Beamformer RAKE” structure and performance analysis

of the proposed receiver in terms of the uncoded BER under different channel condi-

tions; and

• A characterization of the performance of fixed-step closed-loop power control under

different operating scenarios.

Chapter 2

Description and Modeling of Wireless

Channels

The wireless channel in mobile radio poses a great challenge as a medium for reliable high

speed communications. When a radio signal is transmitted in a wireless channel, the wave

propagates through a physical medium and interacts with physical objects and structures

therein, such as buildings, hills, streets, trees, and moving vehicles. The collection of ob-

jects in any given physical region describes the propagation environment. The propagation

of radio waves through this environment is a complicated process that involves diffraction,

refraction, and multiple reflections. In order to analyze the performance of wireless com-

munication systems, it is necessary to define statistical models that reasonably approximate

the propagation environment. Many statistical models for scalar (single antenna) channels

have been reported. In order to analyze performance of recently proposed adaptive antenna

array techniques, it is also necessary to develop statistical channel models for the vector

channel (multiple antennas) case.

In this chapter, we characterize the wireless propagation environment. We also develop

a statistical and time-variant wireless vector channel model that is based on the physical

propagation environment.

13

Chapter 2. Description and Modeling of Wireless Channels 14

2.1 Fundamentals of Radio Propagation

A typical channel model in mobile radio communication in an urban area usually involves

an elevated fixed base-station antenna (or multiple antennas), a line-of-sight (LOS) propaga-

tion path followed by many non-LOS reflected propagation paths, and an antenna mounted

on the mobile or the portable unit. In most situations, because of natural and man-made

structures located between the mobile and the base-station in the propagation environment, a

direct LOS propagation path may not exist. An illustration of such an environment is shown

in Figure 2.1.

The mobile radiation pattern illuminates all the local structures and buildings surround-

ing the mobile that are within a few hundred wavelengths from the mobile. In addition to

the local scatterers around the mobile, there are dominant reflectors such as large buildings,

hills, towers, and other structures. These dominant reflectors couple energy from the mo-

bile and/or the local scattering structure to the base station which gives rise to a propagation

path between the base-station and the mobile. In many instances there may exist more than

one propagation path, and this situation is referred to as multipath propagation. The propa-

gation path or paths change with the movement of the mobile unit and/or the movement of

its surroundings in the propagation environment.

Even the smallest and slowest movement causes time-variant multipath , resulting in a

random time-varying signal at the base-station. As an example, assume a stationary user

near a busy highway. Although the mobile user is stationary , parts of the environment are

moving at 60 mph (miles per hour). The vehicles on the highway become ”moving reflec-

tors” for the radio signal. The movement of the mobile user and/or the movement of the sur-

roundings in the propagation environment causes a Doppler spread in the received signal,

which describes the rate of variation in the received signal level. In addition, each propa-

gation path arriving at the base-station has its own time delay and angle of arrival (AOA),

which causes delay spread and angle spread in the received signal.


Dominant Reflector

Dominant Reflector

MobileLocal Scatterers

Base Station

Figure 2.1: Illustration of the wireless propagation environment.

In general, the mobile radio propagation in such environments is characterized by three

partially separable effects: path loss, multipath fading, and shadowing. Path loss is a func-

tion of the distance between the mobile and the base-station. As discussed earlier, multiple

signal reflections arrive at the base-station each with its own phase, which causes destruc-

tive and constructive interference. The resulting variations in the signal amplitude, called

multipath fading, vary over distances proportional to the signal wavelength; thus, this type

of signal fading is referred to as fast fading. When the number of multipath components is

large, the central limit theorem can be invoked to model the fast fading by a filtered complex

Gaussian process. Multipath fast fading is described by its envelope fading (flat fading),

Doppler spread (time-selective fading), time-delay spread (frequency selective fading), and

angle spread (space selective fading).

In addition to the multipath fast fading, the LOS and reflected paths may be attenuated

by large obstructions such as large buildings and hills that are positioned between the mobile


hr

ht

θθ

d

d

`

r

r

Figure 2.2: Direct and indirect paths on a flat-terrain environment.

and base station. This type of fading, which is called shadowing, varies with distances that

are proportional to the sizes of the buildings (or the obstructions in general), and is thus

referred to as slow fading. Most empirical studies show that the variations in signal level

due to the slow fading follow a log-normal distribution.

2.1.1 Propagation Path Loss

A measure of interest in radio propagation is the path loss, which is defined as the ratio

between the received power Pr and the transmitted power Pt

Ld = Pr

Pt(2.1.1)

Consider the propagation of a radio wave in a flat-terrain environment. The transmitted sig-

nal may reach the receiving antenna in several ways:

• Through a direct LOS path.

• Through an indirect path consisting of the radio wave reflected by the ground.

• Through an indirect path consisting of a surface wave.


Let τ be the time delay between the direct LOS path and the reflected path. If we assume

that τ is much smaller than the inverse bandwidth of the transmitted signal B−1 and if we

neglect the effect of surface wave attenuation1, then the received power Pr is [21]

Pr = PtGtGr

(λ

2πd

)2 ∣∣1 + Re j�φ∣∣2 (2.1.2)

where Gt and Gr are the gains of the transmitting and receiving antennas, respectively (note

that we assumed that GtGr is the same for both the direct LOS path and the reflected path),

λ is the wavelength, R is the reflection coefficient of the ground, and �φ is the phase dif-

ference between the direct LOS path and the reflected path

�φ = 2π(r + r − d)λ

= 2πλ

[(ht + hr

d

)2

+ 1

]1/2

−[(

ht − hr

d

)2

+ 1

]1/2 (2.1.3)

where ht and hr are the heights of the transmitting and receiving antennas, respectively.

Equation (2.1.2) has been shown to agree very closely with the measurements in [22]. The

ground reflection coefficient R is given by

R = sin θ− Zsin θ+ Z

(2.1.4)

where

Z =

√εr − cos2 θ/εr for vertical polarization

√εr − cos2 θ for horizontal polarization

(2.1.5)

and εr is the dielectric constant of the ground, which for earth or road surfaces is approxi-

mately that of a pure dielectric (εr = 15). For large d, we have

�φ ≈ 4πhthr

λd(2.1.6)

Also, in this case the grazing angle θ ≈ 0◦ , and therefore R ≈ −1. Moreover, R tends to

−1 for frequencies above 100 MHz and incidence angles less than 10◦, irrespective of the

1This is a valid approximation for antennas located more than a few wavelengths above the ground.


polarization [23]. In this case (2.1.2) becomes

Pr ≈ PtGtGr

(hthr

d2

)2

(2.1.7)

Thus, in the asymptotic limit of large d, the received power falls off inversely with d4. In

[22], plots of (2.1.2) as a function of distance illustrate this asymptotic behavior. Up to a

certain critical distance dc, the wave experiences constructive and destructive interference

of the two rays. At distance dc, the final maxima is reached, after which the signal power

falls off inversely with d4. If we average out the local maxima and minima, the resulting

average power loss can be approximated by dividing the power loss curve into two regions.

For d < dc, the average power fall off with distance corresponds to free space (i.e. d2). For

d > dc, the average power fall off with distance is approximated by the fourth power law in

(2.1.7).

In practice, the attenuation of radio signals is greater than that of free space and less

than that of free space with a perfectly reflecting ground. Measurements [24] show that for

frequencies around 100 MHz, the attenuation versus distance curve has a slope comparable

to that of free space with additional loss depending on the environment. For example for a

suburban environment around 100 MHz, the attenuation curve follows free space with an

additional 8.5 dB loss. In [25], it was found that at a carrier frequency of 910 MHz in an

urban environment, the attenuation loss in dB has a steeper slope than that of free space

but not as steep as with a fourth power law. Measurements at 800 MHz [26, 27] also show

two distinctly different propagation loss slopes before and after the critical distance dc. For

d < dc, the slope is slightly less than 2 while for d > dc the slope is close to 4.

The two-ray model described above is a simple model that characterizes signal propaga-

tion in isolated areas with few reflectors such as rural roads or highways. It requires infor-

mation only about antenna heights. A more complicated model for urban area transmissions

is the dielectric canyon or ten-rays model developed by Amitay [28]. This model assumes

rectilinear streets (such as in downtown Manhattan) with buildings along both sides of the


street and transmitter and receiver well below the tops of the buildings. The building-lined

streets act as a dielectric canyon to the propagating signal.

The power fall off in both the dielectric canyon model and urban measurements [22, 29,

30] is proportional to d−2, even at relatively large distances. Moreover, the fall off exponent

is relatively insensitive to the transmitter height, as long as the transmitter is significantly

below the building skyline. This fall off with distance squared is due to the dominance of

the wall-reflected rays, which decay as d−2, over the combination of the LOS and ground-

reflected rays (two-path model above), which decays as d−4.

2.1.2 Multipath Fast Fading

The statistical model for the fast fading of the received signal level is based on a physical

propagation environment consisting of a large number of isolated scatterers with unknown

locations and reflection properties. Let the transmitted signal be

x(t) = s(t) · e j(2π f t+φo) (2.1.8)

where s(t) is the complex baseband signal with bandwidth B, f is the carrier frequency, and

φo is an arbitrary initial phase. Without loss of generality we will assume that φo is zero. If

we assume that the mobile is moving at speed v and there is no direct LOS, and if we ignore

the receiver additive white Gaussian (AWGN) noise, the corresponding received signal at

the base station is the sum of all multipath components [21, 31]

y(t) = AL∑

i=1

Ris(t − τi)ej2π[( f+ fd cosψi)t− f τi] (2.1.9)

where A includes the effects of distance loss and antenna gains (here we assumed that the

propagation distance spread�r = maxi

ri −mini

ri is much less than the propagation distance

ri for all i and, therefore, that the attenuation with distance is the same for each component.

This is true in general if all reflectors are within the vicinity of the mobile). For the ith mul-

tipath component, R2i is the fraction of the incoming power in the ith path, τi = ri/c is the


multipath delay where ri is the propagation distance and c is the speed of light, and fd cosψi

is the Doppler shift where fd = v/λ is the maximum Doppler shift and ψi is the direction

of the ith scatterer with respect to the mobile velocity vector. The Doppler spread fm is

given by 2 fd. These parameters vary with time. As noted in [21], the process y(t) is wide-

sense stationary with respect to ensemble averages. It is not stationary with respect to time

averages, however, and thus nonergodic. But the difference between time and ensemble av-

erages decreases as the number of paths L becomes large; thus the statistical properties will

be computed on the basis of ensemble averages. Throughout the rest of this section, we will

assume that A = 1. If we assume that the multipath delay spread defined as

T = maxi

τi − mini

τi (2.1.10)

is much less than the inverse bandwidth of the signal (T � B−1), i.e. we assume that s(t)

is a narrowband signal, then s(t − τi) ≈ s(t − τo) where τo ∈ [mini

τi,maxi

τi] [31]. Then,

we can rewrite (2.1.9) as

y(t) ≈ s(t − τo) ·(

L∑i=1

Riejφi(t)

)· e j2π f t (2.1.11)

where φi(t) = 2π( fd cosψit − f τi). The phases φi(t) modulo 2π can be modeled as i.i.d.

random variables uniformly distributed over [0,2π] [32]. Analytical results based on this

assumptions agree with measurement results in [32].

The equivalent lowpass received signal is

y(t) ≈ s(t − τo) ·(

L∑i=1

Riejφi(t)

)(2.1.12)

In addition to the time delay, the received signal in (2.1.12) differs from the original trans-

mitted signal by the complex scale factor in the parentheses. Let

β(t) =L∑

i=1

Riejφi(t) = α(t)e jφ(t) (2.1.13)


Since y(t) is the response of an equivalent lowpass channel to the lowpass signal s(t), it fol-

lows that the equivalent lowpass channel is described by the time-variant impulse response

[33]

h(t; τ) = δ(τ − τo)α(t)ejφ(t) (2.1.14)

If we assume that the Ri are i.i.d. and independent of the φi, then the first and second mo-

ments of β(t) are

E{β(t)} = 0 (2.1.15)

E{β(t)β∗(t + ν)} =∑

i

E{R2i }e j2π fd cosψiν (2.1.16)

If we also assume that, in addition to being i.i.d., the Ri have bounded variance, then β(t)

will approach a complex Gaussian random variable as the number of scatterers L becomes

large [34]. In this case α(t) has a Rayleigh distribution [33]

f (α) = 2ασ2

exp

(−α2

σ2

), α ≥ 0 (2.1.17)

where σ2 = E{α2}. Therefore the variation of the received signal envelope is Rayleigh,

which has also been confirmed by measurements [35, 36]. Now, if the direct LOS is not

obstructed, then α(t) will have a Rician distribution [33]

f (α) = 2ασ2

exp

(−α2 +µ2

σ2

)Io

(2αµσ2

), α ≥ 0 (2.1.18)

where µ2 is the average power in the direct LOS and In(.) is the modified Bessel function

of the n-th order [37].

Note that the real and imaginary parts of β(t) are independent,

E{Re{β(t)}Im{β(t)}} =L∑

i=1

∫ 2π

0

12π

cos(φi(t)) sin(φi(t)) dφi(t)

= 0 (2.1.19)

Note: A more general distribution for the fast fading amplitude is given by the Nakagami

distribution [32]

f (α) = 2mmα2m−1

�(m)σ2· exp

(−mα2

σ2

)(2.1.20)


where σ2 = E{α2} and m = σ4/E{(α2 − σ2)2} and �(.) is the gamma function. This distri-

bution is general in the sense that none of the above assumptions has to hold. When the scat-

tering process generates merely diffuse wave field, then m ≈ 1 [32] and the Nakagami dis-

tribution in (2.1.20) is identical to the Rayleigh distribution in (2.1.17). When a direct com-

ponent is present, the Nakagami distribution approximates the Rice distribution in (2.1.18)

with m > 1.

It remains now to determine the time-frequency correlation behavior of β(ω, t), where

β(ω, t) is given in (2.1.13). In order to do that, let fT(τ) be the probability density function

of the time delay τi, where fT(τ) is nonzero for 0 ≤ τ < ∞ and zero otherwise. In Ap-

pendix A, it is shown that the time-frequency correlation of β(ω, t) and β(ω+�ω, t + ν)

is given by

ρβ(�ω, ν) = E{β(ω, t)β∗(ω+�ω, t + ν)} (2.1.21)

= Jo(ωdν) · FT( j�ω) (2.1.22)

where Jn(.) is the Bessel funcion of the first kind of order n [37] and FT( j�ω) is the Fourier

transform of fT(τ). Interpretation of some measured data [21] indicates that the path time

delay τi can be modeled as an exponential random variable. That is

fT(τ) = 1

Texp

(− τ

T

), τ ≥ 0 (2.1.23)

where T is mean time delay. In this case, we have

FT( j�ω) = 1 − j�ωT

1 + (�ωT )2(2.1.24)

and

ρβ(�ω, ν) = Jo(ωdν) · 1 − j�ωT

1 + (�ωT )2(2.1.25)

To a good approximation [21, 23, 38], the envelope correlation ρα(�ω, ν) is equal to

the squared magnitude of the complex signal correlation, i.e..

ρα(�ω, ν) = Cov{α(ω, t)α∗(ω+�ω, t + ν)}√Var{α(ω, t)α∗}Var{(ω+�ω, t + ν)} (2.1.26)


Time (Symbol Period Ts)

0 200 400 600 800 1000

Fadi

ng M

agni

tude

(dB

)

-20

-15

-10

-5

0

5

10

Figure 2.3: Magnitude of a Rayleigh fading channel: fd = 80 Hz and Ts = 104.2µs.

ρα(�ω, ν) � J2o (ωdν) · |FT( j�ω)|2 (2.1.27)

Figure 2.3 shows the magnitude for a simulated multipath Rayleigh fading channel. The

mobile is assumed to be moving such that fd = 80 Hz. The transmitted signal is a balanced

QPSK signal with symbol rate of 9600 symbols/sec. Figure 2.4 shows the amplitude time-

correlation ρβ(0, ν)= Jo(2π fdν) as a function of fdν (which is the number of wavelengths

traveled by the mobile over ν sec).

As the signal bandwidth B increases so that T ≈ B−1, the approximation s(t − τi) ≈s(t − τo) for τo ∈ [min

iτi,max

iτi] is no longer valid. Then, the received signal is a sum

of copies of the original signal, each is delayed in time by τi and phase shifted by φi. For

wideband signals, the channel response can be approximated using Turin’s model [36] if the

incoming paths form subpath clusters. In this model, paths that are approximately the same


Delay ( in wavelength traveled)

0 1 2 3 4 5

Am

plitu

de C

orre

lati

on ρ

(ν)

-0.4

-0.2

0.2

0.4

0.6

0.8

0.0

1.0

Figure 2.4: Fast fading amplitude correlation vs. delay (in wavelengths traveled).

length ( |τi − τ j| < B−1 ) are not resolvable at the receiver. Thus, they are combined into a

single path. If we assume a finite number of resolvable paths, then the received signal can

be written as

y(t) =L∑

l=1

x(t − τl )αl(t)ejφl (t) (2.1.28)

where L in this case represents the number of resolvable paths or subpath clusters, and αl,

φl, and τl are the amplitude, phase, and delay of each resolvable path. The complex gains

βl(t) = αl(t)e jφl (t) are independent complex Gaussian processes. Again, the equivalent

lowpass channel is described by the time-variant impulse response [33]

h(t; τ) =L∑

l=1

δ(τ − τl)αl(t)ejφl (t) (2.1.29)


As the mobile moves in the environment, the number and position of scatterers contributing

to the received signal will change and, therefore, the time delay τl, the amplitude fade αl,

the phase φl, and the number of resolvable paths L will also change with time and, there-

fore, in addition to specifying their first order statistics, one also needs to specify their time

correlation properties. The time correlation properties for the complex path gain βl(t) are

also described by the correlation function in (2.1.22). As suggested by measurements [39],

the time delays τl are characterized by a Poisson or modified Poisson process. More details

on characterizing the above parameters for mobile radio and indoor wireless channels can

be found in [39-42].

2.1.3 Log-Normal Slow Fading

The signal fading described in the previous section results from out-of-phase combining of

different multipath components. Since these phases change by π degrees every half wave-

length, the signal amplitude changes rapidly over very short distances as the mobile moves

in the propagation environment (approximately every foot for 900 MHz signals). If we aver-

age out these local variations due to multipath over large distances, the local mean will also

change with distance due to two effects: the propagation path loss with distance described

earlier, and the changing size and geometry of the surrounding buildings and obstacles that

attenuate radio waves as it propagates through the environment. Measurements are usually

used to predict the power loss with distance due to shadowing [21, 43, 44]. Although these

measurement data depend on the environment in which the measurement was taken, most

empirical studies show that the mean signal level is approximately log-normal, i.e. the dB

value of the mean signal power is Gaussian. The following argument suggests why the log-

normal distribution might be expected [23].

Suppose that the ith obstruction has an attenuation constant ai and thickness �ri. If si−1

is the amplitude of the wave entering this obstruction and si is the wave amplitude after the


obstruction, then

si = si−1 exp (−ai�ri) (2.1.30)

Then, it follows that the signal leaving the nth obstruction is given by

sn = so exp

(−

n∑i=1

ai�ri

)(2.1.31)

If we assume that ai and �ri vary randomly and independently from obstruction to obstruc-

tion, then as the number of obstructions gets large enough (n →∞) we can use the central

limit theorem to show that

x�= −

n∑i=1

ai�ri (2.1.32)

is approximately Gaussian. Therefore, S = 10 log10 s will have a Gaussian distribution with

mean µs and standard deviation σs. However, we should note that in a typical urban prop-

agation scenario, only few obstructions affect the signal propagation. The value of σs de-

pends on the environment and varies from 4 to 12 dB [25, 45-47].

The correlation behavior of the slow fading process is not known in general. However,

measurement data in [47] suggest that S(t) can be modeled as a first order Markov process

with autocorrelation

ρS(τ) = σ2s exp

(−v|τ|

Xc

)(2.1.33)

where v is the mobile velocity and Xc is the decorrelation distance which is a function of the

surrounding obstruction sizes and layout. Values of Xc under different measurement condi-

tions are reported in [47]. However, in a dense propagation environment such as New York

city where building sizes and heights vary considerably, the shadowing process is highly

non-stationary and an independent increments process model is more suitable [48].


2.2 Vector Multipath Channels

To support the development of real time adaptive antenna techniques for wireless applica-

tions, which is the focus of this thesis, multiple antenna channel models (vector channels)

are needed to analyze any such proposed approach and predict its performance. Some vec-

tor channel models have been reported [49, 50]. However, these models are incomplete in

the sense that they do not simultaneously account for channel time variation and fading cor-

relation between antenna elements. This problem has been addressed in [9]. In this section

we will develop a statistical model for multipath wireless vector channels. This new model

will be shown to extend the existing scalar channel models in [21, 31, 36, 39, 41, 51-53]

to the vector channel case. We begin by defining the array response vector and the narrow-

band data model for array signal processing. Then, we develop the statistical channel model

based on the physical description of the propagation environment. In the remainder of this

chapter and throughout the thesis we will use xT and x∗ to denote the transpose and Hermi-

tian transpose of x, respectively.

2.2.1 Array Response Vector

Figure 2.5 shows a wireless communication system employing an antenna array where a

base station with K antennas receives signals from a mobile user. In order to introduce the

array response vector concept, we consider a simple scenario where we have a single user.

We also assume that this user can be represented by a point source and that waves arriving

at the array can be considered planar (i.e. no multipath). Once we establish a model for the

received signal for this case, the general model for multipath and multiple users case can

be simply obtained by the ”superposition” principle. In the following discussion,

we will use the so-called narrowband data model for array signal processing to obtain the

model for the received signal vector at the array. This model inherently assumes that as the

signal wavefront propagates across the array, the envelope of the signal remains essentially


Mobile User

Antenna 1

Antenna 2

Antenna K

Base-Station

Receiver Processor

User Signal

Figure 2.5: A Wireless communication system employing antenna arrays.

unchanged. The term narrowband is used here since the assumption of a slowly varying

signal envelope is most often satisfied when either the signals or the antennas have a band-

width that is small relative to the carrier frequency f . However, this assumption can also

be satisfied by wideband signals, as is the case in CDMA, provided that the frequency re-

sponse of each antenna is approximately flat over the signal bandwidth, and provided that

the propagation time across the array is small compared to the inverse bandwidth of the sig-

nal B−1. Under these assumptions, we can write the complex base-band representation of a

real narrowband received signal at the kth antenna as [54]

xk(t) = Hk(ω)e− jωτk s(t)+ nk(t) (2.2.1)

where s(t) is the complex baseband transmitted signal, Hk(ω) is the frequency response of

the kth antenna, τk is the propagation time delay, nk(t) is the additive white Gaussian noise

(AWGN), and ω = 2π f is the carrier frequency. We can write (2.2.1) as

x(t) = v(�)s(t)+ n(t) (2.2.2)


v(θ)

Array Manifold

Figure 2.6: A one-dimensional array manifold.

where x(t) is the array output vector and n(t) is the additive white Gaussian noise vector.

The vector v(�) ∈ C K×1 is the array response vector given by

v(�) = [H1(ω)e

− jωτ1 H2(ω)e− jωτ2 · · · HK(ω)e

− jωτK]T

(2.2.3)

and is a function of the parameter vector � which might include for example the location of

the mobile in some coordinate system, the signal carrier frequency, polarization angles, etc.

If there are p < K elements (different parameters) in v(�), then� will trace a p-dimensional

surface in C K as � is varied over the parameter space. This surface is referred to as the array

manifold and is denoted mathematically as

A = {v(�) : � ∈ Θ} (2.2.4)

where Θ denotes the set of all possible parameter vectors.

Throughout this thesis, we will assume that we have identical antennas in the array. In

this case Hk(ω) will be the same for all antennas. Also, all antennas will have the same


response in any given direction. In this case, the array response vector is parameterized by

the angular carrier frequency ω and the time delays τ1, τ2, · · · , τk, which can be shown to be

a function of the mobile position with respect to each antenna. To simplify the discussion

we also assume that the mobile is in the far-field and that the mobile and the antenna array

are in the same plane2. Under the above assumptions, the parameter vector contains only

the azimuth θ and the array manifold A is a one-dimensional rope winding through C K as

illustrated in Figure 2.6. In this case, by redefining the signal s(t) (by denoting H(ω)s(t)

as s(t) or by including H(ω) within the channel response), then

v(θ) = [e− jωτa

1 e− jωτa2 · · ·e− jωτa

K]T

(2.2.5)

where the time delays τa1, τ

a2, · · · , τa

K are now with respect to a given reference point and are

functions of the array geometry and θ. For example, for a uniform linear array (ULA) of

identical sensors as illustrated in Figure 2.7, taking the 1st antenna as a reference point, the

time delay τak is given by

τak = (k − 1)

d sinθc

, k = 1 · · · K (2.2.6)

where θ is the angle between the arriving signal and the normal to the array, and d is the

spacing between sensors. In this case the array response vector v(θ) is given by

v(θ) =

1

e− j2πd sin θ/λ

e− j4πd sin θ/λ

...

e− j2(K−1)πd sin θ/λ

. (2.2.7)

2These assumptions may be relaxed at the expense of complicating the array response vector model. Inthe case of near-field and 3-D arrays, three parameters are required to define the parameter vector � which areazimuth, elevation and range.


Mobile

θ

12345d

Figure 2.7: The uniform linear array scenario.

2.2.2 Vector Channel Modeling

In the vector channel model developed below, the mobile to base station link is emphasized.

