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ADAPTIVE EQUALIZATION AND RECEIVER

DIVERSITY FOR INDOOR WIRELESS DATA

COMMUNICATIONS

a dissertation

submitted to the department of electrical engineering

and the committee on graduate studies

of stanford university

in partial fulfillment of the requirements

for the degree of

doctor of philosophy

By

Yumin Lee

August 1997

c Copyright 1997 by Yumin Lee

All Rights Reserved

ii

I certify that I have read this dissertation and that in

my opinion it is fully adequate, in scope and quality, as

a dissertation for the degree of Doctor of Philosophy.

Donald C. Cox(Principal Advisor)




John M. Cio�




Teresa H. Meng

Approved for the University Committee on Graduate

Studies:

iii

Abstract

Multipath propagation is one of the most challenging problems encountered in a

wireless data communication link. It causes signal fading, delay spread, and Doppler

spread, and can greatly impair the performance of a data communication system.

Multipath mitigation techniques such as adaptive decision-feedback equalization (DFE)

and receiver diversity are thus required for low-error-rate, high-speed wireless data

communications. This dissertation examines these techniques for indoor wireless data

communications. Receiver diversity is known to be an e�ective way of coping with

signal fading. However, indoor wireless radio channels exhibit frequency-selective fad-

ing which introduces inter-symbol interference (ISI), therefore receiver diversity alone

cannot yield satisfactory performance, and more sophisticated signal processing tech-

niques are often required. Adaptive equalization, on the other hand, is known to be

an e�ective measure against ISI. However, adaptive equalization alone cannot miti-

gate the e�ect of signal fading. Integration of diversity and adaptive equalization is

therefore desirable for communication systems such as indoor wireless data networks

which operate in a delay-spread multipath fading environment.

In this dissertation, the e�ects of multipath propagation and their impact on a data

communication system are �rst discussed. A exible baseband model is developed

for indoor wireless communication channels. The adaptive DFE is then treated alone

as an approach for mitigating the e�ect of delay spread. Algorithms for updating

the DFE �lter coe�cients are discussed. These algorithms are classi�ed as channel-

estimation-based adaptation (CEBA) and direct-adaptation (DA). While they have

been compared previously in the literature, in this dissertation new results regarding

their relative performance are obtained using computer simulations that are realistic

iv

for wireless communications. Furthermore, an improved training method referred

to as \synthetic training" is developed and shown to be very e�ective in improving

the performance of the DA DFE. A numerical technique known as \regularization" is

also applied to improve the performance of the channel-estimation-based fractionally-

spaced DFE.

Sampling instant and decision delay optimization, which are crucial to the per-

formance of the adaptive DFE, are also investigated for the adaptive DFE. In this

dissertation, the sampling instant is obtained via a two-step approach from the over-

samples of the received signal. The decision delay is next optimized using the a

priori approach or the a posteriori approach. The a priori approach is evaluated us-

ing previously proposed as well as new, ad hoc optimization metrics. The a posteriori

approach, on the other hand, is �rst demonstrated using an \ideal" technique which

is not realizable. A realizable a posteriori optimization technique, referred to as the

multiple decision delay DFE (MDDDFE), is later developed, and shown to achieve a

performance that is very close to the ideal technique.

Paralleling the discussion on the adaptive DFE, receiver diversity is also presented

alone as a mitigation technique against signal fading. Computer simulation is used

to show that, when used alone, receiver diversity can also signi�cantly improve the

performance of a wireless data communication system. The performance improve-

ments achieved by receiver diversity and adaptive DFE are, however, due to di�erent

reasons. It is therefore very desirable to integrate these two techniques.

The integration of combining and selection diversity with the adaptive DFE is dis-

cussed in detail in this dissertation. The maximal ratio combining DFE (MRCDFE)

is a technique for introducing combining diversity into adaptive DFE, while the se-

lection diversity DFE (SDDFE) is a technique for incorporating selection diversity

into adaptive DFE. For the MRCDFE, the branch DFE �lter coe�cients are jointly

optimized using extensions of the CEBA and DA algorithms. Regularization can also

be applied to improve the performance of the fractionally-spaced MRCDFE. While

the MRCDFE is not new, we obtained new results regarding the relative performance

of the CEBA and DA MRCDFE's, which are consistent with the results we presented

for the single-branch case. For the SDDFE, we developed a new selection rule which

v

is referred to as the maximum a posteriori probability (MAP) selection rule. This

rule is proved to be optimal, in the MAP sense, for a SDDFE. Based on the MAP

selection rule, two new selection metrics are derived and evaluated. Simulation results

show that both the MRCDFE and MAP SDDFE greatly outperform the unequalized

diversity receiver and adaptive DFE without receiver diversity. Furthermore, the new

MAP selection metrics signi�cantly outperform conventional metrics for the SDDFE,

and achieve a performance that is only slightly inferior to the MRCDFE. Since the

branch DFE �lter coe�cients are independently optimized for the SDDFE, it is com-

putationally simpler than the MRCDFE. Adaptive MAP SDDFE is, therefore, an

attractive approach for simultaneously mitigating the impact of signal fading, delay

spread, and small amount of Doppler spread.

vi

Acknowledgments

First and foremost, I thank my advisor, Professor Donald Clyde Cox.

My �rst encounter with Professor Cox was in the Autumn Quarter of 1993, when

I requested to join his research group and was told to try again the next quarter.

When I did, I was again told to wait until the Qualifying Exams (the \Quals") are

over. It was not until after the Quals was I admitted into his research group.

Looking back, joining the \Cox Group" is the best thing that has happened to

me in Stanford University. During the past four years I have acquired from Professor

Cox a great deal of technical knowledge, and a rigorous yet practical attitude towards

research. Furthermore, I have learned from him that many problems cannot be solved

without clearly specifying the assumptions and conditions, therefore \it depends" is

often the best answer to many questions. Working with Professor Cox is indeed a

highly enjoyable, inspiring, and rewarding experience. I feel very honored to have

the chance to work with Professor Cox, and sincerely thank him for his guidance and

support throughout my Ph.D. studies at Stanford University.

I am also deeply indebted to the other members of my orals and reading commit-

tees: Professors John M. Cio�, C. Robert Helms, Teresa H. Meng, and Madihally J.

Narasimha. They scrutinized my research results, made sure that there are no mis-

takes, and gave me many valuable comments. Without them this dissertation would

never have materialized.

I also owe my sincerest gratitude to Dr. David E. Borth of Motorola Incorpo-

rated at Schaumburg, Illinois, and to Paci�c Bell at San Ramon, California, for their

generous support over the course of this research. In the past four years Motorola

has provided for my tuition and stipend. Furthermore, Dr. Borth has taken time to

vii

review every progress report and publication that resulted from this research. Paci�c

Bell, on the other hand, has donated computer equipments which were extensively

used to produce simulation results for this dissertation.

This acknowledgment would not be complete without mentioning the former and

current members of my research group: Bora Akyol, Sung Chun, Hideaki Haruyama,

Kerstin Johnsson, Byoung-Jo Kim, Dae-Young Kim, Matthew Kolz, Persefoni Kyritsi,

Derek Lam, Andy Lee, Angel Lozano, Ravi Narasimhan, Tim Schmidl, Mehdi Soltan,

Je� Stribling, Qinfang Sun, Karen Tian, Bill Wong, and Daniel Wong. I have greatly

enjoyed and bene�ted from our stimulating discussions and interactions. I will always

remember the bitter-sweet memories that we share { especially the two hard-disk

crashes and two computer break-in incidents. I certainly am very fortunate to have

the opportunity to know and work with all of you. Now that some of us are graduating

and some are still working towards the Ph.D. degree, I wish you best of luck in your

careers and studies. I am sure our paths will cross again in the future. I am also

sincerely grateful to our former and current assistants { Jenny Beltran, Lily Huan,

and Marli Williams, for their most helpful administrative support.

During my Ph.D. studies, many friends in Stanford University have helped me

in course work and preparation for the Quals. They are Navin Chaddha, Jonathan

Chang, Kenneth Chang, Jiunn-Tsair Chen, Pei-chun Chiang, Jimmy Chuang, Suhas

Diggavi, Min-Chen Ho, Winston Lee, Jenwei Liang, Chun-Yi Liao, Janray Liao, Ming-

Chang Liu, Chung-Li Lu, Hui-Ling Lu, Zartash Uzmi, Yao-Ting Wang, Clive Wu,

Tien-Chun Yang, and many others. Thanks to them I have cleared many hurdles to

reach the present stage.

In closing I would like to mention some of my oldest and best friends: Yea-Hueay

Chang of Northwestern University, Ju-Hsien Kao and Hsing-Chuan Su of Stanford

University, and Hao-Hsuan Chiu of Syracuse University. Yea-Hueay is, and always will

be, a very special part of my life. Ju-Hsien is from a completely di�erent background

{ mechanical engineering, yet he sat through my oral defense without falling asleep.

Hao-Hsuan is a very good sounding board and travel companion. Thank you for your

love, support, encouragement, and friendship.

I dedicate this dissertation to my parents and brother.

viii

Contents

Abstract iv

Acknowledgments vii

1 Introduction 1

1.1 Dissertation Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2 Multipath Propagation 6

2.1 Signal Fading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.2 Delay Spread . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.3 Doppler Spread . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.4 Baseband Indoor Wireless Channel Model . . . . . . . . . . . . . . . 17

2.4.1 Simulated Power-Delay Pro�le . . . . . . . . . . . . . . . . . . 20

2.4.2 Simulated Doppler Spectrum . . . . . . . . . . . . . . . . . . . 20

2.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3 Adaptive Decision-Feedback Equalization 24

3.1 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.2 Inter-Symbol Interference . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.3 Adaptive DFE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3.3.1 Channel Estimation Based Adaptation . . . . . . . . . . . . . 33

3.3.2 Direct Adaptation . . . . . . . . . . . . . . . . . . . . . . . . 37

3.4 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

ix

3.4.1 Synthetic Training for DA DFE . . . . . . . . . . . . . . . . . 41

3.4.2 Comparison of CEBA and DA DFE . . . . . . . . . . . . . . . 46

3.5 CEBA DFE with Regularization . . . . . . . . . . . . . . . . . . . . . 49

3.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

4 DFE Timing Alignment 54

4.1 Sampling Instant Optimization . . . . . . . . . . . . . . . . . . . . . 55

4.2 Decision-Delay Optimization . . . . . . . . . . . . . . . . . . . . . . . 58

4.2.1 A Priori Optimization . . . . . . . . . . . . . . . . . . . . . . 58

4.2.2 A Posteriori Optimization . . . . . . . . . . . . . . . . . . . . 60


4.3.1 Sampling Instant Optimization . . . . . . . . . . . . . . . . . 61

4.3.2 Decision Delay Optimization . . . . . . . . . . . . . . . . . . . 64

4.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

5 Receiver Diversity 71

5.1 Combining Diversity DFE . . . . . . . . . . . . . . . . . . . . . . . . 74

5.1.1 CEBA MRCDFE and Regularization . . . . . . . . . . . . . . 75

5.1.2 DA MRCDFE with Synthetic Training . . . . . . . . . . . . . 80


5.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

6 MAP Selection Diversity DFE 88

6.1 Selection Diversity DFE . . . . . . . . . . . . . . . . . . . . . . . . . 92

6.2 MAP Selection Metric for SDDFE . . . . . . . . . . . . . . . . . . . . 93

6.3 Computation of Selection Metric . . . . . . . . . . . . . . . . . . . . . 96


6.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

7 Multiple Decision Delay DFE 107

7.1 Multiple Decision Delay DFE . . . . . . . . . . . . . . . . . . . . . . 109

7.2 The DFE Controller . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

x


7.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

8 Conclusions 118

8.1 Dissertation Summary . . . . . . . . . . . . . . . . . . . . . . . . . . 118

8.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

A Baseband Equivalent Power-Delay Pro�le 126

B Baseband Equivalent Doppler Spectrum 128

C The Least-Squares Lattice DFE 130

D Optimality of MAP Selection Rule 136

Bibliography 138

xi

List of Tables

2.1 Channel parameters used throughout this dissertation. . . . . . . . . 19

7.1 Average complexity of one- and two-branch MDDDFE using SERDFE

for d = 0:5 and average SNR of 10, 15 and 20 dB. The average com-

plexity without pruning is also shown. The channel has a Gaussian

power-delay pro�le with a rms delay spread of 50 ns and average delay

of 200 ns. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

C.1 The dimensionalities of the variables in the LSL DFE algorithm. . . . 135

xii

List of Figures

2.1 Multipath propagation in indoor environments. The signal transmitted

by the transmitter (T) is attenuated and re ected by the walls and

oors. As a consequence the receiver (R) receives multiple distorted

copies of the transmitted signal. . . . . . . . . . . . . . . . . . . . . . 7

2.2 Normalized plots of the Ricean distribution. The numbers labeled on

the curve denote values of �pP. � = 0 corresponds to the Rayleigh

distribution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.3 (a) Normalized exponential power-delay pro�le. (b) Normalized Gaus-

sian power-delay pro�le. �� = 4S in this plot. . . . . . . . . . . . . . . 13

2.4 Illustration of Doppler shift in the free-space propagation environment.

The receiver moves at a constant velocity v along a direction that forms

an angle � with the incident wave. . . . . . . . . . . . . . . . . . . . 14

2.5 The Doppler spectrum corresponding to uniformly distributed angles

of arrival (see Equation 2.16). . . . . . . . . . . . . . . . . . . . . . . 16

2.6 Simulated power-delay pro�les obtained from 200 replications. (a) The

path delays are exponentially distributed with a standard deviation of

50 ns. (b) The path delays are Gaussianly distributed with a mean of

200 ns and standard deviation of 50 ns. . . . . . . . . . . . . . . . . . 21

2.7 Simulated Doppler spectrum obtained from 200 replications. The Doppler

shift frequencies are generated according to Table 2.1 . . . . . . . . . 22

3.1 Baseband model of a wireless data communication system. . . . . . . 25

xiii

3.2 Examples of jp(t)j. (a) When the channel delay-spread is insigni�cant,

very little ISI is introduced. (b) When the channel delay-spread is

signi�cant, the ISI is severe. . . . . . . . . . . . . . . . . . . . . . . . 30

3.3 Block diagram of an adaptive decision-feedback equalizer. . . . . . . . 31

3.4 The channel-estimation-based adaptation DFE. . . . . . . . . . . . . 37

3.5 Prepending the STS to the received waveform for synthetic training

for the DA DFE. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

3.6 The average BER of the symbol-spaced (K = 1) DA DFE with syn-

thetic training. The channel is assumed to have a Gaussian power-delay

pro�le with a rms delay-spread of 50 ns and average delay of 200 ns.

The average SNR is 15 dB. Di�erent training sequence lengths Q and

STS lengths Qs are used. . . . . . . . . . . . . . . . . . . . . . . . . . 42

3.7 Average BER of the symbol-spaced (K = 1) DA DFE using various

CPR estimate lengths for synthetic training. �1 and �2 are both set

to be equal to the value of the abscissa. The channel has a Gaussian

power-delay pro�le with a normalized delay-spread of 0.5. The rms

delay-spread and average delay are 50 ns and 200 ns, respectively. The

average SNR is 15 dB. Synthetic training with Q = Qs = 15 is used. . 44

3.8 The average BER of the half-symbol-spaced (K = 2) DA DFE with

synthetic training. The channel is assumed to have a Gaussian power-

delay pro�le with a rms delay-spread of 50 ns and average delay of 200

ns. The average SNR is 15 dB. Di�erent training sequence lengths Q

and STS lengths Qs are used. . . . . . . . . . . . . . . . . . . . . . . 44

3.9 Average BER of the half-symbol-spaced (K = 2) DA DFE using vari-

ous CPR estimate lengths for synthetic training. �1 and �2 are both set

to be equal to the value of the abscissa. The channel has a Gaussian

power-delay pro�le with a normalized delay-spread of 0.5. The rms

delay-spread and average delay are 50 ns and 200 ns, respectively. The

average SNR is 15 dB. Synthetic training with Q = Qs = 15 is used. . 45

xiv

3.10 Average BER of CEBA and DA DFE's: (a) as a function of average

SNR at d = 0:5; (b) as a function of d at 15dB average SNR. The

channel is assumed to have a Gaussian power-delay pro�le with a rms

delay-spread of 50 ns and average delay of 200 ns. . . . . . . . . . . 47

3.11 Average BER of the half-symbol-spaced (K = 2) CEBA DFE with

regularization: (a) as a function of average SNR at d = 0:5; (b) as a

function of d at 15dB average SNR. The average BER's CEBA without

regularization and DA with K = 2 are also shown. The channel is

assumed to have a Gaussian power-delay pro�le with a rms delay-

spread of 50 ns and average delay of 200 ns. . . . . . . . . . . . . . . 51

4.1 Block diagram of an adaptive decision-feedback equalizer. . . . . . . . 54

4.2 The average BER of the symbol-spaced DA DFE using di�erent sam-

pling instant optimization techniques. The channel has a Gaussian

power-delay pro�le with a rms delay-spread of 50 ns and average delay

of 200 ns. The average SNR is 15 dB. . . . . . . . . . . . . . . . . . . 62

4.3 The average BER of the half-symbol-spaced DA DFE using di�erent

sampling instant optimization techniques. The channel has a Gaussian

power-delay pro�le with a rms delay-spread of 50 ns and average delay-

spread of 200 ns. The average SNR is 15 dB. . . . . . . . . . . . . . . 64

4.4 Average BER as a function of average SNR for di�erent decision delay

optimization schemes, �xed delay and ideal cases. The channel has

a Gaussian power-delay pro�le with normalized delay-spread d = 0:5.

Symbol-spaced DA DFE's are used. . . . . . . . . . . . . . . . . . . . 65

4.5 Average BER as a function of normalized delay-spread for di�erent

decision delay optimization schemes, �xed delay and ideal cases. The

channel has a Gaussian power-delay pro�le with an average SNR of 15

dB. Symbol-spaced DA DFE's are used. . . . . . . . . . . . . . . . . 66

xv

4.6 Average BER as a function of average SNR for di�erent decision delay

optimization schemes, �xed delay and ideal cases. The channel has

a Gaussian power-delay pro�le with normalized delay-spread d = 0:5.

Half-symbol-spaced DA DFE's are used. . . . . . . . . . . . . . . . . 67

4.7 Average BER as a function of normalized delay-spread for di�erent

decision delay optimization schemes, �xed delay and ideal cases. The

channel has a Gaussian power-delay pro�le with an average SNR of 15

dB. Half-symbol-spaced DA DFE's are used. . . . . . . . . . . . . . 68

5.1 Block diagram for selection diversity. One diversity branch is selected

according to some selection rule. . . . . . . . . . . . . . . . . . . . . 73

5.2 Block diagram for combining diversity. The received signals are am-

pli�ed, co-phased and summed. . . . . . . . . . . . . . . . . . . . . . 74

5.3 The maximal-ratio combining DFE. . . . . . . . . . . . . . . . . . . . 75

5.4 Average BER's of the unequalized diversity combiner (DIV-ONLY),

half-symbol-spaced DA DFE without diversity (DFE-ONLY), and dual

diversity (L = 2) DA and CEBA MRCDFE's: (a) as functions of

average SNR at d = 0:5; and (b) as functions of d at 15dB average

SNR. The channel has a Gaussian power-delay pro�le with a rms delay-


5.5 Average BER's of the half-symbol-spaced DA DFE without diversity

(DFE-ONLY), dual diversity (L = 2) DAMRCDFE, and dual diversity

(L = 2) CEBA MRCDFE with regularization: (a) as functions of

average SNR at d = 0:5; and (b) as functions of d at 15dB average

SNR. The channel has a Gaussian power-delay pro�le with a rms delay-


6.1 Possible structures for the selection diversity DFE: (a) Selection is done

before equalization; (b) Selection is done after equalization. . . . . . . 89

6.2 Possible selection schemes for the SDDFE. . . . . . . . . . . . . . . . 90

6.3 Selection-diversity decision-feedback equalizer. . . . . . . . . . . . . . 90

xvi

6.4 Average BER's of the various selection metrics as functions of the

average SNR. The channel has an exponential power-delay pro�le with

a rms delay-spread of 50 ns. The normalized delay spread d is 0.5. As

noted in the text, the IDEAL case is not realizable. . . . . . . . . . . 100

6.5 Conditional probability of correct branch selection given that exactly

one diversity branch decision is wrong. The channel has an exponential

power-delay pro�le with a rms delay-spread of 50 ns. The normalized

delay-spread d is 0.5. . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

6.6 Average BER's as functions of the normalized delay-spread d. The

channel has an exponential power-delay pro�le with 50 ns rms delay-

spread. The average SNR is 15 dB. The IDEAL case is not realizable

as noted in the text. . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

6.7 Average BER's of the various selection metrics as functions of the

average SNR. The channel has a Gaussian power-delay pro�le with a

rms delay-spread of 50 ns and average delay of 200 ns. The normalized

delay spread d is set to 0.5. The IDEAL case is not realizable as noted

in the text. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

6.8 Average BER's as functions of the normalized delay-spread d. The

channel has a Gaussian power-delay pro�le with a rms delay-spread of

50 ns and average delay of 200 ns. The average SNR is 15 dB. The

IDEAL case is not realizable as noted in the text. . . . . . . . . . . . 105

7.1 Structure of the MDDDFE. . . . . . . . . . . . . . . . . . . . . . . . 109

7.2 Algorithm for pruning redundant DFE's. This algorithm is repeated

once every R symbols. . . . . . . . . . . . . . . . . . . . . . . . . . . 111

7.3 Average BER as a function of average SNR at d = 0:5 for the MD-

DDFE, �xed decision delay, and ideal cases. Half-symbol-spaced DA

MDDDFE's used here. The channel has a Gaussian power-delay pro�le

with a rms delay-spread of 50ns and average delay of 200 ns. . . . . . 114

xvii

7.4 Average BER as a function of normalized delay-spread at average SNR

of 15dB for the MDDDFE, �xed decision delay and ideal cases. Half-

symbol-spaced DA MDDDFE's used here. The channel has a Gaussian

power-delay pro�le with a rms delay-spread of 50ns and average delay

of 200 ns. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

C.1 (a) The block diagram of a LSL DFE. Matrix weights are not explicitly

shown. The block labeled \D" denotes a unit-sample delay; (b) The

block diagram of a LSL stage. Matrix weights are not explicitly shown.

The block labeled \D" denotes a unit-sample delay. . . . . . . . . . . 131

xviii

Chapter 1

Introduction

With recent advances in communications, signal processing, and computer technolo-

gies, the dream of wireless networking for data communication systems has become an

achievable goal. One attractive application of wireless networking is the indoor wire-

less local area network (WLAN). First, since it allows for mobility of users, re-wiring

is unnecessary when a user of a WLAN moves. This can be especially important for

users of portable data terminals. Secondly, since the WLAN operates in indoor en-

vironments, high-speed data transmission is possible without requiring an unrealistic

amount of transmitter power. This is an important aspect for many data applications.

One of the most important building blocks for an indoor WLAN is the wireless

data communication link. For a WLAN to function properly, reliable wireless data

communication links must �rst be established. Multipath propagation is one of the

most challenging problems encountered in a wireless data communication link. In a

multipath propagation environment, the transmitted electro-magnetic signal propa-

gates to the receiver via many di�erent paths. In general, these propagation paths

have di�erent amplitude gains, phase shifts, angles of arrival, and path delays that

are functions of the re ection structure of the environment. The e�ects of multipath

propagation include signal fading, delay spread, and, when there is relative motion

between the transmitter and receiver, Doppler spread. Signal fading refers to the phe-

nomenon that in a multipath propagation environment, the received signal strength

is strongly dependent on the locations of the transmitter and receiver. This is caused

1

CHAPTER 1. INTRODUCTION 2

by the interference between signals propagating through di�erent paths. Delay spread

refers to the spread of the duration of the received signal with respect to the trans-

mitted signal. This is due to the di�erent delays associated with the propagation

paths. Delay spread introduces inter-symbol interference (ISI) in a digital wireless

communication system, which limits the achievable transmission rate. It also causes

di�culties for symbol-timing recovery in a digital demodulator. Doppler spread, on

the other hand, refers to the spread of the frequency spectrum of the received signal

with respect to that of the transmitted signal, when there is relative motion between

the transmitter and the receiver. This is due to the di�erent angles of arrival associ-

ated with the propagation paths. Since the spectrum of the received signal is wider

in frequency than that of the transmitted signal, the multipath propagation channel

is clearly a time-varying system. Adaptive signal processing techniques are there-

fore required to track channel variations for a mobile digital wireless communication

system operating in a multipath propagation environment.

This dissertation discusses the digital signal processing techniques that can be

used to mitigate the impact of multipath propagation on the performance of a point-

to-point single-user indoor high-speed wireless data communication link. Adaptive

decision-feedback equalization is a solution for mitigating delay spread and Doppler

spread. In this technique, adaptive linear discrete-time �nite impulse response (FIR)

�lters are used to process the received signal. The FIR �lters can be optimized directly

using the received signal samples, or indirectly through channel estimation. Since the

adaptive equalizer operates in discrete-time, the sampling instants must be deter-

mined before equalization starts. Furthermore, since FIR �lters are used, a design

parameter referred to as the \decision delay" must also be determined. The optimiza-

tion of the sampling instants and decision delay is referred to as \timing alignment" in

this dissertation. This dissertation compares the performance of di�erent adaptation

techniques using realistic computer simulation, and also investigates di�erent timing

alignment approaches for adaptive decision-feedback equalizers (DFE's).

Receiver diversity, which makes use of multiple receiver antennas, is a technique

used to combat signal fading. If the receiver antennas are spaced far enough apart (on

the order of a half-wavelength), the wireless channels corresponding to these antennas


will be approximately uncorrelated. In this case it is unlikely for both antennas to si-

multaneously experience a deep signal fade. The received signals at the output of the

branch antennas can therefore be combined either by taking their weighted average

(combining diversity) or by simply choosing the \best" (selection diversity). Receiver

diversity has been shown to be very e�ective against fading. However, it alone is not

very e�ective in coping with delay spread, and ways to incorporate adaptive equal-

ization into receiver diversity are highly desirable. This dissertation discusses novel

approaches for introducing diversity into the adaptive DFE in order to simultane-

ously mitigate the three di�culties caused by multipath propagation. In particular,

an optimal selection diversity scheme is derived for adaptive DFE's. Simulation re-

sults show that the new scheme discussed in this dissertation can indeed outperform

conventional selection diversity schemes.

1.1 Dissertation Outline

Chapter 2 of this dissertation discusses multipath propagation. Small-scale e�ects of

multipath propagation are brie y described, and a simple baseband model is devel-

oped for use in subsequent chapters.

The adaptive DFE is discussed in Chapter 3. The DFE can be optimized directly

from the received signal samples, or indirectly through channel estimation. These

two approaches have been compared by previous researchers under certain assump-

tions that are not applicable to wireless data communication links. In this chapter,

we compare the performance of these two approaches using realistic computer simu-

lation over a broad range of wireless channel conditions. Performance enhancement

techniques are also investigated for both approaches.

Chapter 4 provides a treatment of timing alignment for adaptive DFE's. The

timing alignment parameters are de�ned mathematically, and practical approaches

for optimizing these parameters are discussed.

In Chapter 5, the concept of receiver diversity is introduced. It is demonstrated

here that, while diversity alone is very e�ective against fading, it is not very e�ective

in coping with delay-spread. A previously proposed structure to combine receiver


diversity and equalization is also described in this chapter. This approach is referred

to as the maximal-ratio combining DFE (MRCDFE), which introduces combining

diversity into a DFE. Computer simulations are used to compare di�erent adaptation

techniques for the MRCDFE.

Chapter 6 describes a novel approach for incorporating adaptive decision-feedback

equalization into selection diversity. This approach, referred to as the maximum a

posteriori probability (MAP) selection diversity, can be shown to be optimal in the

MAP sense for selection-diversity DFE's. Simulation results show that the proposed

approach is very e�cient. It signi�cantly outperforms conventional selection-diversity

DFE's. Furthermore, the MAP selection diversity DFE performs almost as well as,

but is signi�cantly simpler, than the MRCDFE.

In Chapter 7, the selection diversity technique derived in Chapter 6 is applied to

DFE decision delay optimization. An adaptive DFE structure with multiple decision

delays is also proposed and analyzed. Simulation results show that the proposed

structure can greatly improve the performance of a conventional adaptive DFE.

