PERFORMANCE ANALYSIS OF

ADAPTIVE BLIND EQUALIZATION ALGORITHMS FOR NOISY FIR AND HR

CHANNELS

By

AWWAB QASIM ALTHAHAB

B.S., University of Babylon, Iraq, 2007

A thesis submitted to the

Faculty of the Graduate School of the

University of Colorado at Denver in partial fulfillment

of the requirements for the degree of

Master of Science

Electrical Engineering

2013

This thesis for the Master of Science degree by

Awwab Qasim Althahab

has been approved for the

Electrical Engineering Degree

by

Miloje Radenkovic, Chair

Yiming Deng

Tim Lei

02/21 /2013

n

Althahab, Awwab Qasim (M.S., Electrical Engineering)

Performance Analysis of Adaptive Blind Equalization Algorithms for Noisy FIR and

TTR Channels

Thesis directed by Professor Miloje Radenkovic

ABSTRACT

This thesis addresses the problem of blind adaptive equalization of finite

impulse response (FIR) and infinite impulse response (HR) channels (their

characteristics are unknown) in the fractionally sampling scenario. An adaptive

equalizer is used at the receiver to compensate the time dispersion induced by noisy

communication channels and eliminate the effect of Inter-Symbol-Interference (ISI).

In other words, the overall our system model, which is a cascade connection of the

channel and equalizer, provides nearly an ideal transmission medium that the

information source signals can be sent through. Due to this and rely only on

probabilistic and statistical properties (Second Order Statistics (SOS) which has most

communication channel information) of the received signals, the unknown input

information signals can be recovered successfully.

Various blind adaptive algorithms are discussed throughout this thesis.

Simulation results are presented by evaluating the mean square symbol error (MSE)

of these techniques to study their performance behavior in blind channel equalization

m

concept. These algorithms operate blindly in the practical situation, and they can

achieve a complete adaptation without the aid of a training sequence, desired

response, which is either impractical or very costly. The parameters of equalizer are

updated in a recursive way with each single output measurement. Finally, the

performance comparisons are realized to show which algorithm is more efficient and

robustness to noisy channel model (three classes of channel model are used through

thesis's simulation). The aim of this thesis is to improve the performance of a wireless

communication channel using various blind adaptive equalization algorithms through

computer simulations.

The form and content of this abstract are approved. I recommend its publication.

Approved: Miloje Radenkovic

IV

DEDICATION

To my dear father,

To my affectionate mother,

To my dear sister and brothers,

To my lovely wife and daughters,

who always offer their patience, prayers, support, encouragement and endless love.

ACKNOWLEDGMENT

All thanks are offered to Allah who provides me with the help and inspiration

to be able to complete this thesis.

I wish to express my sincere thanks and deep gratitude to my supervisor Prof.

Dr. Miloje Radenkovic for his kind advice, helpful, valuable suggestions and

continuous encouragement throughout the work for this thesis.

I feel indebted to my family; my gratitude and appreciation are to my father

(Qasim) mother, sister (Zahraa) and my wife (Sarah) who have been a source of

motivation and strength during moments of despair and discouragement, and I want

to thanks them a lot for offering everything to me to reach this point in my life. Their

care and support have been shown in incredible ways recently. I also would like to

acknowledge my brothers (Osama and Ahmed) for their encouragement and

assistance in order to push my research up to this point.

