INFORMATION THEORETIC LIMITS ON

COMMUNICATION OVER MULTIPATH FADING

CHANNELS

by

Richard Buz

A thesis submitted to the Department of Electrical

Engineering in conformity with the requirements

for the degree of Doctor of Philosophy

Queen's University

Kingston, Ontario, Canada

June 1994

Copyright c Richard Buz, 1994

Abstract

While a considerable amount of research into the design of channel codes for use in

fading environments continues to be performed, it is done so without knowledge of

the magnitude of the potential gain yet to be achieved. Due to this circumstance, it is

uncertain as to whether the best known codes are actually any good when compared

to the class of all codes. In an attempt to remedy this situation, ultimate limits

on the rate of reliable communication over multipath fading channels are presented

here. These limits are based on the information theoretic notions of channel capacity

and average mutual information, and provide a benchmark against which to measure

the merit of any channel coding scheme. An idealized channel model is considered

�rst, and through comparison with known results for additive noise channels, the

loss due to amplitude fading is determined. Channels with continuous-valued input

alphabets are considered as well as those based on practical signal constellations.

Information theoretic limits are determined when subject to both average and peak

power constraints, the latter being more relevant to mobile communication. While

multiple antennae are commonly used to combat channel loss, the e�ect of this space

diversity is viewed here from an information theoretic perspective, and the resulting

gain in the rate of reliable data transmission is ascertained. The magnitude of the

potential gain over uncoded modulation achievable through the use of signal shaping

and channel coding is stated.

In conjunction with the in uence of amplitude fading, the performance of any prac-

ticable communication strategy is also a�ected by imperfect channel state estimation

as well as time dispersion of the transmitted signal. Each of these considerations is

addressed separately, through augmentation of the idealized channel model, in order

ii

to ascertain their e�ect on the maximum rate of reliable data transmission. The re-

quirement of channel estimation is demonstrated through calculation of information

theoretic limits for channels in which the state of the fading process is unknown.

Similar results are also obtained for the case of perfect coherent detection with no

knowledge of the fading amplitude. Through comparison with results obtained for the

ideal channel, the losses incurred due to the limitation of practical channel estimation

schemes is determined. The particular methods of channel estimation considered are

pilot tone extraction, di�erentially coherent detection, and the use of a pilot sym-

bol. The e�ect of time dispersion is examined through calculation of the capacity

of a frequency-selective fading channel represented by the two-path Rayleigh model.

An inherent time diversity e�ect is demonstrated, which manifests itself in certain

instances of frequency-selective fading.

iii

Acknowledgements

Most people that I meet are under the impression that a Ph.D. is some type of

honor bestowed upon people with superior intellect. I always tell them that the

process of obtaining a doctorate has little to do with intelligence and is more like a

test of endurance. Although I believe this somewhat facetious statement re ects one

important requirement, there were other equally vital contributing factors involved

in the realization of this thesis.

I am indebted to Dr. Peter McLane for inspiring my interest in communications

and for providing the opportunity to accomplish this project. I appreciate the con�-

dence that he showed in my abilities and the patience he exhibited in waiting for me

to get things done.

Funding of this research was provided in part by both the Natural Sciences and

Engineering Research Council of Canada and the Canadian Institute for Telecom-

munications Research. The Telecommunications Research Institute of Ontario also

contributed �nancial support as well as the use of computer facilities. I would like to

express my gratitude to these organizations for their assistance.

I would like to thank Dr. Norman Beaulieu and Dr. Lorne Campbell for repeat-

edly sharing their insight with me whenever I would drop by unannounced with a

mathematical problem.

I can never thank my parents enough. They exhibited the value of hard work to

me and always encouraged me to set my goals high. It was a comfort to know that

they were always there for me, willing to help whenever I needed them.

Finally, it was William Shakespeare, who through his play \King Lear", instilled

the belief in my mind that it's better to go mad than to give up.

iv

Summary of Notation

Abbreviations

AMI average mutual information

AMPM amplitude modulation / phase modulation

AWGN additive white Gaussian noise

BCM block coded modulation

bps bits per second

CSI channel state information

CR cross constellation

dB decibel

DPSK di�erential phase shift keying

Hz hertz

kbps kilobits per second

kHz kilohertz

LOS line-of-sight

MHz megahertz

mph miles per hour

MSAT mobile satellite

MTCM multiple trellis coded modulation

NASA National Aeronautics and Space Administration

PAR peak-to-average power ratio

pdf probability density function

pmf probability mass function

PSK phase shift keying

v

QAM quadrature amplitude modulation

QPSK quaternary phase shift keying

SNR signal-to-noise power ratio

TCM trellis coded modulation

Symbols and Functions

A signal amplitude

ai weighting coeÆcient

a(t) envelope of transmitted signal

BD Doppler spread of channel

(B)ij entry in covariance matrix

B covariance matrix

C channel capacity

Cxy correlation between random variables x and y

CE Euler's constant

d Euclidean distance

Eb average received energy per bit

Es average received symbol energy

EX average power of a random variable X

E f�g operator denoting statistical expectation

Ei(�) exponential integral function

erf(�) error function

erfc(�) complimentary error function

FW water pouring band

fc carrier frequency

fD Doppler frequency

G(f) inverse of channel SNR function

GF (�) Galois �eld

g(t) baseband pulse

vi

H(f) channel transfer function

H(�) entropy of a random variable

H(�j�) conditional entropy

h(t) channel impulse response

I(�; �) average mutual information

I0(�) modi�ed Bessel function of �rst kind and zero order

IW water pouring band for parallel channels

=f�g imaginary part of enclosed complex number

J(�) Jacobian of coordinate transformation

J0(�) Bessel function of �rst kind and zero order

j square root of -1

K0 threshold for water pouring

L level of diversity

Lf�g Laplace transform operator

M size of signal set

m Nakagami channel parameter

mX mean value of a random variable X

N random noise variable

N0 noise power spectral density

NB length of a block of channel symbols

N(f) general noise power spectral density

n(t) random noise process

P average transmitted power

Pb bit error probability

Ps peak power of constellation

Pr(e) probability of detection error

PX entropy power of a random variable X

p(�) probability density function

p(�j�) conditional probability density function

R random fading amplitude variable

R0 computational cuto� rate

vii

Rc rate of transmission

Rh(�) multipath intensity pro�le of channel

Rx(�) auto-correlation function of a random process x(t)

<f�g real part of enclosed complex-valued expression

ri(t) attenuation along ith propagation path

SX support set of a random variable X

Sh(�) Doppler power spectrum of channel

SX(f) power spectral density of signal x(t)

s(t) transmitted bandpass signal

T duration of channel symbol

T transpose of matrix

Tm multipath spread of channel

Ts duration of signal

u(t) complex envelope of transmitted signal

v velocity of mobile unit

W bandwidth of transmit spectrum

Wp bandwidth of pilot tone extraction �lter

X channel input alphabet

x di�erentially encoded channel symbol

xp pilot symbol

Y channel output alphabet

y(t) baseband signal at receiver

yi(t) signal received by ith antenna

�(�) gamma function or generalized factorial

power ratio

R Rician channel parameter

�J discrepancy in Jensen's inequality

(�f)H coherence bandwidth of channel

(�t)h coherence time of channel

Æ(�) Dirac delta function

�A angle of asymmetry

viii

�(t) phase of transmitted signal

� carrier wavelength

� correlation coeÆcient

�2X variance of a random variable X

�i(t) delay along ith propagation path

� random fading variable

~� estimate of fading variable at receiver

�(t) random fading process

� random fading phase variable

�i(t) phase shift along ith propagation path

'i(t) function from an orthonormal set

angle of incidence

(�) Euler's psi function

second moment of Nakagami fading variable

! radian frequency

d�e smallest integer greater than the enclosed value

ix

Contents

Abstract ii

Acknowledgements iv

Summary of Notation v

List of Tables xiv

List of Figures xvi

1 Introduction 1

1.1 Lessons Learned from the AWGN Channel . . . . . . . . . . . . . . . 2

1.2 State of the Art Coding for Fading Channels . . . . . . . . . . . . . . 5

1.3 Known Applications of Information Theory to Fading Channels . . . 10

1.4 Contributions of Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.5 Presentation Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2 Multipath Fading Channel Models 14

2.1 The Physical Channel . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.2 Amplitude Fading Models . . . . . . . . . . . . . . . . . . . . . . . . 20

2.2.1 Rayleigh Fading . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.2.2 Rician Fading . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.2.3 Shadowed Rician Fading . . . . . . . . . . . . . . . . . . . . . 22

2.2.4 Nakagami Fading . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.3 E�ects of Frequency Dispersion . . . . . . . . . . . . . . . . . . . . . 24

2.3.1 Symbol Interleaving . . . . . . . . . . . . . . . . . . . . . . . . 24

x

2.3.2 Diversity Combining . . . . . . . . . . . . . . . . . . . . . . . 25

2.3.3 Channel State Estimation . . . . . . . . . . . . . . . . . . . . 27

2.4 Frequency-Selective Fading Channels . . . . . . . . . . . . . . . . . . 29

2.4.1 Linear Filter Model . . . . . . . . . . . . . . . . . . . . . . . . 29

2.4.2 Three-Path Model . . . . . . . . . . . . . . . . . . . . . . . . 30

2.4.3 Two-Path Rayleigh Model . . . . . . . . . . . . . . . . . . . . 33

3 Information Theoretic Bounds for Ideal Fading Channels 35

3.1 Information Theoretic Concepts . . . . . . . . . . . . . . . . . . . . . 36

3.2 Channels with Discrete-Valued Input . . . . . . . . . . . . . . . . . . 44

3.2.1 Standard Signal Constellations . . . . . . . . . . . . . . . . . 47

3.2.2 Asymmetric PSK Constellations . . . . . . . . . . . . . . . . . 53

3.3 Capacity of Ideal Fading Channels . . . . . . . . . . . . . . . . . . . 56

3.4 Peak Power Considerations . . . . . . . . . . . . . . . . . . . . . . . . 65

3.4.1 Peak Power Results for Discrete-Valued Input . . . . . . . . . 65

3.4.2 Channel Capacity with a Peak Power Constraint . . . . . . . . 72

3.5 Channels with Space Diversity . . . . . . . . . . . . . . . . . . . . . . 76

3.5.1 E�ect of Diversity on Discrete-Input Channels . . . . . . . . . 79

3.5.2 Capacity of Fading Channels with Space Diversity . . . . . . . 87

3.6 Potential Coding Gain for Fading Channels . . . . . . . . . . . . . . . 94

4 E�ects of Non-Ideal Channel State Information 100

4.1 Requirement of Channel State Estimation . . . . . . . . . . . . . . . 101

4.1.1 Channels with Discrete-Valued Input and No CSI . . . . . . . 101

4.1.2 Channel with Continuous-Valued Input and No CSI . . . . . . 104

4.2 Channels with Phase-Only Information . . . . . . . . . . . . . . . . . 109

4.2.1 Phase-Only Channels with Discrete-Valued Input . . . . . . . 111

4.2.2 Phase-Only Channels with Continuous-Valued Input . . . . . 120

4.3 Realistic Channel Estimation Methods . . . . . . . . . . . . . . . . . 122

4.3.1 Channel Estimation Via Pilot Tone Extraction . . . . . . . . . 126

4.3.2 Channel Estimation Via Di�erentially Coherent Detection . . 133

4.3.3 Channel Estimation Via Pilot Symbol Transmission . . . . . . 140

xi

5 Information Theoretic Bounds for Frequency-Selective Fading Chan-

nels 146

5.1 Representation of Waveform Channels . . . . . . . . . . . . . . . . . 147

5.1.1 Time-Invariant Filter Channels . . . . . . . . . . . . . . . . . 150

5.2 Capacity of the Two-Path Rayleigh Channel . . . . . . . . . . . . . . 152

5.2.1 Properties of the Two-Path Model . . . . . . . . . . . . . . . 154

5.2.2 Capacity and Equalization . . . . . . . . . . . . . . . . . . . . 156

5.3 Time Diversity E�ect . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

5.3.1 Channels with Discrete-Valued Input . . . . . . . . . . . . . . 158

5.3.2 Channels with Continuous-Valued Input . . . . . . . . . . . . 164

6 Conclusion 168

6.1 Summary of Presentation . . . . . . . . . . . . . . . . . . . . . . . . . 168

6.2 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170

6.3 Suggestions for Further Research . . . . . . . . . . . . . . . . . . . . 172

Bibliography 174

A Comment on Results Obtained Through Computer Simulation 180

B Calculation of Channel Capacity for Speci�c Fading Distributions 181

B.1 Capacity of a Rayleigh Fading Channel . . . . . . . . . . . . . . . . . 181

B.2 Capacity of a Nakagami Fading Channel . . . . . . . . . . . . . . . . 182

C Calculation of Asymptotic Loss Due to Speci�c Fading Distributions184

C.1 Asymptotic Loss Due to Rayleigh Fading . . . . . . . . . . . . . . . . 184

C.2 Asymptotic Loss Due to Nakagami Fading . . . . . . . . . . . . . . . 185

D Calculation of Error Probability for Uncoded Modulation in Ideal

Rayleigh Fading 187

D.1 Symbol Error Probability for Uncoded QAM . . . . . . . . . . . . . . 187

D.2 Bit Error Probability for Uncoded QPSK . . . . . . . . . . . . . . . . 190

E Derivation of a PDF Related to the Two-Path Rayleigh Channel 192

xii

Vita 194

xiii

List of Tables

3.1 Minimum SNR Required for Various Rates of AMI on an AWGN Chan-

nel: PSK Constellations . . . . . . . . . . . . . . . . . . . . . . . . . 45

3.2 Minimum SNR Required for Various Rates of AMI on an AWGN Chan-

nel: QAM Constellations . . . . . . . . . . . . . . . . . . . . . . . . . 45

3.3 Minimum SNR Required for Various Rates of AMI on an Ideal Rayleigh

Fading Channel: PSK Constellations . . . . . . . . . . . . . . . . . . 51

3.4 Minimum SNR Required for Various Rates of AMI on an Ideal Rayleigh

Fading Channel: QAM Constellations . . . . . . . . . . . . . . . . . . 51

3.5 Loss of SNR Due to Rayleigh Fading: PSK Constellations . . . . . . 52

3.6 Loss of SNR Due to Rayleigh Fading: QAM Constellations . . . . . . 52

3.7 Average Power Gain Due to Increase in Space Diversity of Rayleigh

Fading Channel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

3.8 Average Power Loss Due to Space Correlation of the Fading Process . 84

4.1 Loss of SNR Due to Phase-Only Information: PSK Constellations . . 114

4.2 Loss of SNR Due to Phase-Only Information: QAM Constellations . . 116

4.3 Loss of SNR Due to Phase-Only Information: Hybrid AMPM Constel-

lations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

4.4 Loss of SNR Due to Non-Ideal CSI: Pilot Tone Estimation with PSK

Constellations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

4.5 Loss of SNR Due to Non-Ideal CSI: Pilot Tone Estimation with QAM

Constellations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

4.6 Loss of SNR Due to Non-Ideal CSI: Di�erentially Coherent Detection

with PSK Constellations . . . . . . . . . . . . . . . . . . . . . . . . . 139

xiv

4.7 Loss of SNR Due to Non-Ideal CSI: Pilot Symbol Transmission . . . . 145

5.1 Gain Due to Time Diversity E�ect on the Two-Path Rayleigh Channel 163

xv

List of Figures

2.1 Multipath Fading Environment . . . . . . . . . . . . . . . . . . . . . 16

2.2 Channel Transfer Function of the Three-Path Model . . . . . . . . . . 32

3.1 AMI of an AWGN Channel: PSK Constellations . . . . . . . . . . . . 42

3.2 AMI of an AWGN Channel: QAM Constellations . . . . . . . . . . . 43

3.3 AMI of an Ideal Rayleigh Fading Channel: PSK Constellations . . . . 49

3.4 AMI of an Ideal Rayleigh Fading Channel: QAM Constellations . . . 50

3.5 AMI of an Ideal Rayleigh Fading Channel: Asymmetric PSK Constel-

lations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

3.6 Capacity of an Ideal Rayleigh Fading Channel . . . . . . . . . . . . . 59

3.7 Capacity of an Ideal Rician Fading Channel . . . . . . . . . . . . . . 61

3.8 Capacity of an Ideal Shadowed Rician Fading Channel . . . . . . . . 63

3.9 Capacity of an Ideal Nakagami Fading Channel . . . . . . . . . . . . 64

3.10 AMI of an Ideal Rayleigh Fading Channel in Terms of Peak Power:

QAM Constellations . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

3.11 Hybrid AMPM Constellations . . . . . . . . . . . . . . . . . . . . . . 69

3.12 AMI of an Ideal Rayleigh Fading Channel in Terms of Average Power:

Hybrid AMPM Constellations . . . . . . . . . . . . . . . . . . . . . . 70

3.13 AMI of an Ideal Rayleigh Fading Channel in Terms of Peak Power:

Hybrid AMPM Constellations . . . . . . . . . . . . . . . . . . . . . . 71

3.14 Bounds on the Capacity of an Ideal Rayleigh Fading Channel with a

Peak Power Constraint . . . . . . . . . . . . . . . . . . . . . . . . . . 77

3.15 AMI of an Ideal Rayleigh Fading Channel with Space Diversity: 8-PSK 82

3.16 AMI of an Ideal Rayleigh Fading Channel with Space Diversity: 16-QAM 83

xvi

3.17 AMI of an Ideal Rayleigh Fading Channel with Diversity=2 and Space

Correlated Fading: 8-PSK . . . . . . . . . . . . . . . . . . . . . . . . 85

3.18 AMI of an Ideal Rayleigh Fading Channel with Diversity=2 and Space

Correlated Fading: 16-QAM . . . . . . . . . . . . . . . . . . . . . . . 86

3.19 Capacity of an Ideal Rayleigh Fading Channel with Space Diversity . 91

3.20 Capacity of Dual Diversity Rayleigh Channel with Space Correlated

Fading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

3.21 Probability of Symbol Error for Uncoded QAM in Rayleigh Fading . 96

3.22 Bit Error Probability in Rayleigh Fading for Various Coded Modulation

Schemes with Rate 2 bits/T . . . . . . . . . . . . . . . . . . . . . . . 97

3.23 Bit Error Probability in Rayleigh Fading for Turbo Codes . . . . . . 99

4.1 AMI of a Rician Fading Channel with No CSI: 8-PSK . . . . . . . . . 105

4.2 AMI of a Rician Fading Channel with No CSI: 16-QAM . . . . . . . . 106

4.3 Upper Bounds on the AMI of a Rician Fading Channel with No CSI:

Gaussian Distributed Input . . . . . . . . . . . . . . . . . . . . . . . 110

4.4 AMI of a Phase-Only Rayleigh Fading Channel: PSK Constellations . 115

4.5 AMI of a Phase-Only Rayleigh Fading Channel: QAM Constellations 117

4.6 AMI of a Phase-Only Rayleigh Fading Channel: Hybrid AMPM Con-

stellations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

4.7 Bounds on AMI for a Phase-Only Rayleigh Channel with Continuous-

Valued Input . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

4.8 AMI of Rayleigh Fading Channel with CSI Provided Via Pilot Tone

Extraction: 8-PSK . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

4.9 AMI of Rayleigh Fading Channel with CSI Provided Via Pilot Tone

Extraction: 16-PSK . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

4.10 AMI of Rayleigh Fading Channel with CSI Provided Via Pilot Tone

Extraction: 16-QAM . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

4.11 AMI of Rayleigh Fading Channel with CSI Provided Via Pilot Tone

Extraction: 32-CR . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

4.12 AMI of Rayleigh Fading Channel with CSI Provided Via Di�erentially

Coherent Detection: 8-PSK . . . . . . . . . . . . . . . . . . . . . . . 137

xvii

4.13 AMI of Rayleigh Fading Channel with CSI Provided Via Di�erentially

Coherent Detection: 16-PSK . . . . . . . . . . . . . . . . . . . . . . . 138

4.14 AMI of Rayleigh Fading Channel with CSI Provided Via Pilot Symbol

Transmission: 8-PSK . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

4.15 AMI of Rayleigh Fading Channel with CSI Provided Via Pilot Symbol

Transmission: 16-QAM . . . . . . . . . . . . . . . . . . . . . . . . . . 144

5.1 AMI for Two-Path Rayleigh Fading Channel with Time Diversity Ef-

fect: 8-PSK . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

5.2 AMI for Two-Path Rayleigh Fading Channel with Time Diversity Ef-

fect: 16-QAM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162

5.3 Capacity of Two-Path Rayleigh Fading Channel with Time Diversity

E�ect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167

xviii

Chapter 1

Introduction

At the present time it's becoming rare to �nd any communications related publication

that does not make reference to the \wireless revolution". This revolution has wit-

nessed the realization of a number of mobile communication systems; some examples

of which are mobile satellite, cellular telephone, and personal communication services.

What these schemes have in common is that the transmission medium for commu-

nication is accurately modelled as a multipath fading channel. Due to a continuing

dramatic increase in the use of these systems, as well as the need for standardiza-

tion of the methods used, a considerable amount of research has been directed at the

development of eÆcient strategies for transmitting data over fading channels.

Prior to the wireless revolution, a signi�cant amount of research was focussed on

achieving reliable communication over telephone lines. These telephone line chan-

nels are well-modelled by the sequential combination of a linear �lter followed by

an additive noise process. The noise is usually described as being a spectrally- at

Gaussian random process, and the transmission medium is referred to as an additive

white Gaussian noise (AWGN) channel. The limits of communication over AWGN

channels were determined �rst, and it was not until four decades later that research

in equalization and channel coding allowed these limits to be approached.

The techniques developed for communication over AWGN channels have been

applied to fading channels; however, performance of these techniques in fading is

signi�cantly inferior to additive noise channel results. Much of the research currently

being performed deals with the modi�cation of known channel coding techniques for

1

use over fading channels. So far some great improvements have been made, but

data rates achievable in fading are still quite modest in comparison to telephone

line channels. Due to the extreme di�erence in reliability between fading and noise

channels, one cannot help asking certain fundamental questions. Must performance in

fading be so inferior to the results obtained for the additive noise channel? Although

a new code may result in an improvement over all other known codes, does this mean

that the new one is actually a good code? Until the limits of communication over

multipath fading channels are determined, these questions cannot be answered.

1.1 Lessons Learned from the AWGN Channel

Prior to 1948, most engineers believed that the noise present on a communication

channel placed a limit on the reliability of transmitting data. No matter what solu-

tions were used to combat the noise, most concurred that this perceived limit could

not be surpassed. These engineers learned that this was not the case after the appear-

ance of Claude Shannon's seminal work [1], which gave birth to the �eld of information

theory. Using Hartley's quantitative measure of information [2] and ideas from sta-

tistical mechanics, Shannon developed a number of momentous results. One of these

results, known as the channel coding theorem, states that under certain conditions

there is no �xed limit on the reliability of communication. Given a particular channel

model, there exists a maximum rate of transmission called the capacity of the chan-

nel. The channel coding theorem states that as long as the transmission rate does

not exceed the channel capacity, there exists some coding scheme which can be used

to achieve an arbitrary degree of reliability. For an ideal AWGN channel bandlimited

to W Hz, and a signal-to-noise power ratio denoted by SNR, the capacity measured

in bits per second is given by the formula [1]

C =W log2 (1 + SNR) bps: (1.1)

When using uncoded QAM signal sets to transmit data over a high-SNR AWGN

channel, a symbol error rate of the order of 10�5�10�6 is achieved at a SNR which is

9 dB greater than that given by the capacity formula [3]. An alternate interpretation

of this result is that there is an additional 3 bps/Hz to be gained in the data rate

2

than what is achieved with uncoded QAM. This is based on an asymptotic estimate

for QAM constellations, which demonstrates that each increase in spectral eÆciency

of 1 bps/Hz requires an additional 3 dB of power [4].

Much of the research related to the high-SNR AWGN channel was driven by the

development of telephone line modems. The telephone channel typically has a SNR of

28 dB to 36 dB or more, and a bandwidth of 2400 Hz to 3200 Hz or more. The results

of information theory indicate that the capacity of a telephone channel is somewhere

in the range of 20 kbps to 30 kbps or more. Despite these numbers, for twenty

years following Shannon's work, the practical limit was believed to be 2400 bps1.

There were two major reasons why the information theoretic capacity was considered

unrealistic. The �rst reason is that the telephone channel is not an ideal �lter channel.

Using the entire bandwidth would cause distortion and intersymbol interference, so

the useable bandwidth was limited to 1200 Hz. The second reason is that coding and

modulation were treated as separate processes. Traditional channel coding introduces

redundancy at a cost of reducing the data rate for a �xed bandwidth. During this

time, a signi�cant amount of research e�ort was expended on the development of

adaptive equalization techniques. The result was an increase in the useable bandwidth

of the telephone channel to 2400 Hz. As a consequence, the existence of a practical

limit of 2400 bps was learned to be untrue. By utilizing the increased bandwidth and

increasing the level of modulation from QPSK to 16-QAM, data rates of 9600 bps

were being attained. The next few years witnessed solutions to other telephone line

impairments, such as echo and phase jitter, but it was expected at the time that the

9600 bps limit would not be surpassed.

With the advent of coded modulation in the late 70's, the 9600 bps limit was

shown to be an erroneous assumption. Imai and Hirakawa [5] are credited with the

discovery of block coded modulation (BCM); however, it was the trellis coded mod-

ulation (TCM) schemes of Ungerboeck [6] which were responsible for the renaissance

experienced in the �eld of coding theory. Rather than separating the processes of

coding and modulation, Ungerboeck attempted to jointly optimize the two. As a

1 In fact, earlier perceived limits were thought to be even lower than 2400 bps.

3

result, simple practical codes were discovered which e�ected gains of 3-4 dB with-

out a reduction in data rate, while up to 6 dB of gain was reached by using more

complex codes. With regard to coded modulation, redundancy is added to the data

by expanding the size of the required signal set. Channel symbols are then assigned

to code sequences in a manner that ensures that the gain in distance between signal

sequences is greater than the power loss incurred due to signal set expansion2. One

method of accomplishing this is Ungerboeck's technique of mapping by set partition-

ing. Modi�ed versions of Ungerboeck's codes were included in modem standards in

the mid 1980's for data rates of up to 14.4 kbps [7].

Although Calderbank and Sloane [8] are credited with introducing the language of

lattices and cosets to coded modulation, it was Forney's comprehensive work on coset

codes [9] that placed coded modulation on a �rm mathematical basis, and allowed

a more profound understanding of the subtleties of code structure. One of Forney's

observations was that the total gain obtained could be separated into two relatively

independent terms. The �rst term is called the shaping gain, and has an ultimate limit

of 1.53 dB. Shaping gain is obtained when spherical multi-dimensional constellations

are utilized rather than cubic ones. The second term is referred to as the coding gain.

With respect to the 9 dB di�erence between the error performance of QAM and the

capacity of a high-SNR AWGN channel, since shaping can theoretically be used to

reduce the gap by about 1.5 dB, the remaining 7.5 dB represents the limit for the

maximum coding gain to be achieved. Forney also observed that after the �rst 5-6 dB

of coding gain is acquired, it is easier to obtain the next 1 dB by shaping rather than

by using a more complex code [10]. The combination of these realizable gains places

practical results at less than 3 dB from the theoretical limit. At present, standards

are being determined for modems to transmit data over telephone channels at rates

of 19.2 kbps or higher. By incorporating precoding, channel coding, and shaping in

modem design, the Codex corporation claims to have achieved reliable transmission

at a rate of 24 kbps [11]. Many of the facts contained in this section have been taken

from [11], which contains a more detailed history of telephone line modems.

Another recently published paper [12], although not related to coded modulation,

2 This power loss refers to the increase in average power required to maintain the minimumdistance between channel symbols when increasing the size of the constellation.

4

presents a traditional coding scheme which yields a bit error probability of 10�5 at

only 0.7 dB from the theoretical capacity of the channel. This scheme uses a parallel

concatenation of two 16-state recursive systematic convolutional codes. An iterative

decoding algorithm is utilized, where performance improves in relation to the number

of iterations carried out.

It seems that even though approaching channel capacity was once considered an

impractical goal, knowledge of this limit must have left the impression on some that

there was always more to be gained. With each increase obtained in the data rate,

the theoretical channel capacity has always been a constant goal to strive for.

1.2 State of the Art Coding for Fading Channels

At present, the reliable high data rate communications available on telephone line

channels cannot be achieved over multipath fading channels. There are a number of

factors which contribute to this limitation. One reason is that there are more potential

sources of signal degradation associated with fading environments. For example,

multipath fading and Doppler spread can have deleterious e�ects on communication.

Another reason is that most error correcting codes have been designed to correct

random errors, whereas on a fading channel, errors tend to occur in bursts. These

error bursts arise during deep fades, which cause the signal to become more susceptible

to noise. Another important consideration for many systems used in fading is the

presence of non-linear distortion. Satellite transponders, as well as most cost eÆcient

ampli�ers, are usually operated in a mode that causes them to have a non-linear

characteristic. In this case, constant envelope signalling methods such as PSK or

continuous phase modulation must be used in order to avoid the e�ects of distortion

and bandwidth expansion caused by non-linearities. In terrestrial microwave systems,

the availability of power permits the use of linear ampli�ers, which in turn allows the

use of higher levels of QAM. It is easier to increase the bandwidth eÆciency of linear

channels, since high level QAM outperforms high level PSK modulation with regard

to average signal power.

In order to increase power eÆciency, traditional coding could be used by trading

5

o� bandwidth or data rate. This is obviously diÆcult for satellite communication

systems, which are already constrained to have a lower data rate due to the use of

constant envelope modulation. At present, however, traditional coding techniques

are the only ones which have actually been used over fading channels. The most

popular coding methods adopted for satellite and deep space communications are

convolutional codes and Reed Solomon codes [13]. The standard convolutional codes

used are of rate 12or 2

3, and usually have 64 or 128 encoder states. The standard Reed

Solomon code used by NASA is a (255, 233) code over GF (28) [14]. Reed Solomon

codes are designed to correct burst errors, while convolutional codes are used to

combat random errors. Sometimes the two techniques are concatenated so that data

is protected from both types of error distribution. The constant envelope modulation

generally used in fading is QPSK. Due to the diÆculty in achieving accurate coherent

transmission over mobile fading channels, di�erential encoding is often used in the

transmitter along with di�erentially coherent detection in the receiver. In this case,

the modulation method is referred to as di�erential phase shift keying (DPSK).

It seems that the satellite channel would be a prime area of application for band-

width eÆcient trellis coded modulation. In fact, this is where most of the initial

research was done in the application of coded modulation to fading channels. In par-

ticular, the mobile satellite (MSAT) systems developed in Canada [15] and the United

States [16] were the �rst particular projects which prompted work in this area. Initial

research was aimed at achieving high quality voice transmission over fading channels.

Telephone line quality digital voice transmission requires an average bit error rate of

10�3. In [15], the study used a rate of 1200 symbols per second to obtain a bit rate

of 2400 bps. The MSAT channel was originally proposed to be bandlimited to about

5 kHz, and data rates of 4800 to 9600 bps were of interest in [16]. In both cases, a

maximum spectral eÆciency of 2 bits/T was considered3. Due to the non-linearity

of the satellite channel, constant envelope QPSK and 8-PSK symbol sets were used.

Computer simulations as well as upper bounds on error probability were presented

to indicate code performance. These results included practical considerations such as

3 The parameter T represents the duration of a channel symbol. The spectral eÆciency in unitsof bps/Hz may be determined through multiplication by the number of channel symbols transmittedper Hz of bandwidth.

6

�nite symbol interleaving, di�erential encoding, and the e�ects of non-ideal channel

state information. While still signi�cantly inferior to AWGN channel results, the use

of TCM resulted in tremendous gains over uncoded schemes for use in fading, with

no reduction in data rate. For the AWGN channel, codes are designed to maximize

the minimum Euclidean distance between symbol sequences. When fading channels

are considered, however, it was discovered that codes with parallel trellis transitions

are inferior to those without, even when the minimum Euclidean distance is greater.

For those codes without parallel transitions, Ungerboeck's 8-PSK codes [6] proved

to be superior. Some small additional gains were obtained in [16] by introducing

asymmetry into the PSK constellations used in order to increase the minimum Eu-

clidean distance between code sequences. For the Canadian MSAT system, despite

the promising performance of TCM, a traditional convolutional code used in com-

bination with QPSK modulation was �nally accepted in the standard. In order to

compensate for the loss in data rate due to coding, the bandwidth was increased to

7500 Hz.

In [17], Divsalar and Simon set out performance criteria for the design of TCM

for fading channels. This was done through examination of an expression which

is an upper bound on the probability of bit error. For high values of SNR, the

most important factor in code design is the minimum time diversity between code

sequences. Time diversity refers to the number of signalling intervals in which symbols

di�er between code sequences. For large values of SNR, the bit error probability

decreases in proportion to (SNR)�L, where L is the minimum time diversity between

all code sequences. This explains why Ungerboeck codes with parallel transitions are

not good for use over fading channels. A diversity of 1 symbol would result in an

inverse linear decrease in error probability with respect to SNR. The second design

criterion is to maximize the minimum product distance between sequences. The

product distance is the product of the non-zero squared Euclidean distances between

corresponding symbols in a pair of code sequences. Bit error probability has an inverse

dependence on the product distance. The third criterion is to maximize the minimum

Euclidean distance between code sequences, which is the most important criterion for

the AWGN channel. The minimum Euclidean distance has a more signi�cant e�ect on

7

error performance when there is a line-of-sight path from transmitter to receiver, or

when the fading process does not appear completely independent between symbols at

the receiver. It is interesting to note that when using traditional coding, maximizing

Hamming distance also maximizes diversity between code sequences. Unfortunately,

this is not the case with coded modulation, which uses a Euclidean distance metric.

Multidimensional QAM constellations were �rst used in trellis codes for the tele-

phone channel [18]. The analog using PSK signal sets was called multiple trellis coded

modulation (MTCM) in [19]. Also presented in [20], MTCM was shown to attain a

greater time diversity than conventional TCM for the same amount of decoding com-

plexity. For example, Ungerboeck's 4-state 8-PSK code has a diversity of 1 symbol

due to the parallel transition in the trellis, but by using a 2� 8-PSK signal set with

the same trellis, the diversity is increased to 2 symbols. Another practical advantage

of MTCM is that codes can be made completely rotationally invariant, whereas this

is not possible with conventional TCM using PSK signal sets [7]. Also, higher data

rates are possible with MTCM when compared to conventional TCM schemes which

use the same constituent PSK constellation.

While signi�cant improvements have been made on the reliability of communica-

tion over fading channels, many of the coding techniques used are just extensions of

Ungerboeck's TCM. Other recently discovered codes indicate that TCM may not be

the most promising method for use in fading. Over a fading channel, time correlation

of the fading process is detrimental to code performance [17]. In order to combat this,

symbol interleaving is usually performed after coding and prior to modulation. By

deinterleaving sequences in the receiver, the fading appears independent from symbol

to symbol and the errors appear more random. Zehavi [21] used the fact that bit in-

terleaving is more e�ective than symbol interleaving to show that conventional TCM

may not be the best approach for fading channels. He used a good convolutional

encoder followed by individual bit interleavers and a signal mapper. A suboptimum

decoding technique was used in the receiver instead of maximum likelihood decod-

ing. Computer simulations can be used to show that over a Rayleigh fading channel,

Ungerboeck's 8-state 8-PSK code achieves a bit error probability of 10�6 at an aver-

age bit SNR of approximately 23.9 dB, while Zehavi's suboptimum code attains the

8

same performance at a SNR of approximately 16.8 dB. This is the best trellis code

result known to date for a rate of 2 bits/T and a complexity of 8 encoder states.