Conclusions are given in Chapter 8.

1.2 Contributions

Chapter 3 reports simulation results for the comparison of the channel-estimation-

based adaptation and direct adaptation DFE's. These results take into account the

randomness of the wireless channel lengths, and thus are more realistic than previous

work in the literature. The synthetic training approach for improving the performance

of an adaptive DFE is also novel. The same analyses are also extended in Chapter 5

to accommodate receiver diversity.

Chapter 4 proposes and evaluates an ad hoc, yet simple, metric for DFE decision

delay optimization. It is also shown here that the a posteriori method for decision

delay optimization can potentially outperform conventional schemes. In the a poste-

riori method, multiple DFE's with di�erent decision delays are used to obtain several

decoded bursts, and the burst containing the fewest errors is chosen as the �nal out-

put. A practically implementable structure, referred to as the multiple decision delay


DFE, is proposed and evaluated in Chapter 7 based on this principle.

In Chapter 6, a novel selection rule is proposed and analyzed for the selection

diversity DFE. In this new scheme, selection is done at the bit-level on a symbol-by-

symbol basis. The new selection rule proposed here, referred to as the MAP selection

rule, selects the branch decision that has the highest a posteriori probability of being

correct. This rule is proved to be optimal, in the MAP sense, in Appendix D. Two

selection metrics are also derived from the MAP selection rule. They are shown to

outperform conventional schemes for selection diversity DFE.

Chapter 2

Multipath Propagation

In a wireless communication channel, the transmitted signal generally propagates

to the receiver antenna through many di�erent paths. This phenomenon, depicted

in Figure 2.1 for indoor environments, is termed multipath propagation. Multipath

propagation is due to the multiple re ections caused by re ectors and scatterers in the

environment. Possible re ectors and scatterers may include mountains, hills and trees

in rural environments, buildings and vehicles in built-up urban environments, or walls

and oors in indoor environments. The receiver antenna will therefore receive multiple

copies of the transmitted signal. Since di�erent versions of the signal propagate

through di�erent paths, they will in general have di�erent attenuation, phase shifts,

time delays and angles of arrival. The receiver antenna output is the sum of the

multiple signal copies weighted by the antenna gain pattern.

Multipath propagation is a complicated phenomenon that is very di�cult to char-

acterize. One common approach is to treat the received signal as a spatial-temporal

random process. The statistics of this random process can be collected from extensive

�eld measurements in selected operation environments. Since the properties of the

received signal are clearly a strong function of the multipath environment, statistical

characterization of the received signal is often done in a two-step process. In the

�rst step, it is assumed that the multipath environment is �xed, and variations of the

received signal are measured for the given multipath environment. The statistics thus

collected are referred to as small-scale variations, because they are usually obtained

6

CHAPTER 2. MULTIPATH PROPAGATION 7

T

R

Figure 2.1: Multipath propagation in indoor environments. The signal transmitted

by the transmitter (T) is attenuated and re ected by the walls and oors. As a

consequence the receiver (R) receives multiple distorted copies of the transmitted

signal.


from measurement data obtained at various locations in a small area. In the second

step, variations of the small-scale statistics are determined from measurements taken

in di�erent multipath environments. These variations are referred to as large-scale

variations, because they are obtained from measurement data taken at various loca-

tions in a large area. In this dissertation, we focus on mitigating the e�ects of the

small-scale variations using digital signal processing techniques. These small-scale

variations, including signal fading, delay-spread and Doppler-spread, are discussed in

the remainder of this chapter. For a treatment of large-scale variations, readers are

referred to [1].

2.1 Signal Fading

Signal fading refers to the rapid change in received signal strength over a small travel

distance or time interval. This occurs because in a multipath propagation environ-

ment, the signal received by the mobile at any point in space may consist of a large

number of plane waves having randomly distributed amplitudes, phases, delays and

angles of arrival. These multipath components combine vectorily at the receiver an-

tenna. They may combine constructively or destructively at di�erent points in space,

causing the signal strength to vary with location.

If the objects in a radio channel are stationary, and channel variations are con-

sidered to be only due to the motion of the mobile, then signal fading is a purely

spatial phenomenon. A receiver moving at high speed may traverse through several

fades in a short period of time. If the mobile moves at low speed, or is stationary,

then the receiver may experience a deep fade for an extended period of time. Reliable

communication can then be very di�cult because of the very low signal-to-noise ratio

(SNR) at points of deep fades.

Extensive �eld measurements have previously been done[2, 3, 4, 5] to characterize

the small-scale spatial distribution of the received signal amplitude in multipath prop-

agation environments. It has been found that for many environments, the Rayleigh

distribution provides a good �t to the signal amplitude measurement in environments


where no line-of-sight or dominant path exists[2, 5, 6]. The probability density func-

tion of the Rayleigh distribution is given by[7]

f(r) =

8<:

r

Pexp

�� r

2

P

�r � 0:

0 otherwise;(2.1)

where P is the parameter of the distribution. A normalized plot of the Rayleigh

probability density function is shown in Figure 2.2. The Rayleigh distribution is

related to the zero-mean Gaussian distribution in the following manner. Let XI and

XQ be two independent, identically distributed (i.i.d.) zero-mean Gaussian random

variables with variance P. The marginal probability density functions of XI and XQ

are given by

f(x) =1p2�P

exp

� x2

2P

!;�1 < x <1: (2.2)

Then the random variable R, de�ned as

R =qX2

I+X2

Q; (2.3)

is distributed according to the Rayleigh probability density function given in Equa-

tion 2.1. The fact that the Rayleigh distribution provides a good �t to the mea-

sured signal amplitudes in a non-line-of-sight environment can be explained as fol-

lows. When a signal is transmitted through a multipath propagation channel, the

in-phase and quadrature-phase components of the received signal are sums of many

random variables. Because there is no line-of-sight or dominant path, these random

variables are approximately zero-mean. Therefore, by the central limit theorem, the

in-phase and quadrature-phases components can be modeled approximately as zero-

mean Gaussian random processes. The amplitude, then, is approximately Rayleigh

distributed.

On the other hand, when line-of-sight paths exist in a multipath propagation

environment, or when there is a dominant re ected path, the Ricean distribution

is a good statistical characterization of the signal amplitude distribution[4, 5]. The

Ricean distribution is related to the Gaussian distribution in a manner similar to the


relationship between the Rayleigh and Gaussian distributions. In particular, let XI

and XQ be independent Gaussian random variables with variance P . Furthermore,

assume that E[XI ] = � and E[XQ] = 0. Then the random variable R, de�ned in

Equation 2.3, is distributed according to the Ricean distribution. Thus, one can see

that when a dominant path exist in a multipath propagation environment, by the

central limit theorem, the signal amplitudes are approximately Ricean distributed

when the number of paths is large. The probability density function of the Ricean

distribution is given by[7]

f(r) =

8<:

r

PI0�r�

P

�exp

�� r

2+�2

2P

�; r � 0;

0 otherwise,(2.4)

where

I0(x) � 1

2�

Z 2�

0exp (x cos �)d� (2.5)

is the zeroth-order modi�ed Bessel function of the �rst kind. Note that there are two

parameters in Equation 2.4. P is the variance of the underlying Gaussian random

variable and � is the amplitude of the line-of-sight or dominant component. Normal-

ized plots of the Ricean distribution with di�erent values of � are shown in Figure 2.2,

in which � = 0 corresponds to the Rayleigh distribution. As � tends to in�nity, the

Ricean distribution converges to a Gaussian distribution.

2.2 Delay Spread

As mentioned previously, in a multipath propagation environment, the received sig-

nal consists of a large number of components having di�erent delays. Consequently,

when a \narrow" pulse is transmitted over a multipath propagation channel, dis-

torted replicas of the transmitted pulse arrive at the receiver at various di�erent

times, making the received signal \wider" in time than the transmitted signal. This

phenomenon is referred to as delay spread. The signi�cance of delay spread depends

on the time-width of the signal relative to that of the channel, hence a quantitative

characterization of the severeness of channel delay-spread is necessary.


rP-1/2

f(r)

P1/

2

0 2 4 6 8 10

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

3.0 5.0

0 (Rayleigh)

1.0

Figure 2.2: Normalized plots of the Ricean distribution. The numbers labeled on the

curve denote values of �pP. � = 0 corresponds to the Rayleigh distribution.

One common measure for characterizing channel delay spread is the power-delay

pro�le[6]. The power-delay pro�le of an environment is obtained through �eld mea-

surements by transmitting a short pulse and measuring the received power as a func-

tion of delay at various locations in a small area. These measurements are then aver-

aged over spatial locations to generate a pro�le of average received signal power as a

function of delay. The second central moment of the power-delay pro�le is referred to

as the root-mean-square (rms) delay-spread[6], and can be used as one quantitative

measure of the severeness of multipath propagation. Typical power-delay pro�les for

both outdoor[4][8]-[14] and indoor[15]-[25] environments can be found in the litera-

ture. For outdoor wireless channels, the rms delay-spreads typically range from 1.5

to 5�s; while for indoor environments, the rms delay-spreads typically range from 10

to 100 ns. It should be kept in mind that the value of the rms delay-spread, just as

any other parameter used to characterize wireless channels, is highly environment-

dependent. It is also dependent on the carrier frequency used for transmission. There

is no universal value that can be applied to every multipath propagation channel. It

is, therefore, extremely important for a wireless communication system to be robust

against variations in channel parameters.


In general, for a wireless digital communication system, the signi�cance of channel

delay spread depends on the relationship between the rms delay-spread of the channel

and the symbol period of the digital modulation[26]. If the rms delay-spread is much

less than the symbol period, then delay spread has little impact on the performance

of the communication system. In this case the shape of the power-delay pro�le is

immaterial to the error performance of the communication system. This condition

is often called \ at-fading." On the other hand, if the rms delay-spread is a sig-

ni�cant fraction of, or greater than, the symbol period, then channel delay spread

signi�cantly impairs the performance of the communication system. Furthermore,

the error performance of the communication system depends on the shape of the

power-delay pro�le. This condition is often referred to as \time-dispersive fading" or

\frequency-selective fading." Since the power-delay pro�le is an empirical quantity

that depends on the operating environment, for computer simulation purposes we

can only postulate functional forms of the pro�le, and vary the parameters of these

functional forms in order to obtain results that are applicable to a broad spectrum

of wireless environments. In this dissertation, we make use of two functional forms.

The �rst is the exponential power-delay pro�le, given by

pe(�) =

8<:

1Sexp

��

S

�; � � 0;

0 otherwise.(2.6)

The second is the Gaussian power-delay pro�le, de�ned as

pg(�) =

8<:

1p2�S

exp�� (�� )2

2S2

�; � � 0;

0 otherwise.(2.7)

These power-delay pro�les are plotted in Figure 2.3(a) and (b). In Equations 2.6

and 2.7, S is the rms delay-spread. �� in Equation 2.7 refers to the average delay

introduced by the channel. Note that, to be precise, pg(�) should be referred to as the

truncated Gaussian power-delay pro�le, because it is the causal part of a Gaussian

function. Furthermore, the rms delay-spread of a multipath propagation channel with

power-delay pro�le described by Equation 2.7 is not equal to S. However, whenever


-1τ S

)p e(τ

S

20 4 6 8 10

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

1

0

(a)

-1τ S0 2 4 6 8 10

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0

)p g(τ

S

(b)

Figure 2.3: (a) Normalized exponential power-delay pro�le. (b) Normalized Gaussian

power-delay pro�le. �� = 4S in this plot.


dv

X Y

α

dcos

( )α

Figure 2.4: Illustration of Doppler shift in the free-space propagation environment.

The receiver moves at a constant velocity v along a direction that forms an angle �

with the incident wave.

Equation 2.7 is used in this dissertation, �� is set to a value that is signi�cantly

larger than S. In this case pg(�) is essentially the same as the Gaussian function

before truncation, and the rms delay spread is essentially equal to S. For the sake of

brevity, we will simply refer to pg(�) as the \Gaussian power-delay pro�le."

2.3 Doppler Spread

When a single-frequency sinusoid is transmitted in a free-space propagation envi-

ronment where there is no multipath propagation, the relative motion between the

transmitter and receiver results in an apparent change in the frequency of the received

signal. This apparent frequency change is called Doppler shift. To analyze this e�ect,

consider the simple scenario shown in Figure 2.4. Assuming that the transmitter is

far away so that plane wave approximations hold at the receiver location, and that

the receiver is moving at a constant velocity v along a direction that forms an angle

� with the incident electro-magnetic wave, then it can be seen that the di�erence

in path lengths traveled by the wave from the transmitter to the mobile receiver at

points X and Y is given by

�l = d cos� (2.8)

= v�t cos�; (2.9)


where �t is the time required for the mobile to travel from X to Y. The phase change

in the received signal due to the di�erence in path lengths is therefore

�� =2��l

�(2.10)

=2�v�t

�cos�; (2.11)

where � is the wavelength. Hence, the apparent change in received frequency, or

Doppler shift, is given by

fd =1

2�

��

�t(2.12)

=v

�cos� (2.13)

=v

cfc cos�: (2.14)

In Equation 2.14, c is the speed of light and fc is the frequency of the transmitted

sinusoid. In going from Equation 2.13 to 2.14, the relationship

c = fc� (2.15)

is used.

It can be seen from Equation 2.14 that Doppler shift is a function of, among other

parameters, the angle of arrival of the transmitted signal. In a multipath propagation

environment in which multiple signal copies propagate to the receiver with di�erent

angles of arrival, the Doppler shift will be di�erent for di�erent propagation paths.

The resulting signal is the sum of the multipath components. Consequently, the

frequency spectrum of the received signal will in general be \wider" than that of the

transmitted signal, i.e. it contains more frequency components than were transmitted.

This phenomenon is referred to as Doppler spread. Since the received signal occupies

a wider band than the transmitted signal, the multipath propagation channel is a

time-varying linear system when there is relative motion. The amount of Doppler

spread, then, characterizes the rate of channel variations.

Doppler spread can be quantitatively characterized by the Doppler spectrum[1].


fcfc fmax- fc fmax+f

K

S(f)

Figure 2.5: The Doppler spectrum corresponding to uniformly distributed angles of

arrival (see Equation 2.16).

The Doppler spectrum is the power spectral density of the received signal when a

single-frequency sinusoid is transmitted over a multipath propagation channel. In a

static environment in which the re ectors stay immobile, the Doppler spectrum is

simply an impulse located at the frequency of the transmitted sinusoid when there

is no relative motion. When there is relative motion, the Doppler spectrum occupies

a �nite bandwidth. The exact shape of the Doppler spectrum depends on the con-

�guration of the re ectors. It can be shown[1] that when the mobile receiver moves

at a constant speed v and the signal power received by the receiver antenna arrives

uniformly from all incident angles in [0; 2�), the Doppler spectrum takes a form of

S(f) =Kr

1��f�fcfmax

�2 ; (2.16)

whereK is a proportionality constant and fmax = (vc)fc is the maximumDoppler shift.

This Doppler spectrum is plotted in Figure 2.5. In reality, however, the exact shape

of the Doppler spectrum can only be obtained by extensive �eld measurements, and

Equation 2.16 is approximately true only in certain environments. The bandwidth of

the Doppler spectrum, or equivalently the maximum Doppler shift fmax, is a measure


of the rate of channel variations. When the Doppler bandwidth is small compared

to the bandwidth of the signal, the channel variations are slow relative to the signal

variations. This is often referred to as \slow fading." On the other hand, when the

Doppler bandwidth is comparable to or greater than the bandwidth of the signal, the

channel variations are as fast or faster than the signal variations. This is often called

\fast fading."

2.4 Baseband Indoor Wireless Channel Model

The focus of this dissertation is on the mitigation of small-scale e�ects due to mul-

tipath propagation in indoor wireless channels. Indoor wireless channels di�er from

the traditional outdoor mobile radio channels in several aspects. First, the multi-

path structure is of a much smaller scale. In other words, the re ectors in indoor

environments are much more closed-in than those in outdoor environments. Con-

sequently the rms delay spread is much smaller for indoor environments. The rms

delay-spreads of indoor wireless channels typically range from 10 to 100 ns. This is

signi�cantly smaller than the typical values of 1.5 to 5 �s in outdoor environments.

Secondly, indoor wireless channels vary very slowly. A mobile moving at 6 km/hour

{ a fast walking speed { results in a maximum Doppler shift of 5 Hz when the carrier

frequency is 900 MHz. Secondary e�ects, such as motion of people and doors being

opened and closed, also contribute to channel variations. However, the variations

due to these secondary e�ects are also very slow[18]. In contrast, in outdoor cellular

environments, a vehicle traveling at 120 km/hour results in a maximum Doppler shift

of 100 Hz when the carrier frequency is 900 MHz, which is signi�cantly higher than

that of indoor wireless channels.

Based on the discussions presented in Sections 2.1-2.3, we have adopted the fol-

lowing baseband model for indoor wireless channels. Assuming that a (baseband

equivalent) signal u(t) is transmitted over an indoor wireless channel, the baseband


representation of the received signal r(t) is given by

r(t) =MX

m=1

amu(t� �m)ej�me�j!c�mej!mt + n(t); (2.17)

where !c and M are the carrier frequency and number of multipath components,

respectively, and j =p�1. famg, f�mg, f�mg and f!mg are the path gains, phases,

delays and Doppler shift frequencies, respectively, of the channel. n(t) is the addi-

tive white Gaussian noise (AWGN). These parameters are assumed to be mutually

independent. The path gains famg are assumed to be i.i.d. according to the Rayleigh

distribution, with the constraint that

E

"MX

m=1

a2m

#= 1: (2.18)

The path phases f�mg are assumed to be i.i.d. uniformly in [0; 2�). We assume that

the Doppler shift frequency of the each path is proportional to the cosine of the

corresponding angle of arrival (see Equation 2.14). Furthermore, we also assume that

the angles of arrival of the propagation paths are i.i.d. uniformly in [0; 2�): As a result,

f!mg are distributed according to the inverse-cosine distribution, whose probability

density function is given by

f(!) =

8><>:

1p(!�!max)2

j!j < !max;

0 otherwise;(2.19)

where !max = 2�fmax: Throughout this dissertation, we will assume that fmax = 5

Hz. The number of paths M is set to 20, and the carrier frequency !c is �xed at

2��900 Mrad/sec. Note that the latter assumption, together with the distribution

of the Doppler frequencies, correspond to a mobile speed of 1.67 meters/sec, which is

reasonable for indoor applications.

Two points are worth noting for this channel model. First, it can be shown

that the power-delay pro�le of this channel model is proportional to the probability

density function of the path delays f�mg. The standard deviation of the path delays


Param Meaning Statistics

!c Carrier Freq. 2��900 Mrad/sec

M Num. of Paths 20

famg Path Gains Rayleigh; E[P

M

m=1 a2m] = 1

f�mg Path Delays Exponential with std. dev. of 50ns; OR

Gaussian with mean 200ns and std. dev. 50ns

f�mg Path Phases Uniform in [0; 2�)

f!mg Doppler Shift Freq. Inverse-cosine in [�2� � 5; 2� � 5] rad/sec

fn(t)g Noise Complex WGN

Table 2.1: Channel parameters used throughout this dissertation.

is therefore the rms delay-spread of the channel. A proof is given in Appendix A. It

is, therefore, possible to simulate wireless channels with arbitrarily given power-delay

pro�les by using the channel model described in this section. As mentioned previously,

in this dissertation we assume that the channel power-delay pro�le is either Gaussian

or exponential. Furthermore, we assume that the indoor wireless channel has a rms

delay spread of 50 ns. For the Gaussian power-delay pro�le, we also assume that

the average delay �� = 200ns, i.e. four times the rms delay-spread. This implies that

the path delays f�mg are either i.i.d. Gaussianly or i.i.d. exponentially distributed

with a standard deviation of 50 ns. For the Gaussian case, the mean is set to 200

ns. Secondly, it can also be shown that the Doppler spectrum is proportional to the

distribution of the Doppler shift frequencies f!mg. This is proved in Appendix B.

Therefore, it is also possible to specify arbitrarily the Doppler spectrum of this channel

model. As mentioned previously, throughout this dissertation we assume that f!mgare distributed according to an inverse-cosine distribution. The Doppler spectrum is,

therefore, proportional to the probability density function given in Equation 2.19.

The meanings and statistics of these parameters are summarized in Table 2.1. It

should be noted that although the channel model and the corresponding parameters

that we have adopted are reasonably realistic, they are used only for the purpose of

evaluating the relative performance of di�erent multipath mitigation techniques. If

the absolute performance in a particular environment is to be predicted, di�erent sets

of parameters, or even a di�erent channel model, should be extracted from extensive


�eld measurements taken in the environment in question.

2.4.1 Simulated Power-Delay Pro�le

In order to obtain a better understanding of our baseband channel model, the power-

delay pro�le of the model is obtained by computer simulation. In our simulation

programs, a one-nanosecond baseband equivalent pulse with unity amplitude is �rst

generated. A baseband wireless channel is next generated according to Table 2.1. The

baseband equivalent pulse is then transmitted through the indoor wireless channel

according to Equation 2.17, with n(t) = 0 for all t. The entire process is repeated

200 times, each time using an independently generated set of channel parameters.

The resulting received power waveforms are then averaged to yield the simulated

power-delay pro�les.

The simulation results are plotted in Figure 2.6(a) and (b). In Figure 2.6(a), the

path delays are generated according to an exponential distribution with a standard

deviation of 50 ns. It can be seen that the resulting channel has an approximately

exponential power-delay pro�le. The measured rms delay-spread is 50.15ns. In Fig-

ure 2.6(b), on the other hand, the path delays are generated according to a Gaussian

distribution with a mean of 200 ns and standard deviation of 50 ns. It can be seen

that the resulting channel has an approximately Gaussian power-delay pro�le. The

measured mean delay is 200.19 ns and measured rms delay-spread is 49.85 ns. It can

therefore be seen that the probability density function of the path delays controls

the power-delay pro�le of the channel model, as mentioned previously and shown

analytically in Appendix A.

2.4.2 Simulated Doppler Spectrum

The Doppler spectrum of our baseband channel model can also be generated using

computer simulations similar to that described in Section 2.4.1. Here, instead of using

a short pulse, a baseband equivalent pulse of �ve-second duration and unity amplitude

is used to obtain the received signal. Note that since the duration of the pulse is much

longer than the rms delay-spread, the shape of the power-delay pro�le (or equivalently,


0 50 100 150 200 250Delay (ns)

0.025

0.005

0.01

0.015

0.02

0

Pow

er (

W)

(a)

0 100 200 300 400 500

0.002

0.004

0.006

0.008

0.01

0.012

0

Pow

er (

W)

Delay (ns)

(b)

Figure 2.6: Simulated power-delay pro�les obtained from 200 replications. (a) The

path delays are exponentially distributed with a standard deviation of 50 ns. (b) The

path delays are Gaussianly distributed with a mean of 200 ns and standard deviation

of 50 ns.


0.1

0

0.2

0.3

0.4

0.5

0.6

0.7

Pow

er D

ensi

ty (

W/H

z)

-10 -5 0 5 10Doppler Frequency (Hz)

Figure 2.7: Simulated Doppler spectrum obtained from 200 replications. The Doppler

shift frequencies are generated according to Table 2.1

the distribution of the path delays) is immaterial. The received baseband equivalent

power spectrum, i.e. the squared magnitude of the Fourier transform of the received

signal, is next computed. The simulated Doppler spectrum is then the average of the

replicated baseband equivalent power spectra. The simulation results are plotted in

Figure 2.7. It can be seen that the simulated Doppler spectrum has a shape similar

to that shown in Figure 2.5. Furthermore, the received signal power is approximately

con�ned to �5 Hz. This is expected because, as mentioned previously and shown

in Appendix B, the baseband equivalent Doppler spectrum of our channel model

is proportional to the distribution of the path Doppler shift frequencies given in

Equation 2.19. However, Equation 2.19 is of the same functional form as the Doppler

spectrum given in Equation 2.16, which is plotted in Figure 2.5. It is therefore

veri�ed that the Doppler spectrum of our channel model can be controlled by properly

specifying the distribution of the path Doppler shift frequencies.


2.5 Summary

Due to the presence of re ectors and scatterers in the environment, the signal trans-

mitted through a wireless radio channel propagates to the receiver antenna via many

di�erent paths. The output of the receiver antenna is, therefore, a sum of many

distorted copies of the transmitted signal. These copies generally have di�erent am-

plitudes, time delays, phase shifts, and angles of arrival. This phenomenon is referred

to as multipath propagation.

The e�ect of multipath propagation can be classi�ed into large-scale and small-

scale variations. Small-scale variations include signal fading, delay spread, and Doppler

spread. Signal fading refers to the rapid change in received signal strength over a small

travel distance or time interval. It occurs because of the constructive and destructive

interference between signal copies. Delay spread refers to the smearing or widening

of a short pulse transmitted through a multipath propagation channel. It happens

because di�erent propagation paths have di�erent time delays. Doppler spread refers

to the widening of the spectrum of a narrow-band signal transmitted through a mul-

tipath propagation channel. It is due to the di�erent Doppler shift frequencies asso-

ciated with the multiple propagation paths when there is relative motion between the

transmitter and receiver. These small-scale e�ects can be quantitatively characterized

using the signal amplitude distribution, power-delay pro�le and rms delay-spread, and

Doppler spectrum. All these characterizations are empirical statistics that must be

obtained using extensive �eld measurements.

Based on the qualitative description of multipath propagation and some empirical

data found in the literature, we have adopted a simple baseband model for indoor

wireless radio channels. This model is described in Section 2.4. The power-delay

pro�le and Doppler spectrum of this channel model can be controlled by properly

specifying the distribution of some model parameters. Simulation results are shown

in Sections 2.4.1 and 2.4.2 to demonstrate the exibility of this channel model. This

channel model will be used in all simulations in this dissertation.

Chapter 3

Adaptive Decision-Feedback

Equalization

As mentioned in Chapter 2, delay spread is one of the e�ects of multipath propagation.

In an environment where delay spread is signi�cant, high-speed data transmission

encounters inter-symbol interference, which can drastically impair the performance

of a data communication system. One technique for mitigating the e�ect of delay

spread is adaptive equalization. An adaptive equalizer is a special form of adaptive

�lter which, when designed and optimized correctly, can remove most of the inter-

symbol interference present in the received signal. In this chapter, we describe in

detail inter-symbol interference and adaptive equalization.

3.1 System Model

Consider the wireless data communication system shown in Figure 3.1. The sys-

tem consists of the digital modulator, wireless channel, and receiver. As shown in

Figure 3.1, the digital modulator consists of a binary data sequence generator and

an uncoded 4-QAM modulator[27]. The binary data sequence generator generates a

binary data stream with equally likely 0's and 1's. Successive symbols in this data

stream are assumed to be independent. This data sequence is coherently modu-

lated onto the carrier by the 4-QAM modulator. Square-root raised-cosine spectral

24

CHAPTER 3. ADAPTIVE DECISION-FEEDBACK EQUALIZATION 25

Data Seq. Generator 4QAM

Wireless channel

(t)n

AWGN

MultipathFading

TimingAlignment

LPFSampler

Receiver

r(t)Output

u(t)

y(t)DFEAdaptive

Modulator

Figure 3.1: Baseband model of a wireless data communication system.

shaping[28] with a roll-o� factor of � is also used to reduce transmission bandwidth.

Mathematically, the baseband equivalent modulated signal u(t) can be expressed as

u(t) =Xi

xiq(t� iT ): (3.1)

In Equation 3.1, xi 2 f1+ j; 1� j;�1+ j;�1� jg are the signal points in the 4-QAMconstellation, T is the symbol period, and q(�) is the square-root raised-cosine pulse,de�ned as

q(t) =1pT

n(1� �)sinc

�(1� �)t

T

�+ �

hsinc

��t

T+

1

4

�cos

��t

T+

�

4

�+ sinc

��t

T�

1

4

�cos

��t

T�

�

4

�io; (3.2)

where

sinc(x) =sin�x

�x: (3.3)

The modulated signal u(t) is transmitted through the wireless channel, which is mod-

eled as a frequency-selective multipath fading channel corrupted by additive white

Gaussian noise (AWGN), as described previously in Chapter 2. At the receiver, the

received signal r(t) is �ltered by a lowpass �lter whose impulse response is also a

square-root raised-cosine pulse with a roll-o� factor of �. It is assumed that the fre-

quency o�set between the oscillators of the transmitter and receiver is negligible. The

�ltered signal is next sampled at appropriate sampling instants at a rate of K samples


per symbol-period. The sampling instants are determined by the \timing alignment"

algorithm[29], which will be described in Chapter 4. The resulting discrete-time signal

is fed into the adaptive decision-feedback equalizer to yield the demodulated output.