As stated earlier, for simplicity, only azimuth angles are considered in the propagation ge-

ometry, but the results can be generalized to three dimensions. It is assumed that the mobile

antenna radiates uniformly in azimuth. The propagation environment under consideration

is densely populated with both natural and man-made structures. An illustration of the vec-

tor channel propagation environment is shown in Figure 2.8. The mobile radiation pattern

illuminates all local scattering structures, or local reflectors, surrounding the mobile that

are within few hundred wavelengths from the mobile. Radiation from these local reflectors

and/or the mobile reaches the base station either directly or by reflection from large reflect-

ing objects in the environment such as large buildings and hills. These objects are termed

dominant reflectors. Scalar channel models do not explicitly include the effects of these


Local Scattering Structure

Mobile

Dominant Reflector #1



Base Station Antenna Array

v

∆∆

Figure 2.8: An illustration of the vector channel.

dominant reflectors. For the vector channel, the angle of arrival of each reflected wave with

respect to the base station coordinate system is determined by the physical position of the

dominant reflector with respect to the base. An illustration of the vector multipath channel

is shown in Figure 2.8. Let s(t) be the complex baseband transmitted signal. The complex

baseband received signal vector at the base station antenna array can be written as

x(t) =L∑

i=1

v(θi) · Riej2π( fd cosψi t− f τi) · s(t − τi) (2.2.8)

where ψi is the direction of the local scatterer with respect to the mobile velocity vec-

tor and θi is the angle of arrival of the ith signal path which is also the angular position of

the ith dominant reflector (or local scatterer) with respect to the base station coordinate sys-

tem. v(θi) is the K × 1 array response vector for signals arriving in the ith wavefront and

is defined by (2.2.5). R2i is the fraction of the incoming power in the ith path.

In general, signals arrive at the receive antennas mainly from one given direction. For


example, in rural or suburban areas, a high base station antenna array typically has a line-

of-sight path to the mobile, with local scattering around the mobile generating signals that

arrive mainly within a given range of angles or beamwidth. We assume that all signals from

the mobile arrive at the base station antenna array uniformly within ±� of the mean angle

of arrival θ [50]. That is, we assume that

f�(θi) =

12� −�+ θ ≤ θi ≤ �+ θ,

0 otherwise(2.2.9)

The value of � is called the angle spread (around the mean angle of arrival). We may point

out that, although this assumption is not based on any measured data, according to [55],

analytical results obtained based on this assumption were shown to be consistent with mea-

surement data in [21].

Assuming that s(t) is a narrowband signal (T << B−1), then s(t− τi)≈ s(t− τo)where

τo ∈ [mini

τi,maxi

τi] and we can write the lowpass received signal vector as

x(t) ≈ s(t − τo) ·(

L∑i=1

v(θi) · Riejφi(t)

)(2.2.10)

where φi = 2π( fd cosψit − f τi). As before, the φi modulo 2π are assumed to be i.i.d. over

[0,2π]. We define the complex channel vector a(t) as

a(t) =L∑

i=1

v(θi)Riejφi(t) (2.2.11)

Similar to the scalar channel case, the lowpass vector channel is described by the time-

variant impulse response

h(t; τ) = δ(τ − τo) · a(t) (2.2.12)

For a large number of incoming paths, the complex channel vector will approach a zero

mean complex Gaussian random vector, that is

a(t) ∼ N (0,Ra) (2.2.13)


where Ra is the channel vector covariance defined as

Ra = E{a(t)a∗(t)} (2.2.14)

and is a function of the angular frequency ω, the mean angle of arrival θ, the angle spread

�, and the spacing between sensors and their geometry.

Most of the adaptive antenna arrays processing techniques, which are the focus of this

work, depend on the correlation of the received signals at the array. The resulting corre-

lation matrices will play a key role in this work, making it worthwhile to characterize and

study their structure, how this structure expresses and describes the propagation environ-

ment and signals propagating in that environment, and what structure is ”naturally” present

(induced by stationarity assumptions, for example). Therefore, we need to study the space-

time-frequency correlation behavior of the vector channel. In Appendix A.2, it is shown

that the space-time-frequency correlation matrix of a(ω1, t) and a(ω2, t + ν) is given by

Ra(ω1, ω2, ν) = E{a(ω1, t)a∗(ω2, t + ν)} (2.2.15)

= Jo(ωdν) · FT( j�ω) · Rs (2.2.16)

where Rs is the array spatial correlation matrix defined as

Rs = 12�

∫ �+θ

−�+θ

v(ω1, ϑ)v∗(ω2, ϑ)dϑ (2.2.17)

and v(ω, θ) is defined in (2.2.5). Similar to Ra, Rs is a function of the angular frequencies

ω1 and ω2, the mean angle of arrival θ, the angle spread �, and the spacing between sensors

and their geometry. For ω1 = ω2 = ω we will write Ra as Ra(ω, ν). In Appendix A.2, it is

shown that the real and imaginary parts of Rs(m, n), the signal correlation between the mth

and nth antenna elements, are given by

Re{Rs(m, n)} = Jo(zmn)+ 2∞∑

l=1

J2l(zmn) cos(2l(θmn + δmn))sinc(2l�) (2.2.18)

Im{Rs(m, n)} = 2∞∑

l=0

J2l+1(zmn) sin((2l + 1)(θmn + δmn))sinc((2l + 1)�) (2.2.19)


mnθ

dmn

r

rm

rnmnθr

dmn /2 dmn /2

Reference Point

Sensor m Sensor n

Figure 2.9: Model geometry.

where θm,n is the mean angle of arrival measured with respect to the normal to the line joining

the two sensors as shown in Figure 2.9 and zm,n and δm,n are defined in (A.2.13).For ω1 =ω2 = ω and ν = 0, we have Ra = Rs = Ra, δmn = 0, and zm,n = ωdm,n/c where dm,n is the

distance between the mth and nth sensors.

Figures 2.10 and 2.11 show the spatial envelope correlation ρs = |Rs(m, n)| for θ = 0◦

and θ = 30◦, respectively, and for various angle spreads � based on the expressions in

(2.2.18) and (2.2.19). Figure 2.10 shows that, as � decreases, the first zero in the correla-

tion occurs at a larger antenna spacing. Specifically, the first zero occurs for d/λ ≈ 30/�.

When the signal arrives from a direction other than broadside, as in Figure 2.11, the antenna

spacing for low correlation increases and the envelope correlation is never zero for all val-

ues of θ �= 0◦ and � < 180◦ ( ρs is zero when Re{Rs(m, n)} and Im{Rs(m, n)} have zero

crossings at exactly the same spacing).

The space-time array correlation matrix Ra(ω, ν) provides a full characterization of the


Antenna Separation (d / λ)

0 1 2 3 4 5

Spat

ial E

nvel

ope

Cor

rela

tion

ρs

0 .0

0.2

0.4

0.6

0.8

1.0

1.2

∆ = 0ο

∆ = 5ο

∆ = 15ο

∆ = 60ο

θ = 0ο

Figure 2.10: Spatial envelope correlation vs. antenna spacing: mean AOA θ = 0◦

dynamics of the channel vector a(t). First, we rewrite Ra(ω, ν) as follows

Ra(ω, ν) = ρ(ν) · Rs (2.2.20)

where ρ(ν) is the time correlation part, which under the assumptions stated above, was

shown to be Jo(ωdν). As time progresses, the channel vector a(t) amplitude and direction

in C K will change. While the time correlation part of Ra describes how fast a(t) changes,

with the rate of change being proportional to ωd, the spatial correlation part Rs character-

izes the complexity of change of a(t) (in terms of the dimension of the subspace spanned

by a(t)) in C K . This complexity is proportional to the angle spread �. By (2.2.11), a(t) is

a linear combination of the path vectors v(θ1), v(θ2), · · · ,v(θL), and hence it tends to lie

in the subspace spanned by them. For example, if the angle spread is small, all path vectors


Antenna Separation (d / λ)

0 1 2 3 4 5

Spat

ial E

nvel

ope

Cor

rela

tion

ρs

0 .0

0.2

0.4

0.6

0.8

1.0

1.2

∆ = 0ο

∆ = 5ο

∆ = 15ο

∆ = 60ο

θ = 30ο

Figure 2.11: Spatial envelope correlation vs. antenna spacing: mean AOA θ = 30◦

will point approximately to the same direction. Therefore, the fluctuations in a(t) will be

mostly in magnitude only and for a large percentage of time it will be in a certain preferred

direction. In this case Rs will have off-diagonal correlation terms that are a large fraction

of the diagonal entries. In this case, we have nonspace selective fading. For a large angle

spread, the path vectors will span the full space and therefore the a(t) does not exhibit any

preferred direction and the changes of a(t) in C K will be in both amplitude and direction.

The off-diagonal entries in Rs will become very small compared to the diagonal elements.

In this case, we have space selective fading.


0

2

4

6

80

50100

150200

−8

−6

−4

−2

0

2

4

Space (d / λ)

Sig

nal L

evel

(dB

)

Time (Symbol Period T s )

Figure 2.12: Space-Time Fading: fd = 50 Hz, � = 0◦, and Ts = 208.3µs.

A measure of the complexity of a(t) variability can be obtained by considering the eigen-

value spectrum of Rs. Each eigenvalue is a measure of how strongly a(t) points in the cor-

responding eigendirection in C K . If the eigenvalues are all approximately equal, then a(t)

spans the full space. If only few of the eigenvalues are large compared to the others, the

variability of a(t) will tend to be confined to the subspace spanned by the corresponding

eigendirections.

To demonstrate the effect of angle spread and Doppler spread on the channel vector dy-

namics, the above channel model was simulated. The mobile was assumed to be moving

at 37.5 mph which corresponds to a maximum Doppler shift of 50 Hz at 900 MHz carrier

frequency. The transmitted signal is a balanced QPSK signal with a symbol rate of 4800


0

2

4

6

80

50100

150200

−10

−5

0

5

Space (d / λ)

Time (Symbol Period Ts )

Sig

nal L

evel

(dB

)


symbols/sec. We considered a base station with a ULA with aperture size 8λ. Figures 2.12,

2.13, and 2.14 plot the received signal level across the array over a period of 200 symbols for

angle spreads of 0◦, 3◦, and 40◦ respectively. For the zero angle spread case in Figure 2.12,

we notice that the signal level is the same across the array. This is because, with zero angle

spread, all paths contributing to the received signal at the array arrive from the same direc-

tion and have the same relative phase at different points across the array and therefore will

add up (either constructively or destructively) in the same way. In this case, we have time-

selective fading only. For angle spread � = 3◦ in Figure 2.13, we notice that, for any given

instant of time, the signal level across the array will vary. In this case, different paths con-

tributing to the received signal arrive from different directions and their relative phases will


0

2

4

6

80

50100

150200

−20

−15

−10

−5

0

5

Space (d / λ )

Time (Symbol Period T s )

Sig

nal L

evel

(dB

)


be different, and, therefore they will add up either constructively or destructively at each

point across the array depending on the relative phase relationship. As � increases, as in

Figure 2.14, the signal level across the array will change more rapidly. In this case, we have

space-time selective fading.

As in the scalar channel case, for wideband signals, the vector channel response can be

approximated using Turin’s subpath model described in the previous Section. In this case,

the equivalent lowpass vector channel is described by the time-variant impulse response

h(t; τ) =L∑

l=1

δ(τ − τl)al(t) (2.2.21)

where L is the number of resolvable paths or subpath clusters. The lth channel vector al(t),


as defined in (2.2.11), is itself a linear combination of a large number of path vectors that

are approximately the same length. Each channel vector al(t) is characterized by its mean

angle of arrival θl and angle spread�l and the corresponding array spatial correlation matrix

Rs(�l, θl). Under the same assumptions above, we can show that the channel vectors for

two different resolvable paths are independent, i.e.

E{al(t)a∗k(t + ν)} =

Jo(ωdν) · Rs(�l, θl) if l = k,

0 otherwise(2.2.22)

2.2.3 Path Amplitudes and Power-Delay Profile

As discussed earlier, the short term time-averaged signal power has a log-normal distribu-

tion. In other words, the mean square value of the fast fading on the lth vector channel Sl

is itself a random variable that has a log-normal distribution. Empirical data [40, 39] show

that the slow fading amplitudes can be correlated from path to path. In this case, we have

Sl = 10ξl10 and ξl ∼ N (µs, σ

2s ) l = 1 · · · L (2.2.23)

� ∼ N (�ξ,Rξ), � = [ξ1 ξ2 · · · ξL] (2.2.24)

and µs is the area mean and σs varies between 4-12 dB depending on the degree of shad-

owing. In general, the average power output of the channel is not constant with time delay.

The average propagation loss increases with time delay resulting in a reduction in average

path power as path delay τl increases. The effect is accounted for by the deterministic power

delay profile p(τl ). Therefore, we rewrite the time-variant impulse response of the vector

channel as

h(t; τ) =L∑

l=1

√�l · δ(τ − τl)al(t) (2.2.25)

where �l = Sl · p(τl ).

For the vector channel impulse response defined above, we have al(t) and �l modeled

as random variables. As such we define the power delay profile of the vector channel P (τ)


as

P (τ) = E{h∗(t; τ)h(t; τ)} (2.2.26)

=L∑

l=1

K�l · δ(τ − τl ) (2.2.27)

where K = E{a∗l (t)al(t)} and �l = E{�l}. Here, we used the fact that al(t) and ak(t) are

independent for l �= k. By estimating P (τ) for any given channel, one can estimate the rms

delay spread of the channel τrms as follows. If we model P (τ) as a probability distribution

by normalizing it with∫∞

0 P (τ)dτ, then τrms is the standard deviation of τ with ”probability

distribution” P (τ) [56]. That is

τrms =

√√√√∫∞0 τ2P (τ)dτ∫∞

0 P (τ)dτ−(∫∞

0 τP (τ)dτ∫∞0 P (τ)dτ

)2

(2.2.28)

=

√√√√∑Ll=1 2�lτ

2l∑L

l=1 �l

−(∑L

l=1 2�lτl∑Ll=1 �l

)2

(2.2.29)

Analysis of measurement data [5] shows that for some propagation environments, P (τ) can

be approximated by a decaying exponential of the form P0 · e−τ/τrms where P0 is the total

output power of the channel.

2.3 Cellular Channels

As we mentioned earlier, in cellular systems the coverage area is divided into cells where in

each cell only one user is assigned to a particular radio channel (where a channel is either a

time slot, a frequency band or a specific code). In order to determine the spatial reuse of ra-

dio channels, data rates, and system layouts, models for signal propagation in each cell type

are required. Propagation measurements for different cellular channel types have been well

documented in [25, 32, 35, 56-58]. Based on those results, we now describe the propagation

environment for different types of cells based on their size in the following sections.


2.3.1 Macro Cells

Macro cells correspond to cells where the base station is placed on top of tall buildings or

towers and transmits enough power to cover several miles. The physical propagation en-

vironment in macro cells is characterized by near-ground irregularities (buildings, terrains,

etc). The propagation path length can be up to several miles. Macro cells can be classified

into different channel types: urban, suburban, and rural.

In urban channels, the propagation environment consists of a high density of scatterers

(buildings) irregularly distributed and shaped. The size of each scatterer can be very large

compared to the wavelength λ. Due to the high density of scatterers, signals will experi-

ence heavy shadowing. The measured impulse response profiles in urban areas are typically

concentrated within a delay window of about 1-1.5 µs. Because of strong attenuation and

shadowing, contributions at large delays are rare and no direct path exists. The recorded

impulse response results from diffraction and scattering due to structures in the immediate

neighborhood of the mobile (within a few hundred meters). The fast fading of each resolv-

able path is well described by the Rayleigh fading model. The number of resolvable paths

in the impulse response depends on the bandwidth used. For narrowband systems, only one

peak will be apparent. For wideband systems more than one peak appears.

In suburban channels, we have a less irregular environment because of open areas and

lower density of strong scatterers. The measured impulse response in suburban areas shows

that the direct path occurs with higher probability due to less shadowing. The measurement

studies in [25, 35, 56, 57] showed that the delay spread rarely exceeds 1-1.5 µs. However,

the measurement studies in [32, 58] show that particularly difficult locations (those with

long unblocked paths to significant scatterers such as city skylines or mountains) can offer

delay spreads of up to 15-20 µs, with significant multipath components at delays of about


100-120 µs. The number of peaks in the impulse response will depend on the specific ter-

rain. The fast fading is well described by the Nakagami fading model with m ranging be-

tween 1 and 15.

For rural or open areas, very few buildings or scatterers exist and therefore we have ei-

ther low or no shadowing at all. The delay spread is on the order of 0.1-0.3 µs. The impulse

response in this case typically consists of one peak. The fast fading is also well described

by the Nakagami fading model with m up to 100.

2.3.2 Micro Cells

Micro cells correspond to small cells with high user density (business and industrial areas,

etc.). The base station antenna is usually below rooftops. The propagation path length is a

few hundred meters. In micro cells there are two types of propagation: LOS and non-LOS

propagation.

The LOS propagation in micro cells is well modeled by the dielectric canyon model de-

scribed in [28]. The measured impulse response consists of a strong first path and small

components from wall reflections within 1.5µs. In non-LOS propagation, the signal reaches

the receiver due to coupling with side streets and multiple reflections from buildings. The

measured impulse response consists of many paths within 500 ns. Ray tracing techniques

can be used to model non-LOS propagation. These require detailed information about build-

ings and street layout, dielectric properties etc in the cell of interest and the resulting model

applies for that particular cell only.

2.3.3 Pico or Indoor Cells

Pico cells [5] are used to cover indoor areas inside buildings with various dimensions, con-

struction, arrangement, and furniture. The scattering process in pico cells is three dimen-

sional and is confined. The measured impulse response in pico cells resembles that of the


micro cells. If a LOS path is present, the measured impulse response consists of a strong

first path and few smaller components due to wall reflections within 50-100 ns. In non-LOS

propagation, the measured impulse response will include many paths that are incident from

all directions and within 100-150 ns.

2.4 Summary

The first half of Chapter 2 is a brief overview of radio channel modeling and describes some

problems associated with a wireless radio channel. The models described therein are for

scalar channels. In the second half of Chapter 2 we develop a statistical vector channel

model based on the physical propagation environment and its statistical properties. Such

a model is necessary in order to analyze and design adaptive beamforming techniques with

antenna arrays at the base station that will be introduced in the next chapter.

Chapter 3

Adaptive Beamforming with Antenna

Arrays

The performance of digital mobile radio communication systems is limited by signal fading

and interference from other co-channel users [59-65]. Both these effects can be reduced by

the use of antenna arrays at the base station with the appropriate signal processing and com-

bining of the received signals [11, 12, 60-67]. Specifically, by optimally combining antenna

outputs, adaptive antennas can reduce the multipath fading and permit the use of the spatial

dimension to suppress co-channel interference. The operation of optimally combining the

array output is called beamforming or spatial filtering.

An adaptive array system consists of an array of spatially distributed antennas and an

adaptive signal processor that generates a weight vector for combining the array output. Fig-

ure 3.1 shows a block diagram of an adaptive array system. The exact structure of the signal

processor is dependent on the amount of information that is available or can be estimated

at the base station. This information includes the type of modulation and signaling format,

the number of resolvable signal paths that are received at the base station, the direction of

arrival and time delay of each path signal, availability of reference or training signals, and

the complexity of the propagation environment.

46

Chapter 3. Adaptive Beamforming with Antenna Arrays 47

w1

w2

wK

Adaptive SignalProcessor

Σ arrayoutput

Available Informations

y(t)

x (t)

x (t)

x (t)

1

2

K

Mobile

sensor 1

sensor 2

sensor K

Figure 3.1: Block diagram of an adaptive array system

Several algorithms have been proposed in the array signal processing literature to de-

sign adaptive beamformers to separate multiple co-channel signals based on the availabil-

ity of prior spatial or temporal information. The traditional spatial algorithms combine high

resolution direction-finding (DF) techniques such as MUSIC, ESPRIT, and WSF (weighted

subspace fitting) [68-71] with optimum beamforming to estimate the signal waveforms [66,

72]. However, these techniques require certain assumptions on the number of signal wave-

fronts arriving at the base station and on the complexity of the propagation environment,

which restricts their applicability in a wireless mobile communications setting. Other tech-

niques use reference or training signals to find the optimum adaptive beamformer [73-75].

In the recent past, several property-restoral techniques have been developed that exploit the

temporal or spectral structure of communication signals, while assuming no prior spatial in-

formation. These techniques take advantage of signal properties such as constant modulus

(CM)[76-78, 67], discrete alphabet [79-82], self-coherence [83], and high order statistical

properties [84].


In this chapter we will review some of the previous techniques for adaptive beamform-

ing. Then, we will derive a novel technique for estimating the vector channel and the corre-

sponding adaptive beamformer for CDMA wireless systems. In this technique, we perform

code-filtering at each antenna for each user in the system. We exploit the eigenstructure

of the pre- and post-correlation array covariance matrices to estimate the channel vector

and derive the corresponding adaptive beamformer. We also extend this technique to the

case where we have multipath propagation. The resulting overall receiver structure is called

Beamformer-RAKE. Our approach is a blind technique in the sense that it does not require

any training signals, although it assumes the perfect knowledge of the spreading code for

each user. Also, it does not require any assumptions on the signal propagation and is there-

fore suitable for different propagation settings.

3.1 Adaptive Beamforming Techniques

We consider a typical mobile radio communication link as described in Chapter 2 with an

elevated base station with an antenna array of K elements. The received signal at the base

is due to reflections from all scattering structures in the propagation environment. Let so(t)

be the complex envelope of the transmitted baseband signal and ao be the corresponding

channel vector. Assuming that the delay spread is much less than the inverse bandwidth

of the signal, we can use (2.2.10) and write the received signal vector at the base station

antenna array as

x(t) = so(t)ao + i(t)+ n(t) (3.1.1)

where

i(t) =N∑

i=1

si(t)ai (3.1.2)

represents the composite of the N cochannel interferers and n(t) = nc(t) + jns(t) is the

additive noise vector, which is modeled as a complex Gaussian random vector with zero


mean and covariance

E{n(t)n∗(t)} = σ2nI (3.1.3)

where I is the K × K identity matrix and σ2n is the antenna noise variance. Equation 3.1.3

implies that the noise is spatially white. The noise vectors nc(t) and ns(t) are both low-

pass white Gaussian random processes. In addition, the noise vector n(t) is assumed to be

temporally white, i.e

E{n(t1)n∗(t2)} = σ2nIδ(t1 − t2) (3.1.4)

This modeling of the noise vector does not take into account the effect of low pass filtering.

This filtering will be considered in the performance analysis results presented in Chapter 6.

Let wo represent the desired weight vector for linearly combining the array outputs.

From Figure 3.1, the output of the beamformer y(t) is given by

y(t) = w∗ox(t) (3.1.5)

where (.)∗ denotes the Hermitian transpose. As we mentioned above, there are several tech-

niques to estimate wo. A brief review of some of these techniques follows.

3.1.1 Direction-Finding Based Beamforming

The direction-finding based beamforming techniques are based on the assumptions that the

angle spread � is either zero or relatively small such that all the unresolvable paths that

contribute to the received signal will essentially arrive from the same direction. In such

case the channel vector as defined in (2.2.11) becomes

ai ≈ v(θi) ·(

L∑l=1

Rlejφl (t)

)= βi(t) · v(θi) (3.1.6)

where θi is the angular position of the ith source with respect the base station. In this case,

we rewrite the received signal vector x(t) as

x(t) = so(t)v(θo)+N∑

i=1

si(t)v(θi)+ n(t) (3.1.7)


where si(t) = βi(t) · si(t). Let σ2i = E{si(t)s∗i (t)}. A reasonable assumption to make is that

the signals so(t), s1(t), · · · , sN(t) are uncorrelated. All DF-based techniques use the array

output x(t) and the knowledge of the array manifold A to get an estimate of the directions of

arrival (DOA) θo, θ1, · · · , θN and the corresponding estimates of the array response vectors

v(θo),v(θ1), · · · ,v(θN ). A number of high resolution techniques such as MUSIC, ESPRIT,

and WSF can be used to estimate the DOAs.

A reasonable strategy is to find the best weight vector to optimally combine the array

outputs under some suitable criterion. Minimization of mean squared error (MMSE), max-

imization of signal to interference-plus-noise ratio (SINR), and maximum likelihood (ML)

have been widely used as optimization criteria, and in all these cases the optimal weight

vector turns out to be a function of signal strengths of the desired and undesired signals,

their directions of arrival, and their covariances. Before we proceed to derive the optimal

weight vector under any of those criteria, we first define some of the statistical quantities

for the signal vector model in (3.1.7). Let u(t) = i(t)+ n(t) be the total undesired signal

vector. We define the array covariance Rxx and the undesired signal vector covariance Ruu

as

Rxx = E{x(t)x∗(t)} =N∑

i=o

σ2i v(θi)v∗(θi)+ σ2

nI (3.1.8)

Ruu = E{u(t)u∗(t)} =N∑

i=1

σ2i v(θi)v∗(θi)+ σ2

nI (3.1.9)

Maximization of SINR Beamformer

The weight vector wo can be optimized by maximizing the SINR at the output of the

beamformer. First, we rewrite (3.1.7) as

x(t) = so(t)v(θo)+ u(t) (3.1.10)

This gives the output SINR to be

(SI N R)o = E{|w∗oso(t)v(θo)|2}

E{|w∗ou(t)|2} = σ2

o|w∗ov(θo)|2

w∗oRuuwo

(3.1.11)


Using Schwarz’ inequality in (3.1.11), we get

(SI N R)o = σ2o|(R1/2

uu wo

)∗ (R−1/2

uu v(θo)) |2

w∗oRuuwo

(3.1.12)

≤ σ2ov∗(θo)R−1

uu v(θo) (3.1.13)

�= (SI N R)max (3.1.14)

with the equality achieved for

wSI N R = ζ · R−1uu v(θo) (3.1.15)

where ζ is any nonzero complex constant.