Another area of practical interest in the �eld of coded modulation, that has been

receiving much attention lately, is the design of multi-level codes [5]. The interest

in this code structure is due to the fact that a multi-level code can be detected by

a suboptimum staged decoding procedure, which trades a small loss in performance

with a large reduction in decoding complexity. This is accomplished by partitioning

the code into a chain of subcodes, then sequentially decoding each of these simple

subcodes subject to the decoding decision made for the previous subcode in the chain.

One such scheme, presented by Lin [22], transmits approximately 2 bits/T by using

8-PSK modulation combined with two levels of Reed Solomon codes. Over a Rayleigh

fading channel, this code attains a bit error rate of 10�6 at an average bit SNR of

approximately 12 dB. This appears to be the best known BCM performance to date

for transmission at a rate of 2 bits/T over a Rayleigh channel.

A very recent publication contains a channel coding scheme based on the com-

bination of binary error correcting turbo-codes with higher levels of PSK and QAM

modulation [23]. In addition to being simpler to implement than TCM, these codes

are claimed to outperform trellis codes over both AWGN and Rayleigh fading chan-

nels. A bit error rate of 10�5 is realized at a bit SNR of 6.5 dB for a spectral eÆciency

of 2 bits/T using a 16-QAM constellation. For the same error rate at a spectral ef-

�ciency of 3 bits/T with 16-QAM, the required SNR is 9.6 dB. In order to attain a

rate of 4 bits/T , a 64-QAM signal set is used. A SNR of 11.7 dB is required to attain

a bit error rate of 10�5 with this code. These are the best known results to date for

transmission over a Rayleigh fading channel.

Most of the coding schemes presented have a maximum rate of 2 bits/T . With

the constant demand for larger data rates, new research is turning toward higher

levels of modulation for use in fading. In particular, the use of 16-QAM and 16-PSK

signal sets are being studied to �nd the most promising scheme for transmitting data

at a rate of 3 bits/T [23, 24]. Naturally, this rate will also soon be inadequate and

higher levels of modulation will eventually be investigated. See [25] for an up to date

overview of the use of coded modulation in fading, as well as a list of references.

9

1.3 Known Applications of Information Theory to

Fading Channels

A surprisingly small amount of work has been performed in order to determine in-

formation theoretic limits on communication over multipath fading channels. One

of the earliest known results is due to Kennedy [26], and can also be found stated

in [27]. Kennedy considered orthogonal signalling over a channel with no bandwidth

constraint. It was discovered that the unlimited diversity diminishes the e�ect of

fading to the point where the capacity of a fading channel is the same as that of an

AWGN channel.

The capacity of a bandlimited discrete-time fading channel was considered by both

Ericson [28] and Lee [29]. In both cases, it was assumed that fading is independent

with respect to discrete-time signalling intervals, and also that the state of the fading

process is known at the receiver. The �nal result was a form similar to that of an ideal

AWGN channel, except that the SNR is dependent upon the particular value of the

fading process, and the expression obtained must be averaged over fading statistics.

An explicit expression for the capacity of a Rayleigh fading channel was presented

in [29].

The computational cuto� rate has been considered for channels based on discrete-

valued constellations. Although the cuto� rate is not an ultimate limit, it is considered

to be a practical limit for many known coding techniques, and usually re ects the

general behavior of the channel capacity. In [30], the cuto� rate of the channel was

obtained under the assumption of independent fading and knowledge of the channel

state at the receiver. The e�ects of time-correlation of the fading process on the cuto�

rate were examined in [31].

The capacity of fading channels subject to a �nite decoding delay constraint was

considered in [32]. It was stated that for stringent constraints on decoding delay, the

Shannon capacity did not exist. It was suggested that an appropriate information

theoretic performance measure is given by the capacity versus outage probability

characteristics. References to other works related to multipath fading channels are

also stated in [32]

10

1.4 Contributions of Thesis

The following list is a synopsis of the signi�cant contributions presented in this thesis.

1. The capacity of an ideal fading channel is determined. An explicit expression is

derived for the case of Rayleigh fading as well as certain instances of Nakagami

fading.

2. Limits on reliable transmission are determined for ideal fading channels when

standard signal constellations are used. Results are also presented for certain

special constellations.

3. A benchmark is obtained for measuring code performance over fading channels,

and the potential gains available through channel coding and signal shaping are

stated.

4. Information theoretic limits are determined for the case of when a peak power

constraint is placed on the signal.

5. Potential gains available through the use of space diversity are stated from an

information theoretic perspective.

6. In addition to the loss experienced due to amplitude fading, the loss due to

non-ideal channel state information is ascertained. In particular, these losses

are determined for systems in which practical channel estimation schemes are

utilized.

7. The capacity of a frequency-selective fading channel represented by the two-path

Rayleigh model is determined.

8. A time diversity e�ect occurs for certain instances of frequency-selective fading,

and the resulting gains are quanti�ed here in terms of information theoretic

measures.

11

1.5 Presentation Outline

Most scienti�c work begins with the derivation of a model of the phenomenon being

studied. Mathematical models of multipath fading channels are presented in Chapter

2. The general model is based upon an interpretation of the physical e�ects of the

channel on a transmitted signal. Simplifying assumptions are applied to obtain a

static amplitude fading model which isolates the e�ects of fading. The models become

more complex as the e�ects of both frequency and time dispersion on the signal are

introduced.

In Chapter 3, fundamental concepts of information theory are stated, then used to

determine the limits of communication over ideal fading channels. The cases of inter-

est are categorized as to whether the channel input is discrete-valued or continuous-

valued. Communication rates possible with discrete-valued signal sets are ascertained

�rst. In order to determine the loss due to fading, these bounds are compared with

those obtained for the AWGN channel. While results of this type are usually presented

only in terms of average symbol energy, both average and peak power constraints are

considered. The interest in peak power results is due to the physical limit on the

peak transmitter power available in mobile communications. Channel capacity is also

investigated for a number of amplitude fading models when a continuous-valued input

is subject to either a peak or average power constraint. In practical mobile systems,

antenna diversity is commonly used to reduce the possibility of channel loss. The

e�ect of this space diversity on communication rate is examined from an informa-

tion theoretic viewpoint. The investigation of ideal fading channels is summarized by

inspecting the gains possible through the use of signal shaping and channel coding.

The e�ect of non-ideal channel state information on the rate of reliable trans-

mission is considered in Chapter 4. In order to justify the requirement of channel

state estimation, information theoretic limits are calculated for channels where the

state of the fading process is not known at all at the receiver. A speci�c model for

non-ideal channel state information, which has appeared in channel coding literature,

is the case of perfect coherent detection with no knowledge of the fading amplitude.

Based on this assumption, average mutual information is determined for the case of

discrete-valued constellations, while an upper bound on this quantity is calculated

12

for channels with a continuous-valued input. The remainder of the chapter exam-

ines channels for which fading information is provided by some realistic estimation

method. The estimation schemes considered are the use of a pilot tone, di�erentially

coherent detection, and the use of a pilot symbol. Relevant parameters are derived

for each of these estimation techniques in order to determine the maximum rate of

reliable data transmission for such communication systems.

Determining ultimate limits on the rate of reliable transmission over channels

subject to frequency-selective fading is the focus of Chapter 5. Since the response

of a frequency-selective fading channel shapes the spectrum of the transmitted sig-

nal, a continuous waveform channel must be considered rather than the discrete-time

channel model used in previous chapters. Due to this eventuality, the presentation

begins with a discussion of waveform channels in general, which leads to the state-

ment of an appropriate de�nition of average mutual information and capacity. The

particular frequency-selective fading model examined is that of the two-path Rayleigh

channel. Certain characteristics of the response of the two-path channel are stated,

which can be used to simplify the calculation of channel capacity. The capacity of

a frequency-selective fading channel subject to certain practical design constraints is

then determined. For certain instances of frequency-selective fading, a time diversity

e�ect occurs at the channel output. By capitalizing on this diversity e�ect, an im-

provement in the reliability of data transmission may be realized. The magnitude of

this potential gain is quanti�ed here by determining the average mutual information

for discrete-valued signal sets, as well as the capacity for channels with a continuous-

valued input. Results are presented for di�erent values of power distribution between

the individual beams of the channel model.

Chapter 6 contains a compendium of the principal results presented in the the-

sis. Suggestions for further research are listed, as well as a number of miscellaneous

questions which remain unanswered.

13

Chapter 2

Multipath Fading Channel Models

When transmitting information over a multipath fading channel, the signal undergoes

amplitude fading, dispersion in both time and frequency, as well as corruption by ad-

ditive noise. Describing this channel mathematically can result in some quite complex

equations which can make any further analysis of a communication system improb-

able. In order to avoid such a situation, engineers often make assumptions in order

to simplify the mathematical model. The result is a representation that accurately

re ects the relevant e�ects of the channel on signals, but also allows further analysis

of the system to be tractable. A given fading channel is characterized not only by

its physical layout, but also by communication parameters such as carrier frequency,

signalling rate, and velocity of the mobile unit. Depending on the particular commu-

nication scenario, di�erent sets of simplifying assumptions are possible, resulting in

a number of di�erent potential models for a given physical channel. In this chapter,

the most common fading channel models that have been used in the development of

modern communication systems are described, as well as the simplifying assumptions

on which the models are based.

2.1 The Physical Channel

A general fading model is derived here based upon an interpretation of the physical

e�ects of the channel on a transmitted signal. A more detailed discussion of this

model is given by Kennedy in [26, pp. 9-67] and by Proakis in [33, pp. 455-463]. The

14

general form of a bandpass signal with carrier frequency fc is

s(t) = a(t) cos(2�fct+ �(t)) (2.1)

where transmitted information is carried in the amplitude a(t), the phase �(t), or

both. It is assumed that the signal bandwidth is much smaller than the carrier

frequency. Equivalently, the signal may be expressed in the form

s(t) = <fu(t) exp(j2�fct)g (2.2)

where <f�g is an operator denoting the real part of the enclosed complex-valued

expression, and u(t) is the complex envelope or baseband equivalent of s(t). Suppose

s(t) is transmitted through a typical mobile communication channel as shown in

Figure 2.1. The signal at the mobile receiver will be composed of a number of re ected

signals, as well as a possible line-of-sight (LOS) signal. Along the ith path, the signal

will experience an attenuation ri(t) and a delay �i(t), both of which vary with time

due to movement of the mobile unit. Thus, the received multipath signal can be

expressed as a summation of the received signals over all propagation paths, and

takes the form Xi

ri(t)s(t� �i(t)): (2.3)

Substituting equation (2.2) into the summation given in (2.3) results in the expression

<("X

i

ri(t) exp(�j2�fc�i(t))u(t� �i(t))

#exp(j2�fct)

): (2.4)

In addition to the multipath e�ects, the signal is also corrupted by an additive white

Gaussian noise process with a complex envelope n(t). From the expression in (2.4),

the equivalent low pass received signal is

y(t) =Xi

ri(t) exp(�j2�fc�i(t))u(t� �i(t)) + n(t): (2.5)

The �rst term on the right hand side of equation (2.5) is just a discrete convolution

of the baseband signal u(t) with the time-varying channel impulse response

h(� ; t) =Xi

ri(t) exp(�j2�fc�i(t))Æ(� � �i(t)): (2.6)

15

LOS path

scattered paths

Figure 2.1: Multipath Fading Environment

16

With regard to the impulse response h(� ; t), convolution is performed with respect

to the variable � , while dependence upon the variable t illustrates the time-varying

nature of the channel.

The fading exhibited by the multipath channel can be understood much more

clearly by considering the e�ects on an unmodulated carrier. This is accomplished

by setting u(t) = 1 for all values of t in equation (2.5), so that the received signal can

be expressed as

y(t) =Xi

ri(t) exp(j�i(t)) + n(t) (2.7)

where �i(t) = �2�fc�i(t). In this case, the received signal is just the sum of a number

of two-dimensional vectors, or phasors, having length ri(t) and phase �i(t) along

the ith path. The change in the attenuation ri(t) is not very signi�cant over short

periods of time; however, the phase �i(t) will change by 2� radians whenever the delay

�i(t) changes by f�1c , which is usually a very small number. It is assumed that the

attenuation and delay associated with di�erent paths are independent of each other.

The number of paths can usually be assumed to be large, so by applying the central

limit theorem, this sum of independent and identically distributed random processes

looks like a complex-valued Gaussian random process. This example illustrates the

reason why amplitude fading occurs. Since the received signal consists of the sum of a

large number of time-varying vectors, sometimes they add constructively causing an

increase in signal level, and sometimes they add destructively causing the amplitude

of the received signal to be very small. Since the attenuation factors ri(t) do not

change much over short periods of time, this fading phenomenon is mainly due to the

time variation of the phases �i(t).

In order to illustrate the dispersive e�ects of the channel, the auto-correlation

of the channel impulse response h(� ; t) shown in (2.6) is examined. The response

is modelled as a complex-valued Gaussian random process, so the auto-correlation

function is

Rh(�1; �2; t1; t2) =1

2E fh(�1; t1)h�(�2; t2)g : (2.8)

It is assumed that the random process is wide sense stationary so that the auto-

correlation depends only on the time di�erence �t = t2 � t1. It is also assumed

that the scattering is uncorrelated, which means that the attenuation and phase

17

shifts associated with di�erent path delays are uncorrelated. This means that the

expression in (2.8) will equal zero unless �1 = �2. Incorporating these assumptions

into equation (2.8) results in the auto-correlation function being expressed in the form

Rh(� ; �t), where the parameter � is equal to both �1 and �2. By letting �t = 0, the

multipath intensity pro�le Rh(�) is obtained. This is the average power output of the

channel as a function of the delay � . Let Tm be the range of � over which Rh(�) is

essentially non-zero. The parameter Tm is called the multipath spread of the channel,

and is the average length of time over which the energy of a very narrow pulse, or

impulse, is dispersed. If Tm happens to be larger than the signalling interval T , then

the energy of the transmitted symbol will be dispersed over other signalling intervals

causing intersymbol interference to appear in the received waveform.

An alternate approach to investigating the e�ects of time dispersion is to �nd the

auto-correlation of the channel transfer function H(f ; t), which is just the Fourier

transform of h(� ; t) taken with respect to the variable � . By using the same assump-

tions that were used to simplify equation (2.8), the auto-correlation function obtained

is

RH(f1; f2; t1; t2) =1

2E fH(f1; t1)H

�(f2; t2)g (2.9)

or equivalently RH(�f ; �t), where �f = f2� f1 is the di�erence in frequency. Once

again by letting �t = 0, the resulting auto-correlation function is denoted as RH(�f).

De�ne (�f)H to be the range of frequencies over which RH(�f) is essentially non-

zero. This is called the coherence bandwidth of the channel, and can be estimated as

(�f)H � T�1m . Depending on the frequency spectrum of the signal, the channel can

be classi�ed as belonging to one of two categories. If (�f)H is large in comparison

with the bandwidth of the transmitted signal, then the spectrum of the signal is not

a�ected. In this case the channel is called at fading or frequency non-selective. If

(�f)H is small in comparison to the bandwidth of the transmitted signal, then the

spectrum of the signal will be altered by the channel transfer function, resulting in a

distortion of the waveform. The received signal may exhibit the e�ects of intersymbol

interference due to this distortion. In this case, the channel is called frequency-

selective.

Since the mobile unit is usually in motion, the lengths of the propagation paths

18

are constantly changing, and this results in frequency dispersion of the transmitted

signal. Taking the Fourier transform of the auto-correlation function Rh(� ; �t) with

respect to �t and integrating over � , the power spectral density function obtained is

Sh(fD) =Z 1

�1

Z 1

�1Rh(� ; �t) exp(�j2�fD�t)d�td�: (2.10)

This is called the Doppler power spectrum of the channel, and the parameter fD is

called the Doppler frequency. Thus, the time variation of the channel results in a

Doppler spreading of the signal spectrum as well as a possible spectral shift. Let BD

be the set of fD over which Sh(fD) is essentially non-zero. This number is called

the Doppler spread of the channel, and indicates the range of frequency over which

the energy in a transmitted signal may be dispersed. The coherence time of the

channel is de�ned as (�t)h � B�1D , and measures the length of time over which the

fading process is considered to be correlated. A channel with a large BD has a small

coherence time and is called fast fading, since the channel response varies quickly

with time. A slow fading channel will have a large coherence time and experience

fades of longer duration.

For a signal with wavelength � which arrives at an angle with respect to the

direction of a vehicle travelling at a velocity v, the Doppler frequency is

fD =v

�cos : (2.11)

Frequency dispersion is detrimental to practical communication systems, so in anal-

ysis, the largest possible Doppler frequency is often used to represent the worst case.

The maximum Doppler frequency is simply

fDmax =����v����� : (2.12)

This model of a multipath fading channel is based on a physical interpretation

of the transmission medium, and while very general, can usually be simpli�ed when

analyzing a particular communication system. This requires additional assumptions

which depend on the particular system. A number of speci�c amplitude fading models

are presented next for non-dispersive channels, and can be augmented to include the

e�ects of both frequency and time dispersion.

19

2.2 Amplitude Fading Models

In order to focus on amplitude fading, non-dispersive channels are considered here.

Although no channel will be completely free of dispersion, sometimes the amount is

so small and the e�ects so negligible that the assumption of no dispersion is valid.

When no time dispersion is present, the channel will be at fading and the received

signal will not be distorted. When frequency dispersion is relatively insigni�cant, the

channel is described as slow fading, because variation of the fading process is limited

by the Doppler power spectrum. Since the random fading process �(t) changes very

slowly, it can be represented as a random variable � over any given signalling interval.

Using this assumption with a particular value � of the fading variable �1, the received

signal over a short period of time can be expressed in the form

y(t) = �u(t) + n(t): (2.13)

As was indicated in the previous section, the e�ect of fading is modelled as a complex-

valued Gaussian variate � with mean m� and variance �2�. The probability density

function (pdf) for this distribution is

p�(�) =1

2��2�exp

�j� �m�j2

2�2�

!: (2.14)

Due to the slow variation of the channel, it is sometimes assumed that the phase

can be tracked perfectly in the receiver after fading occurs. Since � is a complex-

valued Gaussian random variable, the phase � = arg � will be modelled as a uniform

random variable � with pdf

p�(�) =

8><>:

12�

�� < � � �

0 otherwise: (2.15)

The statistical distribution of the envelope r, which is the magnitude of �, takes

on di�erent forms depending on the particular fading scenario. In some cases it is

desirable to focus on the fading amplitude separately from the phase, since processes

in the receiver such as equalization or diversity combining operate on the complex

1 Random variables are indicated by uppercase symbols, whereas a particular value of the randomvariable is represented by the corresponding lower case symbol. The same convention is used herefor random processes.

20

envelope of the received signal after fading e�ects on the phase have been corrected.

Various statistical models for amplitude fading are described next.

2.2.1 Rayleigh Fading

In urban areas, a mobile unit is usually surrounded by many large buildings as well

as other vehicles. In this case it is unlikely that a LOS path exists, and the received

signal consists only of scattered components. With no LOS path, the complex-valued

Gaussian fading variable will have a zero mean. Let r = j�j be the envelope of thefading variable. By setting m� = 0 and performing a change of variables in equation

(2.14), the pdf of the random envelope R is derived to be

pR(r) =r

�2�exp

� r2

2�2�

!for r � 0 (2.16)

where it is understood here, and throughout the thesis, that any pdf such as pR(r)

which is de�ned for r � 0 takes on a value of zero for r < 0. This is called the Rayleigh

distribution, and usually represents the most severe case of amplitude fading. When

performing analysis, it is sometimes convenient to set EfR2g = 1 so that the average

power attenuation is unity. This is possible since the actual average loss due to long

term fading and free-space attenuation can be factored out, and the performance

of a communication system can be studied in the context of a short term fading

environment [34, pp. 169-171]. Usually messages are short in comparison to variations

in long term fading, so it is the short term models that are of interest. To set the

average power attenuation to unity, let �2� = 12in the pdf given by equation (2.16).

The resulting form of the Rayleigh density is

pR(r) = 2r exp��r2

�for r � 0: (2.17)

Since Rayleigh fading usually represents the worst case of fading, it has been used

as a basis for a number of fading channel studies [21, 35]. Also, the Rayleigh pdf

is simpler to manipulate mathematically than those describing other fading models,

which makes it favorable for use in analysis.

21

2.2.2 Rician Fading

In residential and rural areas, the lack of large surrounding structures allows for a

LOS component to exist in the received signal. In this case, the complex-valued

Gaussian fading process no longer has a zero mean. Let � = jm�j, where m� is the

mean value of �. By letting r = j�j and performing a change of variables in equation

(2.14), the envelope of the fading variable R will have the probability density

pR(r) =r

�2�exp

�r

2 + �2

2�2�

!I0

r�

�2�

!for r � 0 (2.18)

where I0(�) is the modi�ed Bessel function of the �rst kind and of zero order [16]. In

this case, the fading envelope is said to have a Rician distribution. The ratio of the

power in the LOS component to that in the scattered components is called the Rician

channel parameter, and is determined by the relation

R =�2

2�2�: (2.19)

When R = 0 there is no power in the LOS component, and the model reduces to

that of a Rayleigh channel. When R = 1, all the power is in the LOS component

and no scattered components exist, so the model looks like a simple AWGN channel.

In order to set EfR2g = 1, as was done for the Rayleigh channel, it is required that

�2 + 2�2� = 1. This is obtained by setting �2 = R1+ R

and 2�2� = 11+ R

in equation

(2.18). The resulting form of the Rician pdf is

pR(r) = 2r(1+ R) exp��r2(1 + R)� R

�I0

�2rq R(1 + R)

�for r � 0: (2.20)

The Rician fading model has been used in studies of mobile satellite communication

in the United States [16].

2.2.3 Shadowed Rician Fading

In some cases the LOS component of the received signal is not constant, but is prone

to shadowing e�ects from foliage, buildings, or uneven terrain. In this case, the

magnitude of the LOS component is assumed to be a random variable Z, described

by a lognormal distribution with pdf

pZ(�) =1

�q2��2Z

exp

�(ln � �mZ)

2

2�2Z

!for � > 0: (2.21)

22

Conditioned on the �xed value of Z = �, the fading amplitude will have a Rician

density as shown in equation (2.18). To incorporate the e�ects of shadowing, it is

required that the Rician pdf be averaged over the distribution of Z. The resulting

pdf for this channel can be expressed in the form

pR(r) =r

�2�

1q2��2Z

Z 1

0

1

�exp

�(ln � �mZ)

2

2�2Z� (r2 + �2)

2�2�

!I0

r�

�2�

!d� for r � 0

(2.22)

which is obtained by averaging equation (2.18) over the pdf given in (2.21). Loo has

shown that this model �ts the experimental data measured for the Canadian land

mobile satellite channel. In [36] he presents values for �2Z , mZ , and �2� which were

obtained from experimental measurements, and that correspond to di�erent degrees

of shadowing.

2.2.4 Nakagami Fading

The Nakagami fading model, described by the Nakagami-m distribution, is an approx-

imation that links two di�erent fading distributions. The �rst is the Rician fading

model, which is also known as the Nakagami-n distribution, and spans the range from

Rayleigh fading to the AWGN channel. It was stated that the Rician distribution de-

scribes the envelope of a complex-valued Gaussian random variable with components

that are uncorrelated and have equal variance as well as non-zero mean values. This

is a chi distribution with two degrees of freedom [37]. The other distribution that is

included in the Nakagami model is known as the Nakagami-q distribution, and spans

the range from one-sided Gaussian fading to Rayleigh fading. The q-distribution de-

scribes the envelope of a complex-valued Gaussian random variable with uncorrelated

components that have unequal variance and a mean value of zero. The q-distribution

describes fading that is even more severe than the Rayleigh channel. The Nakagami

model approximates both of these distributions with the pdf

pR(r) =2mmr2m�1

�(m)mexp

��mr2�

for r � 0 (2.23)

where m is the Nakagami channel parameter, is equal to the second moment of

R, and �(�) is the well known gamma function or generalized factorial. Crepeau [38]

23

describes the Nakagami distribution as a central chi distribution generalized to a non-

integral number of degrees of freedom. When m = 12the model describes one-sided

Gaussian fading, which is the case where only one faded component of the complex-

valued signal is available at the output of the channel. When m = 1 the Rayleigh

model is obtained, and when m = 1 the channel is non-fading. Unfortunately,

this model is not as intuitive on a physical basis as the Rician model and has no

accompanying phase distribution. However, this model is gaining popularity, since

in many cases it matches measured data better than the Rician model [39, 40]. The

pdf for the Nakagami model is also easier to use in mathematical analysis than the

Rician pdf. To set the average power attenuation to unity in this model, let = 1,

which results in the modi�ed pdf

pR(r) =2mmr2m�1

�(m)exp

��mr2

�for r � 0: (2.24)

2.3 E�ects of Frequency Dispersion

The amplitude fading models presented in the previous section remain valid for a at

fading channel even if the amount of frequency dispersion due to the channel increases.

For the digital cellular phone system proposed in North America, carrier frequencies

will be in the 900 MHz frequency band. If vehicle speeds from 0 to 100 mph are

considered, then from equation (2.12), resulting Doppler frequencies can range from

approximately 0 to 130 Hz. This is still very small compared to the signal bandwidth

of 30 kHz. So for most cases of interest, the fading amplitude will remain essentially

constant over a symbol interval. The e�ects of frequency dispersion in uence the

practical design of a communication system, which in turn a�ects the models that

are used to describe the channel. The practical considerations examined here are

symbol interleaving, diversity combining, and channel state estimation.

2.3.1 Symbol Interleaving

When a fade of long duration occurs during transmission, a sequence of symbols

are made susceptible to additive noise, resulting in a corresponding long burst of

errors. In order to break up bursts of errors, interleaving is often used. Interleaving

24

can be performed either on channel symbols or on individual bits. Although bit

interleaving is more e�ective in making bit errors appear random, symbol interleaving

is more popular since it can make the channel appear e�ectively memoryless. The

performance of receivers, such as those that use the Viterbi algorithm, is improved

when the fading channel appears memoryless. Analysis is also simpli�ed, since a

joint probability density describing the channel could be factored into a product of

the marginal densities.

The simplest way to understand the e�ect of interleaving is to examine the opera-

tion of a block interleaver. In the transmitter, a block interleaver reads symbols into

a two-dimensional array by rows, and then transmits the symbols by columns. In the

receiver, a deinterleaver is used to put the symbols back in sequence. If a fade of long

duration were to occur, then the deinterleaved sequence would experience random

symbol errors rather than a long burst of errors. In reality, interleaving introduces

a delay which increases the time required for transmission. In some cases, such as

digital voice transmission, this delay must be kept minimal. The size of the inter-

leaver required is related to the coherence time of the channel. The column length of

a block interleaver should in fact be large enough so that the transmission time for a

column of symbols is at least one signalling interval longer than the coherence time

of the channel.

In [41], a convolutional interleaver is used in the study of mobile satellite channels.

A convolutional interleaver uses less memory elements than a block interleaver, but

the symbols are not equally separated. Wei [42] proposes that the interleaver be

matched to the coding scheme so that interleaving is more e�ective in a practical

communication system. This means that the same performance could be achieved

with a shorter interleaving delay. When performing analysis, it is common to assume

symbol interleaving is ideally in�nite. This assumption allows a memoryless channel

model to be used.

2.3.2 Diversity Combining

When a mobile unit comes to a stop, the time variation of the channel ceases. If

this should happen during the occurrence of a deep fade, then the channel would

25

e�ectively be lost until the mobile unit started moving again. In order to compensate

for such an occurrence, diversity is commonly used. Diversity may be obtained by

repeating transmission, as in the case of time diversity, or by transmitting on di�erent

carriers, as in the case of frequency diversity. These types of diversity, however, do

not provide any bene�t when the mobile unit stops during the occurrence of a deep

fade. The bene�cial form to use in order to reduce the possibility of channel loss

is space diversity, which means that L di�erent antennae are used to receive the

signal. This method is also more eÆcient, since the same throughput and bandwidth

is used as a non-diversity system. If two antennae are separated by a distance d, then

the correlation coeÆcient between the Gaussian fading processes received by the two

antennae is given by [34]

�� = J0

2�d

�

!(2.25)

where J0(�) is a Bessel function of the �rst kind and of zero order, and � is the wave-

length of the signal. By spacing the antennae so that the minimum separation is 0:5�,

the fading processes received on any pair of antennae will be essentially uncorrelated2.

If one antenna were to receive a severely faded signal, it would be less likely that all

L would also be experiencing a deep fade at the same time. Thus, the probability

of losing the channel when the mobile unit is stationary is signi�cantly reduced. For

a 900 MHz carrier, � is approximately 13 inches. So a case of practical interest for

the digital cellular phone system would be the use of two antennae separated by a

distance of approximately 6-7 inches.

After adjusting the received signals over each antenna so that they are cophased,

the receiver combines them to obtain

y(t) =LXi=1

aiyi(t) (2.26)

where the ai are weighting factors, and yi(t) is the complex envelope of the received

signal for the ith antenna. Maximal-ratio combining sets the ai so that the carrier-to-

noise ratio is maximized. This is achieved when ai =rini, where ri is the envelope of

2 When results are obtained for systems with antenna diversity, the correlation coeÆcient usuallyappears in the form j��j2. Although �� 6= 0 when d = 0:5�, the value of j��j2 is suÆciently negligibleat this point, and the fading processes are considered to be essentially uncorrelated from a practicalviewpoint.

26

the fading process experienced by the ith antenna, and ni is the additive noise which

corrupts the signal received by the ith antenna. Although this technique is optimum,

it requires additional complexity in order to estimate the weighting factors ai, as the

ri cannot be directly obtained at the receiver. Equal-gain (i.e. constant ai) combining

is slightly worse in performance, but can be implemented with an inexpensive phase-

locked summing circuit [43]. After adjusting the phases for each antenna, the receiver

simply adds together the complex envelopes of the received signals. In this case, ai = 1

for all i in equation (2.26). Selective diversity combining is the simplest technique to

implement, but has the worst performance. This technique merely selects the best

signal from all L antennae. With regard to equation (2.26), if the jth path has the

strongest carrier-to-noise ratio, then aj = 1, and ai = 0 for all the other paths.

2.3.3 Channel State Estimation

In order to transmit information reliably over a fading channel when using uncoded

modulation or a technique such as TCM, some information must be obtained about

the fading variable �. In coding research, it is usually assumed that some information

about � is given without any mention of how this information is obtained. Divsalar

and Simon [16] assumed two di�erent cases for channel state information (CSI). In the

case of ideal CSI, the receiver is assumed to know the fading variable completely. For

a slow fading channel, it is reasonable to assume that a fairly good channel estimate

can be obtained. The other extreme case was termed no CSI. This means that there

is no information about the fading envelope, but it is assumed that the phase of the

fading process can be tracked. In this case, knowledge of the phase constitutes the

CSI.

If the fading is modelled as a complex-valued Gaussian random process, then

the actual fading variable � and the estimate ~� can be modelled as being jointly

Gaussian. It is assumed that the mean value of these variables are equal, and that

the correlation coeÆcient � is dependent upon the particular estimation scheme used.

The auto-correlation of the fading process is given by [34]

R�(�) = �2�J0 (2�fD�) (2.27)

27

where fD is the Doppler frequency. Any channel estimation scheme that involves a

time delay between the occurrence of � and the availability of ~� will yield a correlation

coeÆcient that is dependent upon the amount of frequency dispersion.

One practical channel estimation scheme involves the transmission of a pilot tone

along with the data signal. Since this tone is always known, the fading variable can

be determined by observing the amplitude and phase of the tone at the receiver.

The spectrum of the data signal should contain a spectral null in order to facilitate

extraction of the tone. The channel estimator consists of a bandpass pilot extraction

�lter in the receiver with bandwidth Wp. The bandwidth must be large enough to

pass the fading process without distortion, so it is assumed that Wp � 2BD. In [35],

it is stated that the squared magnitude of the correlation coeÆcient is

j�j2 = 1

1 +�1+

�(WpT )

��EsN0

��1 (2.28)

where is the ratio of the energy in the pilot tone to that in the data signal, T is the

duration of a signalling interval, and �Es represents the total average received symbol

energy in both the data and the pilot tone. By examining this expression, it is evident

that the quality of the channel estimate is dependent upon the SNR as well as the

Doppler spread.

Di�erential phase shift keying is a desirable type of modulation in environments

where phase estimation is diÆcult. If channel symbols xi are modelled as complex

numbers with unit magnitude, the di�erential encoding process can be described by

the complex multiplication xi = xi�1xi, where xi is the di�erentially encoded channel

symbol transmitted at time i. Transmission of the phase di�erences results in the

received symbol

yi = �ixi + ni = �ixi�1xi + ni: (2.29)

A noisy estimate of �xi�1 may be obtained by using ~�i = �i�1xi�1 + ni�1, which

is simply the channel symbol received during the previous signalling interval. The

resulting correlation coeÆcient has squared magnitude [35]

j�j2 = J20 (2�fDT )

1 +�

�EsN0

��1 : (2.30)

28

The time di�erence T between the observation of �i and �i�1 causes frequency dis-

persive e�ects to be involved, while the additive noise results in the dependence upon

signal-to-noise ratio.

A �nal method of channel estimation is to insert a known channel symbol every

NBth signalling interval. This allows a noisy version of � to be determined over

that particular time slot, and this can be used as a channel estimate for the next

NB � 1 symbols. Due to the increasing time duration between transmission of the

pilot symbol and the successive data symbols in a block of size NB, the e�ect of

frequency dispersion can be summarized as follows. As larger Doppler frequencies are

encountered, the rate of time variation of the channel increases. This in turn causes

the reliability of the channel estimate to diminish quickly. A natural response would

be to reduce the value of NB by transmitting pilot symbols at shorter time intervals.

As a consequence of this action, lower data rates would have to be used.

2.4 Frequency-Selective Fading Channels

The at fading model may be applied to many narrowband systems, since it is likely

that the transmit spectrum is not altered by the channel. For wider bandwidth

systems carrying digital data, however, the e�ects of time dispersion become more

prominent, and the channel may be more accurately modelled as being frequency-

selective. Since this is a likely scenario for modern mobile communications, it is

desirable to have an analytic model in order to study the e�ects of such a channel on

the performance of a communication system.

2.4.1 Linear Filter Model

In [44], Turin et. al. used the linear �lter model as a basis for experimental measure-

ments of an urban microwave radio propagation environment. The resulting linear

�lter was

h(t) =1Xi=0

Skiri exp(j�i)Æ(t� �i) (2.31)

with the following statistical description of the �lter parameters. As usual, it was

conjectured that the phases �i can be described as independent uniform random

29

variables. Since the phase is sensitive to small changes in path length, this assumption

is usually accepted. If �0 is the arrival time of the LOS path, then it was determined

that the set of di�erences f�i � �0g1i=1 form a modi�ed Poisson sequence on the time

interval (0;1), with a time-varying arrival rate. Experimental results indicated that

the later arriving paths �t this assumption more closely. The amplitudes ri were

modelled as being lognormal with a time-varying mean and variance, as well as a

dependence upon the path delays f�i � �0g. It appears that these amplitudes re ect

long-term fading e�ects, as indicated by the lognormal assumption. A short-term

fading model would use amplitudes ri that are either Rayleigh or Rician distributed.