3.2 Inter-Symbol Interference

In a digital communication system, a sequence of narrow pulses is used to repre-

sent a stream of information symbols, as described in Section 3.1. As mentioned

in Chapter 2, due to delay spread caused by multipath propagation, each of these

narrow pulses widens in time when transmitted through a wireless channel. As a

consequence, successive pulses interfere with each other, causing inter-symbol inter-

ference (ISI). To see this, consider the received signal r(t) in Figure 3.1. Ignoring the

channel variations and using quasi-static approximations1, the received signal can be

expressed as

r(t) = h u(t) + n(t); (3.4)

where

h(t) =MX

m=1

amej�me�j!c�m�(t� �m); (3.5)

and \" denotes linear convolution, de�ned by

f g(t) =

Z 1

�1f(t0)g(t� t0)dt0: (3.6)

In Equation 3.5, �(�) is the Dirac delta function. Substituting Equation 3.1 into

Equation 3.4, we have

r(t) =Xi

xiq h(t� iT ) + n(t): (3.7)

1This approximation is reasonably accurate for indoor wireless radio channels.


As shown in Figure 3.1, the received signal is passed through a �lter with impulse

response q(�). Thus, the output of the �lter is

y(t) = r q(t) (3.8)

=Xi

xiq h q(t� iT ) + n q(t): (3.9)

Finally, the signal y(t) is sampled at proper instants to yield the discrete-time signal

yj. Assume, for the time being, that K = 1 sample is taken per symbol period. Then,

we have

yj = y(jT + �); j = 0; 1; 2; : : : (3.10)

=Xi

xiq h q ((j � i)T + �) + n q(jT + �); (3.11)

where � is the time at which the �rst sample is taken. Let

p(t) = q h q(t) (3.12)

and

pi = p(iT + �); (3.13)

ni = n q(iT + �); (3.14)

then

yj =Xi

xipj�i + nj (3.15)

= p0xj +Xi6=j

xipj�i + nj: (3.16)

It can thus be seen that if pj = 0 for all j 6= 0, then the �ltered received signal

yj is simply a scaled version of the transmitted symbol corrupted by additive �ltered

Gaussian noise. However, in general with delay spread pj 6= 0 even when j 6= 0,

thus the discrete-time signal yj contains not only the contribution from the \current"


symbol xj (given by the �rst term, p0xj, in Equation 3.16), but also contributions

from \previous" and \future" symbols (P

i6=j xipj�i) and �ltered Gaussian noise (nj).

The second term in Equation 3.16 is the ISI. It is the weighted sum of previous and

future symbols, with weights taken from fpig. pi is an important quantity that plays

the key role in characterizing ISI, and will be referred to as the channel pulse response

(CPR) in this dissertation. As can be seen in Equation 3.12 and 3.13, the CPR is

simply the sampled version of the response of the channel to the transmitted pulse

after being �ltered by the receive �lter.

The above analysis can be extended to the case where more than one sample is

take per symbol-period. Assuming that the signal y(t) is sampled at a rate of K

samples per symbol-period, where K is assumed to be an integer. These samples are

denoted as

yj;k = y(jT + � � kT

K) (3.17)

=Xi

xip

(j � i)T + � � kT

K

!+ n q

jT + � � kT

K

!; (3.18)

where j = 0; 1; : : :, and k = 0; 1; : : : ; (K � 1). Let

pi;k = p

iT + � � kT

K

!; (3.19)

and

ni;k = n q

iT + � � kT

K

!; (3.20)

then

yj;k =Xi

xipj�i;k + nj;k (3.21)

for k = 0; 1; : : : ; (K � 1). We can combine Equations 3.19, 3.20 and 3.21 into a more


compact form given by

yj =

26666664

yj;0

yj;1

: : :

yj;K�1

37777775

(3.22)

=Xi

xipj�i + nj (3.23)

= p0xj +Xi6=j

xipj�i + nj: (3.24)

where

pi =

26666664

pi;0

pi;1

: : :

pi;K�1

37777775; (3.25)

and

nj =

26666664

nj;0

nj;1

: : :

nj;K�1

37777775: (3.26)

It can be seen that Equations 3.24 is the vector version of Equations 3.16. The K�1

column vectors pi are the \vectorized" CPR; when K = 1 they degenerate into the

form de�ned in Equation 3.13. In this dissertation, pi will simply be referred to as

the CPR. The exact dimension of the vector is clear from the context.

We can see from Equations 3.13 and 3.19 that the CPR consists of samples of

p(t) de�ned in Equation 3.12. When the channel delay-spread is small, p(t) is ap-

proximately a raised-cosine pulse with a roll-o� factor of �. In this case p(t) can be

sampled so that jpij � 0 for i 6= 0, and the contribution of ISI in Equations 3.16

and 3.24 is relatively small. An example of this case is shown in Figure 3.2(a). On

the other hand, when the channel delay-spread is signi�cant compared to the symbol

period, p(t) has a broader main-lobe and slower roll-o� than the raised-cosine pulse.

As a consequence, jpij is signi�cant even for some values of i 6= 0. An example of this


1.4

1.2

1

0.8

0.6

0.4

0.2

06 8 10 12 14 16 18

Symbol Period

Mag

nitu

de o

f p(

t)

(a)

108 12 14 16 18 20 22 24

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

Symbol Period

Mag

nitu

de o

f p(

t)

(b)

Figure 3.2: Examples of jp(t)j. (a) When the channel delay-spread is insigni�cant,

very little ISI is introduced. (b) When the channel delay-spread is signi�cant, the ISI

is severe.


Tap-Wt.Adapt

xj

+FeedforwardFilter

FeedbackFilterDFE

zj∆y

j+y

j

AlignmentTiming

FilterReceiver r(t)

τ ∆Advance

Sample∆y(t)

Figure 3.3: Block diagram of an adaptive decision-feedback equalizer.

case is shown in Figure 3.2(b). We can then see from Equations 3.16 and 3.24 that ISI

introduces signi�cant additional unwanted components into the signal yj, and thus

limits the achievable transmission rate of the wireless data communication system.

If high-speed, low-error-rate data transmission is desired in a multipath propagation

environment, special signal processing must be performed to mitigate the impairment

of ISI. Adaptive equalization[30, 31, 32, 33] is one such technique. An equalizer is

a special form of discrete-time �lter which processes yj in order to reduce the e�ect

of ISI. In this dissertation, we make use of the decision-feedback equalizer (DFE),

because it has been shown to have superior performance compared to other types

of equalizers and lower computational complexity than the maximum likelihood se-

quence estimator (MLSE)[33]. Readers are referred to [33] for a discussion of MLSE

as well as other forms of equalization.

3.3 Adaptive DFE

A DFE is a nonlinear equalizer that is particularly useful for channels with severe

amplitude distortion. A block diagram of the DFE is shown in Figure 3.3. The

equalized signal is the sum of the outputs of the feedforward and feedback �lters of

the equalizer. The feedforward �lter minimizes the ISI contributed by future sym-

bols. Decisions made on the equalized signal are fed back via the feedback �lter. If

the past decisions are correct, then the ISI contributed by the past symbols can be

canceled approximately by properly setting the coe�cients of the feedback �lter. In


this dissertation, we use linear �nite-impulse-response (FIR) feedforward and feed-

back �lters with Nf and Nb taps, respectively. Linear FIR �lters are chosen to make

the adaptation algorithms simpler[34]. The DFE can then be regarded as an esti-

mator which uses the linear combination of the components of the �ltered received

signal samples yj+�;yj+��1; : : : ;yj+��Nf+1 and past decisions x̂j�1; x̂j�2; : : : ; x̂j�Nb

to estimate the transmitted symbol xj. The integer � is the \decision delay" of the

DFE[29, 35, 36, 37]. It re ects the signal spread introduced by multipath propaga-

tion and controls the trade-o� between time diversity and ISI. The performance of

the DFE is very sensitive to the choice of �, especially when the number of taps in

the feedforward �lter is small. Optimization of � is discussed in Chapter 4.

The DFE �lter coe�cients are jointly optimized according to some optimization

criterion. For the equalization of wireless channels, it is desirable to adapt, or re-

compute, the DFE �lter coe�cients on a symbol-by-symbol basis. First, wireless

channels are time-varying channels, therefore the optimal DFE �lter coe�cients vary

with time. Secondly, although indoor wireless channels vary very slowly, they are ran-

dom channels that are di�erent in di�erent environments. Therefore, the DFE must

be self-synthesizing, so that the �lter coe�cients can automatically adapt to the en-

vironment in which the DFE operates. DFE's whose �lter coe�cients are optimized

on a symbol-by-symbol basis are called adaptive DFE's. Mathematically, the output

(equalized signal) z(j; k) of an adaptive DFE at time j using the �lter coe�cients

obtained at time k can be expressed as2

z(j; k) =

Nf�1Xi=0

w�i;kyj+��i +

NbXi=1

b�i;kx̂j�i; (3.27)

where wi;k are K � 1 feedforward coe�cient vectors and bi;k are the feedback coe�-

cients, both optimized at time k. Equation 3.27 can be expressed in a more compact

form given by

z(j; k) = [w�k;b�

k]Yj+�; (3.28)

2\*" denotes the Hermitian transposition.


where

wk =

26666664

w0;k

w1;k

� � �wNf�1;k

37777775; (3.29)

bk =

26666664

b1;k

b2;k

� � �bNb;k

37777775; (3.30)

and

Yj+� =

266666666666666666664

yj+�

yj+��1

: : :

yj+��Nf+1

x̂j�1

x̂j�2

: : :

x̂j�Nb

377777777777777777775

: (3.31)

The �lter coe�cients wk and bk are jointly optimized using adaptation approaches

to be described in Section 3.3.1 and 3.3.2. The \one-step output" zj = z(j; j � 1) is

used to produced the detected symbol x̂j:

3.3.1 Channel Estimation Based Adaptation

The channel estimation based adaptation (CEBA) is an indirect approach for opti-

mizing the DFE �lter coe�cients. In this approach, an estimate of the CPR is �rst

obtained using channel estimation algorithms, e.g. the recursive least-squares (RLS)

algorithm[38, 42]. The DFE �lter coe�cients are next computed as a function of

these CPR estimates. This process is repeated once every symbol-period. It should


be noted that, �rst, knowledge of the transmitted symbol is required for any non-

blind CPR estimation method. This knowledge is, of course, not available at the

receiver. One common approach around this problem is to estimate the CPR in a

decision-directedmanner. A sequence of known symbols, called the training sequence,

is transmitted before the information symbols. This sequence is used to obtain a rea-

sonably accurate initial estimate of the CPR. During the information symbols, the

decisions (demodulated output) are assumed to be correct and used to re�ne the ini-

tial CPR estimate. Secondly, for a wireless channel, the exact number of taps in the

actual CPR is an unknown random quantity. Without knowledge of this quantity,

we can only obtain a �xed-length estimate of the CPR. We denote the �xed-length

CPR estimate as fp̂t;��1 � t � �2g, where �1 and �2 are �xed non-negative integers.

The maximum length of the CPR estimate is limited by the length of the training

sequence. As will be shown later, because the length of the CPR estimates is �xed,

and in general di�erent from the actual number of channel taps, the performance of

CEBA is sensitive to variations in channel delay-spread.

At every symbol period, after the CPR estimates are obtained, the DFE �lter

coe�cients are computed to minimize the approximate mean-square error (MSE) at

the DFE output (input of the slicer), de�ned as

MSE(wk;bk) = Ejxj � [w�k;b�

k]Yj+�j2: (3.32)

The solution to this minimization problem is given by the set of linear equations[38]

nEhYj+�Y

�j+�

io 24 wk

bk

35 = E

hx�jYj+�

i: (3.33)

Equation 3.33 involves second-order statistics of the received and transmitted signal.

Approximations of these statistics can be obtained from the CPR estimates as follows.


First, assuming that the past decisions are correct, we have3

Yj+� = ~PXj+�1 +

24 Nj+�

0Nb

35 ; (3.34)

where

~P =

24 P

0Nb;�1+1 INb0Nb;�2+Nf�Nb�1

35 ; (3.35)

Xj+�1 =

26666664

xj+�1

xj+�1�1

: : :

xj��2�Nf+1

37777775; (3.36)

Nj+� =

26666664

nj+�

nj+��1

: : :

nj+��Nf+1

37777775; (3.37)

0a denotes an a� 1 column vector of zeros, INbdenotes an identity matrix of size Nb,

and 0a;b denotes an a�b block of zeros. In Equation 3.35, P is a NfK�(�1+�2+Nf )

matrix composed ofNf rows of column vectors, each withK components. Speci�cally,

P is de�ned as

P =

26666664

p̂��1+� p̂��1+1+� � � � p̂�2+� 0K 0K � � � 0K

0K p̂��1+� p̂��1+1+� � � � p̂�2+� 0K � � � 0K

� � � � � � � � � � � � � � � � � � � � � � � �0K 0K � � � 0K p̂��1+� p̂��1+1+� � � � p̂�2+�

37777775:

(3.38)

Thus,

EhYj+�Y

�j+�

i= ~P

nEhXj+�X

�j+�

io~P� + E

8<:24 Nj+�

0Nb

35 h N�

j+� 0�Nb

i9=; ; (3.39)

3This is actually only an approximation because of the inaccuracy in estimating the CPR.


and

Ehx�jYj+�

i= ~PE

hx�jXj+�

i: (3.40)

Making use of the assumption that successive transmitted symbols are independent

of each other, we have

EhXj+�X

�j+�

i= 2I�1+�2+Nf

; (3.41)

thus

EhYj+�Y

�j+�

i= 2

24 PP� +RN H

H� INb

35 ; (3.42)

where

H = P

26664

0�1+1;Nb

INb

0�2+Nf�Nb�1;Nb

37775 ; (3.43)

and

RN =1

2EhNj+�N

�j+�

i: (3.44)

Furthermore,

Ehx�jXj+�

i= 2e�1; (3.45)

where e�1 denotes a (�1+ �2+Nf )� 1 column vector with a one in the �1-th position

and zeros elsewhere. Thus,

Ehx�jYj+�

i= 2~Pe�1 : (3.46)

Substituting Equations 3.42 and 3.46 into Equation 3.33, we have

24 PP� +RN H

H� INb

3524 wk

bk

35 = ~Pe�1 (3.47)

It can be seen from Equation 3.35 that the last Nb elements of the column vector ~Pe�1

are zeros. Therefore the DFE �lter coe�cients, optimal within the approximations,


DFE coeffComputation

ChannelEstimation

Training Symbolsyj

jxDFE

Figure 3.4: The channel-estimation-based adaptation DFE.

in Equation 3.47 are given by4

wk = (PP� �HH� +RN)�1� (3.48)

bk = �H�wk; (3.49)

where

� = Pe�1: (3.50)

The corresponding minimum MSE (MMSE) is

MSEmin;k = E jxjj2 � [w�k;b�

k]

24 PP� +RN H

H� INb

3524 wk

bk

35 (3.51)

= E jxjj2 �w�k(PP� �HH� +RN)wk (3.52)

A block diagram of the CEBA DFE is shown in Figure 3.4.

3.3.2 Direct Adaptation

In the direct adaptation (DA) approach, the DFE �lter coe�cients are computed

directly from the received signal samples and past decisions based on a least-squares

4It should again be noted that one set of CPR estimates is obtained for every symbol period.

Therefore in the previous derivations, all quantities, except for RN and e�1 , are actually functions of

time and should be computed once per symbol period. However, to simplify notations we explicitly

show the time-dependence only for the DFE coe�cients.


criterion. The exponentially-weighted square output error (WSE) Jk, de�ned as

Jk =kX

j=0

�k�j jz(j; k)� xjj2 ; (3.53)

is minimized at every time k. In Equation 3.53 the \forgetting factor" � is a positive

real number less than but close to 1. As in any other non-blind adaptive algorithm,

a training sequence is required to initialize the DA DFE. When evaluating Equa-

tion 3.53, only those xj that correspond to the training sequence are known to the

receiver. Whenever xj is unknown Jk is evaluated from Equation 3.53 by substituting

x̂j for xj. Since the training sequence consists of symbols known both to the receiver

and the transmitter, it contains no information and is a transmission overhead. It is

therefore desirable to minimize the number of training symbols needed to maintain

an acceptable performance. A simple technique for synthesizing additional training

symbols from the available training sequence will be presented later in this section.

The DA approach di�ers from the CEBA approach in that it is purely determin-

istic. The least-squares criterion in Equation 3.53 does not involve any statistics.

Consequently, no knowledge is assumed about the received signal statistics when

computing the DFE �lter coe�cients. Recall that in the CEBA approach, channel

estimation was done as an intermediate step; the CPR estimates are used to obtain

approximations of the second-order statistics of the received signal. Here, in the DA

approach, channel estimation is not necessary in the computation of the DFE �lter

coe�cients because no statistics are used. As will be shown later by simulation, di-

rectly adapting the DFE �lter coe�cients without going through channel estimation

makes the DA approach more robust against variations in channel delay-spread.

Numerous recursive algorithms[38] are available to minimize Jk. In this disserta-

tion we use the exponentially-weighted least-squares lattice (LSL) algorithm[39, 40]

because of its superior numerical stability and computational e�ciency[40]. A brief

description of the LSL DFE algorithm is given in Appendix C.


Initialization by Synthetic Training

As mentioned previously, the training sequence is necessary to initialize the adaptive

DFE. In this dissertation we assume that a sequence of Q pseudo-random (PN) QAM

symbols are transmitted at the beginning of a burst. The corresponding received

signal samples are used to initialize the adaptive DFE. In general, the average BER

performance of an adaptive DFE can be improved by increasing Q. This performance

gain is, however, achieved at the expense of increased transmission overhead (more

symbols to transmit) as well as receiver processing power (more training). Reduction

of overhead while maintaining the performance gain is therefore desirable.

In order to achieve this goal, we have proposed and evaluated a scheme, referred

to as synthetic training, in which a \synthetic training sequence" (STS) is generated

at the receiver through channel estimation, and combined with the actual transmitted

training sequence for DFE initialization. In this scheme, the CPR estimates are �rst

generated from the actual training sequence using the least-squares approach. After

the CPR estimates are obtained, the STS sj can be computed as

sj =�2X

i=��1

p̂iqj�i; j = ��1;��1 + 1; : : : ; Qs + �2 � 1; (3.54)

for l = 1; 2; : : : ; L, where fqi; i = 0; 1; : : : ; (Qs�1)g is one cycle of PN QAM symbols.

Note that Equation 3.54 is similar to Equation 3.23. sj thus correspond to the

response to qj of a discrete-time vector channel whose impulse response is p̂t. If

the AWGN is small compared to the signal component, and the length of p̂t is long

enough to cover the actual CPR, then sj will have statistical properties that are very

similar to those of the actual transmitted training sequence.

After the STS sj is obtained, it is prepended to the received waveform yj. The

resulting waveform, ~yj, is used as the input for the adaptive DFE. This procedure is

shown in Figure 3.5. Mathematically, we have

~yj =

8>>><>>>:sj; j = ��1;��1 + 1; : : : ; (Qs � �1 � 1);

sj + yj�Qs; j = Qs � �1; Qs � �1 + 1; : : : ; Qs + �2 � 1;

yj�Qs; j = Qs + �2; Qs + �2 + 1; : : : :

(3.55)


Qs Qs+ν2 Qs

Qs+ν2

��

��

−ν −ν1 10 Q+

ActualTraining

DataSTS

-1

Time

Figure 3.5: Prepending the STS to the received waveform for synthetic training for

the DA DFE.

The proposed scheme e�ectively lengthens the training sequence from Q to Q+Qs

without incurring any additional transmission overhead. It does, however, require ad-

ditional receiver processing. Furthermore, the STS in general has somewhat di�erent

statistical properties than the actual training sequence. Therefore the e�ectiveness

of this approach must be veri�ed.

3.4 Simulation Results

The performance of the DA and CEBA algorithms is evaluated using bit-by-bit com-

puter simulation for both symbol-spaced (K = 1) and half-symbol-spaced (K = 2)

DFE's. In each experiment, one set of channel parameters is �rst independently gen-

erated according to Table 2.1, and a burst of (150+Q) 4-QAM symbols is generated

and transmitted over the wireless channel according to Equation 2.17. A square-root

raised cosine �lter with a roll-o� factor of � = 0:35 is used as the transmit �lter.

The �rst Q transmitted symbols are composed of one cycle of pseudo-random (PN)

sequence5. The PN sequence is assumed to be known to the receiver and used for

timing alignment, synthetic training for the DA approach, and obtaining initial CPR

estimates for the CEBA approach. The remaining 150 symbols are assumed to be

the information symbols.

At the receiver, the received signal is �rst �ltered with a low-pass �lter with a

35% square-root raised-cosine impulse response. Timing alignment is then performed

using the transmitted PN training sequence to optimize the symbol-timing � and

decision delay �. The optimization procedure for these parameters will be described

5This implies that Q = 2n � 1, where n is a positive integer.


in detail in Chapter 4. Note that in this chapter, timing alignment is performed before

equalization starts, therefore the same optimization procedures for � and � are used

for both the CEBA and DA approaches. After the timing alignment parameters are

acquired, the DFE �lter coe�cients are adapted using the DA and CEBA approaches.

For the CEBA approach, the PN training sequence is used to obtaining initial CPR

estimates. During the remaining 150 symbols, the CPR estimates are updated once

per symbol period using the received signal and past decisions, and the DFE �lter

coe�cients are computed from the CPR estimates on a symbol-by-symbol basis as

described in Section 3.3.1. For the DA approach, the PN training is used to generate

a STS of length Qs. The STS is prepended to the received signal burst, and the DFE

�lter coe�cients are adapted once per symbol period using the signal samples and past

decisions as described in Section 3.3.2. For either approach, the number of decision

errors is tabulated for the 150 information symbols. The entire process is repeated

15,000 times. After that, it is continued until 3,000 bit errors are accumulated or

7,500,000 symbols are transmitted, whichever occurs �rst. In all experiments the

feedforward sections of the adaptive DFE span four symbol periods. The feedback

section has three taps spaced at the symbol period.6

3.4.1 Synthetic Training for DA DFE

Figure 3.6 shows the average bit-error rate (BER) for the symbol-spaced (K = 1) DA

DFE with synthetic training. The average BER is de�ned as the ratio of the number

of accumulated bit errors to the total number of transmitted bits. Here the channel

is assumed to have a Gaussian power-delay pro�le with a rms delay-spread of 50ns

and average delay of 200ns. In Figure 3.6, the average BER is shown as a function of

the normalized delay spread d, de�ned as

d =rms delay-spread

symbol period: (3.56)

6Though choices of DFE parameters were empirically chosen, they are tested using simulations

over a broad range of channel conditions.


10-1

10-3

10-2

Unequalized

0.5 0.75 1.0

Ave

rage

BE

R

Normalized Delay-Spread0.25

Q=15,Qs=3

Q=15,Qs=15

Q=15,Qs=7

Q=15,Qs=0

Q=31,Qs=0

Figure 3.6: The average BER of the symbol-spaced (K = 1) DA DFE with synthetic

training. The channel is assumed to have a Gaussian power-delay pro�le with a rms

delay-spread of 50 ns and average delay of 200 ns. The average SNR is 15 dB. Di�erent

training sequence lengths Q and STS lengths Qs are used.

For an rms delay-spread of 50ns and 4-QAM modulation, d = 0:25 and d = 1:0

correspond to data rates of 10 Mb/s and 40 Mb/s, respectively. The average SNR of

in this �gure is �xed at 15 dB. It is de�ned as the ratio of the signal power, averaged

over the Rayleigh distributed multipath fading, to the mean Gaussian noise power

at the output of the receive �lter (see Equations 2.17 and 2.18). For our simulation

setup the average SNR is numerically equal to the ratio of transmitted energy per bit

to the two-sided power spectral density of the AWGN. �1 = �2 = 3 (a total of 7 taps)

is used here for estimating the CPR for synthetic training purposes.

Di�erent combinations of actual training sequence length (Q) and STS length

(Qs) are tested. It can be seen from Figure 3.6 that without using STS (Qs = 0),

increasing the length of the actual training sequence from Q = 15 to Q = 31 results in

a 20% to 50% reduction in average BER. This is achieved at the price of both increased

transmission overhead and receiver processing. However, it can also be seen that with

Q �xed at 15, the DFE performance can be improved without introducing additional

transmission overhead by using a STS. The performance improves as the STS length

increases. In particular with (Q;Qs) = (15; 15) (a total of 30 \e�ective" training


symbols), the DFE performance is as good as using Q = 31 without STS. In other

words, STS is capable of achieving the same performance improvement as additional

actual training symbols without incurring any transmission overhead.

The performance of unequalized 4-QAM modulation is also plotted in Figure 3.6

as a baseline for comparison. It can be seen that without equalization, the BER

increases monotonically as d increases. This is expected because as d increases, the

ISI becomes more severe and reliable communications becomes more di�cult without

equalization. On the other hand, the average BER of the DFE decreases as d increases

from 0.25 to 0.5. As d increases beyond 0.5, the average BER increases with d. This

implies that as d increases from 0.25 to 0.5, the advantage of having more time

diversity outweighs the disadvantage of increased ISI. However, as d increases beyond

0.5, the length of the FIR DFE's becomes insu�cient and the increased residual ISI

degrades the performance. It can also be seen that the performance of the DFE is,

in general, signi�cantly superior to the unequalized case. At d = 1:0, for example,

the average BER of the DFE is lower than that of the unequalized case by almost an

order of magnitude.

The average BER of the symbol-spaced DFE is plotted in Figure 3.7 for di�erent

CPR estimate lengths. Here �1 and �2 are set to be equal and various di�erent

values are used. The channel is assumed to have a Gaussian power-delay pro�le with

normalized delay-spread d = 0:5, and the average SNR is set to 15 dB. The lengths

Q and Qs of the actual and synthetic training sequences, respectively, are both set

to 15. It can be seen that the average BER decreases only marginally as �1 and �2

increase beyond 3. There are two explanations for this observation. First, the number

of taps in the CPR estimate is limited by the length Q of the actual training sequence.

With Q �xed, the CPR estimate becomes increasing inaccurate as �1 and �2 increase.

Secondly, with a normalized delay-spread of 0.5, �1 = �2 = 3 is adequately long as

far as synthetic training is concerned. Increasing �1 and �2 beyond 3 does not make

the statistics of the STS any more reliable.

Similar plots for the half-symbol-spaced (K = 2) DFE are shown in Figures 3.8 and

3.9. The average BER's for various combinations of (Q;Qs) are shown in Figure 3.8 as

a function of d, while the average BER for various values of �1 and �2 (with �1 = �2)


10-1

10-3

10-2

Ave

rage

BE

R

ν 1 ( )ν 2

3 4 52

Figure 3.7: Average BER of the symbol-spaced (K = 1) DA DFE using various CPR

estimate lengths for synthetic training. �1 and �2 are both set to be equal to the value

of the abscissa. The channel has a Gaussian power-delay pro�le with a normalized

delay-spread of 0.5. The rms delay-spread and average delay are 50 ns and 200 ns,

respectively. The average SNR is 15 dB. Synthetic training with Q = Qs = 15 is

used.

10-1

10-3

10-2

Unequalized

0.5 0.75 1.0

Ave

rage

BE

R

Normalized Delay-Spread0.25

Q=15,Qs=3

Q=15,Qs=15

Q=15,Qs=7

Q=15,Qs=0

Q=31,Qs=0

Figure 3.8: The average BER of the half-symbol-spaced (K = 2) DA DFE with

synthetic training. The channel is assumed to have a Gaussian power-delay pro�le

with a rms delay-spread of 50 ns and average delay of 200 ns. The average SNR is 15

dB. Di�erent training sequence lengths Q and STS lengths Qs are used.


10-1

10-3

10-2

Ave

rage

BE

R

ν 1 ( )ν 2

3 4 52

Figure 3.9: Average BER of the half-symbol-spaced (K = 2) DA DFE using various

CPR estimate lengths for synthetic training. �1 and �2 are both set to be equal

to the value of the abscissa. The channel has a Gaussian power-delay pro�le with a

normalized delay-spread of 0.5. The rms delay-spread and average delay are 50 ns and

200 ns, respectively. The average SNR is 15 dB. Synthetic training with Q = Qs = 15

is used.


are shown in Figure 3.9. In all cases the channel is assumed to have a Gaussian

power-delay pro�le and the average SNR is set at 15 dB. We can see that the general

conclusions for the symbol-spaced (K = 1) case also hold in these plots.