Maximum Likelihood Beamformer

The derivation of the maximum likelihood optimum beamformer requires the assump-

tion that the composite of the interferers i(t) is Gaussian so the total undesired signal vector

u(t) = i(t)+ n(t) will have a multivariate Gaussian distribution. We define the likelihood

function of the input signal vector as

L(x(t)) = fU(x(t)) = 1|πRuu|e

−[x(t)−s(t)v(θo)]∗R−1

uu [x(t)−s(t)v(θo)] (3.1.16)

The ML optimum beamformer is obtained by solving for the estimate of s(t) that maximizes

(3.1.16) or equivalently its logarithm. Taking the partial derivative of log (L(x(t)) and set-

ting the result to zero yields

0 = ∂ log (L(x(t))∂s(t)

= −2v∗(θo)R−1uu x(t)+ 2s(t)v∗(θo)R−1

uu v(θo) (3.1.17)

It immediately follows that the estimate of s(t) that maximizes log (L(x(t)) is given by

s(t) =(R−1

uu v(θo))∗

x(t)v∗(θo)R−1

uu v(θo)= w∗

MLx(t) (3.1.18)

Thus, the ML optimum weight vector has the form

wML = R−1uu v(θo)

v∗(θo)R−1uu v(θo)

(3.1.19)


Comparing (3.1.19) with (3.1.15), we notice that the ML beamformer also maximizes the

SINR.

MMSE Beamformer

The MSE beamformer minimizes the error e(t) between the beamformer output w∗ox(t)

and the desired signal s(t). Thus

e(t) = s(t)− w∗ox(t) (3.1.20)

and the mean squared error ε(wo) is given by

ε(wo) = E{|e(t)|2} = σ2o − Re{2w∗

orxs − w∗oRxxwo} (3.1.21)

where rxs = E{x(t)so(t)}. Taking the partial derivative of ε(wo) with respect to wo and set-

ting the result to zero yields

0 = ∂ε(wo)

∂wo= −2(Rxxwo − rxs) (3.1.22)

or

wMSE = R−1xx rxs (3.1.23)

This is often referred to as the optimum Wiener solution [85]. With the assumption that

the signals so(t), s1(t), · · · , sN(t) are uncorrelated, rxs = σ2ov(θo) and the use of the matrix

inversion lemma [86], gives

wMSE = σ2o

1 + v∗(θo)R−1uu v(θo)

· R−1uu v(θo) (3.1.24)

In this case, we see that the MMSE beamformer wMSE is a scalar multiple of the SINR beam-

former wSI N R in (3.1.15). This is only natural, since for Gaussian measurements, the MMSE

and the ML criterion lead to the same estimator [87].

Given S snapshots of the received signal vector at the array x(1),x(2), · · · ,x(S) and the

estimates of the array response vectors V(�) = [v(θo),v(θ1), · · · ,v(θN )], it was shown in


[72] that the maximum likelihood (ML) estimates of Rxx and Ruu are given by

Rxx = 1S

S∑l=1

xlx∗l (3.1.25)

Ruu = Rxx − σ2ov(θo)v∗(θo) (3.1.26)

where σ2o is the ML estimate of the desired signal power and is given by the (1,1) element

of

Rss = V†(�)(Rxx − σ2nI)V†∗(�) (3.1.27)

and

V†(�) =(

V∗(�)V(�))−1

V∗(�) (3.1.28)

is the pseduo-inverse of V(�). Furthermore, σ2n is the ML estimate of the noise variance

and is given by

σ2n =

1K − N − 1

· Tr{

P⊥V

Rxx

}(3.1.29)

where P⊥V= I−V(�)

(V∗(�)V(�)

)−1V∗(�) is the orthogonal projection into the null space

of V(�) and Tr(.) denotes the trace operator.

Based on the above covariance and array response vector estimates, the optimum weight

vector estimates according to the different performance measures discussed above are sum-

marized in Table 3.1.

While DF-based beamforming techniques are analytically more tractable, they suffer

from several drawbacks that may limit their applicability in a wireless setting. First, all tech-

niques start with a DOA estimation step that involves an eigen-decomposition and one or

more multidimensional, non-linear optimizations, which may be a difficult and time con-

suming task. Also, the DOA step requires knowledge of the array manifold and is very

sensitive to errors in this knowledge. The presence of multipath in urban or suburban en-

vironments, where the condition of zero or relatively small angle spread � does not hold,

makes knowledge of the array manifold unreliable. Moreover, a key assumption in all DOA


Performance Measure Optimum Beamformer Estimate

SINR wSI N R = R−1uu v(θo)

ML wML = R−1uu v(θo)

v∗(θo)R−1uu v(θo)

MMSE wMSE = σ2o

1 + v∗(θo)R−1uu v(θo)

· R−1uu v(θo)

Table 3.1: Optimum weight vector for SINR, ML, and MMSE performance measures

estimation techniques is that the number of signal wavefronts including cochannel interfer-

ence signals must be less than the number of antennas in the array, a fact that restricts their

applicability in a wireless setting.

3.1.2 Beamforming Based on Training-Signals

An attractive alternative to DF-based beamforming techniques assumes the availability of

a set of so-called reference or training signals, one for each user. The training signal based

techniques offer computationally inexpensive estimates of the signal waveforms, without

requiring a preceding DOA estimation step. The advantage of this method is that knowl-

edge of the DOA or the array manifold is not necessary. Moreover, this technique does not

make any assumption about the multipath angle spread and it does not place any structural

constraints on the antenna array itself.

Given a reference signal d(t), the weight vector can be chosen to minimize the mean

squared error between the beamformer output and the reference signal. The resulting opti-

mum beamformer solution is the Wiener-Hopf solution in (3.1.23),

wMSE = R−1xx rxd (3.1.30)


where rxd = E{x(t)d(t)}. Therefore, given the snapshots of the received signal at the ar-

ray x(1),x(2), · · · ,x(S) and the corresponding reference signal d(1), d(2), · · · , d(S), the

estimate of the beamformer weight vector is

wMSE = R−1xx rxd (3.1.31)

where Rxx is given by (3.1.25) and rxd is similarly defined as

rxd = 1S

S∑l=1

xl · d(l) (3.1.32)

The weight vector estimate can be calculated by a number of different techniques such

as LMS [88] or Direct Matrix Inversion (DMI) [85]. The technique of training-signals based

beamforming was studied for use with the digital mobile radio system IS-54 in [74]. In this

approach, the base station uses the synchronization sequence in each block as a reference

signal and uses DMI as in (3.1.31) for weight acquisition in each block.

The use of training signals requires prior carrier and symbol recovery, which is made

difficult by the presence of co-channel interference. In addition, sending a training sig-

nal along with the information signal reduces the payload efficiency of the system. Also,

training-signals based beamforming techniques are not applicable in systems where train-

ing or synchronization signals are not available such as in the IS-95 CDMA standard.

3.1.3 Signal-Structure-Based Beamforming

In signal-structure based beamforming techniques, the base station adaptive processor

exploits the temporal and/or spectral structure and properties of the received signal to con-

struct the beamformer. The signal property is damaged by the presence of interference and

the adaptive beamformer attempts to restore the signal property at its output and thus au-

tomatically reduces interference. Examples of property restoral beamforming techniques

include the constant modulus algorithm (CMA), which is used to correct FM, PSK, FSK,

and QAM signals on the basis of low modulus variations. A stronger signal constraint than


FM Signal 2

FM Signal 1

s (t)2

s (t)1

s (t)1^

wBeamformer

signal corruptedby interference

property restored

Figure 3.2: Constant modulus property restoral beamformer

that in the CMA algorithms is the finite alphabet (FA) property of digital signals. In ad-

dition to the low modulus variation, this approach uses the digital modulation and channel

coding structure, and the base station processor attempts to construct the beamformer by fit-

ting the FA model to the underlying data model described in (3.1.1). Another class of signal-

structure based beamforming is based on spectral self-coherence. Most communication sig-

nals are correlated with frequency-shifted and possibly conjugated versions of themselves

for certain discrete values of frequency shift. This spectral self-coherence is commonly in-

duced by periodic gating, mixing, and multiplexing operations at the transmitter.

As an example of the property-restoral beamforming techniques, a brief description of

the CMA-based beamforming follows. Because of multipath propagation and the presence

of interference, the received signal at the array output will exhibit amplitude fluctuation al-

though the transmitted signal has a constant envelope. As shown in Figure 3.2, the objective

of the CMA beamformer is to restore the array output y(t) to a constant envelope signal on

the average. This can be accomplished by choosing the weight vector w to minimize a cost

function J(y(t)) that provides a measure of the amplitude fluctuation. An example of such


a cost function is the mean squared amplitude fluctuation

ε(t) = E

{∣∣∣∣y(t)− y(t)|y(t)|

∣∣∣∣2}

(3.1.33)

Other cost functions may be used. Note that the noise tolerance and the convergence be-

havior of the CMA will depend on the choice of the cost function. The update equation of

the weight vector w is [76, 77]

w(t + 1) = w(t)−µe(t)∗x(t) (3.1.34)

e(t) = y(t)− y(t)|y(t)| (3.1.35)

where µ is the step constant. Equation (3.1.34) is very similar to the LMS algorithm. How-

ever, the CMA beamformer does not require any reference signal and uses the array output

y(t) itself to generate the reference signal.

The signal-structure based beamforming techniques are very robust against different

propagation conditions. No knowledge of the array manifold or DOAs is required. How-

ever, a drawback that limits the applicability of signal structure-based beamforming meth-

ods for wireless applications is that the convergence and capture characteristics are still not

well understood.

3.2 Adaptive Beamforming for Wireless CDMA

In CDMA wireless systems, each user within a cell modulates the information signal with

a wideband semi-orthogonal coding sequence. Although all users use the same frequency

band at the same time, the base station can separate each of the received signals by separately

decoding each spreading sequence. However, since the codes are semi-orthogonal, the users

within a cell interfere with each other (intracell interference) and with other users in other

cells (intercell interference). Both the intercell and intracell interference power are reduced

by the spreading gain of the code [89].


The different beamforming techniques discussed in the previous section are not suitable

for CDMA systems for the following reasons. First, as we mentioned above, all users in a

CDMA wireless system are cochannel and their number may easily exceed the number of

antennas. Moreover, due to multipath propagation and the fact that each transmission path

may contain direct, reflected and diffracted paths at different time delays, the array manifold

may be poorly defined. Therefore DF-based beamforming techniques are not applicable.

Also, no training or reference signals are present in the mobile to base link [8]. Therefore,

reference-signal based techniques cannot be used. In this section we propose a new space-

time processing framework for adaptive beamforming with antenna arrays in CDMA. This

approach may be seen as a signal-structure based technique since, as will be seen later, it

exploits both temporal and spatial structure of the received signal

3.2.1 CDMA Signal Models

A useful starting point is to develop a model for the total received signal vector at the base

station antenna array. We assume that the cell site alone uses an antenna array to receive

and transmit signals from and to mobiles. No antenna arrays are assumed for mobiles. In

order to simplify the derivation of the optimum beamformer, we will consider a single cell

system, although we can easily follow the same approach in the case of a multicell system,

as has been reported in [11]. Let N denote the number of users in the cell. We shall assume

that the mobile transmitter uses balanced DQPSK spreading as shown in Figure 3.3 where

the same differentially encoded information bit is used in both the in-phase and quadrature

channels. Then, we can write the transmitted signal from the ith user as

si(t) =√

Pi · bi(t)(cI

i (t) cos(ωct)+ cQi (t) sin(ωct)

)(3.2.1)

where Pi is the ith user’s transmitted power per dimension (i.e. the total transmitted power

is 2Pi), ωc is the angular carrier frequency, bi(t) is the differentially encoded bit of duration

Tb, and cIi (t) and cQ

i (t) are the in-phase and quadrature spreading codes used by the ith user,


differentialencoder

Informationbits a (I) ωc tcos

(Q)a ωc tsin

s (t)ib (t)i

Figure 3.3: Mobile transmitter block diagram

respectively. In our analysis, the spreading code cIi (t) is represented by

cIi (t) =

∞∑h=−∞

cIi,h p(t − hTc) (3.2.2)

where ci,h are assumed to be i.i.d. random variables taking values±1 with equal probability,

Tc is the chip duration, and p(t) is the chip pulse shape. p(t) can be any time-limited wave-

form. Here we assume that p(t) is rectangular although our results can be easily extended

for any time-limited waveform. The processing gain is defined as G = Tb/Tc.

We assume the multipath vector channel model defined in (2.2.21). We also assume

that the channel parameters vary slowly as compared to the bit duration Tb, so that they are

constant over several bit durations. After down-converting to baseband, we can write the

complex baseband received signal vector from the i-th user at the K element antenna array

as

xi(t) =L∑

l=1

√Pl,ibi(t − τl,i)ci(t − τl,i)e

jφl,i al,i (3.2.3)

where√

Pl,i is the average power per dimension from the ith user in the lth multipath com-

ponent, ci(t) = cIi (t)+ jcQ

i (t), φl,i = ωcτl,i, and L is the number of multipath components.

The total received signal at the cell site is the sum of all the users’ signals plus noise and is

given by

x(t) =N∑

i=1

L∑l=1

xl,i(t)+ n(t) (3.2.4)


3.2.2 Code-Filtering Approach for Adaptive Beamforming

We will start by considering the simple scenario where we have a single propagation path,

i.e, L = 1. In the following section, we will extend the approach developed below to the

L > 1 case. In a wireless CDMA system, a large number of users distributed around the

base station communicate simultaneously with the base station at almost the same power

level. Therefore, with an antenna array with a few elements, adaptive beamforming cannot

null out cochannel users for any given user, though it will, of course, maximize the SINR.

We propose the following approach for estimating the channel vector for the ith user ai and

the corresponding beamformer weight vector wi. Without loss of generality, let us assume

that we want to estimate the channel vector of the 1st user. Let τ1 be the corresponding time

delay, which is assumed to be perfectly estimated (later in this chapter, we will show how

to use the adaptive beamforming algorithm derived below to also estimate the time delays

of the multipath components). Then, we can write the total received signal as

x(t) =√P1b1(t − τ1)c1(t − τ1)e

jφ1a1 + i(t)+ n(t) (3.2.5)

where

i(t) =N∑

i=2

√Pibi(t − τi)ci(t − τi)e

jφiai (3.2.6)

is the composite of multiple access interference (MAI). In our approach, the antenna outputs

are correlated with c1(t − τ1) to yield one sample vector per bit. The post-correlation signal

vector for the nth bit is given by

y1(n) = 1√Tb

∫ nT+τ1

(n−1)T+τ1

x(t)c∗1(t − τ1)dt (3.2.7)

= 2√

Tb P1b1(n)ejφ1a1 + i1 + n1 (3.2.8)

where

n1 = 1√Tb

∫ nT+τ1

(n−1)T+τ1

n(t)c∗1(t − τ1) dt (3.2.9)


is the undesired component due to thermal noise and

i1 =N∑

i=2

√Pi I1,ie

jφiai (3.2.10)

is the undesired component due to the multiple access interference, where I1,i is defined as

I1,i = 1√Tb

∫ nT+τ1

(n−1)T+τ1

bi(t − τi)ci(t − τi)c∗1(t − τ1) dt (3.2.11)

In Chapter 6, we show that n1 is a zero mean complex Gaussian vector with covariance

2σ2nTcI and that I1,i, i = 2 · · · N are i.i.d with zero mean and variance 4Tc

1. Using (3.2.5)

and (3.2.6), we can show that the short term covariance of the pre-correlation signal vector

x(t) is given by

Rxx = E{x(t)x∗(t)} = 2P1a1a∗1 +

N∑i=2

2Piaia∗i + σ2

nI (3.2.12)

Similarly, using (3.2.8), (3.2.9), and (3.2.10), we can show that the short term covariance of

the post-correlation signal vector y1(n) is given by

Ryy,1�= 1

2TcE{y1(n)y∗

1(n)} = 2GP1a1a∗1 +

N∑i=2

2Piaia∗i + σ2

nI (3.2.13)

Let u1 = i1 + n1 be the total undesired component due to thermal noise and multiple access

interference. Then, the short term interference plus noise covariance is

Ruu,1 = E{u1u∗1} =

N∑i=2

2Piaia∗i + σ2

nI (3.2.14)

Note that we can rewrite both Rxx and Ryy,1 as

Rxx = 2P1a1a∗1 + Ruu,1 (3.2.15)

Ryy,1 = 2GP1a1a∗1 + Ruu,1 (3.2.16)

= 2(G − 1)P1a1a∗1 + Rxx (3.2.17)

1In this analysis we included the effect of ideal lowpass filtering after down conversion to baseband.


We can easily see there is a processing gain, equal to G, in power for the desired user as

compared to the interference plus noise.

Assuming we have adequate data to obtain near asymptotic covariance estimates Rxx

and Ryy,1, we can estimate the channel vector a1 as the generalized eigenvector correspond-

ing to the largest eigenvalue of the Hermitian-definite matrix pencil Ryy,1 − ηRxx [90]. In

particular, we solve the following generalized eigenvalue problem to estimate a1:

Ryy,1a1 = ηRxxa1 (3.2.18)

Similarly, the interference-plus-noise covariance Ruu,1 can be estimated as

Ruu,1 = GG − 1

(Rxx − 1

GRyy,1

)(3.2.19)

In the previous section, it was shown that the beamformer that will maximize the SINR has

the form

w1 = ζR−1uu,1a1 (3.2.20)

Since ζ is arbitrary and does not affect the SINR, we will set ζ = 1. Therefore, given the

weight vector w1, the beamformer output corresponding to the nth bit is

z1(n) = w∗1y1(n) = 2

√Tb P1b1(n)e

jφ1w∗1a1 + w∗

1i1 + w∗1n1 (3.2.21)

and the corresponding signal to interference-plus-noise ratio (SINR) γ1 is

γ1 = 2GP1 · a∗1R−1

uu a1 (3.2.22)

A more computationally attractive approach to find the weight vector w1 that will maxi-

mize the SINR is to solve for w1 directly (i.e. no intermediate step to estimate a1). This can

be done as follows. First, we rewrite the post-correlation signal vector y1 as

y1(n) = s1(n)+ i1 + n1 (3.2.23)

where s1(n) = 2√

Tb P1e jφ1a1. We can write the SINR at the output of the beamformer as

γ1(w) = w∗Rss,1ww∗Ruu,1w

(3.2.24)


GG-1 xxuu

uu

yyG-1R = R - R1

w = . R a-1

ζ

xx yyR , R )= principal eigenvector of (a

w1

LPF

j tωe

LPF

j tωe

LPF

j tωe 1

1

1

c (t- )

c (t- )

c (t- )

τ

τ

τ

1

1

1

x y

weight vector w

w2

*

*

Kw *

w y*

1

1

∫

∫

∫

Beamformeroutput

Figure 3.4: CDMA receiver-beamformer block diagram

where Rss,1 is the covariance of s1(n) signal component defined as

Rss = 12Tc

E{s1(n)s∗1(n)} (3.2.25)

The goal is to find w1 that will maximize γ1(w), that is

w1 = maxw

γ1(w) = maxw

w∗Rss,1ww∗Ruu,1w

(3.2.26)

Let η1 = γ1(w1), so that

η1 = w∗1Rss,1w1

w∗1Ruu,1w1

(3.2.27)

and

w∗1(Rss,1 − ηRuu,1)w1 = 0 (3.2.28)


It is clear from (3.2.28) that the pair (η,w1) is the solution corresponding to the largest

eigenvalue of the generalized eigenvalue problem

Rss,1w = ηRuu,1w (3.2.29)

Now, we note that

Ryy,1 = Rss,1 + Ruu,1 (3.2.30)

and the problem

(Rss,1 + Ruu,1)w = ηRuu,1w (3.2.31)

is equivalent to

Rss,1w = (η− 1)Ruu,1w (3.2.32)

Therefore, we can find the weight vector w1 directly by solving for the eigenvector corre-

sponding to the largest eigenvalue of the generalized eigenvalue problem

Ryy,1w = ηRuu,1w (3.2.33)

The proposed code-filtering approach for adaptive beamforming can be summarized as

follows

• Perform code filtering at each array element for each user in the cell.

• Given the pre- and post-correlation vectors’ samples x(1), x(2), · · ·, x(S) and y(1),

y(2), · · ·, y(S), estimate the array covariance matrices Rxx and Ryy,1, assuming that

we have enough samples (at least 2K uncorrelated snapshots [91]) to obtain quasi-

asymptotic covariance estimates.

• Estimate the channel vector a1 as the generalized eigenvector corresponding to the

largest generalized eigenvalue of the matrix pair (Ryy,1,Rxx).

• Estimate the interference-plus-noise covariance Ruu,1 according to (3.2.19).

• The optimum weight vector can be found according to (3.2.20).


despreading filter(correlator)

channel matchedfilter

(chip rate)

channelsounder

channel impulseresponse

decisioninput

Figure 3.5: RAKE receiver block diagram

3.3 CDMA Beamforming with Multipath

In this section, we will extend the code-filtering approach for adaptive beamforming de-

scribed above to the case where we have multipath propagation. CDMA is usually applied

to multipath channels where the delay spread of the channel is significantly larger than a

symbol period. The wider the bandwidth of the spread signal, the more resolvable the in-

dividual path components are in time. By combining different components, signal strength

fluctuation due to Rayleigh fading is reduced. On such channels intersymbol interference

(ISI) is not a problem after matched filtering and despreading. In the presence of multipath

and additive white Gaussian noise, the optimal receiver for CDMA modulation is the famil-

iar RAKE receiver, originally proposed by Price and Green in [51] and studied in [92-95].

A simplified block diagram for the RAKE receiver shown in Figure 3.5. First, the input is

despread by a filter matched to the spreading sequence, creating a response that has peaks

where the channel response has peaks. Then the following transversal filter, matched to the

channel by a channel sounder, reinforces the contributions from various peaks. If nonco-

herent detection is required due to the lack of a phase reference or a pilot signal, then the

channel sounder does not give phase information, and the transversal filter can implement

a different combining law. For coherent detection, the RAKE receiver is effectively a filter

matched to the scattered (multipath) signal, at the spreading sequence (chip) rate.


τ1 ∫

τ2 ∫

c (t)

c (t)

c (t)

τL ∫

scanningcorrelator

select delays

Combinerdecision

input

~*

~*

~*

Figure 3.6: Multipath CDMA RAKE receiver

The RAKE receiver can be simplified by implementing only a small number of taps in

the chip-rate matched filter, as shown in Figure 3.6. A scanning correlator identifies the

delays at which the most significant peaks occur, and only at those time delays is the received

signal despread and combined. The performance of this structure is optimum for the case of

discrete multipath when the number of paths equals to the number of correlators. In the case

of a diffuse multipath, this structure performs significantly worse than the RAKE receiver

in Figure 3.5, because then a significant fraction of the transmitted energy is lost unless a

large number of correlators is used [96].

Thus, in a conventional multipath receiver for CDMA, temporal code filtering is used

to resolve paths that are separated in time by more than a chip interval Tc and to estimate

the multipath structure of the channel (in terms of the time delay and the complex path

strength of each resolvable path). Then, the signal is passed through a RAKE correlator

that is matched to the channel output and that coherently/incoherently combines correla-

tors outputs. In this way the temporal structure of the multipath received signal is exploited


efficiently. We refer to this as a 1-D or temporal RAKE since only temporal structure is

exploited.

In general, different multipath components arrive at the receiver not only with different

time delays, but also from different directions and therefore each multipath component has

a different channel vector al,1. A single antenna receiver can not exploit the spatial structure

of the received signal. With antenna arrays, the spatial dimension can be used to efficiently

resolve and combine different multipath components. A new approach that we describe be-

low is to identify both the temporal and spatial structure of the individual paths arriving at

the receiver and then construct a space-time receiver that we call a Beamformer-RAKE. This

receiver is matched to the desired signal while maximally rejecting the interfering signals.

3.3.1 Space-Time Matched Filter

Consider the output of the antenna array for a single user case and additive white Gaussian

noise:

x(t) =L∑

l=1

√Plb(t − τl)c(t − τl)e

jφl al + n(t) (3.3.1)

In order to derive the optimal space-time matched filter for the received signal, we consider

the likelihood function of the received signal, conditioned on the knowledge of all param-

eters. First, let sl(t) =√

Plb(t − τl)c(t − τl)e jφl al. The likelihood function of the received

multipath signal can be written via the Cameron-Martin formula [97] as

L({x(t);−∞ < t < ∞}) = C · exp{!(b(t))/σ2

n

}(3.3.2)

where C is an arbitrary constant and !(b(t)) is defined as

!(b(t)) = 2 · Re

{∫ ∞

−∞

L∑l=1

s∗l (t)x(t)dt

}−∫ ∞

−∞

∣∣∣∣∣L∑

l=1

sl(t)

∣∣∣∣∣2

dt (3.3.3)

The objective is to select the bits b(t) that will maximize ! and therefore will maximize

the likelihood function. Assuming that the user sends M information bits, the first integral


yields

∫ ∞

−∞

L∑l=1

s∗l (t)x(t)dt =L∑

l=1

√Ple

− jφl

∫ ∞

−∞b(t − τl)c

∗(t − τl)a∗l x(t) dt (3.3.4)

=M∑

n=1

b(n)L∑

l=1

√Ple

− jφl zl(n) (3.3.5)

where z1(n), z2(n), · · · , zL(n) are the matched filter outputs synchronously sampled with

respect to each path signal and

zl(n) =∫ nTb+τl

(n−1)Tb+τl

c∗(t − τl)a∗l x(t) dt , l = 1, · · · , L (3.3.6)

The second integral in ! in (3.3.3) does not depend on the received signal at the array.