In [45], the short-term amplitudes were modelled by the Nakagami distribution. The

factor S is a lognormal variable with a �xed mean and variance, and is also a result

of long-term fading. The ki are to be determined empirically, but can be set so that

ki = 1 for all i. In this case, S would represent long-term fading e�ects common to

all paths. As shown in [45], this model is suitable for computer simulation, but it is

too complex to be used in mathematical analysis. In this case, it is desirable to have

a simpler model that still re ects the relevant behavior of the channel.

2.4.2 Three-Path Model

In order to create a useful analytic model based on multiple propagation paths, some

simpli�cation must occur. Rummler [46] proposed using a model consisting of three

propagation paths. The parameters of the model are then judiciously chosen to �t

the experimental channel measurements. The resulting model was shown to be indis-

tinguishable from the more complex channel model within the accuracy of existing

measurement techniques.

The model assumes the arrival of three paths at the receiver. The �rst path has

an amplitude of 1, the second an amplitude of r1 and a delay of �1 with respect to

the �rst path, and the third has an amplitude of r2 and a delay of �2 with respect to

the �rst path. It is assumed that �2 > �1, and also that (!2 � !1)�1 � 1, where !2 is

the highest frequency in the band and !1 is the lowest. The impulse response of this

model is

h(t) = Æ(t) + r1Æ(t� �1) + r2Æ(t� �2): (2.32)

30

By applying the Fourier transform to h(t), the transfer function of the channel is

determined to be

H(!) = 1 + r1 exp(�j!�1) + r2 exp(�j!�2): (2.33)

The vector that results from summing the �rst two components of H(!) will have

amplitude a and phase �. Since the value of �1 is assumed to be very small, the term

!�1 can be considered to be essentially constant for all ! in the frequency band, which

allows the value of � to be treated as being constant. If !0 is the radian frequency

at which H(!) takes on its minimum value, then this will occur when the third term

in (2.33) opposes the sum of the �rst two, or in other words when � = !0�2 � �. By

setting r2 = ab, the transfer characteristic becomes

H(!) = a exp(�j(!0�2 � �)) + ab exp(�j!�2): (2.34)

Finally, by rotating H(!) through an angle � = !0�2 � � and setting �2 = � , the

transfer function is expressed in the form

H(!) = a [1� b exp (�j (! � !0) �)] : (2.35)

A plot of H(!) taken from [46] is shown in Figure 2.2. The parameter a is called

the scale parameter, and sets the overall gain of H(!). The variable b is called the

shape parameter, and controls the shape of H(!) as well as the depth of the notches.

In Figure 2.2, the notch depth is 20 log�1+b1�b�. The spacing between the minima of

H(!) is ��1, where � is the delay di�erence of the channel. By changing the various

parameters of H(!), one can see that notches can be included at a number of fre-

quencies within the transmit spectrum. For notches located out of band, a variety

of slopes can be obtained. It was stated in [46] that the path parameters overspecify

the channel. That is, within the accuracy of available measurement techniques, the

parameters are not unique for a given transfer characteristic. In order to specify a

unique model, it is required that one of the parameters a, b, �, or � , be �xed. Although

�xing the delay di�erence � does not intuitively seem reasonable, doing so results in

the most e�ective model. The value of � can be set to Tm, the multipath spread of

the channel. The remaining parameters must be given a statistical characterization.

31

35

30

25

20

15

-20l

og|H

(f)|

-50 0 50 100 150 200 250

f - fo (MHz)

Figure 2.2: Channel Transfer Function of the Three-Path Model

32

In [46], the model was used to describe a microwave radio channel. The scale pa-

rameter a was modelled as being lognormal conditioned on b, which indicates that

long-term fading e�ects are included. The shape parameter was given a distribution

of the form (1� b)2:3, and !0 was distributed uniformly at two levels. Unfortunately,

statistics that would describe a mobile fading channel have not been determined using

this model. However, by using this model as a basis and making some assumptions

based on an interpretation of the physical e�ects of Rayleigh fading, a simple model

for a frequency-selective mobile channel is obtained.

2.4.3 Two-Path Rayleigh Model

The two-path Rayleigh model is used to model frequency-selective e�ects on mobile

channels [47, 48]. Justi�cation for the use of this model is based on the close �t of the

simple three-path model to fading channels, as well as the lack of a LOS component in

many mobile channels. The two-path model can be derived similarly to the three-path

model by removing the LOS path. This results in the channel impulse response

h(t) = �1Æ(t� �1) + �2Æ(t� �2) (2.36)

where �1 and �2 are modelled as independent complex-valued Gaussian fading vari-

ables with zero mean. It may be assumed that the variances of the random variables

�1 and �2 are �2�1

= 12(1+ )

and �2�2=

2(1+ ), respectively. This ensures that the sum

of the two fading processes has a unity gain. The delays are set so that �1 = 0 and

�2 = � , where once again � is the delay di�erence of the channel.

When a channel is at fading, the e�ect of fading on the signal can be resolved

into a scaling of the amplitude of the transmitted symbol in addition to a phase

rotation. In this case, analysis can be performed in terms of the channel symbols and

without regard to the baseband pulse used. When the channel is frequency-selective,

intersymbol interference occurs in the received signal and the baseband pulse can no

longer be ignored. In this case the baseband signal is expressed in the form

u(t) =Xi

xig(t� iT ) (2.37)

where xi is the channel symbol transmitted during the ith signalling interval, and g(t)

33

is the baseband pulse used. The shape and duration of g(t) has a signi�cant e�ect on

communication over frequency-selective fading channels.

A number of models have been presented, which are used to represent the e�ect

that mobile communication channels have on signals. These models can also be

used as a basis for determining the limits that multipath fading channels impose on

communication.

34

Chapter 3

Information Theoretic Bounds for Ideal

Fading Channels

In this chapter, results from the �eld of information theory are utilized in order to

determine the fundamental limits of communication over multipath fading channels.

Channel capacity and average mutual information are two upper bounds on trans-

mission rate which are used to determine the losses incurred due to fading. In order

to isolate the e�ects of fading from other factors, such as time dispersion and Doppler

spread, an idealized channel model is used. This idealized model is the same one that

is generally used in channel coding research. The limits obtained here can also be

used as a benchmark for code performance. This eliminates the necessity of having to

use the performance of other known codes, or the AWGN channel performance of the

code in question, as a basis for judging its merit. Constraints on both average and

peak power are considered. Physical restrictions on the size of mobile transmitters

place a limit on the peak power available, and it is this constraint which is presumably

more relevant to mobile communications. Also related to peak power performance is

the peak-to-average power ratio of the constellation. This number gives an indication

of the e�ect of signal dependent perturbations, such as non-linear distortion, on the

performance of the system. In order to reduce the possibility of channel loss, multiple

antennae are often used in mobile systems. The e�ect of this space diversity on com-

munication rate is examined from an information theoretic viewpoint. Finally, the

gains possible through the use of shaping and channel coding are determined through

comparison of channel capacity with the performance of uncoded modulation.

35

3.1 Information Theoretic Concepts

Average mutual information and channel capacity are two information theoretic

bounds on the maximum communication rate over a channel. The rate is a max-

imum in the sense that if it is exceeded, then the probability of a detection error

occurring in the receiver cannot be made arbitrarily small. Some of the key de�-

nitions required from the �eld of information theory are stated here along with the

channel coding theorem. More detail can be found in texts by Gallager [27] or Cover

and Thomas [49].

In order to simplify notation, the probability density function of a random variable

X will be denoted by p(x) rather than using a form such as pX(x). The symbol x

serves here both as a label for the random variable X, as well as an argument for

the pdf. The pdf's p(x) and p(y) are not to be considered the same unless indicated.

This notation is assumed from now on for all probability densities and probability

mass functions. In addition, although the following de�nitions are stated in terms

of probability density functions, the obvious extensions to discrete distributions and

probability mass functions will be assumed to be understood.

A mathematical de�nition of a general communication channel is required in order

to interpret information theoretic results in a communications context.

De�nition 3.1 A communication channel is de�ned to consist of an input alphabet

X , an output alphabet Y, and a transition probability assignment p(yjx). The input

and output alphabets can be either discrete-valued or continuous-valued.

A case of practical interest is when X is a discrete-valued alphabet, such as a signal

constellation, and Y is continuous-valued, representing the set of all possible corrupted

channel outputs. The transition probabilities p(yjx) will depend on the channel model

used. For the fading channel models presented in the previous chapter, additional

information is available at the output of the channel in the form of the estimate ~�

of the fading variable �. Since it is desirable to use this additional information in

determining which symbol was transmitted, the transition probabilities for fading

channels will be of the form p(yjx; ~�).The entropy of a random variable X can be interpreted as a measure of the

36

uncertainty in determining the value of X, or equivalently, as the average amount of

information it contains.

De�nition 3.2 If the random variable X has a pdf p(x), then the entropy H(X) is

de�ned as

H(X) = �Ep(x) flog p(x)g (3.1)

where Ep(x) denotes expectation taken over p(x).

This de�nition assumes that both the pdf p(x) as well as the average value of log p(x)

exist, which is likely for cases of practical interest. For a discrete-valued random

variable, the entropy is always positive and absolute. For a continuous-valued random

variable, however, the entropy is taken relative to the coordinate system used and may

take on negative values. When X is a continuous-valued random variable, H(X) is

sometimes referred to as di�erential entropy. Usually the logarithm used is assumed

to have base 2, which causes the entropy to be measured in units of bits. Entropy

can also be de�ned for a set of random variables.

De�nition 3.3 If a set of random variables X = (X1; : : : ; Xn) are speci�ed by a

joint pdf p(x1; : : : ; xn), then the joint entropy of the set is de�ned as

H(X) = �Ep(x1;:::;xn) flog p(x1; : : : ; xn)g (3.2)

where Ep(x1;:::;xn) denotes expectation taken over p(x1; : : : ; xn).

Sometimes knowledge of one random variable a�ects the uncertainty associated

with another random variable. In this case, the entropy can be conditioned on the

known variable.

De�nition 3.4 If random variables X and Y have a joint pdf p(x; y), then the con-

ditional entropy of X given knowledge of Y is de�ned as

H(XjY ) = �Ep(x;y) flog p(xjy)g (3.3)

where Ep(x;y) denotes expectation taken over p(x; y).

37

Conditioning can only reduce entropy, and H(XjY ) is always less than or equal to

H(X) [49, p. 27]. The case where equality holds is when the random variables are

statistically independent. Conditional entropy can also be expanded to sets of random

variables, resulting in a conditional joint entropy. For example, the joint entropy of

the set X = (X1; : : : ; Xn) given knowledge of the set Y = (Y1; : : : ; Ym) is

H(XjY ) = �Ep(x1;:::;xn;y1;:::;ym) flog p(x1; : : : ; xnjy1; : : : ; ym)g (3.4)

where Ep(x1;:::;xn;y1;:::;ym) denotes expectation taken over the joint probability density

function p(x1; : : : ; xn; y1; : : : ; ym).

Average mutual information is a measure of the amount of information that two

random variables have in common.

De�nition 3.5 Suppose X and Y have a joint pdf p(x; y) and marginal pdf's p(x)

and p(y), respectively. The average mutual information (AMI) between X and Y is

de�ned as

I(X;Y ) = Ep(x;y)

(log

p(x; y)

p(x)p(y)

!)(3.5)

where Ep(x;y) denotes expectation taken over p(x; y).

By using Bayes' theorem and properties of the logarithm, equation (3.5) can be

rewritten in the form

I(X;Y ) = H(X)�H(XjY ): (3.6)

The conditional entropy H(XjY ) is commonly referred to as the equivocation of the

channel when X and Y represent the channel input and output, respectively. Average

mutual information can also be interpreted as the average reduction in uncertainty

of X given knowledge of the value of Y .

In the context of a communication system, the random channel output y is a

corrupted version of the transmitted symbol x, and must be used to determine the

value of x that was transmitted. For example, if the constellation X consists of 8

equally likely symbols, then the entropy H(X) of the input can be calculated to be

equal to log2 8 = 3 bits. If y is a good estimate of x, then the 3 bits of uncertainty

can essentially be eliminated. In other words, knowing the value of the variable Y

gives a good estimate of the variable X, and on average 3 bits/T can be transmitted

38

over the channel with a small probability of error. If y is not such a good estimate of

x, then knowledge of the value of Y does not reduce the uncertainty in the value of

X to such a great degree. In this case, less than 3 bits must be transmitted every T

seconds when using the constellation along with some coding scheme, otherwise the

number of errors occurring in the receiver cannot be controlled.

Notice from equation (3.5) that average mutual information is symmetric with

respect to the arguments. That is, I(X;Y ) = I(Y ;X), and knowledge of the random

variable X reduces the uncertainty in Y by the same amount as knowledge of Y re-

duces the uncertainty in X. This symmetry results in an alternate form for expressing

average mutual information as a di�erence of entropies, and is given by the expression

I(X;Y ) = H(Y )�H(Y jX): (3.7)

Another fact of signi�cance is that for continuous-valued random variables, even

though the individual entropies may take on negative values, the average mutual

information is always positive and does not depend on the particular coordinate sys-

tem used [1].

Rather than being limited to single random variables, average mutual information

can also be de�ned between sets of random variables. For instance, the AMI between

the sets of random variables X = (X1; : : : ; Xn) and Y = (Y1; : : : ; Ym) is de�ned by

the equation

I(X;Y ) = Ep(x1;:::;xn;y1;:::;ym)

(log

p(x1; : : : ; xn; y1; : : : ; ym)

p(x1; : : : ; xn)p(y1; : : : ; ym)

!)(3.8)

which is just an extension of equation (3.5). Also, the AMI between two sets of random

variables can be conditioned on another set of known random variables. For example,

the conditional average mutual information between the sets of random variables

X = (X1; : : : ; Xn) and Y = (Y1; : : : ; Ym) given knowledge of the set Z = (Z1; : : : ; Zl)

is

I(X;Y jZ) = E

(log

p(x1; : : : ; xn; y1; : : : ; ymjz1; : : : ; zl)

p(x1; : : : ; xnjz1; : : : ; zl)p(y1; : : : ; ymjz1; : : : ; zl)

!)(3.9)

where expectation is taken over the joint pdf p(x1; : : : ; xn; y1; : : : ; ym; z1; : : : ; zl).

Channel capacity is the ultimate limit for reliable transmission of data over a given

communication channel.

39

De�nition 3.6 The capacity C of a communication channel is de�ned as

C = maxp(x)

I(X;Y ) (3.10)

where maximization is taken over all possible distributions of the input alphabet.

Given a constraint on average transmitted power of the form EfjXj2g � Es, the

capacity of an AWGN channel is given by the well known equation [49, p. 242]

C = log2

�1 +

EsN0

�bits=T (3.11)

where N0 is the average power of the additive noise, and T is the time required

to transmit a channel symbol. The transmission medium here is represented as a

discrete-time channel, and the unit of bits/T is equivalent to bps/Hz1. If the channel

is ideal and has a bandwidth of W Hz, then the capacity can also be expressed in the

form

C = W log2

�1 +

EsN0

�bps: (3.12)

The capacity of the AWGN channel is achieved by using a continuous-valued input

alphabet that has a Gaussian distribution with zero mean and variance �2X = Es2.

This can be proven using the well-known result that given a constraint on average

power, a Gaussian distribution maximizes entropy [49, p. 234].

The importance of average mutual information and channel capacity can be sum-

marized by the channel coding theorem. This theorem states that as long as the

communication rate Rc is less than the channel capacity C, the probability of detec-

tion error Pr(e) can be made arbitrarily small by using long enough code sequences.

This can be stated more rigorously as follows.

Theorem 3.1 For every � > 0 and rate Rc < C, there exists a code consisting of

d2NBRce codewords of length NB, such that Pr(e) < �. Conversely, if there exists a

code consisting of d2NBRce codewords of length NB that results in Pr(e) ! 0, then

Rc < C.

1 Equivalence of the units of bits/T and bps/Hz is based on the assumption that no more thanW symbols per second can be transmitted in a bandwidth of W Hz [11].

40

For a proof of the theorem see [49, pp. 199-202]. If Rc < I(X;Y ) then Rc < C, so

for schemes using practical constellations with a given input probability assignment

p(x), the maximum communication rate can always be bounded from above by the

resulting average mutual information, and the channel coding theorem applies.

Ungerboeck [6] calculated AMI curves for the AWGN channel using practical

PSK and QAM constellations. For the AWGN channel, the received symbol can be

represented in the form y = x + n, where x is the transmitted symbol, and n is

the additive noise. The random noise variable N has a complex-valued Gaussian

distribution with zero mean and variance �2N . It is assumed that the symbol x comes

from a M -point constellation and has a uniform discrete probability mass function

(pmf) given by p(xi) =1M

for i = 1; : : : ;M . Given knowledge of the value X = xi,

the conditional pdf of the received symbol Y can be determined by performing a

translation of the noise variable N . The resulting pdf is

p(yjxi) = 1

2��2Nexp

�jy � xij2

2�2N

!: (3.13)

Given p(yjxi) and p(xi), the joint density p(xi; y) can be determined from the law of

total probability. The pdf's and pmf obtained can then be used in equation (3.5) to

obtain an expression for average mutual information in the form

I(X;Y ) = log2M � 1

M

MXi=1

Ep(n)

8<:log2

24 MXj=1

exp

�jxi � xj + nj2 � jnj2

2�2N

!359=; : (3.14)

Since a base 2 logarithm is used, the average mutual information has units of bits/T .

In equation (3.14), Ep(n) f�g denotes expectation taken over the pdf p(n) of the noise

variable N . This expression can be evaluated by means of a computer simulation.

What is meant by computer simulation is an evaluation of the statistical expectation

through Monte Carlo averaging of the enclosed expression. The reliability of those

results presented in the thesis which were obtained through computer simulation is

discussed in Appendix A.

Ungerboeck's AMI curves are shown in Figures 3.1 and 3.2 for PSK and QAM

constellations, respectively, along with the capacity curve for the AWGN channel.

The standard constellations considered in the thesis are the same as those used

in [6]. The term QAM is used here as a general label for the AMPM constellations

41

Figure 3.1: AMI of an AWGN Channel: PSK Constellations

42

Figure 3.2: AMI of an AWGN Channel: QAM Constellations

43

considered in [6], which includes both square and cross-constellations, as well as the

8-AMPM signal set. Ungerboeck interpreted the AMI curves by observing that 2

bits/T can be transmitted with uncoded QPSK, obtaining an error probability of

10�5 when the SNR is 12.9 dB. However, by using an 8-PSK signal set, error-free

transmission of 2 bits/T is theoretically possible at a SNR of 5.8 dB. In addition, only

1 dB of further improvement can be obtained by reaching capacity. This is the basis

of Ungerboeck's heuristic design rule for trellis coding, which states that twice the

number of signal points as required for uncoded modulation can be used to obtain

most of the coding gain. The minimum values of SNR required to attain various

rates of AMI over an AWGN channel using standard PSK and QAM constellations

are contained in Tables 3.1 and 3.2, respectively. These results are used to determine

the losses incurred due to multipath fading.

3.2 Channels with Discrete-Valued Input

The same bounds used for the AWGN channel are implemented here in order to

determine the limits on reliable communication over fading channels when a discrete-

valued signal set is used. In this chapter, the fading channel is assumed to be ideal

in the sense characterized by the following de�nition.

De�nition 3.7 An ideal fading channel is one in which: i) The fading variables �

are known at the receiver. ii) The fading variables � are independent with respect to

time. iii) The transmitted pulse is not distorted or dispersed in time. iv) The fading

variables � remain essentially constant over an entire symbol interval.

An ideal fading channel satis�es those assumptions which are usually made in coding

research in order to simplify analysis. The �rst assumption is that the channel state

information is ideal. The second assumption indicates that ideally in�nite interleaving

is possible, which results in a memoryless channel model. The third assumption

means that the channel is at fading, and the �nal assumption indicates that the

channel is slow fading compared to the symbol rate. When this idealized model

is used, the in uence of time dispersion and Doppler spread are non-existent, and

this enables one to focus on the e�ects of amplitude fading alone. In general, if

44

Number of Signal Points

4 8 16 32

4.5 - - - 20.5 dB

4 - - - 17.4 dB

3.5 - - 14.6 dB 14.5 dB

AMI 3 - - 11.5 dB 11.5 dB

(Bits/T ) 2.5 - 8.8 dB 8.6 dB 8.6 dB

2 - 5.8 dB 5.8 dB 5.8 dB

1.5 3.4 dB 3.0 dB 3.0 dB 3.0 dB

1 0.2 dB 0.1 dB 0.1 dB 0.1 dB

Table 3.1: Minimum SNR Required for Various Rates of AMI on an AWGN Channel:

PSK Constellations

Number of Signal Points

4 8 16 32 64

5.5 - - - - 18.2 dB

5 - - - - 16.1 dB

4.5 - - - 14.8 dB 14.4 dB

4 - - - 12.7 dB 12.6 dB

AMI 3.5 - - 11.6 dB 10.8 dB 10.8 dB

(Bits/T ) 3 - - 9.3 dB 9.0 dB 9.0 dB

2.5 - 7.9 dB 7.2 dB 7.1 dB 7.1 dB

2 - 5.3 dB 5.1 dB 5.0 dB 5.0 dB

1.5 3.4 dB 2.9 dB 2.8 dB 2.7 dB 2.7 dB

1 0.2 dB 0.1 dB 0.1 dB 0.0 dB 0.0 dB

Table 3.2: Minimum SNR Required for Various Rates of AMI on an AWGN Channel:

QAM Constellations

45

any of these assumptions are not satis�ed, then it becomes more diÆcult to achieve

reliable communication over the channel. This is the reason that this fading model is

considered to be ideal.

The average mutual information for an ideal fading channel is derived here for

the case of when the input alphabet is a discrete-valued signal set and the output is

continuous-valued. Suppose there are M points in the discrete-valued signal constel-

lation with a priori probability p(xi). The entropy of the channel input is determined

from the expression

H(X) = �MXi=1

p(xi) log p(xi): (3.15)

At the output of the channel, both of the values y = �x+n and � are obtained. The

a posteriori probability that the particular value of X = xi was transmitted, given

knowledge of the variables Y and �, is determined by the pmf

p(xijy; �) = p(yjxi; �)p(xi)p(yj�) : (3.16)

The conditional entropy of the channel input given knowledge of the channel output

Y and the fading variable � is

H(XjY;�) =MXi=1

ZSY

ZS�p(y; xi; �) log

"p(yjxi; �)p(xi)

p(yj�)

#d�dy (3.17)

where SY and S� are the respective support sets for the variables Y and �. The

average mutual information can now be determined from the di�erence of entropies

I(X;Y j�) = H(X)�H(XjY;�): (3.18)

By factoring the joint pdf into p(y; xi; �) = p(yjxi; �)p(xi)p(�) and using the fact that

p(yj�) =PMj=1 p(yjxj; �)p(xj), the conditional entropy of the channel input given

knowledge of the channel output and fading variable can be expressed in the form

H(XjY;�) =MXi=1

ZSY

ZS�p(yjxi; �)p(xi)p(�) log

"p(yjxi; �)p(xi)PMj=1 p(yjxj; �)p(xj)

#d�dy: (3.19)

Assuming that the constellation has an equiprobable a priori distribution, the discrete

pmf of the input is p(xi) =1M

for i = 1; : : : ;M . Using this in equation (3.15) results

46

in H(X) = logM . Given knowledge of the variables X and �, the channel output

will have a statistical distribution described by the pdf

p(yjxi; �) = 1

2��2Nexp

�jy � �xij2

2�2N

!(3.20)

which is obtained from the pdf of the additive noise through translation of the variable

n. By substituting the required density and mass functions into equation (3.19), the

conditional entropy takes the form

H(XjY;�) = (3.21)

1

M

MXi=1

ZSY

ZS�p(yjxi; �)p(�) log

24 MXj=1

exp

�jy � �xjj2 � jy � �xij2

2�2N

!35 d�dy:

After making the substitution y = �xi + n into equation (3.21), averaging can be

performed over p(n) rather than p(yjxi; �). Using this fact, the average mutual infor-mation in bits/T can be expressed as

I(X;Y j�) = log2M � 1

M

MXi=1

Ep(n)p(�)

8<:log2

24 MXj=1

exp

�j�(xi � xj) + nj2 � jnj2

2�2N

!359=;

(3.22)

where Ep(n)p(�) denotes expectation taken over p(n)p(�). This statistical expectation

can be evaluated by means of computer simulation.

3.2.1 Standard Signal Constellations

The AMI is determined here for an ideal Rayleigh fading channel when standard

signal constellations are used as input to the channel. The average received symbol

energy is de�ned as �Es = Efj�Xj2g. When considering a signal constellation X , the

energy and power of a symbol x are usually assumed to both equal jxj2, and the terms

energy and power can then be interchanged. This may be based on the underlying

assumption that a matched �lter is used in the receiver, and that the pulse used to

transmit the symbol is assumed to be scaled so that the energy has a value of 1.

In order to evaluate equation (3.22), it is assumed that Efj�j2g = 1 and that the

constellation has been normalized so that EfjXj2g = 1. The SNR is then determined

by the value of �2N .

47

Figure 3.3 contains the AMI curves for when standard PSK constellations are

used as input to the channel, while Figure 3.4 illustrates the case for when QAM

constellations are used. In Rayleigh fading, the AMI curves do not rise as quickly

to a �nal value of log2M as in the AWGN case. Also, even when the average SNR

is as large as 40 dB, the average mutual information is still a little less than the �nal

value of log2M . This is re ected in the error performance of uncoded modulation.

In the case of an AWGN channel, bit error curves have a quickly decreasing waterfall

characteristic. Over fading channels, however, the error curves are almost linear and

decrease very slowly in comparison to noise channels. In [30] and [31], similar results

are presented in terms of the computational cuto� rate R0 of the channel. The cuto�

rate is a limit derived for the speci�c technique of sequential decoding, and must not

be surpassed in order to ensure that the average computational load remains �nite.

Since other known decoding techniques also become impractical at rates above R0, it

has been touted for some time as being a practical limit on the rate of reliable data

transmission. Due to the transitory nature of the term `practical' when applied to

communications technology, it is preferable to determine the ultimate limit without

regard to any speci�c decoding technique.

The minimum values of average SNR required to attain various rates of AMI are

shown in Table 3.3 for PSK signal sets and in Table 3.4 for QAM. When using coded

modulation, less than log2M bits/T are transmitted with a constellation consisting

ofM symbols, so it is of interest to look at the loss due to fading for lower data rates.

Table 3.5 compares the loss in SNR due to multipath fading when PSK constellations

are used as input to the channel. Similar results for QAM constellations are shown

in Table 3.6. This loss is the di�erence in SNR where theoretically error-free trans-

mission is possible at the given data rate. For example, the AMI curve for 16-QAM

indicates that error free transmission at a rate of 3 bits/T is possible for an average

SNR greater than or equal to 12.3 dB, whereas a SNR of 9.3 dB is required on the

AWGN channel. Thus, when using a 16-QAM constellation, the loss due to fading at

a rate of 3 bits/T is 3.0 dB.

In order to interpret the results presented, �rst consider constellations which are

equal in size and have comparable values of AMI. When considering transmission over

48

Figure 3.3: AMI of an Ideal Rayleigh Fading Channel: PSK Constellations

49

Figure 3.4: AMI of an Ideal Rayleigh Fading Channel: QAM Constellations

50

Number of Signal Points

4 8 16 32

4.5 - - - 24.6 dB

4 - - - 20.3 dB

3.5 - - 18.6 dB 17.0 dB

AMI 3 - - 14.4 dB 14.0 dB

(Bits/T ) 2.5 - 12.7 dB 11.1 dB 11.0 dB

2 - 8.4 dB 8.0 dB 8.0 dB

1.5 6.7 dB 4.9 dB 4.9 dB 4.9 dB

1 1.8 dB 1.4 dB 1.4 dB 1.4 dB

Table 3.3: Minimum SNR Required for Various Rates of AMI on an Ideal Rayleigh

Fading Channel: PSK Constellations

Number of Signal Points

4 8 16 32 64

5.5 - - - - 23.4 dB

5 - - - - 19.7 dB

4.5 - - - 19.8 dB 17.2 dB

4 - - - 16.1 dB 15.0 dB

AMI 3.5 - - 16.2 dB 13.4 dB 13.0 dB

(Bits/T ) 3 - - 12.3 dB 11.1 dB 11.0 dB

2.5 - 12.1 dB 9.5 dB 9.0 dB 8.9 dB

2 - 7.9 dB 6.9 dB 6.7 dB 6.7 dB

1.5 6.7 dB 4.6 dB 4.3 dB 4.2 dB 4.2 dB

1 1.8 dB 1.3 dB 1.2 dB 1.1 dB 1.1 dB

Table 3.4: Minimum SNR Required for Various Rates of AMI on an Ideal Rayleigh

Fading Channel: QAM Constellations

51

Number of Signal Points

4 8 16 32

4.5 - - - 4.1 dB

4 - - - 2.9 dB

3.5 - - 4.0 dB 2.5 dB

AMI 3 - - 2.9 dB 2.5 dB

(Bits/T ) 2.5 - 3.9 dB 2.5 dB 2.4 dB

2 - 2.6 dB 2.2 dB 2.2 dB

1.5 3.3 dB 1.9 dB 1.9 dB 1.9 dB

1 1.6 dB 1.3 dB 1.3 dB 1.3 dB

Table 3.5: Loss of SNR Due to Rayleigh Fading: PSK Constellations

Number of Signal Points

4 8 16 32 64

5.5 - - - - 5.2 dB

5 - - - - 3.6 dB

4.5 - - - 5.0 dB 2.8 dB

4 - - - 3.4 dB 2.4 dB

AMI 3.5 - - 4.6 dB 2.6 dB 2.2 dB

(Bits/T ) 3 - - 3.0 dB 2.1 dB 2.0 dB

2.5 - 4.2 dB 2.3 dB 1.9 dB 1.8 dB

2 - 2.6 dB 1.8 dB 1.7 dB 1.7 dB

1.5 3.3 dB 1.7 dB 1.5 dB 1.5 dB 1.5 dB

1 1.6 dB 1.2 dB 1.1 dB 1.1 dB 1.1 dB

Table 3.6: Loss of SNR Due to Rayleigh Fading: QAM Constellations

52

an AWGN channel, the use of QAM constellations results in a lower average SNR

requirement compared to PSK constellations for a given value of AMI. An example of

this is illustrated by comparing 16-PSK versus 16-QAM for an AMI rate of 3 bits/T ,

where 16-QAM is 2.2 dB better. The loss due to fading is generally greater for QAM

constellations, in fact it is always greater for higher values of AMI. However, even

with the fading loss included, QAM remains the superior choice. At an AMI rate of

4 bits/T , the 32-CR signal set loses 3.4 dB due to fading while 32-PSK only loses 2.9

dB. However, the SNR values required to attain an AMI of 4 bits/T are 16.1 dB and

20.3 dB for 32-CR and 32-PSK, respectively.

Additional insight may be gained by examining the fading loss experienced at

di�erent values of AMI when a 2k-point constellation is used. For 2k-point PSK

constellations, the fading loss seems to converge to approximately 4.1 dB, 2.9 dB, and

2.5 dB for AMI values of k�0:5, k�1, and k�2, respectively. For the same respective

values of AMI, the losses for a 2k-point QAM constellation appear to converge to

values of 5.2 dB, 3.6 dB, and 2.4 dB. A fading loss value of 2.5 dB is signi�cant, and

in the next section it is shown that this is the asymptotic loss in channel capacity due

to Rayleigh fading. An AMI value of k � 2 with a 2k-point signal set corresponds to

constellation expansion by a factor of 4. While Ungerboeck showed that most of the

gain available when transmitting over an AWGN channel is achieved by doubling the

size of the required constellation, expansion by a factor of 4 is required to minimize

the theoretical loss due to Rayleigh fading. From a practical standpoint, a doubling

of the size of the signal set is still adequate for PSK constellations, since increasing

the expansion factor from 2 to 4 only results in an additional 0.4 dB saving. On the

other hand when QAM constellations are used, increasing the expansion factor from

2 to 4 results in an additional gain of approximately 1.2 dB.

3.2.2 Asymmetric PSK Constellations

Asymmetric PSK constellations have been considered for use in trellis coded modula-

tion [16]. A symmetric constellation consists of M points that have equal magnitude

and are spaced at equal intervals of 2�M

radians apart on a circle. An asymmetric

PSK constellation consists of M points that have equal magnitude, but which are

53

alternately spaced on a circle by intervals of �A and 2�M� �A radians. The angle �A

can take on any value between 0 and 2�M

radians. In order to bene�t from the asym-

metry introduced into the signal set, the trellis code design procedure is followed as

with standard constellations, but the angle �A is left as a variable. Once the code

is designed by standard procedure, the minimum distance between code sequences

can be expressed in terms of �A. This expression for minimum distance can then

be maximized with respect to �A in order to optimize the code for an AWGN chan-

nel. In Rayleigh fading, the asymmetry would likely be used to optimize the product

distance, which has a more dramatic e�ect on code performance than the minimum

Euclidean distance. Due to these potential gains, it is of interest to determine the

e�ect that asymmetry of the input has on the average mutual information of the

channel.

Figure 3.5 shows the AMI curves for the ideal Rayleigh fading channel when

asymmetric QPSK and 8-PSK constellations are used as input. A symmetric 8-PSK

constellation is obtained by setting the angle �A to 45o. When transmitting at a rate

of 2 bits/T with 8-PSK, setting �A to 30o results in a 0.6 dB loss compared to the AMI

curve obtained for the symmetric constellation. By further reducing �A to a value

of 15o, the loss increases to 2.5 dB. A symmetric QPSK constellation is obtained by

setting the angle �A to 90o. When transmitting data at a rate of 1 bit/T with QPSK,

a 0.6 dB loss compared to the symmetric case is observed when �A is set to 60o. This

loss increases to 2.9 dB when �A is further reduced to 30o. Therefore, while the use of

asymmetric constellations improves the performance of some known code structures,

the results here indicate that there is more to be gained by using symmetric PSK

constellations for coded modulation.

For the results presented in this section, two observations may be made regarding

how the choice of signal constellation a�ects AMI. The �rst is that the number of

distinct signal points determines the maximum value of AMI. For aM -point constella-

tion this is obviously log2M . The second observation is that the minimum Euclidean

distance between signal points determines how quickly the maximum value of AMI is

reached with respect to SNR. For asymmetric PSK signal sets, the maximum value of

AMI remains the same, but as the minimum distance between signal points decreases,

54

Figure 3.5: AMI of an Ideal Rayleigh Fading Channel: Asymmetric PSK Constella-

tions

55

so does the rate at which the AMI approaches the maximum value. Based on these

observations, it is evident that the average mutual information of a channel speci�ed

by an equiprobable PSK input alphabet is actually the channel capacity. However,

this is not necessarily the case with an arbitrary discrete-valued input alphabet.

3.3 Capacity of Ideal Fading Channels

In [1], Shannon obtained an expression for the capacity of an AWGN channel when

a constraint is placed on the average power of the channel input. A similar result

is obtained here for the case of an ideal fading channel. As in the AWGN case,

it is assumed that the channel has continuous-valued input and output alphabets.