3.4.2 Comparison of CEBA and DA DFE

Figure 3.10(a) shows the average bit-error rate (BER) for symbol-spaced (K = 1) and

half-symbol-spaced (K = 2) DA and CEBA DFE's. The average BER is shown as a

function of the average SNR. The channel is again assumed to have a Gaussian power-

delay pro�le with a normalized delay-spread of 0.5. Q = 15 PN symbols are used for

training. For the DA DFE, a STS with Qs = 15 symbols is used for synthetic training.

�1 = �2 = 3 is used for estimating the CPR for DA DFE synthetic training and CEBA

DFE �lter coe�cient adaptation. Only average SNR's greater than or equal to 10 dB

are simulated, because it had previously been found that the average BER's for lower

average SNR's are too high to be of interest to data communications[43]. It can be

seen that at d = 0:5, the di�erence between the average BER performance of symbol-

spaced DA and CEBA DFE's is insigni�cant. This is consistent with previously

reported results[44]. However, here we �nd that DA is slightly better than CEBA,

while it was found in [44, 45] that the reverse is true. The reason for this discrepancy is

that in this dissertation we did not assume knowledge of the exact number of channel

taps. In our simulations we generated the received signal in continuous time; while in

[44, 45] discrete-time equivalent channels with known �xed lengths are used to directly

generate the signal component of the received waveform samples. It can also be seen

that for d = 0:5, the di�erence between half-symbol-spaced (K = 2) and symbol-

spaced (K = 1) DA DFE's is also slight. This is because d = 0:5 is relatively small

for the DFE structure used here. When d is increased, the performance di�erence

between symbol-spaced and half-symbol-spaced DA DFE's becomes more signi�cant.

Finally, it can be seen that the CEBA DFE with K = 2 is signi�cantly worse than all

the other cases. In fact going from K = 1 to K = 2 incurs a signi�cant performance

loss for the CEBA DFE.

Figure 3.10(b) shows the average BER as a function of normalized delay-spread d


10 15 20Average SNR (dB)

10-1

10

10

10

-3

-4

-2

Ave

rage

BE

RCEBA (K=2)

CEBA (K=1)

DA (K=2)

DA (K=1)

(a)

10-1

10-3

10-2

0.25 0.5 0.75 1.0

DA (K=1)Ave

rage

BE

R

Normalized Delay-Spread

CEBA (K=2)

CEBA (K=1)

DA (K=2)

(b)

Figure 3.10: Average BER of CEBA and DA DFE's: (a) as a function of average

SNR at d = 0:5; (b) as a function of d at 15dB average SNR. The channel is assumed

to have a Gaussian power-delay pro�le with a rms delay-spread of 50 ns and average

delay of 200 ns.


at an average SNR of 15 dB. For the DA approach, it can be seen that as d increases

from 0.25 to 0.75, performance improvement of K = 2 (half-symbol-spaced) over

K = 1 (symbol-spaced) becomes increasingly signi�cant. This is because when d

is small, the channel is close to a at-fading channel, and relatively few taps are

required to handle the small amount of ISI. However, as d increases, the ISI caused

by multipath propagation becomes more severe, and more degrees of freedom in the

DFE are necessary.

For the CEBA approach, on the other hand, K = 2 is much worse than K = 1

for all d. This leads us to conclude that, for the argument for DA DFE's to hold

here, there must be another factor that seriously degrades the performance of K = 2

CEBA DFE. This factor is identi�ed in Section 3.5, where a remedial technique is

described.

It can also be seen that, for both K = 1 and K = 2, DA outperforms CEBA for

all cases simulated here, with larger performance gains at higher d. This is because

the CEBA DFE is more sensitive to variations in d than the DA DFE. CEBA is a

stochastic approach in which the second-order statistics of the signal are computed

using �xed-length CPR estimates. For small d, with high probability the actual

number of channel taps is smaller than or comparable to the number of estimated

CPR taps. The CPR estimates are therefore unbiased with high probability, and the

estimated second-order statistics are accurate for most channel realizations. However,

as d increases, the actual CPR becomes stochastically longer, therefore the CPR

estimates become biased because of insu�cient length, and the estimated second-order

statistics are inaccurate. For the CEBA approach, therefore, the increase of average

BER with d re ects both insu�cient CPR estimate length and insu�cient number

of taps in the DFE �lters to handle the ISI. The DA approach, on the other hand,

is a deterministic approach in which nothing is assumed about the statistics of the

signal. Therefore the performance degradation as d increases re ects only insu�cient

number of taps in the DFE �lters. We can thus conclude that the DA approach is

more robust than the CEBA approach with respect to delay-spread variations.


3.5 CEBA DFE with Regularization

In the previous section, the performance of the half-symbol-spaced CEBA DFE is

much worse than the symbol-spaced CEBA DFE. The performance loss of K = 2

over K = 1 is as large as 2 dB for the high-SNR case. To understand this behavior,

one should refer to the MMSE solution in Equations 3.48 and 3.49. It can be shown

that RP = PP� �HH� is singular if NfK > Nf + �1 + �2 �Nb: For our simulations

we have Nf = 4, Nb = 3 and �1 = �2 = 3. Therefore if K � 2, RP will be singular.

RN , on the other hand, is the autocorrelation matrix of the additive Gaussian noise

after sampling (see Equation 3.44). Since the impulse response of the receive �lter is

a truncated square-root raised-cosine pulse with symbol period T , the autocorrelation

function of the noise component at the output of the receive �lter (before sampling)

is approximately a raised-cosine pulse with symbol period T . Therefore, if only one

sample is taken per symbol period (K = 1), the sampled noise would be approximately

white, and RN is very close to diagonal. In this case RP + RN is likely to be

well-conditioned7, even when RP is singular. However, if the noise is over-sampled

(K > 1), the discrete-time power spectrum of the sampled noise will be very small in

the region(1+�)

K� < j!j < �, where � is the roll-o� factor of the receive �lter (� = 0:35

in our simulations) and ! is the discrete-time frequency. RN , as a consequence, will

be ill-conditioned[38], and RP +RN is likely to be ill-conditioned if RP is singular.

This leads to inaccuracies in solving for the DFE �lter coe�cients for the K = 2 case,

which in turn results in poor BER performance as we have observed in the previous

section.

To remedy the situation, we propose optimizing the fractionally-spaced CEBA

DFE using a \regularized" criterion[42, 46]. \Regularization" or \leakage"[47, 48, 49,

50, 51] is a technique that is used in the context of adaptive �ltering to improve the

stability of numerical algorithms. In this technique, the cost function is augmented by

a term which in some way bounds the energy in adaptive �lter response. The resulting

adaptive algorithms then try to �nd a compromise setting between minimizing the

7A matrix is ill-conditioned if the ratio of the maximum singular value to the minimum singular

value is much greater than 1. It should be kept in mind that here RP is a random matrix.


original cost function and containing the energy of the �lter settings. This technique

has also been applied to restrict the energy contained in the DFE feedback �lter in

order to reduce the e�ect of error-propagation[52]. In this dissertation, we will apply

regularization to the DFE feedforward �lter in order to improve the performance of

the fractionally-spaced DFE. Mathematically, instead of minimizing the approximate

mean-square error (MSE) as de�ned in Equation 3.32, we minimize a generalized

mean-square error, de�ned as

GMSE(wk;bk) = Ejxj � z(j; k)j2 +w�k�wk; (3.57)

where � is a positive-de�nite weighting matrix. The second term in Equation 3.57

prevents the magnitude of the optimal �lter coe�cients from becoming too large, as

often happens in an ill-conditioned problem. Note that � = 0 is equivalent to the

ordinary mean-square error criterion, and� =1I, where I is the identity matrix, will

result in the solution wk = 0. It can be shown[42] that the solution that minimizes

GMSE(�; �) is given by

wk = (PP� �HH� +RN +�)�1� (3.58)

bk = �H�wk (3.59)

It should be noted that Equation 3.58 is very similar to Equation 3.48. The only

di�erence is that in Equation 3.58 (PP� � HH� + RN) is perturbed by � before

it is inverted. Therefore we are guaranteed a well-conditioned matrix even when

(PP� �HH� +RN) is ill-conditioned.

Computer simulations were performed to evaluate the proposed regularization

approach for K = 2 CEBA DFE's, with � = 0:005I. The simulation method is as

described earlier in this section. The results of the regularized half-symbol-spaced

CEBA DFE are plotted in Figure 3.11 as functions of average SNR at d = 0:5 and

d at average SNR of 15 dB. Some results for DA DFE and CEBA DFE without

regularization are also repeated for comparison. It can be seen that regularization is

very e�ective in improving the performance for half-symbol-spaced CEBA DFE's. For


10-1

10-4

10-3

10-2

DA (K=2)

10 15 20

Ave

rage

BE

R

Average SNR (dB)

CEBA (K=2)

CEBA/REG (K=2)

CEBA (K=1)

(a)

10-1

10-3

10-2

Ave

rage

BE

R

0.25 0.5 0.75 1.0Normalized Delay-Spread

CEBA (K=2)

CEBA (K=1)CEBA/REG (K=2)

DA (K=2)

(b)

Figure 3.11: Average BER of the half-symbol-spaced (K = 2) CEBA DFE with

regularization: (a) as a function of average SNR at d = 0:5; (b) as a function of

d at 15dB average SNR. The average BER's CEBA without regularization and DA

with K = 2 are also shown. The channel is assumed to have a Gaussian power-delay

pro�le with a rms delay-spread of 50 ns and average delay of 200 ns.


example, with d = 0:5, at high average SNR regularization improves the performance

of the K = 2 CEBA DFE by 2 dB. Furthermore, the half-symbol-spaced (K = 2)

regularized CEBA DFE outperforms the symbol-spaced CEBA DFE for 0:25 � d �1:0. However, regularized CEBA remains sensitive to variations in d. As mentioned

previously, this is because �xed-length CPR estimates are used to synthesize the

regularized DFE coe�cients. When d is large the estimate is biased, resulting in

DFE �lter coe�cients that are signi�cantly inferior to the DA solution.

3.6 Summary

Inter-symbol interference (ISI), caused by delay spread in a multipath propagation

environment, is one of the limiting factors in high-speed wireless data transmission.

Adaptive equalization is a digital signal processing technique that can be used to

reduce the e�ect of ISI. Among the multitude of adaptive equalization techniques,

the decision-feedback equalizer (DFE) is an attractive candidate for indoor wireless

data communications.

Adaptive decision-feedback equalization using recursive least-squares (RLS) algo-

rithms can be classi�ed into channel-estimation-based adaptation (CEBA) and direct

adaptation (DA). In CEBA, the channel pulse response (CPR) is �rst estimated using

the RLS algorithm, and the DFE coe�cients are computed as functions of the CPR

estimates. In DA, the DFE �lter coe�cients are directly computed from the received

signal samples using the RLS algorithm. In both cases a training sequence is required

to obtain reasonable initial values for the DFE �lter coe�cients. For the DA DFE,

we propose and analyze a synthetic training technique which e�ectively lengthens

the training sequence without incurring any additional transmission overhead. Sim-

ulation results show that this technique can work as good as lengthening the actual

training sequence if the sequence is long enough for the channel delay encountered.

In this dissertation, we have found that 1) DA slightly outperforms CEBA for

symbol-spaced DFE's when the amount of channel delay-spread is moderate; 2)

DA signi�cantly outperforms CEBA for fractionally-spaced cases regardless of delay-

spread; and 3) CEBA is more sensitive to the variations in delay-spread, therefore


both symbol-spaced and fractionally-spaced CEBA DFE's become signi�cantly infe-

rior to the DA DFE as the delay-spread is increased. These results are not entirely

consistent with previously reported results[44, 45]. The reason for the discrepancy

is that, to make our simulations realistic, we did not assume the knowledge of the

actual length of the channel. Therefore when the delay-spread is large for a random

realization of a channel, the CPR estimates used in the CEBA approach are biased,

resulting in poor performance.

We have also found that for fractionally-spaced CEBA DFE's, not only are the

CPR estimates biased, the minimummean-square error (MSE) solution (Equation 3.48)

is also ill-conditioned. To remedy that, we can use a regularization, or leakage, tech-

nique which minimizes a generalized MSE, as described in Equations 3.57, 3.58, and

3.59. Simulation results show that this technique is very e�ective. In particular, a gain

as much as 2dB in average SNR can be achieved by regularization. However, CEBA

with regularization remains more sensitive to variations in delay-spread than the DA

approach. The DA approach is therefore more robust with respect to delay-spread

variations.

Chapter 4

DFE Timing Alignment

It was shown in Chapter 3 that adaptive decision-feedback equalization is an e�ective

technique to use against the impairment of ISI. However, in applying decision-feedback

equalization, the received signal must �rst be sampled at the \best" sampling instants

with respect to the received symbols. If a �nite impulse response (FIR) DFE is used,

as in this dissertation, then a decision delay must also be determined. While the

choice of sampling instants and decision delay is not crucial for equalizers with many

taps, the performance of FIR DFE's with small number of taps is sensitive to the

choice of these parameters[36]. We refer to the optimization of sampling instants and

DFE decision delay as the \timing alignment" problem.

A block diagram of an adaptive DFE is shown in Figure 4.1. As mentioned

previously, the �ltered received signal y(t) is sampled at a rate of K samples per

Tap-Wt.Adapt

xj

+FeedforwardFilter

FeedbackFilterDFE

zj∆y

j+y

j

AlignmentTiming

FilterReceiver r(t)

τ ∆Advance

Sample∆y(t)

Figure 4.1: Block diagram of an adaptive decision-feedback equalizer.

54

CHAPTER 4. DFE TIMING ALIGNMENT 55

symbol-period, where K is assumed to be an integer. The sampling period is therefore

Ts =T

K; (4.1)

where T is the symbol-period. We can denote these samples as

rj = y(jTs + �); j = 0; 1; : : : ; (4.2)

where � is the sampling instant, or symbol-timing, parameter. It is the time at which

the �rst sample is taken. Note that rj in Equation 4.2 and yj;k in Equation 3.17

describe the same discrete-time signal using di�erent indexing schemes. In particular,

we have

yj;k = rK(j+1)�k�1; (4.3)

for j = 0; 1; : : : and k = 0; 1; : : : ; (K � 1). The adaptive DFE then uses the linear

combination of the components of the signal samples yj+�;yj+��1; : : : ;yj+��Nf+1,

de�ned in Equation 3.22, and past decisions x̂j�1; x̂j�2; : : : ; x̂j�Nbto estimate the

transmitted symbol xj. The integer � is the \decision delay" of the DFE[29, 35, 36,

37]. The timing alignment problem, then, refers to the optimization of the parameters

� and �.

4.1 Sampling Instant Optimization

As mentioned previously, Ts-spaced samples of the �ltered received signal y(t) are fed

into the equalizer, where Ts is the tap-spacing of the feedforward sections of the DFE.

The sampling instant � , de�ned in Equation 4.2, re ects the delay introduced by the

channel as well as the band-edge behavior of the frequency spectrum of rj when there

is aliasing, and must be chosen carefully in order to achieve acceptable bit error rate

(BER) performance.

In our approach the sampling instant is acquired by �rst over-sampling y(t). �

is next optimized in two steps. In the �rst step a coarse value t0 is obtained based

on the correlation in time between the sampled received signal and the transmitted


training sequence. This value re ects the average delay introduced by the multipath

propagation channel, and thus will be referred to as the \sampling delay." A �ne-

tuning adjustment �0, where 0 � �0 < Ts, is next determined based on frequency-

domain calculations. This value controls the band-edge behavior of the sampled

received signal, and will be referred to as the \sampling phase." The resulting value

for � is then

� = t0 + �0: (4.4)

Mathematically, let

~rl = y

�lTs

L

�; l = 0; 1; : : : ; (4.5)

be the over-samples of the received signal, where L is the over-sampling factor for

timing alignment. The complex time correlations �m between the transmitted train-

ing sequence and the received signal are then calculated at M -over-sample shifts.

Speci�cally, we have

�m =Q�1Xj=0

x�j~rmM+jL; m = 0; 1; : : : ; dLt0;max

MTse; (4.6)

where Q is the length of the training sequence, M is the resolution in over-samples

to which t0 is estimated, and t0;max is the maximum channel delay expected1. The

resulting value for the sampling delay then corresponds to the position, accurate to

MTs

L, of the correlation peak between the received signal and transmitted training

sequence, i.e.

t0 = m0

MTs

L; (4.7)

where

m0 = argmax fj�mjg : (4.8)

Given t0, the sampling phase is obtained based on frequency-domain calculations.

Let p̂(l)k

denote the Ts-spaced channel pulse response (CPR) estimate corresponding

1In this dissertation we assume that the presence of a data burst is detected with 100% accuracy.

Only the delay introduced by the channel and the appropriate sampling phase remain to be found.

The data burst can be detected by, for example, comparing the correlation peak to a preset threshold.


to sampling delay m0(MTs

L) and sampling phase l(Ts

L), i.e. an estimate of the CPR

de�ned as

p(l)k= p(kTs +m0

MTs

L+ l

Ts

L); k = : : : ;�1; 0; 1; : : : ; (4.9)

where p(t) is de�ned in Equation 3.12. Furthermore, let P̂ (l)(!) denote the discrete-

time Fourier transform of p̂(l)k. Finally, de�ne the quantity SER(l)

1 as

SER(l)1 =

Z 2�

0ln

�N0

2+��P̂ (l)(!)

��2� d!; l = 0; 1; : : : ; (L� 1): (4.10)

The estimated sampling phase then corresponds to the candidate phase with the

highest SER(l)1 , i.e.

�0 = l0Ts

L; (4.11)

where

l0 = argmaxnSER(l)

1

o: (4.12)

The reason for using the metric in Equation 4.10 for sampling phase optimization

is that it is proportional to the logarithm of the maximum output signal-to-MSE

ratio (SER) achievable by a DFE with an in�nite number of taps in the feedforward

and feedback sections[34]. If a particular value of the sampling phase results in poor

performance for an in�nite-length DFE, the performance of a FIR DFE will very

likely be even worse. Therefore, using SER(l)1 can, in principle, preclude the sampling

phases that are likely to result in poor performance.

It should be noted that �rst, the CPR estimate p̂(l)k

can be obtained using least-

squares methods[38]. Furthermore, as described in Chapter 3, a �xed-length CPR

estimate fp̂t;��1 � t � �2g is used. Secondly, the frequency-domain spectra P̂ (l)(!)

can be obtained from p̂(l)kusing the fast Fourier transform (FFT). The spectra P̂ (l)(!)

di�er from each other only in the region where aliasing occurs. Therefore when 1Ts

is

greater than twice the highest frequency component of the received signal spectrum,

SER(l)1 will be equal for all l and �0 cannot be obtained using the proposed method.

This typically happens for a fractionally-spaced DFE (K > 1) where the feedforward

�lter is spaced at less than a symbol period. However, our simulation results show


that in this case sampling phase has an insigni�cant e�ect on the average BER per-

formance. Therefore if a fractionally-spaced DFE is used, only the sampling delay

t0 needs to be optimized. �0 can be set to an arbitrary value after t0 is obtained.

This is because for fractionally-spaced DFE's, the �ltered received signal is sampled

at a frequency higher than the Nyquist rate. Therefore there is no aliasing and the

spectrum of the sampled signal behaves the same in the band-edge, regardless of the

value of �0. Thirdly, for the case when 1Ts

is less than twice the highest frequency

component of the received signal spectrum, the integration in Equation 4.10 needs

to be carried out only over the region where aliasing occurs. This property could

be exploited to reduce the number of mathematical operations required to evalu-

ate SER(l)1 . For example, for square-root raised-cosine spectral shaping with roll-o�

factor � and symbol-spaced DFE, the integration needs to be carried out only for

! 2 [(1� �)�; (1 + �)�]. Finally, instead of computing p̂(l) for every l, we could �rst

estimate the unaliased channel frequency response P̂ (!). The spectra P̂ (l)(!) can

then be obtained by aliasing P (!).

4.2 Decision-Delay Optimization

The performance of the adaptive DFE is very sensitive to the choice of the decision

delay �, especially when number of taps in the feedforward �lter is small. � re ects

the signal spread introduced by multipath propagation and controls the trade-o�

between time diversity and ISI. In general � can be optimized in an a priori or a

posteriori fashion.

4.2.1 A Priori Optimization

Conventionally the value of the decision delay for a DFE is determined in an a priori

manner, i.e. before the DFE operation starts, based on statistical computations. The

optimization metric is usually a function of the CPR estimates. After the decision de-

lay is chosen, the DFE �lter coe�cients can be adaptively computed using algorithms

described in Chapter 3, and the data burst can be decoded. Several researches have


been devoted to e�ciently optimizing the decision delay[29, 35, 37, 53]. One natural

approach is to choose the decision delay that corresponds to the highest SER at the

output of the DFE. The output SER of a DFE is given by

SERDFE(�) =1

MSEmin;k

; (4.13)

where MSEmin;k is given by Equation 3.52. Note that MSEmin;k is implicitly a function

of �. Furthermore, computation of SERDFE(�) requires inverting a Nf�Nf complex

matrix. Al-Dhahir and Cio�[37] developed a fast algorithm for computing MSEmin;k

which can be used to e�ciently optimize �.

SERDFE(�) is the optimal (in the MMSE sense) metric to use if the CPR is known

exactly at the receiver. However, since only an estimate of the CPR is available for

computing SERDFE(�), the decision delay thus obtained is optimal only to within the

accuracy of the CPR estimates. In this case, an approximate, yet computationally

simpler, ad hoc criterion may su�ce to yield a decision delay that corresponds to

a near-optimal BER performance. One such ad hoc optimization metric that we

propose and analyze in this dissertation is the output SER achievable by a FIR

linear equalizer[33] with Nf taps and decision delay �. The tap-weights of this linear

equalizer are set so that they match the CPR within the span of the equalizer. This

metric, denoted as SERLE(�), is de�ned as

SERLE(�) =

��PNf�1m=0 kp̂��mk2

��2P�2

s=��1s6=�

��PNf�1m=0 p̂

��mp̂s�m

��2 +PNf�1s=0

PNf�1m=0 p̂

��mRm�sp̂��s

; (4.14)

where

Rm�s = E [nsn�m] (4.15)

is the noise correlation matrix. For a given channel, Nf and �, it can be shown that

SERLE(�) � SERDFE(�), hence SERLE(�) is in fact a lower bound of SERDFE(�).

Since no matrix inversion is involved, SERLE(�) is easier to compute than SERDFE(�),

especially when the noise is white and the length of the CPR estimate is long. As will

be shown by computer simulation, the two metrics can achieve comparable average


BER performance.

4.2.2 A Posteriori Optimization

A priori optimization of the decision delay based on CPR estimates is suitable for

continuous data transmission over a relatively static channel. For bursty mobile com-

munication applications such as a time division multiple access (TDMA) based indoor

wireless data network, data are transmitted in short blocks. Because of the mobile

nature of wireless communications, the communication channel may be di�erent from

block to block. In this case the \optimal" decision delay that minimizes the over-

all average BER is not only channel-dependent but also data-dependent. Therefore,

for short-burst wireless communications it may be necessary to actually decode the

transmitted data burst using several di�erent decision delays, and choose the \best"

output. This is the a posteriori approach for decision delay optimization. One possi-

ble brute-force a posteriori approach is to store the samples of the received signal and

perform multiple equalization operations using all reasonable values for �. Cyclic

redundancy coding (CRC) checksums could be used to detect errors and the one with

the smallest number of errors is chosen as the �nal output. Although apparently

this approach will yield the \optimal" average BER performance, the computational

cost associated with multiple equalizations will be undesirably high. Furthermore,

this approach relies on the CRC checksum to determine an optimal choice �. This

may require a powerful code and, as a result, incur more transmission overhead. An

e�cient way to reduce the complexity and transmission overhead of the a posteriori

optimization approach while maintaining a low average BER remains to be investi-

gated. The discussion of such techniques is deferred until Chapter 7.


The performance of the timing alignment algorithms presented in this chapter is

evaluated using bit-by-bit computer simulation for both K = 1 (symbol-spaced) and

K = 2 (half-symbol-spaced) DA DFE's. In each experiment, one set of channel


parameters is �rst generated according to the distributions tabulated in Table 2.1.

A burst of 165 4-QAM symbols is next generated and transmitted over the wireless

channel according to Equation 2.17. A square-root raised cosine �lter with a roll-

o� factor of � = 0:35 is used both as the transmit and receive �lters. The �rst 15

transmitted symbols are composed of one cycle of pseudo-random (PN) sequence.

They are assumed to be known to the receiver, and are used for timing alignment as

well as synthetic training to initialize the adaptive DFE. The CPR estimate used for

timing alignment and synthetic training have parameters �1 = �2 = 3. A synthetic

training sequence (STS) with 15 symbols is used for synthetic training as described

in Chapter 3. During the 150 information symbols, the coe�cients of the DA DFE

are updated using the received signal and past decisions, as described previously in

Chapter 3. The number of decision errors is tabulated for these 150 symbols. The

entire process is repeated 15,000 times. After that, it is continued until 3,000 bit

errors are accumulated or 7,500,000 symbols are transmitted, whichever occurs �rst.

As in Chapter 3, in all experiments the feedforward sections of the adaptive DFE

span four symbol periods. The feedback section has three taps spaced at the symbol

period.

4.3.1 Sampling Instant Optimization

The average BER's of the symbol-spaced (K = 1) DA DFE are shown in Figure 4.2 as

functions of the normalized delay-spread, de�ned in Equation 3.56. As in Chapter 3,

the average BER is de�ned as the ratio of the number of accumulated bit errors

to the total number of transmitted bits. Here the channel is assumed to have a

Gaussian power-delay pro�le with a rms delay-spread of 50 ns and average delay of

200 ns. The average SNR is �xed at 15 dB in this �gure. It is de�ned as the ratio

of the signal power, averaged over the Rayleigh distributed multipath fading, to the

mean Gaussian noise power at the output of the receive �lter (see Equations 2.17 and

2.18). For our simulation setup the average SNR is numerically equal to the ratio

of transmitted energy per bit to the two-sided power spectral density of the AWGN.

An over-sampling factor of L = 16 is used, and the decision delay is optimized using


10-3

10-2

10-1

100

τ0t 0τ = +

0tτ =

0.25 0.5 0.75 1.0

Ave

rage

BE

R


τ = 0

τ = τ _

Figure 4.2: The average BER of the symbol-spaced DA DFE using di�erent sampling

instant optimization techniques. The channel has a Gaussian power-delay pro�le with

a rms delay-spread of 50 ns and average delay of 200 ns. The average SNR is 15 dB.

SERLE(�), de�ned in Equation 4.14. Four curves are shown in Figure 4.2. The curve

labeled with \� = 0" is the average BER of the DA DFE using t0 = �0 = 0 for all

d. Recall that t0 and �0, de�ned in Section 4.1, are the \sampling delay" and the

\sampling phase," respectively. The curve labeled with \� = ��" is the average BER of

the DA DFE using t0 = �� and �0 = 0, where �� is the mean path delay, or equivalently,

the �rst moment (mean) of the power-delay pro�le. The curve labeled with \� = t0"

is the BER of the DA DFE using values of t0 obtained using the approach outlined

in Section 4.1 and �0 = 0. Finally, the curve labeled with \� = t0+ �0" is the BER of

the DA DFE with both t0 and �0 optimized as described in Section 4.1.

One can see from Figure 4.2 that optimizing the sampling delay alone greatly

improves the BER performance, especially when d is large. It should be noted that

� is measured relative to the instant at which the data burst is detected, which is

assumed to be at t = 0 (the time at which transmission begins) in this dissertation.