Therefore, the sufficient statistic for the detection of bits b(n), n = 1 · · · M, is zl(n), l =1 · · · L, obtained by a linear operation on the received signal vector x(t).

Equation (3.3.6) means that the sufficient statistic is obtained by passing the noisy re-

ceived signal vector through a spatial matched filter A∗ = [a1 a2 · · ·aL]∗, which is equiva-

lent to conventional beamforming [91], followed by a bank of L matched filters. The lth

matched filter is matched to the code waveform in the lth path c(t − τl). In general, we

can use a more selective spatial matched filter. This is important if the noise is not spatially

white and if interference is present. In this case, the lth sufficient statistic is given by

zl(n) =∫ nTb+τl

(n−1)Tb+τl

c∗(t − τl)w∗l x(t) dt , l = 1, · · · , L (3.3.7)

where the spatial matched filter, or the beamformer weight vector, wl for the lth path is cho-

sen so that w∗l x(t) is a good estimate of the signal received in the lth path. In the previous

section, we showed that wl is given by

wl = ζR−1uu,lal (3.3.8)

where Ruu,l is the covariance matrix of the sum of all of the undesired signals.


τ

τ

τ

1

2

L

∫

∫

c (t- )

c (t- )

c (t- )

∫

~*

~*

~*

w1*

beamformer

w2*

beamformer

wL*

beamformer

spatialmatched filter

temporalmatched filter

1z (n)

2z (n)

Lz (n)

x

x

x

(t)

(t)

(t)

Figure 3.7: Space-Time Matched Filter

3.3.2 Beamformer-RAKE Receiver Structure

The outputs of the space-time matched filter shown in Figure 3.7 can be combined (coher-

ently or incoherently depending on whether a pilot signal, from which a phase reference

can be obtained, is available or not) and the output of the combiner is passed to a decision

device. This overall receiver structure is shown in Figure 3.8 and is called Beamformer-

RAKE, since the resulting structure is equivalent to a beamformer front-end followed by a

conventional temporal RAKE receiver.

Although the idea of the RAKE receiver was first proposed for single access channels


Spatial

Beamformer RAKE

"Beamformer-RAKE"

Mobile

Temporal

Combiner

Figure 3.8: Beamformer-RAKE receiver structure

with additive white Gaussian noise, with orthogonal codes or codes with low crosscorrela-

tions they can be used in CDMA multiple access systems [8, 98]. However, in this case the

receiver structure, although simple to implement, is not optimum and other structures based

on multiuser detection have been proposed [99-102].

In order to construct the Beamformer-RAKE receiver, we use the code filtering approach

described in the previous section for each resolvable multipath component as follows. First,

we recall the total received signal vector as

x(t) =N∑

i=1

L∑l=1

√Pl,ibi(t − τl,i)ci(t − τl,i)e

jφl,i al,i + n(t) (3.3.9)

Without loss of generality, we consider the 1st user and assume that the time delays τl,1, l =1 · · · L are perfectly known. Later in this section, we will show how to use the code filtering

approach to estimate the time delays τl,1. For the nth bit, the post-correlation signal vector

for the lth multipath component is given by

yl,1(n) = 1√Tb

∫ nT+τl,1

(n−1)T+τl,1

x(t)c∗1(t − τl,1)dt (3.3.10)

yl,1(n) = 2√

Tb Pl,1b1(n)ejφl,1 al,1 + il,1 + nl,1 (3.3.11)


where

nl,1 = 1√Tb

∫ nT+τl,1

(n−1)T+τl,1

n(t)c∗1(t − τl,1) dt (3.3.12)

is the undesired component due to thermal noise and

il,1 =L∑

k=1k �=l

√Pk,1 Il,1,k,1e

jφk,1 ak,1 +N∑

i=2

L∑k=1

√Pk,i Il,1,k,ie

jφk,i ak,i (3.3.13)

is the undesired component due to multiple access interference (MAI) plus self interference

(SI), where Il,1,k,i is defined as

Il,1,k,i = 1√Tb

∫ nT+τ1

(n−1)T+τ1

bi(t − τk,i)ci(t − τk,i)c∗1(t − τl,1) dt (3.3.14)

Let ul,1 = il,1 + nl,1 be the total undesired (all array signals other than 1st user’s lth path

signal) component. We can easily show that Rxx, Ryy,l,1, and Ruu,l,1 are given by

Rxx = 2Pl,1al,1a∗l,1 + Ruu,l,1 (3.3.15)

Ryy,l,1 = 2GPl,1al,1a∗l,1 + Ruu,l,1 (3.3.16)

Ruu,l,1 =L∑

k=1k �=l

2Pk,1ak,1a∗k,1 +

N∑i=2

L∑l=1

2Pl,ial,ia∗l,i + σ2

nI (3.3.17)

Hence, we can use the same approach developed in the previous section to find wl,1 as

wl,1 = ζR−1uu,l,1al,1 (3.3.18)

Therefore, the corresponding beamformer output, or decision statistic, for the lth path is

zl,1(n) = w∗l,1yl,1(n) = 2

√Tb Pl,1b1(n)e

jφl,1 w∗l,1al,1 + w∗

l,1ul,1 (3.3.19)

and the corresponding path signal to interference-plus-noise ratio is

γl,1 = 2GPl,1 · a∗l,1R−1

uu,l,1al,1 (3.3.20)

It remains now to show how to estimate the time delays τl,1, l = 1 · · · L. We recall that

in order to find the channel vector al,1 we solve for the principal eigenvector of the ma-

trix pencil Ryy,l,1al,1 − ηRxxal,1 and the corresponding eigenvalue ηmax is a good estimate


for the average signal power Pl,1 in that path. Based on this observation, we can estimate

the path amplitudes and time delays by computing the maximum eigenvalue ηmax(τ) of the

matrix pencil Ryy,1(τ)al,1 − ηRxxal,1 as τ changes over the range of possible time delays

[τmin, τmax]. The values of τ that correspond to the local maxima of ηmax(τ) are good esti-

mates of the time delays τl,1, l = 1 · · · L. The value of each of those local maxima is also a

good estimate of the corresponding average path strength. In practice, each local maximum

is compared to a preset threshold to determine whether this local maximum corresponds to

a signal path or not and also to determine the number of existing signal paths L that will be

used in the subsequent detection stage. The threshold level needs to be optimized in order to

maximize system performance [94]. If the threshold level is smaller than the optimum level,

then some time delays corresponding to noise-plus-interference only will be taken as time

delays corresponding to resovable paths which will lead to performance degradation [33].

On the other hand, if the threshold level is larger than the optimum level, the performance

will also degrade as thresholding becomes more likely to reject some time delays corre-

sponding to resolvable paths as well as time delays corresponding to noise-plus-interference

only.

3.3.3 A Simulation Example

In order to evaluate the performance of the code-filtering approach for beamforming, we

considered the following simulation experiment. We considered a base station with a ULA

of 5 omni-directional antennas with antenna spacing λ/2. We assumed DQPSK spreading

with processing gain G of 128. We considered 25 mobiles randomly distributed in azimuth

around the base station with a uniform distribution over [0,2π]. We also assumed that 3

propagation paths are received from each mobile and that the total received power from

each user is the same for all users. Each path was modeled as a single planar wavefront

with average strength different for each. In this case, the pre-correlation average SINR is


−18.7 dB. Also, the channel vector al,1 is

al,1 = αl,1

1

e− j2πd sin θl,1/λ




(3.3.21)

where αl,1 is the average path strength. Although this simulation scenario is simple and

does not fully simulate the vector channel model described in Chapter 2, it enables us to see

the ability of the approach to estimate different parameters of the multipath vector channel

accurately where other approaches would fail.

Path # 1 2 3θl,1 90◦ 45◦ 135◦

τl,1 0 4Tc 8Tc

αl,1 1 0.65 0.8

θ 89.43◦ ± 1.28 42.71◦ ± 1.88 137.88◦ ± 2.35

α 1 0.66 ± 0.029 0.81 ± 0.027

Table 3.2: Estimated multipath parameters

For the first user, for each multipath component we considered the values of the angle of

arrival θl,1, relative time delay τl,1, and average path strength αl,1 as shown in Table 3.2. The

pre and post-correlation signal vectors x and yl,1 corresponding to 10 transmitted bits were

used to estimate the array covariances Rxx and Ryy,l,1. Using Rxx and Ryy,l,1, we estimated

the channel vector al,1. From this estimate, the angle of arrival θl,1 can be estimated as

θl,1 = sin−1

(λ

2πd� al,1(2)

)(3.3.22)

where al,1(2) is the second element in al,1. The mean and standard deviation of the esti-

mated angles of arrival over 500 runs are also shown in Table 3.2. The Table also shows


Relative delay τ (chip periodTc)

0 2 4 6 8

η max

(τ)

0 .0

0.2

0.4

0.6

0.8

1.0

1.2

mean {ηmax}

mean {ηmax} + σ

mean {ηmax} - σ

Figure 3.9: Estimated multipath profile

the mean and standard deviation of the estimated average path strength. These numbers

show that with the code-filtering approach, we are able to estimate the channel vector and

the multipath parameters accurately. Note that under this scenario with 25× 3 different ar-

riving paths, estimating different parameters for each arriving signal would be impossible if

we use any of the above mentioned DF techniques. In addition, the multipath delay profile

for the 1st user was estimated using the approach described above. The estimated profile is

shown in Figure 3.9. As we can see, the peaks of the estimated profile occur at values of τ

that correspond to the actual delays.


τ

τ

τ

1

2

L

∫

∫

c (t- )

c (t- )

c (t- )

∫

~*

~*

~*

w1*

beamformer

w2*

beamformer

wL*

beamformer

x

x

x

(t)

(t)

(t)

decisiondevice

DPSKdemod.

DPSKdemod.

DPSKdemod.

Figure 3.10: Balanced DQPSK Beamformer-RAKE receiver with incoherent combining

3.4 ”Beamformer-RAKE” Receiver Examples

In this section we describe two ”Beamformer-RAKE” receiver examples and discuss their

average probability of error performance. The first one is for balanced DQPSK spreading

with incoherent combining. The second example is for balanced QPSK spreading with co-

herent combining.

Figure 3.10 shows a Beamformer-RAKE receiver structure for balanced DQPSK with

post detection equal gain diversity combining. After the space-time matched filter, each

signal path decision statistic zl,1(n) is fed into a DPSK demodulator. The decision variable

is the sum of all demodulator outputs. Analysis results for CDMA communication systems

employing DPSK modulation are reported in [98, 103-105]. If we assume that the multiple

access interference is Gaussian, then the bit error probability as a function of the SINR per


bit γb is given by [33]

Pb(γb) = 122L−1

e−γb

L−1∑l=0

Cl · γ lb (3.4.1)

where for the receiver structure in Figure 3.10 γb is given by

γb =L∑

l=1

γl,1 = 2GL∑

l=1

Pl,1 · a∗l,1R−1

uu,l,1al,1 (3.4.2)

and

Cl = 1l!

L−1−l∑k=0

(2L − 1

k

)(3.4.3)

The probability of error derived above is a conditional probability since it depends on

γb. Therefore, to get the average bit error probability, we need to average this conditional

probability over the statistics of γb. First, let us assume that the average signal power per

path is the same for all paths. That is, we assume Pl,1 = P, l = 1 · · · L. If the total unde-

sired signal vector ul,1 can be modeled as a spatially white Gaussian random vector2, then

Ruu,l,1 = σ2I where σ2 is the total interference-plus-noise power. In this case we have

γb = 2GPσ2

L∑l=1

|al,1|2 (3.4.4)

In Chapter 6 we show that for zero angle spread, i.e. �= 0, γb has a χ2 probability density

function (pdf) with 2L degrees of freedom

fγb(γ) =γL−1

(γK)L(L − 1)!e−γ/γK (3.4.5)

where γ = 2GP/σ2 is the average SINR per path per antenna and K is the number of an-

tennas in the array. In this case, we can easily show that the average bit error probability Pe

is given by

Pe =∫ ∞

0Pb(γ) fγb(γ) dγ (3.4.6)

2Simulation results in Chapter 6 show that this assumption holds if the code length G is large and if thetotal number of path signals N · L is large, with a uniform angle of arrival distribution.


Lγ, Average Antenna SINR/bit, dB

5 10 15 20 25 30

Pe

, P

roba

bilit

y of

Bit

Err

or

10-7

10-6

10-5

10-4

10-3

10-2

10-1

K =1 K=2K=4

Figure 3.11: Pe for balanced DQPSK with incoherent combining and � = 0◦

Pe =L−1∑l=0

Cl

22L−1(L − 1)!(γK)L

∫ ∞

0e−γ

(γK+1γK

)γL−1+l dγ (3.4.7)

= 122L−1(L − 1)!(1 + γK)L

L−1∑l=0

Cl · (L + l − 1)!

(γK

1 + γK

)l

(3.4.8)

Also, in Chapter 6 we show that for large angle spread γb has a χ2 pdf with 2K L degrees of

freedom

fγb(γ) =γK L−1

(γ )K L(K L − 1)!e−γ/γ (3.4.9)

Therefore, we can easily show that in this case the average bit error probability Pe is



5 10 15 20 25 30

Pe

, P

roba

bilit

y of

Bit

Err

or

10-10

10-9

10-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

K =1 K=2K=4

Figure 3.12: Pe for balanced DQPSK with incoherent combining and large �

given by

Pe =∫ ∞

0Pb(γ) fγb(γ) dγ (3.4.10)

=L−1∑l=0

Cl

22L−1(K L − 1)!(γ )K L

∫ ∞

0e−γ

(γ+1γ

)γK L−1+l dγ (3.4.11)

= 122L−1(K L − 1)!(1 + γ )K L

L−1∑l=0

Cl · (K L + l − 1)!

(γ

1 + γ

)l

(3.4.12)

The probability of bit error performance of the Beamformer-RAKE receiver with bal-

anced DQPSK and incoherent combining is shown in Figures 3.11 and 3.12 for angle spread

� = 0◦ and large � respectively. The probability of bit error Pe in (3.4.8) and (3.4.12) is

plotted as a function of the average antenna SINR/bit, defined as Lγ, for K = 1, 2, and 4.


τ

τ

τ

1

2

L

∫

∫

c (t- )

c (t- )

c (t- )

∫

~*

~*

~*

w1*

beamformer

w2*

beamformer

wL*

beamformer

x

x

x

(t)

(t)

(t)

P e1-jφ

1½

P e2-jφ

2½

P eL-jφ

L½

decisiondevice

Figure 3.13: Balanced QPSK Beamformer-RAKE receiver with coherent combining

We assumed that L = 2. For the zero angle spread case in Figure 3.11, we notice the reduc-

tion in the required antenna SINR to achieve certain probability of bit error performance as

the number of antennas increases. This reduction is proportional to the number of sensors

and is due to the beamforming operation only. This is due to the fact that with zero angle

spread, the same path signal will undergo the same fading at all antennas and, therefore, the

array will not provide any space diversity. For the case of large angle spread, the signal fad-

ing at different antennas (that are sufficiently separated) will be independent. Therefore, in

addition to the beamforming gain, the array will provide a space diversity gain. The gain

due to space diversity can be seen from the Pe plot in Figure 3.12. As we can see, with 2

antennas, an additional gain of 5.5 dB due to space diversity can be obtained.

Figure 3.13 shows a Beamformer-RAKE receiver structure for balanced QPSK spread-

ing with maximal ratio diversity combining and coherent detection. After the space-time

matched filter, each path signal is weighted by the complex (conjugate) amplitude of that

path√

Pl,1e− jφl,1 . The weighted path signals are then added together and fed to a decision


device. For coherent detection with maximal ratio combining, in addition to the beamformer

weight vector wl,1 the receiver needs also an estimate of the path amplitude√

Pl,1 and the

phase e jφl,1 . The path amplitudes can be estimated using the approach described above. The

phase information e jφl,1 is lost in the process of computing Ryy,l,1. Therefore, for coherent

detection we assume that a pilot sequence is available. In this case, the method described

in [89] can be used to estimate the phase e jφl,1 . Analysis of CDMA communication systems

with QPSK spreading are reported in [33, 89, 106]. The bit error probability as a function

of γb is [33]

Pb(γb) = 12

erfc(√

γb

)(3.4.13)

where

erfc(x) = 2√π

∫ ∞

xe−t2

dt (3.4.14)

and γb is given by (3.4.2). If we make the same assumption that the total undesired com-

ponent is white then for zero angle spread γb has the probability density function in (3.4.5)

and the average bit error probability Pe is [33]

Pe =∫ ∞

0Pb(γ) fγb(γ) dγ (3.4.15)

=∫ ∞

0

γL−1

(γK)L(L − 1)!e−γ/γK 1√

π

∫ ∞

√γ

e−x2dx dγ (3.4.16)

= 1√π

∫ ∞

0e−x2

∫ x2

0

γL−1

(γK)L(L − 1)!e−γ/γK dγ dx (3.4.17)

=(

1 −µ1

2

)L L−1∑l=0

(L − 1 + l

l

)(1 +µ1

2

)l

(3.4.18)

where µ1 =√

γK1+γK . For large angle spread, γb has the pdf in (3.4.9). In this case, the aver-

age bit error probability Pe is

Pe =∫ ∞

0Pb(γ) fγb(γ) dγ (3.4.19)

=∫ ∞

0

γK L−1

(γ )K L(K L − 1)!e−γ/γ 1√

π

∫ ∞

√γ

e−x2dx dγ (3.4.20)



5 10 15 20 25 30

Pe

, P

roba

bilit

y of

Bit

Err

or

10-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

K =1 K=2K=4

Figure 3.14: Pe for balanced QPSK with coherent combining and � = 0◦

Pe = 1√π

∫ ∞

0e−x2

∫ x2

0

γK L−1

(γ )K L(K L − 1)!e−γ/γ dγ dx (3.4.21)

=(

1 −µ2

2

)K L K L−1∑l=0

(K L − 1 + l

l

)(1 +µ2

2

)l

(3.4.22)

where µ2 =√

γ

1+γ.

The probability of bit error performance of the Beamformer-RAKE receiver with bal-

anced QPSK and coherent combining with L = 2 is shown in Figures 3.14 and 3.15. We also

plot Pe for both angle spread � = 0◦ and large � and as a function of the average antenna

SINR/bit Lγ. These figures also show the performance improvement due to both beam-

forming gain and space diversity gain provided by the array.



5 10 15 20 25 30

Pe

, P

roba

bilit

y of

Bit

Err

or

10-10

10-9

10-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

K =1 K=2K=4

Figure 3.15: Pe for balanced QPSK with coherent combining and large �

3.5 Summary

In this chapter, we reviewed different beamforming techniques and the conditions under

which they are applicable. We also discussed the reasons why those techniques are not ap-

plicable for wireless CDMA systems. We then introduced our space-time approach for es-

timating the channel vector and optimum beamforming weight vector. We derived a space-

time receiver that we call a ”Beamformer-RAKE” to exploit both temporal and spatial struc-

ture of the received multipath signal to maximize performance. In Chapter 4, we will look

at techniques to implement the beamforming approach described above.

Chapter 4

Beamforming for Time-Variant

Channels

In the previous chapter, we introduced the code-filtering approach for adaptive beamform-

ing for CDMA signals. We assumed that the channel remains constant or varies very slowly

over several symbol periods so that we have enough samples to obtain quasi-asymptotic ar-

ray covariance estimates. However, as the mobile moves in the propagation environment,

the direction of the mobile with respect to the base station will change. Also, the effective

scatterers contributing to the received signal at the base station antenna array will change.

This will be also true for other interfering users. This means that both the array covariances

and the channel vector for different signals will change with time. This motivates the need

for recursive method for computing and tracking the optimal weight vector solution that is

easy to implement and is adaptive to the time-varying nature of the channel. In this chapter,

we will introduce a method for recursively tracking the optimum weight vector.

Tracking the optimum beamforming weight vector solution obtained earlier involves

tracking the principal eigenvector of the pre- and post-correlation array covariance matrix

pair and tracking the undesired signal vector covariance. The problem of tracking eigenvec-

tors when the matrices are slowly varying with time, turns out to be central to many adaptive

83

Chapter 4. Beamforming for Time-Variant Channels 84

antenna array processing techniques and some numerical methods have already been pro-

posed [107-109]. Some methods take into account the fact that the array covariance matrix

varies by a rank one update from one time step to the next in order to speed up computa-

tional time [90, 110]. In addition to this fact, in the method that we will describe shortly, we

also make use of the inherent time correlation structure to further speed up computation.

4.1 Recursive Adaptive Beamforming

As we mentioned earlier, when the signal environment frequently changes because of the

nonstationary desired and undesired signals, the adaptive beamformer must continuously

update the weight vector to match the changing environment. The approach that we describe

below employs direct implementation of the optimum weight vector solution in (3.3.18), i.e.

estimating the unknown undesired signal array covariance Ruu,l,1 and the channel vector

at each sampling instant and implementing the optimal solution directly. This approach of

direct implementation of the optimum solution, although more computationally expensive

than other approaches, has better convergence behavior [91, 85].

We begin by recalling the optimum weight vector solution in (3.3.18)

wl,1 = ζR−1uu,l,1al,1 . (4.1.1)

We have shown that the undesired signal covariance can be estimated as

Ruu,l,1 = GG − 1

(Rxx − 1

GRyy,l,1

)(4.1.2)

and that the channel vector al,1 can be also estimated as the principal eigenvector of the

generalized eigenvalue problem

Ryy,l,1al,1 = ηRxxal,1 (4.1.3)


The constant ζ does not affect the beamformer output SINR and, therefore, we will not con-

sider its value. Therefore, to find the optimum weight vector at each sampling instant we

need an estimate for the channel vector al,1(n) and the undesired signal covariance Ruu,l,1.

4.1.1 Recursive Estimation of the Channel Vector

We now describe a recursive procedure to estimate the channel vector ak,1. Consider the

problem defined in (4.1.3). First, let us assume that the pre-correlation signal vector covari-

ance Rxx is equal to I. In this case, power method recursion [90]

cm+1 = Ryy,l,1cm

|Ryy,l,1cm| (4.1.4)

is known to converge so that

limm→∞

|Ryy,l,1cm| = ηmax (4.1.5)

limm→∞

cm = al,1 (4.1.6)

and

Ryy,l,1al,1 = ηmaxal,1 (4.1.7)

where ηmax is the maximum eigenvalue of Ryy,l,1. Since Rxx can be any arbitrary positive

definite matrix, the idea is to decompose Rxx into the product of two matrices

Rxx = R∗/2xx R1/2

xx (4.1.8)

where R1/2xx is called the square root of the matrix Rxx. Using (4.1.3) and (4.1.4), we have

R−∗/2xx Ryy,l,1R−1/2

xx R1/2xx al,1 = ηR1/2

xx al,1 (4.1.9)

Defining the matrix Bl,1�= R−∗/2

xx Ryy,l,1R−1/2xx and the vector el,1

�= R1/2xx ak,1, we can rewrite

(4.1.9) as

Bl,1el,1 = ηel,1 (4.1.10)


el,1(n) = POWER(Bl,1(n), el,1(n − 1))f

• Initialization:

1. co(n) = el,1(n − 1)

2. c1(n) = Bl,1(n)co(n)/|Bl,1(n)co(n)|3. m = 1

• Computation:

1. while (|Bl,1(n)cm(n)− Bl,1(n)cm−1(n)|/|Bl,1(n)cm(n)| ≥ ε) dof

(a) cm+1(n) = Bl,1(n)cm(n)/|Bl,1(n)cm(n)|(b) m → m + 1

g

2. el,1(n) = cm(n)

g

Table 4.1: Power recursion for estimating principal eigenvector

and hence the recursion defined in (4.1.4) can be used to estimate (ηmax, el,1,max). Then, the

array response vector al,1 is given by

al,1 = R−1/2xx el,1,max (4.1.11)

As we mentioned earlier, the array covariance matrices are functions of time and need to

be estimated for every symbol n from samples of the pre- and post-correlation signal vectors

x(n) and yyy,l,1(n), respectively, where yyy,l,1(n) is defined as

yl,1(n) = 1√2Ts

yl,1(n) (4.1.12)

Therefore, for every symbol the estimates Ryy,l,1(n) and Rxx(n) are used in place of Ryy,l,1

and Rxx.


The above recursion for estimating the channel vector al,1 for each symbol can be sum-

marized as in Table 4.1. The convergence behavior of the above recursion depends on the

initial guess of the eigenvector co(n) and on the ratio |ηmax/η2|where η2 is the second largest

eigenvalue [90]. Due to the inherent time correlation of the channel vector, the previously

estimated eigenvector el,1(n − 1) is a close estimate to el,1(n) and, therefore, is used as

the initial guess to speed up convergence. For example, for a mobile moving at 75 mph

and symbol rate of 4800 symbols per second, the time correlation between the elements of

the channel vector of two consecutive symbols is 0.995. Also, the ratio |ηmax/η2| is lower

bounded by the average SINR per path1. In the current CDMA IS-95 standard, the closed

loop power control holds the SINR at the output of the RAKE combiner such that 1% FER

( frame error rate) is achieved. This translates to average SINR per path of 10.5 dB, 5.7

dB, and 2.8 dB for 1, 2, and 4 independent paths used for combining, respectively [111]. In

general, the lower the average SINR per path the more the number of iterations required for

convergence.