Since all of the random variables involved are continuous-valued, the average mutual

information de�ned in equation (3.5) is expressed in the form

I(X;Y j�) =ZSY

ZSX

ZS�p(y; x; �) log

"p(yjx; �)p(yj�)

#d�dxdy: (3.23)

By using properties of the logarithm, the average mutual information can be rewritten

as a di�erence of entropies, and expressed in the form

I(X;Y j�) = H(Y j�)�H(Y jX;�): (3.24)

Since y = �x + n, the entropy of Y given knowledge of X and � is the same as the

entropy H(N) of the noise variable N . Using this result in equation (3.24) yields the

modi�ed expression

I(X;Y j�) = H(Y j�)�H(N): (3.25)

A result that is well known in information theory is that if a complex-valued Gaussian

variable has a variance of �2, then it has an entropy of log(2�e�2) [49, p. 225]. This

fact can be used to determine the entropy of the additive noise variable to be

H(N) = log(2�e�2N ): (3.26)

In order to obtain an expression for H(Y j�), it is necessary to �nd the pdf p(yj�).This can be accomplished by starting with the channel input x, which for the moment

56

will be assumed to have a Gaussian distribution given by the pdf

p(x) =1

2��2Xexp

� jxj

2

2�2X

!: (3.27)

Let a new random variable Z be de�ned as the product �X. The conditional pdf of

Z given knowledge of the value of � is

p(zj�) = 1

2� j�j2 �2Xexp

� jzj22 j�j2 �2X

!(3.28)

which is obtained from p(x) by scaling the random variable X. Since the channel

output is y = z + n, the conditional pdf of Y given knowledge of � and N is

p(yj�; n) = 1

2� j�j2 �2Xexp

� jy � nj22 j�j2 �2X

!(3.29)

and is obtained by performing a translation on the random variable Z. Finally, the

conditional probability of the channel output Y given knowledge of the fading variable

� is determined by evaluating the expressionRSNp(yj�; n)p(n)dn, which in this case

takes the form

p(yj�) =ZSN

1

2� j�j2 �2Xexp

� jy � nj22 j�j2 �2X

!1

2��2Nexp

� jnj

2

2�2N

!dn: (3.30)

This is just a convolution of two exponential functions. Using Fourier transform

relations [50], it can be shown that

ZSN

1

2��21exp

�jy � nj2

2�21

!1

2��22exp

�jnj

2

2�22

!dn =

1

2�(�21 + �22)exp

� jyj22(�21 + �22)

!:

(3.31)

By setting �21 = j�j2�2X and �22 = �2N in equation (3.31), the desired pdf is determined

to be

p(yj�) = 1

2��j�j2 �2X + �2N

� exp0@� jyj2

2�j�j2 �2X + �2N

�1A : (3.32)

Since p(yj�) is a Gaussian pdf with variance j�j2�2X+�2N , the entropy can immediately

be written as

H(Y j�) = Ep(�)nlog

h2�e

�j�j2�2X + �2N

�io: (3.33)

Finally, the average mutual information expressed in units of bits/T can be deter-

mined from the di�erence of entropies in equation (3.25) to be

I(X;Y j�) = Ep(�)

(log2

1 + j�j2 �

2X

�2N

!): (3.34)

57

This looks similar to the expression for the capacity of an AWGN channel with an

average power constraint. The di�erence is that the SNR is scaled by j�j2, and the

expression is averaged over the pdf of the fading variable. Given this result, the

following theorem may be stated.

Theorem 3.2 The capacity of an ideal fading channel with an average input power

constraint of E fjXj2g � 2�2X is C = Ep(�)

�log2

�1 + j�j2 �2X

�2N

��bits/T , and is

achieved with an input that has a zero mean Gaussian distribution with variance

�2X .

Proof: Scaling a variable will also cause the variance or power to be scaled, so by

�xing the value of the fading variable to be � = � and assuming the average input

power to be Es = 2�2X , the received signal power will be 2j�j2�2X regardless of the

probability distribution of the input. Since a Gaussian input maximizes the average

mutual information subject to an average power constraint, using the result obtained

for the AWGN channel yields the inequality

I(X;Y j� = �) � log

1 + j�j2 �

2X

�2N

!: (3.35)

Suppose the fading variable has a pdf given by p(�). Since p(�) � 0 for all values of

�, then

p(�)I(X;Y j� = �) � p(�) log

1 + j�j2 �

2X

�2N

!(3.36)

and ZS�p(�)I(X;Y j� = �)d� �

ZS�p(�) log

1 + j�j2 �

2X

�2N

!d�: (3.37)

This means that

I(X;Y j�) � Ep(�)

(log

1 + j�j2 �

2X

�2N

!): (3.38)

Therefore, by choosing the logarithm in this expression to have base 2, the capacity

expressed in units of bits/T is C = Ep(�)

�log2

�1 + j�j2 �2X

�2N

��, and is achieved with

a Gaussian distributed input. 2

The capacity of a Rayleigh fading channel is shown in Figure 3.6 along with the

capacity of an AWGN channel. The capacity of an ideal Rayleigh channel can be

58

Figure 3.6: Capacity of an Ideal Rayleigh Fading Channel

59

expressed in the form

C = � (log2 e) exp

0@" �Es

N0

#�11AEi0@�

"�EsN0

#�11A bits=T (3.39)

where Ei(�) is the exponential integral function. The derivation of this expression is

shown in Appendix B.1. As the SNR increases, the separation between the capacity

curves for the Rayleigh fading and AWGN channels increases until it appears to

reach a �xed value where the curves seem to be parallel. This �xed di�erence in

SNR between the curves can be interpreted as the maximum loss in capacity due to

amplitude fading. Let CN = log(1+SNRN) be the capacity of an AWGN channel and

let CF = Ep(�)flog(1+j�j2SNRF )g be the capacity of a fading channel. To ensure thatEp(�)fj�j2SNRFg = SNRF , it is assumed that Efj�j2g = 1. For small values of SNR,

it is easy to see that CN � CF . Suppose that for large values of SNR the two capacities

are equal, and estimate this by setting log(SNRN ) � Ep(�)flog(j�j2SNRF )g. This

expression can be rearranged into the form

SNRF

SNRN

= exp��Ep(�)fln j�j2g

�: (3.40)

This is the asymptotic loss in SNR due to amplitude fading, and can be expressed

in decibels as 10 log10(SNRFSNRN

). For a Rayleigh fading channel, the asymptotic loss is

shown in Appendix C.1 to be eCE (2.51 dB), where CE is Euler's constant.

Channel capacity can also be calculated for the other amplitude fading models

discussed in the previous chapter. Due to the complexity of the Rician and shadowed

Rician probability densities, an explicit expression for channel capacity is diÆcult to

obtain for these models. For the results presented here, averaging over the fading

distribution was accomplished by means of computer simulation. Capacity curves for

the Rician fading channel model are shown in Figure 3.7. When the Rician channel

parameter takes on a value of R = 5 dB, the asymptotic loss due to fading is 1.14 dB,

while for R = 10 dB the loss is 0.42 dB. The capacity of an ideal shadowed Rician

fading channel is illustrated in Figure 3.8. The asymptotic loss due to fading is 1.04

dB, 1.29 dB, and 2.51 dB for light, average, and heavy shadowing, respectively. The

heavily shadowed channel appears to be equivalent to the Rayleigh channel in terms

of capacity, while the lightly shadowed channel is almost indistinguishable from the

60

Figure 3.7: Capacity of an Ideal Rician Fading Channel

61

Rician channel with R = 5 dB. The results for the Nakagami fading model are shown

in Figure 3.9. For integer values of m, the capacity of the Nakagami fading channel

can be expressed in the form

C = (log2 e)(�m)m

�(m)

"�EsN0

#�m d

ds

!m�1 �es

sEi(�s)

�bits=T (3.41)

where s = m�

�EsN0

��1. The derivation of this expression is shown in Appendix B.2.

The asymptotic loss due to Nakagami fading is calculated in Appendix C.2 to be

me� (m), where (�) is Euler's psi function. It should be noted that this expression

is valid for all values of m > 0 and is not restricted to integer values of m. The

asymptotic loss due to Nakagami fading is 1.17 dB for m = 2, 1.60 dB for m = 1:5,

and 5.52 dB for m = 0:5. The case where m = 0:5 is for single-sided Gaussian

fading. This case has been proposed for some indoor wireless environments where

fading is even more severe than the Rayleigh case. For mobile communications, and

the work that follows, the Rayleigh channel serves to model the most severe loss due

to multipath fading.

In [29], the expression for the capacity of an ideal Rayleigh fading channel stated

in (3.39) was presented. However, rather than evaluating the capacity expression,

an estimate was used instead. Although the estimate asymptotically approaches the

capacity for high values of SNR, it is always less than the true channel capacity. Based

on the estimate, it was stated that the loss due to amplitude fading is greater for lower

values of SNR, which is not true. In addition, this paper illustrates a misguided view

toward channel capacity in the presence of fading that seems to be prevalent. Results

are often taken for the case of the AWGN channel with the SNR being treated as a

random variable. By averaging over the statistics of the random SNR, an `average

capacity' is obtained. Capacity should be viewed as an upper limit on an average

quantity, not vice versa. In this case it is fortunate that when the fading variable is

known at the receiver, the end results are the same.

62

Figure 3.8: Capacity of an Ideal Shadowed Rician Fading Channel

63

Figure 3.9: Capacity of an Ideal Nakagami Fading Channel

64

3.4 Peak Power Considerations

The results obtained thus far have been presented in terms of average received symbol

energy. For a mobile fading channel it may be more relevant to determine the results

in terms of peak power. The main reason for this is due to the compact size of

mobile transmitters. This implies that the batteries used will also be limited in size,

which in turn restricts the peak power available. Another reason for interest in both

peak and average power results is due to the non-linearities introduced by ampli�ers.

These non-linearities cause the signal to experience amplitude and phase distortion,

which results in an intermodulation of the signal spectrum. When using a constant

envelope signalling method, such as in the case of PSK modulation, the e�ect of the

non-linearity manifests itself as a scaling of the amplitude combined with a constant

phase shift and can easily be compensated for by the receiver. Multilevel QAM used

over a non-linear channel su�ers a performance loss due to the number of di�erent

amplitude levels and resulting distortion.

The peak-to-average power ratio (PAR) of a signal constellation can be used as

a crude measure of the sensitivity of the signal set to non-linearities present in the

channel. The PAR is the ratio of the peak power value used in the constellation to

the average power of all the signal points. The extreme values of PAR are exhibited

by PSK type constellations with a PAR of 1, to say, a Gaussian distributed input

where the PAR is in�nite. To achieve a greater spectral eÆciency and minimize the

e�ect of non-linear distortion, large constellations which maintain an ample distance

between signal points and have small values of PAR are desired. With regard to

average mutual information, smaller values of PAR translate into a decrease in the

di�erence between peak and average power results.

3.4.1 Peak Power Results for Discrete-Valued Input

The average mutual information of an ideal Rayleigh fading channel with discrete-

valued input and continuous-valued output is presented here in terms of peak power.

If a constellation has a peak power value given by Ps, then the average received peak

power at the receiver is �Ps = Ep(�)fj�j2Psg, which takes into account the e�ect of

65

fading on power level. For the purpose of analysis, it is assumed that the fading has

a unity gain so that �Ps = Ps. Therefore, the quantity �Ps will simply be referred to as

peak power. The ratio of the peak and average power is given by PAR = PsEs.

The AMI curves speci�ed with respect to average symbol energy can be used to

present the results in terms of peak power, since 10 log10�

�PsN0

�can be factored into the

sum 10 log10(PAR) + 10 log10�

�EsN0

�, which amounts to translating the average energy

curves away from the origin by a value of 10 log10 (PAR). Since the PAR is 1 for PSK

constellations, the average mutual information curves presented in terms of�EsN0

remain

the same when presented in terms of�PsN0. The AMI curves shown in Figure 3.10 are for

the case of when equiprobable QAM constellations are used as input to the channel.

The 8 and 16-point QAM constellations both have a PAR of 1.8, which results in a

2.6 dB shift between the average and peak power results. The 32-CR constellation

has a PAR of 1.7, which is smaller than that of 16-QAM since the signal set is more

circular in shape. The most noticeable result is for 64-QAM with a PAR of 2.33,

which translates into a 3.68 dB di�erence between peak and average results. With

any square QAM constellation, the PAR asymptotically approaches a value of 3 as

the size of the signal set grows [51]. Thus, the maximum di�erence between peak and

average power results will be 4.77 dB for QAM.

Due to the di�erent slopes of the AMI curves associated with particular signal sets,

there exist crossover points where one constellation is better for lower values of AMI

while the other is superior above the crossover point. As an illustration of this point,

consider the standard 16-point constellations. When signalling at a rate that is close

to 4 bits/T , 16-QAM is still superior to 16-PSK with respect to peak power. However,

for an AMI of 3 bits/T , the PSK constellation has a 0.5 dB advantage over QAM

with respect to peak power. This gain increases to 1.4 dB when an AMI of 2 bits/T

is considered. The crossover point for the 32-point constellations is at approximately

2.7 bits/T , where PSK is better for the lower data rates. The fact that 32-CR is

more circular in shape than standard QAM causes the crossover point to occur at

a lower value of AMI. It is interesting to note that using a 32-CR constellation to

transmit 3 bits/T is more promising with respect to both peak and average power

when compared to the standard 16-point signal sets. Increasing past 32 signal points

66

Figure 3.10: AMI of an Ideal Rayleigh Fading Channel in Terms of Peak Power: QAM

Constellations

67

requires that QAM constellations be considered. In this case, a constellation which is

more circular in shape will have a signi�cant e�ect on the PAR. By using a continuous

approximation, it is easy to show that the PAR of a circle is 23that of a square of

equal area, which amounts to a saving of 1.77 dB in peak power through the use of

shaping.

While PSK constellations possess the ultimate desirable value of PAR, the perfor-

mance of these constellations is generally inferior to that of QAM due to the proximity

of the signal points in Euclidean space. When considering non-linear channels, how-

ever, this situation may be reversed. Considering these facts, it may be of interest to

�nd a constellation that trades o� some of the average power advantage of standard

QAM constellations in order to get a better PAR. In order to do this, a constellation

that compromises some of the bene�ts of both PSK and QAM can be used. Such a

constellation would have more amplitude levels than PSK but fewer than standard

QAM. For a given average symbol power, the signal points would be spaced further

apart than PSK but closer together than QAM. An 8 and 16-point version of such a

hybrid AMPM constellation is shown in Figure 3.11 [33]. For the 8-point constellation

the PAR is 1.58. The PAR of the 16-point constellation is 1.43, which is lower than

that of 16-QAM. A similar version of this constellation with 32 points has a PAR of

1.28.

The average mutual information curves for an ideal Rayleigh channel which uses

these AMPM signal sets as input are shown in Figure 3.12 in terms of average power,

while Figure 3.13 contains the results in terms of peak power. When comparing the

8-point signal sets, 8-PSK is superior by approximately 1.5 to 2.0 dB with respect to

peak power at all rates of AMI in the range of 1 to 2.5 bits/T . When considering

average power, 8-PSK is no more than 0.5 dB worse for the same range of AMI. For

the 16-point signal sets and rates of AMI in the range of 1 to 3.5 bits/T , the hybrid

AMPM constellation is up to 2.0 dB better than 16-PSK with respect to average

power, and no more than 0.5 dB worse than 16-QAM. With regard to peak power

results for this range of AMI, the hybrid AMPM constellation is roughly equivalent to

16-PSK, the crossover point being at approximately 2.8 bits/T . For rates of AMI in

the range of 1 to 4.5 bits/T , the 32-point hybrid constellation is up to 3.0 dB better

68

d1

d2

d1

d2

Figure 3.11: Hybrid AMPM Constellations

69

Figure 3.12: AMI of an Ideal Rayleigh Fading Channel in Terms of Average Power:

Hybrid AMPM Constellations

70

Figure 3.13: AMI of an Ideal Rayleigh Fading Channel in Terms of Peak Power:

Hybrid AMPM Constellations

71

than 32-PSK with respect to average symbol energy, and up to 1.9 dB better with

respect to peak power at higher values of AMI. The peak power crossover point with

32-PSK is at approximately 2.5 bits/T . When compared with the 32-CR constellation

for the same rates of AMI, the hybrid constellation is no more than 1.8 dB worse with

respect to average energy, and less than 0.6 dB inferior with respect to peak power.

The peak power crossover point with 32-CR is at about 2.9 bits/T , where the hybrid

constellation is up to 1 dB better at low values of AMI. Considering these results, the

hybrid AMPM signal sets are an attractive alternative to the standard constellations

for use over fading channels. By adjusting the number of amplitude levels and the

spacing between them, these constellations can be tailored to provide the desired level

of performance.

3.4.2 Channel Capacity with a Peak Power Constraint

The problem of determining the absolute limit on the rate of communication over an

ideal fading channel with a peak power constraint is diÆcult to solve. In order to

calculate the capacity, one may try to use the average mutual information expression

given in equation (3.25). Since H(N) does not depend on the probability distribution

of the input, it is only necessary to choose p(x) to maximize H(Y j�) subject to a

peak power constraint. Unfortunately, this does not work out as nice mathematically

as in the case of when an average power constraint is imposed. An alternate approach

is to calculate upper and lower bounds on the channel capacity. In order to do this,

results which were stated in [1] are utilized.

The average power of a random variable X is de�ned to be EX = E fjXj2g.Shannon de�nes the entropy power PX of a random variable X to be the power of

a Gaussian random variable with entropy H(X). If X actually happens to be a

Gaussian random variable, then EX = PX . Using the fact that H(X) = ln�ePXwhen X is Gaussian, the entropy power of X can be expressed in the form

PX =1

�eH(X)�1: (3.42)

For a �xed value of average power, a Gaussian distributed random variable maximizes

entropy. So in general for any random variable X, EX � PX , where equality holds

72

only if X is Gaussian. The results required in order to calculate bounds on channel

capacity are stated here.

Theorem 3.3 Let Z and N be two independent random variables with average power

EZ and EN , respectively. Assume that at least one of the variables has a zero mean.

If Y = Z + N , then the entropy power of the variable Y can be bounded from above

by

PY � EZ + EN (3.43)

where equality holds only if Z and N are both Gaussian. This inequality will be

referred to as the entropy power bound.

Proof: Since Z and N are independent, the variance of Y will equal the sum of the

variances of Z and N . Therefore, EY � EZ + EN where equality holds when either

Z or N have a zero mean. Since the entropy power PY is less than or equal to the

average power EY , equation (3.43) follows. Also, PY = EY when Y is Gaussian,

which means that Z and N must also be Gaussian for equality in (3.43) to hold. 2

Theorem 3.4 Let Z and N be two independent random variables with entropy power

PZ and PN , respectively. If Y = Z + N , then the entropy power of the variable Y

can be bounded from below by

PY � PZ + PN (3.44)

where equality holds only if Z and N are both Gaussian. This inequality will be

referred to as the entropy power inequality.

Proof: The proof of this theorem is quite involved. Shannon used variational methods

in [1] to show that for given values of PZ and PN , the entropy power has a stationary

point when Z and N are both Gaussian. Shannon's approach, however, did not

account for the possibility that other distributions might yield an equal or lower

value of PY . The �rst rigorous proof of this theorem is credited to Stam [52], and an

improved version is due to Blachman [53]. 2

Over a fading channel, the received symbol Y is the sum of two independent

random variables, the �rst being the product �X of the fading process and channel

symbol, and the second being the noise process N . Suppose for the moment that the

73

value of the fading variable is �xed at � = �. Since � is assumed to be known at the

receiver, the entropy power of Y conditioned on � can be expressed as

PY j� =1

�eH(Y j�=�)�1: (3.45)

From equation (3.43), this entropy power can be bounded by PY j� � EZj�+EN where

the random variable Z is equal to the product �X. The equality in this expression is

met for a �xed value of � when Y is Gaussian with variance 12(EZj� + EN). This is

possible only when, for a �xed value of �, both Z and N are Gaussian with variances

of 12EZj� and 1

2EN , respectively. Using this inequality, the entropy can be bounded

from above by

H(Y j� = �) � lnh�e(EZj� + EN)

i: (3.46)

Using the fact that I(X;Y j� = �) = H(Y j� = �) � H(N) and H(N) = ln�eEN ,

the average mutual information is bounded from above by

I(X;Y j� = �) � ln

1 +

EZj�EN

!: (3.47)

Given a peak power constraint on the input, the upper bound on AMI can be ex-

pressed as

I(X;Y j� = �) � ln

1 +

PZj�EN

!(3.48)

where PZj� is the peak power of Z for a �xed value of �. This inequality is obtained

from the fact that average power is always less than or equal to peak power, and from

the monotonicity of the logarithm. By setting PZj� = j�j2Ps, where Ps is the peakpower of the channel input x, and representing the average noise power by EN = N0,

the upper bound to capacity is obtained by averaging equation (3.48) over the pdf

p(�). The resulting upper bound on channel capacity with a peak power constraint

is

CU = Ep(�)

�log2

�1 + j�j2 Ps

N0

��bits=T: (3.49)

This expression is very similar to that for the channel capacity given an average power

constraint. If the peak power of the input is �xed at Ps, then the average power will

be less than or equal to Ps, and the capacity will be less than that obtained with a

Gaussian input that has a variance of 12Ps. Using the results in Appendix B.1, the

74

upper bound on capacity for an ideal Rayleigh channel with a peak power constraint

is

CU = � (log2 e) exp

0@" �Ps

N0

#�11AEi0@�

"�PsN0

#�11A bits=T: (3.50)

A lower bound on channel capacity may be obtained through use of the entropy

power inequality PY j� � PZj� + PN . By substituting in the appropriate expressions

for entropy power, the inequality becomes

1

�eH(Y j�=�)�1 � 1

�eH(Zj�=�)�1 +

1

�eH(N)�1: (3.51)

This inequality can be rearranged into the form

H(Y j� = �) � ln�eH(Zj�=�) + eH(N)

�: (3.52)

In order to get a tight bound, it is desirable to choose the input distribution so that

H(Zj� = �) is maximized. Given a peak power constraint of jXj2 � Ps, the entropy

H(Zj� = �) is maximized by choosing the input to be uniformly distributed on a

circle of radiuspPs. This is accomplished by setting p(zj�) = 1

�j�j2Ps for jzj � j�jpPs,which results in the entropy H(Zj� = �) = ln�j�j2Ps. By substituting this into

equation (3.52) along with the entropy expression for the noise variable N , the lower

bound on the entropy of Y conditioned on a �xed value of � becomes

H(Y j� = �) � ln��j�j2Ps + �eEN

�: (3.53)

Using this result, the lower bound on average mutual information for a �xed value of

� is

I(X;Y j� = �) � ln�1 + j�j2 Ps

eEN

�: (3.54)

By averaging this inequality over the fading variable and converting the logarithm to

base 2, the resulting lower bound takes on the form

CL = Ep(�)

�log2

�1 + j�j2 Ps

eN0

��bits=T: (3.55)

By making use of the results in Appendix B.1, the lower bound on capacity for an

ideal Rayleigh channel with a constraint on peak power is determined to be

CL = � (log2 e) exp

0@" �Ps

eN0

#�11AEi0@�

"�PseN0

#�11A bits=T: (3.56)

75

The upper and lower bounds obtained here for the ideal Rayleigh fading channel

are plotted in Figure 3.14. The capacity of an ideal Rayleigh fading channel subject to

a peak input power constraint is located somewhere in between these two curves. For

large values of SNR, the power gap between the two bounds is a factor of e, which is

equivalent to approximately 4.34 dB. In terms of achievable data rate, this discrepancy

is no more than 1.44 bits/T . If one could obtain an average mutual information curve

for a speci�c input distribution, then the resulting AMI curve could be used to replace

the lower bound obtained here and reduce the gap between the limits on capacity.

3.5 Channels with Space Diversity

Diversity combining is often used in mobile communication systems in order to reduce

the possibility of outage or loss of channel. This is usually accomplished by means of

space diversity, where L antennae are used to receive a given transmitted signal. If

one of the antennae should experience a deep fade in signal level, then the likelihood

that all of them are also experiencing a deep fade at the same time will diminish as

L increases. The focus of this section is the e�ect of space diversity on the capacity

and average mutual information of a multipath fading channel.

It is assumed that when a symbol x is transmitted, the L symbols yk = �kx + nk

for k = 1; : : : ; L are received over parallel channels. Since these parallel channels are

considered to be ideal, it is also assumed that the receiver has knowledge of the fading

variables �k for k = 1; : : : ; L. In the case of maximal-ratio combining, estimates of

the fading variables are actually used in calculating the weighting coeÆcients. Given

knowledge of the additional received symbols over other antennae, the entropy of the

random variable X should decrease, resulting in a proportional increase in average

mutual information. The form of the average mutual information that is of interest

here is

I(X;YLj�L) = E

(log

p(x; y1; : : : ; yLj�1; : : : ; �L)p(x)p(y1; : : : ; yLj�1; : : : ; �L)

)(3.57)

where YL is used to denote the set of random variables (Y1; : : : ; YL) representing

the received symbols, �L denotes the set of random fading variables (�1; : : : ;�L),

and expectation is taken over the joint pdf p(x; y1; : : : ; yL; �1; : : : ; �L). A collection of

76

Figure 3.14: Bounds on the Capacity of an Ideal Rayleigh Fading Channel with a

Peak Power Constraint

77

symbols such as YL = (Y1; : : : ; YL) will be viewed here either as an ordered L-tuple of

symbols or as a 1� L matrix. This is often done in linear algebra, and the intended

interpretation will be de�ned by the context in which the symbol is used. The average

mutual information can also be expressed as the di�erence of entropies

I(X;YLj�L) = H(YLj�L)�H(YLjX;�L): (3.58)

By factoring the conditional pdf speci�ed in equation (3.4) into product form and

using properties of the logarithm, a joint entropy can be expressed as a sum. Doing

so for the entropies in equation (3.58) results in the expression

I(X;YLj�L) =LXk=1

H(YkjYk�1;�k)�LXk=1

H(YkjYk�1; X;�k) (3.59)

which can also be written as

I(X;YLj�L) =LXk=1

I(Yk;XjYk�1;�k): (3.60)

In these expressions, Yk represents the subset of k symbols (Y1; : : : ; Yk), and �k

represents the subset of k fading variables (�1; : : : ;�k). Finally, equation (3.60) can

be written in the form

I(X;YLj�L) = I(Y1;Xj�1) +LXk=2

I(Yk;XjYk�1;�k): (3.61)

Since average mutual information is always greater than or equal to zero, and the

AMI due to a single diversity channel is given by the term I(Y1;Xj�1), the in-

crease in average mutual information due to diversity is given by the expressionPLk=2 I(X;YkjYk�1; �k).

In order to compare the results for di�erent levels of diversity, it is necessary to

de�ne an average received signal-to-noise ratio. This is de�ned here as the sum of the

average SNR observed on each antenna. For the kth antenna, the average received

signal power is de�ned to be �Esk = Efj�kj2jXj2g, or equivalently �Esk = 2�2�kEs. The

average received noise power is assumed to be the same for all antennae, since any

di�erence can always be absorbed into the fading variable �k. For additive white

Gaussian noise, the received noise power is simply N0 = 2�2N . By taking the ratio

of these two power terms, the average SNR for the kth antenna is determined to be

78

�EskN0

=�2�k

Es

�2N

. In order to compare the fading levels experienced by each antenna, it

is assumed that �2�k = ak�2� for k = 1; : : : ; L. Using this assumption, the average

received SNR is expressed as

�EsN0

=LXk=1

ak�2�Es�2N

: (3.62)

The only case investigated here is for when all ak are equal to 1. If a mobile unit is

constantly in motion, then on average it is reasonable to assume that the channels

associated with the individual antennae are equally good. In this case, the average

received SNR is�EsN0

=L�2�Es�2N

: (3.63)

3.5.1 E�ect of Diversity on Discrete-Input Channels

The e�ect of diversity on the average mutual information of a multipath fading chan-

nel is examined here for the case of when a discrete-valued signal constellation is used

as an input alphabet. The average mutual information can be written as a di�erence

of entropies in the form

I(X;YLj�L) = H(X)�H(XjYL;�L): (3.64)

Suppose the constellation has a uniform a priori distribution over M points. The pmf

which describes this is p(xi) =1M

for i = 1; : : : ;M , and the entropy of the signal set is

H(X) = logM . The conditional entropy H(XjYL;�L) in the expression is obtained

by averaging the logarithm of the conditional pmf p(xijy1; : : : ; yL; �1; : : : ; �L) over

the joint pdf p(xi; y1; : : : ; yL; �1; : : : ; �L). By using Bayes' theorem and the law of

total probability, the conditional pmf of X given Y1; : : : ; YL and �1; : : : ;�L can be

expressed as

p(xijy1; : : : ; yL; �1; : : : ; �L) = p(y1; : : : ; yLjxi; �1; : : : ; �L)p(xi)PMj=1 p(y1; : : : ; yLjxj; �1; : : : ; �L)p(xj)

: (3.65)

After substituting in the pmf for X and factoring the joint density in this expression,

the resulting form of the pmf is

p(xijy1; : : : ; yL; �1; : : : ; �L) =QLk=1 p(ykjyk�1; : : : ; y1; xi; �1; : : : ; �L)PM

j=1

QLl=1 p(yljyl�1; : : : ; y1; xj; �1; : : : ; �L)

: (3.66)

79

If the symbols x and �k are known at the receiver, then the symbol yk = �kx+nk will

not depend on any of the other received symbols or fading variables obtained from

the other antennae. The pmf in equation (3.66) can therefore be simpli�ed to

p(xijy1; : : : ; yL; �1; : : : ; �L) =QLk=1 p(ykjxi; �k)PM

j=1

QLl=1 p(yljxj; �l)

: (3.67)

The constituent pdf's in this expression can be derived from the pdf of the additive

white Gaussian noise as was done in equation (3.20), and take the form

p(ykjxi; �k) = 1

2��2Nexp

�jyk � �kxij2

2�2N

!: (3.68)

By substituting these pdf's into equation (3.67) and simplifying, the resulting prob-

ability mass function is

p(xijy1; : : : ; yL; �1; : : : ; �L) =exp

��1

2

�PLk=1

jyk��kxij2�2N

��PMj=1 exp

��1

2

�PLl=1

jyl��lxj j2�2N

�� : (3.69)

The conditional entropy of X given YL and �L is then determined to be

H(XjYL;�L) =1

M

MXi=1

E

8<:log

24 MXj=1

exp

�

LXk=1

jyk � �kxjj2 � jyk � �kxij22�2N

!359=;(3.70)

where expectation is taken over the joint pdf p(y1; : : : ; yL; �1; : : : ; �Ljxi). Since the

received symbol on each antenna is yk = �kx+nk for k = 1; : : : ; L, by substituting this

into equation (3.70), averaging can be performed over the distribution of the noise

variables Nk rather than the conditional distribution of the Yk given knowledge of X

and the �k. By using this fact, the resulting form of the average mutual information

in bits/T is

I(X;YLj�L) = log2M� 1

M

MXi=1

E

8<:log2

24 MXj=1

exp

�

LXk=1

j�k(xi � xj) + nkj2 � jnkj22�2N

!359=;

(3.71)

where expectation is taken over p(n1) : : : p(nL)p(�1; : : : ; �L).

In order to ascertain the e�ects of space diversity on the AMI of channels with

discrete-valued input, the expression in equation (3.71) was evaluated by computer

simulation for the case of an ideal Rayleigh fading channel. The input alphabets

80

Diversity=2 Diversity=3

8-PSK 16-QAM 8-PSK 16-QAM

3.5 - 2.5 dB - 0.8 dB

3 - 1.6 dB - 0.5 dB

AMI 2.5 2.2 dB 1.2 dB 0.6 dB 0.3 dB

(Bits/T ) 2 1.4 dB 0.9 dB 0.4 dB 0.3 dB

1.5 0.9 dB 0.7 dB 0.3 dB 0.3 dB

1 0.6 dB 0.6 dB 0.2 dB 0.1 dB

Table 3.7: Average Power Gain Due to Increase in Space Diversity of Rayleigh Fading

Channel

considered were 8-PSK and 16-QAM constellations. To ensure that the average SNR

is equal for all antennae and that the fading has a unity gain, it is assumed that all the

noise variables Nk have variance �2N and that the variances �2�k of the fading variables

�k all have a value of 12L. By ensuring that E fjXj2g = 1, the SNR is completely

speci�ed by the value of �2N .

The �rst case considered is that of uncorrelated fading between individual anten-

nae. Figure 3.15 shows the AMI curves for when 8-PSK is used over an ideal Rayleigh

channel with a space diversity of 1, 2, and 3. Figure 3.16 shows the curves for the

case of when a 16-QAM constellation is used. The average power gains due to an

increase in the level of space diversity are shown in Table 3.7 for various rates of AMI.

These gains counteract a signi�cant portion of the loss due to fading experienced on

single diversity channels. By comparing Table 3.5 with the results in Table 3.7, one

can see that the loss due to fading for 8-PSK at the indicated rates of AMI is in the

range of 1.3 to 3.9 dB for a single diversity channel, from 0.7 to 1.7 dB for a dual

diversity channel, and roughly 0.5 to 1.1 dB for a diversity of three antennae. The

ranges of loss due to fading for 16-QAM are observed by comparing the results in

Table 3.6 with those in Table 3.7, and are determined to be 1.1 to 4.6 dB, 0.5 to 2.1

dB, and 0.4 to 1.3 dB for a diversity of one, two, and three antennae, respectively.

The gains available in going from 2 to 3 antennae are less than those obtained when

increasing from 1 to 2 antennae. For example, at an AMI rate of 2 bits/T with an

81

Figure 3.15: AMI of an Ideal Rayleigh Fading Channel with Space Diversity: 8-PSK

82

Figure 3.16: AMI of an Ideal Rayleigh Fading Channel with Space Diversity: 16-QAM

83

j��j2 = 0:3 j��j2 = 0:6 j��j2 = 0:9

8-PSK 16-QAM 8-PSK 16-QAM 8-PSK 16-QAM

3.5 - 0.4 dB - 1.0 dB - 2.0 dB

3 - 0.3 dB - 0.7 dB - 1.3 dB

AMI 2.5 0.4 dB 0.3 dB 0.9 dB 0.6 dB 1.8 dB 1.0 dB

(Bits/T ) 2 0.3 dB 0.2 dB 0.7 dB 0.5 dB 1.2 dB 0.8 dB

1.5 0.2 dB 0.1 dB 0.5 dB 0.3 dB 0.8 dB 0.6 dB

1 0.1 dB 0.1 dB 0.3 dB 0.3 dB 0.5 dB 0.5 dB

Table 3.8: Average Power Loss Due to Space Correlation of the Fading Process

8-PSK constellation, 1.4 dB is gained by increasing the diversity from 1 to 2 anten-

nae, but only 0.4 dB is gained by increasing the diversity from 2 to 3. By increasing

the level of space diversity from 1 to 2 antennae, there is roughly 0.5 to 2.5 dB to

be gained when using these M -point signal sets at rates of AMI in the range of 1 to

log2M � 0:5 bits/T . A further increase to 3 antennae yields an additional 0.1 to 0.8

dB of gain. The rate at which the gains diminish indicate that a diversity of 4 would

probably not be of practical interest.