Setting � to 0 is thus equivalent to taking the �rst sample of the �ltered received

signal immediately after the data burst is detected. If the data burst is detected at

t = 0 as assumed here, the performance of the �nite-length DFE will be poor unless


the feedforward �lter of the DFE has many taps. This is because if the �rst sample

is taken at t = 0, the feedforward �lter of the DFE does not capture most of the

energy of the CPR, especially when d is large. For small d, the delay introduced by

multipath propagation is statistically small, and the \best" sampling instant will be

close to t = 0. However, as d increases, the delay introduced by multipath propagation

also increases, and tracking the multipath propagation delay becomes increasingly

important. In reality the data burst can only be detected at some time t > 0. In this

case the time at which the �rst sample is taken will be closer to the optimal choice than

is presented in Figure 4.2. However, it is still important to optimize the parameter

� to re ect multipath propagation delay. The curves \� = t0" and \� = ��" are two

techniques for tracking the multipath propagation delay. For the curve \� = t0",

the sampling delay t0 is optimized using the method described in Section 4.1, while

for \� = �� ," the power-delay pro�le of the channel is assumed to be known and t0

is set to be equal to the �rst moment of the power-delay pro�le. In both cases the

sampling phase �0 is not optimized. It can be seen from Figure 4.2 the two methods

yield almost identical average BER performance, and they all outperform \� = 0"

signi�cantly when d is large. However, explicit knowledge of the mean channel delay

is not required for our method described in Section 4.1.

By comparing the curves \� = t0" and \� = ��" to \� = t0 + �0", one can also see

that optimizing both t0 and �0 gives additional performance gain over optimizing t0

alone. As previously mentioned, �0 controls the band-edge behavior of the sampled

received signal. For symbol-spaced DA DFE's, the sampling rate (one sample per

symbol-period) is less than the Nyquist sampling frequency, hence aliasing occurs in

the sampling process. It is, therefore, important to choose a suitable sampling phase

�0 to eliminate nulls in the frequency region in which aliasing occurs. As can be seen

from Figure 4.2, choosing �0 according to the optimization metric SER1 provides

some performance gain.

Similar curves for the half-symbol-spaced (K = 2) DA DFE are shown in Fig-

ure 4.3. Since the sampling rate (two samples per symbol-period) is higher than the

Nyquist rate, there is no aliasing in the sampled received signal. As mentioned pre-

viously, in this case we can just use a �xed value for the sampling phase �0. Two


10-3

10-2

10-1

100

Ave

rage

BE

R

0tτ = +0.5Ts

0tτ =


τ = 0

Figure 4.3: The average BER of the half-symbol-spaced DA DFE using di�erent

sampling instant optimization techniques. The channel has a Gaussian power-delay

pro�le with a rms delay-spread of 50 ns and average delay-spread of 200 ns. The

average SNR is 15 dB.

such values are shown in Figure 4.3: �0 = 0 and �0 = 0:5Ts, where Ts is the sam-

pling period. It can be seen that these two values yield the same BER performance.

Furthermore, as in the symbol-spaced case, optimizing t0 alone results in a signi�-

cant performance gain. This can be explained using the same argument as presented

earlier in this section.

4.3.2 Decision Delay Optimization

The performance of the decision delay optimization approaches is also evaluated using

computer simulation. The channel is again assumed to have a Gaussian power-delay

pro�le with a rms delay-spread of 50 ns and average delay of 200 ns. For each burst,

� = t0+�0 is optimized as described in Section 4.1 using an over-sampling factor of L =

16. For the a priori optimization approach, SERDFE(�) de�ned in Equation 4.13 and

SERLE(�) de�ned in Equation 4.14 are simulated. For the a posteriori optimization

approach, multiple decision-feedback equalizations are performed using all possible

values of decision delay to yield multiple decoded bursts. The burst with the smallest


DFE/SER LE

10-1

10-4

10-3

10-2

DFE/SER DFE

Ave

rage

BE

RIDEAL

FIXED

Average SNR2010 15

Figure 4.4: Average BER as a function of average SNR for di�erent decision delay

optimization schemes, �xed delay and ideal cases. The channel has a Gaussian power-

delay pro�le with normalized delay-spread d = 0:5. Symbol-spaced DA DFE's are

used.

number of bit errors is then used as the �nal output to tabulate the average BER. Note

that this scheme is not realizable because it requires the knowledge of the transmitted

data symbols at the receiver. A �xed-decision-delay case in which � = 2 is also

simulated to provide a baseline for comparison.

Figure 4.4 shows the average BER for the various decision-delay optimization

schemes as a function of the average SNR with d = 0:5. The curve labeled as \IDEAL"

corresponds to the BER of the unrealizable, a posteriori approach. It can be seen

that at d = 0:5 with average SNR greater than 10dB, SERDFE(�) always outperforms

SERLE(�) as a criterion for decision delay optimization. This shows that with a

50% normalized delay-spread, a 7-tap CPR estimate is adequately accurate as far

as decision delay optimization is concerned. The gap between the two curves is

insigni�cant for low average SNR and approximately 2dB for high average SNR. This

is because for low average SNR the performance is primarily noise-limited and for

high average SNR the performance is primarily ISI-limited. Since the decision delay

controls the trade-o� between time-diversity and ISI, it is more important when the

performance is ISI-limited. It can also be seen that both criteria greatly outperform


0.25 0.5 0.75 1.0

10-1

10-2

10-3

010

SER LE

DFESER


Ave

rage

BE

RFIXED

IDEAL

Figure 4.5: Average BER as a function of normalized delay-spread for di�erent de-

cision delay optimization schemes, �xed delay and ideal cases. The channel has a

Gaussian power-delay pro�le with an average SNR of 15 dB. Symbol-spaced DA

DFE's are used.

the �xed case in which no decision delay optimization is performed, and that both are

signi�cantly inferior to the ideal case in which the number of bit errors is minimized

for each data burst. The latter observation can be explained as follows. For the a

priori decision delay optimization approach, the decision delay is optimized based

on an estimate of the channel before the transmitted data is decoded. However, for

short-burst transmissions the optimal value for decision delay is not only dependent

on the channel, but also dependent on the data burst itself. Therefore an a posteriori

approach such as the ideal case would outperform the a priori approach in terms of

average BER because by actually decoding the data the burst-dependence property

of optimal decision delay is taken into account.

The average BER's of the decision delay optimization approaches are shown in

Figure 4.5 as functions of d with the average SNR �xed at 15 dB. It can be seen

that for all but the �xed decision delay case, the average BER �rst decreases then

increases as d is increased from 0.25. While this phenomenon had been observed

in previous research[54] and Chapter 3 of this dissertation, here we �nd that this

is true only when the trade-o� between time diversity and ISI is handled properly


10-1

10-4

10-3

10-2

DFE/SER DFE

DFE/SER LE

Ave

rage

BE

R

IDEAL

FIXED

Average SNR2010 15

Figure 4.6: Average BER as a function of average SNR for di�erent decision delay

optimization schemes, �xed delay and ideal cases. The channel has a Gaussian power-

delay pro�le with normalized delay-spread d = 0:5. Half-symbol-spaced DA DFE's

are used.

by carefully optimizing the decision delay. The �gure also suggests that SERDFE(�)

outperforms SERLE(�) as an optimization criterion only for d < 0:7. For d > 0:7

SERLE(�) achieves a better average BER performance. This is because as d increases

beyond 0.7 a 7-tap CPR estimate is no longer adequate for timing alignment, and the

sensitivity of SERDFE(�) to CPR estimation inaccuracy begins to show. SERDFE(�)

is inherently more sensitive to CPR estimation inaccuracy because matrix inversion

is involved in its computation. Finally, the ideal case outperforms the realizable

schemes in all values of d at 15 dB average SNR. This suggests the potentially superior

BER performance of the a posteriori optimization approach. Although the scheme

simulated here is unrealizable, a practically realizable a posteriori approach will be

described in Chapter 7 of this dissertation.

Similar BER plots for the half-symbol-spaced (K = 2) DA DFE are shown in

Figures 4.6 and 4.7. The BER's are shown as functions of the average SNR in

Figure 4.6 for d = 0:5, and as functions of d in Figure 4.7 at 15 dB average SNR.

The channel is assumed to have a Gaussian power-delay pro�le. General conclusions

obtained for the K = 1 case also apply to these plots.


0.25 0.5 0.75 1.0

010

10-1

10-2

10-3

10-4

SER LE

DFESER


Ave

rage

BE

R

IDEAL

FIXED

Figure 4.7: Average BER as a function of normalized delay-spread for di�erent de-

cision delay optimization schemes, �xed delay and ideal cases. The channel has a

Gaussian power-delay pro�le with an average SNR of 15 dB. Half-symbol-spaced DA

DFE's are used.

4.4 Summary

Timing alignment refers to the optimization of sampling instants, or symbol-timing,

and DFE decision-delay. This process can be regarded as a procedure to \align"

the DFE with the channel pulse response. In this dissertation, timing alignment is

performed through channel estimation. The sampling instant, � , is �rst optimized in

two steps. In the �rst step, a coarse value t0 is obtained based on the correlation in

time between the sampled receive signal and the transmitted training sequence. This

value re ects the average delay introduced by the multipath propagation channel,

and is referred to as the \sampling delay." A �ne-tuning adjustment �0 is next

determined based on frequency-domain calculations. This value controls the band-

edge behavior of the sampled received signal and is referred to as the \sampling

phase." The resulting value for � is then t0 + �0.

After � is acquired, the decision delay � is next optimized. � controls the trade-

o� between time-diversity and ISI. It can be optimized in an a priori or a posteriori

fashion. For a priori optimization, the value of the decision delay for a DFE is


determined based on statistical computations before the DFE operation starts. After

the decision delay is chosen, the DFE �lter coe�cients can be adaptively computed

using algorithms described in Chapter 3, and the data burst can be decoded. Two

metrics can be used for a priori optimization: SERDFE(�), de�ned in Equation 4.13,

and SERLE(�), de�ned in Equation 4.14. SERDFE(�) is approximately the maximum

SER at the output of a DFE with decision delay �. This quantity can be computed

using e�cient algorithms such as that developed by Al-Dhahir and Cio�[37]. It is the

optimal (in the MMSE sense) metric to use if the CPR is known exactly at the receiver.

However, since only an estimate of the CPR is available for computing SERDFE(�),

the decision delay thus obtained is optimal only to the within the accuracy of the

CPR estimates. In this case, an approximate, yet computationally simpler, ad hoc

metric may su�ce to yield a decision delay that corresponds to a near-optimal BER

performance. SERLE(�) is one such ad hoc metric. It is approximately the output

SER achievable by a FIR linear equalizer with decision delay � and Nf taps matched

to the CPR within the span of the equalizer.

For a posteriori optimization, the data burst is decoded using several di�erent

decision delays, and the \best" output is chosen. This yields better average BER per-

formance for short-burst transmissions over wireless channels where the average BER

is not only channel-dependent but also data-dependent. However, the computational

complexity associated with multiple equalizations is undesirably high. Furthermore, a

mechanism is required to determine which decoded output contains the fewest errors.

A selection mechanism, as well as an e�cient way to reduce computational complex-

ity while maintaining superior performance, are therefore desirable. The discussion

of such techniques is deferred until Chapter 7.

Simulation results show that it is very important to optimize the sampling delay

t0 in order to compensate for the delay introduced by multipath propagation. If a

symbol-spaced DFE is used, optimizing the sampling phase �0 can achieve signi�cant

additional performance gain. Furthermore, it is also very important to optimize the

decision delay �. It is shown by simulation that for a priori optimization, SERDFE(�)

provides better performance for smaller values of the normalized delay-spread d. On

the other hand, SERLE(�) is a better metric to use for larger d. However, the


di�erence between these two approaches is small (within 1 dB for d = 0:5), therefore

the relative simplicity of SERLE(�) justi�es its use even when d is small. Finally,

it is also observed that the unrealizable, \ideal" a posteriori approach simulated

here outperforms all a priori approaches. This provides the incentive for further

investigation, as will be presented in Chapter 7.

Chapter 5

Receiver Diversity

Adaptive DFE has been shown in Chapter 3 to be an e�ective technique to use

against performance degradation due to inter-symbol interference (ISI). However,

when the SNR of a particular realization of the channel is low, the performance of

the adaptive DFE will be severely impaired by the additive white Gaussian noise

(AWGN). Therefore, when the channel delay-spread is moderate so that the adaptive

DFE is capable of suppressing the ISI with high probability, most transmission errors

are due to signal fading. Unfortunately the adaptive DFE is not capable of mitigating

signal fading, therefore for channels with moderate delay-spread, the performance of

the adaptive DFE is ultimately AWGN- and fading-limited.

Diversity, on the other hand, is a technique commonly used in wireless communi-

cation systems to combat signal fading in a at-fading environment[1]. It uses more

than one radio transmission channel to convey the same message in order to reduce

the e�ect of multipath propagation. The di�erent channels are referred to as diversity

branches. If several replicas of the information-carrying signal are received over mul-

tiple channels that have independent propagation parameters, then there is a good

likelihood that at least one of these received signals will not be in a fade at any given

time instant, thus making it possible to deliver adequate signal level to the receiver.

Diversity can greatly reduce the transmitter power required to achieve a certain per-

formance, because without diversity, the transmitter has to deliver a high power level

to protect the radio link against channel realizations with very low SNR's.

71

CHAPTER 5. RECEIVER DIVERSITY 72

There are many di�erent techniques[55, 57] for providing diversity branches with

independent channel statistics. For example, diversity can be achieved using multiple

receiver antennas separated in space (space diversity), having di�erent polarizations

(polarization diversity), or with di�erent beam angles (angle diversity). These tech-

niques are collective known as \antenna diversity" or \receiver diversity." They do

not require additional frequency spectrum resource. For space diversity, the receiver

antennas are separated by a suitable distance so that the corresponding diversity

channels are su�ciently decorrelated. The spacing required to obtain decorrelation

depends on the multipath propagation environment. If the multipath signals arrive

uniformly from all directions, then antenna spacing on the order of a half-wavelength

is adequate[55]. However, if the multipath angle spread is small, larger spacing may

be necessary[55]. Polarization diversity, on the other hand, uses multiple receiver

antennas with di�erent polarizations. The signal can be transmitted either with a

single polarization or in di�erent polarizations. Bergmann and Arnold[56] have ob-

served signal levels received on orthogonal polarizations are largely uncorrelated in

some environments, thus providing diversity gain. Finally, in situations where the

angles of arrival have a wide distribution, as in the case of many indoor wireless

channels, signals collected from multiple non-overlapping beams have approximately

uncorrelated statistics. Angle diversity makes use of this property, and has been

utilized in indoor WLAN's to achieve substantial increase in data throughput[58].

This dissertation focuses on the processing of the multiple signals received from

the diversity branch channels, rather than on the techniques per se used to establish

the diversity branches. Therefore, we assume that multiple receiver antennas are

appropriately con�gured such that the parameters associated with di�erent diversity

channels are independent of each other. We also assume that the information-carrying

signal is transmitted using one carrier frequency, therefore no additional frequency

spectrum resource is required. We further assume that the multiple antennas are

mounted on the same receiver, with the aim of mitigating the small scale e�ects of

multipath propagation. This is often called \microscopic diversity."

One key issue for diversity techniques is the way in which the multiple signals

received through diversity branches are combined. Conventional approaches include


Rcv Sig 1

τ(1)

τ(L)

xj

Rcv Sig L

Filter 1Rcv

Filter LRcv

TimingRecovery

Select"Best"

Figure 5.1: Block diagram for selection diversity. One diversity branch is selected

according to some selection rule.

selection diversity and combining diversity. These approaches are designed and often

analyzed for at-fading channels[1]. In selection diversity, one \best" branch is se-

lected according to some selection rule to yield the combiner output. This approach

is shown in Figure 5.1. In combining diversity, the received signals are �rst each am-

pli�ed by an appropriate gain. They are then co-phased so that the resulting signals

are in-phase. These co-phased signals are �nally summed to produce the combiner

output. A block diagram for combining diversity is shown in Figure 5.2. Depending

on the techniques for setting the ampli�er gains, combining diversity can be further

classi�ed as equal-gain diversity and maximal-ratio combining[1]. In equal-gain di-

versity, all ampli�er gains are set to the same value. In maximal-ratio combining

diversity, on the other hand, the gains are chosen so that the SNR at the combiner

output is maximized.

In frequency-selective, or time-dispersive, environments such as high-speed indoor

wireless data communications channels, multipath propagation causes both signal

fading and ISI. It will be shown later in this chapter that while diversity itself is

capable of combating ISI, it is not as e�ective as the adaptive DFE, especially when

the channel delay-spread is signi�cant. Integration of diversity and adaptive DFE is

therefore desirable for communication systems such as indoor wireless data networks

which operate in delay-spread multipath fading environments. The resulting receiver


Rcv Sig 1

τ(1)

τ(L)

A 1

xj

Rcv Sig L

Filter 1Rcv

Filter LRcv

TimingRecovery

A L

Co-phase

Figure 5.2: Block diagram for combining diversity. The received signals are ampli�ed,

co-phased and summed.

structure is referred to as \adaptive diversity DFE," which can be further classi�ed

as \combining diversity DFE" and \selection diversity DFE." Combining diversity

DFE's are presented in this chapter, while selection diversity DFE's will be discussed

in Chapter 6.

5.1 Combining Diversity DFE

As previously mentioned, in a multipath fading environment where channel delay-

spread is signi�cant, it is desirable to extend conventional diversity techniques and

incorporate them into the adaptive DFE. A receiver structure for integrating combin-

ing diversity and adaptive DFE is shown in Figure 5.3. This structure, �rst proposed

by Monsen[59, 60], comprises one feedforward �lter in each diversity branch and a

common �lter for decision feedback. Since the ISI in di�erent diversity branches are

correlated, the �lters must be jointly optimized to take the ISI correlation into ac-

count. Both the channel-estimation-based adaptation (CEBA) and direct adaptation

(DA) algorithms, presented in Chapter 3, can be extended to accommodate receiver

diversity. For the CEBA approach, the mean-square error (MSE) between transmit-

ted symbol and output of the DFE (input of the slicer) is minimized. For the DA

approach, the weighted-square error (WSE) between transmitted symbol and output


Rcv Sig 1y

j

AlignmentTiming

τ(1)

yj

τ(L)

∆y

j+

∆y

j+

FFF L FBF

xjz(k,k-1)

(1)∆ -sampleAdvance

∆(1)

∆(L)

(L)∆ -sampleAdvance

FFF 1Adapt.

Tap-Wt.(1)

(L)

Rcv Sig L

Filter 1Rcv

(1)

Filter LRcv

(L)

Figure 5.3: The maximal-ratio combining DFE.

of the DFE is minimized. Since both approaches attempt to minimize some measure

of the amount of unwanted component at the DFE output, this structure can be re-

garded as the equalized counterpart of maximal-ratio diversity combining. It will thus

be referred to as maximal-ratio combining decision-feedback equalizer (MRCDFE). If

the lengths of the feedforward �lters are set to 1 and that of the feedback �lter is set

to 0, then the MRCDFE degenerates into the conventional maximal-ratio diversity

combiner.

5.1.1 CEBA MRCDFE and Regularization

It is fairly straightforward to extend the CEBA algorithm of Chapter 3 to accommo-

date receiver diversity. Mathematically, the samples of the �ltered received signal of

the l-th diversity branch can be expressed as

y(l)j;k

= y(l)(jT + � (l) � kT

K) (5.1)

=Xi

xip(l)

(j � i)T + � (l) � kT

K

!+ n(l) q

jT + � (l) � kT

K

!; (5.2)

where j = 0; 1; : : :, k = 0; 1; : : : ; (K�1), and l = 1; 2; : : : ; L, with L being the number

of diversity branches. K is the number of samples taken per symbol period, and is


assumed to be an integer. p(l)(�) in Equation 5.2 is the response, after being �ltered

by the receiver �lter, of the l-th branch channel to the transmitted pulse. n(l)(�) isthe AWGN in the l-th channel. � (l) in Equations 5.1 and 5.2 is the sampling instant

of the l-th diversity branch. Let

y(l)j

=

26666664

y(l)j;0

y(l)j;1

: : :

y(l)j;K�1

37777775

(5.3)

=Xi

xip(l)j�i + n

(l)j

(5.4)

where

p(l)i =

26666664

p(l)i;0

p(l)i;1

: : :

p(l)i;K�1

37777775; (5.5)

n(l)j=

26666664

n(l)j;0

n(l)j;1

: : :

n(l)j;K�1

37777775; (5.6)

p(l)i;k

= p(l) iT + � (l) � kT

K

!; (5.7)

and

n(l)i;k

= n(l) q

iT + � (l) � kT

K

!: (5.8)

Here \" denotes linear convolution de�ned in Equation 3.6. Then the output (equal-ized signal) z(j; k) of the adaptive DFE at time j using the �lter coe�cients obtained

at time k can be expressed as

z(j; k) =LXl=1

Nf�1Xi=0

w(l)�i;ky(l)

j+�(l)�i +NbXi=1

b�i;kx̂j�i; (5.9)


where w(l)i;k

are K � 1 feedforward coe�cient vectors for the l-th branch and bi;k are

the feedback coe�cients, all optimized at time k. �(l) is the decision delay for the

l-th branch. Equation 5.9 can be expressed alternatively as

z(j; k) = [W�k;b�

k]Yj; (5.10)

where

Wk =

26666664

w(1)k

w(2)k

: : :

w(L)k

37777775; (5.11)

bk =

26666664

b1;k

b2;k

� � �bNb;k

37777775; (5.12)

w(l)k=

26666664

w(l)0;k

w(l)1;k

� � �w

(l)Nf�1;k

37777775; (5.13)

and

Yj =

266666666666666666664

Y(1)

j+�(1)

Y(2)

j+�(2)

: : :

Y(L)

j+�(L)

x̂j�1

x̂j�2

: : :

x̂j�Nb

377777777777777777775

: (5.14)


In Equation 5.14, Y(l)

j+�(l) is given by

Y(l)

j+�(l) =

266666664

y(l)

j+�(l)

y(l)

j+�(l)�1

� � �y(l)

j+�(l)�Nf+1

377777775: (5.15)

For the CEBA approach, the channel pulse responses (CPR) for each diversity branch

are �rst estimated as described in Chapter 3. At every symbol period, after the CPR

estimates are obtained, the DFE �lter coe�cients are computed to minimize the

approximate MSE at the DFE output (input of the slicer), de�ned as

MSE(Wk;bk) = Ejxj � [W�k;b�

k]Yjj2: (5.16)

Note that Equation 5.16 is identical in form to Equation 3.32, therefore the corre-

sponding solution is similar to the solution for the single-branch case presented in

Chapter 3. Following the same derivations given in Chapter 3, it is straightforward

to show that the optimal DFE coe�cients are given by

Wk = (PP� �HH� +RN)�1� (5.17)

bk = �H�Wk (5.18)

where

P =

26666664

P(1)

P(2)

� � �P(L)

37777775; (5.19)

H = P

26664

0�1+1;Nb

INb

0�2+Nf�Nb�1;Nb

37775 ; (5.20)


RN = E

0BBBBBB@

26666664

n(1)j

n(2)j

� � �n(L)j

37777775hn(1)�j

;n(2)�j

; � � � ;n(L)�j

i1CCCCCCA; (5.21)

and

� = Pe�1: (5.22)

In Equation 5.19, P(l) are NfK � (�1 + �2 + Nf) matrices composed of Nf rows of

column vectors, each with K components. Speci�cally, P(l) is de�ned as

P(l) =

26664

p̂(l)

��1+�(l)

p̂(l)

��1+1+�(l)

� � � p̂(l)

�2+�(l)

0K 0K � � � 0K

0K p̂(l)

��1+�(l)

p̂(l)

��1+1+�(l)

� � � p̂(l)

�2+�(l)

0K � � � 0K

� � � � � � � � � � � � � � � � � � � � � � � �

0K 0K � � � 0K p̂(l)

��1+�(l)

p̂(l)

��1+1+�(l)

� � � p̂(l)

�2+�(l)

37775 ;

(5.23)

where 0K refers to a K�1 block of zeros. In Equation 5.20, 0a;b denotes an a�b block

of zeros, and INbdenotes an identity matrix of size Nb. H is, therefore, a NfKL�Nb

matrix. In Equation 5.21, n(l)j

are K � 1 vectors de�ned as

n(l)j=

26666664

n(l)j;0

n(l)j;1

� � �n(l)j;K�1

37777775

(5.24)

for l = 1; 2; : : : ; L. Finally, in Equation 5.22, e�1 is a (�1+�2+Nf )�1 column vector

with a 1 in the �1-th position and 0 elsewhere.

It can be shown that RP = PP��HH� is singular if NfKL > Nf + �1+ �2�Nb:

RN , on the other hand, is ill-conditioned if the noise is over-sampled (K > 1), as

explained earlier in Chapter 3. In the case when both RP and RN are ill-conditioned,

RP+RN is likely to be ill-conditioned, and regularization may be necessary to improve

the performance of the MRCDFE. As in Chapter 3, the regularized MRCDFE �lter

coe�cients approximately minimizes the generalized MSE, de�ned as

GMSE(Wk;bk) = Ejxj � z(j; k)j2 +W�k�Wk (5.25)


where � is a positive-de�nite weighting matrix. The solution of this minimization

problem is given by

Wk = (PP� �HH� +RN +�)�1� (5.26)

bk = �H�Wk (5.27)

5.1.2 DA MRCDFE with Synthetic Training

For the DA approach, the LK � 1 vector

~Yj =

26666664

y(1)

j+�(1)

y(2)

j+�(2)

: : :

y(L)

j+�(L)

37777775

(5.28)

and the past decisions (transmitted symbols if available) are used to compute Wk

and bk using the least-squares lattice (LSL) algorithm given in Appendix C. Note

that the feedforward and feedback stages have dimensionalities KL and KL + 1,

respectively. If synthetic training is used, a synthetic training sequence (STS) can

�rst be produced for and prepended to the received signal samples from each branch

channel, as described in Chapter 3.


The performance of the DA and CEBA MRCDFE are evaluated using bit-by-bit

computer simulation for both symbol-spaced (K = 1) and half-symbol-spaced (K =

2) cases. In each experiment, L sets of channel parameters are �rst independently

generated according to the previously mentioned distributions, where L is the number

of branch channels. A burst of 165 4-QAM symbols is generated and transmitted over

the wireless diversity channels according to Equation 2.17. The diversity channels are

assumed to be independent and have Gaussian power-delay pro�les with rms delay-

spreads of 50 ns and average delays of 200 ns. A square-root raised-cosine �lter with


a roll-o� factor of � = 0:35 is used as the transmit �lter. The �rst 15 transmitted

symbols are composed of one cycle of PN sequence, and are assumed to be known to

the receiver. This sequence is used for equalizer initialization, timing alignment and

synthetic training. The remaining 150 symbols are assumed to be the information

symbols.

At the receiver, the received signals are each �ltered by a lowpass �lter with a


independently on each of the L �ltered signals using the approach outlined in Chap-

ter 4. SERLE(�), de�ned in Equation 4.14, is used as the criterion for decision delay

optimization. After the timing alignment parameters are acquired, the received sig-

nal samples are fed through the MRCDFE to yield the demodulated output. It is

assumed that there is no frequency o�set between the oscillators of the transmitter

and receiver. The DFE �lter coe�cients are jointly optimized in a decision-directed

manner using the CEBA and DA approaches. For the CEBA approach, 7-tap CPR

estimates are used for each branch, i.e. �1 = �2 = 3. For the DA approach, a STS with

15 symbols is used for synthetic training. The number of decision errors is tabulated

for the 150 information symbols. The entire process is repeated 15,000 times. After

that, it is continued until 3,000 bit errors are accumulated or 7,500,000 symbols are

transmitted, whichever occurs �rst. In all experiments the feedforward sections of the

adaptive DFE span four symbol periods. The feedback section has three taps spaced

at the symbol period.