vi

TABLE OF CONTENTS

CHAPTER

I. INTRODUCTION................................................... 1

1.1 Introduction and Background Information to Blind Adaptive Equalizer

...............................................................1

1.2 Thesis Overview..........................................4

1.3 Mathematical Notations................................... 6

II. MODEL OF COMMUNICATION SYSTEM AND MATHEMATICAL

FRAMEWORK...................................................... 10

2.1 Introduction............................................ 10

2.2 Channel Model........................................... 12

2.3 Vector Representation................................... 16

2.4 Overall System Model with Equalizer..................... 18

III. BLIND ADAPTIVE CHANNEL EQUALIZATION METHODS ...................22

3.1 Adaptive Filter Theory...................................22

3.2 Adaptive Equalization....................................23

3.3 Blind Adaptive Equalization............................. 24

3.4 Blind SIMO Channel Equalization Methods Using Second Order

Statistics (SOS).........................................24

vii

3.5 Cyclic Statistics

25

3.6 Direct Blind MMSE Equalizers..................................27

3.7 Methods for Blind Equalizer Calculation.......................29

3.7.1 Recursive Blind Adaptive Using Recursive Least Square

Algorithm (RLS)........................................29

3.7.2 Recursive Blind Adaptive Using Cyclic Least Mean Square

Algorithm (cyclic LMS)................................ 31

3.7.3 Recursive Blind Adaptive Using Least Mean Square

Algorithm (LMS)........................................33

3.7.4 Unbiased Blind Adaptive Using Gradient Projection

Technique..............................................36

3.7.5 Recursive Blind Adaptive Using Recursive Extended Least

Square Algorithm (RELS)............................... 38

IV. PERFORMANCE OF ADAPTIVE BLIND EQUALIZATION

ALGORITHMS AND SIMULATIONS ........................................45

4.1 Performance Evaluations.......................................46

4.1.1 Mean Square Error (MSE)................................46

4.1.2 Symbols Constellation Plots............................47

4.2 Simulations...................................................48

4.2.1 Experiment la..........................................48

4.2.2 Experiment lb......................................... 52

4.2.3 Experiment 2...........................................55

4.2.4 Experiment 3a......................................... 57

viii

4.2.5 Experiment 3b.................................. 58

4.2.6 Experiment 4a...................................62

4.2.7 Experiment 4b.................................. 64

4.2.8 Experiment 5a.................................. 66

4.2.9 Experiment 5b.................................. 68

V. CONCLUSION...................................................72

REFERENCES.......................................................74

IX

LIST OF FIGURES

FIGURE

2.1 General model of a communication system.............................. 10

2.2 Basic elements in a digital communication system..................... 11

2.3 Fractionally sampled (P = 2 or more) communication system............ 13

2.4 SIMO multichannel model (P = 2) ..................................... 15

2.5 Multichannel system models with equalizer g(n) ...................... 18

3.1 SIMO channel Model (HR)...............................................40

3.2 Predictor Based Equalizer.............................................40

4.1 Two-ray multipath channel, the magnitude of impulse response h(n) ....49

4.2 The zeros of h(ri), and the subchannels zeros hQ(ji)and h^ri) ........50

4.3 Constellation plot of the received sequence for 16 QAM modulation....51

4.4 Equalized symbol-eye diagram, using RLS algorithm

with the raised-cosine channel.......................................51

4.5 Mean-square symbol error of RLS algorithm.............................52

x

4.6 The magnitude and zeros of impulse response h(n),

second class of channel model........................................53

4.7 Eye diagram of the received signal for 16 QAM modulation.............. 54

4.8 Equalizer output symbols, using RLS algorithm

with the second class of channel model...............................54

4.9 Mean-square symbol error of RLS algorithm............................. 55

4.10 Eye diagram of equalizer output, using cyclic LMS

with the raised-cosine channel........................................56

4.11 Mean-square symbol error of cyclic LMS ............................... 56

4.12 Equalized eye diagram, using LMS algorithm with the raised-cosine channel

.............................................................................57

4.13 Mean-square symbol error of LMS algorithm..............................58

4.14 Eye diagram of equalizer output, using LMS algorithm

with the second class of channel model...............................59

4.15 Mean-square symbol error of LMS algorithm..............................59

4.16 Mean-square symbol error, Comparison using raised-cosine channel......60

4.17 Mean-square symbol error, Comparison using second class of channel model

......................................................................61

4.18 Eye diagram of equalizer output, using LP method

with the braised-cosine channel.......................................62

4.19 Mean-square symbol error of LP method..................................63

xi

4.20 Eye diagram of equalizer output, using LP algorithm

with the second class of channel model...............................64

4.21 Mean-square symbol error of LP algorithm..............................65

4.22 Eye diagram of received signal for 16 QAM modulation..................67

4.23 Eye diagram of equalizer output, using RELS algorithm with the third

class of channel model (HR)..........................................67

4.24 Mean-square symbol error of RELS algorithm............................68

4.25 Scatter plot of equalized symbols, using RELS algorithm

with the presence of modeling error using (HR) channel...............69

4.26 Sample mean square error of RELS algorithm............................70

4.27 Mean-square symbol error, Comparison for those five algorithms

with the second class of channel model...............................71

xii

CHAPTER I

INTRODUCTION

1.1 Introduction and Background Information to Blind Adaptive Equalizer

Communication systems have been growing during the past 40 years, and

their applications have been used in many electronic products. In these days, radios,

televisions, mobiles, and computer terminals with Internet comprise an essential

element in our life. They all have ability of providing fast communications from

every corner of the globe. We notice that there is an almost unlimited or endless

amount of applications relying on the use of communication systems.