4.1.2 Time-Update of Covariance Estimates

It remains now to obtain the time updates for the estimates of the array covariance matrices

R−1/2xx (n), Ryy,l,1(n), and R−1

uu,l,1(n). Our goal is to track the time varying channel so that the

covariance estimates at each sampling instant closely approximate the second order statis-

tics of the observed channel process. When the channel process is slowly time-varying with

respect to the symbol rate, a fading memory update can be used to estimate the covariance

matrices at each sampling instant [112]. In this case, given the pre- and post-correlation sig-

nal vector x(n) and yl,1(n), the time-update equations for Rxx(n), Ryy,l,1(n), and Ruu,l,1(n)

can be written as

Rxx(n) = µRxx(n − 1)+ x(n)x∗(n) (4.1.13)

1In fact, if we assume that the total undesired signal is spatially white, we can easily show that |ηmax/η2| =1 + SINR.


Ryy,l,1(n) = µRyy,l,1(n − 1)+ yl,1(n)y∗l,1(n) (4.1.14)

Ruu,l,1(n) = µRuu,l,1(n − 1)+ GG − 1

x(n)x∗(n)− 1G − 1

yl,1(n)y∗l,1(n) (4.1.15)

where µ is a positive constant between 0 and 1 and is called the forgetting factor. The use of

the forgetting factor µ < 1 is used to ensure that the data in the distant past are forgotten in

order to afford the possibility of statistical variation of the observed received signal vector

as the mobile moves in the propagation environment. Simulation results presented at the

end of this chapter show that a typical value of µ should range between 0.8 and 1.

However, we need the time update for the inverse square root matrix R−1/2xx (n) and not

Rxx(n). In order to find the time update for the inverse square root, consider a K × K Her-

mitian positive definite matrix Q defined as

Q = µP + pp∗ (4.1.16)

where P is a K × K Hermitian positive definite matrix and p is a K × 1 vector. Our goal is

to find Q−1/2 given P−1/2 and p. Using the matrix inversion lemma [86], we can show that

Q−1 is given by

Q−1 = (µP + pp∗)−1 (4.1.17)

= µ−1P−1 −µ−2 P−1pp∗P−1

1 +µ−1p∗P−1p(4.1.18)

= µ−1P−1/2P−∗/2 −µ−2 P−1/2P−∗/2pp∗P−1/2P−∗/2

1 + µ−1p∗P−1/2P−∗/2p(4.1.19)

Let f �= µ−1/2P−∗/2p and β�= 1 + f∗f. Then, we can rewrite Q−1 as

Q−1 = µ−1P −µ−1 P−1/2ff∗P−∗/2

β(4.1.20)

= µ−1/2P−1/2P−∗/2µ−1/2 − µ−1/2P−1/2ff∗P−∗/2µ−1/2

β(4.1.21)

= µ−1/2P−1/2

(I − 1

βff∗)

P−∗/2µ−1/2 (4.1.22)

Now, we can easily verify that(I − 1

βff∗)= (I − γff∗) (I − γff∗)∗ (4.1.23)


P−1/2(n) = SR UPDATE(P−1/2(n − 1),p, µ)f

1. f = µ−1/2P−∗/2(n − 1)p

2. g = µ−1/2P−1/2(n − 1)f

3. β = 1 + f∗f

4. γ = 1β+√

β

5. P−1/2(n) = µ−1/2P−1/2(n − 1)− γgf∗

g

Table 4.2: Inverse square root time-update

where γ = 1β+√

β. Hence, Q−1 can be rewritten as

Q−1 = µ−1/2P−1/2 (I − γff∗) (I − γff∗)∗ P−∗/2µ−1/2 (4.1.24)

Therefore, it follows from (4.1.8) and (4.1.24) that the inverse square root matrix Q−1/2

given P−1/2 and p is given by

Q−1/2 = µ−1/2P−1/2 (I − γff∗) (4.1.25)

This is summarized in the SR UPDATE procedure shown in Table 4.2. Therefore we can

write the time-update for the inverse square root R−1/2xx (n) as

R−1/2xx (n) = SR UPDATE(R−1/2

xx (n − 1),x(n), µ) (4.1.26)

Similarly, for the undesired signal vector we need the time-update of the matrix inverse

R−1uu,l,1(n) not Ruu,l,1(n). This time-update can be obtained by using the time-update expres-

sion for Ruu,l,1(n) in (4.1.15) and the matrix inversion lemma [86]. First we note that the

time-update of Ruu,l,1(n) in (4.1.15) is a rank-2 update. We start by rewriting Ruu,l,1(n) as

Ruu,l,1(n) = Kl,1(n)− 1G − 1

yl,1(n)y∗l,1(n) (4.1.27)


where

Kl,1(n) = µRuu,l,1(n − 1)+ GG − 1

x(n)x∗(n) (4.1.28)

Hence, using the matrix inversion lemma, we can easily show that the time-update of the

matrix R−1uu,l,1(n) is given by the two-step update

K−1l,1(n) = µ−1R−1

uu,l,1(n − 1)−µ−2 GG − 1

R−1uu,l,1(n − 1)x(n)x∗(n)R−1

uu,l,1(n − 1)

1 +µ−1 GG−1x∗(n)R−1

uu,l,1(n − 1)x(n)(4.1.29)

R−1uu,l,1(n) = K−1

l,1(n)+1

G − 1

K−1l,1(n)yl,1(n)y∗

l,1(n)K−1l,1(n)

1 − 1G−1 y∗

l,1(n)K−1l,1(n)yl,1(n)

(4.1.30)

The above time-update for R−1uu,l,1(n) involves subtraction. While R−1

uu,l,1(n) is theoreti-

cally Hermitian positive definite (being the inverse of a Hermitian positive definite matrix),

accumulation of numerical and quantization errors with time-update may lead to the loss of

Hermitian positive definiteness. This can be avoided if we update the inverse square root

R−1/2uu,l,1(n) and since R−1

uu,l,1(n) = R−1/2uu,l,1(n)R

−∗/2uu,l,1(n), it remains Hermitian positive definite

regardless of any numerical or quantization errors in the inverse square root. Even in the

presence of those errors, the numerical conditioning of the inverse square root R−1/2uu,l,1(n) is

generally much better than that of R−1uu,l,1 [112]. Hence, using (4.1.27) and (4.1.27) we can

write the two-step time-update for R−1/2uu,l,1(n) as

K−1/2l,1 (n) = SR UPDATE(R−1/2

uu,l,1(n − 1), a1 · x, µ) (4.1.31)

R−1/2uu,l,1(n) = SR UPDATE(K−1/2

l,1 (n), a2 · yl,1,1) (4.1.32)

where a1 =√

GG−1 and a2 = j√

G−1.

As we mentioned in Chapter 2, the dynamics of the channel vector depend on the speed

of the mobile v, or equivalently the maximum Doppler shift fd, and the angle spread � in

the corresponding received signal vector. Therefore, as we will see later in this chapter, we

should expect that the value of µ that will give the best tracking performance will be a func-

tion of the channel dynamics, i.e. a function of ( fd,�) and hence µ should be a function


• Initialize

δ = a small positive constant,

R−1/2xx (0) = R−1/2

uu,l,1(0) = δ−1/2I,Ryy,l,1(0) = δI,

el,1(0) = 1,

• For each instant of time, n = 1,2, · · · , computef

– New Data: get x(n) and yl,1(n).

– Time-Update:

1. if n = 1,2µ(n) = 1

else

µ(n) = |a∗l,1(n − 2)al,1(n − 1)|

|al,1(n − 2)||al,1(n − 1)|2. R−1/2

xx (n) = SR UPDATE(R−1/2xx (n − 1),x(n), µ(n))

3. K−1/2l,1 (n) = SR UPDATE(R−1/2

uu,l,1(n − 1), a1 · x(n), µ(n))

4. R−1/2uu,l,1(n) = SR UPDATE(K−1/2

l,1 (n), a2 · yl,1(n),1)

5. Ryy,l,1(n) = µ(n)Ryy,l,1(n − 1)+ yl,1(n)y∗l,1(n)

6. Bl,1(n) = R−1/2xx (n)Ryy,l,1(n)R

−∗/2xx (n)

– Weight Computation:

7. el,1(n) = POWER(Bl,1(n), el,1(n − 1))

8. al,1(n) = R−∗/2xx (n)el,1(n)

9. wl,1(n) = R−1/2uu,l,1(n)R

−∗/2uu,l,1(n)al,1(n)

g

Table 4.3: Beamforming algorithm summary


of time µ(n). In general, the value of µ(n) needs to be optimized to satisfy two competing

requirements. The value of µ(n) should be as close as possible to 1 so that the effective

number of samples used in estimating the array covariances is large enough to obtain quasi-

asymptotic covariance estimates and better noise averaging. On the other-hand the value of

µ(n) should be small enough to track the channel dynamics.

Finding an expression for µ(n) that will give the best tracking performance for al,1(n)

(which is estimated as the principal generalized eigenvector of Rxx(n) and Ryy,l,1(n)) might

not be feasible. However, certain approximations can be made to find a close estimate for

µ(n). One such approximation is to take µ(n) to be equal to the normalized inner product

ρl,1(n) between last two estimates al,1(n − 1) and al,1(n − 2) defined as

ρl,1(n) =|a∗

l,1(n − 2)al,1(n − 1)||al,1(n − 2)||al,1(n − 1)| (4.1.33)

which is a good measure of how fast the channel vector is changing with time.

4.1.3 Algorithm Summary

In Table 4.3, we present a summary of the adaptive algorithm for estimating the channel

vector al,1(n) and the corresponding optimum weight vector wl,1(n) given the pre- and post-

correlation signal vectors x(n) and yl,1(n), respectively, including initial conditions and

the recursions that are involved in the computation. The two building blocks for the algo-

rithm are the recursive procedure for estimating the channel vector POWER and the inverse

square root time-update procedure SR UPDATE.

For the initial value of the covariance matrices, we use an initialization similar to that

used for the recursive least squares algorithm (RLS) [112]. In this case, we set the initial

value of the covariance matrices to

Rxx(0) = Ryy,l,1(0) = Ruu,l,1(0) = δI (4.1.34)

where δ is a small positive constant. Correspondingly, the initial value for R−1/2xx (0) and


Step # Number of Flops

1 3K2 4K2 + 4K3 4K2 + 4K4 3K2 + 3K

5+6 11K2 + 2K7 O(K2)

8 K2

9 K2

Table 4.4: Floating point operations count

R−1/2uu,l,1(0)

R−1/2xx (0) = R−1/2

uu,l,1(0) = δ−1/2I (4.1.35)

The initial value of the principal eigenvector el,1(0) is set to 1, where 1 is a K × 1 vector

whose elements are all equal to 1. This choice ensures that the initial guess used in the power

recursion has a component in the direction of el,1(1) and therefore will ensure convergence.

Table 4.4 gives the number of floating point operations (flops) for each step of the algo-

rithm shown in Table 4.3. From this Table we can easily see that the overall complexity of

the algorithm is O(K2).

The number of flops is taken to be the number of multiplication operation involved. The

reason is that the instruction set of many of the current generation digital signal processors,

such as the TMS320C30 by Texas Instruments, include a multiply-and-accumulate instruc-

tion that will implement calculations such as a · b + c in one operation. For the sake of

simplicity, we made no distinction between real and complex numbers. For instance, the

product of a complex K × 1 vector and a scalar (either real or complex) will be said to

require K (complex) flops. In calculating the number of flops required for step 5 and 6,


Maximum Doppler Frequency fd (Hz)

0 20 40 60 80 100

Mis

mat

ch L

oss

ρ (d

B)

-1.8

-1.6

-1.4

-1.2

-0.8

-0.6

-0.4

-0.2

-2.0

-1.0

0.0

∆ = 0ο

∆ = 5ο

∆ = 10ο

∆ = 20ο

∆ = 40ο

∆ = 60ο

Forgetting Factor µ = 0.98

Figure 4.1: Mismatch loss as a function of fd and � for µ = 0.98

we considered the rank-1 update structure for both R−1/2xx (n) and Ryy,l,1(n) in (4.1.14) and

(4.1.26). If we were to directly implement steps 5 and 6, then the number of flops required

is 2K3 + 2K2. By considering the rank-1 update structure, the number of flops required is

11K2 + 2K, which implies a reduction in the required number of flops for K ≥ 5. For step

7, the number of required flops will depend on the number of iterations required for con-

vergence. The use of el,1(n − 1) as an initial guess for el,1(n) will minimize the number of

iterations required and, therefore, will speed up convergence.



0 20 40 60 80 100

Mis

mat

ch L

oss

ρ (d

B)

-1.2

-0.8

-0.6

-0.4

-0.2

-1.0

∆ = 0ο

∆ = 5ο

∆ = 10ο

∆ = 20ο

∆ = 40ο

∆ = 60ο

Forgetting Factor µ = 0.95

Figure 4.2: Mismatch loss as a function of fd and � for µ = 0.95

4.2 Simulation Results

To study the performance of the above algorithm, we considered a base station with a 5

element uniform linear array with sensor spacing 0.5λ. The multiple access plus self inter-

ference signal vector il,1 is modeled as an additive Gaussian noise that is both temporally

and spatially white. Simulation results in Chapter 6 show that this assumption holds if the

code length G is large and if the total number of path signals N · L is large with uniform an-

gle of arrival distribution. In this case Ruu,l,1 = σ2I, and the optimum beamforming weight

vector is wl,1 = al,1.

We considered one multipath component arriving at the array with mean angle of arrival


-0.8

-0.7

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0.80

0.85

0.90

0.950

2040

6080

100

Mism

atch Loss ρ (dB

)

Forgetting Factor µ

Maximum Doppler Frequency f d

Angle Spread ∆ = 5o

Figure 4.3: Mismatch loss as a function of fd and µ for � = 5◦

of θ = 90◦ and average SNR = −18dB. The processing gain G is 128. We assumed bal-

anced DQPSK spreading with the same information bits in both the in-phase and quadrature

channels. The bit rate was assumed to be 9.6 kbps. In each simulation run, a block of 100

bits was generated and differentially encoded. The differentially encoded data was spread

in the in-phase and quadrature channels using two different spreading codes. The channel

was generated using the model described in Chapter 2. We considered different cases with

different values of angle spread �, maximum doppler frequency fd, and forgetting factor

µ. The algorithm in Table 4.3 was used to estimate the channel vector al,1.

To study the performance of the algorithm, we considered the mean beamformer output



0 20 40 60 80 100

Forg

etti

ng F

acto

r µ

0.75

0.80

0.85

0.90

0.95

1.00

1.05

∆ = 0ο

∆ = 5ο

∆ = 10ο

∆ = 20ο

∆ = 40ο

∆ = 60ο

Figure 4.4: Optimum forgetting factor µ as a function of fd and �

SINR normalized to the maximum output SINR (output SINR when al,1 = al,1, i.e. no errors

in estimating the channel vector). This mean normalized SINR is given by

ρ = E

{20 log10

|a∗l,1al,1|

|al,1||al,1|}

(4.2.1)

This quantity represents the loss in the average beamformer output SINR due to errors in

the channel vector estimate al,1 and is therefore called the mismatch loss.

Some of the simulation results are shown in Figures 4.1-4.5. Figure 4.1 shows the mis-

match loss as function of fd and � for µ = 0.98. We notice that for low doppler frequency,

the loss at low angle spread is higher than the loss at high angle spread, which can be ex-

plained by the reduction in space diversity gain provided by the array. At high Doppler fre-

quency, the situation is reversed. The reason for this is that at high Doppler frequency and



0 20 40 60 80 100

Mis

mat

ch L

oss

ρ (d

B)

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0.0

∆ = 0ο

∆ = 5ο

∆ = 10ο

∆ = 20ο

∆ = 40ο

∆ = 60ο

Figure 4.5: Mismatch loss as a function of fd and � with optimum forgetting factor

high angle spread, the channel variations are much faster than the rate at which array sample

covariances are updated (which is a function of the forgetting factor µ). In this case more

and more incorrect samples will be used in estimating the array sample covariances which

will lead to errors in the estimated channel vector. On the other hand, at high Doppler fre-

quency and low angle spread, the channel vector tends to fluctuate only in magnitude, and

the fluctuations in direction (phase) are very small and hence the errors in the estimated array

response vector will be reflected mainly in its magnitude, which will not affect the beam-

former output SINR. Figure 4.2 is similar to Figure 4.1 but with forgetting factor µ = 0.95.

We notice that there is a slight increase in the mismatch loss at low Doppler frequencies

while at high Doppler frequencies we notice a reduction in the mismatch loss.


As we mentioned earlier, the forgetting factor µ that will give the best tracking perfor-

mance is a function of fd and �. With this optimum value of the forgetting factor µ, the

mismatch loss will be minimized. This is clearly illustrated in Figure 4.3 which shows the

mismatch loss as a function of fd and µ for � = 5◦. The optimum forgetting factor µ for

different values of fd and � was obtained via simulations and is shown in Figure 4.4. The

corresponding loss is shown in Figure 4.5.

4.3 Summary

In this chapter, we presented an adaptive algorithm for estimating the channel vector and the

corresponding weight vector. The algorithm is based on the code-filtering approach pre-

sented in the previous chapter. The algorithm consists of two basic building blocks. The

first is a recursive procedure ”POWER” for tracking and estimating the channel vector.

The second block is a procedure ”SR UPDATE” for time-updating the inverse square roots

R−1/2xx (n) and R−1/2

uu,l,1(n). The overall algorithm complexity is O(K2). Simulation results

show that with optimum time-update (i.e. if we use optimum forgetting factor), we can track

the channel vector within 0.5 dB of the true value even in severe propagation environments.

Chapter 5

Overview of the IS-95 CDMA Standard

CDMA is a modulation and multiple access scheme based on spread spectrum communi-

cations. Proponents of the CDMA technology cite several potential advantages over the

traditional FDMA AMPS and the IS-54 TDMA approaches [113-115]. First, statistics of

telephone conversations suggest that in a typical full-duplex two-way voice conversation,

the duty cycle of each voice is less than 35%. Exploiting the voice activity in either FDMA

or TDMA systems might be hard to implement because of the time delays associated with

reassigning channel resources during speech pauses. In CDMA, it is possible to reduce the

transmission rate when there is no speech, and thereby substantially reduce interference to

other users. This reduction in interference power can be transformed to either an increase

in system capacity or a reduction in average mobile transmit power [113]. However, we

should note that a statistical multiplexing protocol based on packet reservation multiple ac-

cess (PRMA) has been proposed to make use of the speech activity in TDMA systems [116,

117].

Another potential advantage for CDMA is that special frequency reuse plans are not nec-

essary. Since users are separated by using different codes on the entire system bandwidth

100

Chapter 5. Overview of the IS-95 CDMA Standard 101

simultaneously, there is no need to avoid co-channel interference by requiring different fre-

quency usage in neighboring cells. In fact the concept of a cell in terms of frequency plan-

ning is no longer necessary.

The CDMA IS-95 standard uses time and path diversity to mitigate the effect of fre-

quency selective multipath fading. Time diversity is obtained by the use of forward error

correction (FEC) and interleaving. Path diversity is inherently provided by the CDMA ap-

proach by spreading the signal over a wide bandwidth (1.25 MHz). Such a signal with wide

bandwidth will resolve the multipath components and, thus, provide the receiver with sev-

eral independently fading signal paths. This path diversity is exploited by the use of a RAKE

receiver to combine different multipath components. The use of a wideband signal for trans-

mission may be also viewed as just another method for obtaining frequency diversity [33].

In a CDMA system every user is a source of interference to every other user, which

makes mobile station transmit power control a key element in its current implementation.

The current CDMA standard uses several power control techniques (open loop and closed

loop power control) to optimize the system performance. In addition, the base station uses

three sectored antennas (each covers 120◦ of the azimuth) to reduce the multiple access in-

terference and thereby increase system capacity.

Field test results of the IS-95 cellular system are reported in [118, 111]. This test was

conducted in August 1993 in the San Diego area and included four base stations for a total

of eight sectors and 86 class I mobile units. Capacity estimates from this field test show

that the IS-95 radio capacity/sector is 11.4 times better than that of AMPS. This translates

to Erlang capacity/sector 16.4 times higher than that of AMPS. However, we should note

that in [119], in commenting on a similar previous field test experiment, Cox cites several

potential capacity-reducing factors that were not exercised in this experiment.

In the following sections we will briefly describe the different signal processing func-

tions used for forward and reverse link modems.


ConvolutionalEncoder

and Repetition

Interleaver

Long CodeGenerator

User LongCode Mask

× × +

User WalshCode

Pilot WalshCode

SpeechCoder

I-ChannelShort Code

Q-ChannelShort Code

× ×

×

+

×

BasebandFilter

BasebandFilter

cos( t)ωc

sin( t)ωc

I(t)I

Q(t)Q

a (t)(I)

a (t)(Q)

c (t)i

W (t)i

W (t)p

Figure 5.1: CDMA IS-95 forward link waveform generation

5.1 CDMA Forward Link

In the proposed CDMA standard the forward link uses a combination of frequency division,

pseudorandom code division, and orthogonal signal multiple access techniques. Frequency

division is employed by dividing the available cellular spectrum into nominal 1.25 MHz

bandwidth channels.

A simplified description of the signal processing functions for transmission from the

base station to the mobile unit is shown in Figure 5.1. The underlying data rate for the sys-

tem is 9600 bits/sec which represents the speech coder rate of 8550 bits/sec augmented by

error correction coding which is tailored to the speech-coding technique used. (The speech

coder actually detects speech activity and changes data rate to lower values during quiet

intervals, but 9600 bits/sec is the maximum error-protected data rate). The 9600 bits/sec

bit stream is segmented into 20-msec blocks and then further convolutionally encoded to

provide the capability of error correction and detection at the receiver. The convolutional


encoder used has a constraint length k = 9 and a code rate r of one-half. This will bring the

data rate to 19.2 kbits/sec. The convolutional encoding is followed by interleaving over a

20-msec interval for burst error protection due to fast fading in the radio channel. The 19

kbits/sec output of the interleaver is then modified by the use of the so-called long code,

which serves as a privacy mask.

The modified stream is then encoded for spread-spectrum transmission using binary or-

thogonal Walsh codes [33] of dimension 64. This will produce 64-fold spreading of the data

stream, resulting in a transmission rate of 1.2288 Mchips/sec. The structure of the Walsh

code provides 64 orthogonal sequences, and one of the 64 sequences is assigned to a mo-

bile unit during call setup. In this way, 64 orthogonal ”channels ”can be established on the

forward link by the CDMA encoding on the same carrier frequency. After Walsh encoding

, the spread data stream is separated into I and Q streams, each of which is modified by a

unique ”short code” of length 32768. The resulting spread spectrum stream is carried over

the air interface with filtered QPSK modulation.

All signals transmitted from a base station in a particular CDMA radio channel share

the same set of 64 Walsh codes and the same pair of short codes. However, signals from

different base stations are distinguished by time offsets from the basic short code which

allows CDMA signals from each base station to be uniquely identified. Different signals

transmitted from a given base station in a particular CDMA radio channel are distinguished

at the mobile receiver by the orthogonal Walsh code. The orthogonality of the Walsh codes

provides near perfect isolation between the multiple signals transmitted by the same base

station. However, the presence of multipath propagation will partially destroy this orthog-

onality.

An important aspect of the forward link waveform generated at the base station is the use

of a pilot signal that is transmitted by each base station and is used as a coherent carrier ref-

erence for demodulation by all mobile receivers. The pilot is transmitted a relatively higher

power level (approximately 20% of total forward link power budget [113, 120]) than other


types of signals which allows for tracking of the carrier phase. The pilot channel signal is

unmodulated by information and uses the zeros Walsh function. Thus the pilot signal simply

consists of the quadrature pair of short codes. The mobile receiver can obtain synchroniza-

tion with the nearest base station without prior knowledge of the identity of the base station

by searching out the entire length of the short code. The strongest signal’s time offset cor-

responds to the time offset of the short code of the base station to which the mobile has the

best propagation channel (which is often the nearest base station). After synchronization,

the pilot signal is used as a coherent carrier phase reference for demodulation of the other

signals from this base station.

The mobile receiver uses a limited number of correlators in a RAKE structure for de-

modulating the signals from the base station. The receiver uses the pilot signal to estimate

the phase and the amplitude for each of the tracked multipath components and to coherently

combine them.

5.2 CDMA Reverse Link

The CDMA reverse link also employs PN spread spectrum modulation using the same short

code as that used for the forward link. Here, however, all mobiles use the same code phase

offset. Signals from different mobiles are distinguished at the base station by the use of a

very long (242 − 1) PN sequence with a user address determined time offset. Because every

possible time offset is a valid address, an extremely large address space is provided.

The data rate of the system is also 9600 bits/sec. The transmitted digital information

stream is segmented into 20-msec blocks and then further convolutionally encoded using a

code of rate 1/3 and constraint length k = 9. This will bring the data rate to 28.8 kbits/sec.

The encoded information bits are then interleaved over the 20 ms block. The interleaved

information is then grouped into symbol groups (or code words) of 6 bits. These code words


UserLong Code

M-ary OrthogonalWalsh Modulator

W(t)Interleaver

ConvolutionalEncoder

andRepetition

SpeechCoder

Long CodeGenerator

Long CodeMask c (t)i

Power Control

ωcsin( t)

ωccos( t)

+

Q(t)

I(t)

(Q)a (t)

(I)a (t)

cT / 2Delay

I-ChannelShort Code

Q-ChannelShort Code

Q

I BaseBandFilter

BaseBandFilter

S/PbitsJ

Figure 5.2: CDMA IS-95 reverse link waveform generation

are used to select one of 64 different orthogonal Walsh functions for transmission. At the

output of the Walsh modulator, the chip rate is 307.2 kchips/sec. The final signal processing

elements perform the direct-sequence spreading functions. First, the modulation symbols ,

or Walsh functions, are spread by using the mobile-specific long code at a rate of 1.2288

Mchips/sec, i.e., 256 chips per modulation symbol. The data stream is then split into I and

Q streams where it is modified with the short code pair. The resulting spread spectrum signal

is then carried over the air interface with a filtered O-QPSK (Offset-QPSK) modulation.