Figure 5.4 shows the average BER of the dual-diversity (L = 2) symbol-spaced

(K = 1) and half-symbol-spaced (K = 2) DA and CEBA MRCDFE's. The average

BER's of the half-symbol-spaced (K = 2) DA DFE without diversity (DFE-ONLY)

and the dual-branch combining diversity receiver without equalization (DIV-ONLY)

are also shown as baselines for comparison. The average BER's are shown as func-

tions of the average SNR in Figure 5.4(a). Here the average SNR refers to the ratio

of the signal power, averaged over the Rayleigh distributed multipath fading, to the

mean Gaussian noise power at the output of the receive �lter for each branch. For

our simulation setup the average SNR is numerically equal to the ratio of transmitted

energy per bit to the two-sided power spectral density of the AWGN. The normalized


10-1

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15 2010Average SNR (dB)

DA (L=2,K=2)

DA (L=2,K=1)

CEBA (L=2,K=1)

DIV-ONLY

DFE-ONLY(DA,K=2)CEBA (L=2,K=2)

(a)

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DA (L=2,K=2)

CEBA (L=2,K=1) DA (L=2,K=1)

DIV-ONLY(DA,K=2)

DFE-ONLYCEBA (L=2,K=2)

(b)

Figure 5.4: Average BER's of the unequalized diversity combiner (DIV-ONLY), half-

symbol-spaced DA DFE without diversity (DFE-ONLY), and dual diversity (L = 2)

DA and CEBA MRCDFE's: (a) as functions of average SNR at d = 0:5; and (b) as

functions of d at 15dB average SNR. The channel has a Gaussian power-delay pro�le

with a rms delay-spread of 50 ns and average delay of 200 ns.


delay-spread d, de�ned in Equation 3.56, is at 0.5. For the DIV-ONLY case, the sig-

nals received from the diversity branches are co-phased and combined with weights

proportional to the received power averaged over the 165-symbol burst. Comparing

the DIV-ONLY case to the DFE-ONLY case, one can see that at d = 0:5, diver-

sity outperforms adaptive DFE when the average SNR is lower than 15 dB. This

is because when the average SNR is low, the performance is primarily AWGN- and

fading-limited. As mentioned previously, receiver diversity is very e�ective in a noise-

limited environment. Adaptive DFE, on the other hand, does not perform well against

AWGN. Furthermore, when the average SNR is low, decision errors occur often and

the error propagation associated with decision-feedback and decision-directed adap-

tation causes the DFE �lter coe�cients to diverge from the optimal values. When

the average SNR is higher than 15 dB, adaptive DFE outperforms the unequalized

diversity receiver. This is because in the high average SNR region, performance is

limited by the ISI. While receiver diversity is very e�ective against signal fading, it

is not as e�ective as the adaptive DFE in combating ISI.

Among the various MRCDFE's, it can be seen that both the symbol-spaced (K =

1) DA and CEBA MRCDFE with two branches (L = 2) outperform the DFE-ONLY

case by approximately 6�7 dB. Their performance gain over the DIV-ONLY case is

in excess of 10 dB. For the half-symbol-spaced (K = 2) case, on the other hand,

dual-diversity (L = 2) provides a gain of 7�8 dB over the DFE-ONLY case when the

DA approach is used to update the DFE �lter coe�cients. The performance gain over

the DIV-ONLY case is again more than 10 dB. However, when the CEBA approach

is used, the performance gain of the dual-diversity MRCDFE over the DFE-ONLY

case is only 2�3 dB. In fact, dual-diversity CEBA MRCDFE is slightly inferior to

the DIV-ONLY receiver when the average SNR is low.

The average BER's of the approaches shown in Figure 5.4(a) are plotted in Fig-

ure 5.4(b) as functions of the normalized delay-spread, with the average SNR �xed

at 15 dB. We can see that for d < 0:5, diversity combining outperforms DFE. This

is because for small values of d, the ISI introduced by multipath propagation is in-

signi�cant, and transmission errors are mainly caused by signal fading. Since receiver

diversity is more e�ective in combating fading, DIV-ONLY outperforms DFE-ONLY


for low values of d. For values of d beyond 0.5, DFE-ONLY outperforms DIV-ONLY.

This is because as d increases, the multiple signals become more separated in delay,

thus causing a signi�cant amount of ISI. An adaptive DFE with su�cient number

of �lter taps is capable of resolving and combining constructively the multiple signal

copies, thus bene�ting from the time diversity implicit in multipath propagation. An

unequalized diversity combiner, on the other hand, is not as e�cient in coping with

ISI. Therefore the DFE-ONLY outperforms DIV-ONLY when d is large.

Among the MRCDFE's, the dual-diversity (L = 2) symbol-spaced (K = 1) DA

and CEBA MRCDFE's achieve average BER's that are almost two orders of mag-

nitude lower than that of the DFE-ONLY case. Compared to the DIV-ONLY case,

dual-diversity symbol-spaced DA and CEBA MRCDFE's achieve average BER's that

are between one to two orders of magnitude lower. The dual-diversity (L = 2)

half-symbol-spaced (K = 2) DA MRCDFE also attains a much better performance

than the DIV-ONLY and DFE-ONLY cases. However, the BER of the dual-diversity

(L = 2) half-symbol-spaced (K = 2) CEBA MRCDFE is less than one order of

magnitude lower than that of DFE-ONLY only when d < 0:8. For d > 0:8, the dual-

diversity half-symbol-spaced CEBA MRCDFE performs worse than the DFE-ONLY

case. Furthermore, the dual-diversity half-symbol-spaced CEBA MRCDFE is inferior

to the DIV-ONLY receiver for values of d close to 0.25 and 1.0. The poor performance

of the fractionally-spaced CEBA MRCDFE is due to the poor numerical properties of

the MMSE solution, as explained in Chapter 3. As will be shown later, this problem

can be remedied using regularization.

It can also be seen from Figure 5.4 that for the dual-diversity (L = 2) cases, the

general conclusions obtained in Chapter 3 regarding the relative performance of the

CEBA and DA approaches also apply. In particular, we �nd that 1) DA slightly out-

performs CEBA for symbol-spaced MRCDFE's when the amount of channel delay-

spread is moderate; 2) DA signi�cantly outperforms CEBA for fractionally-spaced

cases regardless of delay-spread; and 3) CEBA is more sensitive to the variations

in delay-spread, therefore both symbol-spaced and fractionally-spaced CEBA MR-

CDFE's become signi�cantly inferior to the DA MRCDFE if the delay-spread is

increased. These observations can be explained using arguments similar to those


presented in Chapter 3.

The average BER of the dual-diversity (L = 2) half-symbol-spaced (K = 2)

CEBA MRCDFE with regularization is shown in Figure 5.5. In Figure 5.5(a), the

BER's are shown as functions of the average SNR, with normalized delay-spread

d = 0:5. In Figure 5.5(b), the BER's are shown as functions of the normalized delay-

spread, with an average SNR of 15 dB. The BER's of the dual-diversity symbol-

spaced (K = 1) CEBA MRCDFE and half-symbol-spaced DA and CEBA MRCDFE

without regularization are also repeated here. The BER of the DFE-ONLY case is

also shown. It can again be seen that regularization is very e�ective in improving the

performance for half-symbol-spaced CEBA MRCDFE's. For example, with d = 0:5,

at high average SNR regularization improves the performance of the K = 2 CEBA

MRCDFE by 6 dB1. With regularization, the performance gain of the dual-diversity

(L = 2) CEBA MRCDFE over the DFE-ONLY case is 7�8 dB at d = 0:5. At 15 dB

average SNR, the average BER of the dual-diversity (L = 2) CEBA MRCDFE with

regularization is approximately two orders of magnitude lower than the DFE-ONLY

case for 0:25 < d < 0:75. However, regularized CEBA MRCDFE remains sensitive to

variations in d. As mentioned previously, this is because �xed-length CPR estimates

are used to synthesize the regularized DFE coe�cients. When d is large the estimate

is biased, resulting in DFE �lter coe�cients that are signi�cantly inferior to the DA

solution.

5.3 Summary

High-speed indoor wireless data networks encounter both signal fading and channel

delay-spread. Adaptive DFE has been shown in Chapter 3 to be very e�ective against

the performance degradation due to the ISI caused by channel delay-spread. How-

ever, when the average SNR is low, the performance of an adaptive DFE is limited

by signal fading. Receiver diversity, on the other hand, is a technique commonly

used in wireless communication systems to combat signal fading. Diversity uses more

1A gain of 2dB was obtained for the DFE-ONLY case in Chapter 3.


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15 2010Average SNR (dB)

DA (L=2,K=2)

(L=2, K=2)CEBA/REG

CEBA (L=2,K=2)

CEBA (L=2,K=1)

DFE-ONLY(DA,K=2)

(a)

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DA (L=2,K=2)

CEBA/REG(L=2,K=2)CEBA (L=2,K=1)

CEBA (L=2,K=2)

DFE-ONLY(DA,K=2)

(b)

Figure 5.5: Average BER's of the half-symbol-spaced DA DFE without diversity

(DFE-ONLY), dual diversity (L = 2) DA MRCDFE, and dual diversity (L = 2)

CEBA MRCDFE with regularization: (a) as functions of average SNR at d = 0:5;

and (b) as functions of d at 15dB average SNR. The channel has a Gaussian power-

delay pro�le with a rms delay-spread of 50 ns and average delay of 200 ns.


than one radio transmission channel to convey the same message. The e�ect of sig-

nal fading can be mitigated if the signals from the diversity branches are properly

combined. Although diversity is very e�ective against signal fading, it alone cannot

yield satisfactory performance in a time-dispersive, or frequency-selective, fading en-

vironment. Integration of diversity and adaptive equalization is therefore desirable

for communication systems such as indoor wireless data networks which operate in a

delay-spread multipath fading environment.

Receiver diversity can be classi�ed according to the techniques used to combine

the branch signals. The simplest form of diversity combining is selection diversity.

In selection diversity, one branch is selected as the combiner output according to

some selection rule. Another diversity approach is combining diversity, in which the

weighted sum of the co-phased branch signals is used as the combiner output. These

techniques were designed and often analyzed for at-fading environments, and must

be modi�ed to accommodate equalization in order to yield satisfactory performance

in a time-dispersive fading environment. The integration of combining diversity and

adaptive DFE can be achieved using a structure referred to in this dissertation as

the maximal-ratio combining DFE (MRCDFE). This structure comprises multiple

feedforward �lters and a common feedback �lter, which are jointly optimized using

the CEBA or DA algorithms. The extension of these algorithms are described in this

chapter.

Simulation results show that the MRCDFE can achieve a signi�cant performance

gain over both the adaptive DFE without diversity and the unequalized diversity re-

ceiver. In particular, for 50% normalized delay-spread, a gain of 6�8 dB is achieved

over the adaptive DFE without diversity. The gain over the unequalized diversity

receiver is in excess of 10 dB. The relative performance of the CEBA and DA MR-

CDFE's are also analyzed, and it is found that the general conclusions obtained in

Chapter 3 also apply. Furthermore, just as in Chapter 3, half-symbol-spaced CEBA

MRCDFE encounters numerical di�culties in computing the minimum mean-square

error (MMSE) �lter coe�cients. Regularization is shown to be an e�ective remedy

for this situation, and is very e�ective in improving the performance of fractionally-

spaced CEBA MRCDFE's.

Chapter 6

MAP Selection Diversity DFE

The maximal ratio combining DFE (MRCDFE) presented in Chapter 5 integrates

adaptive DFE and combining diversity. While it is very e�ective in mitigating the

e�ects of multipath propagation, the MRCDFE is computationally intensive because

the branch DFE �lter coe�cients are jointly optimized. Integration of adaptive DFE

and selection diversity, on the other hand, provides a simpler alternative because

the branch DFE's can be individually optimized if selection diversity is used. A

wide variety of approaches exist for integrating selection diversity and equalization.

For example, selection diversity could be performed at the input or output of the

equalizer[61], as shown in Figure 6.1. Furthermore, selection could be performed on

a per-frame or per-symbol basis[61], as shown in Figure 6.2. The metric for selec-

tion is also an extremely important design parameter, and di�erent choices have been

investigated[62, 63, 64]. In general the \best" choice will depend, among other fac-

tors, on the characteristics of the channel, adaptive equalization algorithm, equalizer

structure, and tolerable computational complexity. For adaptive DFE's operating on

indoor high-speed wireless radio links, symbol-by-symbol selection at the DFE output

(input of the decision device) is attractive. This approach, shown in Figure 6.3, will

be referred to as the selection diversity DFE (SDDFE). First, indoor high-speed wire-

less radio links generally have low mobile speed and short symbol period. Thus the

channels are approximately time-invariant. This simpli�es the design of the selection

metric. Secondly, the �lter coe�cients of an adaptive DFE are updated at the end

88

CHAPTER 6. MAP SELECTION DIVERSITY DFE 89

Rcv Sig 1

∆y

j+

DFE xjAdvanceSample∆

Rcv Sig L

Filter 1Rcv

Filter LRcv

Select

AlignmentTiming

yj

τ ∆

(a)

DFE L

DFE 1Rcv Sig 1

yj ∆

yj+

AlignmentTiming

τ(1) ∆(1)

yj ∆

yj+

τ(L) ∆(L)

Select xj



Rcv Sig L

Filter 1Rcv

(1)(1)

Filter LRcv

(L)(L)

(b)

Figure 6.1: Possible structures for the selection diversity DFE: (a) Selection is done

before equalization; (b) Selection is done after equalization.


Per-frame selection Symbol-by-symbol selection

Figure 6.2: Possible selection schemes for the SDDFE.

Rcv Sig 1y

j

AlignmentTiming

τ(1) ∆(1)

yj

τ(L) ∆(L)



zk(1)

zk(L)

FinalDecision

θk(L)

θk(1)

FFF 1

FFF L

Feedback L

Feedback 1

SelectionMetric 1

SelectionMetric L

Tap-Wt.Adapt. L

Tap-Wt.Adapt. 1

Per-symSelect

∆y

j+

∆y

j+

Rcv Sig L

Filter 1Rcv

(1)

Filter LRcv

(L)

(1)

(L)

Figure 6.3: Selection-diversity decision-feedback equalizer.


of every symbol period. In this case symbol-by-symbol selection diversity can, with

a properly designed selection rule, dynamically select the equalizer which will yield a

lower probability of error. Selection on a per-frame basis, in contrast, locks on to one

particular branch DFE even when it is outperformed by other branch DFE's some

time after the selection is done. Thirdly, symbol-by-symbol selection makes possi-

ble the feedback and adaptation structure shown in Figure 6.3. More speci�cally, at

the end of every symbol period the �nal (and hence more reliable) decision is fed

back through the feedback sections to all branch DFE's. It is also used to adapt

the DFE �lter settings for all diversity branches. This should somewhat mitigate the

e�ect of error propagation in decision-directed adaptive DFE's. It was also shown

that while the MRCDFE is theoretically very e�cient in suppressing both ISI and

signal-fading, when some timing alignment parameters are not chosen properly, the

MRCDFE may in fact have a poorer performance than certain selection diversity

combining approaches if the number of training symbols is limited[43, 63]. Finally,

SDDFE has the merit of having lower complexity than MRCDFE, especially when a

simple selection metric is used[62].

In this chapter, we investigate the SDDFE for the direct adaptation (DA) least-

squares lattice (LSL) DFE. While this structure has been proposed and analyzed

by previous researchers[61, 62], in this chapter selection is performed at the bit-

level instead of the symbol-level. We also develope a new selection metric based

on a maximum a posteriori probability (MAP) selection rule, which is shown to be

optimal for a SDDFE in the MAP sense. Simulation results show that the new

method developed in this chapter is very e�ective. In particular, a gain of as high as

2 dB in average signal-to-noise ratio (SNR) over a conventional scheme is achieved.

The new scheme is also compared to the MRCDFE, and the results show that the

new approach can achieve almost the same performance as the MRCDFE with lower

computational complexity.


6.1 Selection Diversity DFE

The selection diversity DFE (SDDFE) is shown in Figure 6.3. It consists of L branch

DFE's, a selection metric estimator, a selection combiner, and a decision device. At

the end of every symbol period the output of one branch DFE is selected according

to the estimated selection metric. Note that selection is done at the bit-level, i.e.

the in-phase (real part) and the quadrature-phase (imaginary part) components of

the SDDFE output are independently selected and may come from di�erent branch

DFE's. The selected output is passed on to the decision device to obtain the de-

modulated symbol. This �nal and hence more reliable decision is then used as the

input to the feedback sections of the DFE's and to adapt the DFE �lter settings in a

decision-directed manner. The branch DFE's are individually adapted using the DA

algorithm of Chapter 3.

SDDFE has been proposed and analyzed in several previous researches[61, 62, 63].

However, our approach presented here di�ers from the previously proposed approaches

in two signi�cant aspects. First, in our approach selection is done at the bit-level in-

stead of symbol-level as previously proposed. Intuitively this should result in a better

performance provided that an e�ective metric for selection is used. Secondly, the

selection metric used in this dissertation is derived from the maximum a posteriori

(MAP) selection rule. Conceptually, the a posteriori probabilities of correctly de-

tecting the in-phase (quadrature-phase) information symbol component conditioned

solely on the real part (imaginary part) of one branch DFE output are �rst estimated

for all L diversity branches. The real part (imaginary part) of the branch DFE output

with the highest a posteriori probability of begin correct is then used to produce the

in-phase (quadrature-phase) component of the �nal decision. This is considerably dif-

ferent from the previously described approaches[61, 62, 63], where selection is based

on the average di�erence between the input and output of the decision device. In

fact, as proved in Appendix D, the MAP selection rule is optimal for a SDDFE in

the MAP sense, i.e. it maximizes the conditional probability of making a correct �nal

decision given all the outputs of the branch DFE's. Furthermore, as will be shown in

Section 6.2, the selection metric resulting from the MAP selection rule is very simple


under a certain assumption on the distribution of the ISI.

6.2 MAP Selection Metric for SDDFE

The mathematical derivation for the MAP selection metric will be presented in this

section. The derivations presented here are equally applicable for the real (in-phase)

and imaginary (quadrature-phase) components. For signal constellations such as

square QAM, the same formula for selection metric can be applied for the in-phase

and quadrature-phase components. In the general case di�erent formulae will have

to be used for di�erent components. The extension to these cases are, however,

straightforward. In order to simplify notations, in this chapter we use bold-face

symbols to denote complex quantities, while plain-text symbols are used to denote

generically the real or imaginary part of the complex quantity being considered. For

example, the transmitted information symbol xk is a complex quantity which takes

values of f1 + j; 1 � j;�1 + j;�1 � jg. xk is therefore the real or imaginary part ofxk and takes values of f�1;+1g. It should be kept in mind that all quantities in this

chapter are scalar quantities.

Consider L independent diversity branches equalized by the SDDFE as shown

in Figure 6.3. The a posteriori probability of correct detection for the l-th DFE

conditioned solely on its output can be expressed as

P(l)k(z

(l)k) = Prob

hx̂(l)k= xkjz(l)k was observed

i; (6.1)

where xk is the transmitted symbol at time k, x̂(l)k

is the decision of the l-th branch

DFE based on its output z(l)k

according to some decision rule. Using Bayes' rule

Equation 6.1 can be rewritten as

P(l)k(z

(l)k) =

24f (l)k

(z(l)kjxk = x̂

(l)k)

f(l)k(z

(l)k)

35� Prob[xk = x̂

(l)k]

= �(l)k� Prob[xk = x̂

(l)k]; (6.2)


where f(l)k(�j�) and f

(l)k(�) are the conditional and marginal probability density func-

tions (pdf) of z(l)k, respectively, and �

(l)k

is the ratio of the two. The MAP selection

rule selects as the �nal decision the output of the branch with the highest a posteriori

probability of being correct. Throughout this dissertation we will assume xk to be

uniformly distributed, therefore Prob[xk = x̂(l)k] is the same for all l. Hence the MAP

selection rule can equivalently be expressed as

x̂k = x̂(l)k

if �(l)k� �

(m)k

for all m 6= l. (6.3)

Assuming that the fedback decisions are error-free, the equalizer output z(l)k

of the

l-th diversity branch can be expressed as

z(l)k

= �(l)kxk + �

(l)k

(6.4)

where

�(l)k= Re

8<:Ehz(l)kx�k

iEhjxkj2

i9=; (6.5)

and �(l)k

is the noise component which is uncorrelated with xk, i.e.

E[xk�(l)k] = 0 (6.6)

For a multipath channel with AWGN, �(l)k

consists of the residual ISI, cross-talk

between the in-phase or quadrature-phase component, and �ltered Gaussian noise.

In general the exact distribution of �(l)k

is, if at all possible, very di�cult to obtain.

However, we will assume �(l)kto be zero-mean Gaussian random variables with variance

�(l)2k

. One reason for making this assumption is that for a static channel and time-

invariant DFE, �(l)k

is a sum of several zero-mean uncorrelated random variables, and

its distribution will be approximately Gaussian if the number of summands is large.

The multipath fading channel that is being considered here is, though not static,

very slowly varying. The Gaussian assumption greatly simpli�es the mathematics.

Then, as will be shown later, the resulting metric for selection proves to be e�ective

in signi�cantly reducing the average BER, further justifying the assumption.


Under the assumption of Gaussian distributed �(l)k, it is straightforward to derive

a formula for �(l)k. In particular, we have

f(l)k

�z(l)kjxk = x̂

(l)k

�=

1p2��

(l)k

exp

0B@�

�z(l)k� �

(l)kx̂(l)k

�22�

(l)2k

1CA (6.7)

For the 4-QAM modulation that is being considered here, we have

x̂(l)k=

8<: 1 if z

(l)k� 0

�1 if z(l)k

< 0: (6.8)

Therefore,

f(l)k

�z(l)k

�=

1p2��

(l)k

26412exp

0B@�

�z(l)k� �

(l)k

�22�

(l)2k

1CA+

1

2exp

0B@�

�z(l)k+ �

(l)k

�22�

(l)2k

1CA375 ; (6.9)

thus

�(l)k=

2

1 + e�2�(l)

k

; (6.10)

where

�(l)k=

�(l)k

��z(l)k

��(l)2k

: (6.11)

Therefore the MAP selection rule for the new SDDFE is

x̂k = x̂(l)k

if �(l)k� �

(m)k

; for all m 6= l. (6.12)

where �(l)k

is the selection metric de�ned in Equation 6.11. Note that this metric

depends on the statistical parameters �(l)k

and �(l)2k

which, as will be discussed in

Sections 6.3, can be easily computed during the adaptation process.


6.3 Computation of Selection Metric

In order to make use of the MAP selection rule, the parameters �(l)k

and �(l)2k

must

be estimated. For simplicity we will omit the superscript \(l)" since it is understood

that these parameters are independently estimated for each diversity branch. We will

also use the notations �k and �2kto denote the statistical parameters and �̂k and �̂2

k

to represent the estimates of the statistical parameters.

To obtain an estimate for �2k, we start with the lowpass-�ltered squared one-step

output error (FSE), de�ned as

Ek = LPFk

hjzj � xjj2

i; (6.13)

where zj and xj are the output of the DFE and transmitted symbol, respectively,

and the operator LPFk [gj] denotes the output at time k of a lowpass �lter with

input signal gj. This quantity is simple to calculate and has previously been used

alone[62] as the selection metric for SDDFE. In this dissertation, a �rst-order single-

pole lowpass �lter with coe�cient 0.5 is used. This number is chosen empirically based

on some preliminary simulations on SDDFE using FSE alone as the selection metric1.

As will be shown by simulation, using FSE alone as the selection metric yields worse

average BER performance than the metrics developed in this dissertation.

Since Ek consists of the weighted sum of quantities which are correlated with the

signal xk, it cannot be used directly as an estimate of �2k, and �k must be estimated

�rst. A candidate for the decision-directed estimate of �k would be

�̂k = Re

8<:LPFk

hzjx

�j

iLPFk

hjxjj2

i9=; (6.14)

1Note that the same �lter coe�cient is used for the FSE-based MAP selection metric which will

be discussed later. While this coe�cient is not optimized for the MAP metric, it was observed that

varying the low-pass �lter coe�cient does not change signi�cantly the average BER of the FSE-based

MAP metric.


Then we would have

�̂2k=

1

2

nEk � (1� �̂k)

2LPFk

hjxjj2

io: (6.15)

However, since the FSE is not minimized at every time k by the LSL algorithm,

Equations 6.14 and 6.15 cannot be further simpli�ed and explicit evaluation of both

estimates is required.

In order to eliminate the need to estimate both �k and �2k, we make use of the

weighted-square error (WSE) Jk de�ned in Equation 3.52. Speci�cally, let

�̂k = Re

24Pk

j=0 �k�jz(j; k)x�

jPk

j=0 �k�j jxjj2

35 (6.16)

and

�̂2k=

1

2

24Jk � (1� �̂k)

2kX

j=0

�k�j jxjj235 ; (6.17)

where z(j; k) was de�ned in Section 3.1. As previously mentioned Jk is minimized

at every k by the LSL algorithm, therefore it can be shown by the orthogonality

principle[38] that

�̂k =

Pk

j=0 �k�j jxjj2 � JkP

k

j=0 �k�j jxjj2

(6.18)

and

�̂2k=

�̂kJk

2; (6.19)

thus

�k =2 jzkjJk

: (6.20)

It can be seen from Equation 6.20 that this metric for diversity selection depends only

on Jk and jzkj, therefore the need to evaluate Equations 6.18 and 6.19 is eliminated.

Three important points are worth noting. First, the evaluation of Equations 6.13,

6.14 and 6.15 requires the knowledge of the transmitted symbol xk at the receiver. Of

course this information is unavailable except during training. Therefore, whenever xk

is unknown, these equations are evaluated with xk replaced by x̂k. Secondly, while the


WSE Jk could be evaluated explicitly from Equation 3.52, in the LSL algorithm Jk

is computed recursively in a decision-directed manner, and can in fact be considered

as a by-product of adaptation. Thirdly, a multiplicative factor that only depends

on � and k should be introduced in Equation 6.19 in order to make the estimate

approximately unbiased. However, since this factor is common to all branches, we

have simply set it to 1 for all k, i.e. we have normalized it out of all branches since

comparisons are among branches only.


The performance of the proposed new selection diversity scheme is evaluated using

bit-by-bit computer simulation. The number of diversity branches L is �xed at 2

although the approach applies to any number of diversity branches. In all experiments

the feedforward �lter of the adaptive DFE has 8 taps, spaced at half the symbol-

period. The feedback �lter has 3 taps spaced at the symbol period. Only the half-

symbol-spaced DA DFE is simulated because it was observed in Chapter 3 and 5 to

have the best BER performance over a wide range of channel conditions. In each

experiment, L sets of channel parameters are �rst independently generated according

to the previously mentioned distributions, and a burst of 165 4-QAM symbols is

generated and transmitted over the diversity channels according to Equation 2.17. A

square-root raised cosine �lter with a roll-o� factor of � = 0:35 is used as the transmit

�lter. The �rst 15 transmitted symbols are composed of one cycle of PN sequence,

and are assumed to be known to the receiver. This sequence is used for equalizer

initialization, timing alignment, and synthetic training. The remaining 150 symbols

are assumed to be the information symbols.

At the receiver, the received signals are each �ltered by a lowpass �lter with a


independently on each of the L �ltered signals using the approach outlined in Chap-

ter 4. SERLE(�), de�ned in Equation 4.14, is used as the metric for decision delay

optimization. After the timing alignment parameters are acquired, the received sig-

nal samples are fed through the SDDFE to yield the demodulated output. It is


assumed that there is no frequency o�set between the oscillators of the transmitter

and receiver. The DFE �lter coe�cients are optimized in a decision-directed manner

using the DA approach. A synthetic training sequence (STS) with 15 symbols is also

used for synthetic training. The number of decision errors is tabulated for the 150

information symbols. The entire process is repeated 15,000 times. After that, it is

continued until 3,000 bit errors are accumulated or 7,500,000 symbols are transmitted,

whichever occurs �rst.

The three selection metrics simulated are: 1) the FSE de�ned in Equation 6.13;

2) the FSE-based MAP metric (FSE/MAP) de�ned in Equations 6.14 and 6.15; and

3) the WSE-based MAP metric (WSE/MAP) de�ned in Equation 6.20. In addition,

an \ideal" selection scheme is also simulated in which a decision error occurs only

when all diversity branch decisions are wrong. This is not implementable since it

assumes knowledge of the symbol transmitted, but it represents the \best" possible

performance for any selection diversity scheme. The performance of the two-branch

MRCDFE as well as DFE-only (L = 1) and diversity-only (unequalized) cases are

also simulated in order to provide a baseline for comparison. For the unequalized

case two-branch selection diversity is performed using the received signal power as

selection metric.

Figure 6.4 shows the average BER for the various selection metrics as a func-

tion of the average SNR. As in Chapter 5, the average SNR is numerically equal to

the ratio of transmitted energy per bit to the two-sided power spectral density of

the AWGN. Here the channel is assumed to have an exponential power-delay pro-

�le with a rms delay-spread of 50 ns2. The normalized delay-spread d, de�ned in

Equation 3.56, is set at 0.5. It can be seen from this �gure that in the region sim-

ulated, the MAP metrics signi�cantly outperform the FSE metric. In particular, at

high SNR the FSE/MAP metric achieves a gain of approximately 0.75 dB in average

SNR over the FSE metric. Furthermore, WSE/MAP achieves an additional gain of

0.5�1 dB over FSE/MAP, and is 1�2 dB better than FSE. The di�erence between

the FSE/MAP and WSE/MAP metrics lies in the method by which �k and �2kare

2Recall that, for the channel model used, this implies that the path delays are generated from an

exponential distribution with a standard deviation of 50 ns.