Any digital communication system involves transmission of analog signals

through channels. These channels are usually dispersive mediums that introduce

delay or memory in the received signal, which in turn spreads the symbols over time.

This spreading of symbols induces a distortion known as inter-symbol-interference

(ISI) [11], This kind of distortion is undesirable because it makes communication

systems less reliable and in order to maintain our systems reliable performance, it

must be removed at the receiver by equalization. In many high speed data rates

bandlimited digital communication systems, it is shown that ISI appears in all

communication channels such as twisted pairs cables, coaxial cables, fiber optics,

satellite, microwave and radio channels. This interference between the adjacent

symbols corrupts the detection of those symbols since their spreading can corrupt the

1

time spacing. Increasing ISI causes to increase the error probability at the receiver.

Therefore, under strong ISI, a simple memoryless decision device may not be able to

recover the original data sequence [7],

ISI needs to be suppressed at the receiver to keep our systems in reliable

performance. Equalizers are used as a solution at the receiver to compensate the

distortion (ISI) occurring during transmission. Equalizers try to extract the

transmitted symbol sequence by countering the effect of ISI. Therefore, they improve

the probability of correct symbol detection [16],

One of the earliest channel equalization methods is achieved by sending a

training sequence to estimate the linear filtering dispersion characteristics of the

nonideal channel (channel identification). Note that the nonideal channel

characteristics are usually not known priori, and the training sequence was already

known at the receiver. The training sequence is required to be sent periodically since

an adaptive channel equalizer relies on it, which occupies much of the bandwidth,

resulting lower communication link efficiency. For this reason, a self-recovering

approach that does not need a training sequence is invented. Such approach is termed

'blind'. This idea of self recovering blind equalization was first introduced by Sato in

1975 [7],

In recent few years, many different blind algorithms have been introduced,

and it has been growing to the idea of the so-called 'blind problem' [4], [5], [14], [17]

[28] and [29], In general, blind processing can be defined as a digital processing of

2

unknown signals that are sent through a linear channel without any knowledge of its

characteristics and additive noise. In wireless communications systems, the blind

signal processing plays an important role in combating frequency selective fading and

inter-symbol interference (ISI) [12], Blind adaptive equalization methods treat the

problem of recovering the transmitted input sequence by using only the received

output signal without knowledge of the channel characteristics. Since then, blind

channel equalization has obtained great research interest because of the fact that

receivers start equalization with no need of transmitter assistance (sending training

signals). Consequently, a better channel bandwidth efficiency is obtained.

3

1.2 Thesis Overview

This thesis considers the problem of blind adaptive equalization in

communication systems. It starts presenting some introduction to general

communication systems and their applications in our daily life. The introduction also

explains how the distortion (ISI) is generated and due to its effects on our system's

performance, it has to be removed by using equalization. It introduces one of the

earliest equalization algorithms (using training sequences) and its drawbacks with

respect to valuable performance and throughput, causing to develop the so-called

blind equalization techniques. Definition of blind processing and its advantages is

also demonstrated in a simple way. Chapter 2 describes the channel model used

throughout this thesis. Fractionally sampling scenario is considered in our channel

model that is SIMO scheme. It also presents the complete system model (channel

with linear equalizer) and their vector forms representation that will be useful for

derivations the blind algorithms that are discussed in chapter 3. This chapter ends

with discussing the conditions to achieve a perfect equalization.

Chapter 3 begins with the definition of adaptive filter and its applications,

including channel equalization. It introduces blind adaptive SIMO channel

equalization method (using only second order statistics (SOS) of the received signals

to design the equalizer) such as Direct Blind MMSE Equalizers. This chapter also

presents the calculation methods for blind equalizer including derivations of

Recursive Least Square (RLS), cyclic Least Mean Square (cyclic LMS), Least Mean

4

Square (LMS), Linear Prediction (using GP technique) and Recursive Extended Least

Square (RELS).