Note that the use of the Walsh function in generating the reverse link waveform is dif-

ferent from that on the forward link. On the forward link, the Walsh function is determined

by the channel that is assigned to each mobile while on the reverse link the Walsh function

is determined by the information being transmitted. The use of Walsh function modulation

on the reverse link is a simple way of obtaining 64-ary modulation. This is the best way

of providing a high quality link over a fading channel with low SINR where a pilot phase

reference cannot be provided [89].

Figure 5.3 shows a block diagram for the base station receiver. The base station uses


ΣSearchers

Incoherent RAKE

1

L

1

2

6AntennaOutputs

M WalshSequence

Correlators

M WalshSequence

Correlators

Figure 5.3: CDMA Base station receiver: IS-95 approach

dual antenna diversity in each sector to provide path diversity. The antennas are widely

separated such that independent fading at each antenna is obtained. For any given mobile,

this structure implements a noncoherent RAKE receiver with a limited number of corre-

lators for demodulating the L strongest multipath components received on all base station

antennas. Noncoherent reception is used since, as we mentioned earlier, a pilot signal is not

available on the reverse link. Time delay information of the L strongest multipath compo-

nents is obtained by the front-end searcher. The front-end searcher has a larger number of

scanning correlators that continuously scans all six antenna outputs for multipath signals

received from the mobile. Tracked multipath components are rank ordered based on their

energy and the L strongest are used for combining (current implementation of cell modem

ASICs searches for 8 paths and uses the strongest 4 for combining [8, 111]).


5.3 Summary

This chapter provides a brief overview of the CDMA IS-95 cellular standard. We briefly

described different signal processing functions used for both the forward and reverse link

modems. In the next chapter, we will apply the adaptive beamforming and ”Beamformer-

RAKE” ideas presented in Chapter 3 to CDMA wireless systems that are based on the IS-95

standard and propose an antenna array-based base station receiver architecture and study its

performance.

Chapter 6

CDMA Base Station Receiver with

Antenna Arrays

In Chapter 3 we derived a space-time ”Beamformer-RAKE” receiver that exploits the spa-

tial structure in the received multipath signal in addition to the temporal structure to pro-

vide a more efficient combining of paths. We showed that for coherent detection, a pilot

signal is still needed for the ”Beamformer-RAKE” to estimate the phase information for

each multipath signal. However, transmitting a pilot in each mobile’s signal, whose power

is greater than the data-modulated portion of the signal, reduces efficiency to less than 50%

[89]. Instead, either differential phase shift keying (DPSK) which does not require phase co-

herence or M-ary orthogonal modulation with noncoherent reception should be used. For

M > 8, where M is the number of orthogonal signals, orthogonal modulation is better than

DPSK [33, 89]. Therefore, the CDMA IS-95 standard that was briefly described in the pre-

vious chapter uses 64-ary orthogonal modulation for the reverse link. We also mentioned

in the previous chapter that in the current implementation of the CDMA standard dual an-

tenna diversity is used at the base station to provide path diversity and that the L strongest

paths are incoherently combined. In this approach, the base station treats all diversity el-

ements equally. In other words, in selecting the L strongest paths for combining, the base

108

Chapter 6. CDMA Base Station Receiver with Antenna Arrays 109

ΣIncoherent RAKE

1

L

Beamformerfor 1st path

Beamformerfor L-th path

Searcher

Beamformer-RAKE Receiver

Base Station Antenna Array

M WalshSequence

Correlators

M WalshSequence

Correlators

AntennaArrayOutput

Figure 6.1: CDMA Base station receiver: beamforming approach

station receiver does not make any distinction as to whether different paths came from the

same antenna or different antennas. Also, signals received at two or more different antennas

due to the same multipath component wavefront are considered different multipath signals.

Clearly, this approach does not utilize any spatial structure in the received multipath signal.

This spatial structure can be utilized by the use of the ”Beamformer-RAKE” receiver.

In this case, the base station will have an antenna array of K elements in each sector. The

proposed base station receiver structure is similar to that of the IS-95 system except that the

front-end searcher now searches for vector multipath components (by using the approach

outlined in Chapter 3). In addition, each multipath demodulator will be preceded by a beam-

former. A simplified overview of the ”Beamformer-RAKE” base station receiver is shown

in Figure 6.1.

In this chapter, we will present an overall base station receiver architecture for CDMA

with M-ary orthogonal modulation based on the ”Beamformer-RAKE” structure and study

its performance in terms of the uncoded bit error rate as a function of number of users and


number of antennas. In this analysis, we shall make use of the power control performance

results that will be presented later in Chapter 7. We start by developing a model for the

received signal vector at the base station array. Then, we will describe the proposed re-

ceiver structure and its different signal processing blocks. This will be followed by a de-

tailed performance analysis for the proposed receiver and numerical and simulation results.

The analysis in this chapter makes use of analysis results for DS/CDMA with M-ary orthog-

onal modulation but without antenna arrays that appeared in [121-125].

6.1 Received Signal Vector Model

The mobile transmitter block diagram is shown in Figure 5.2. The binary data at the output

of the interleaver are grouped into groups of J = log2 M bits. Each group is mapped into one

of M orthogonal Walsh sequences W(t). The resulting signal is then spread using the user’s

long PN code ci(t). The signal is further multiplied in both I and Q channels by the short

PN codes a( I)(t) and a(Q)(t) respectively. The PN modulated Q channel signal is delayed

by half a chip period Tc/2. The two spread signals are up-converted to radio frequency for

transmission. The power of the transmitted signal is adjusted according to both the open

and closed loop power control mechanisms (see Chapter 7). Then we can write the signal

transmitted by the i-th mobile as

si(t) = ψi

√Pi

(W (h)(t)ci(t)a

( I)(t) cos(ωct)+W (h)(t − To)ci(t − To)a

(Q)(t − To) sin(ωct))

0 ≤ t ≤ Tw (6.1.1)

where Pi is the transmitted power per symbol per dimension, Tw is the symbol period, ωc

is the carrier angular frequency, To = Tc/2 is the time offset between the I and Q channels,

and finally ψi is a Bernoulli random variable that models the voice activity of the ith user

(we assume that a user will be on with probability ν and will be off with probability 1− ν).


W (h)(t) is the hth orthogonal Walsh function, h = 1, · · · , M. Let the processing gain be

G = Tw/Tc . For simplicity of notation, we shall denote the product of the user’s PN code

and the I or Q channel PN code as

c( I)i (t) = ci(t)a

( I)(t) and c(Q)i (t) = ci(t)a

(Q)(t)

To simplify our analysis, the PN codes c( I)i (t) and c(Q)

i (t) are represented by [125, 106]

c( I)i (t) =

∞∑r=−∞

c( I)i,r p(t − kTc) (6.1.2)

c(Q)i (t) =

∞∑r=−∞

c(Q)i,r p(t − kTc) (6.1.3)

where c( I)i,r and c(Q)

i,r are assumed to be i.i.d. random variables taking values ±1 with equal

probability, and p(t) is the chip pulse shape, which can be any time-limited waveform. Here

we assume that p(t) is rectangular although our results can be easily extended for any time-

limited waveform.

We assume the multipath vector channel model in (2.2.25). To simplify our analysis, we

assume that we have a constant deterministic power-delay profile and that the log-normal

slow fading is the same for all multipath components. We also assume that the channel pa-

rameters vary slowly as compared to the symbol duration Tw so that they are constant over

several symbol durations. Therefore, after downconverting to baseband, we can write the

K × 1 complex baseband received signal vector for the i-th user as

xi(t) =√

Si Piψi

Li∑l=1

c(h)i (t − τl,i)ejφl,i al,i (6.1.4)

where Si represents the log-normal shadowing experienced by the ith user, φl,i = ωcτl,i and

Li is the number of multipath component for the ith user, al,i is the K × 1 channel vector of

the base station antenna array to signals in the l-th path from the ith mobile, and c(h)i (t) is

defined as

c(h)i (t) = W (h)(t)c( I)i (t)+ jW (h)(t − To)c

(Q)i (t − To) (6.1.5)


Let N be the number of cochannel mobiles. The total received signal at the cell site is

the sum of all users’ signals plus noise and is given by

x(t) =N∑

i=1

Li∑l=1

xl,i(t)+ n(t) (6.1.6)

The vector n(t)= nc(t)+ jns(t) is the K ×1 additive Gaussian noise vector with zero mean

and covariance

E{n(t1)n∗(t2)} = σ2nI · δ(t1 − t2) (6.1.7)

where σ2n is the noise variance per antenna.

6.2 Receiver Model

The block diagram of the base station antenna array receiver is shown in Figure 6.2. It has

a ”Beamformer-RAKE” structure where several multipath components are tracked in both

time and space. After down-converting to baseband, the outputs of the LPF are fed into a

bank of M Walsh correlators shown in Figure 6.3. Assuming that the hth Walsh symbol

was transmitted, where h = 1, · · · , M, the pre-correlation and post-correlation signal vec-

tors x(t) and y(h)l,i are used to estimate the channel vector al,i and the corresponding optimum

beamforming weight vector wl,i for the lth multipath component of the ith mobile from the

pre-correlation and post-correlation array covariances Rxx and Ryy,l,i1 using the code filter-

ing approach derived in Chapter 3.

1As we have seen earlier, all matrices and vectors are functions of time. However, for simplicity of nota-tion, we will drop the time index in the rest of this chapter.

Chapter

6.C

DM

AB

aseStation

Receiver

with

Antenna

Arrays

113

x

z

w , w , ... , w1 2 L

z1

(M)

z1

(1)

ET

K

1

MWalsh Sequence

Correlators

MWalsh Sequence

Correlators

MWalsh Sequence

Correlators

MWalsh Sequence

Correlators

L parallel demodulators

L parallel demodulators

Optimum Beamformingand Incoherent

RAKE Combining

Optimum Beamformingand Incoherent

RAKE Combining

Weight VectorEstimation

weight vectors

select post-correlationsignal vector

M-ary Decoder,Deinterleaver,

andViterbi Decoder

Measure FrameError Rate (FER)

Closed LoopPower Control

Algorithm

Up/DownPower Control

Command

Threshold

Select Indexof Maximum

data

pre-correlationsignal vector

post-correlationsignal vector

Down-converterand LPF

Down-converterand LPF

Figure 6.2: Base station receiver block diagram

Chapter

6.C

DM

AB

aseStation

Receiver

with

Antenna

Arrays

114

c (t)1

a (I)

a (Q)

a (Q)

a (I)

WM

W1

W1

WM

LPF

cos( t)

sin( t)

ω

ω

c

c

-

+

+

+

z (m)(1)

1,1

z (m)(1)

L,1

z (m)(1)

k,1

z (m)(M)

k,1

z (m)(M)

1,1

z (m)(M)

L,1

LPF

∫

∫

∫

∫

Figure 6.3: Correlators


For each multipath component, we have M different post-correlation signal vectors y(h)l,i ,

h = 1, · · · , M. The vectors y(h)l,i are fed to an optimum beamformer. The outputs of the

L1 beamformers for the hth Walsh function w∗l,iy

(h)l,i are then fed into an incoherent RAKE

combiner. The output of the incoherent RAKE combiner z(h)i is the decision variable for the

hth Walsh function. The beamformer and the incoherent RAKE combiner for the hth Walsh

function are shown in Figure 6.4.

In order to update the post-correlation array covariance Ryy,l,i (that will be used in es-

timating al,i and wl,i), the receiver needs the post-correlation vector y(h)l,i corresponding to

the true transmitted Walsh symbol W (h)(t). However, at this stage the receiver has no prior

knowledge of which post-correlation vector y(n)l,i is the right one. Here, the receiver relies on

the inherent correlation of the multipath vector channel and the assumption that the chan-

nel remains almost constant over several symbol periods. In this case, the receiver uses a

delayed update of Ryy,l,i (and hence delayed estimation of the channel vector and the opti-

mum beamforming weight vector). This is done by using the decision on the current Walsh

symbol h to select the post-correlation vector y(h)l,i to update Ryy,l,i and obtain the optimum

weight vector wl,i. This weight vector wl,i will be used for beamforming for the next sym-

bol.

The decision variables z(1)i , · · · , z(M)i at the output of the incoherent RAKE are then fed

to an M-ary decoder, deinterleaver, and Viterbi convolutional decoder. Without loss of gen-

erality let us assume that the 1st user is the desired user and let τk,1 be the time delay of the

kth tracked multipath which is assumed to be estimated perfectly and k = 1, · · · , L1. Then,

we can write the post-correlation signal vector y(n)l,1 for the kth tracked multipath component

for the first user as

y(n)k,1 = 1√

Tw

∫ τk,1+Tw

τk,1

x(t)c(n)∗1 (t − τk,1)dt (6.2.1)

= d(n)k,1 + u(n)

k,1 if n = h (6.2.2)

= u(n)k,1 if n �= h (6.2.3)


where

d(n)k,1 = 2

√TwS1 P1e jφk,1 ak,1 (6.2.4)

u(n)k,1 = m(n)

k,1 + s(n)k,1 + n(n)k,1 (6.2.5)

and d(n)k,1 is the desired signal vector, m(n)

k,1 is the multiple access interference (MAI) signal

vector, s(n)k,1 is the self-interference (SI) signal vector due to other multipath components of

the 1st user, and n(n)k,1 is due to the AWGN. Let Ai =

√Si Piψi. Also, for the kth tracked

multipath let the optimum beamforming weights determined using the previously estimated

Walsh symbol be wk,1. For an equal gain combining incoherent RAKE, the nth decision

variable of the 1st user corresponding to the nth Walsh symbol is given by [33]

z(n)1 =L1∑

k=1

z(n)k,1 =L1∑

k=1

|w∗k,1y(n)

k,1|2 n = 1, · · · , M (6.2.6)

Now, to select which post-correlation signal vector y(n)l,1 should be used in estimating the

post-correlation array covariance Ryy,l,i, a hard decision is made on which Walsh symbol

was transmitted

h = arg maxn=1,···,M

(z(n)1

)(6.2.7)

However, for the data a symbol-by-symbol M-ary decoder is used [89]. Both approaches

yield exactly the same decisions for the M-ary symbol and both are optimal (i.e. a maxi-

mum likelihood rule) for an AWGN channel. Since the MAI is not necessarily Gaussian,

this decision rule is actually not optimal. However, when the number of cochannel users is

large, the multiple access interference can be modeled as Gaussian noise and therefore this

decision rule can be used. The primary reason for using the symbol-by-symbol approach

for the data is to provide improved performance with error-correcting codes by using soft

decision decoding [33, 89].

Note also that we cannot use the output after the convolutional decoding and deinter-

leaving to select the post-correlation signal vector. The reason for this is that we will have

to wait for a decision to be made on the current symbol and convolutionally encode and


*1,1

w

*w2,1

*wL,1

Σ

z(n)

1,1

z (n)

L,1

z (n)

Time Align

τ 1

Time Alignτ 2

Time Alignτ L

1z (n)

2,1

2| . |

2| . |

2| . |

Figure 6.4: Optimum Beamforming and Incoherent RAKE

interleave again. By the time this process is over (which is at least twice the time of one

frame of bits), the channel would have changed and the estimated channel vector and the

channel vector of the new symbol will be quite different. This will lead to a degradation in

the beamformer output SINR.

6.3 Signal Statistics

In order to derive the uncoded bit error probability, we need to derive the statistics of the

decision variables z(1)1 , z(2)1 · · · , z(M)

1 . First, we will examine the different terms in z(n)k,1, i.e.

the multiple access interference signal vector due to other cochannel users m(n)k,1, the self in-

terference signal vector due to the user own multipath components s(n)k,1, and the noise vector

n(n)k,1.


6.3.1 Noise Analysis

The noise term n(n)k,1 is given by

n(n)k,1 = 1√

Tw

∫ τk,1+Tw

τk,1

n(t)c(n)∗1 (t − τk,1) dt (6.3.1)

= (n(n), I I

k,1 + n(n),QQk,1

)+ j(−n(n), I Q

k,1 + n(n),QIk,1

)(6.3.2)

where n(n), I Ik,1 , n(n),QQ

k,1 , n(n), I Qk,1 , and n(n),QI

k,1 are defined as

n(n), I Ik,1 = 1√

Tw

∫ τk,1+Tw

τk,1

nc(t)c( I)i (t − τk,1)W

(n)(t − τk,1) dt (6.3.3)

n(n),QQk,1 = 1√

Tw

∫ τk,1+Tw

τk,1

ns(t)c(Q)i (t − To − τk,1)W

(n)(t − To − τk,1) dt (6.3.4)

n(n), I Qk,1 = 1√

Tw

∫ τk,1+Tw

τk,1

nc(t)c(Q)i (t − To − τk,1)W

(n)(t − To − τk,1) dt (6.3.5)

n(n),QIk,1 = 1√

Tw

∫ τk,1+Tw

τk,1

ns(t)c( I)i (t − τk,1)W

(n)(t − τk,1) dt (6.3.6)

For n(n), I Ik,1 , we have

n(n), I Ik,1 = 1√

Tw

∫ τk,1+Tw

τk,1


(n)(t − τk,1) dt (6.3.7)

= 1√Tw

G−1∑b=0

∫ τk,1+(1+b)Tc

τk,1+bTc


(n)(t − τk,1) dt (6.3.8)

= 1√Tw

G−1∑b=0

∫ τk,1+(1+b)Tc

τk,1+bTc

±nc(t) dt (6.3.9)

We can easily show that n(n), I Ik,1 is a Gaussian random vector with zero mean and covariance

σ2n

2 I. Similarly, we can show that n(n),QQk,1 , n(n), I Q

k,1 , and n(n),QIk,1 are all uncorrelated zero mean

Gaussian random vectors with the same covariance. Hence n(n)k,1 is a zero mean Gaussian

random vector with covariance 2σ2nI.


6.3.2 Self and Multiple Access Interference Analysis

The self interference due to other multipath components is given by

s(n)k,1 =L1∑l=1l �=k

1√Tw

∫ τk,1+Tw

τk,1

xl,1(t)c(n)∗1 (t − τk,1) dt (6.3.10)

= A1

L1∑l=1l �=k

I (n)k,1,l,1ejφl,1 al,1 (6.3.11)

= A1

L1∑l=1l �=k

[(I (n), I Ik,1,l,1 + I (n),QQ

k,1,l,1

)+ j(−I (n), I Q

k,1,l,1 + I (n),QIk,1,l,1

)]e jφl,1al,1 (6.3.12)

where I(n), I Ik,1,l,i, I (n),QQ

k,1,l,i , I (n), I Qk,1,l,i , and I (n),QI

k,1,l,i are defined as

I (n), I Ik,1,l,i = 1√

Tw

∫ τk,1+Tw

τk,1

W (h)(t − τl,i)c( I)i (t − τl,i)

W (n)(t − τk,1)c( I)1 (t − τk,1) dt (6.3.13)

I (n),QQk,1,l,i = 1√

Tw

∫ τk,1+Tw

τk,1

W (h)(t − To − τl,i)c(Q)i (t − To − τl,i)

W (n)(t − To − τk,1)a(Q)

1 (t − To − τk,1) dt (6.3.14)

I (n), I Qk,1,l,i = 1√

Tw

∫ τk,1+Tw

τk,1

W (h)(t − τl,i)c( I)i (t − τl,i)

W (n)(t − To − τk,1)c(Q)

1 (t − To − τk,1) dt (6.3.15)

I (n),QIk,1,l,i = 1√

Tw

∫ τk,1+Tw

τk,1

W (h)(t − To − τl,i)c(Q)i (t − To − τl,i)

W (n)(t − τk,1)c( I)1 (t − τk,1) dt (6.3.16)

Also, we can write the MAI due to other users’ signals as

m(n)k,1 =

N∑i=2

Li∑l=1

1√Tw

∫ τk,1+Tw

τk,1

xl,i(t)c(n)∗1 (t − τk,1) dt (6.3.17)

=N∑

i=2

Li∑l=1

Ai I(n)k,1,l,ie

jφl,i al,i (6.3.18)

=N∑

i=2

Li∑l=1

Ai

[(I (n), I Ik,1,l,i + I (n),QQ

k,1,l,i

)+ j(−I (n), I Q

k,1,l,i + I (n),QIk,1,l,i

)]e jφl,i al,i . (6.3.19)


Let

W (h)(t − τl,i)a( I)i (t − τl,i) =

∞∑r=−∞

q( I)i,r p(t − rTc − τl,i) (6.3.20)

where q( I)i,r = W (h)

r c( I)i,r . It follows from (6.1.2) and (6.1.3) that q( I)

i,r , r = −∞, · · · ,∞ is an

i.i.d. binary random sequence taking values ±1 with equal probability. Hence, it follows

that I (n), I Ik,1,l,i is zero mean. Using Equation (6.3.20), we can rewrite I(n), I I

k,1,l,i as

I (n), I Ik,1,l,i = 1√

Tw

G−1∑b=0

q( I)1,b

∫ τk,1+(b+1)Tc

τk,1+bTc

p(t − bTc − τk,1)

∞∑r=−∞

q( I)i,r p(t − rTc − τl,i) dt

= 1√Tw

G−1∑b=0

q( I)1,b

[q( I)

i,b−1Rp(βk,1,l,i)+ q( I)i,b Rp(Tc − βk,1,l,i)

](6.3.21)

where βk,1,l,i = τl,i − τk,1 modulo-Tc and Rp(s) is the partial auto correlation of the chip

waveform defined as

Rp(s) =∫ s

op(t)p(t + Tc − s) dt 0 ≤ s ≤ Tc (6.3.22)

For rectangular pulses, Rp(s) = s. For asynchronous networks, a reasonable assumption is

that βk,1,l,i are independent and uniformly distributed over [0, Tc]. Let

Fb = q( I)i,b−1Rp(βk,1,l,i)+ q( I)

i,b Rp(Tc − βk,1,l,i) (6.3.23)

Using the results in [106], we can show that {Fb}b=0,···,G−1 are independent random vari-

ables. Hence

Var{

I (n), I Ik,1,l,i

} = 1Tw

G−1∑b=0

Var {Fb} (6.3.24)

= 1Tw

G−1∑b=0

E{

R2p(βk,1,l,i)+ R2

p(Tc − βk,1,l,i)}

(6.3.25)

= 1T2

c

∫ Tc

0

[β2

k,1,l,i + (Tc − βk,1,l,i)2]

dβk,1,l,i (6.3.26)

= 23

Tc (6.3.27)

Similarly, we can also show that I(n), I Ik,1,l,i,I

(n),QQk,1,l,i , I (n), I Q

k,1,l,i , and I (n),QIk,1,l,i are all zero mean uncor-

related random variables with the same variance given by (6.3.27).


Remark: In deriving the variance of I(n), I Ik,1,l,i, we used the assumption that the chip pulse

shape is rectangular. In reality the channel is bandlimited (due to low pass filtering follow-

ing the down converter) and the received signal cannot be a square wave. Under the condi-

tion that the same amount of energy is received regardless of the channel used, the received

signal through a bandlimited channel will have a higher peak value, resulting in a higher

level of MAI interference due to larger fluctuations. In [124, 106] it was shown that if the

bandlimited channel has ideal low pass filter characteristics with a bandwidth B = 1/Tc,

then, we would have

Var{I (n), I Ik,1,l,i} = Tc (6.3.28)

Similarly, covariance of n(n)k,1 will then be

Var{n(n)

k,1

} = 2Tcσ2nI (6.3.29)

2

The total interference vector i(n)k,1 = m(n)k,1 + s(n)k,1 is modeled as a zero mean complex Gaus-

sian random vector with covariance I(n)k,1 = E{i(n)k,1i(n)∗k,1 }. Although, this assumption does not

always hold for CDMA analysis, it was shown in [106] that it is valid for large G. Moreover,

simulation results presented later in this chapter show that for large N · L, if we assume that

the angles of arrival of the multipath components are uniformly distributed over the sector,

the total interference vector i(n)k,1 will be spatially white. In this case

I(n)k,1 = 2Tcσ2I I (6.3.30)

where σ2I is given by

σ2I = C ·

[N∑

i=2

νLi E{Si Pi} + (L1 − 1)E{S1 P1}]

(6.3.31)

and C is a constant equal to 2 for a bandlimited channel and 43 for a rectangular pulse shape.

For the remainder of our analysis we will assume the case of a bandlimited channel, i.e.


C = 2. The covariance of the interference-plus-noise vector u(n)k,1 is then given by

R(n)uu,k,1 = σ2I (6.3.32)

where σ2 = 2Tc(σ2I + σ2

n).

6.3.3 Decision Statistics

Consider the pre-correlation and post-correlation signal vectors x(t) and y(h)k,1. With the as-

sumption that the MAI is spatially white, the optimum beamforming weights can be shown

to be

wk,1 = ζak,1 (6.3.33)

where ζ is some arbitrary constant (that does not change the beamformer output SINR). For

simplicity of the analysis, we set ζ = 1/√

a∗k,1ak,1.