12 14 16 18 2010Average SNR

10-6

10-5

10-4

10-3

10-2

10-1

100

Ave

rage

BE

R FSE/MAPWSE/MAP

MRCDFE

IDEAL

FSE

Figure 6.4: Average BER's of the various selection metrics as functions of the average

SNR. The channel has an exponential power-delay pro�le with a rms delay-spread of

50 ns. The normalized delay spread d is 0.5. As noted in the text, the IDEAL case

is not realizable.

estimated. FSE/MAP makes use of time-averaged (lowpass-�ltered) squared one-step

output error to obtain estimates of these parameters (see Equations 6.13, 6.14, and

6.15). However, since the adaptive DFE is a time-varying �lter, the squared one-step

output error in Equation 6.13 is a non-stationary random process. Therefore the FSE

obtained from Equation 6.13 is not an accurate estimate of the mean-square error

(MSE). On the other hand, WSE/MAP makes use of the WSE, Jk, de�ned in Equa-

tion 3.52. As shown in Equation 3.52, at every time k, Jk is the \time average" of

squared output error in estimating xj, j = 0; 1; : : : ; k, using the DFE settings of time

k. This is a better estimate of the true MSE since, conceptually, the DFE settings

are \frozen" when computing the time average in Equation 3.52, making the error

power process being averaged stationary. The additional gain of 0.5�1 dB achieved

by WSE/MAP over FSE/MAP justi�es this interpretation.

It can also be seen from Figure 6.4 that MRCDFE achieves a lower average BER

than SDDFE for all average SNR's simulated, regardless of selection metric. This is

di�erent from previously reported results[43], where it was found that MRCDFE is

inferior to SDDFE with FSE/MAP or WSE/MAP metrics when the average SNR is


high. There are two reasons for this apparent discrepancy. First, in this disserta-

tion the timing alignment parameters (sampling instant and DFE decision delay) are

better optimized. This shows that MRCDFE is very sensitive to the choice of these

parameters. Second, a better training sequence is used in this dissertation. This im-

plies that, with parameters jointly optimized, the length of the training sequence for

MRCDFE as well as the training sequence itself must be carefully chosen. Although

MRCDFE outperforms the SDDFE approaches, from Figure 6.4 we can see that the

performance gap between MRCDFE and SDDFE with WSE/MAP metric is less than

0.5 dB, while the computational complexity of the latter is signi�cantly lower[62]. For

example, when L = 2 branches are used, the half-symbol-space (K = 2) DA MR-

CDFE requires 4- and 5-dimensional lattice stages, while the SDDFE requires two

DFE's, each with 2- and 3-dimensional lattice stages. In this case the MRCDFE re-

quires approximately four times the computation of the SDDFE[40]. This increase in

computational requirements makes the latter an attractive alternative to MRCDFE.

It is also evident in Figure 6.4 that all practically implementable diversity ap-

proaches are signi�cantly inferior to the unrealizable \ideal" selection diversity. In

particular, the WSE/MAP and MRCDFE approaches are approximately 2�2.5 dB,

while the FSE/MAP and FSE approaches 4�4.5 dB, inferior to the ideal selection

diversity. This signi�cant performance gap is due to the fact that the �nal decision is

used to cancel the postcursor ISI as well as to adapt the DFE �lter tap weights. For

the ideal selection diversity case, the �nal decision is correct if any branch decision is

correct. Therefore, if at the current time at least one branch decision is correct, not

only is the current �nal decision guaranteed to be correct, but the chance of having

at least one correct branch decision in the subsequent symbols is also high. Similarly,

for the practically implementable cases, when some branch decisions are wrong, it is

not only likely to make a decision error in the current symbol, but also possible to

have no correct branch decision in subsequent symbols once a decision error is made.

This regenerative e�ect (error propagation) of decision-feedback and decision-directed

adaptation widens the performance gap between the ideal and non-ideal cases.

In order to further investigate the relative performance of the various metrics

for SDDFE, we also tabulated the probability of correct bit detection (i.e. branch


12 14 16 18 2010Average SNR

96

98

100

94

82

90

92

88

86

84

FSE

FSE/MAP

WSE/MAPPr

ob. o

f C

orre

ct D

ecis

ion

(%)

Figure 6.5: Conditional probability of correct branch selection given that exactly

one diversity branch decision is wrong. The channel has an exponential power-delay

pro�le with a rms delay-spread of 50 ns. The normalized delay-spread d is 0.5.

selection) given that exactly one of the diversity branch decisions is wrong. This

conditional probability re ects the ability of the selection metric to correctly assess

the reliability of the diversity branch decisions. The results for the various selection

metrics are shown in Figure 6.5 as a function of the average SNR. Note that by de�-

nition this probability is always 100% for the ideal case, and that it is not applicable

to MRCDFE because the branch DFE's in MRCDFE are jointly optimized. It can be

seen that, in comparison to the FSE metric, MAP metrics provide better indications

of the reliability of the diversity branch decisions, with WSE/MAP being better than

FSE/MAP, as was inferred from the error performance earlier.

The average BER of the various diversity DFE's are also shown in Figure 6.6

as a function of the normalized delay-spread with the average SNR set to 15 dB.

The channel is still assumed to have an exponential power-delay pro�le. The aver-

age BER's for the DFE-only (L = 1) and diversity-only (unequalized) cases are also

shown for comparison. Several observations are noteworthy on this �gure. First, at

15dB SNR, with 0:25 � d � 0:6, two-branch unequalized selection diversity outper-

forms one-branch DFE. However, as d increases beyond 0.6, one-branch DFE is more


10-6

10-5

10-4

10-3

10-2

10-1

100

Ave

rage

BE

R

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0Normalized Delay-Spread

MRCDFEWSE/MAP

DFE-ONLY DIV-ONLY

IDEAL

FSE/MAP

FSE

Figure 6.6: Average BER's as functions of the normalized delay-spread d. The channel

has an exponential power-delay pro�le with 50 ns rms delay-spread. The average SNR

is 15 dB. The IDEAL case is not realizable as noted in the text.

e�cient in combating ISI. Second, incorporating diversity greatly enhances the aver-

age BER performance of the DFE. In particular, the average BER for the practically

implementable diversity DFE's is in general an order of magnitude lower than the

DFE-only case. Third, the relative average BER performance of the various diver-

sity DFE approaches at other values of d are consistent with that observed for the

d = 0:5 case shown in Figure 6.4. It can also be seen that for all equalized cases the

average BER decreases a little as d is increased from 0.25 to 0.75. As d increases

beyond 0.75, the average BER increases with d. This implies that as d increases from

0.25 to 0.75, the advantage of having more time diversity outweighs the disadvantage

of increased ISI. However, as d increases beyond 0.75, the length of the FIR DFE's

becomes insu�cient and the increased residual ISI degrades the performance. This

trend is somewhat inconspicuous because at 15 dB average SNR, the AWGN also

plays an important role in limiting the performance of the DFE's. At higher average

SNR's this trend is more obvious[29]. Finally, for the unequalized case, the average

BER increases monotonically with d, as one would expect.

The same experiments are also performed using channels with Gaussian power-

delay pro�les. Here the power-delay pro�le, or equivalently, the distribution of the


12 14 16 18 2010Average SNR

10-6

10-1

10-2

10-3

10-4

10-5

Ave

rage

BE

R

IDEAL

FSE/MAP

FSE

WSE/MAP

Figure 6.7: Average BER's of the various selection metrics as functions of the average

SNR. The channel has a Gaussian power-delay pro�le with a rms delay-spread of 50

ns and average delay of 200 ns. The normalized delay spread d is set to 0.5. The

IDEAL case is not realizable as noted in the text.

path delays, is assumed to be the causal part of a shifted Gaussian function centered

at t = 200 ns and with a rms delay-spread of 50 ns. Figure 6.7 shows the average

BER's for the selection diversity DFE's as a function of the average SNR while the

normalize delay-spread is �xed at 0.5. Figure 6.8 shows the average BER as a

function of d at 15 dB average SNR. It can be seen that the general conclusions

drawn for the exponential power-delay channels also hold for Gaussian power-delay

channels. However, here average BER's for the equalized cases are in general slightly

lower than those for the exponential case. This is because the frequency spectrum of

the Gaussian power-delay pro�le has a steeper roll-o� in frequency than that of the

exponential power-delay pro�le. Therefore, with the same normalized delay-spread,

a channel with exponential power-delay pro�le appears to be closer to a at-fading

channel than the Gaussian pro�le case. An equalized receiver with su�ciently long

�lters would therefore be more e�ective on a channel with Gaussian power-delay

pro�le because of the higher level of time diversity the channel provides.


0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0Normalized Delay-Spread

10-5

10-4

10-2

10-3

IDEAL

WSE/MAP

FSE/MAPFSE

Ave

rage

BE

R

Figure 6.8: Average BER's as functions of the normalized delay-spread d. The channel

has a Gaussian power-delay pro�le with a rms delay-spread of 50 ns and average delay

of 200 ns. The average SNR is 15 dB. The IDEAL case is not realizable as noted in

the text.

6.5 Summary

A symbol-by-symbol bit-level selection diversity scheme is developed for the adaptive

least-squares lattice DFE. In this scheme selection is performed at the bit-level, with

the aim of selecting as the �nal output the diversity branch decision with the highest

a posteriori probability of being correct. This selection rule is shown to be optimal

in Appendix D for a selection diversity DFE in the maximum a posteriori probability

(MAP) sense, i.e. it maximizes the conditional probability of making a correct �nal

decision given all the outputs of the branch DFE's. A simple selection metric is then

derived for 4-QAM under certain assumptions. This selection metric depends on two

statistical parameters to be estimated in a decision-directed manner on a symbol-by-

symbol basis. Two such estimation formulae are derived based on the lowpass-�ltered

squared one-step output error (FSE/MAP, see Equations 6.14 and 6.15) and weighted-

square output error (WSE/MAP, see Equation 6.20), respectively. Computer simula-

tion is performed for the two-branch diversity (L = 2) case, and the average BER's of

the proposed new schemes are compared to those of a previously proposed approach


which uses FSE alone (Equation 6.13) as the selection metric. The results show that

the new selection metrics developed in this dissertation outperform the conventional

FSE approach by as much as 2 dB in average SNR. Furthermore, the WSE/MAP and

FSE/MAP metrics have better ability to correctly assess the reliability of diversity

branch decisions. It is also found that the WSE/MAP scheme can perform almost

as good as the maximal-ratio-combining DFE (MRCDFE). Because the WSE/MAP

scheme has lower computational complexity than the MRCDFE approach, it is an

attractive alternative for practical implementation.

Chapter 7

Multiple Decision Delay DFE

The adaptive DFE is shown in Chapter 3 to be very e�ective against inter-symbol in-

terference (ISI) introduced by multipath propagation. Furthermore, receiver diversity

can be introduced into the adaptive DFE to signi�cantly enhance its performance, as

shown in Chapters 5 and 6. In applying these algorithms, the �ltered received signals

must �rst be sampled at proper sampling instants. A decision delay must also be

determined since �nite impulse response (FIR) �lters are used in the adaptive DFE.

It has been shown in Chapter 4 that the performance of the adaptive DFE is sen-

sitive to the choice of these parameters. The optimization of sampling instants and

decision delay is referred to as \timing alignment," and was discussed in Chapter 4

of this dissertation.

As previously mentioned, the value of the decision delay for a DFE can be de-

termined using the a priori or a posteriori methods. For the a priori method, the

decision delay is determined based on statistical computations using channel esti-

mates before the data burst is decoded. The channel pulse response (CPR) is �rst

estimated from the transmitted training sequence. The optimal decision delay is then

determined based on some optimization metric which is usually a function of the CPR

estimates. The DFE �lter coe�cients are next determined using adaptive algorithms.

Two optimization metrics, namely SERDFE(�) and SERLE(�), are described in Sec-

tion 4.2.1 for decision delay optimization. SERDFE(�), de�ned in Equation 4.13, is

107

CHAPTER 7. MULTIPLE DECISION DELAY DFE 108

the signal-to-mean-square-error-ratio (SER) at the output of a DFE with decision de-

lay �. SERLE(�), on the other hand, is the output SER achievable by a FIR linear

equalizer with Nf taps and decision delay �. The tap-weights of this linear equalizer

are set so that they match the CPR within the span of the equalizer. SERLE(�) is

de�ned in Equation 4.14.

The a posteriori optimization approach, on the other hand, decodes the data burst

using several values for � and chooses the \best" output. In Chapter 4, we described

a brute-force a posteriori approach in which multiple equalization operations are per-

formed using all reasonable values for �. The decoded burst with the fewest errors is

used as the �nal output. Simulation results show that this approach yields a signi�-

cantly lower average bit-error rate (BER) than the a priori optimization approach for

short-burst data communications over random wireless channels. Unfortunately, the

approach we simulated is not practically implementable because it assumes knowl-

edge of the transmitted symbols at the receiver. A key ingredient of the a posteriori

decision delay optimization approach is a practical method for selecting the decision

delay that corresponds to the fewest number of bit errors. A MAP selection metric is

proposed in Chapter 6 as a measure of the reliability of diversity branch DFE's. It is

natural to apply this metric to the problem of a posteriori optimization of the DFE

decision delay.

The multiple decision delay decision-feedback equalizer (MDDDFE) is proposed

and analyzed in this chapter. The MDDDFE is a combination of the a priori and

a posteriori approaches for decision delay optimization. In this approach, initial

candidate decision delays that are likely to result in few bit errors are �rst carefully

chosen. Multiple DFE's, each with a di�erent decision delay, are then used to decode

the data burst. At the end of every symbol period one branch output is selected

based on some reliability calculations without assuming knowledge of the transmitted

symbols. The selected output is used to produced the demodulated symbol which in

turn is fed back through the DFE feedback �lters as well as used to adapt the DFE

�lter coe�cients in a decision-directed manner. Redundant branch DFE's are also

dynamically detected and pruned so that the overall complexity is reduced. As will

be shown by simulation, this scheme can achieve better average BER performance


FilterReceiver

yj

zk(1)

zk(N)

θ(1)

SymbolTiming

CPREstimate

∆(1)-sample

∆(N)-sample

r(t)

k

θk(N)

DDG

FFF 1

Feedback N

Reliab. N

Control

Reliab. 1Tap-Wt.Adapt. 1

FinalDecision

Tap-Wt.Adapt. N

Feedback 1

FFF N

τ

Advance

Advance

Figure 7.1: Structure of the MDDDFE.

than the conventional single-decision-delay DFE approach without introducing too

much computational overhead.

7.1 Multiple Decision Delay DFE

The proposed multiple decision delay DFE is a combination of the a priori and a

posteriori approaches for decision delay optimization. It uses statistical calculations,

as in the a priori approach, to determine possible optimal values of � before the

decoding process. Multiple DFE's with di�erent decision delays are then used to

compensate for the uncertainty in decision delay optimization, as in the a posteriori

approach. As shown in Figure 7.1, a MDDDFE consists of a decision delay generator

(DDG), N independent DFE's, a DFE controller and a decision device. Based on

the CPR estimates, the DDG generates N distinct initial candidate decision delays

that are likely to result in few bit errors. In our algorithm, these N initial candidates

correspond to the top N decision delay choices based on the optimization metrics

SERDFE(�) or SERLE(�). The DFE's are then independently trained with the train-

ing sequence, each using a di�erent decision delay. In the decoding phase, at the end

of every symbol period the DFE controller selects the most reliable output from the


N DFE's. This selection is done at the bit-level and the resulting output is passed

on to the decision device. The decoded information symbol is used as the common

input to the DFE feedback �lters and also to adapt the DFE's in a decision-directed

manner. During a burst the DFE controller also prunes redundant DFE's using a

simple algorithm to be discussed in Section 7.2. This is done to avoid unnecessary

computations.

The proposed MDDDFE is very similar in structure to the selection diversity

DFE (SDDFE) discussed in Chapter 6. In both receiver structures, multiple DFE's

are used, and the demodulated symbol is based on one DFE output selected at the

bit-level on a per-symbol basis. However, several di�erences between the two are note-

worthy. First, SDDFE was proposed to combine receiver diversity and equalization.

In MDDDFE, however, there is only one wireless channel between the transmitter and

receiver. Multiple DFE's are used in MDDDFE to compensate for the uncertainty

in decision delay optimization. Therefore, only one radio-frequency (RF) front-end is

necessary for MDDDFE. Since a substantial part of the cost of a receiver lies in the

RF front-end, a MDDDFE with N DFE's will cost signi�cantly less than a diversity

receiver with N branches. Secondly, in addition to acting as the selection device, the

controller in MDDDFE also prunes redundant DFE's during a data burst. This was

not done in SDDFE. As will be shown by our simulation results, pruning of redundant

DFE's reduces the number of computations required to decode a data burst. Pruning

is helpful largely due to the fact that, in spite of di�erent decision delays, the multiple

DFE's in a MDDDFE operate on the same received signal.

7.2 The DFE Controller

As mentioned earlier, during the decoding phase the function of the DFE controller

is twofold. First, at the end of every symbol period the output of one DFE is selected

and passed on to the the decision device. A complex reliability measure, de�ned as

�(i)k

=

��Re hz(i)k

i��+ j��Im h

z(i)k

i��J(i)k

; for all surviving DFE i, (7.1)


Y

N

Y

N

Delete min

Γ ?

ReliabilityCompute Ave.

Retain Max

Retain Allmax/min>

Discrepancy?

Figure 7.2: Algorithm for pruning redundant DFE's. This algorithm is repeated once

every R symbols.

is used as the metric for selection. In Equation 7.1 z(i)k

and J(i)k

are the output and

weighted square error (WSE), respectively, of the the i-th DFE at the k-th symbol

period, and j =p�1. This measure is based on the MAP selection metric proposed

in Chapter 6, and re ects the a posteriori probability of a branch decision being

correct. As in Chapter 6, the selection is done at the bit-level, i.e. at every k the real

(imaginary) part of the branch DFE output with the highest Reh�(i)k

i(Im

h�(i)k

i) is

used to produce the in-phase (quadrature-phase) component of the �nal decision.

The second function of the DFE controller is to prune redundant branch DFE's.

During the training phase all branch DFE's are retained and trained using the training

sequence. During the decoding phase a simple algorithm, shown in Figure 7.2, is

repeated every R symbols. In this algorithm the average reliability over the past S

symbols, de�ned as

h�(i)ki = 1

S

kXs=k�S+1

�Re

h�(i)s

i+ Im

h�(i)s

i�; for all surviving DFE i, (7.2)

is �rst calculated for each branch DFE. If all branch DFE's yield the same decisions for

the past R symbols, then it is concluded that only one DFE is necessary. In this case


the branch DFE with the highest h�(i)ki is retained and the rest are discarded. If there

is any discrepancy in branch decisions for the past R symbols, then maxinh�(i)

kiois

compared to mininh�(i)

kio. If the maximum average reliability is more than � times

greater than the minimum average reliability, then the branch DFE with the minimum

average reliability is discarded. Otherwise all surviving branches are retained. This

procedure is repeated once every R symbols until only one DFE is left, or until the end

of the data burst. In this dissertation R will be referred to as the \check interval,"

S as the \averaging interval," and � the \reliability threshold." It will be shown

by simulation that with a proper choice of these parameters, the DFE controller is

capable of reducing the amount of computations with very little performance loss.


The performance of the MDDDFE is evaluated using bit-by-bit computer simulations

identical to those outlined in Section 4.3. In each experiment, one set of channel

parameters is �rst generated according to the distributions tabulated in Table 2.1

A burst of 165 4-QAM symbols is next generated and transmitted over the wireless

channel according to Equation 2.17. A square-root raised cosine �lter with a roll-o�

factor of � = 0:35 is used as the transmit �lter. The �rst 15 transmitted symbols

are composed of one cycle of pseudo-random (PN) sequence. They are assumed to

be known to the receiver, and are used for timing alignment as well as synthetic

training to initialize the adaptive DFE. The remaining 150 symbols are assumed to

be information symbols. The channel is assumed to have a Gaussian power-delay

pro�le with a rms delay-spread and average delay of 50ns and 200 ns, respectively.

At the receiver, the received signal is �ltered by a lowpass �lter with a 35%

square-root raised-cosine impulse response. Symbol-timing (sampling instants) is next

acquired using the method described in Chapter 4. After symbol-timing is acquired,

the received signal samples are fed through the MDDDFE to yield the demodulated

output. A 7-tap CPR estimate is used for timing alignment and synthetic training, i.e.

�1 = �2 = 3 are used. A synthetic training sequence (STS) with 15 symbols is used

for synthetic training. During the 150 information symbols, the coe�cients of the


DFE are updated using the received signal samples and past decisions, as described

previously in Chapter 3. The number of decision errors is tabulated for these 150

symbols. The entire process is repeated 15,000 times. After that, it is continued until

3,000 bit errors are accumulated or 7,500,000 symbols are transmitted, whichever

occurs �rst. As in Chapter 4, in all experiments the feedforward �lters of the adaptive

DFE span four symbol periods. The feedback �lters have three taps spaced at the

symbol period.

The K = 2 (half-symbol-spaced) DA MDDDFE is simulated using SERDFE(�)

de�ned in Equation 4.13 and SERLE(�) de�ned in Equation 4.14 for decision de-

lay generation. Both N = 1 and N = 2 are simulated. Note that the N = 1

case (\DFE/SERDFE" and \DFE/SERLE") is equivalent to a conventional DFE with

proper decision delay optimization, which was simulated in Chapter 4. In fact, the

results for N = 1 shown in this section are directly replicated from Chapter 4. For the

N = 2 case (\MDD/SERDFE" and \MDD/SERLE"), the check interval R, averaging

interval S, and reliability threshold � are empirically set to 25 symbols, 10 symbols,

and 3 dB, respectively.

Figure 7.3 shows the average BER's for the various MDDDFE schemes withN = 1

and N = 2 as functions of the average signal-to-noise ratios (SNR's). As in previous

chapters, the average SNR is numerically equal to the ratio of transmitted energy

per bit to the two-sided power spectral density of the AWGN. The normalized delay-

spread d, de�ned in Equation 3.56, is set to 0.5. The �xed decision delay and ideal

a posteriori cases are also repeated here as a baseline for comparison. In the �xed

decision delay case, � is �xed at 2. For the ideal case, multiple decision-feedback

equalization is performed using all possible values for decision delay to yield multi-

ple decoded bursts. The burst with the smallest number of bit errors is then used

as the �nal output to tabulate BER. Note, as before, that this ideal case is not re-

alizable because it assumes knowledge of the transmitted symbols. It can be seen

that with N = 1, SERDFE(�) outperforms SERLE(�); but for N = 2, SERDFE(�)

and SERLE(�) yield comparable performance. Furthermore, at low average SNR

the MDDDFE schemes achieve the average BER of the ideal case, while at high

average SNR the MDDDFE schemes are only 1dB inferior to the ideal case. This


10-1

10-4

10-3

10-2

DFE/SER DFE

MDD/SER DFE

DFE/SER LE

MDD/SER LE

Ave

rage

BE

RFIXED

Average SNR2010 15

IDEAL

}N=2

}N=1

Figure 7.3: Average BER as a function of average SNR at d = 0:5 for the MDDDFE,

�xed decision delay, and ideal cases. Half-symbol-spaced DA MDDDFE's used here.

The channel has a Gaussian power-delay pro�le with a rms delay-spread of 50ns and

average delay of 200 ns.

shows that MDDDFE is indeed an e�ective scheme for DFE timing alignment. The

burst-dependence property is exploited by combining the a priori and a posteriori

optimization approaches to yield a gain of approximately 3dB over the DFE/SERLE

approach.

The MDDDFE approaches are simulated at average SNR of 15 dB with d varying

from 0.25 to 1.0. The results are shown in Figure 7.4. The average BER's for the

�xed decision delay and ideal cases are again repeated here. It can be seen that

�rst, N = 2 signi�cantly outperforms N = 1 for both SERDFE(�) and SERLE(�).

Secondly, the general conclusions obtained in Chapter 4 for N = 1 also apply for

N = 2. In particular, for small values of d, SERDFE(�) outperforms SERLE(�); but as

d increases beyond a certain value, SERLE(�) becomes superior to SERDFE(�). This

is because as d increases, a 7-tap CPR estimate becomes inadequate in length and the

sensitivity of SERDFE(�) to CPR estimation inaccuracy begins to show. SERDFE(�)

is inherently more sensitive to CPR estimation inaccuracy because matrix inversion

is involved in its computation. Finally, the ideal case outperforms the MDDDFE

schemes in all values of d at 15 dB average SNR. The ideal scheme, however, is not


0.25 0.5 0.75 1.0

010

10-1

10-2

10-3

10-4

DFESERSER LE


Ave

rage

BE

R

IDEAL

FIXED

}N=2

}N=1

Figure 7.4: Average BER as a function of normalized delay-spread at average SNR

of 15dB for the MDDDFE, �xed decision delay and ideal cases. Half-symbol-spaced

DA MDDDFE's used here. The channel has a Gaussian power-delay pro�le with a

rms delay-spread of 50ns and average delay of 200 ns.

realizable because it assumes knowledge of transmitted symbols at the receiver.

Table 7.1 lists the average complexity of the MDDDFE for di�erent average SNR

with d = 0:5. Only the MDDDFE using SERDFE(�) for DDG is shown. The average

complexity is de�ned as the ratio of the total number of DFE operations (including

training and decoding phases) to the total number of information symbols transmit-

ted (excluding training symbols). The average complexity without pruning is also

shown. It can be seen that pruning the redundant DFE's as proposed in this disser-

tation signi�cantly reduces the amount of computational overhead. Furthermore, the

With pruning Without pruning

10 dB 15 dB 20 dB

N = 1 1.10 1.10 1.10 1.10

N = 2 1.68 1.63 1.61 2.20

Table 7.1: Average complexity of one- and two-branch MDDDFE using SERDFE for

d = 0:5 and average SNR of 10, 15 and 20 dB. The average complexity without

pruning is also shown. The channel has a Gaussian power-delay pro�le with a rms

delay spread of 50 ns and average delay of 200 ns.


average complexity is slightly higher at low average SNR. At low average SNR bit er-

rors occur often due to signal fading and AWGN. This results in a higher probability

of inconsistency within a check interval.

7.4 Summary

A multiple decision delay DFE (MDDDFE) is developed to compensate for the uncer-

tainty in decision delay optimization. This scheme can be regarded as the combination

of the a priori optimization approach, in which the decision delay is optimized based

only on channel estimates before decoding the transmitted data, and a posteriori op-

timization approach, in which multiple DFE's with di�erent decision delays are used.

In this scheme the �ltered received signal is �rst over-sampled. Symbol-timing is then

acquired using the over-sampled signal based on channel pulse response (CPR) esti-

mates. The received signal samples corresponding to the proper symbol-timing are

then fed into the MDDDFE. The decision delay generator (DDG) in the MDDDFE

then generates N initial candidate decision delays that are likely to result in the

fewest bit errors. Two di�erent CPR-based metrics, SERDFE(�) and SERLE(�), are

de�ned in Section 4.2.1, and used to generate these N initial candidates. SERDFE(�),

de�ned in Equation 4.13, is the approximate signal-to-mean-square-error-ratio (SER)

at the output of a DFE with decision delay �. SERLE(�), on the other hand, is

the approximate output SER achievable by a FIR linear equalizer with Nf taps and

decision delay �. The tap-weights of this linear equalizer are set so that they match

the CPR within the span of the equalizer. SERLE(�) is de�ned in Equation 4.14.

After the initial candidate decision delays are determined, N DFE's using these

candidate decision delays are then independently trained. During the decoding phase

the DFE controller selects a branch DFE output using the MAP selection metric

proposed in Chapter 6, and passes it on to the decision device. The decoded output is

then used as the common input to the DFE feedback �lters and to adapt the DFE �lter

coe�cients. The selection is done at the bit-level on a per-symbol basis. Redundant

DFE's are also pruned during the decoding phase in order to reduce unnecessary

computation. A simple pruning algorithm is also presented in this dissertation.