In order to determine the e�ects of correlation between the fading processes ex-

perienced by the individual antennae, computer simulations were performed for the

case of a dual diversity system. This stochastic occurrence is referred to here as

space correlation of the fading processes. Figure 3.17 shows the AMI curves for when

an 8-PSK constellation is used as input to the channel, while Figure 3.18 has the

results for when 16-QAM is used. The simulations were run with j��j2 taking on

values of 0.3, 0.6, and 0.9, where �� is the space correlation coeÆcient of the two

fading variables. Table 3.8 shows the loss in average power due to space correlation

of the fading processes for various rates of AMI. When the amount of correlation is

small, this loss is negligible. When the fading processes are heavily correlated with

j��j2 = 0:9, the loss in average power for the case of 8-PSK or 16-QAM is roughly 0.5

to 2 dB for values of AMI in the range of 1 to log2M � 0:5 bits/T . So even when the

fading processes experienced by the individual antennae are moderately correlated,

say with j��j2 less than 0.5, using a space diversity of 2 can still result in a signi�cant

84

Figure 3.17: AMI of an Ideal Rayleigh Fading Channel with Diversity=2 and Space

Correlated Fading: 8-PSK

85

Figure 3.18: AMI of an Ideal Rayleigh Fading Channel with Diversity=2 and Space

Correlated Fading: 16-QAM

86

increase in the average mutual information when discrete-valued signal sets are used

for transmission over an ideal Rayleigh fading channel.

3.5.2 Capacity of Fading Channels with Space Diversity

The e�ect of space diversity is examined here for channels with a continuous-valued

input alphabet. The set of received symbols yL = (y1; : : : ; yL) can be expressed as

a vector equation yL = x�L + nL, where the scalar x is the transmitted symbol,

�L = (�1; : : : ; �L) is a vector of the fading variables associated with each antenna,

and nL = (n1; : : : ; nL) is a vector of the noise variables associated with each antenna.

The average mutual information between the transmitted symbol X and the vector of

received symbols YL, given knowledge of the �L, can be expressed as the di�erence

of entropies

I(X;YLj�L) = H(YLj�L)�H(YLjX;�L): (3.72)

The �rst term considered in this expression is the conditional entropy H(YLjX;�L).

If the variables X and �L are known, then the joint entropy of YL conditioned on

these variables is the same as the joint entropy of the vector of noise variables NL.

The general form of a L-dimensional complex-valued Gaussian density, which is used

here to describe the additive noise vector, is

p(nL) =1

(2�)L detBNexp

��1

2n�LB

�1N nTL

�(3.73)

where BN is a L � L covariance matrix with entries (BN)ij = 12EfNiN

�j g. Since

the noise variables are assumed to be independent, BN is a diagonal matrix with

the non-zero entries being equal to (BN )ii = �2N . By computing the value of the

entropy H(NL) = �Ep(nL)flog p(nL)g, the entropy of YL conditioned on X and �L

is determined to be

H(YLjX;�L) = logh(2�e)L detBN

i: (3.74)

As in the case of a single random variable, given a constraint on the second order

moments of a set of L complex-valued random variables, the entropy is maximized

by a L-dimensional complex-valued Gaussian distribution. In order to calculate the

value of the entropy H(YLj�L), it is �rst necessary to determine the pdf p(yLj�L).

87

This is accomplished by starting with the pdf of the additive noise vector given in

equation (3.73). Since yk = �kx + nk for k = 1; : : : ; L, the conditional pdf of Y L

given knowledge of �L and X is

p(yLj�L; x) = 1

(2�)L detBNexp

��1

2(yL � x�L)

�B�1N (yL � x�L)

T�

(3.75)

which is obtained from p(nL) by performing a vector translation. For the moment, it

is assumed that the input alphabet has a Gaussian distribution. The desired pdf can

then be obtained by evaluating the integral

p(yLj�L) =ZSXp(yLj�L; x)p(x)dx: (3.76)

By substituting in the expressions for the required probability densities, this integral

can be written explicitly as

p(yLj�L) = AZSX

exp

�1

2

"1

�2X+ ��LB

�1N �

TL

#jxj2

!exp

�<nxy�LB

�1N �

TL

o�dx

(3.77)

where A = 1(2�)L+1�2

XdetBN

exp��1

2y�LB

�1N yTL

�. Transforming the variable x into polar

coordinates, where � = jxj and � = arg x, equation (3.77) can be expressed in the

form

p(yLj�L) = AZ 1

0� exp

�1

2

"1

�2X+ ��LB

�1N �

TL

#�2!

(3.78)

�Z 2�

0exp

��<

ny�LB

�1N �

TL

ocos � � �=

ny�LB

�1N �

TL

osin �

�d�d�

where <f�g and =f�g denote the real and imaginary parts of the enclosed complex-

valued expression, respectively. By making use of relation 3.937 2. taken from [54],

the integral with respect to � can easily be solved as

Z 2�

0exp

��<

ny�LB

�1N �

TL

ocos � � �=

ny�LB

�1N �

TL

osin �

�d� = 2�I0

����y�LB�1N �

TL

��� �� :(3.79)

By changing the variable � topu in equation (3.78), and using the result in equation

(3.79), the pdf can be expressed in the form

p(yLj�L) = �AZ 1

0exp

�1

2

"1

�2X+ �

�LB

�1N �

TL

#u

!I0����y�LB�1

N �TL

���pu� du: (3.80)

88

By viewing this integral as the Laplace transform of I0����y�LB�1

N �TL

���pu�, it is evalu-ated to be [55, p. 197 (14)]

p(yLj�L) = 2��2XA1 + �2X�

�LB

�1N �

TL

exp

0B@ �2X

���y�LB�1N �

TL

���22�1 + �2X �

�LB

�1N �

TL

�1CA : (3.81)

By substituting in the value of A and simplifying, the desired pdf is determined to

be

p(yLj�L) = 1

(2�)L detBY j�exp

��1

2y�LB

�1Y j�y

TL

�(3.82)

which is a Gaussian distribution. The covariance matrix BY j� consists of the entries

(BY j�)ij = �i��j�

2X for i 6= j, and (BY j�)ii = j�ij2�2X +�2N along the diagonal. The de-

terminant of this matrix is detBY j� = (1+�2X ��LB

�1N �

TL ) detBN . Since the resulting

distribution is Gaussian, the entropy of Y L conditioned on �L can be immediately

written as

H(YLj�L) = Ep(�L)

nlog

h(2�e)L detBN

�1 + �2X�

�LB

�1N �

TL

�io: (3.83)

By substituting the entropies obtained into equation (3.72), the average mutual in-

formation is determined to be

I(X;YLj�L) = Ep(�L)

nlog

�1 + �2X�

�LB

�1N �

TL

�o: (3.84)

Recalling the results for the single diversity channel, the pdf p(ykj�k) will be Gaussianonly if p(x) is. If this is not the case, then the marginal densities p(ykj�k) as well asthe joint density will not be Gaussian. Since this is the distribution that maximizes

H(YLj�L) subject to a constraint on the second order moments, capacity is achieved

for an ideal fading channel with space diversity by a Gaussian distributed input.

Thus, equation (3.84) speci�es the capacity of the channel. Assuming that the average

received SNR is the same for each antenna, the capacity can be written in bits/T as

C = Ep(�L)

(log2

1 +

�2X�2N

LXk=1

j�kj2!)

: (3.85)

Averaging over p(�L) in this capacity expression can be explicitly evaluated for the

case of Rayleigh fading. Since the fading variables in this expression occur separately

as terms in a summation, a new random amplitude fading variable R can be de�ned

89

where R2 =PLk=1 j�kj2. The capacity can then be expressed as an average over the

pdf of R, as given by the equation

C = Ep(r)

(log2

1 + r2

�2X�2N

!): (3.86)

This expression is the same as that derived for the case of a single antenna. Since

the �k are independent Gaussian variables with zero mean, the pdf of the variable R

is a central chi distribution with 2L degrees of freedom. By constraining the second

moment of the variable so that E fR2g = 1, this pdf can be written in the form

p(r) =2LLr2L�1

�(L)exp

��Lr2

�for r � 0 (3.87)

which is the same as the Nakagami pdf in equation (2.23) with m = L and = 1.

Therefore, an ideal Rayleigh fading channel with space diversity is equivalent to an

ideal Nakagami fading channel. Using the results of Appendix B.2, equation (3.86)

is evaluated to be

C = (log2 e)(�L)L�(L)

"�EsN0

#�L d

ds

!L�1 �es

sEi(�s)

�bits=T (3.88)

where s = L�

�EsN0

��1. For a dual diversity channel, the closed form expression for

capacity is

C = (log2 e)

241 +

0@2

"�EsN0

#�1� 1

1A exp

0@2

"�EsN0

#�11AEi0@�2

"�EsN0

#�11A35 bits=T:

(3.89)

For a diversity of three antennae, the capacity is expressed in the form

C = �1

2(log2 e)

hs� 3 + (s2 � 2s+ 2) exp (s)Ei (�s)

ibits=T (3.90)

where the parameter s is equal to 3�

�EsN0

��1. Finally, for the case L = 4, the channel

capacity is determined to be

C =1

6(log2 e)

hs2 � 4s+ 11 +

�s3 � 3s2 + 6s� 6

�exp (s)Ei (�s)

ibits=T (3.91)

where s = 4�

�EsN0

��1.

These results are plotted in Figure 3.19 along with the capacity for the AWGN

channel. Also included is the capacity of the Rayleigh channel with no space diversity.

90

Figure 3.19: Capacity of an Ideal Rayleigh Fading Channel with Space Diversity

91

By using the results presented in Appendix C, the asymptotic power loss due to

amplitude fading compared to the AWGN channel is determined to be Le� (L), and

the gain achieved through the use of space diversity is eCE�Le� (L). A dual diversity

system results in a 1.33 dB asymptotic gain, and is no more than 1.17 dB from the

AWGN channel results. For a system which uses three antennae, the asymptotic

diversity gain is 1.74 dB, and the channel capacity is 0.76 dB from the AWGN results.

For the case of four antennae, the asymptotic diversity gain is 1.94 dB, and the

distance from the AWGN channel capacity is 0.56 dB. As L ! 1, the e�ect of

amplitude fading diminishes to the point where capacity is the same as that of an

AWGN channel.

The expression for channel capacity given by equation (3.85) was also evaluated

by means of computer simulation for the case of a dual diversity channel with space

correlated fading. The results obtained for di�erent levels of correlation are shown in

Figure 3.20. These results once again indicate that even when the fading processes

experienced by the individual antennae are moderately correlated, the resulting loss

is small. When the space correlation coeÆcient is set so that j��j2 = 0:3, the loss

compared to the uncorrelated case is 0.2 dB. For a value of j��j2 = 0:6 the loss is

approximately 0.6 dB, and when j��j2 = 0:9 the loss is roughly 1.0 dB.

A result was presented in [34] which was intended to represent the capacity of

an ideal Rayleigh fading channel with space diversity. Although the end results are

similar to those derived here, the formulation of the problem was di�erent. The ex-

pression used for capacity was that of a AWGN channel. The SNR was modelled

as being a random variable based on maximal ratio combining, with statistics de-

termined by the level of diversity. The pdf describing the SNR was a chi-squared

distribution essentially equivalent to the pdf shown in equation (3.87). Although no

explicit expression for the channel capacity was given, the necessary averaging was

performed through computer simulation, and the curves presented were equivalent to

those derived here.

Examination of the results shown in this section indicate that a signi�cant gain in

capacity is obtained by using a dual diversity system. From a practical standpoint,

however, it is not likely that the small additional potential gains associated with

92

Figure 3.20: Capacity of Dual Diversity Rayleigh Channel with Space Correlated

Fading

93

higher levels of diversity would warrant the increase in complexity and cost. The

actual number of antennae used in any system will likely be governed by the desired

outage probability.

3.6 Potential Coding Gain for Fading Channels

Up to this point, the limits on reliable communication have been determined for

a number of ideal fading channel models. In this section, these limits are used to

determine the magnitude of the coding gain that can be expected over a fading

channel. The expression for the capacity of an ideal fading channel is

C = Ep(�)nlog2

�1 + j�j2SNR

�obits=T: (3.92)

At high values of SNR, the channel coding theorem states that an arbitrarily small

probability of error can be realized as long as the data rate Rc in bits/T satis�es the

relation

Rc < Ep(�)nlog2

�j�j2SNR

�o: (3.93)

Since a rate ofRc bits/T requires a constellation of size 2Rc, a signal-to-noise ratio nor-

malized with respect to the constellation size can be de�ned as SNRnorm = 2�RcSNR.

By using this de�nition, equation (3.93) can be rearranged into the form

SNRnorm > 2�Ep(�)flog2(j�j2)g: (3.94)

This result indicates that an arbitrarily small probability of error can be attained as

long as the normalized SNR is greater than the asymptotic fading loss of the channel.

Since the fading process is assumed to have a unity gain, for the ideal Rayleigh channel

this means that SNRnorm > eCE , which is equivalent to 2.51 dB.

This capacity bound can be compared to the symbol error probability of uncoded

QAM in order to determine the potential coding gain available. The probability of

symbol error is closely estimated by the expression

Pr(e) = 1� 2p1 + � �1

�1� 2

�arctan

�q1 + � �1

��(3.95)

where � = 32SNRnorm. This expression is derived in Appendix D.1 and is actually

an upper bound, but becomes more accurate as the size of the signal set grows.

94

The symbol error probability for uncoded QAM is compared to the capacity bound

in Figure 3.21. Due to the almost inverse linear nature of this error probability, the

distance with respect to SNR between the error probability curve and channel capacity

increases dramatically as Pr(e) gets smaller. The ultimate shape gain for the AWGN

channel is approximately 1.53 dB, and is obtained by using a Gaussian distributed

input rather than a square constellation of equal average energy. Since the average

received symbol energy is Efj�Xj2g = 2�2�Es, amplitude fading has no e�ect on

shape gain, so it is the same for a fading channel as it is for the AWGN channel. If the

1.53 dB of shape gain is subtracted from the di�erence between the channel capacity

and the error probability curve for QAM, the remainder represents the potential

coding gain over uncoded QAM. At a symbol error rate of 10�3, the potential coding

gain is about 23.3 dB. For small values of Pr(e) this increases dramatically; roughly

10 dB for every decrease in error probability by a power of 10. At a symbol error

rate of 10�6 the potential gain is approximately 53.3 dB, and increases to 73.3 dB for

Pr(e) = 10�8.

The potential coding gain for fading channels is extremely large compared to the

AWGN channel. In general, much of this gain is realized by known bandwidth eÆcient

coding techniques, but the performance of these known codes is still signi�cantly far

from capacity. It may be of interest to see where the best known codes stand against

capacity results. Since most research has focussed on achieving spectral eÆciencies

of 2 bits/T using an 8-PSK constellation, these results are examined here. The AMI

curve for 8-PSK in Rayleigh fading indicates that 2 bits/T can be achieved with an

arbitrarily small probability of error for an average bit SNR greater than 5.4 dB.

The bit SNR is 3 dB less than the symbol SNR for a rate of 2 bits/symbol. This

bound on AMI is compared to the bit error performance of a number of coding

schemes in Figure 3.22. The schemes considered are Ungerboeck's 8-state 8-PSK

trellis code [6], Zehavi's suboptimal 8-state 8-PSK code [21], and Lin's multi-level

block coded scheme [22]. Also included is the bit error probability of uncoded QPSK

modulation with Gray coding. In Appendix D.2 this is shown to be given by the

95

Figure 3.21: Probability of Symbol Error for Uncoded QAM in Rayleigh Fading

96

Figure 3.22: Bit Error Probability in Rayleigh Fading for Various Coded Modulation

Schemes with Rate 2 bits/T

97

expression

Pb =1

2

26641� 1r

1 +h�EbN0

i�13775 (3.96)

where�EbN0

is the average SNR per bit. At a bit error rate of 10�6, the distance

between the AMI curve for 8-PSK and the bit error curve corresponding to QPSK is

roughly 48.6 dB. Ungerboeck's code reduces this distance to 18.5 dB. Zehavi's code

is approximately 11.4 dB away from capacity at this error rate, and Lin's code is

approximately 6.6 dB away. Even the best known result leaves 6.6 dB of potential

gain, which is signi�cant. In general, there is much more to be gained at higher

spectral eÆciencies when larger signal constellations are considered.

Figure 3.23 contains the bit error probability curves for the recently published

turbo codes designed for use in Rayleigh fading [23]. The curve labelled TC1 is for

the code which transmits 2 bits/T using a 16-QAM constellation. The corresponding

capacity curve for 16-QAM at this rate is labelled C1. For a bit error probability

of 10�5, the turbo code is 4.4 dB from capacity. The bit error curve labelled TC2

is for the turbo code which transmits 3 bits/T using a 16-QAM constellation. The

corresponding capacity curve for this rate using 16-QAM is labelled C2. This code

attains a bit error probability of 10�5 at a SNR which is 5.1 dB from capacity. The

�nal error curve labelled TC3 is for the code which transmits 4 bits/T using 64-QAM.

The corresponding capacity curve for a rate of 4 bits/T using 64-QAM is labelled C3.

For a bit error rate of 10�5, this code is also 5.1 dB from capacity. These are the best

known coding results to date.

98

Figure 3.23: Bit Error Probability in Rayleigh Fading for Turbo Codes

99

Chapter 4

E�ects of Non-Ideal Channel State

Information

Having determined the achievable rates of data transmission over ideal fading chan-

nels, the next logical step is to consider practical limitations to this idealization and

the resulting consequence to reliable communication. In the previous chapter, a fad-

ing channel was de�ned to be ideal if it satis�ed four conditions. In this chapter it

is assumed that the fading channel model still satis�es all of these conditions except

for the postulate concerning channel state information. This means that the value

of the fading variable � is no longer assumed to be available at the receiver. This

assumption of non-ideal channel state information (CSI) re ects the reality of hav-

ing to estimate information about the fading process using some practical scheme

at the receiver. Rather than determining the fading variable � exactly, an estimate

is obtained. The requirement of a channel state estimate is demonstrated through

calculation of average mutual information for discrete-valued constellations, as well

as an upper bound on this quantity for continuous-valued input alphabets. These

numbers turn out to be quite small when no information is available about the fading

process and most of the received energy is due to scattered signal components. The

�rst instance of non-ideal CSI investigated is for the case of perfect phase information

with no amplitude information. This model is based upon the assumption of perfect

coherent detection with no attempt made to determine the fading level. The AMI for

standard constellations is obtained and compared to the ideal channel results in order

to determine the amount of loss which results when the fading amplitude is ignored.

100

Bounds on average mutual information are also calculated for the phase information

case when a continuous-valued input is used. In order to investigate the e�ects of

practical channel estimation schemes, the true value of � and the channel estimate

are modelled as jointly Gaussian random variables. The variance of the estimate, as

well as the correlation coeÆcient between the true and estimated values, are depen-

dent upon the particular method used. The schemes considered here are the use of a

pilot tone, di�erentially coherent detection, and the use of a pilot symbol. The loss

incurred due to the limitations of these estimation techniques is ascertained through

calculation of AMI for channels using standard constellations.

4.1 Requirement of Channel State Estimation

In this section, the need for channel state estimation is exhibited through calculation

of information theoretic quantities. It has been demonstrated that reduced knowledge

of the fading variable � results in a performance loss of certain types of coded and

uncoded modulation [16]. It is conceivable, however, that a scheme may exist which

achieves reliable communication at an arbitrary rate when no CSI is used in detection.

4.1.1 Channels with Discrete-Valued Input and No CSI

Consider the AMI when a discrete-valued constellation is used as input to a fading

channel, and where the value of the fading variable � is unknown at the receiver.

The AMI between the input X and the output Y can be expressed as a di�erence

of entropies by the equation I(X;Y ) = H(X) � H(XjY ), where H(X) = logM for

an equiprobable input distribution. In order to determine the entropy H(XjY ), thepmf p(xijy) is required. This probability mass function can be obtained by starting

with the pdf of the fading variable �. A Rayleigh fading channel is considered �rst,

which means that � will be a zero mean complex-valued Gaussian random variable

with pdf

p(�) =1

2��2�exp

� j�j

2

2�2�

!: (4.1)

101

If a new variable Z is de�ned as being equal to the product �X, then the pdf of Z

given knowledge of the value of X can be determined from equation (4.1) to be

p(zjxi) = 1

2�jxij2�2�exp

� jzj22jxij2�2�

!: (4.2)

This is obtained by scaling the random variable � by the complex number xi. Since

the channel output can be written as the sum y = z + n, the pdf of the variable Y

given knowledge of X and N is

p(yjxi; n) = 1

2�jxij2�2�exp

� jy � nj22jxij2�2�

!(4.3)

which is obtained from equation (4.2) by performing a translation on the variable z.

The complex-valued noise variable N is described by a zero mean Gaussian distri-

bution with variance �2N . As was shown in the previous chapter, evaluation of the

integralRSNp(yjxi; n)p(n)dn is simply a convolution of the two density functions. In

this case, application of equation (3.31) results in the conditional pdf

p(yjxi) = 1

2�(�2�jxij2 + �2N)exp

� jyj22(�2�jxij2 + �2N )

!: (4.4)

Using Bayes' theorem in conjunction with the law of total probability, and the as-

sumption that p(xi) =1M

for i = 1; : : : ;M , one may write the desired pmf in the

form

p(xijy) = p(yjxi)PMj=1 p(yjxj)

: (4.5)

The entropy H(XjY ) = �Ep(xi;y) flog p(xijy)g can then be determined from the ex-

pression

H(XjY ) = Ep(xi;y)

(log

"PMj=1 p(yjxj)p(yjxi)

#): (4.6)

By substituting the pdf given by (4.4) into equation (4.6) and simplifying, this be-

comes

H(XjY ) = (4.7)

1M

PMi=1

RSYp(yjxi) log

�PMj=1

�2�jxij2+�2N�2�jxjj2+�2N

exp�� jyj2�2�(jxij2�jxj j2)

2(�2�jxij2+�2N )(�2�jxj j2+�2N )

��dy:

Averaging over p(yjxi) in this expression can be replaced by expectation taken over

p(�)p(n). This is accomplished by setting y = �xi+n in equation (4.7), which results

102

in the conditional entropy being expressed as

H(XjY ) = 1

M

MXi=1

E

8<:log

24 MXj=1

�2�jxij2 + �2N�2�jxjj2 + �2N

exp

� j�xi + nj2�2�(jxij2 � jxjj2)2(�2�jxij2 + �2N)(�

2�jxjj2 + �2N)

!359=; :

(4.8)

The form of this entropy expression lends insight into the general magnitude of

the achievable rates of AMI for Rayleigh fading channels with discrete-valued input

and no information about the fading variable. For the ideal fading channel, this

entropy is a function of the Euclidean distances between signal points. With no CSI

available at the receiver, the entropy depends upon the di�erence of the magnitudes of

the various symbols. For constant envelope modulation, such as in the case of PSK,

the symbol magnitudes are all equal. In this case the equivocation of the channel

is H(XjY ) = logM , which is the same as the entropy H(X) of the input, and the

resulting AMI of the channel is I(X;Y ) = 0. Therefore, observation of the channel

output without any CSI does not reduce the average uncertainty of which PSK symbol

is transmitted, and on average, no information is transmitted over the channel.

The AMI was also determined through computer simulation for the case of when

certain multi-level constellations are used as input to the channel. Even at an average

received SNR of 40 dB, the AMI achievable using 16-QAM is approximately 0.29

bits/T , while a rate of 0.32 bits/T is attained by using a 32-CR constellation.

Examination of equation (4.8) reveals that the use of multi-level signal sets cannot

improve the AMI to a very large degree. For a �xed value of X = xi, consider the

term log2

�PMj=1 exp

�� j�(xi�xj)+nj2�jnj2

2�2N

��, which comes from the expression given in

equation (3.22) for the equivocation of an ideal fading channel. For large values of

SNR, all the exponential terms tend to zero except for the one in which xj = xi,

which has a value of 1. Therefore, the logarithm approaches a value of zero, which in

turn causes the equivocation of the channel to vanish. The corresponding expression

for a channel with no CSI is log2

�PMj=1

�2�jxij2+�2N�2�jxj j2+�2N

exp�� j�xi+nj2�2N (jxij2�jxj j2)

2(�2�jxij2+�2N )(�2�jxj j2+�2N )

��. In

this case, not only the term in which xj = xi will yield a value of 1, but so will any

other term involving a signal point with the same magnitude as xi. For those signal

points which have a di�erent magnitude than that of xi, the resulting term in the

summation over j will approach a value of approximately jxij2jxj j2 exp

�1� jxij2

jxj j2�for large

values of SNR. Subject to an average power constraint of the form 1M

PMi=1 jxij2 = 1,

103

one can see that the equivocation of a Rayleigh fading channel with no CSI cannot

be made arbitrarily small. As the size of the constellation is made larger, the signal

points become increasingly crowded together. This causes larger numbers of channel

symbols to have relatively equivalent magnitudes.

Due to the low potential data rates for a Rayleigh channel with no CSI, it is of

interest to determine similar results for the case of when a LOS component exists

in the received signal. If the pdf shown in equation (4.1) is given a non-zero mean

m�, then by following the same procedure shown for the Rayleigh case, the AMI of

a Rician channel expressed in units of bits/T is determined to be

I(X;Y ) = log2M� (4.9)

1M

PMi=1E

�log

�PMj=1

�2�jxij2+�2N�2�jxj j2+�2N

exp��1

2

�j�xi�m�xj+nj2�2�jxjj2+�2N

� j(��m�)xi+nj2�2�jxij2+�2N

����:

For a non-zero mean m�, there is a dependence upon the weighted distance between

discrete channel symbols. As the Rician channel parameter R spans the range from

0 to1, the AMI curves span the range delimited by the Rayleigh and AWGN results.

The expression for AMI in equation (4.9) was evaluated by computer simulation

for di�erent values of R. The resulting AMI curves for when an 8-PSK signal set is

used as input to the channel are shown in Figure 4.1. Similar results for 16-QAM are

shown in Figure 4.2. In order to interpret these results, one may view the receiver as

using the LOS component of the signal for synchronization, but completely ignoring

the e�ect of the scattered multipath components. When R = 5 dB, up to 1.9 bits/T

can be transmitted using an 8-PSK constellation, while a maximum rate of 2.4 bits/T

occurs for the case of 16-QAM. A Rician channel parameter of R = 10 dB allows

maximum rates of 2.7 bits/T and 3.4 bits/T to be attained using 8-PSK and 16-QAM

signal sets, respectively.

4.1.2 Channel with Continuous-Valued Input and No CSI

An upper bound on average mutual information may also be calculated in order to

determine the achievable rate of communication without CSI. An upper bound to the

AMI I(X;Y ) = H(Y )�H(Y jX) is determined here for a Rayleigh fading channel with

a continuous-valued input. The �rst step in accomplishing this is to bound the term

104

Figure 4.1: AMI of a Rician Fading Channel with No CSI: 8-PSK

105

Figure 4.2: AMI of a Rician Fading Channel with No CSI: 16-QAM

106

H(Y ) from above. The entropy power bound was presented in the previous chapter,

and stated that the entropy power of a sum of two independent random variables is

less than or equal to the sum of the average power of the individual variables. Since

the channel output symbol can be expressed as Y = Z +N , where EfjZj2g = 2�2�Es

and EfjN j2g = 2�2N , the entropy power bound can be used here to write

PY =1

�eH(Y )�1 � 2�2�Es + 2�2N : (4.10)

This expression can also be stated in the form

H(Y ) � lnh2�e

��2�Es + �2N

�i: (4.11)

In order for the equality in (4.11) to hold, there must exist a random variable X

with pdf p(x) and second moment 12Es such that the product Z = �X is a Gaussian

variable. The existence of such a distribution for X has not been determined here.

In order to determine an upper bound on �H(Y jX), or equivalently a lower bound

to H(Y jX), the entropy power inequality will be invoked. For a conditional entropy

it is obvious that

H(Y jX) � Ep(x)nlnheH(ZjX=x) + eH(N)

io: (4.12)

This is true because the entropy power inequality can be applied when X is �xed,

and the averaging necessary to obtain H(Y jX) does not a�ect the inequality. In [53],

a relation referred to as \the log cosh inequality" can be used to further show that

H(Y jX) � Ep(x)nlnheH(ZjX=x) + eH(N)

io� ln

heH(ZjX) + eH(N)

i: (4.13)

As stated previously, the entropy of the additive Gaussian noise is H(N) = ln 2�e�2N .

By examining the pdf given by equation (4.2), one can see that p(zjx) is Gaussian,so the conditional entropy of Z given knowledge of X can be written as

H(ZjX) = Ep(x)nln�2�e�2�jxj2

�o= ln

�2�e�2�

�+ Ep(x)

nln jxj2

o: (4.14)

Since the natural logarithm is a concave function, one can apply Jensen's inequal-

ity [49, p. 25] to obtain the relation

Ep(x)nln jxj2

o� ln

hEp(x)

njxj2

oi(4.15)

107

where equality holds only when jxj2 is constant for all values of x. This inequality

can also be expressed in the form

Ep(x)nln jxj2

o= lnEs ��J (4.16)

where �J is a non-negative constant which is independent of the energy in the signal.

To show that this is true, one need only scale the variable x used in equation (4.15).

The scale factor then manifests itself as an identical additive constant on both sides

of the inequality. By subtracting this additive term from both sides, one obtains

the original inequality. Substituting the relation given by (4.16) into equation (4.14)

results in the entropy expression

H(ZjX) = ln�2�e1��J�2�Es

�: (4.17)

The lower bound on the conditional entropy of Y given knowledge of X is obtained

by substituting the required entropies into (4.13). This results in the relation

H(Y jX) � lnh2�e

�e��J�2�Es + �2N

�i: (4.18)

By substituting the bounds shown in (4.11) and (4.18) into the di�erence of entropies

expression, the resulting upper bound on AMI is determined to be

IU = log2

24 1 +

�EsN0

1 + exp (��J)�EsN0

35 bits=T (4.19)

where�EsN0

=�2�Es�2N

. For large values of SNR, IU � �J log2 e. Therefore, no matter how

much energy is used, the capacity cannot exceed a certain constant value.

Unfortunately, the entropy power relations result in very weak bounds for certain

distributions. Therefore, the expression in (4.19) cannot be used to obtain a good

bound on capacity. It is possible, however, to calculate the value of �J for certain

speci�c distributions which result in a small limiting value of AMI. When p(x) is

Gaussian, the number �J is equal to Euler's constant, which results in the AMI

being bounded to less than 0.83 bits/T . When X has a uniform distribution over a

circle centered at the origin, this constant equals �J = 1 � ln 2, which bounds the

AMI to less than 0.44 bits/T .

108

If the Gaussian fading variable is assumed to have a non-zero mean m�, and the

upper bound in equation (4.19) is rederived, then this bound as applied to the Rician

fading channel takes the form

IU = log2

24 1 +

�EsN0

1 + exp(��J )1+ R

�EsN0

35 bits=T: (4.20)

This upper limit on AMI is plotted in Figure 4.3 for various values of the Rician

channel parameter R. The value of �J used is that derived for a Gaussian distributed

input. As R !1, this bound becomes increasingly precise. For large values of SNR,

this bound will be approximately IU � (log2 e)�J + log2(1 + R). When a Gaussian

distributed input is used, the AMI is less than 2.9 bits/T for R = 5 dB and less than

4.3 bits/T for R = 10 dB.

As an illustration of how the bound in equation (4.19) can be made arbitrarily

poor, consider the following case [56]. De�ne the random variable � to equal jXj2. Thediscrepancy in Jensen's inequality can be represented in this case by the expression

�J = lnhEp(�) f�g

i� Ep(�) fln �g : (4.21)

Now if � has a lognormal distribution described by the pdf

p(�) =1q

2��2��exp

�(ln � �m�)

2

2�2�

!for � > 0 (4.22)

then Ep(�) f�g = exp�m� +

�2�2

�and Ep(�) fln �g = m�. In this case, the expression

in (4.21) evaluates to �J =�2�2, which can be made arbitrarily large by choosing �2� to

be arbitrarily large. Any power constraint can be applied independently by choosing

an appropriate value for m�. For example, to set E f�g = 1, choose m� = ��2�2.

Thus, even though this result indicates that capacity is limited to being less than

some constant value, the bound is extremely weak in some cases.

4.2 Channels with Phase-Only Information

The results of the previous section show that for a Rayleigh fading channel with no

CSI available at the receiver, most of the information in the signal amplitude and

109

Figure 4.3: Upper Bounds on the AMI of a Rician Fading Channel with No CSI:

Gaussian Distributed Input

110

all the information in the phase is lost. In this section it is assumed that the phase

of the fading process is known at the receiver, but that the amplitude is not. This

model will be referred to as a channel with phase-only information or a phase-only

channel. A somewhat practical realization of this model would be a coherent receiver

that can track the variations of the phase of the fading process. Since no e�ort is

made to determine the fading amplitude, the implementation of such a receiver would

be less complex, and this would likely result in the system being less expensive. The

phase-only information assumption was referred to as a channel with no CSI in the

work of Simon and Divsalar [16]. Information theoretic bounds on communication

rate are determined �rst for channels using discrete-valued input, then for channels

with continuous-valued input.

4.2.1 Phase-Only Channels with Discrete-Valued Input

The AMI of a fading channel with discrete-valued input and phase-only information is

determined here by evaluating the familiar di�erence of entropies expression, which in

this case is written I(X;Y j�) = H(X)�H(XjY;�). As usual, an equiprobable input

distribution is assumed over a M -point constellation, which results in the entropy of

the channel input beingH(X) = logM . In order to determine the entropyH(XjY;�),an expression for the pmf p(xijy; �) must be found. This probability mass function is

obtained here by �rst �nding the pdf p(yjxi; �), then applying Bayes' theorem. The

pdf of the additive noise variable N will be used here as a point of departure in the

derivation of p(yjxi; �). Starting with the probability density function

p(n) =1

2��2Nexp

� jnj

2

2�2N

!(4.23)

and setting y = �xi + n, the conditional pdf of Y given knowledge of � and X can

be determined by performing a translation on n. The resulting pdf is

p(yjxi; �) = 1

2��2Nexp

�jy � �xij2

2�2N

!: (4.24)

Knowledge of the fading variable � means knowledge of both the magnitude variable

R and the phase variable �. By setting � = rej� in equation (4.24), the pdf can be

111

assumed to be conditioned on both R and �, which results in the density function

being expressed as

p(yjxi; r; �) = 1

2��2Nexp

�jy � rej�xij2

2�2N

!: (4.25)

This can also be written in the form

p(yjxi; r; �) = 1

2��2Nexp

� jyj

2

2�2N

!exp

�jxij

2

2�2Nr2!exp

0@<

nye�j�x�i

o�2N

r

1A : (4.26)

The motivation for writing the pdf in this fashion is the desire for evaluating the

integral p(yjxi; �) =Rp(yjxi; r; �)p(r)dr, which eliminates the dependence on the

fading amplitude. Under the assumption of Rayleigh fading, the statistics of the

amplitude R are described by the pdf

p(r) =r

�2�exp

� r2

2�2�

!for r � 0: (4.27)

The desired probability density function can be placed in the form

p(yjxi; �) = (4.28)

12��2

��2N

exp�� jyj2

2�2N

� R10 r exp

���jxij22�2

N

+ 12�2

�

�r2�exp

�<fye�j�x�ig

�2N

r�dr:

In order to simplify notation in the evaluation of the integral in this expression,

the following substitutions are made. Let the �rst variable be �i =�2�jxij2+�2N

2�2��2N

and

de�ne the second variable to be �i =jyj2+jxij2�jy�ej�xij2

2�2N

. The subscripts on �i and �i

are used to indicate the dependence of these parameters upon a particular channel

symbol xi. Ignoring this dependence for the moment, the integral in equation (4.28)

can be expressed as Z 1

0r exp

���r2 + �r

�dr: (4.29)

By completing the square and factoring, this integral becomes

exp

�2

4�

! Z 1

0r exp

0@��

r � �

2�

!21A dr: (4.30)

Making the substitution t = r � �2�

breaks this into the sum

exp

�2

4�

!Z 1

� �2�

t exp���t2

�dt+

�

2�exp

�2

4�

!Z 1

� �2�

exp���t2

�dt: (4.31)

112

The �rst integral in this expression is evaluated to equal 12�, while the second can

be placed in a more familiar form. By letting s =p2�t, the second integral in this

expression can be written

�p2�

(2�)3=2

Z 1

� �p2�

1p2�

exp

�s

2

2

!ds: (4.32)

This integral represents the area under a Gaussian pdf with zero mean and unit

variance taken over the interval�� �p

2�;1

�. This fact allows the expression in (4.32)

to be written in the form

�p2�

(2�)3=2

"1

2+Z �p

2�

0

1p2�

exp

�s

2

2

!ds

#: (4.33)

Making the �nal substitution of variables u = sp2results in

�

2

r�

�exp

�2

4�

!"1 +

Z �

2p�

0

2p�exp

��u2

�du

#: (4.34)

The remaining integral is erf�

�2p�

�, where erf(�) is the well-known error function.