Define the beamformer output for the k-th multipath component of the 1st user w∗k,1y(n)

k,1

as U (n)k,1

U (n)k,1 = 2A1

√Tw|ak,1|2e jφk,1 + w∗

k,1u(n)k,1 for n = h , (6.3.34)

= w∗k,1u(n)

k,1 for n �= h (6.3.35)

where |ak,1| = √a∗

k,1ak,1. We can easily show that V (n)k,1 = w∗

k,1u(n)k,1 is a zero mean complex

Gaussian random variable with variance σ2. For simplicity of notation, let L1 = L. Then,

the decision variables for the first user are

z(n)1 =L∑

l=1

|U (n)l,1 |2 , n = 1, · · · , M (6.3.36)

From [33], and conditioned on A1 and al,1, l = 1 · · · L, we can show that for n = h, z(n)1 has

a non-central χ2 distribution with 2L degrees of freedom and non-centrality parameter

E = 4A21Tw

L∑l=1

|al,1|2 (6.3.37)


The non-centrality parameter E is the symbol energy. For n �= h, z(n)1 has a χ2 distribution

with 2L degrees of freedom. Therefore we can write the conditional probability density

function of z(n)1 as

fz(n)1(z|γs) =

1σ2

(z

σ2γs

) L−12

e−γsσ2+z

σ2 IL−1

(√4γszσ2

)n = h,

1σ2L�(L)z

L−1e−z/σ2n �= h

(6.3.38)

where IL(.) is the modified Bessel function of the L-th order defined earlier in Section 2.1.2,

�(.) is the Gamma function, and

γs = Eσ2

(6.3.39)

We may recognize γs as the symbol energy to interference-plus-noise ratio.

Remark: In this analysis, we used the assumption that the channel vector remains constant

over two symbol periods. Also, we assumed that ak,1 is estimated perfectly. In reality the

channel is time varying and the array covariances are estimated using few samples. This

will lead to errors in the estimated channel vector and hence a reduction in the symbol en-

ergy γs, as shown in Chapter 4. Therefore, the analysis results obtained here can be regarded

as an upper bound on the system performance. 2

6.4 Probability of Error Analysis

In this section we derive the uncoded bit error probability with hard decision. In our anal-

ysis, we will use some of the results of the power control performance study that will be

presented in the next chapter. To derive the probability of error, without loss of generality,

let us assume that h = 1, i.e. the first Walsh symbol W1(t) is transmitted. Then the proba-

bility of symbol error is


PM(γs) = 1 − Pc (6.4.1)

= 1 − P(z(2)1 < z(1)1 , z(3)1 < z(1)1 , · · · , z(M)

1 < z(1)1 ) (6.4.2)

= 1 −∫ ∞

0

[P(z(2)1 < z|z(1)1 = z)

]M−1fz(1)1

(z|γs)dz (6.4.3)

and

P(z(2)1 < z|z(1)1 = z) =∫ z

0fz(2)1

(x)dx (6.4.4)

= 1 − e−z/σ2L−1∑l=0

1l!

( zσ2

)l(6.4.5)

Finally, the corresponding bit error probability Pb(γs) is given by

Pb(γs) = 2J−1

2J − 1PM(γs) (6.4.6)

The symbol error probability and the corresponding bit error probability derived above

are conditional probabilities and are functions of γs, the symbol energy to interference plus

noise ratio. γs itself is a function of the channel vectors for the multipath components re-

ceived from the 1st user a1,1, · · · ,aL,1, shadowing and path loss S1, and the 1st user trans-

mitted power P1. Also note that because of power control (both open loop and closed loop),

P1, S1, and a1,1, · · · ,aL,1 are generally dependent variables. The dependency among these

variables is in general a function of the maximum Doppler shift fd of the first user. From

the simulation results in Chapter 7 and the results in [126, 127], a reasonable assumption

is that the combination of open loop and closed loop power control is perfect in eliminat-

ing the slow fading due to shadowing and path loss. Based on the mobile speed (or fd), we

consider the following different cases.

6.4.1 Low Doppler Frequency

At low Doppler frequencies ( fdTw ≤ 0.00125) and high diversity orders, fast closed loop

power control can eliminate most of the channel variation due to multipath fast fading. In


the case of ideal power control, γs is a fixed quantity and is given by

γs = γ · L · K (6.4.7)

where γ is symbol energy to interference plus noise ratio per path per antenna. Also, the den-

sity function of z(n)1 for n = h given in (6.3.38) becomes an unconditional density. Therefore,

the symbol error probability is

PM = 1 −∫ ∞

0

[1 − e−z/σ2

L−1∑l=0

1l!

( zσ2

)l]M−1

· 1σ2

(z

σ2γs

) L−12

e−γsσ2+z

σ2 IL−1

(2

√γszσ2

)dz (6.4.8)

However, due to the delay in the control loop, finite step size by which the mobile can in-

crease or decrease its power, and errors on the downlink, power control can not be ideal (see

Chapter 7). Therefore the symbol error probability obtained above needs to be averaged

over the probability density function of γs, which is not known. However, an approxima-

tion to the bit error probability can be obtained as follows. First, let Cv denote the coefficient

of variation of γs, defined as

Cv =√

Var{γs}E{γs} (6.4.9)

The average symbol error probability, denoted by PM, is an expectation of a real function

of a random variable, namely γs, so that

PM = E {PM(γs)} =∫ ∞

0PM(γ) fγs (γ) dγ (6.4.10)

where PM(γs) is given by (6.4.8) and fγs(γ) is the probability density function of γs. The

objective here is to avoid carrying out the integration in computing (6.4.10), which requires

the knowledge of fγs(γ). Here, we use the results of [128] which are outlined below to

obtain PM. PM(γs) is expanded in terms of a Taylor’s series expansion, so that

PM(γs)= PM(γs)+ (γs − γs)P′M(γs)+ (γs − γs)

2

2!P′′

M(γs)+· · ·+ (γs − γs)n

n!Pn

M(γs)+ Rn

(6.4.11)


where γs is mean value of γs, PnM(γs) is the nth derivative of PM(γs) (assuming they exist),

and Rn is a remainder that vanishes as n becomes large. Taking expectations of both sides

of (6.4.11) and ignoring terms beyond the second order term yields

PM ≈ PM(γs)+σ2γ

2P′′

M(γs) (6.4.12)

where σγ is the standard deviation of γ. We can easily see that σγ also represents the power

control error. However, in order to evaluate (6.4.12), we need the second derivative of

PM(γs), which is very difficult to obtain. Instead, we expand PM(γs) in terms of central

differences [37]; then taking expectation and ignoring terms beyond the second order term

yields

PM ≈ PM(γs)+σ2γ

2PM(γs + h)+ PM(γs − h)− 2PM(γs)

h2(6.4.13)

Choosing h = √3σγ is shown to give good accuracy [128]. The results in [125] show that

this approximation is accurate for low coefficient of variations Cv. Therefore, a reasonable

approximation of the symbol error probability is

PM ≈ 23

PM(γs)+ 16

PM(γs +√

3σγ )+ 16

PM(γs −√

3σγ) (6.4.14)

where γs is the mean symbol energy to interference-plus-noise ratio and σγ is the power

control error. Then, the corresponding bit error probability for small Cv (less than 0.3 [125])

is

Pb = 2J−1

2J − 1PM (6.4.15)

6.4.2 High Doppler Frequency

For high Doppler frequency and/or long loop delay, the fading statistics of the received sig-

nal after power control remain the same as those of the multipath fast fading with only per-

fect average power control (see Chapter 7). In this case we have

γs = γ

L∑l=1

|al,1|2. (6.4.16)


The distribution of γs depends on the angle spread� through al,1. We consider the following

three cases.

• Small angle spread - For zero (or relatively small) angle spread�, the channel vector

of the lth multipath component can be expressed as

al,1 ≈ αl,1vl,1 (6.4.17)

where αl,1 is a zero mean complex Gaussian random variable and for a ULA vl,1 is a

Vandermonde vector [91] given by

vl,1 = [1 e− jπ sin θl,1 D/λ · · ·e− jπ sin θl,1 D(K−1)/λ]T (6.4.18)

In this case, we can show that γs has a χ2 distribution with 2L degrees of freedom.

That is

fγs(γ) =γL−1

(γK)L(L − 1)!e−γ/(γK) (6.4.19)

Hence, it can be shown that the corresponding unconditional probability density func-

tion of z(n)1 for h = n is (see Appendix B)

fz(n)1(z) = zL−1

σ2L(1 + γK)L(L − 1)!e−z/(σ2(1+γK)) (6.4.20)

• Large angle spread - For large angle spread, the elements of al,1 becomes uncorre-

lated and henceL∑

l=1|al,1|2 is a sum of K L i.i.d. random variables having a χ2 distribu-

tion with 2 degrees of freedom. Therefore, γs is distributed as a χ2 random variable

with 2K L degrees of freedom

fγs (γ) =γK L−1

(γ )K L(K L − 1)!e−γ/γ (6.4.21)

The corresponding unconditional probability density function of z(n)1 for h = n is (see

Appendix B)

fz(n)1(z) =

(K−1)L∑l=0

RlzK L−1+l

(K L − 1 + l)!e−z/(σ2(1+γ)) (6.4.22)


where

Rl = 1(σ2(1 + γ ))K L−l

((K − 1)L

l

)(γ

1 + γ

)(K−1)L−l( 11 + γ

)l

(6.4.23)

• Other values of angle spread - For other values of �, we can easily show (see Ap-

pendix B) that the symbol energy to interference plus noise ratio is

γs =L∑

l=1

K∑i=1

γli|uli|2 (6.4.24)

where u11 · · · uLK are i.i.d. zero mean complex Gaussian random variables and γli =γλli,

{λl,i

}i=1,···,K are the eigenvalues of Rs,l,1, the spatial correlation matrix of the 1st

mobile’s lth multipath component defined in (2.2.17). Let {γli}l=1···L,i=1···K be equal

to {γi}i=1···K L. Also, we assume that the γli’s are distinct (this is true if the angles of

arrival are sufficiently different). Then, γs is distributed as [33]

fγs(γ) =K L∑i=1

πi

γie−γ/γi (6.4.25)

where

πk =LK∏i=1i�=k

γk

γk − γi, k = 1, · · · , LK (6.4.26)

and the corresponding unconditional probability density function of z(n)1 for h = n is

(see Appendix)

fz(n)1(z) =

K L∑i=1

πi

{e−z/(σ2(1+γi))

(σ2(1 + γi))$ g(z)

}(6.4.27)

where

g(z) = zL−2

σ2(L−1)(L − 2)!e−z/σ2

(6.4.28)

and $ denotes the convolution operation.

Using the unconditional pdf of z(n)1 for h = n in Equations (6.4.20), (6.4.22), and (6.4.27),

the average symbol error probability is given by

PM = 1 −∫ ∞

0

[1 − e−z/σ2

L−1∑l=0

1l!

( zσ2

)l]M−1

fz(1)1(z)dz (6.4.29)

and the corresponding average bit error probability is given by Equation (6.4.15).


∆

∆

θ d d

d

d=8λ

Figure 6.5: Simulation scenario

6.5 Numerical and Simulation Results

First, we study the accuracy of the approximation that the MAI signal vector can be as-

sumed to be a spatially white complex Gaussian random vector. The base station receiver in

Figure 6.2 was simulated. In our simulation we assumed that the processing gain G = 256,

L = 4, N = 40, M = 64 and ν= 0.375. We also assumed ideal power control. We assumed

that the base station has 3 sectors, each with a 5 element ULA as shown in Figure 6.5. The

angle of arrivals {θk,i} were assumed to be uniform over [0,120◦]. The angle spreads {�k,i}were assumed uniform over [0,60◦]. The results of 10000 post-correlation signal vectors

were used to estimate the statistics of the MAI signal vector. Figures 6.6 and 6.7 show the

empirical PDF of both the I and Q component of the MAI at the first antenna. From both

figures we can see the validity of the Gaussian approximation. Also, the spatial correlation


Interference Signal Level

-20 -10 0 10 20

PDF

0.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

SimulationsNormal Fitting

Figure 6.6: I-channel: first antenna interference distribution

matrix of the MAI vector R(n)uu,k,1 was estimated as

R(n)uu,k,1 = 10−2 ×

100.43 0.62 + 0.29i 2.63 + 0.04i 2.12 + 0.03i 2.04 + 1.13i

0.62 − 0.29i 100.39 −1.98 − 0.06i 0.90 + 0.09i 1.02 − 0.27i

2.63 − 0.04i −1.98 + 0.06i 100.55 −0.39 + 0.11i 1.11 − 1.15i

2.12 − 0.03i 0.90 − 0.09i −0.39 − 0.11i 100.55 −1.56 + 0.19i

2.04 − 0.13i 1.02 + 0.27i 1.11 + 1.15i −1.56 − 0.19i 100.27

The Frobenius norm [90] of the error ‖e‖F = ‖R(n)uu,k,1− I‖F was estimated as 0.0058 which

also shows that the MAI can be assumed to be spatially white.

Next, we study the system bit error probability. For the low fd case, we used the power


Interference Signal Level

-20 -10 0 10 20

PDF

0.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08SimulationsNormal Fitting

Figure 6.7: Q-channel: first antenna interference distribution

control simulation results presented in Chapter 7. In this simulation, we assumed that sym-

bol rate is 4800 symbols/sec and that the mobile can increase/decrease its transmit power

by 0.5 dB at a time and that the power control command was sent every 1.25 msec. We as-

sumed zero additional loop delay and a forward link error rate of 0.05. Figure 6.8 shows

the bit error probability for the case of ideal power control and for the case of power control

with fd = 5 Hz.

For the ideal power control case, the probability of error was computed using (6.4.6)

and (6.4.8). For the power control case with fd = 5 Hz, the approximation in (6.4.14) was

used. The resulting Pb is plotted for L = 4 and K = 1, K = 3, and K = 5. Note that this

is independent of the angle spread � since with fd = 5 Hz, the fading is slow enough to


N, Number of Users

50 100 150 200

Pb

, Pro

babi

lity

of

Bit

Err

or

10-5

10-4

10-3

10-2

10-1

K = 1, Cv = 0.084

K = 3, Cv = 0.077

K = 5, Cv = 0.077

Power Control, fd = 5 Hz

Ideal Power Control

Figure 6.8: Pb for fd = 5 Hz and closed loop power control.

be tracked by the power control loop for all values of �. This can be seen from the power

control simulation results presented in the next chapter.

If we assume that the required bit error rate is ≤ 10−3, then for K = 1 the maximum

number of users is 29 for ideal power control and 28 for power control with fd = 5 Hz.

For K = 3, these numbers go up to 85 users and 82 respectively. This shows the improved

performance due to beamforming.

For the high fd case, as mentioned earlier, the distribution of γs , and hence the distri-

bution of z(n)1 for n = h, depends on the angle spread �. Figure 6.9 shows the pdf of z(n)1

for different values of � and with ideal power control. From this figure we can see that

the higher the angle spread is, the closer to the ideal power control case the pdf becomes.


Decision Variable z1(1)

0 20 40 60 80 100

Prob

abili

ty D

ensi

ty F

unct

ion,

f(z)

0.00

0.01

0.02

0.03

0.04

0.05

0.06

Ideal Power ControlFading, ∆ = 0ο

Fading, ∆ = 3ο

Fading, ∆ = 60ο

K = 5, L= 4, γs = 13.82 dB

Figure 6.9: PDF of z(n)1 for n = h at high fd.

This can be explained as follows. At zero angle spread, the received signal in any multipath

component will experience the same fading at all antennas and the antenna array will not

provide any space diversity for this multipath component. As the angle spread increases,

the signal fading at different antennas becomes more and more uncorrelated which leads to

less variations in the RAKE output.

Figures 6.10, 6.11, and 6.12 show the bit error probability for � = 0◦, � = 3◦, and

� = 60◦ for a high maximum Doppler frequency fd (high enough such that the statistics

of the received signal after power control remain the same as those of the multipath fast

fading). For � = 0◦ we can see that for Pb = 10−3 the maximum number of allowable mo-

bile reduces to (compared to the perfect power control case) 10, 29, and 55 for K = 1, 3,


N, Number of Users

20 40 60 80 100 120 140 160 180 200

Pb

, Pro

babi

lity

of

Bit

Err

or

10-5

10-4

10-3

10-2

10-1

K = 1 K = 3K = 5

Angle Spread ∆ = 0ο

Figure 6.10: Pb for high fd and � = 0◦, and power control.

and 5, respectively, which corresponds to a 65% reduction in system capacity. This capacity

reduction is due to the multipath fading which was not eliminated by the closed loop power

control.

With a single antenna, the statistics of γs does not depend on the angle spread. Therefore,

the maximum number of allowable mobiles for K = 1 is the same at 10 mobiles per cell for

any value of angle spread. However, for angle spread �= 3◦ this number goes up to 44 and

90 mobiles for K = 3 and K = 5, respectively. This is due to the additional diversity gain

provided by the array. For � = 60◦, these number goes even higher to 58 and 110, which

is, again, due to the space diversity gain provided at large angle spreads. The results of

Figures 6.10, 6.11, and 6.12 are summarized in Table 6.1. This table gives the % reduction


N, Number of Users

20 40 60 80 100 120 140 160 180 200

Pb

, Pro

babi

lity

of

Bit

Err

or

10-5

10-4

10-3

10-2

10-1

K = 1K = 3K = 5



in capacity at Pb = 10−2 and Pb = 10−3 relative to the case with ideal power control. We

make the following observation. For a fixed angle spread, increasing the number of antennas

will provide more space diversity gain. Similarly, for a fixed number of antennas, increasing

the angle spread will provide more space diversity gain.


N, Number of Users

20 40 60 80 100 120 140 160 180 200

Pb

, Pro

babi

lity

of

Bit

Err

or

10-5

10-4

10-3

10-2

10-1

K = 1K = 3K = 5



Reduction at Pb = 10−2 Reduction at Pb = 10−3K

� = 0◦ � = 3◦ � = 60◦ � = 0◦ � = 3◦ � = 60◦

1 50% 50% 50% 65.5% 65.5% 65.5%3 50% 34.6% 19.6% 65.5% 48.8% 31.8%5 50% 26.5% 13.8% 65.5% 36.9% 21.0%

Table 6.1: Percent reduction in capacity at Pb = 10−2 and Pb = 10−3 for high Doppler fre-quency


6.6 Summary

In this chapter, we proposed an antenna array-based base station receiver architecture for

wireless CDMA systems. The receiver structure is based on the ”Beamformer-RAKE” idea

derived earlier in Chapter 3. We also studied the performance of the proposed base station

receiver. In our performance analysis we used some of the results in the next chapter on

power control performance. The average uncoded bit error probability was evaluated as

a function of the number of mobiles, number of antennas, and angle spread for different

power control scenarios. An improvement in the system performance that is proportional

to the number of antennas is observed. Additional improvement is obtained due to space

diversity gain at high angle spread.

Chapter 7

Performance of Power Control in

CDMA

It is very desirable to maximize the capacity of the CDMA mobile telephone system in terms

of the number of simultaneous users that can be served in a given system bandwidth. To

achieve high capacity, the CDMA system employs reverse link power control. The objec-

tive of the mobile transmitter power control process is to control the transmit power of each

mobile operating within the cell so that its signal arrives at the cell site receiver with the

minimum required signal-to-interference ratio. In this case the system capacity will be max-

imized [89]. Capacity will be maximized by power control where several asynchronous

users are communicating with the same base station and share the same bandwidth because

interuser interference caused by non-zero cross-correlation between different codes adds up

on a power basis. The performance of the radio link of any given user degrades as the num-

ber of users in the system increases.

In the case of no power control, if a mobile’s signal is received at the cell site with a

too low value of received power, a high level of interference is experienced by this mobile

and its bit error rate performance will be too high to permit high quality communications

between the mobile and its cell site. On the other hand, if the received power is too high,

138

Chapter 7. Performance of Power Control in CDMA 139

the performance of this mobile is acceptable, but interference to all other mobiles that are

sharing the same channel is increased, and may result in unacceptable performance to other

mobiles.

In fading radio environments, the system capacity is in general limited by multipath fad-

ing, shadowing and path loss, and multiple access interference. The system capacity can be

improved by using several techniques including diversity combining, forward error correct-

ing codes (FEC) and interleaving, and power control [126, 129].

Two different control mechanisms may be used for power control which are open loop

power control (OLPC) and closed loop power control (CLPC). In OLPC, disparities caused

mostly by different path loss and shadowing effects are adjusted individually by each mo-

bile by controlling its transmitted power according to the received forward link signal level.

OLPC will be affected by the multipath fast fading dynamics [130] since the forward and re-

verse link propagation losses and multipath fast fading are not symmetric, particularly when

their center frequencies are widely separated. Errors or inaccuracy in open loop power con-

trol will cause fluctuations of the signal to interference ratios, the effects of which cannot

be corrected by coding and interleaving. Therefore, OLPC alone is not enough and a more

rapidly acting closed loop power control (CLPC) must be used. In CLPC, the base station

monitors the received signal power from each user and compares it to a preset threshold.

Depending on the comparison result, a one bit up/down command is sent on the forward

link to the user at a rate higher than the rate of multipath fading [115]. The current IS-95

CDMA system, uses a combination of OLPC and CLPC for power control on the reverse

link.

Previous work on power control in CDMA includes the work in [127, 131, 132]. The

work in [127, 131] considered two approaches for CLPC: CLPC based on signal strength

and CLPC based on signal to interference-plus-noise ratio SINR. CLPC based on SINR may

results in improved system performance over CLPC based on signal strength [131]. How-

ever, previous studies considered antenna diversity at the base station as a source of path


diversity only. With the adaptive beamforming approach proposed in the previous chapter,

the signal statistics of each multipath component at the output of the beamformer will not

follow a Rayleigh distribution and will be affected by the presence of CLPC and other sys-

tem parameters such as angle spread. In this chapter we will study the performance of signal

strength-based CLPC of a single cell CDMA system.

7.1 CDMA Reverse Link Open Loop Power Control

In open loop power control, each mobile attempts to estimate the path loss from the base

station to the mobile. As we mentioned earlier, in the CDMA forward link, all the base

stations in a region transmit a pilot signal on the same frequency. The mobile measures the

power level of both the pilot from the cell site to which it is connected and the sum of all

cell site signals receivable at the mobile. The forward link signal strength at the mobile is

used by the mobile to adjust its own transmitted power; the stronger the received signal, the

lower the mobile’s transmitted power. Reception of a strong signal from cell site indicates

that the mobile is either close to the cell site or has an unusually good path to the cell site.

This means that relatively less mobile transmitter power is required to produce a nominal

received power at the base station.

The major benefit of open loop power control, which is analog in nature and has about

85 dB of dynamic range [133], is to provide for a very rapid response over a period of just a

few microseconds for cases of sudden change in the channel condition. It adjusts the mobile

transmit level and thus prevents the mobile transmitter power from exceeding some thresh-

old with respect to the forward link received power level.


7.2 CDMA Reverse Link Closed Loop Power Control

In cellular telephony, the full-duplex radio channel is provided by using one frequency band

for transmission from the cell site to the mobile and a different frequency band for transmis-

sion from the mobile to the cell sites. This frequency separation has very important impli-

cations for the power control process: it causes multipath fading on the forward and reverse

links to be independent. This is because the 45 MHz frequency separation between the two

links greatly exceeds the coherence bandwidth of the channel. This means that a mobile can

not measure the path loss of the reverse link by measuring the path loss on the forward link.

This measurement technique, which is used for open loop control, usually provides the cor-

rect average transmit power, but additional provisions must be made for the effects of the

asymmetric Rayleigh fading. Therefore, the mobile transmitter power is also controlled by

its cell site. Each cell site demodulator measures the received signal power Pm (or SINRm in

case of SINR based CLPC) from each mobile. The measured Pm is compared to the desired

power level Pd for that mobile and a power adjustment command is sent accordingly.

The cell site power control command signals the mobile to nominally increase or to de-

crease the mobile power by a predetermined amount �p. The rate of the power control

command must be high enough to permit tracking the Rayleigh fading on the reverse link.

It is important that the latency in estimating the power control signal and the transmission

process be kept small so that the channel conditions will not change significantly before the

control bit can be received and acted upon.

The system controller residing at the MTSO provides each cell site controller with a

value of the Pd (or SINRd) to be used for each individual mobile based on the error rate

performance of that mobile. This level is passed to the channel controller where it is used

to determine whether to command a particular mobile to increase or decrease its transmitted

power. This overall mechanism is called ”closed loop power control”.


1

Control CommandDecision

TransmitPower

Variation Due to Channeland Receiver Processing

Return ChannelError

1

Desired Level

x

T

d

p∆p

ie

i

i

(dB)

(dB)

(dB)

(dB)

Loop Delay

Integrator

PpkT

Step Size

Figure 7.1: Feedback Power Control Model

7.3 Closed Loop Power Control Model

The purpose of the power control loop is to maintain the received signal level for any

given mobile at the required nominal level in an environment where the propagation loss

varies significantly, but slowly enough that the power control mechanism, including inher-

ent delays, can track those variations.

To study the closed loop power control, we use the feedback model shown in Figure 7.1.

As we mentioned earlier, we consider CLPC based on signal strength. In this model all

the quantities are expressed in dB. Let us assume that a power control bit is sent every TP

symbols requesting an incremental change in the transmitted power where Tp is the power

control sampling period. The mobile transmitted signal power p j during the jth period is

updated by a fixed step �p every Tp symbols. During the jth period, the received signal

power at the base station’s RAKE combiner output is pj + x j where x j represents the vari-

ation in signal power due to channel losses and receiver processing. The received signal


power is estimated using the algorithm described below and compared to a preset thresh-

old d. Based on the error e j = p j + x j − d, a power control bit is generated and sent to the

mobile requesting an increase or decrease by �p.