The proposed scheme is evaluated using bit-by-bit computer simulation for frequency-

selective fading indoor wireless communication channels that have a Gaussian power-

delay pro�le with a rms delay-spread of 50 ns and average delay of 200 ns. Various

values of normalized delay-spread and average SNR are simulated. Simulation results

show that on this type of channel the proposed scheme is capable of optimizing de-

cision delay not only against the channel but also against the transmitted burst for

short-burst communications. In particular, for a normalized delay-spread of 0.5, a

MDDDFE with two branches (N = 2) using a simple formula for DDG can perform

almost as good as an \ideal", but unrealizable, scheme in which all possible decision

delays are tested and the decoded burst with the fewest errors is used as the �nal out-

put. Our simulation results also show that pruning redundant DFE's is an e�ective

method of reducing computational complexity.

Chapter 8

Conclusions

8.1 Dissertation Summary

Multipath propagation is one of the most challenging problems encountered in a wire-

less data communication link. It causes signal fading, inter-symbol interference (ISI),

and, when there is relative motion between the transmitter and the receiver, Doppler

spread. For high-speed indoor wireless data communications with data rates greater

than 10 Mb/s, signal fading and ISI are the main factors that signi�cantly degrade

the average bit error rate (BER) performance. This dissertation investigates digital

signal processing techniques for multipath mitigation. We mainly focus on e�cient

techniques for mitigating signal fading and ISI, since the Doppler spread is insigni�-

cant for indoor environments and existing adaptive algorithms can be directly applied

without special modi�cations. Receiver diversity is investigated as a technique to use

against signal fading, while adaptive equalization is presented as a technique for com-

bating ISI. New techniques for integrating these two techniques are also developed

to simultaneously combat signal fading and ISI. Since the Doppler spread is insignif-

icant, the recursive least-squares (RLS) algorithms and variants thereof are used for

initial acquisition and channel tracking.

Chapter 2 of this dissertation discusses multipath propagation. Small-scale e�ects

of multipath propagation are described, and a simple baseband model is developed

118

CHAPTER 8. CONCLUSIONS 119

for use in subsequent chapters. It is shown in Appendices A and B that the power-

delay pro�le and Doppler spectrum of this channel model can be easily controlled

by appropriately specifying the statistical distribution of some model parameters.

Simulation results are shown to verify the exibility of our channel model.

Chapter 3 describes the adaptive decision-feedback equalizer (DFE). An adaptive

DFE can be optimized, on a symbol-by-symbol basis, using the channel-estimation-

based adaptation (CEBA) or direct adaptation (DA) algorithms. In the CEBA al-

gorithm, the channel pulse response (CPR) is �rst estimated, and the DFE �lter

coe�cients are then computed from the CPR estimates. In the DA algorithm, the

DFE �lter coe�cients are directly computed from the received signal samples using

the least-squares lattice (LSL) algorithm without going through channel estimation.

Both approaches can provide signi�cant performance gain over the unequalized 4-

QAM receiver. In both algorithms, a sequence of training symbols is necessary for

initializing the DFE �lter coe�cients. Since the training symbols are known both to

the transmitter and receiver, they do not convey information and are considered as

transmission overhead. \Synthetic training" is developed here for the DA approach to

reduce transmission overhead. In this approach, a synthetic training sequence (STS)

is generated at the receiver, and used together with the actual training sequence for

DFE �lter coe�cient initialization. Simulation results show that STS is very e�ective

in improving the performance of DA DFE's without incurring additional transmis-

sion overhead. In particular, the performance of a DA DFE using a training sequence

and STS of 15 symbols each (i.e. 30 \e�ective" training symbols) has essentially the

same performance as that of a DA DFE using a training sequence of 31 symbols.

This shows that synthetic training is capable of improving the performance without

incurring any additional transmission overhead.

The performance of the CEBA and DA DFE's are also compared using computer

simulation in Chapter 3. It is found that when a �xed-length CPR estimator is

used, the CEBA DFE is more sensitive to variations in channel delay-spread than

the DA DFE. In other words, when the delay-spread is small, the CEBA and DA

approaches have comparable performance. However, as the delay-spread increases,

the CEBA DFE becomes signi�cantly inferior to the DA DFE. These observations


are new and do not entirely agree with those reported in the literature. The reason

for the discrepancies is that in this dissertation, we did not assume knowledge of

the true CPR length. Therefore for channel realizations with a CPR longer than

the �xed-length CPR estimator, the second-order statistics obtained from the CPR

estimate are inaccurate, thus causing the resulting CEBA DFE �lter coe�cients to

deviate substantially from the optimal values. For the DA approach, however, the

�lter coe�cients are not obtained from the CPR estimates. It is, therefore, more

robust with respect to channel delay-spread variations.

We further �nd that a fractionally-spaced CEBA DFE encounters numerical prob-

lems which greatly degrade its performance. In fact, the half-symbol-spaced CEBA

DFE performs 1�2 dB worse than the symbol-spaced CEBA DFE when the channel

has a Gaussian power-delay pro�le with a normalized delay spread of 0.5. A nu-

merical technique, known as \regularization" or \leakage," can be used to remedy

these problems. In our simulations, we �nd that regularization improves the perfor-

mance of the half-symbol-spaced CEBA DFE by 2 dB. However, the CEBA DFE

with regularization remains sensitive to channel delay-spread variations.

In applying the CEBA or DA DFE algorithms, the �ltered received signal must

�rst be sampled at proper sampling instants. If a �nite-impulse-response (FIR) DFE

is used, the decision delay must also be pre-determined. The optimization of the

sampling instants and DFE decision delay is referred to as the \timing alignment"

problem, and is discussed in Chapter 4 of this dissertation. In our approach, the

sampling instants are optimized using a two-step approach. In the �rst step, the

\sampling delay" is obtained using the complex time-correlation between the se-

quence of the received signal samples and the transmitted training sequence. In the

second step, the \sampling phase" is obtained from the CPR estimates using the fast

Fourier transform (FFT). The sampling instant is then the sum of the sampling delay

and sampling phase. After the sampling instants are acquired, the decision delay is

optimized using the a priori or a posteriori methods. For the a priori method, the

decision delay is optimized based on some statistical computations using the CPR

estimates before the burst is decoded. Two optimization metrics are evaluated in


this dissertation. The �rst is SERDFE(�), which is the approximate signal-to-mean-

square-error ratio (SER) for a DFE with decision delay �: This metric has been

investigated in the past and fast algorithms exist for its computation. The second is

SERLE(�), which is the approximate SER for a FIR linear equalizer with decision

delay � and �lter coe�cients matched to the CPR within the span of the equalizer.

This metric is an ad hoc metric proposed in this dissertation, and is simpler to com-

pute than SERDFE(�). Both metrics are functions of the CPR estimates and are thus

a�ected by any inaccuracy in channel estimation. For the a posteriori method, multi-

ple equalization operations are done using di�erent decision delays, and the decoded

burst that contains the fewest errors is selected as the �nal output.

Simulation results show that it is very important to optimize both the sampling

instants and decision delay. Optimizing the sampling delay alone improves the ro-

bustness of the DFE with respect to channel delay-spread variations. Optimizing

both the sampling delay and sampling phase provides additional performance gain

for symbol-spaced DFE's. Furthermore, with the sampling instants optimized, a

priori decision delay optimization using SERDFE(�) or SERLE(�) provides a perfor-

mance gain in excess of 5 dB at high average SNR over a �xed-decision-delay DFE

when the channel has a Gaussian power-delay pro�le with a normalized delay-spread

of 0.5, with SERDFE(�) being better than SERLE(�) in terms of the average BER.

However, as the normalized delay-spread increases, the sensitivity of SERDFE(�) to

CPR estimation inaccuracy begins to show, and SERLE(�) becomes superior.

Simulation results also show that the a posteriori method for decision delay opti-

mization signi�cantly outperforms the a priori method. In particular, a performance

gain of almost 3 dB is achieved at high average SNR when the channel has a Gaus-

sian power-delay pro�le with a normalized delay-spread of 0.5. Unfortunately, the

a posteriori method simulated in this chapter assumes knowledge of the transmit-

ted symbols at the receiver, and is thus unrealizable. However, an implementable a

posteriori optimization approach is developed in Chapter 7.

Chapter 5 discusses receiver diversity. While adaptive equalization is very e�ec-

tive against the e�ect of delay-spread, it cannot mitigate the e�ect of signal fading.


Receiver diversity, on the other hand, is an e�ective technique for combating sig-

nal fading, but is less e�ective in combating delay-spread. This chapter discusses

an approach for integrating combining diversity and adaptive DFE. The resulting

structure is referred to as the maximal-ratio combining DFE (MRCDFE). The MR-

CDFE is an adaptive DFE with multiple feedforward �lters and one feedback �lter

which are jointly optimized. Both the CEBA and DA algorithms can be extended

for the MRCDFE, and the synthetic training algorithm developed in Chapter 3 as

well as regularization can also be applied. The DA MRCDFE is found to be more

robust with respect to channel delay-spread variations than the CEBA MRCDFE.

This observation is consistent with the results for the single-branch cases, and can

be explained using arguments similar to those presented in Chapter 3. Furthermore,

as in Chapter 3, regularization is found to be very e�ective in improving the perfor-

mance of the half-symbol-spaced CEBA MRCDFE. A gain of 6 dB is obtained at

high average SNR using regularization. Simulation results show that, in general, the

MRCDFE can indeed simultaneously reduce the e�ects of signal fading and delay

spread. For two-branch diversity, for example, a performance gain in excess of 10

dB is achieved at high average SNR over the unequalized diversity receiver when the

channel has a Gaussian power-delay pro�le with a normalized delay-spread of 0.5.

The performance gain over the DA DFE without receiver diversity is approximately

8 dB at high average SNR.

In Chapter 6, the integration of selection diversity and adaptive equalization is

investigated. The resulting structure is referred to as the selection diversity DFE

(SDDFE). The SDDFE consists of multiple DFE's that are independently optimized.

At the end of every symbol period, one branch decision is selected as the �nal decision

based on a novel selection rule. The selection is done at the bit-level, and the �nal

decision is used as the common input to the DFE feedback �lters, and also to adapt

the DFE �lter coe�cients. The new selection rule developed in this dissertation is

referred to as the maximum a posteriori (MAP) selection rule. According to this rule,

the branch decision that has the highest a posteriori probability of being correct is

selected as the �nal decision. It is proved in Appendix D that the MAP selection rule

is optimal, in the MAP sense, for a SDDFE. Two selection metrics are derived based


on the MAP selection rule under a certain assumption. Simulation results show that,

at high average SNR, selection based on the new MAP metrics yields a performance

gain of approximately 2 dB over conventional SDDFE methods for channels having

the Gaussian and exponential power-delay pro�les with normalized delay-spreads of

0.5 Furthermore, the probability of correct branch selection also indicates that the new

selection metrics derived in chapter are indeed good measures of the reliability of the

branch DFE's. The average BER's of the MAP SDDFE's are comparable to those

of the MRCDFE in Chapter 5. However, since the branch DFE �lter coe�cients

are independently optimized for the SDDFE, the computational complexity of the

SDDFE is lower than that of the MRCDFE whose branch DFE �lter coe�cients are

jointly optimized. The MAP SDDFE is therefore an attractive method for integrating

receiver diversity with adaptive equalization.

In addition to receiver diversity, the MAP selection metric developed in Chap-

ter 6 can also be applied to the a posteriori optimization of DFE decision delay. It is

demonstrated in Chapter 4 that a posteriori decision delay optimization can poten-

tially outperform the a priori method. A key ingredient of the a posteriori method

is the ability to assess the reliability of candidate decisions without assuming knowl-

edge of the transmitted symbols. The multiple decision delay DFE (MDDDFE) is

developed in Chapter 7 based on this principle. The MDDDFE has similar structure

as the SDDFE. It consists of the decision delay generator (DDG), DFE controller,

and multiple DFE's that are independently optimized. The DDG �rst generates ini-

tial candidate values for the decision delay using metrics de�ned in Chapter 4 for a

priori optimization. Multiple DFE's are then used to equalize the burst, each with

a di�erent decision delay. At every symbol-period, one branch decision is selected

using the MAP selection metric, as in the SDDFE. The selected branch decision is

used as the common input to the DFE feedback �lters, and also to adapt the DFE

�lter coe�cients. The DFE controller also detects and prunes redundant branches

in order to reduce unnecessary computation. The MDDDFE can be regarded as a

combination of the a priori and a posteriori methods for decision delay optimization.

Simulation results show that it can achieve an average BER very close to that of the

unrealizable a posteriori case simulated in Chapter 4. The performance gain over


the a priori optimization is approximately 2�3 dB. Our results also show that prun-

ing is very e�ective in reducing the computational overhead associated with multiple

equalization operations.

8.2 Future Work

The average BER is used as the performance measure for the various digital signal

processing techniques discussed in this dissertation. The average frame error-rate and,

more generally, the statistical distribution of the BER's are also useful performance

measures, and should be investigated in the future.

The LSL algorithm used in this dissertation has the merits of high modularity,

computational e�ciency, and numerical stability. However, it is still computation-

ally intensive for practical low-power implementations. Furthermore, for short-burst

transmissions, numerical stability may not be a serious concern. Other variants of

the recursive least-squares (RLS) algorithm should be investigated. The combination

of the RLS and least-mean-square (LMS)[65] algorithms should be studied. The RLS

algorithm, or variants thereof, can be used for initial acquisition of the DFE �lter

coe�cients. The LMS algorithm can then be used for tracking the channel variations

and updating the DFE coe�cients.

In addition to multipath mitigation techniques, error control mechanisms are also

necessary for establishing a reliable wireless communication link. Error detecting

codes[66] and automatic request for retransmission (ARQ) protocols[67] should be

investigated in the future. The structure of the data frame should be re-designed to

accommodate these error control mechanisms.

Throughout this dissertation, the frequency o�set between the transmitter and

receiver oscillators is assumed to be negligible. The performance of the techniques

discussed in this dissertation should also be evaluated in the presence of frequency

o�set between the transmitter and receiver oscillators. When frequency o�set is

present, the channel has a wider \e�ective" Doppler spread and, equivalently, varies

at a higher rate. Depending on the amount of the frequency o�set, the increased


Doppler spread can be handled by using a smaller value for the forgetting factor1 (see

Appendix C) for the LSL algorithm when the o�set is \small", or by compensating for

the frequency o�set[68] before the received signal is equalized if the o�set is \large."

1A forgetting factor of � = 0:98 is empirically chosen for this dissertation.

Appendix A

Baseband Equivalent Power-Delay

Pro�le

Let the transmitted baseband equivalent signal be a rectangular pulse with duration

�t, i.e.

u(t) =

sP

�t�

�t

�t

�; (A.1)

where P is the total energy of the pulse and

�(t) =

8<: 1 jtj < 1

2

0 otherwise.(A.2)

Assuming that there is no AWGN, the received signal is

r(t) =MX

m=1

amu(t� �m)ej�me�j!c�mej!mt (A.3)

=

sP

�t

MXm=1

am�

�t� �m

�t

�ej�me�j!c�mej!mt; (A.4)

where the meanings of the parameters M , !c, am, �m, �m, and !m are tabulated in

Table 2.1. Thus, the power-delay pro�le is

126

APPENDIX A. BASEBAND EQUIVALENT POWER-DELAY PROFILE 127

p(t) � lim�t!0

Ehjr(t)j2

i(A.5)

= lim�t!0

P

�tE

24��MX

m=1

am�

�t� �m

�t

�ej�me�j!c�mej!mt

��235 (A.6)

= lim�t!0

P

�t

MXm=1

E

�a2m�

�t� �m

�t

��(A.7)

= PMX

m=1

Eha2m

ilim�t!0

E

�1

�t�

�t� �m

�t

��: (A.8)

Let g(�) denote the marginal probability density function of the path delays �m, we

have, for all m,

lim�t!0

E

�1

�t�

�t� �m

�t

��= lim

�t!0

Z 1

�1

1

�t�

�t� �m

�t

�g(�m)d�m (A.9)

= g(t): (A.10)

Therefore

p(t) = PE

"MX

m=1

a2m

#g(t): (A.11)

In other words, the baseband equivalent power-delay pro�le is proportional to the

probability density function of the path delays g(�), with the proportionality constantbeing PE

hPM

m=1 a2m

i. If we let E

hPM

m=1 a2m

i= 1 as in Equation 2.18, then the

baseband equivalent power-delay pro�le is

p(t) = Pg(t): (A.12)

Appendix B

Baseband Equivalent Doppler

Spectrum

Let the transmitted baseband equivalent signal be a rectangular pulse with amplitude

A and duration duration �t, i.e.

u(t) = A�

�t

�t

�; (B.1)

where �(�) is de�ned in Equation A.2. Assuming that there is no AWGN, the received

signal is

r�t(t) =MX

m=1

amu(t� �m)ej�me�j!c�mej!mt (B.2)

= AMX

m=1

am�

�t� �m

�t

�ej�me�j!c�mej!mt; (B.3)

where the meanings of the parameters M , !c, am, �m, �m, and !m are tabulated in

Table 2.1. Now, let �t approach 1, we have

r(t) � lim�t!1

r�t(t) (B.4)

= AMX

m=1

amej�me�j!c�mej!mt: (B.5)

128

APPENDIX B. BASEBAND EQUIVALENT DOPPLER SPECTRUM 129

The power spectral density of r(t) is, by de�nition, the Doppler spectrum of the

channel. Since r(t) is simply a sum of complex sinusoids with random frequencies, its

power spectral density is

S(f) = A2E

"MX

m=1

a2m�(f � fm)

#(B.6)

= A2MX

m=1

Eha2m

iE [�(f � fm)] ; (B.7)

where fm = !m

2�and �(�) is the Dirac delta function. Let h(�) denote the marginal

probability density function of the path Doppler shift frequencies fm, we have, for all

m,

E [�(f � fm)] =

Z 1

�1�(f � fm)h(fm)dfm (B.8)

= h(f); (B.9)

where the last equality follows from the sifting property of the Dirac delta function.

Thus,

S(f) = A2E

"MX

m=1

a2m

#h(f): (B.10)

In other words, the baseband equivalent Doppler spectrum of the channel is propor-

tional to the probability density function h(�) of the path Doppler shift frequencies,

with the proportionality constant being A2EhP

M

m=1 a2m

i. If we let E

hPM

m=1 a2m

i= 1

as in Equation 2.18, then the baseband equivalent Doppler spectrum is

S(f) = A2h(f): (B.11)

Appendix C

The Least-Squares Lattice DFE

The least-squares lattice (LSL) DFE used in this dissertation was proposed by Ling

and Proakis[39, 40]. A block diagram of a LSL DFE with Nf taps in the feedforward

�lter and Nb taps in the feedback �lter is shown in Figure C.1. Assuming that Nf �Nb, the LSL DFE consists of (Nf �Nb � 1) \feedforward" stages, one \transitional"

stage, and Nb � 1 feedback stages. The feedforward stages are multi-channel LSL

stages[41] with dimensionality K, where the integer K is the number of samples

taken per symbol-period. The feedback stages are multi-channel LSL stages with

dimensionality K + 1. The transitional stage is a special stage that serves as the

interface between the K-dimensional feedforward stages and (K + 1)-dimensional

feedback stages. The inputs to the DFE are the sampled received signal y(t) and the

transmitted symbol (previous decision) x(t). This algorithm minimizes the weighted

square error (WSE) which is de�ned in Chapter 3 and repeated here for convenience1:

JNf�1(t) =tX

j=0

�t�j jx̂(j; t)� x(j)j2 ; (C.1)

In Equation C.1, the \forgetting factor" � is a positive real number less than but close

to 1, and x̂(j; t) is the estimate of the transmitted symbol x(j) obtained using the

�lter coe�cients of time t. The output of the LSL DFE is an estimate x̂(t) = x̂(t; t�1)

1All quantities in this appendix are discrete-time signals. Subscripts are used as order indices,

and parenthetic arguments are used as time indices. m and t are integers in all equations.

130

APPENDIX C. THE LEAST-SQUARES LATTICE DFE 131

+

+ +

+ + + +

+ ++ + + + +

N bN f- -1 N bN f

- N bN f- +1 N f-1

+ +

+

f0(t)

0(t)b

e (t)1D

21f1(t)

1(t)b

f2(t)

2(t)b

x (t)

e (t)2 e (t)3

y(t)

x(t)

(a)

+

+

f (t)m-1

b (t)m-1

f (t)m

b (t)m

D

(b)

Figure C.1: (a) The block diagram of a LSL DFE. Matrix weights are not explicitly

shown. The block labeled \D" denotes a unit-sample delay; (b) The block diagram of

a LSL stage. Matrix weights are not explicitly shown. The block labeled \D" denotes

a unit-sample delay.


of the transmitted symbol x(t).

The time- and order-update recursions for the LSL DFE are as follows.

Feedforward stages (1 � m � (Nf �Nb� 1)) and feedback stages ((Nf �Nb+1) �m � (Nf � 1)):

fm(t) = fm�1(t)�K�m(t� 1)R�b

m�1(t� 2)bm�1(t� 1) (C.2)

bm(t) = bm�1(t� 1)�Km(t� 1)R�fm�1(t� 1)fm�1(t) (C.3)

Km(t) = �Km(t� 1) + �m�1(t� 1)bm�1(t� 1)f�m�1(t) (C.4)

�m(t) = �m�1(t)� j�m�1(t)j2b�m�1(t)R�bm�1(t)bm�1(t) (C.5)

Rfm(t) = Rf

m�1(t)�K�m(t)R�b

m�1(t� 1)Km(t) (C.6)

= �Rfm(t� 1) + �m(t� 1)fm(t)f

�m(t) (C.7)

Rbm(t) = Rb

m�1(t� 1)�Km(t)R�fm�1(t)K

�m(t) (C.8)

= �Rbm(t� 1) + �m(t)bm(t)b

�m(t) (C.9)

x̂m(t) = x̂m�1(t) + k�m(t� 1)R�b

m�1(t� 1)bm�1(t) (C.10)

em(t) = x(t)� x̂m(t) (C.11)

= em�1(t)� k�m(t� 1)R�b

m�1(t� 1)bm�1(t) (C.12)

km(t) = �km(t� 1) + �m�1(t)bm�1(t)e�m�1(t) (C.13)

Jm(t) = Jm�1(t)� k�m(t� 1)R�b

m�1(t� 1)km(t� 1) (C.14)

Transitional stage (m = Nf �Nb):

f (1)m(t) = fm�1(t)�K�

m(t� 1)R�b

m�1(t� 2)bm�1(t� 1) (C.15)

b(1)m(t) = bm�1(t� 1)�Km(t� 1)R�f

m�1(t� 1)fm�1(t) (C.16)

b(2)m(t) = em�1(t� 1)� kb�

m(t� 1)R�f

m�1(t� 1)fm�1(t) (C.17)

fm(t) =

24 f (1)

m(t)

em(t� 1)

35 (C.18)

bm(t) =

24 f (1)m

(t)

b(2)m(t)

35 (C.19)

Km(t) = �Km(t� 1) + �m�1(t� 1)bm�1(t� 1)f�m�1(t) (C.20)


�m(t) = �m�1(t)� j�m�1(t)j2b�m�1(t)R�bm�1(t)bm�1(t) (C.21)

Rfm(t) = �Rf

m(t� 1) + �m(t� 1)fm(t)f

�m(t) (C.22)

Rbm(t) = �Rb

m(t� 1) + �m(t)bm(t)b

�m(t) (C.23)

x̂m(t) = x̂m�1(t) + k�m(t� 1)R�b

m�1(t� 1)bm�1(t) (C.24)

em(t) = x(t)� x̂m(t) (C.25)

= em�1(t)� k�m(t� 1)R�b

m�1(t� 1)bm�1(t) (C.26)

km(t) = �km(t� 1) + �m�1(t)bm�1(t)e�m�1(t) (C.27)

kbm(t) = �kb

m(t� 1) + �m�1(t� 1)fm�1(t)e

�m�1(t� 1) (C.28)

Jm(t) = Jm�1(t)� k�m(t� 1)R�b

m�1(t� 1)km(t� 1) (C.29)

The initial and boundary conditions are as follows.

Boundary conditions (m = 0):

f0(t) = y(t) (C.30)

b0(t) = y(t) (C.31)

Rf0(t) =

tXn=0

�t�ny(n)y�(n) + �t�I (C.32)

=

8<: y(0)y�(0) + �I; t = 0;

�Rf0(t� 1) + y(t)y�(t); t > 0:

(C.33)

Rb0(t) = Rf

0(t) (C.34)

�0(t) = 1 (C.35)

x̂0(t) = 0 (C.36)

e0(t) = x(t) (C.37)

J0(t) =tX

n=0

�t�njx(n)j2 (C.38)

=

8<: jx(0)j2; t = 0;

�Jm(t� 1) + jx(t)j2; t > 0:(C.39)


Initial conditions (t = �1; 0):

Rbm(�1) = �I (C.40)

fm(0) =

8>>><>>>:y(0); 0 � m � Nf �Nb � 1;24 y(0)x(0)

35 ; Nf �Nb � m � Nf � 1:

(C.41)

bm(0) =

8<: y(0); m = 0;

0; m 6= 0:(C.42)

Km(0) = 0 (C.43)

�m(0) = 1 (C.44)

Rfm(0) =

8>>><>>>:y(0)y�(0) + �I; 0 � m � Nf �Nb � 1;24 y(0)x(0)

35 h y�(0) x�(0)

i+ �I; Nf �Nb � m � Nf � 1:

(C.45)

Rbm(0) =

8<: y(0)y�(0) + �I; m = 0;

�I; m 6= 0:(C.46)

km(0) = 0 (C.47)

� is a small positive real number in the above equations. The dimensionalities of

the variables in the LSL DFE algorithm are tabulated in Table C.1.


Variable Feedforward Transitional Feedback

fm(t) K � 1 (K + 1)� 1 (K + 1)� 1

bm(t) K � 1 (K + 1)� 1 (K + 1)� 1

f (1)m(t) K � 1

b(1)m(t) K � 1

b(2)m(t) Scalar

Km(t) K �K K �K (K + 1)� (K + 1)

�m(t) Scalar Scalar Scalar

Rfm(t) K �K (K + 1)� (K + 1) (K + 1)� (K + 1)

Rbm(t) K �K (K + 1)� (K + 1) (K + 1)� (K + 1)

x̂m(t) Scalar Scalar Scalar

em(t) Scalar Scalar Scalar

km(t) K � 1 K � 1 (K + 1)� 1

kbm(t) K � 1

Jm(t) Scalar Scalar Scalar

Table C.1: The dimensionalities of the variables in the LSL DFE algorithm.

Appendix D

Optimality of MAP Selection Rule

In this Appendix, we will prove that the MAP selection rule described in Section 6.1

is optimal for a given SDDFE in the MAP sense, i.e. it maximizes the conditional

probability of making a correct �nal decision given all the outputs of the branch

DFE's.

Let1 z(l); l = 1; 2; : : : ; L, and x̂(l); l = 1; 2; : : : ; L, denote the branch DFE out-

puts and corresponding branch DFE decisions, respectively, where L is the number of

diversity branches. Furthermore, let x̂ denote the �nal decision for a given SDDFE us-

ing some selection rule. Conditioned on the branch DFE outputs z(l); l = 1; 2; : : : ; L,

we have

x̂ = x̂(l) with probability p(l) (D.1)

where

p(l) = Probhbranch l is selectedjz(1); z(2); : : : ; z(L)

i: (D.2)

Note that p(l) is a function of z(1); z(2); : : : ; z(L). Hence,

1Here we will omit the time index k since it is understood that selection is done on a symbol-by-

symbol basis.

136

APPENDIX D. OPTIMALITY OF MAP SELECTION RULE 137

Probhx̂ correctjz(1); : : : ; z(L)

i=

LXl=1

p(l)Probhx̂(l) correctjz(1); : : : ; z(L)

i(D.3)

=LXl=1

p(l)Probhx̂(l) correctjz(l)

i(D.4)

� maxl

nProb

hx̂(l) correctjz(l)

io; (D.5)

where the last inequality follows from the fact thatP

L

l=1 p(l) = 1. The upper bound

in Equation D.5 can be achieved by choosing a selection rule such that

p(l) =

8<:

1 if l = argmaxnProb

hx̂(j) is correctjz(j)

i; j = 1; 2; : : : ; L

o0 otherwise.

(D.6)

However, Equation D.6 describes precisely the MAP selection rule, i.e. select (with

probability 1) the branch decision with the highest conditional probability of being

correct. Therefore the MAP selection rule is optimal for a given SDDFE in the MAP

sense, i.e. it maximizes the conditional probability of making a correct �nal decision

given all the outputs of the branch DFE's.

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