Referring back to the integral in equation(4.28), it is evaluated to be

1

2�

"1 +

�

2

r�

�exp

�2

4�

!"1 + erf

�

2p�

!##: (4.35)

Therefore, the conditional pdf of Y given knowledge of X and � is

p(yjxi; �) = 1

2�(�2�jxij2 + �2N )exp

� jyj

2

2�2N

!"1 +

�i2

s�

�iexp

�2i4�i

!"1 + erf

�i

2p�i

!##:

(4.36)

Using this density function along with Bayes' theorem and the law of total probability,

one may obtain the pmf p(xijy; �) = p(yjxi;�)PM

j=1p(yjxj ;�)

. This can then be used to evaluate

the conditional entropy

H(XjY;�) = (4.37)

1M

PMi=1

RRp(yjxi; �)p(�)log

2664PM

j=1

(�2�jxij2+�2N )

�1+

�j2

q��j

exp

��2j

4�j

�h1+erf

��j

2p�j

�i�

(�2�jxj j2+�2N )

�1+

�i2

p��i

exp

��2i

4�i

�h1+erf

��i

2p�i

�i�3775d�dy:

By making the substitution y = rej�xi+n in this expression, averaging over the joint

density p(yjxi; �)p(�) can be replaced by expectation taken over the joint pdf p(�)p(n)

113

Number of Signal Points

4 8 16 32

4.5 - - - 0.0 dB

4 - - - 0.1 dB

3.5 - - 0.1 dB 0.1 dB

AMI 3 - - 0.1 dB 0.2 dB

(Bits/T ) 2.5 - 0.3 dB 0.2 dB 0.2 dB

2 - 0.3 dB 0.3 dB 0.3 dB

1.5 0.5 dB 0.5 dB 0.4 dB 0.4 dB

1 0.7 dB 0.6 dB 0.6 dB 0.6 dB

Table 4.1: Loss of SNR Due to Phase-Only Information: PSK Constellations

of the fading variable and the additive noise variable. The AMI for a discrete-valued

input channel with phase-only information can then be expressed in units of bits/T

as

I(X;Y j�) = log2M� (4.38)

1M

PMi=1Ep(�)p(n)

8>><>>:log2

2664PM

j=1

(�2�jxij2+�2N )

�1+

�j2

q��j

exp

��2j

4�j

�h1+erf

��j

2p�j

�i�

(�2�jxj j2+�2N )

�1+

�i2

p��i

exp

��2i

4�i

�h1+erf

��i

2p�i

�i�37759>>=>>;

where averaging over p(�)p(n) can be easily accomplished by computer simulation.

Since �4�

gets very large for small values of SNR, the term exp��2j4�2j

� �2i4�2i

�should be

factored out in equation (4.38) in order to prevent numerical over ow during computer

simulation.

The expression for AMI in equation (4.38) was evaluated for three types of sig-

nal constellation. The results for PSK constellations are shown in Figure 4.4. The

curves for the case of phase-only information appear to be identical to those for the

ideal channel. Table 4.1 contains the loss in average SNR for various rates of AMI.

In general, the loss for PSK constellations is roughly 0.6 dB at a rate of 1 bit/T

and decreases for higher rates of AMI. At small values of SNR, the unknown fading

amplitude acts in conjunction with the additive noise to degrade performance. This

causes the AMI to be slightly lower than that achieved over the ideal channel, but

114

Figure 4.4: AMI of a Phase-Only Rayleigh Fading Channel: PSK Constellations

115

Number of Signal Points

16 32 64

5.5 - - 13.6 dB

5 - - 9.0 dB

4.5 - 7.2 dB 6.7 dB

4 - 5.1 dB 5.2 dB

AMI 3.5 5.7 dB 3.9 dB 3.9 dB

(Bits/T ) 3 3.1 dB 2.9 dB 3.0 dB

2.5 2.0 dB 2.0 dB 2.2 dB

2 1.5 dB 1.4 dB 1.6 dB

1.5 1.0 dB 1.0 dB 1.1 dB

1 0.8 dB 0.8 dB 0.9 dB

Table 4.2: Loss of SNR Due to Phase-Only Information: QAM Constellations

this e�ect vanishes with increasing SNR.

The standard QAM constellations are the second category of signal set considered.

The AMI curves are shown in Figure 4.5 for various sizes of signal constellation. In

this case there is a dramatic di�erence compared to the ideal fading channel. The

loss in SNR due to lack of knowledge of the fading amplitude is stated in Table 4.2

for various rates of AMI. In general, the loss at 1 bit/T is approximately 1 dB and

increases for higher rates of AMI. The losses at a rate of log2M � 1 bits/T for a

M -point constellation are 5.7 dB, 5.1 dB, and 9.0 dB for M equal to 16, 32, and 64,

respectively. Another noticeable e�ect is that the AMI curves tend to a maximum

value which is less than log2M . This maximum value is in fact approximately equal

to the entropy of the phase of the discrete-valued constellation. For the 16-QAM

signal set, there are 12 di�erent phase values, 8 of which occur with a probability of

0.0625 and 4 of which occur with a probability of 0.125. The entropy of the phase is

easily calculated to be 3.5 bits/T . This can be compared to the AMI of the channel

for a SNR of 40 dB, which equals 3.73 bits/T . Similarly, the entropy of the phase of

the 32-CR and 64-QAM signal sets is 4.75 bits/T and 5.5 bits/T , respectively. The

respective values of AMI for these constellations at a SNR of 40 dB is 4.84 bits/T and

116

Figure 4.5: AMI of a Phase-Only Rayleigh Fading Channel: QAM Constellations

117

Number of Signal Points

8 16 32

4.5 - - 3.2 dB

4 - - 2.8 dB

3.5 - 2.4 dB 2.3 dB

AMI 3 - 2.1 dB 1.7 dB

(Bits/T ) 2.5 1.6 dB 1.7 dB 1.2 dB

2 1.4 dB 1.3 dB 0.8 dB

1.5 1.1 dB 0.9 dB 0.7 dB

1 0.9 dB 0.8 dB 0.67 dB

Table 4.3: Loss of SNR Due to Phase-Only Information: Hybrid AMPM Constella-

tions

5.57 bits/T for 32-CR and 64-QAM. Using these constellations to transmit log2M

bits/T over a channel with phase-only information should result in a noticeable error

oor in performance.

Due to the large losses incurred for high values of AMI when using QAM constel-

lations, it is of interest to determine the achievable rates when the hybrid AMPM

signal sets introduced in section 3.4.1 are used as input to the channel. The AMI

curves for channels using these constellations are shown in Figure 4.6. The loss in

SNR for various �xed values of AMI are shown in Table 4.3. For the 8-point constel-

lations, when the losses due to lack of knowledge of the fading variable are considered,

8-PSK is superior by roughly 0.2 to 0.8 dB for values of AMI in the range of 1 to

2.5 bits/T . The 16-point constellations are more or less equivalent. For rates of AMI

in the range of 1 to 3.5 bits/T , 16-PSK is no more than 0.4 dB better. For the case

of 32 signal points, the two types of constellation are essentially indistinguishable for

rates of AMI in the range of 1 to 4.5 bits/T , with the PSK constellation being no

more than 0.1 dB better.

In practice, it is unlikely that the phase of a 32-PSK constellation could be tracked

e�ectively by a coherent receiver. So for any practical system which attempts to track

the phase but ignores the amplitude of the fading process, the spectral eÆciency will

118

Figure 4.6: AMI of a Phase-Only Rayleigh Fading Channel: Hybrid AMPM Constel-

lations

119

likely be limited to less than 4 bps/Hz. To surpass this rate, it appears that an e�ort

must be made to determine the amplitude of the fading process as well as the phase.

4.2.2 Phase-Only Channels with Continuous-Valued Input

The focus now turns to phase-only information channels with a continuous-valued

input. Since channel capacity is not easily calculated, an upper bound on AMI will

be used instead. By writing the average mutual information of the channel in the

form I(X;Y j�) = H(Y j�) �H(Y jX;�), an upper bound is obtained by separately

bounding each of the entropies in this expression then adding the results. When the

channel output symbol is expressed as the sum y = rej�x + n, the entropy power

bound can be used to limit the entropy power of Y to being less than the sum of

the average power of each term. Notice that EfjRej�Xj2g = 2�2�Es regardless of the

value of �. Using this along with EfjN j2g = 2�2N , which is the average power of the

additive noise, the entropy power bound may be written as

H(Y j�) � lnh2�e

��2�Es + �2N

�i: (4.39)

To obtain a lower bound on H(Y jX;�), the entropy power inequality will be usedin the form

H(Y jX;�) � lnheH(ZjX;�) + eH(N)

i: (4.40)

Since H(N) is known to be ln 2�e�2N , only H(ZjX;�) needs to be determined, where

the variable Z is de�ned to be equal to the product �X. In order to calculate this

entropy, the following result will be used. If � is a complex-valued random variable

and X = x is a constant, then the entropy power of the product �X is [49, p. 233]

H(�X) = H(�) + log jxj2: (4.41)

This is easily proven by examining the expression for entropy and using a change of

variables. With knowledge of this result one can write H(ZjX;�) = H(�XjX;�),and this yields the form

H(ZjX;�) = H(�j�) + Ep(x)nlog jxj2

o: (4.42)

120

If X happens to be a random variable, then equation (4.41) is also averaged over p(x),

which yields the result in equation (4.42). By using Jensen's inequality as was done

in the previous section, one can make the substitution Ep(x) flog jxj2g = log e��JEs,

where �J is the magnitude of the discrepancy in the inequality. Since the fading

variable is � = Rej�, then if the phase � is known at the receiver, all of the uncertainty

is in the fading amplitude R. Therefore, the substitution H(�j�) = H(R) can be

made, where the entropy H(R) is that of a Rayleigh variable. Using the fundamental

de�nition of entropy, one can substitute in the pdf p(r) and calculate

H(R) = 1 + ln��p2+CE

2(4.43)

where CE is Euler's constant. An adjustment to this entropy must be made before

substituting it back into equation (4.42). As was stated earlier, the entropy of a

continuous-valued random variable is dependent upon the coordinate system used.

The expression in equation (4.42) is taken with respect to a Cartesian coordinate

system, whereas the entropy in (4.43) is evaluated in a polar coordinate system. By

using the de�nition of entropy and performing a change of variables, the entropyH(R)can be stated in rectangular coordinates by subtracting the term Ep(r;�)

nlogJ

�polrect

�ofrom the result in (4.43). In this expression, J

�polrect

�represents the Jacobian of the

transformation. In particular J�polrect

�= 1

rand Ep(r;�)

nlog 1

r

o= ln 1p

2��+ CE

2. So in

rectangular coordinates, the entropy of the fading amplitude R is H(�j�) = ln e�2�.

Referring back to equation (4.42), the entropy may be written

H(ZjX;�) = ln e1��J�2�Es: (4.44)

The lower bound on the entropy H(Y jX;�) can be determined from equation (4.40)

to be

H(Y jX;�) � ln

"2�e

e��J

2��2�Es + �2N

!#: (4.45)

Using this result, the upper bound on average mutual information can be written in

bits/T as

I(X;Y j�) � IU = log2

24 1 +

�EsN0

1 + exp(��J)2�

�EsN0

35 (4.46)

where�EsN0

=�2�Es�2N

.

121

This upper bound on AMI is plotted in Figure 4.7 for the case of a Gaussian

distributed input. Also shown is the capacity curve for the ideal Rayleigh fading

channel. The AMI for this continuous-valued input asymptotically approaches a

maximum value which is less than that achieved by some discrete-valued input chan-

nels. This result seems surprising at �rst, since for most practical channel models a

continuous-valued input can be used to attain higher values of AMI. However, upon

further examination of the situation, this result makes perfect sense. Consider the

upper bound in (4.46) as the signal-to-noise ratio gets large. This is estimated to be

log2 2�+�J log2 e. The second term of this sum was obtained for the case of no CSI,

and is an upper bound on the AMI when only the signal amplitude is considered. The

�rst term is the entropy of a uniformly distributed phase variable, and is therefore an

upper bound on the AMI transmitted through the phase of the signal. For the case

of a M -PSK constellation, by choosing M large enough and by using a suÆciently

high SNR, the equivocation of the channel can be made very small, which results in

the AMI approaching a value of log2M . For a continuous-valued phase distribution,

a maximum entropy of log2 2� bits/T is achieved by a uniform distribution. If the

equivocation of the channel is made very small, the maximum AMI in the phase can

be no greater than log2 2� bits/T .

4.3 Realistic Channel Estimation Methods

The remainder of this chapter will deal with fading channels in which the CSI is

provided at the receiver by some practical channel estimation method. The statistics

of the channel estimate are speci�ed by a model which was used in [35] for the purpose

of analyzing the performance of TCM. It is assumed that the fading variable � is a

complex-valued Gaussian random variable. Since the focus will be on a Rayleigh

fading channel, the pdf of � will be the same as that stated in equation (4.1). Let

~� be an estimate of � determined by some realistic channel estimation scheme. The

general form of the channel estimate for the methods considered here can be written

as ~� = ��� + n, where � is a complex-valued scale factor, n is additive noise, and ��

is the value taken on by the fading process at a time duration � prior to when ~� is

122

Figure 4.7: Bounds on AMI for a Phase-Only Rayleigh Channel with Continuous-

Valued Input

123

determined. As usual, the noise is assumed to be Gaussian with a zero mean. Due to

this form, it is reasonable to assume that ~� is also Gaussian with the same zero mean

value as �. If a LOS component exists in the signal, then ~� will have a non-zero

mean that is scaled by �. Based on these assumptions, the channel estimate ~� will

be assumed to be described by the pdf

p(~�) =1

2��2~�exp

�j

~�j22�2~�

!: (4.47)

It is also assumed that � and ~� are jointly Gaussian random variables. The joint pdf

describing this is

p(�; ~�) =1

(2�)2(1� j�j2)�2��2~�exp

� 1

2(1� j�j2)

" j�j2�2�

� 2<f���~��g���~�

+j~�j2�2~�

#!:

(4.48)

The parameter � is the correlation coeÆcient between � and ~�, and is de�ned as

� =C�~�

�2��2~�

, where C�~� = 12En�~��

o. If the value of the random variable ~� is known,

then the conditional pdf of � given ~� = ~� is

p(�j~�) = 1

2�(1� j�j2)�2�exp

0@� j� � ���~�

�~�j2

2(1� j�j2)�2�

1A (4.49)

which is obtained by dividing the joint pdf given in equation (4.48) by the pdf in

equation (4.47). This density function can be used to determine the pdf p(yjx; ~�),which in turn is used to calculate the AMI of the channel. If a new random variable

Z is de�ned to equal the product �X, then given knowledge of X and ~�, the pdf of

Z is

p(zjx; ~�) = 1

2�(1� j�j2)jxj2�2�exp

0@� jz � ���~�x

�~�j2

2(1� j�j2)jxj2�2�

1A (4.50)

which is obtained by scaling the variable �. Since the channel output symbol is

y = z + n, then also given knowledge of the value of N , the pdf of Y is

p(yjx; ~�; n) = 1

2�(1� j�j2)jxj2�2�exp

0@� jy �

���~�x�~�

� nj22(1� j�j2)jxj2�2�

1A : (4.51)

Evaluation of the integralRSNp(yjx; ~�; n)p(n)dn is once again a convolution between

two complex-valued Gaussian pdf's. When this is evaluated, the resulting pdf is

p(yjx; ~�) = 1

2�[(1� j�j2)jxj2�2� + �2N ]exp

0@� jy � ���~�x

�~�j2

2[(1� j�j2)jxj2�2� + �2N ]

1A : (4.52)

124

This probability density function can be used in the calculation of AMI for a chan-

nel with discrete-valued input. Assuming that the distribution of the input alpha-

bet is uniform with p(xi) = 1M

for i = 1; : : : ;M , then the entropy of the input

is simply H(X) = logM . The entropy of the input given knowledge of the out-

put and channel state estimate is H(XjY; ~�) = �E flog p(xijy; ~�)g, where the pmf

p(xijy; ~�) =PMj=1

p(yjxj ;~�)p(yjxi;~�) is obtained from p(yjxi; ~�) through application of Bayes'

theorem and the law of total probability. Substituting equation (4.52) into this en-

tropy expression results in

H(XjY; ~�) = (4.53)

1M

PMi=1

RS~�

RSYp(yjxi; ~�)p(~�) log

"PMj=1

[(1�j�j2)jxij2�2�+�2N ]

[(1�j�j2)jxj j2�2�+�2N ]exp

jy����~�xi

�~�j2

2[(1�j�j2)jxij2�2�+�2N ]

� jy����~�xj�~�

j2

2[(1�j�j2)jxj j2�2�+�2N ]

1A35 dyd~�:

By making the substitution y = �xi + n in equation (4.53), the integration can be

replaced by expectation taken over p(�; ~�)p(n), and can be accomplished by means

of computer simulation. By substituting the required entropies into the di�erence

equation I(X;Y j~�) = H(X)�H(XjY; ~�), the AMI of the channel can be written in

units of bits/T as

I(X;Y j~�) = log2M� (4.54)

1M

PMi=1E

(log2

"PMj=1

[(1�j�j2)jxij2�2�+�2N ]

[(1�j�j2)jxj j2�2�+�2N ]exp

j�xi+n����~�xi

�~�j2

2[(1�j�j2)jxij2�2�+�2N ]

� j�xi+n����~�xj�~�

j2

2[(1�j�j2)jxj j2�2�+�2N ]

1A359=; :

This is the general form of the AMI of a channel with CSI provided by a practical

channel estimation method. When considering a particular type of channel estimation

scheme, slight modi�cations to equation (4.54) may be required. Speci�c channel state

estimation techniques are considered next, where expressions are determined for the

parameters j�j2 and �2~�, as well as the required modi�cations to equation (4.54).

125

4.3.1 Channel Estimation Via Pilot Tone Extraction

One possible method of obtaining an estimate of the fading variable � is to transmit

a known pilot tone along with the data stream. The state of the channel can then

be estimated by observing the e�ect that fading has on the amplitude and phase of

the tone. Only the case of a single pilot tone is considered here, and it is assumed

that the frequency spectrum of the data signal can be shaped in order to create a null

in the center of the transmit spectrum. This is required so that the pilot tone may

be extracted at the receiver. The null is placed in the center of the transmit band

since amplitude and phase characteristics tend to be more stable in this area. It is

also assumed that the bandwidth Wp of the pilot extraction �lter is large enough to

pass the fading process without distortion. This requires that Wp be at least twice

the value of the maximum Doppler frequency. With regard to the analysis performed

here, it is assumed that Wp = 2fD. It is assumed that the received signal is passed

through two disjoint �lter responses. The pilot tone extraction �lter considered is an

ideal �lter with unity gain over the frequency band of width Wp. The data portion

of the signal is assumed to be passed through an ideal �lter with unity gain over the

band W �Wp, where W is the total transmit bandwidth.

For each symbol interval of duration T , the baseband signal A + xg(t) is trans-

mitted, where A is the amplitude of the pilot tone, x is the data symbol, and g(t) is

the transmitted pulse. Let EX be the energy in the data portion of the signal. Over

a symbol interval of duration T , the energy expended in the pilot tone is of the form

A2T . De�ne = A2TEX

to be the ratio of the energy used in transmission of the pilot

tone to that used in the data bearing portion of the signal. The fraction of the total

energy spent on pilot tone transmission is 1+

, and the fraction spent on data is 11+

.

Let Es = (1 + )EX be the total symbol energy used on both data and pilot tone.

The signal arriving at the receiver will be of the form �A+ �xg(t) + n(t), where � is

the value of the fading variable, and n(t) is a sample function of a zero mean additive

noise process, which is assumed to be Gaussian with a constant power spectral density

of N0 over the transmit band. After �ltering and sampling, the output of the pilot

tone extraction �lter will be �A+n1, where n1 has a second moment of 2Wp�2N . The

126

correlation coeÆcient � is calculated from the equation

� =En�~��

orE fj�j2gE

nj~�j2

o : (4.55)

Using ~� = �A + n1 as the channel estimate, it is easy to show En�~��

o= 2�2�A,

E fj�j2g = 2�2�, and Enj~�j2

o= 2�2�A

2+2Wp�2N . By substituting these results back

into equation (4.55), the squared norm of the correlation coeÆcient can be written

j�j2 = 1

1 +2Wp�2NA2

: (4.56)

Since A2 = 1+

EsT, equation (4.56) can also be expressed in the form

j�j2 = 1

1 + 1+ WpT

��EsN0

��1 (4.57)

where�EsN0

=�2�Es�2N

.

In order to determine the AMI of the channel, the variance of the estimate is also

required. The value of 12Enj~�j2

ohas already been determined, however, this is not

the same as �2~� as required by equation (4.54). The variances used in determining

AMI are required in units of watts/Hz, whereas the value of Enj~�j2

ois given in watts.

Thus, the quantity Enj~�j2

omust be normalized by the bandwidth of the transmit

spectrum. By observing this fact, the variance �2~� can be obtained by evaluating

�2~� =Enj~�j2

o2W

= �2�A2T +WpT�

2N : (4.58)

Using the relation A2 = 1+

EsT, this variance can also be written as

�2~� =

1 + �2�Es +WpT�

2N (4.59)

which is the form that is used here to evaluate equation (4.54). When determining

the correlation coeÆcient �, normalization of the variables is an unnecessary consid-

eration, since this coeÆcient is immune to scaling.

Before calculating the AMI for a given channel, one more consideration needs to

be addressed. Consider the term �x + n2, which is the received faded data symbol

plus additive noise. The noise term n2 is independent of n1, since the �lters are

127

assumed to be non-overlapping. Due to the fact that part of the whole bandwidth is

required for pilot tone extraction, the variance of the variable n2 is calculated to beW�Wp

W�2N = (1�WpT )�

2N . Therefore, for the case of pilot tone transmission, equation

(4.54) must be modi�ed by replacing �2N with (1�WpT )�2N . The AMI for a channel

with pilot tone estimation can now be calculated by using the parameters determined

here in the equation

I(X;Y j~�) = log2M� (4.60)

1M

PMi=1E

(log2

"PMj=1

[(1�j�j2)jxij2�2�+(1�WpT )�2N ]

[(1�j�j2)jxj j2�2�+(1�WpT )�2N ]exp

j�xi+n����~�xi

�~�j2

2[(1�j�j2)jxij2�2�+(1�WpT )�2N ]

� j�xi+n����~�xj�~�

j2

2[(1�j�j2)jxj j2�2�+(1�WpT )�2N ]

1A359=; :

A normalized fading bandwidth fDT taking on values of 0.01, 0.03, and 0.06 was

considered for the results presented here. Through computer simulation of equation

(4.60), it was shown that for a given fDT , the AMI was relatively insensitive over a

large range of values to variations in the power distribution parameter . In [35], it

was stated that an optimum value of exists for a given fDT when considering error

performance of TCM. It was suggested that setting to 0.1, 0.25, and 0.3 for fDT

taking on respective values of 0.01, 0.03, and 0.06, resulted in optimum performance.

Since the values of suggested are in the range in which the AMI is roughly equal to

the maximum value, they were used for the computer simulations performed in order

to obtain the results presented here.

The AMI curves for the case of 8-PSK are shown in Figure 4.8, while Figure 4.9

contains similar results for a 16-PSK constellation. Simulations were also performed

for 16-QAM and 32-CR constellations, and the results are shown in Figure 4.10 and

Figure 4.11, respectively. The AMI curves seem to exhibit the same behavior for

all constellations considered, and on observation of the results, two relationships are

evident. The �rst is the e�ect of SNR on the channel estimate. This is evident through

the decreasing gap, for increasing values of SNR, between the results for ideal CSI and

those obtained here for pilot tone extraction. As the SNR becomes greater, the CSI

approaches ideal and the AMI approaches a value of log2M . The second e�ect is that

of frequency dispersion, which manifests itself as an increasing discrepancy compared

128

Figure 4.8: AMI of Rayleigh Fading Channel with CSI Provided Via Pilot Tone

Extraction: 8-PSK

129

Figure 4.9: AMI of Rayleigh Fading Channel with CSI Provided Via Pilot Tone

Extraction: 16-PSK

130

Figure 4.10: AMI of Rayleigh Fading Channel with CSI Provided Via Pilot Tone

Extraction: 16-QAM

131

Figure 4.11: AMI of Rayleigh Fading Channel with CSI Provided Via Pilot Tone

Extraction: 32-CR

132

fDT = 0:01 fDT = 0:03 fDT = 0:06

8-PSK 16-PSK 8-PSK 16-PSK 8-PSK 16-PSK

3.5 - 1.1 dB - 1.7 dB - 2.2 dB

3 - 1.1 dB - 1.7 dB - 2.2 dB

AMI 2.5 1.2 dB 1.1 dB 1.8 dB 1.7 dB 2.2 dB 2.3 dB

(Bits/T ) 2 1.2 dB 1.2 dB 1.8 dB 1.8 dB 2.3 dB 2.4 dB

1.5 1.4 dB 1.3 dB 1.9 dB 1.9 dB 2.6 dB 2.4 dB

1 1.5 dB 1.5 dB 2.1 dB 2.1 dB 2.8 dB 2.8 dB

Table 4.4: Loss of SNR Due to Non-Ideal CSI: Pilot Tone Estimation with PSK

Constellations

to the ideal case corresponding to an increase in Doppler frequency. The magnitude

of the loss compared to a fading channel with ideal CSI is stated for various values of

AMI in Table 4.4 for the PSK constellations, and in Table 4.5 for the case of when

QAM signal sets are used. When a 2k-point signal set is considered for data rates in

the range of 1 to k � 0:5 bits/T , the loss for a normalized Doppler frequency of 0.01

is in the range of 1 to 1.5 dB. As the Doppler frequency increases to 0.03 so does the

loss, which takes on values in the range of 1.5 to 2 dB. For a fDT of 0.06, the loss

compared to the ideal case is in the range of 2 to 2.8 dB.

Although the results presented here indicate that pilot tone extraction is a promis-

ing method of channel state estimation, in practice it is not a very popular method

to use. One reason is due to the extra complexity necessary to transmit and extract

the pilot tone. Another reason is the requirement of shaping the spectrum in order to

create a null for the placement of the pilot tone. As a solution to this second problem,

a method which utilizes two pilot tones placed at the edges of the transmit band is

considered in [57] for use in mobile satellite communications.

4.3.2 Channel Estimation Via Di�erentially Coherent Detec-

tion

Di�erentially coherent detection combined with DPSK modulation is commonly used

in fading environments where time variation of the channel makes estimation of the

133

fDT = 0:01 fDT = 0:03 fDT = 0:06

16-QAM 32-CR 16-QAM 32-CR 16-QAM 32-CR

4.5 - 1.0 dB - 1.5 dB - 2.0 dB

4 - 1.0 dB - 1.5 dB - 2.0 dB

3.5 1.1 dB 1.1 dB 1.6 dB 1.6 dB 2.1 dB 2.1 dB

AMI 3 1.1 dB 1.1 dB 1.6 dB 1.6 dB 2.1 dB 2.1 dB

(Bits/T ) 2.5 1.1 dB 1.1 dB 1.7 dB 1.6 dB 2.1 dB 2.1 dB

2 1.2 dB 1.2 dB 1.8 dB 1.7 dB 2.2 dB 2.2 dB

1.5 1.2 dB 1.2 dB 1.8 dB 1.8 dB 2.3 dB 2.3 dB

1 1.4 dB 1.5 dB 2.0 dB 2.1 dB 2.6 dB 2.7 dB

Table 4.5: Loss of SNR Due to Non-Ideal CSI: Pilot Tone Estimation with QAM

Constellations

134

carrier phase diÆcult. This signalling method can also be viewed as a scheme which

provides information about the fading variable �. Only PSK constellations are dealt

with here, although di�erential encoding of the phase of multi-level constellations has

also been considered [58].

Let xi denote a symbol from a PSK constellation. It is assumed here that the

amplitude is normalized so that jxij = 1. The process of di�erentially encoding the

phase of the channel symbol can be interpreted as a multiplication of two complex

numbers. The di�erentially encoded symbol during the ith transmission interval is

xi = xi�1xi (4.61)

where xi is the PSK symbol to be transmitted during the ith signalling interval, and

xi�1 is the di�erentially encoded symbol transmitted during the previous interval.

During transmission, the symbol xi is a�ected by fading and further corrupted by

additive noise. The received channel symbol is

yi = �ixi + ni: (4.62)

Substituting equation (4.61) into equation (4.62) yields

yi = �ixi�1xi + ni: (4.63)

In this case, it is desired to estimate the quantity �ixi�1. On examination of equation

(4.62), it is observed that yi�1 can serve as a noisy estimate of �ixi�1. Therefore, the

channel state estimate for �ixi�1 is

~�i = �i�1xi�1 + ni�1: (4.64)

The variance of the estimate ~�i is given by 12Enj~�ij2

o, and can be shown to equal

�2~� = �2� + �2N : (4.65)

The quantity 12En�i

~��iois equal to R�(T )x

�i�1, where R�(T ) is the autocorrelation

function of the fading process evaluated for a time di�erence of the channel symbol

duration T . For land mobile radio, the autocorrelation function is modelled by the

equation [29]

R�(�) = �2�J0(2�fD�) (4.66)

135

where fD is the Doppler frequency, and J0(�) is a Bessel function of the �rst kind

and of zero order. Using this result in equation (4.55) allows one to calculate the

squared magnitude of the correlation coeÆcient between the fading variable � and

the estimate ~� to be

j�j2 = J20 (2�fDT )

1 +�

�EsN0

��1 (4.67)

where�EsN0

is de�ned here to equal�2��2N

.

The required parameters necessary to evaluate equation (4.54) have been deter-

mined. However, this expression for AMI can be somewhat simpli�ed. Since it is

assumed that PSK constellations are used, the magnitude jxij of the channel symbolsis the same for all points of the constellation for the system under consideration. It

is also assumed in this case that jxij = 1 for all i. Using this fact, the expression for

AMI can be simpli�ed to

I(X;Y j~�) = log2M� (4.68)

1

M

MXi=1

Ep(�;~�)p(n)

8<:log2

24 MXj=1

exp

0@ j�xi + n� ���~�xi

�~�j2 � j�xi + n� ���~�xj

�~�j2

2[(1� j�j2)�2� + �2N ]

1A359=; :

This expression for AMI was evaluated by means of computer simulation for two

levels of PSK constellation. Results for an 8-PSK signal set are shown in Figure 4.12,

while Figure 4.13 contains the curves for 16-PSK. Three di�erent values were con-

sidered for the normalized Doppler frequency fDT . These values are 0.01, 0.03, and

0.06. On examination of the AMI curves, it is evident at low values of SNR that

the loss is mostly due to the noise term added to the estimate. As the SNR gets

larger, the signi�cant e�ect is due to Doppler spread. As the Doppler frequency in-

creases, the maximum achievable AMI decreases noticeably from a value of log2M .

For normalized Doppler frequencies of 0.01, 0.03, and 0.06, the AMI curves for 8-PSK

reach maximum values of approximately 2.97, 2.80, and 2.36 bits/T , respectively.

The respective values of maximum AMI for the case of 16-PSK are 3.90, 3.38, and

2.35 bits/T . The loss in SNR required for error free transmission compared to the

ideal CSI case is shown in Table 4.6 for various rates of average mutual information.

Considering rates in the range of 1 to k � 0:5 bits/T using a 2k-point constellation,

and a Doppler frequency of fDT = 0:01, the loss is roughly 3.2 to 3.7 dB. When

136

Figure 4.12: AMI of Rayleigh Fading Channel with CSI Provided Via Di�erentially

Coherent Detection: 8-PSK

137

Figure 4.13: AMI of Rayleigh Fading Channel with CSI Provided Via Di�erentially

Coherent Detection: 16-PSK

138

fDT = 0:01 fDT = 0:03 fDT = 0:06

8-PSK 16-PSK 8-PSK 16-PSK 8-PSK 16-PSK

3.5 - 3.7 dB - - - -

3 - 3.3 dB - 6.0 dB - -

AMI 2.5 3.3 dB 3.2 dB 4.9 dB 4.3 dB - 16.2 dB

(Bits/T ) 2 3.2 dB 3.2 dB 3.8 dB 3.8 dB 6.4 dB 6.2 dB

1.5 3.4 dB 3.3 dB 3.6 dB 3.6 dB 4.7 dB 4.7 dB

1 3.7 dB 3.7 dB 3.8 dB 3.8 dB 4.3 dB 4.3 dB

Table 4.6: Loss of SNR Due to Non-Ideal CSI: Di�erentially Coherent Detection with

PSK Constellations

139

fDT = 0:03, the losses are in the range of 3.6 to 4.9 dB for 8-PSK, and from 3.6 to

6.0 dB for 16-PSK. For a normalized Doppler frequency of 0.06, the loss for 8-PSK is

roughly 4.3 to 6.4 dB, while the 16-PSK results show a loss in the range of 4.3 to 16.2

dB. Although the use of di�erential detection is much simpler to implement than the

use of a pilot tone, the performance of the resulting communication system is much

more sensitive to frequency dispersive e�ects.