However, although the base station makes the decision to increase or decrease the re-

ceived signal power, the up/down command must still be transmitted on the down link to the

mobile so that it may increase or decrease its transmit power level accordingly. If this com-

mand is received in error, the opposite action will take place. Moreover, the power control

bit is sent unprotected, since the usual long delay due to coding and interleaving is incon-

sistent with a fast acting power control. Therefore, to include the effects of the forward link

errors on the power control performance, we will assume that the power control bit will be

received incorrectly with probability π, where π is the probability of error for the forward

link. Errors on the forward link are assumed to be independent of those on the reverse link.

This is a reasonable assumption since the multipath fast fading processes on both links are

independent. In addition, this model accounts for extra loop delay kTp which accounts for

the two-way signal propagation delay and the time delay involved in generating, transmit-

ting, and executing the power control command.

We can summarize the closed loop power control model described above as follows

p j+1 = p j −�p · ε j−k ·Q (e j−k) (7.3.1)

Q (e j−k) = 1 if ei−k ≥ 0

−1 if ei−k < 0(7.3.2)

ε j−k = 1 with probability 1 − π

−1 with probability π(7.3.3)

ei−k = pi−k + xi−k − d (7.3.4)

The random variable ε represents the forward link errors. We can rewrite (7.3.1) in terms

of the received signal power Ej = x j + p j as

E j+1 = E j −�p · ε j−k ·Q (e j−k)+[x j+1 − x j

](7.3.5)


This equation is the control equation of the first-order discrete-time feedback control loop

shown in Figure 7.1. The quantity x j+1 − x j represents the incremental change in the prop-

agation loss. There is no obvious way to solve (7.3.5) in its current form. A simplifying as-

sumption could be used by considering the time-continuous limit and using Fokker-Planck

techniques [134]. However, it is more useful to simulate (7.3.5), and from this we can draw

several conclusions about the CLPC performance. In the next section, we will present some

of our simulation results.

It remains to be shows how to estimate the received signal power at the base station.

Because of the orthogonality of the M-ary signals used for modulation on the reverse link,

on the average at a relatively high SINR the maximum of the decision variables z(1)1 , z(2)1 ,

· · ·, z(M)

1 corresponds to the signal power plus interference-plus-noise power. The remaining

M − 1 variables correspond to interference-plus-noise power only. A reasonable assump-

tion to make is that the fast fading remains almost constant over Tp symbols in any power

control sampling period. Therefore, if we make a hard decision to select the maximum of

z(1)1 , z(2)1 , · · ·, z(M)

1 , then we can estimate the short term interference-plus-noise and signal

powers over the Tp symbols in a power control sampling period as

µ2 = 1Tp(M − 1)

Tp∑i=1

M∑n=1

n�=nmax

z(n)1 (i) (7.3.6)

ET = 1Tp

Tp∑i=1

maxn=1:M

{z(n)1 (i)} −µ2 (7.3.7)

The decision variables z(1)1 , z(2)1 , · · · , z(M)

1 are also passed to a symbol-by-symbol M-ary

decision device followed by a deinterleaver and a Viterbi convolutional decoder. The re-

ceiver uses the output bit sequence and other information from the convolutional decoder

(such as the branch metrics) to get an estimate of the frame error rate (FER). Based on this

estimated FER, the receiver selects the appropriate threshold level to be used. Figure 7.2

shows a block diagram for the power control algorithm.


Λ(1)

Λ(2)

Λ(J)

b b . . . b(2)(1) (J)

ThresholdET

(1)

(2)

(M)

M-ary

Decision

Symbol-by-Symbol. . . .

J = log M

Convolutional

Decoder

z

z

z

Measure FER

µ2 =1 Σ

j=1ΣM

max

1 max= Σj=1 M

{T

TET

Power Control

Up/Down

Command

Signal Strength Based Power Control Algorithm

T (M-1)

pT

p1

1

1

1

12µ}

z (j)

z (j)p

(n)

n=1n=n

(n)

2

Deinterleaver

Figure 7.2: Power Control Algorithm

7.4 Simulation Results

To study the CLPC performance, a single cell CDMA system was simulated. We assumed

the same base station antenna array model used in the previous chapter. Signals received at

the base station were generated using the channel and transmitted signal models described in

Chapters 2 and 6, respectively. The uncoded bit rate was assumed to be 9.6 kbits/sec. With

a rate 1/3 convolutional code and M-ary orthogonal modulation with M = 64, the symbol

rate is 4800 symbols/sec. The processing gain G is 256. We assumed that a power control

command is sent every 1.25 ms, i.e every 6 Walsh symbols (in which case Tp = 6). Unless

otherwise specified, we assumed that the total loop delay (including power measurement)

is 1 Tp, the forward link error rate π is 0.05, the power step size �p is 0.5 dB (these are the

same values used in [127, 131]), and that the angle spread � is zero.


Symbol Period Tw

0 200 400 600 800 1000

Sign

al L

evel

(dB

)

-5

0

5

10

15

Multipath Fast Fading

Power-Controlled Signal

(a) Signal waveforms

Normalized Signal Level (dB)

-4 -2 0 2 4

Pr

{Sig

nal L

evel

≤ A

bsci

ssa}

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Power-Controlled Signal LevelMultipath Fast Fading

(b) Signal level distributions

Figure 7.3: Power-controlled received signal vs. simulated Rayleigh fading: fd = 5 Hz,K = 5, L = 4.


Symbol Period Tw

0 200 400 600 800 1000

Sign

al L

evel

(dB

)

-5

0

5

10

15

Multipath Fast FadingPower-Controlled Signal

(a) Signal waveforms

Normalized Signal Level (dB)

-4 -2 0 2 4

Pr

{Sig

nal L

evel

≤ A

bsci

ssa}

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Power-Controlled Signal LevelMultipath Fast Fading

(b) Signal level distributions

Figure 7.4: Power-controlled received signal vs. simulated Rayleigh fading: fd = 100 Hz,K = 5, L = 4.


Maximum Doppler Shift fd (Hz)

20 40 60 80 100

Pow

er C

ontr

ol E

rror

σE (

dB)

0

1

2

3

4

5

6

L = 1L = 2L = 4

Figure 7.5: Power control error vs. number of paths: K = 5.

First, we study the effect of closed loop power control on the received signal level statis-

tics. Figures 7.3 and 7.4 show the RAKE output received signal level waveform and the es-

timated distribution versus the simulated multipath fast fading for fd = 5 Hz and fd = 100

Hz, respectively. From these two figures we can easily see that closed loop power control

eliminates most of the channel variations due to fast fading at low fd, while at high fd the

received signal statistics after power control remain almost the same as that of the simulated

multipath fast fading with only average power control. This result is consistent with the re-

sults in [127]. In addition, we may note that in [127, 115], it was suggested that for high fd

the signal level distribution can be approximated by a log-normal distribution.



20 40 60 80 100

Pow

er C

ontr

ol E

rror

σE (

dB)

0

1

2

3

4

∆p = 0.25 dB∆p = 0.5 dB∆p = 1 dB

Figure 7.6: Power control error vs. power step size: L = 2,K = 5.

Figure 7.5 shows the effect of the number of RAKE branches L on the power control

error σE. It can be seen from this figure that σE is reduced with higher diversity order. We

can also note that with high diversity order and low fd, CLPC can eliminate most of the

channel variation. Figure 7.6 shows the effect of step size �p on the power control error

σE for �p = 0.25, 0.5, and 1 dB. We can see that σE is lower with large step size �p and

high fd, while at low fd, a small step size provides more precise control. The reason is that

at high fd the fading rate is too high and a large step size is necessary to track the fading.

On the other hand, at low fd the fading rate is slow enough to allow the control loop to

track the fading with a small step size. Figure 7.7 shows the effect of loop delay kTp for



20 40 60 80 100

Pow

er C

ontr

ol E

rror

σE (

dB)

0

1

2

3

4

Loop Delay = 1 Tp

Loop Delay = 2 Tp

Loop Delay = 3 Tp

Figure 7.7: Power control error vs. loop delay: L = 2,K = 5.

k = 1,2, and 3. We can see from the figure that the longer the loop delay is, the poorer σE.

Figure 7.8 shows the effect of return channel error rate π on σE and it suggests the CLPC is

robust against return channel errors. Again, these results are consistent with the results in

[127, 132].

Next we look at the effect of angle spread and number of antennas in the array. As in the

previous chapter, we assumed that the array size is 8λ and the sensors are placed as a ULA.

In Figures 7.9 and 7.10 we plot the power control error σE against fd for different values of

� for K = 5 and 9. From these figures we can make the following observations. With zero

angle spread �, the number of antennas has no effect on the CLPC error σE. This is due



20 40 60 80 100

Pow

er C

ontr

ol E

rror

σE (

dB)

0

1

2

3

4

π = 0.05π = 0.1

Figure 7.8: Power control error vs. forward link error rate: L = 2, K = 5.

to the fact that with zero angle spread �, the received signal in any multipath component

will have the same fading at each antenna. Thus, the antenna array will not provide any

space diversity for this multipath component. Also, for a given number of antennas K, as

the angle spread � increases, the gain due to space diversity increases which will lead to a

reduction in the power control error σE (i.e. better CLPC performance).



20 40 60 80 100

Pow

er C

ontr

ol E

rror

σE (

dB)

0

1

2

3

∆ = 0ο

∆ = 5ο

∆ = 10ο

∆ = 40ο

∆ = 60ο

Figure 7.9: Power control error vs. angle spread: L = 4, and K = 5.

7.5 Summary

In this chapter, we studied the closed loop power control performance of the reverse link

in wireless DS/CDMA through a combination of analysis and discrete event simulation.

The closed loop power control mechanism is modeled as a first-order discrete time feed-

back control loop. Based on this model, we simulated the closed loop power control on the

communication link from a CDMA mobile to the base station. Several conclusions can be

drawn from our simulation results. Our simulation results show that with high diversity or-

der (either path or space) and low Doppler frequencies ( relative to the power control rate),



20 40 60 80 100

Pow

er C

ontr

ol E

rror

σE (

dB)

0

1

2

3

∆ = 0ο

∆ = 5ο

∆ = 10ο

∆ = 40ο

∆ = 60ο

Figure 7.10: Power control error vs. angle spread: L = 4, and K = 9.

closed loop power control can eliminate most of the channel variations, and with small an-

gle spread adaptive beamforming with antenna arrays has a little effect on the power control

error.

Chapter 8

Conclusions

The main focus of this thesis has been the development and application of advanced array

signal processing techniques to CDMA wireless systems that have a practical implementa-

tion complexity and achieve high performance levels. This work has been prompted by the

current thrust in wireless communication technology to look for new approaches and tech-

nologies to improve spectrum efficiency and to be able to support the projected capacity

demands with the introduction of new personal communication services.

8.1 Thesis Summary

The second half of Chapter 2 and Chapters 3, 4, 6, and 7 presented the thesis contributions.

In our work, we focused on the mobile to base station link and assumed that only the base

station uses an antenna array for transmission and reception of signals to and from the mo-

bile. The first half of Chapter 2 provides a brief overview of wireless radio channels and

statistical models for scalar (single antenna) channels. In the second half of this chapter, we

develop a statistical vector (multiple antennas) channel model based on the physical prop-

agation environment and derive its statistical properties. This is necessary in order analyze

the new array signal processing and beamforming techniques proposed later on in the thesis.

154

Chapter 8. Conclusions 155

In Chapter 3, we start by reviewing general techniques for adaptive beamforming and

discuss the reasons why such techniques are not suitable for CDMA wireless mobile sys-

tems. We then present a new space-time code-filtering approach for channel vector esti-

mation and optimum beamforming. This approach exploits both the temporal and spatial

information in the received signal. A key result in this thesis is the derivation of the space-

time matched filter receiver for multipath signals over AWGN channels. This receiver con-

sists of a front-end spatial matched filter, or beamformer, followed by a temporal matched

filter. Based on the space-time matched filter, we construct a Beamformer-RAKE receiver

for CDMA signals. Performance analysis results of the Beamformer-RAKE receiver show

improved bit error rate performance due to the exploitation of the spatial dimension in the

received signal. In Chapter 4, we consider the time-variant nature of the wireless channel

and develop a recursive algorithm for tracking the channel vector and estimating the beam-

former weight vector. The algorithm has a reasonable computational complexity of O(K2)

per path per user. Simulation results show that the proposed algorithm can track the channel

vector closely even in severe propagation conditions.

Chapters 6 and 7 look at the application of the array signal processing techniques derived

in the previous chapters to the existing CDMA IS-95 cellular standard which we briefly de-

scribe in Chapter 5. In Chapter 6 we propose an overall base station receiver architecture

based on the Beamformer-RAKE structure and describe different signal processing func-

tions. We then study the performance of the proposed base station receiver in terms of the

uncoded bit error probability as a function of loading (number of users) and number of an-

tennas. Under the propagation conditions considered, the analysis results show a perfor-

mance improvement in terms of the number of users that can be supported due to the use

of antenna arrays and the associated signal processing. These analysis results make use of

the results of Chapter 7, where we look at modeling and performance of closed loop power

control. Many conclusions can be drawn from the simulation results described in Chap-

ter 7. The most important result is that beamforming reduces the power control error when

Chapter 8. Conclusions 156

the angle spread is non-zero.

8.2 Future Directions

Although this thesis has answered several questions related to the application of array signal

processing techniques to CDMA wireless systems, several questions remain open including:

• The Beamformer-RAKE receiver in Chapter 3 is not an optimum structure when mul-

tiple access interference is present. Therefore, optimal Beamformer-RAKE structures

which include multiuser detection and interference cancellation is an interesting fu-

ture research topic.

• The performance analysis of the proposed base station receiver for wireless CDMA

did not include the effects of forward error correction and interleaving, nonuniform

power-delay profiles, and the effect of channel vector estimation errors on the overall

receiver performance. Extension of this performance analysis to include these effects

will also be another interesting future research direction.

• The base station must also beamform on the forward link in order to effectively im-

prove overall system performance. However, since both forward and reverse links

use different frequency bands and, therefore, the multipath vector channels for both

links are quite different, the base station cannot use the reverse link beamformer for

forward link transmission. Hence, forward link beamforming is also an interesting

and important future research direction.

Appendix A

Multipath Fading Correlation

A.1 Fading Correlation for Scalar Channels

In this Appendix, we derive the time-frequency correlation of the multipath fast fading for

scalar channels. First, we recall the definition of β(ω, t) from (2.1.13) as

β(ω, t) =L∑

i=1

Riejφi(t) (A.1.1)

where φi = ωd cosψit −ωτi and the φi modulo 2π are assumed to be i.i.d. and uniform over

[0,2π]. We assume that the angular position of the ith scatterer ψi is uniformly distributed

over [0,2π]. We also assume that the time delays τi τi are i.i.d. with probability density

function fT(τ), where fT(τ) is nonzero for 0 ≤ τ <∞ and zero otherwise. Then, the time-

frequency correlation of β(ω, t) can be derived as follows:

ρβ(ω1, ω2, t, t + ν) = E{β(ω1, t)β∗(ω2, t + ν)} (A.1.2)

= E

{L∑

i=1

L∑l=1

Ri Rlej(φi(ω1,t)−φl(ω2,t+ν))

}(A.1.3)

The average will vanish unless i = l. In this case φi(ω1, t)− φi(ω2, t + ν) = ωd cosψiν−�ωτi, where �ω = ω1 − ω2. Therefore

ρβ(ω1, ω2, t, t + ν) = ρβ(�ω, ν) (A.1.4)

157

Appendix A. Multipath Fading Correlation 158

ρβ(ω1, ω2, t, t + ν) =∑

i

E{R2i }e j(ωd cosψiν−�ωτi) (A.1.5)

E{R2i } represents the average fraction of incoming power in the ith path, which can be rewrit-

ten as

E{R2i } = σ2 f&(ψi) fT(τ)dψidτi (A.1.6)

where σ2 is the total radiated from the mobile, which will be assumed to equal 1, and the

term f&(ψi) fT(τi)dψidτi represents the average fraction of incoming power within dψi of

angle ψi and within dτi of the time τi. If we assume that L is large (i.e. L → ∞), then we

can express the sum in (A.1.5) with integrals, independent of i,

ρβ(�ω, ν) = 12π

∫ 2π

0

∫ ∞

0e j(ωd cosψν−�ωτ) fT (τ)dψdτ (A.1.7)

= 12π

∫ 2π

0e j(ωd cosψν)dψ ·

∫ ∞

0e− j�ωτ fT(τ)dτ (A.1.8)

= Jo(ωdν) ·∫ ∞

0e− j�ωτ fT(τ)dτ (A.1.9)

= Jo(ωdν) · FT( j�ω) (A.1.10)

where FT(s) is the characteristic function of the time delay τ defined as

FT(s) =∫ ∞

0e−sτ fT(τ) dτ (A.1.11)

We notice that FT( j�ω) is also the Fourier transform of the probability density function

fT(τ).

A.2 Fading Correlation for Vector Channels

In this Appendix, we derive the space-time-frequency correlation for the multipath vector

channel. The derivation follows the same steps and reasoning used in the preceding ap-

pendix. Again, we start by recalling the definition of the channel vector

a(ω, t) =L∑

i=1

v(θi)Riejφi(t) (A.2.1)


In addition to the assumptions used in the preceding Appendix, we also use the assumption

from (2.2.9) that the angle of arrival of the ith path θi is uniformly distributed over [−�+θ,�+ θ] where θ is the mean angle of arrival and 2� is the angle spread. Then, we can

write Ra, the space-time-frequency correlation matrix of the channel vector, as

Ra(ω1, ω2, ν) = E{a(ω1, t)a∗(ω2, t + ν)} (A.2.2)

=∑

i

E{R2i }e j(ωd cosψiν−�ωτi) · v(ω1, θi)v∗(ω2, θi) (A.2.3)

The average fraction of incoming power in the ith path E{R2i } is now redefined as

E{R2i } = f&(ψi) fT (τ) f�(ϑi)dψidτidϑi (A.2.4)

where the term f&(ψi) fT(τ) f�(ϑi)dψidτidϑi now represents the average fraction of in-

coming power form all paths that arrive within dϑi of angle ϑi within dψi of angle ψi and

within dτi of the time τi.

Again, as L → ∞, then we can express the sum in (A.2.3) with integrals, independent

of i,

Ra(ω1, ω2, ν) = 12π

∫ 2π

0

∫ ∞

0e j(ωd cosψν−�ωτ) fT(τ)dψdτ

× 12�

∫ �+θ

−�+θ

v(ω1, ϑ)v∗(ω2, ϑ)dϑ (A.2.5)

= Jo(ωdν) · FT( j�ω) · Rs (A.2.6)

where Rs is defined in ( 2.2.17). To evaluate the matrix Rs, we consider the model geometry

in Figure 2.9 and evaluate the (m, n) element of Rs as follows

Rs(m, n) = 12�

∫ �+θ

−�+θ

vm(ω1, ϑ)v∗n(ω2, ϑ)dϑ (A.2.7)

vm(ω1, ϑ) is the phase shift for a plane wave propagating with angular frequencyω1 between

the mth sensor and the reference point. Similarly, vn(ω2, ϑ) is the phase shift for a plane

wave propagating with angular frequency ω2 between the nth sensor and the reference point.


Using the definition in (2.2.5), we can show that

vm(ω1, ϑ)v∗n(ω2, ϑ) = exp

{− j

(ω1rm

c− ω2rn

c

)}(A.2.8)

where c is the speed of light and rm and rn are given by

rm = dmn

2sinϑmn + dr

mn sin(ϑrmn + ϑmn) (A.2.9)

rn = drmn sin(ϑr

mn + ϑmn)− dmn

2sinϑmn (A.2.10)

Let ζ = (ω1rm − ω2rn)/c and define

z1 = ω1 + ω2

2cdmn + ω1 − ω2

2cdr

mn cosϑrmn (A.2.11)

z2 = ω1 − ω2

2cdr

mn sinϑrmn (A.2.12)

We can easily show that ζ = z1 sinϑmn + z2 cosϑmn = zmn sin(ϑmn + δmn) where

zmn =√

z21 + z2

2 and δmn = cos−1 z1

zmn. (A.2.13)

Therefore, we have

Rs(m, n) = 12�

∫ �+θ

−�+θ

vm(ω1, ϑ)v∗n(ω2, ϑ)dϑ (A.2.14)

= 12�

∫ �+θ

−�+θ

e− jzmn sin x dx (A.2.15)

= 12�

∫ �+θ

−�+θ

(cos(zmn sin x)− j sin(zmn sin x)) dx (A.2.16)

where θ = θmn + δmn. Now, by making use of the well-known series representation

cos(zmn sin x) = Jo(zmn)+ 2∞∑

l=1

J2l(zmn) cos(2lx) (A.2.17)

sin(zmn sin x) = 2∞∑

l=0

J2l+1(zmn) sin((2l + 1)x) (A.2.18)

we can integrate (A.2.16) to get

Re{Rs(m, n)} = Jo(zmn)+ 2∞∑

l=1

J2l(zmn) cos(2l(θmn + δmn))sinc(2l�) (A.2.19)

Im{Rs(m, n)} = 2∞∑

l=0

J2l+1(zmn) sin((2l + 1)(θmn + δmn))sinc((2l + 1)�) (A.2.20)


Note that Rs is Hermitian i.e Rs(m, n) = R∗s (n,m).

Appendix B

Probability Distributions

To derive the unconditional probability density functions in (6.4.20), (6.4.22), and (6.4.27),

first we recall the characteristic function representation of the conditional density in (6.3.38)

[33]

fz(n)1(z|γs) = 1

2πj

∮exp{sz − γsσ

2s/(σ2s + 1)}(σ2s + 1)L

ds (B.0.1)

Then the unconditional density of z(n)1 is given by

fz(n)1(z) =

∫ ∞

0fz(n)1

(z|γs) fγs(γ) dγ

= 12πj

∮esz

(σ2s + 1)L

{∫ ∞

0e−γσ2s/(σ2s+1) fγs (γ) dγ

}ds

= 12πj

∮esz

(σ2s + 1)LFγs

(σ2s

σ2s + 1

)ds

(B.0.2)

where Fγs (.) is the characteristic function of γs. Then,

• For small angle spread, fγs (γ) is given by Equation (6.4.19) and

fz(n)1(z) = 1

2πj

∮esz

(σ2s + 1)L

(σ2s + 1)L

(σ2(1 + γK)s + 1)Lds

= 12πj

∮esz

(σ2(1 + γK)s + 1)Lds

162

Appendix B. Probability Distributions 163

= zL−1

σ2L(1 + γK)L(L − 1)!e−z/(σ2(1+γK)) (B.0.3)

• For large angle spread, fγs(γ) is given by Equation (6.4.21) and

fz(n)1(z) = 1

2πj

∮esz

(σ2s + 1)L

(σ2s + 1)K L

(σ2(1 + γ )s + 1)K Lds

= 12πj

∮esz (σ2s + 1)(K−1)L

(σ2(1 + γ )s + 1)K Lds (B.0.4)

Now we have

Fz(s) = (σ2s + 1)(K−1)L

(σ2(1 + γ )s + 1)K L

=(K−1)L∑

l=0

Rl(s + 1/(σ2(1 + γ )

)K L−l (B.0.5)

where

Rl = 1(σ2(1 + γ ))K L−l

1l!

dl

dsl(σ2s + 1)(K−1)L

∣∣∣∣s= −1

(σ2 (1+γ ))

= 1(σ2(1 + γ ))K L−l

((K − 1)L

l

)(γ

1 + γ

)(K−1)L−l( 11 + γ

)l

(B.0.6)

It follows that

fz(n)1(z) =

(K−1)L∑l=0

RlzK L−1+l

(K L − 1 + l)!e−z/(σ2(1+γ)) (B.0.7)

• For other values of angle spread, first we need to derive the density function for γs

itself. We have

γs = γ

L∑l=1

|al,1|2 (B.0.8)

Let al,1 = R∗/2s,l,1ul where Rs,l,1 is the K × K spatial correlation matrix of the array for

al,1 and ul is a K × 1 zero mean complex Gaussian random vector with covariance

matrix I. Since Rs,l,1 is Hermitian, we can rewrite Rs,l,1 as

Rs,l,1 = UlΛlU∗l (B.0.9)

Appendix B. Probability Distributions 164

where Ul is orthogonal and Λl is a diagonal matrix of the eigenvalues of Rs,l,1. Let

Λl = diag{λl1 λl2 · · ·λlK}. Then we can write |al,1|2 as

|al,1|2 = u∗l UlΛlU∗

l ul

= u∗l Λlul

=K∑

i=1

λli|uli|2 (B.0.10)

where ul = U∗l ul is also a zero mean complex Gaussian random vector with covari-

ance I. Then, we can rewrite γs as

γs =L∑

l=1

K∑i=1

γli|uli|2 (B.0.11)

where γli = γλli and, therefore, γs has the density function in Equation (6.4.25). It

follows that

fz(n)1(z) = 1

2πj

∮esz

(σ2s + 1)L

{K L∑l=i

πiσ2s + 1

σ2(1 + γi)s + 1

}ds

=K L∑l=i

πi1

2πj

∮esz

(σ2s + 1)L−1

1σ2(1 + γi)s + 1

ds

=K L∑i=1

πi

{e−z/(σ2(1+γi))

(σ2(1 + γi))$

zL−2

σ2(L−1)(L − 2)!e−z/σ2

}

=K L∑i=1

Bi

{e−z/(σ2(1+γi)) − e−z/σ2

L−2∑l=0

(γi

1 + γi

)l zl

σ2l l!

}(B.0.12)

where

Bi = πi

γiσ2

(γi + 1γi

)L−2

(B.0.13)

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Documents

Thesis Adaptive Antennas For CDMA WIRELESS Networks