4.3.3 Channel Estimation Via Pilot Symbol Transmission

As a �nal method of obtaining an estimate of the fading process of the channel, the

transmission of a pilot symbol along with the data is considered. This method is

simple in concept. For a block of NB signalling intervals, the �rst transmission is a

known pilot symbol xp. Since this symbol is known at the receiver, it allows a noisy

estimate of the fading variable � to be obtained during the �rst signalling interval of

the block. This estimate is then used to approximate the fading variable for the next

NB � 1 received channel symbols. The channel estimate for a block of size NB is

~� = �xp + n: (4.69)

A complication arises in analyzing this method of channel estimation, in that

the model parameters are dependent upon the time delay between when the pilot

symbol is transmitted and when the channel estimate is used. For the moment this

consideration will be ignored, and the correlation coeÆcient between the estimate

and the fading variable is determined as a function of the delay. Suppose the fading

estimate is used l symbol intervals after it is obtained. The term 12En�l

~��ois

calculated to be x�pR�(lT ), where R�(lT ) is the autocorrelation function of the fading

process evaluated for a time di�erence of lT . The Bessel function model stated in

equation (4.66) for the case of di�erential detection will be used again here. Assuming

the pilot symbol to have an amplitude A, the quantity 12Enj~�j2

ois determined to

be equal to �2�A2 + �2N . Using these facts in conjunction with equation (4.55), the

squared magnitude of the correlation coeÆcient between the channel estimate ~� and

140

the fading variable � at a time of lT symbol intervals later is determined to be

j�lj2 = J20 (2�lfDT )

1 +�2N

�2�A2

: (4.70)

Let EX be the average energy of those symbols carrying information. De�ne the

parameter = A2

(NB�1)EX to be the ratio of the energy used in the pilot symbol to

that used in the data portion of the transmission. If Es is the average energy per

symbol of a block of NB symbols, of which one is a pilot symbol and the other are

data symbols, it is a simple matter to show that

A2 = NB

1 + Es: (4.71)

By using this in equation (4.70), the squared norm of the correlation coeÆcient is

expressed as

j�lj2 = J20 (2�lfDT )

1 +�1+ NB

� ��EsN0

��1 (4.72)

where�EsN0

=�2�Es�2N

. The variance �2~� is simply equal to �2�A2 + �2N , as shown above.

Using the result in (4.71), and the assumption that the average symbol energy is

normalized so that Es = 1, this variance can be determined from the equation

�2~� =

NB

1 +

!�2� + �2N : (4.73)

All that needs to be dealt with is the time variation of the model. In order to do

this, the AMI will be considered as an average over long sequences of symbols. The

de�nition of average mutual information can be expressed as

I(X;Y j~�) = lim�!1

1

�NB[H(X�NB)�H(X�NB jY�NB ; ~��NB)] (4.74)

where, for example, X�NB represents a sequence of �NB symbols Xl. Under the

assumption of ideal interleaving, a memoryless channel model is obtained, and the

expression in (4.74) can be written

I(X;Y j~�) = lim�!1

1

�NB

24�NBXl=1

I(Xl;Ylj~�l)

35= lim

�!11

�NB

24�NBXl=1

H(Xl)��NBXl=1

H(XljYl; ~�l)

35 :

(4.75)

141

For a block of �NB symbols, the additive terms in equation (4.75) may be divided

into groups of size �. In e�ect, this can be viewed as an average over NB parallel

channels. One of these channels will be used for the transmission of the pilot symbol

xp. Since xp is known, the AMI over this particular channel is zero. For each of

the other parallel channels, the AMI between Xl and Yl conditioned on ~�l may be

determined, where the delay between the fading variable and the estimate is lT . This

allows the AMI to be expressed

I(X;Y j~�) = lim�!1

1

NB

"NXl=1

I(Xl;Ylj~�l)

#=

1

NB

NB�1Xl=1

hH(Xl)�H(XljYl; ~�l)

i: (4.76)

Assuming an equiprobable input distribution, and using the expression forH(XjY; ~�)stated in equation (4.53), the AMI for a channel in which the fading estimate is

obtained through pilot symbol transmission is determined to be

I(X;Y j~�) = NB � 1

NBlog2M� (4.77)

1NBM

PNB�1l=1

PMi=1E

(log2

"PMj=1

[(1�j�lj2)jxij2�2�+�2N ]

[(1�j�lj2)jxj j2�2�+�2N ]exp

j�xi+n��l��~�xi

�~�j2

2[(1�j�lj2)jxij2�2�+�2N ]

� j�xi+n��l��~�xj

�~�j2

2[(1�j�lj2)jxj j2�2�+�2N ]

1A359=; :

The expression for AMI in equation (4.77) was evaluated for a number of di�erent

cases. The constellations considered were 8-PSK and 16-QAM, and the AMI curves

for these signal sets are contained in Figure 4.14 and Figure 4.15, respectively. The

same values of normalized Doppler frequency were considered as for the previous

methods of channel estimation, namely fDT taking on values of 0.01, 0.03, and 0.06.

A block size of NB = 5 was chosen for analysis. This value was used in [59] in

combination with TCM for transmission over a Rician fading channel. An optimum

value of the parameter exists for a given value of NB. In general, as long as jxpj2is of the order of EX , the AMI is essentially near the maximum value. A slight

improvement occurs when jxpj2 = 2EX or more, and a value of = 0:5 was used in

the simulations performed because of this. If jxpj2 is made extremely large, then the

AMI will start to decrease due to the fact that most of the energy is being used to

transmit the pilot symbol rather than the data.

142

Figure 4.14: AMI of Rayleigh Fading Channel with CSI Provided Via Pilot Symbol

Transmission: 8-PSK

143

Figure 4.15: AMI of Rayleigh Fading Channel with CSI Provided Via Pilot Symbol

Transmission: 16-QAM

144

fDT = 0:01 fDT = 0:03 fDT = 0:06

8-PSK 16-QAM 8-PSK 16-QAM 8-PSK 16-QAM

2.5 - 7.2 dB - - - -

AMI 2 8.3 dB 5.5 dB - 11.5 dB - -

(Bits/T) 1.5 5.7 dB 4.8 dB 10.7 dB 7.1 dB - -

1 5.0 dB 4.5 dB 6.6 dB 5.7 dB 27.3 dB 10.3 dB

Table 4.7: Loss of SNR Due to Non-Ideal CSI: Pilot Symbol Transmission

The results for pilot symbol detection indicate that the losses are quite severe

compared to other estimation methods, and the e�ect of Doppler spread is clear. The

maximum achievable AMI tends to a value which is signi�cantly less than log2M

bits/T for a M -point constellation. For a normalized Doppler frequency fDT of 0.01,

the 8-PSK results tend to a value of 2.27 bits/T , while the 16-QAM curve levels

out at 2.99 bits/T . As the Doppler frequency increases to 0.03, the AMI is limited

to approximately 1.70 bits/T for 8-PSK, and 2.21 bits/T for 16-QAM. Maximum

values of 1 bit/T and 1.36 bits/T are reached for a normalized Doppler frequency of

0.06 using 8-PSK and 16-QAM, respectively. Table 4.7 contains the loss in required

SNR for error free transmission compared to the ideal CSI case when pilot symbol

transmission is used. These losses are quite large, and indicate the sensitivity of the

channel estimation method to Doppler e�ects. Additional simulations were performed

in order to determine the best value ofNB for a given value of fDT . When fDT = 0:01,

values of NB in the range of 5 to 8 yield comparable results. When fDT = 0:03, a

block length NB set to 4 yields the best result. For fDT = 0:06, a value of NB = 3

yields the best result. By choosing NB = 3 when fDT = 0:06, the maximum AMI

can be increased by approximately 0.3 bits/T compared to the results presented here

for NB = 5.

Pilot symbol transmission was considered in [59] and [60] for use in combination

with TCM. The results presented indicate that this method is promising when a

LOS component exists. Results presented here indicate that for a Rayleigh channel,

this method is only good for systems which experience small amounts of frequency

dispersion.

145

Chapter 5

Information Theoretic Bounds for

Frequency-Selective Fading Channels

When the e�ects of time dispersion on the performance of a communication system

become prominent, a fading channel is more accurately modelled as being frequency-

selective. In this chapter, the e�ect of time dispersion on the rate of reliable data

communication is investigated through calculation of information theoretic quantities

for frequency-selective fading channels. A signi�cant di�erence in comparison to the

at fading case is that rather than considering only channel symbols, the actual wave-

forms used for communication must be taken into account when performing analysis.

As a basis for the results presented here, a de�nition of average mutual information

and capacity for waveform channels is discussed �rst. The two-path Rayleigh fading

model is used here to account for the e�ect of time dispersion on communication.

Following an examination of the properties of the two-path model, the capacity of a

bandlimited frequency-selective Rayleigh fading channel is determined.

For certain cases of frequency-selective fading, the channel exhibits a time diversity

e�ect. Since the use of diversity usually in uences the performance of a communica-

tion system in a positive manner, the e�ect of this time diversity on the achievable

rate of reliable data transmission is determined for channels with discrete-valued in-

put. This is accomplished through calculation of the average mutual information of

the channel. Channels with continuous-valued input are also considered, and limits

on communication rate are likewise determined.

146

5.1 Representation of Waveform Channels

The channel models considered up to now have been discrete with respect to the

variable of time. For such channels, the statistical behavior of a random signal is de-

scribed by a joint probability distribution de�ned at speci�c instants of time, usually

at equally spaced intervals of the channel symbol duration T . When time dispersion

is present in a communication medium, the energy associated with any given trans-

mitted symbol is spread out over more than just the original pulse duration. Even

when the output of the channel is sampled at intervals of T seconds, the received

symbol is also dependent upon the behaviour of the transmitted signal between sam-

pling instants. Due to this eventuality, the waveforms chosen for communication play

an important role in determining the performance of the system. Thus, any analytic

model which is applied must be based on waveform channels which are continuous

with respect to time.

The �rst problem encountered for such a model is in de�ning the probability

of a continuous waveform. In order to completely describe the stochastic behavior

of a random process, a rule is required which allows the speci�cation of a joint pdf

describing the process observed at any given �nite set of times. An alternate solution,

which is more bene�cial analytically, is to represent signals by a series expansion of

a set of orthonormal waveforms. If a waveform can be completely speci�ed by such a

series, then the signal can be characterized by a joint pdf over the coeÆcients of the

expansion. For example, suppose that any given waveform x(t) from a particular set

X(t) de�ned over the time interval (0; Ts) can be uniquely represented in the form

x(t) =1Xi=1

xi'i(t) (5.1)

where the 'i(t) are a set of orthonormal functions, and the xi are coeÆcients of

the series expansion. Such a family of waveforms can be described for the �rst NB

values of xi by the joint pdf p(x1; : : : ; xNB). If this joint pdf were to completely specify

each waveform in the set, then the class of functions would be described as having NB

degrees of freedom. It is assumed that any sample function y(t) of the channel output

process Y (t) can also be uniquely expanded into a series of orthonormal functions

147

given by

y(t) =1Xi=1

yi'i(t) (5.2)

where the 'i(t) are a set of orthonormal functions which are not necessarily the same

as the set of 'i(t). If the statistics of the �rst NB coeÆcients of this expansion are

described by the joint pdf p(y1; : : : ; yNB), then one may de�ne the AMI between the

two continuous-time random processes X(t) and Y (t) to be

I(X(t);Y (t)) = limNB!1

�Ep(x1;:::;xNB ;y1;:::;yNB )

(log

p(x1; : : : ; xNB ; y1; : : : ; yNB)

p(x1; : : : ; xNB)p(y1; : : : ; yNB)

!):

(5.3)

Using this de�nition of AMI, the capacity of the channel per unit time can then be

de�ned as

C = limTs!1

1

Tssup I(X(t);Y (t)) (5.4)

where the supremum is taken over all possible sets of input waveforms which can be

described by the series expansion. If such a value of C exists, then the information

rate must be kept less than C in order to attain an arbitrary degree of reliability.

In addition, it is noted that examples of channels can be constructed for which such

a value of C exists, but for which arbitrarily low probability of transmission error

cannot be realized at rates below C [27].

Such a model for a continuous-time channel can easily be made mathematically

unstable with regard to convergence of series, uniqueness of the series expansion,

and modelling of the additive noise process. In order to circumvent these problems,

certain constraints are often imposed upon the system. Fortunately, the type of

restrictions imposed to stabilize the model are not arti�cial in nature, but re ect

practical constraints which are encountered in actual communication systems. The

usual constraints assumed to be placed on the system are a �nite limit on the value

of transmitted energy or average power, as well as limits on the time duration of the

signal and the bandwidth of the system.

In order to allow the expansion of a random process into a series of orthonormal

functions, any sample function x(t) of the process X(t) under consideration can usu-

ally be formulated in a manner that allows it to be categorized as being a L2 function.

148

This class of functions has the property that

Z 1

�1jx(t)j2dt <1 (5.5)

which bounds the energy of the signal. If x(t) is expanded into the series given by

equation (5.1), and the set of orthonormal functions 'i(t) are complete over the class

of functions X(t), then the energy can also be expressed as

1Xi=1

jxij2 =Z 1

�1jx(t)j2dt: (5.6)

which is commonly known as Parseval's equation. This relationship allows one to

specify the energy of the signal in terms of the series expansion.

It is usually assumed that the signal x(t) is constrained to some �nite time interval

of length Ts seconds and that the signal takes on a value of zero outside of this interval.

From basic Fourier transform theory, a signal which is �nite in time will be spread

out over the entire frequency domain [50]. In order to also impose a limit on the

bandwidth occupied by the random process, it is assumed that x(t) is passed through

an ideal �lter bandlimited from �W2to W

2Hz. Such an assumption re ects a practical

design consideration which also helps in making the model mathematically tractable.

Since the signal is bandlimited to W2Hz, the sampling theorem states that such a

signal can be uniquely speci�ed by its values taken at intervals of 1W

seconds [33]. By

observing the fact that the signal is also limited in time to an interval of Ts seconds,

one can see that it has WTs degrees of freedom.

Consider an ideally bandlimited additive noise channel where the transmitted sig-

nal is corrupted by a white Gaussian noise process. The received signal is expressed

in the form y(t) = x(t) + n(t), where x(t) is the transmitted signal, and n(t) is a

sample function of the additive noise process. Suppose that the class of functions

considered has NB degrees of freedom. Such a continuous-time channel can then be

represented through orthonormal expansion as NB parallel discrete-time channels,

where the received symbol over the ith channel is yi = xi + ni. The variables ni in

this expression are the coeÆcients of an orthonormal expansion of the noise process,

and are assumed to be modelled as zero mean complex-valued Gaussian variables

with an average power of N0. Subject to a constraint on input power of the form

149

EnR Ts

0 jx(t)j2dto� PTs, the average power of the discrete-time symbols xi can be de-

rived from equation (5.6) to be PTsNB

, where it is assumed that the power is distributed

equally among the parallel channels. The capacity for each parallel channel is

Ci = log�1 +

PTsNBN0

�bps=Hz (5.7)

which is the classical result for a discrete-time AWGN channel. By adding the con-

tributions from each individual parallel channel, the total capacity per unit time of

the waveform channel is determined to be

C =NB

Tslog

�1 +

PTsNBN0

�bps: (5.8)

Since it is assumed that the signal is bandlimited to W2Hz and time limited to Ts

seconds, the class of signals has NB = WTs degrees of freedom, and equation (5.8)

becomes

C =W log�1 +

P

WN0

�bps (5.9)

which is the capacity of an ideal bandlimited AWGN channel.

For more on the modelling of waveform channels, see [27], which contains more

mathematical detail as well as references to works which deal with some of the prob-

lems encountered with this model.

5.1.1 Time-Invariant Filter Channels

While the assumption of an ideally at transfer characteristic for a channel re ects

the imposed bandwidth constraint, the response of any real channel will likely not

be at across all frequencies in the transmit bandwidth. A more accurate model of

a bandlimited channel would include an arbitrary transfer characteristic. In order to

accomplish this, the response of the channel can be modelled as a linear time-invariant

�lter. Based on this idea, the channel can be represented so that the output signal is

y(t) =Zx(�)h(t� �)d� + n(t) (5.10)

where x(t) is the transmitted signal, h(t) is the impulse response of the �lter describing

the channel, and n(t) is a sample function of the additive noise process. It is also

assumed that the input is limited in time to a duration of Ts seconds and has an

150

average power of P watts. In [27], it is shown that the input can be represented by an

orthonormal expansion of a set of functions 'i(t), and the output can be represented

by an orthonormal expansion of a set of functions 'i(t), where 'i(t) is the response

of the �lter h(t) to an input waveform 'i(t). The stochastic behavior of the system

can then be represented by considering the coeÆcients of the expansion. Since only

the �nal result is required here, a heuristic explanation is given for the derivation of

the capacity for a linear time-invariant �lter channel with additive noise.

In order to state the capacity of a linear �lter channel with additive noise, a

description is given here in terms of what occurs in the frequency domain. Assume

that the �lter describing the channel has a frequency transfer characteristic H(f).

The power spectral density function of the input signal X(t) is SX(f), and that of

the additive noise process N(t) is N(f). The additive noise considered here need not

be a white noise process. Suppose that the channel bandwidth W is divided up into

frequency bands of width �f , such that the channel response is essentially at over

each sub-band. Each sub-band is assumed to be centered at a frequency of fi =iTs

for integer values of i. By representing any given input x(t) as a Fourier series with

fundamental frequency 1Ts

and series coeÆcients xi, the output of the ith parallel

discrete-time channel can be approximated to be yi = xijH(fi)j+ni. It is noted here

that the Fourier series representation is not the best orthonormal expansion to use

when considering mathematical limits; however, it is used here only as a tool for the

purpose of illustration. For a mathematically rigorous approach, prolate spheroidal

wave functions are more appropriate [27]. If the output of the ith parallel channel is

normalized by jH(fi)j, then it is equivalent to an additive noise channel where the

variance of the noise is N(fi)jH(fi)j2 . Using the results for the ideal channel, the capacity of

one of these parallel channels which is bandlimited to �f and centered at a frequency

of fi is

C(fi) = log2

"1 +

SX(fi)jH(fi)j2N(fi)

#�f bps (5.11)

where SX(fi) is the amount of transmitted power allocated to the ith channel.

In order to optimally distribute the transmitted power over the entire bandwidth

W , a result which is obtained for parallel discrete-time AWGN channels can be

151

used [27, p. 343]. The optimal distribution of power over all bands is

SX(fi) =

8><>:K0 � N(fi)

jH(fi)j2 for i 2 IW0 for i =2 IW

(5.12)

where IW =ni j fi 2 W; N(fi)

jH(fi)j2 < K0

o, and K0 is chosen so that

Pi2IW SX(fi) = P .

By using this result along with equation (5.11), the capacity of a channel with an

arbitrary �lter response is approximated to be

C � Xi2IW

log

"K0jH(fi)j2N(fi)

#�f: (5.13)

By letting �f ! 0, this discrete approximation becomes continuous in the limit, and

the capacity of a �lter channel with noise is determined to be

C =Zf2FW

log

"K0jH(f)j2N(f)

#df (5.14)

where FW =nf 2 W j N(f)

jH(f)j2 < K0

o, and the level K0 is set so that the total trans-

mitted power is limited toRf2FW SX(f)df = P . The power spectral density of the

capacity achieving input is

SX(f) =

8><>:K0 � N(f)

jH(f)j2 for f 2 FW0 for f =2 FW

: (5.15)

This distribution of the input power is often given a \water-�lling" interpretation.

The meaning of this statement can be explained as follows. If the function N(f)jH(f)j2 is

considered to be the bottom of a vessel, and an amount P of water is poured into this

vessel, the liquid would naturally be distributed in a manner described by equation

(5.15). If it is assumed that jH(f)j = 1 over the band from �W2

to W2

and zero

elsewhere, and that the noise is white with a constant power spectral density of N0,

then application of equation (5.14) would yield the result shown in equation (5.9) for

the ideal �lter channel.

5.2 Capacity of the Two-Path Rayleigh Channel

It would be nice if the results obtained for the time-invariant �lter case could also be

used for time-varying channel responses. In [27], mathematical development of the

152

orthonormal expansion of functions does not depend on the time-invariance of the

�lters. In fact, time-varying �lters are used in the model in order to limit the time

duration of strictly bandlimited signals. In order to use these known results, a number

of assumptions are made here. Suppose that the time-varying �lter can be described

stochastically by some type of probability distribution p(h(t)). The term p(h(t)) is

used to denote the general statistics of the random process h(t). If it is possible for

this response to be determined, then a conditional AMI I(X(t);Y (t)jh(t)) can be

de�ned. Since h(t) is assumed to always be known, this AMI can be determined for

a �xed response of h(t), then averaged over the statistics describing h(t) in order to

determine the conditional AMI.

The two-path Rayleigh channel is described by the impulse response

h(t) = �1(t)Æ(t) + �2(t)Æ(t� �) (5.16)

and is completely speci�ed by �1(t), �2(t), and � . The usual complex-valued fading

variables are written here as random processes in order to illustrate the time-varying

nature of the channel. Since � is assumed to be �xed, the desired quantity to be

determined is I(X(t);Y (t)j�1(t);�2(t)). In order to achieve capacity, given the value

of �1(t) and �2(t) at any �xed time, the spectrum of the input should be that achieved

by the water pouring argument stated in the previous section. As time progresses and

�1(t) and �2(t) change, the spectrum should also be adjusted in order to maintain the

water pouring characteristic. Since no other spectrum results in a greater AMI, this

result yields the capacity of the channel. The assumption that �1(t) and �2(t) are

always known is equivalent to having ideal CSI. In a practical system, this assumption

may be realized by an equalizer which is able to track changes in the channel.

If the response of a time-varying channel is always known, then the capacity may

be determined. In order to obtain actual numbers, however, one must overcome the

diÆculty of averaging over the statistics of h(t). It would be nice if the problem could

be formulated in the same manner as was done for the time-invariant case, in which

random channel responses could be generated and the capacity could be determined

for each �xed response. The capacity could then be obtained from the average of

these results. It is reasonable to assume that h(t) is an ergodic random process and

that the time average is the same as the ensemble average taken with respect to

153

the variables �1 and �2. This could be used to verify the results mathematically.

Rather than attempting this diÆcult approach, a more practical argument is used

here. Another way of looking at this situation is by implementing the assumption of

perfect interleaving. Let the transmitted waveform be represented as

x(t) =Xi

xi(t) (5.17)

where the xi(t) are pulses which are transmitted every T seconds. These pulses

need not be �nite in duration, although this is necessarily the case for a real system.

It is reasonable to believe that if the xi(t) are L2 functions and overlap of these

functions does not cause the the signal to become unbounded, then the signal may

be represented in this manner. If ideal interleaving is used, then at the receiver

these pulses look as though they were a�ected by independent channel responses.

The main point here is that in order to perform averaging over the statistics of the

channel response, one may randomly generate values of �1 and �2 and perform a water

pouring of signal energy for each given channel. The average result of this random

water pouring yields the capacity of the channel.

5.2.1 Properties of the Two-Path Model

In order to apply the water pouring of signal energy to randomly generated channel

responses, there are certain obstacles to overcome. For each randomly generated

channel response, the band of frequencies FB over which the signal spectrum is non-

zero as well as the parameter K0 are variable. This makes it diÆcult to develop an

algorithm to evaluate the capacity through computer simulation.

Fortunately, the two-path model has a high degree of symmetry, and parameters

which are easily calculated from the given values of �1, �2, and � . It is assumed that

the additive noise is a white noise process with a constant power spectral density of

N0 over the bandwidth of interest. The channel SNR function is jH(f)j2N0

, and it is

the inverse of this quantity into which the signal energy is poured. For the two-path

Rayleigh channel, this is

G(f) =N0

�1 + �2 cos 2��f + �3 sin 2��f(5.18)

154

where �1 = j�1j2 + j�2j2, �2 = 2<f�1��2g, and �3 = �2=f�1��2g. The function G(f)is periodic with period ��1. By di�erentiating G(f) with respect to f , the critical

points are determined to occur at

fcrit =1

2��arctan

�3

�2+

i

2�(5.19)

where i is an integer. The value of �2 speci�es whether the value obtained with i = 0

results in a maximum or minimum value of G(f). If �2 is positive, then the maximum

points occur at values of fmax =1

2��arctan �3

�2+ i

�and the minimum points occur at

fmin = 12��

arctan �3�2

+ i�+ 1

2�. If �2 should happen to be negative, then the roles

of fmax and fmin stated here are reversed. The maximum value taken on by G(f) is

N0

j�1j2+j�2j2�2j�1jj�2j , and the minimum value is N0

j�1j2+j�2j2+2j�1jj�2j . This information about

G(f) can be used to create an algorithm to optimally distribute the signal spectrum

for a randomly generated G(f).

In order to perform a computer simulation, the total power of the signal is assumed

to be set to P = 1. The SNR is then determined by the value of the noise power

WN0, whereW2is the bandwidth of the channel. The channel responses are obtained

by randomly generating values of �1 and �2. The delay spread � is chosen to equal

some �xed value. The channel is strictly bandlimited from �W2

to W2

Hz. The

capacity achieving spectrum for a given channel response can be determined as follows.

Since the maximum value of N0

jH(f)j2 can be easily determined, the parameter K0 can

initially be set to this value. IfR W

2

�W2

K0� N0

jH(f)j2df is less than 1, then K0 is chosen to

equal 1W

�1 +

R W2

�W2

N0

jH(f)j2df�, and the set of frequencies FW is the entire bandwidth.

By applying equation (5.14), the capacity conditioned on �1 and �2 is determined

through numerical integration. If setting K0 to the maximum value of N0

jH(f)j2 results

inR W

2

�W2

K0 � N0

jH(f)j2df > 1, then the transmit spectrum must be broken up, and

FW is no longer a continuous set of frequencies. The �rst step is to determine the

location of the maximum points within the bandwidth, as well as the two maxima just

outside either band edge. A bisection algorithm can then be applied to the range of

frequencies between maxima. For each iteration, a new range of FW and value of K0

is determined, and the resulting total average power is reevaluated. This is continued

until the total transmit power is suÆciently close to 1. When this is accomplished, a

155

�nal value of K0 is determined as well as the set of frequencies FW . This is then used

to evaluate equation (5.14).

5.2.2 Capacity and Equalization

The random water pouring algorithm was performed for various values of delay spread

and power distribution between the individual paths of the model. In all cases, the

capacity was equivalent to that of an ideal Rayleigh fading channel. Only a minor

variation occurred at low values of SNR.

Such a consequence might have been guessed by applying Price's result. In [61],

he showed that the capacity of an arbitrary linear �lter channel with additive noise

is the same at high SNR as an ideal �lter channel. In addition, it was shown that

the capacity could be realized at high SNR by adaptive equalization as performed

through the use of precoding or decision feedback equalization. By extending these

results to the case of a fading channel, the capacity for any given �lter response is

determined to be the same as that of a channel with a at transfer characteristic. The

only di�erence is the varying level of attenuation of the equalized channel. Averaging

over randomly generated �lters yields the result for the at fading channel. As long

as variations in the channel can be tracked, use of decision feedback equalization or

precoding should place the capacity of a frequency-selective channel at no more that

2.51 dB from the capacity of an ideal AWGN channel, which is the loss due to fading

experienced by an ideal Rayleigh channel.

5.3 Time Diversity E�ect

One of the drawbacks of the waveform channel model used in the previous section

is that the channel output resulting from a given input of duration Ts is restricted

to be observed over essentially the same length of time. As a consequence of this

model characteristic, the e�ect of intersymbol interference between channel symbols

is essentially ignored. As was mentioned previously, a time diversity e�ect sometimes

manifests itself when time dispersion is present on the channel. In order to analyze

the e�ects of this time diversity, it is necessary to think in terms of discrete signalling

156

intervals. Partitioning the time continuum in this manner allows one to determine

the e�ect that a waveform transmitted during a given interval has on the waveform

observed during subsequent intervals at the receiver.

A particular class of signalling waveforms is investigated here in order to illustrate

the e�ect of this inherent time diversity on the rate of reliable data transmission. In

order to partition the continuous waveform into discrete signalling intervals, the �rst

assumption made is that channel symbols are transmitted at a rate of one symbol

every T seconds, and that received channel symbols are obtained by sampling the

output of the waveform channel at the same rate. This is a reasonable assumption to

make in that any practical system can be viewed at certain points as being discrete

in time. The joint probability distribution between the input and output symbol se-

quences, however, is still dependent upon the continuous-time waveform channel. The

second assumption made is that the signalling waveforms are restricted to the class

of Nyquist pulse waveforms. This means that every information bearing symbol is

transmitted over the waveform channel by means of a continuous-time pulse satisfying

Nyquist's criterion [33]. When a Nyquist pulse is transmitted over a non-distorting

channel with no intersymbol interference, and the received signal is properly synchro-

nized and sampled at a rate of 1Tsamples per second, then the in uence of the symbol

shows up in only one sample. This is due to the fact that a Nyquist pulse takes on

a value of zero at intervals which are integer multiples of T seconds from the time of

occurrence of the peak value.

Consider the consequence of transmitting such a waveform over the two-path

Rayleigh channel when the delay spread is an integer multiple of T seconds. Without

loss of generality, the impulse response for a multipath spread of T seconds is

h(t) = �1Æ(t) + �2Æ(t� T ): (5.20)

Since Nyquist signalling is assumed, each received channel symbol will be of the form

y = �1x + �2~x + n (5.21)

where �1x is a faded version of the information symbol, �2~x is an intersymbol inter-

ference term, and n is additive noise.

157

Any given transmitted symbol x will appear in exactly two received symbols. This

is modelled here by considering two received symbols y1 and y2, where

y1 = �11x + �12~x1 + n1 (5.22)

and

y2 = �21~x2 + �22x+ n2: (5.23)

With regard to y1, the symbol x is the intended information symbol, and in y2 it

appears as an intersymbol interference term. Under the given circumstances, it is of

interest to determine the average mutual information between the transmitted symbol

X and the received random variables Y1 and Y2. In addition, the ideal CSI assumption

is made here. In other words, the value of the fading variables �11, �12, �21, and �22

are all assumed to be known at the receiver. It is also assumed that the terms ~X1

and ~X2 are known at the receiver. Such an assumption can be justi�ed by imagining

the use of di�erential encoding in a manner similar to that performed in the case of

duobinary signalling. Based on all of the above conditions, it is desired to evaluate

the AMI expression of the form I(X;Y1; Y2j�11;�12;�21;�22; ~X1; ~X2).

5.3.1 Channels with Discrete-Valued Input

Examination of the inherent time diversity e�ect on frequency-selective fading chan-

nels is accomplished here through calculation of AMI for channels with discrete-valued

input. In this case, the AMI may be expressed as a di�erence of entropies in the form

I(X;Y1; Y2j�11;�12;�21;�22; ~X1; ~X2) = H(X)�H(XjY1; Y2;�11;�12;�21;�22; ~X1; ~X2)

(5.24)

where H(X) = logM for a M -point signal set with a uniform a priori probability

assignment. In order to determine the entropyH(XjY1; Y2;�11;�12;�21;�22; ~X1; ~X2),

the pmf p(xijy1; y2; �11; �12; �21; �22; ~x1; ~x2) must be determined. As a starting point,

this pmf can be expressed in terms of other more easily obtainable probability density

functions. By applying Bayes' theorem along with the law of total probability, the

desired pmf can be obtained from the expression

p(xijy1; y2; �11; �12; �21; �22; ~x1; ~x2) = p(y1; y2; �11; �12; �21; �22; ~x1; ~x2jxi)p(xi)PMj=1 p(y1; y2; �11; �12; �21; �22; ~x1; ~x2jxj)p(xj)

:

(5.25)

158

Since p(xi) =1M

for i = 1; : : : ;M , this term can be removed from both the numerator

and denominator of the expression. By further factoring the pdf's involved, one

obtains

p(xijy1; y2; �11; �12; �21; �22; ~x1; ~x2) = p(y1jxi; �11; �12; ~x1)p(y2jxi; �21; �22; ~x2)PMj=1 p(y1jxj; �11; �12; ~x1)p(y2jxj; �21; �22; ~x2)

:

(5.26)

Some of the conditioning variables were removed from the pdf's in this expression,

since the random variables described by the pdf's shown are independent of further

conditioning on these additional variables. By starting with the pdf of the additive

noise variable

p(n) =1

2��2Nexp

� jnj

2

2�2N

!(5.27)

and using equations (5.22) and (5.23), the pdf's

p(y1jxi; �11; �12; ~x1) = 1

2��2Nexp

�jy1 � �11xi � �12~x1j2

2�2N

!(5.28)

and

p(y2jxi; �21; �22; ~x2) = 1

2��2Nexp

�jy2 � �22xi � �21~x1j2

2�2N

!(5.29)

are obtained by performing a translation of the variable n. By substituting these

probability density functions into equation (5.26), the desired pmf is expressed in the

form

p(xijy1; y2; �11; �12; �21; �22; ~x1; ~x2) =exp

�� jy1��11xi��12~x1j2+jy2��22xi��21~x2j2

2�2N

�PMj=1 exp

�� jy1��11xj��12~x1j2+jy2��22xj��21~x2j2

2�2N

� :(5.30)

The entropy expression required to determine the AMI can be obtained by evaluating

H(XjY1; Y2;�11;�12;�21;�22; ~X1; ~X2) = �E fp(xijy1; y2; �11; �12; �21; �22; ~x1; ~x2)g, inwhich the expectation is taken with respect to the joint density of all the variables

involved. By using the pmf shown in equation (5.30), this is determined to be

H(XjY1; Y2;�11;�12;�21;�22; ~X1; ~X2) = (5.31)

1M

PMi=1E

�log

�PMj=1 exp

�jy1��11xi��12~x1j2�jy1��11xj��12~x1j2

2�2N

+ jy2��22xi��21~x2j2�jy2��22xj��21~x2j22�2

N

���

159

where the expectation operator shown in this expression is taken over the joint pdf

p(y1; y2; �11; �12; �21; �22; ~x1; ~x2jxi). The values for the received channel symbols y1

and y2 given by equations (5.22) and (5.23) can be substituted back into this entropy

expression. Doing so yields a conditional entropy which can be used in equation (5.24)

to obtain an expression for the average mutual information. The AMI between the

transmitted symbol x and the received symbols y1 and y2 is determined to be

I(X;Y1; Y2j�11;�12;�21;�22; ~X1; ~X2) = log2M� (5.32)

1M

PMi=1E

�log

�PMj=1 exp

�� 1

2�2N

[j�11(xi � xj) + n1j2 � jn1j2

+j�22(xi � xj) + n2j2 � jn2j2])]g :

The expectation operator in this expression is taken with respect to the joint pdf

p(�11)p(�22)p(n1)p(n2). In addition, this expression does not depend upon the vari-

ables �12, �21, ~x1, or ~x2. The fading variables �11 and �22 in this expression are

zero mean complex-valued Gaussian variables with variances of �2�11= 1

2(1+ )and

�2�22=

2(1+ ), respectively. The parameter is equal to the power ratio

�2�22�2�11

. If it is

assumed that E fjXj2g = 1, then the value chosen for �2N de�nes the signal-to-noise

ratio.

Computer simulations were performed in order to evaluate equation (5.32). The

case for when an 8-PSK constellation is used as input to the channel is shown in

Figure 5.1, while the results for 16-QAM are contained in Figure 5.2. When the

parameter is equal to 1, the results are equivalent to those obtained for an ideal

Rayleigh channel with dual antenna diversity. The potential gains due to this time

diversity are shown in Table 5.1 for di�erent rates of AMI and di�erent power distri-

butions between the individual beams of the two-path model. When = 1 and the

two signal paths are equal in strength, the gains are in the range of 0.6 to 2.5 dB for

AMI rates of 1 to log2M � 0:5 bits/T . When = 0:1 there is a 10 dB di�erence in

the received power distribution between the individual paths. For the same rates of

AMI, the gains in this case are in the range of 0.2 to 1.3 dB. When = 0:01 there is

a 20 dB di�erence between the individual paths, and the gains due to time diversity

are negligible at this point.

160

Figure 5.1: AMI for Two-Path Rayleigh Fading Channel with Time Diversity E�ect:

8-PSK

161

Figure 5.2: AMI for Two-Path Rayleigh Fading Channel with Time Diversity E�ect:

16-QAM

162

= 0:01 = 0:1 = 1

8-PSK 16-QAM 8-PSK 16-QAM 8-PSK 16-QAM

3.5 - 0.2 dB - 1.3 dB - 2.5 dB

3 - 0.1 dB - 0.8 dB - 1.6 dB

AMI 2.5 0.2 dB 0.1 dB 1.1 dB 0.5 dB 2.2 dB 1.2 dB

(Bits/T ) 2 0.1 dB 0.1 dB 0.6 dB 0.3 dB 1.4 dB 0.9 dB

1.5 0 dB 0.1 dB 0.4 dB 0.3 dB 0.9 dB 0.7 dB

1 0 dB 0.1 dB 0.2 dB 0.2 dB 0.6 dB 0.6 dB

Table 5.1: Gain Due to Time Diversity E�ect on the Two-Path Rayleigh Channel

163

5.3.2 Channels with Continuous-Valued Input

The gains which result from the time diversity e�ect are determined here for the case

of a channel with continuous-valued input. All of the assumptions stated for the

discrete-valued input case are assumed to be valid here. In order to determine the

capacity, the AMI will be expressed as a di�erence of entropies in the form

I(X;Y1; Y2j�11;�12;�21;�22; ~X1; ~X2) = H(Y1; Y2j�11;�12;�21;�22; ~X1; ~X2) (5.33)

�H(Y1; Y2jX;�11;�12;�21;�22; ~X1; ~X2):

The entropyH(Y1; Y2jX;�11;�12;�21;�22; ~X1; ~X2) can immediately be determined to

be equal to log(2�e�2N )2. By examining equation (5.22) and (5.23), one can see that

if the variables �11, �12, �21, �22, ~x1, and ~x2 are all known, the only uncertainty is due

to the noise variables n1 and n2. Since these variables are modelled as independent

Gaussian variates, each with entropy log(2�e�2N), the joint entropy is simply equal to

twice this value.

In order to determine the entropy H(Y1; Y2j�11;�12;�21;�22; ~X1; ~X2), certain re-

sults taken from Chapter 3 can be used after some simpli�cation. Once again by

examining the expressions shown in equation (5.22) and (5.23), it is obvious that the

additive terms �12~x1 and �21~x2 have no e�ect on the entropy of Y1 and Y2 conditioned

on knowledge of these variables. The problem may be restated in terms of determin-

ing the entropy H(Y1; Y2j�11;�22), where y1 = �11x + n1 and y2 = �22x + n2. If the

variable X is assumed to have a Gaussian distribution, then equation (3.82) may be

used to write

p(y1; y2j�11; �22) = 1

(2�)2 detBY j�exp

0B@�1

2[y�1 y

�2]B

�1Y j�

264 y1

y2

3751CA (5.34)

where BY j� is the covariance matrix of this joint Gaussian pdf. Since this pdf is

Gaussian, one may immediately write the entropy as

H(Y1; Y2j�11;�22) = Ep(�11)p(�22)nlog

h(2�e)2 detBY j�

io: (5.35)

In this case,

detBY j� = �2Nh�j�11j2 + j�22j2

��2X + �2N

i: (5.36)

164

Since the pdf yielding H(Y1; Y2j�11;�22) is Gaussian, this means that the entropy is

maximized and that the resulting AMI is actually an expression for the capacity of

the channel. This capacity is achieved with a Gaussian distributed input. Using the

information gathered here, the capacity is determined to be

C = Ep(�11)p(�22)

(log

"1 + (j�11j2 + j�22j2)�

2X

�2N

#): (5.37)

This capacity can also be expressed in the form

C = Ep(r)

(log

"1 + r2

�2X�2N

#)(5.38)

where r =qj�11j2 + j�22j2. The variances of the zero mean Gaussian fading variables

�11 and �22 are �2�11= 1

2(1+ )and �2�22

= 2(1+ )

, respectively. If = 1, then the

capacity is the same as that determined for the dual antenna diversity case for the

ideal Rayleigh channel. If 6= 1, then the probability distribution of R is described

by the pdf

p(r) =2(1 + )r

1�

"exp

��(1 + )r2

�� exp

�(1 + )

r2!#

for r � 0: (5.39)

Details of the derivation of this pdf can be found in Appendix E. In order to determine

an explicit expression for the channel capacity, one must evaluate the expression

Z 1

0

2(1 + )r

1�

"exp

��(1 + )r2

�� exp

�(1 + )

r2!#

ln

"1 + r2

�2X�2N

#dr: (5.40)

Making the substitution of variables t = 1+ r2�2X

�2N

allows this integral to be expressed

in the form

�2N (1 + )

�2X(1� )

"exp

(1 + )�2N

�2X

! Z 1

1exp

�(1 + )�2N

�2Xt

!ln tdt (5.41)

� exp�(1+ )�2

N

�2X

� R11 exp

�� (1+ )�2

N

�2X

t�ln tdt

�:

The remaining integrals may be viewed in terms of the exponential integral function.

Doing so results in the capacity being determined as

C =log2 e

1�

24 exp

0@1 +

"�EsN0

#�11AEi0@�1 +

"�EsN0

#�11A (5.42)

� exp�(1 + )

h�EsN0

i�1�Ei

��(1 + )

h�EsN0

i�1��bits=T

165

where�EsN0

=2(�2�11

+�2�22)�2X

�2N

. The channel capacity is plotted in Figure 5.3 for various

values of . As was mentioned, when = 1 the results are the same as the dual

antenna diversity case, resulting in a gain of approximately 0.44 bits/T at high values

of SNR. When = 0:1 this gain in capacity is approximately 0.22 bits/T , and when

= 0:01 the gain is a negligible 0.05 bits/T . Unless the signal power on both paths

is comparable, there is not much to be gained in capacity through exploitation of this

time diversity e�ect.

166

Figure 5.3: Capacity of Two-Path Rayleigh Fading Channel with Time Diversity

E�ect

167

Chapter 6

Conclusion

6.1 Summary of Presentation

The practical utility of information theoretic limits as applied to realizable commu-

nication systems is argued in Chapter 1 through an examination of the history of the

telephone line channel. It is pointed out that at one time the information theoretic

capacity of an AWGN channel was considered to be grossly optimistic and extremely

impractical. Today, realistic communication systems approach this theoretic result

to within 3 dB or less. It follows that similar calculations made for multipath fading

channels should also be viewed as reasonably realistic limits. If this is not the case for

any particular situation, then the problem can likely be identi�ed in terms of whether

the channel model used is appropriate. A summary of the state of the art in coding

for fading channels is also given.

Chapter 2 focuses on the fading channel models used in research. A generalized

model is presented �rst, which is based on a physical interpretation of what occurs

when a signal is transmitted over a multipath channel. In order to isolate the e�ects of

amplitude fading, an idealized model is used. Amplitude fading statistics are speci�ed

for a number of di�erent fading situations. The in uence of Doppler spread on various

practical aspects of the channel model is discussed. Time dispersive channels are also

considered, and the modelling of this characteristic is examined. A popular model

for a frequency-selective channel is presented, which due to its simplicity, allows

mathematical analysis to be tractable.

168

The third chapter is intended as a reference for further research dealing with the

design of codes for fading channels. Following a summary of some basic concepts

from the �eld of information theory, limits on the maximum rate of reliable data

transmission are determined for an ideal fading channel. This idealized model is

commonly used in coding research, and the results obtained can be used as an ultimate

benchmark against which to compare code performance. The results are established

in terms of both average and peak transmitted power. Various signal constellations

are considered in order to illustrate the possible trade o� between peak and average

power results. Gains available through the use of space diversity are determined in

terms of information theoretic quantities. The chapter concludes with a statement of

the potential coding gain available on the ideal Rayleigh channel, and a summary of

the best known coding techniques to date.

When the ideal fading channel model is used, it is assumed that the state of the

fading process is known at the receiver. In reality, some scheme must be used to

obtain an estimate of the channel characteristics. Chapter 4 deals with determining

the additional losses in information rate incurred due to the practical limitations of

realistic channel estimation schemes. The necessity of estimating the fading process

is examined �rst, and is accomplished through calculation of limits on the rate of

reliable data transmission when no CSI is available at the receiver. Following this,

information theoretic quantities are determined for a number of practical channel esti-

mation schemes. The particular estimation methods considered are perfect coherent

detection, pilot tone extraction, di�erentially coherent detection, and pilot symbol

transmission. The losses due to non-ideal CSI are determined through comparison

with the results obtained for the ideal channel.

The e�ect of time dispersion on the rate of reliable communication is investigated

in Chapter 5. Application of information theoretic concepts to waveform channels

is discussed �rst. These results are then extended to determine the capacity of a

frequency-selective fading channel speci�ed by the two-path Rayleigh model. A time

diversity e�ect occurs for certain instances of frequency-selective fading. The resulting

gain in the rate of reliable data transmission available through exploitation of this

inherent time diversity is determined for a certain class of waveform channels.

169

6.2 Conclusions

The following list is a synopsis of the principal results adduced in the thesis.

1. When designing coded modulation for an ideal Rayleigh fading channel, a sig-

nal set expansion of 2 is still suÆcient when considering PSK constellations.

For higher levels of QAM, however, an expansion factor of 4 may result in a

signi�cant gain.

2. The capacity of an ideal Rayleigh fading channel with an average power con-

straint is

C = �(log2 e) exp0@" �Es

N0

#�11AEi0@�

"�EsN0

#�11A bits=T: (6.1)

This indicates that the loss in SNR due to Rayleigh fading is no more than 2.51

dB, or equivalently, that the loss in capacity is no more than 0.84 bits/T .

3. For positive integer values of m, the capacity of an ideal Nakagami fading chan-

nel is

C = (log2 e)(�m)m

�(m)

"�EsN0

#�m d

ds

!m�1 �es

sEi (�s)

�bits=T: (6.2)

where s = m�

�EsN0

��1. For positive real values of m, the loss in SNR due to

Nakagami fading is no greater than me� (m), where (�) is Euler's psi function.

4. While PSK is superior with respect to peak power, and QAM is better with

respect to average power, hybrid AMPM constellations show promise in perfor-

mance with respect to both types of constraint.

5. Subject to a peak power constraint, the capacity of an ideal Rayleigh fading

channel is bounded from above by

C � �(log2 e) exp0@" �Ps

N0

#�11AEi0@�

"�PsN0

#�11A bits=T (6.3)

and from below by

C � �(log2 e) exp0@" �Ps

eN0

#�11AEi0@�

"�PseN0

#�11A bits=T: (6.4)

170

The discrepancy between these bounds at high values of SNR is 4.34 dB, or

equivalently, a factor of 1.44 bits/T .

6. The use of space diversity reclaims a signi�cant amount of the loss experienced

due to amplitude fading, even when the fading processes experienced by the

individual antennae are moderately correlated. Most of the gain is achieved by

using 2 or 3 antennae.

7. The capacity of an ideal Rayleigh fading channel with space diversity is the same

as that of an ideal Nakagami fading channel, where the Nakagami parameter

m is set to equal the number of antennae. For example, the capacity of a dual

diversity system is

C = (log2 e)

241 +

0@2

"�EsN0

#�1�11A exp

0@2

"�EsN0

#�11AEi0@�2

"�EsN0

#�11A35 bits=T:

(6.5)

8. The potential coding gain over uncoded QAM in Rayleigh fading increases by

10 dB for each reduction in error probability by a power of 10. For example,

the potential gain is 23.3 dB at a symbol error rate of 10�3, 53.3 dB at an error

rate of 10�6, and 73.3 dB for a symbol error probability of 10�8.

9. When no LOS component exists, the fading process must be estimated in order

to facilitate useable data rates. A Rician channel with R � 10 dB can support

practical data rates without CSI.

10. When the constellation under consideration is large, information transmitted

through the signal amplitude is essentially separable from that transmitted

through the phase. When only the phase of the fading process is determined

at the receiver, the information in the signal amplitude is essentially lost, and

only the information in the signal phase is transmitted reliably. For PSK con-

stellations, phase-only information is adequate.

11. When the phase of the fading process can be determined and the amplitude is

ignored, higher data rates are obtained with a discrete-valued input rather than

with a continuous-valued input.

171

12. Channel estimation by means of pilot tone extraction is essentially ideal for high

values of SNR.

13. Channel estimation by means of di�erentially coherent detection is sensitive to

Doppler spread. This sensitivity increases with larger PSK constellations. The

loss due to Doppler spread can be minimized by considering a constellation

expansion factor of 4 when designing coded modulation.

14. Although pilot symbol detection shows promise for use over channels in which a

LOS component exists, it is extremely sensitive to Doppler spread in Rayleigh

fading environments.

15. Subject to the assumption of being able to track changes in the channel, the

capacity of the two-path Rayleigh channel is the same as that of the ideal at

fading Rayleigh channel.

16. The inherent time diversity due to frequency-selective fading results in an in-

crease of the AMI and capacity of certain waveform channels. This gain can be

as great as that achieved by using dual antenna diversity.

6.3 Suggestions for Further Research

Presented here is a list of issues which merit further consideration.

1. Now that limits on the rate of reliable data transmission over fading channels

have been determined, the next obvious step is to apply these results to the

design of coded modulation for fading channels. Constellation expansion by

a factor of more than 2 shows promise for use in various fading situations.

Investigation of the design and performance of codes based on hybrid AMPM

signal sets for use over non-linear channels may be studied.

2. There is still plenty of work to be done on the modelling of fading channels.

For example, the mobile cellular channel is usually described by a patchwork

of known simple models. For mobile communication in urban areas, a Rayleigh

172

model is often used. However, once a mobile unit moves out of the city envi-

ronment, a Rician model is usually assumed. Changing models in this manner

completely alters the characteristics of the channel. It would be nice if a single

system could be based on a single channel model.

3. Better bounds are needed on information theoretic quantities such as AMI and

entropy. Those based on the notion of entropy power are good for probability

distributions that are close to being Gaussian, but become very weak in other

cases. A good bound for an arbitrary probability distribution is desirable.

4. For the case of non-ideal CSI, bounds on AMI were determined. For certain

input probability distributions, however, it was demonstrated that these bounds

become very weak, and consequently cannot be used as a basis for determining

the capacity of such channels. It is still of interest to determine the capacity of a

channel with non-ideal CSI, as well as the distribution of the capacity achieving

input.

173

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179

Appendix A

Comment on Results Obtained Through

Computer Simulation

Throughout the thesis, it is stated that certain entropy expressions were evaluated

by computer simulation. What this means is that the entropy is presented as a

function of certain random variables, and the statistical expectation of the expression

is evaluated through Monte Carlo averaging. Pseudo-random number generators are

used to obtain the random variables. Expectation is usually taken with respect to

some type of Gaussian distribution. A uniform random variate generator based on the

linear congruential method is used to randomly generate numbers between 0 and 1.

These are then transformed into Gaussian variables by using the polar method. The

expression to be averaged is repeatedly evaluated for di�erent randomly generated

numbers, and the results are added together in an accumulator. The �nal average is

obtained by dividing the contents of the accumulator by the total number of trials.

For the majority of the results obtained, the number of trials performed for a

�xed value of SNR was in the range of 100,000 to 1,000,000. As expressions become

more complex and larger constellations are considered, the duration of a computer

simulation becomes prohibitively long. In some cases, only 20,000 trials were used

due to computer time constraints. Con�dence intervals could be determined by using

a method such as that shown in Appendix A of [62]. However, since the �nal result

for the cases considered is always a number between 0 and 14, the criterion that was

used in judging the quality of the simulation was convergence to at least the second

decimal place of the sample mean.

180

Appendix B

Calculation of Channel Capacity for

Speci�c Fading Distributions

Explicit expressions are derived here for the capacity of ideal Rayleigh and Nakagami

fading channels.

B.1 Capacity of a Rayleigh Fading Channel

The capacity of an ideal fading channel expressed in units of bits/T is given by the

expression

C = Ep(�)

(log2

1 + j�j2�

2X

�2N

!)

= (log2 e)Ep(�)

(ln

1 + j�j2�

2X

�2N

!): (B.1)

For a Rayleigh channel, the fading variable is described by the complex-valued Gaus-

sian pdf

p(�) =1

2��2�exp

� j�j

2

2�2�

!: (B.2)

The expression Ep(�)

�ln�1 + j�j2 �2X

�2N

��can be determined for the Rayleigh case by

evaluating the integral

ZS�

1

2��2�exp

� j�j

2

2�2�

!ln

1 + j�j2�

2X

�2N

!d�: (B.3)

181

By transforming to polar coordinates, where the particular value of the complex

fading variable is represented as � = r cos�+ jr sin�, this integral becomes

1

2��2�

Z 1

0

Z 2�

0r exp

� r2

2�2�

!ln

1 + r2

�2X�2N

!d�dr: (B.4)

After integrating over the phase variable �, the remaining integral is

1

�2�

Z 1

0r exp

� r2

2�2�

!ln

1 + r2

�2X�2N

!dr: (B.5)

Making the substitution t = 1 + r2�2X

�2N

places this integral in the form

�2N2�2��

2X

exp

�2N

2�2��2X

! Z 1

1exp

� �2N2�2��

2X

t

!ln tdt: (B.6)

Using the relation [54, p. 602, 4.331 2.]

Z 1

1e��x lnxdx = � 1

�Ei (��) for <f�g > 0 (B.7)

and the fact that�EsN0

=2�2��

2X

�2N

, the integral in (B.6) can be solved to obtain

Ep(�)

(ln

1 + j�j2�

2X

�2N

!)= � exp

0@" �Es

N0

#�11AEi0@�

"�EsN0

#�11A (B.8)

where Ei(�) is the exponential integral function de�ned by

Ei(x) = �Z 1

�x1

te�tdt for x < 0: (B.9)

Thus, the capacity of an ideal Rayleigh fading channel is given by the expression

C = � (log2 e) exp

0@" �Es

N0

#�11AEi0@�

"�EsN0

#�11A bits=T: (B.10)

B.2 Capacity of a Nakagami Fading Channel

Since Nakagami fading lacks an accompanying phase distribution, it may be assumed

to be uniform over the interval (��; �]. This is usually the case, since small changes

in the channel result in signi�cant variations of the phase. The phase distribution is

not important here anyway, since equation (B.1) only depends upon the magnitude

182

of the fading variable. For the Rayleigh fading case, averaging over the amplitude

distribution is given by equation (B.5). By substituting the Nakagami pdf for the

Rayleigh pdf in equation (B.5), one obtains the integral

2mm

�(m)m

Z 1

0r2m�1 exp

��mr2�ln

1 + r2

�2X�2N

!dr: (B.11)

Making the substitution t =�2X

�2N

r2 changes the form of the integral to

mm

�(m)

�2N�2X

!m Z 1

0tm�1 exp

�m�

2N

�2Xt

!ln(1 + t)dt: (B.12)

A solution for this integral has not been found for arbitrary values of m. However,

if m takes on positive integer values, then an explicit expression for channel capacity

may be obtained. By using the Laplace transform property [55]

Lntm�1f(t)

o= (�1)m�1

d

ds

!m�1F (s) (B.13)

where the functions f(t) and F (s) constitute a Laplace transform pair, and the inte-

gral relation [54, p. 603, 4.337 1.]

Z 1

0e��x ln (� + x) dx =

1

�

hln� � e��Ei (���)

ifor j arg�j < �;<f�g > 0

(B.14)

the capacity of an ideal Nakagami fading channel can be expressed in the form

C = (log2 e)(�m)m

�(m)

"�EsN0

#�m d

ds

!m�1 �es

sEi(�s)

�bits=T: (B.15)

In this equation, the parameter s is de�ned to be equal to m�

�EsN0

��1, where the SNR

relation�EsN0

=�2

X

�2N

is used. For m = 1, this expression yields the capacity of the

Rayleigh channel stated in equation (B.10).

183

Appendix C

Calculation of Asymptotic Loss Due to

Speci�c Fading Distributions

Explicit expressions are derived here for the asymptotic loss in SNR due to ideal

Rayleigh and Nakagami fading.

C.1 Asymptotic Loss Due to Rayleigh Fading

The asymptotic loss due to multipath fading is given by the expression

2�Ep(�)flog2 j�j2g: (C.1)

The base 2 logarithm used in the exponent can be factored to obtain

Ep(�)nlog2 j�j2

o= (log2 e)Ep(�)

nln j�j2

o: (C.2)

Averaging over the Rayleigh fading pdf can be accomplished by evaluating the integral

ZS�

1

2��2�exp

� j�j

2

2�2�

!ln j�j2d�: (C.3)

By transforming to polar coordinates, where � = r cos�+jr sin�, the integral becomes

1

2��2�

Z 1

0

Z 2�

0r exp

� r2

2�2�

!ln r2d�dr: (C.4)

Integrating over the phase variable � leaves

1

�2�

Z 1

0r exp

� r2

2�2�

!ln r2dr: (C.5)

184

Making the substitution t = r2 places the integral in the form

1

2�2�

Z 1

0exp

� t

2�2�

!ln tdt: (C.6)

The solution to this integral is obtained by using the relation [54, p. 602, 4.331 1.]

Z 1

0e��x lnxdx = � 1

�(CE + ln�) for <f�g > 0: (C.7)

The parameter CE is Euler's constant, which is de�ned as

CE = �Z 1

0e�t ln tdt � 0:57721566490 : : : (C.8)

By using the relation given in (C.7), the solution obtained for the expression in (C.6)

is

ln�2�2�

��CE: (C.9)

Since it is assumed that 2�2� = 1 so that the fading has a unity gain, then the desired

solution in this case is

Ep(�)nln j�j2

o= �CE (C.10)

and the asymptotic loss due to fading is

2(log2 e)CE = eCE (C.11)

which is equivalent to 2.51 dB.

C.2 Asymptotic Loss Due to Nakagami Fading

The asymptotic loss due to Nakagami fading can be obtained in the same manner

shown in Appendix C.1 for the Rayleigh channel. By using equation (C.5), and

replacing the Rayleigh pdf with the Nakagami pdf, one obtains

Ep(�)nln j�j2

o=

2mm

�(m)m

Z 1

0r2m�1 exp

��mr2�ln r2dr: (C.12)

After making the substitution t = r2, this expression takes the form

mm

�(m)m

Z 1

0tm�1 exp

��mt�ln tdt: (C.13)

185

The relation [54, p. 604, 4.352 1.]

Z 1

0x��1e��x lnxdx =

1

���(�) [ (�)� ln�] for <f�g > 0;<f�g > 0 (C.14)

can be used to evaluate the integral in equation (C.13). Assuming = 1 so that the

fading gain is set to unity, the solution is

Ep(�)nln j�j2

o= (m)� ln(m) (C.15)

where (�) is Euler's psi function, and is de�ned by the expression

(x) =d

dxln �(x): (C.16)

The function �(�) is known as the gamma function or generalized factorial. The

asymptotic loss due to Nakagami fading is

2log2 e[ln(m)� (m)] = me� (m): (C.17)

For the case m = 1, (1) = �CE, which gives the proper value for the Rayleigh

channel. It is important to note that equation (C.17) is not restricted to integer

values of m, but is valid for any m > 0.

186

Appendix D

Calculation of Error Probability for

Uncoded Modulation in Ideal Rayleigh

Fading

Expressions are derived here for the performance of uncoded modulation used over

an ideal Rayleigh fading channel. An approximation to the symbol error probability

is obtained for QAM constellations. An expression for the bit error probability of

uncoded QPSK is also determined.

D.1 Symbol Error Probability for Uncoded QAM

Suppose that the received channel symbol takes the form Y = �X +N , where X is

a point from a QAM constellation, � is a complex-valued Gaussian fading variable,

and N is a sample of a complex-valued zero mean AWGN process. If a particular

symbol X = xi is an interior point of the constellation where the minimum separation

between QAM symbols is a distance d, then the usual decision region for this point

will be a square of side length d centered on xi. When � = �, the e�ect of the

fading variable in the product �X is a scaling of the coordinates of xi by a factor

r = j�j, as well as a rotation of the point through an angle � = arg � about the origin.

If � is known to equal � at the receiver, then the e�ect of the phase rotation can

be compensated for. After adjusting the phase, the decision region for the symbol

X = xi is chosen as a square of side rd centered on rxi. Given the knowledge that

187

� = � and X = xi, a correct decision is made when the additive noise N does not

cause Y = �X + N to fall outside of the corresponding decision region. Due to

the circular symmetry of the Gaussian pdf used to represent the additive noise, the

probability of making a correct decision is

Pr(correctj�; xi) =Z rd

2

� rd2

Z rd2

� rd2

1

2��2Nexp

�n

2r + n2i2�2N

!dnidnr (D.1)

where nr and ni are the real and imaginary parts of a particular value of the complex-

valued noise variable, respectively. Due to the independence of the real and imaginary

components of the noise variable, as well as the symmetry of the pdf about the origin,

equation (D.1) can be expressed in the form

Pr(correctj�; xi) ="2Z rd

2

0

1p2��N

exp

� n2r2�2N

!dnr

#2: (D.2)

By making the change of variables t = nrp2�N

, this integral can be placed in the form

Pr(correctj�; xi) ="Z rd

2p2�N

0

2p�exp

��t2

�dt

#2: (D.3)

This is equivalent to erf2�

rd2p2�N

�, where erf (�) is the error function. The probability

of making a decision error conditioned on � = � and X = xi is simply

Pr(ej�; xi) = 1� erf2

rd

2p2�N

!: (D.4)

For a M -point square QAM constellation, the average energy per channel symbol is

Es =d2(M�1)

6. By making the substitutions d =

q6EsM�1 and �N =

qN0

2in (D.4), one

obtains

Pr(ej�; xi) = 1� erf2

0@vuut 3r2

2(M � 1)SNR

1A (D.5)

where the SNR is EsN0. For large values of M , one can make the approximation

M � 1 � M where M = 2Rc for a rate of Rc bits per symbol. A normalized signal-

to-noise ratio is de�ned as SNRnorm = 2�RcSNR. By letting = 32SNRnorm, the

conditional error probability is expressed

Pr(ej�; xi) = 1� erf2 (rp ) : (D.6)

188

Assuming the constellation points to be equiprobable, averaging over the pmf p(xi)

results in Pr(ej�) = Pr(ej�; xi). In order to remove the dependence upon �, equation

(D.6) must be averaged over the pdf p(�). Assuming a Rayleigh fading channel, this

is accomplished by evaluating the expression

Pr(e) = 1�Z 1

0

r

�2�exp

� r2

2�2�

!erf2 (r

p ) dr: (D.7)

The complementary error function is de�ned as erfc(�) = 1 � erf(�). This relation,

along with the change of variables s = rp , can be used to write equation (D.7) in

the form

Pr(e) = 1� 1

�2�

"Z 1

0s exp

� s2

2�2�

!ds�

Z 1

02s exp

� s2

2�2�

!erfc(s)ds (D.8)

+Z 1

0s exp

� s2

2�2�

!erfc2(s)ds

#:

Making the change of variables u = s2 for the �rst integral in this expression allows

it to be evaluated as

Z 1

0s exp

� s2

2�2�

!ds =

1

2

Z 1

0exp

� u

2�2�

!du = �2� : (D.9)

By using the integral relation [54, p. 678, 6.287 2.], the second integral is evaluated

to be

Z 1

02s exp

� s2

2�2�

!erfc(s)ds = 2�2�

241� 1q

(2�2� )�1 + 1

35 : (D.10)

The integral relation in [54, p. 941, 8.258 2.] can be used to solve the �nal integral

in the form

Z 1

0s exp

� s2

2�2�

!erfc2(s)ds = �2�

2641� 4

�

arctan�q

1 + (2�2� )�1�

q1 + (2�2� )

�1

375 : (D.11)

After substituting the solved integrals back into equation (D.8), the probability of

detection error is determined to be

Pr(e) = 1� 2p1 + � �1

�1� 2

�arctan

�q1 + � �1

��(D.12)

where � = 2�2� . In terms of the normalized SNR, � = 32SNRnorm where the average

received normalized signal-to-noise ratio isSNRnorm = 2�2�SNRnorm.

189

Due to the edge e�ects of the constellation, this is actually an upper bound on

symbol error probability. However, as the constellation gets larger, this expression

becomes increasingly accurate.

D.2 Bit Error Probability for Uncoded QPSK

Assume that X = xi is a QPSK symbol with coordinates (d; d). The decision region

for this symbol is the �rst quadrant of the Cartesian coordinate system in which it is

represented. The e�ect of a given value of the fading variable � = � is to scale the

coordinates by a factor r and rotate the point through an angle � about the origin. If

the fading variable � is known to equal � at the receiver, then the phase rotation can

be compensated for and the decision regions for all QPSK symbols remain the same.

The only di�erence is that the received symbol will be viewed as having coordinates

(rd; rd). Assuming a Gray mapping of the data bits, a single bit error occurs if the

received symbol falls into an adjacent quadrant, and two bit errors occur when the

received symbol falls into the diagonally opposite quadrant. In terms of the pdf p(n)

of the additive noise variable, this can be expressed as

Pr(bit errorj�; xi) =Z �rd

�1

Z 1

�rdp(n)dn+

Z 1

�rd

Z �rd

�1p(n)dn+ 2

Z �rd

�1

Z �rd

�1p(n)dn:

(D.13)

This can be simpli�ed into the form

Pr(bit errorj�; xi) = 2Z �rd

�1

Z 1

�11

2��2Nexp

�n

2r + n2i2�2N

!dnidnr (D.14)

where nr and ni are the real and imaginary parts of a particular value of the complex-

valued noise variable, respectively. The variable ni vanishes after integrating over the

interval (�1;1), and the remaining integral may be expressed in the form

Pr(bit errorj�; xi) = 1� erf

r

sd2

2�2N

!(D.15)

where erf (�) is the error function. The energy per data bit is Eb = d2 and the noise

power is N0 = 2�2N , so equation (D.15) may also be written in the form

Pr(bit errorj�; xi) = 1� erf

r

sEbN0

!: (D.16)

190

Assuming the QPSK symbols occur with equal probability, the symmetry of the

constellation and the Gray mapping ensure that Pr(bit errorj�) = Pr(bit errorj�; xi).For a Rayleigh fading channel, averaging over the fading variable � requires evaluation

of the expression

Pr(bit error) =Z 1

0

r

�2�exp

� r2

2�2�

!"1� erf

r

sEbN0

!#dr: (D.17)

Using the integral relation in [54, p. 678, 6.287 2.], equation (D.17) is evaluated to

be

Pr(bit error) = 1� 1r1�

��EbN0

��1 (D.18)

where�EbN0

=2�2�EbN0

. This expression yields the probability of bit error per QPSK

symbol. In order to represent the average probability of a data bit being in error,

equation (D.18) must be divided by the number of bits per symbol. After performing

this �nal step, the average bit error probability is determined to be

Pb =1

2

26641� 1r

1��

�EbN0

��13775 : (D.19)

191

Appendix E

Derivation of a PDF Related to the

Two-Path Rayleigh Channel

When considering the two-path Rayleigh channel with Nyquist pulse signalling, a

given transmitted symbol X = x will a�ect only two of the received channel sym-

bols when the delay between the two propagation paths is an integer multiple of the

signalling interval T . In one of these output symbols it appears as an additive term

�1x and in the second it appears in the form �2x, where �1 and �2 are independent

complex-valued Gaussian variables with zero mean. The sum of the squared magni-

tudes of these terms is (j�1j2 + j�2j2)jxj2, and it is of interest to �nd the pdf p(r) of

the random variable R, where R is de�ned by the expression

R =qj�1j2 + j�2j2: (E.1)

By de�ning R1 = j�1j and R2 = j�2j to be the magnitudes of the complex-valued

variables, one may state equation (E.1) in the form

R2 = R21 +R2

2 (E.2)

where

p(r1) =r1�2�1

exp

� r212�2�1

!for r1 � 0 (E.3)

and

p(r2) =r2�2�2

exp

� r222�2�2

!for r2 � 0: (E.4)

192

De�ne two new random variables Z1 and Z2 to be equal to R21 and R

22, respectively. By

performing the required transformation of variables, the probability density functions

given in equations (E.3) and (E.4) can be used to obtain

p(z1) =1

2�2�1

exp

� z12�2�1

!for z1 � 0 (E.5)

and

p(z2) =1

2�2�2

exp

� z22�2�2

!for z2 � 0: (E.6)

Let the random variable Y be de�ned to equal the sum Z1 + Z2. Given knowledge

that the value of Z1 is z1, the conditional pdf

p(yjz1) = 1

2�2�2

exp

�s� z1

2�2�2

!for y � z1 (E.7)

results from equation (E.6) by performing a translation of the variable z2. Through

evaluation of the expressionR y0 p(yjz1)p(z1)dz1, the pdf p(y) is obtained. Assuming

that �2�16= �2�2

, this is easily calculated to be

p(y) =1

2(�2�1� �2�2

)

"exp

� y

2�2�1

!� exp

� y

2�2�2

!#for y � 0: (E.8)

The desired pdf is for the random variable R, which is equal topY . By performing

the appropriate change of variables, one obtains

p(r) =r

(�2�1� �2�2

)

"exp

� r2

2�2�1

!� exp

� r2

2�2�2

!#for r � 0: (E.9)

In order to ensure that E fR2g = 1, the variances of the originating Gaussian dis-

tributions are set to �2�1= 1

2(1+ )and �2�2

= 2(1+ )

, where is the power ratio�2�2�2�1

.

Using this in equation (E.9) places the pdf in the form

p(r) =2(1 + )r

1�

"exp

��(1 + )r2

�� exp

�(1 + )

r2!#

for r � 0: (E.10)

When = 1, this pdf is indeterminate. However, by applying L'Hopital's rule,

one obtains the pdf of a chi distribution with four degrees of freedom. This is the

expected result which would have been obtained if it was assumed that �2�1= �2�2

at

the beginning of the derivation of p(r).

193

Vita

Personal Information

Name: Richard Buz

Date of birth: January 10, 1962

Birth place: Sudbury, Ontario, Canada

Education

1994 Ph.D. in Electrical Engineering (Present Program)

Queen's University, Kingston, Ontario, Canada

1989 M.Sc. in Electrical Engineering

Queen's University, Kingston, Ontario, Canada

1987 B.Sc.(Hons.) in Mathematics and Engineering

Queen's University, Kingston, Ontario, Canada

Awards

1987-88 Queen's Graduate Fellowship, Queen's University

1987 Paithouski Prize, Queen's University

194

Work Experience

1989-94 Computer System Manager

Telecommunications Research Institute of Ontario

1987-89 Teaching Assistant

Department of Electrical Engineering, Queen's University

Publications

� R. Buz, \Uniformity of Non-linear Trellis Codes", in Coded Modulation and

Bandwidth-EÆcient Transmission, edited by E. Biglieri and M. Luise, Amster-

dam: Elsevier, 1992.

� R. Buz, \Signal Mapping and Nonlinear Encoder for Uniform Trellis Codes",

Proc. Globecom'90 Conf., pp. 907.1.1-907.1.7, San Diego, CA, Dec. 2-5, 1990.

� L. Berg, R. Buz, P. J. McLane and M. Turgeon, \Design Procedure for Optimum

or Rotationally Invariant Trellis Codes", Proc. ICC'90 Conf., pp. 607-613,

Atlanta, GA, April 16-19, 1990.

� R. Buz and P. J. McLane, \Error Bounds for Multi-Dimensional TCM", Proc.

ICC'89 Conf., pp. 1360-1366, Boston, MA, June 11-14, 1989.

195