Chapter 4 gives simulation results of the equalization techniques to examine

their performance behaviors. One of the performance measures is Mean Square Error

(MSE) that can be used to compare the equalization algorithms. The channel model

classes used through simulations are also described in this chapter. The rest of the

chapter is graphs, results, and comparisons between all of these algorithms with

respect to their MSE using different classes of channel model. Chapter 5 gives a brief

conclusion to the overall concepts included in this thesis.

5

1.3 Mathematical Notations

Wc(t) Continuous-time input information signal

w(n) Input discrete-time sequences

w(n) Equalizer output

m Dirac delta function

Continuous-time impulse response of composite channel

h(n) Discrete-time equivalent of hc(t)

/loOO.hiO) Finite-length impulse responses of the two subchannels

h Vector form of channel impulse response

h,h, Vector form of the two subchannels coefficients

vr(0 Continuous-time additive noise

v(n) Discrete-time equivalent of vc(t)

xc(t) Continuous-time channel output

y,(0 Continuous-time channel output with additive noise

y(n) Received discrete data signal, input to equalizer

yo00,yi(n) Received discrete data signals of the subchannels outputs

6

Discrete-time impulse response of linear equalizer

9(70

g......................Vector form of linear equalizer

go, gi ................Vector forms of the two subequalizers

Ts ....................Pulse duration

* ..................... Convolution

(.)* ..................Complex conjugate

(.)T...................Transpose

(. )H .................Hermition or the conjugate transpose

(.)t...................Pseudoinverse

HT .................... Channel convolution matrix, Sylvester or Block Toeplitz

matrix

^2w(n), C2v(.n)> C2y(n) Correlations of the stationary input, noise, and channel

noisy output

/......................Cost function

A

V/0(JV) ............... The instantaneous approximation to the gradient of the cost

function /0

/(g)...................Prediction error variance

||. || ................Norm or absolute value

7

El] Expected value

Input variance

ffn The channel noise variance

A Forgetting factor

/.i,a The small step size

Y Constraint value

B(q 1),C(qi 1),i4(q1 x).. Polynomials of IIR channels

F(q x) ................. Common factor (minimum phase)

y(i + 1)................One step-ahead prediction of x1(t + 1)

A

y(i + 1)................A posteriori prediction

R(q 1),5(q1 1),Z)((7 x) .. Polynomials of HR equalizer

A

6(i) .....................An estimate of unknown 0O

I ........................Identity matrix

p0 .......................An arbitrary finite positive scalar

MSE.......................Mean square error

8

Roll-off factor

P

rc(t,p)...................Raised-cosine channel

S ........................ A parameter defining the size of the channel modeling

error

i.i.d.....................Independent and identically distributed

FIR, HR...................Finite impulse response, Infinite impulse response

SIMO...................... Single input-multiple output

SOS.......................Second order statistics

QAM.......................Quadrature amplitude modulation

LTI.......................Linear time invariant

9

CHAPTER II

MODEL OF COMMUNICATION SYSTEM AND

MATHEMATICAL FRAMEWORK

2.1 Introduction

A general and basic model for a communication system is comprised of digital

signal that is transmitted by a transmitter through an analog channel to a receiver (see

Figure 2.1).

Source Info. Transmitter Channel Receiver

Received Info.

Figure 2.1 General model of a communication system.

Source information may be either a digital or an analog signal. The transmitter

is fed from source information, which for the purpose of this thesis we assume is a

binary. However, when the information source is an analog signal, transmitter is

extended to have a so-called source encoder to convert the analog signal into a

sequence of binary digits. This sequence is passed through a channel encoder to

introduce a redundancy in the binary information sequence. This redundancy in the

binary information is used at the receiver to overcome the effect of noise and

interference in the transmission signal through the channel. The binary sequence at

the output of the channel encoder is passed to a digital modulator which does the job

of mapping the binary information sequence into waveforms to be sent through

10

communication channels. Figure 2.2 illustrates basic elements in a digital

communication system model. The main digital modulation scheme used in the

analysis and simulations throughout this thesis is a 16-QAM (quadrature amplitude

modulation) since the symbols are usually complex with in-phase and quadrature

components. Non ideal analog media such as telephone cables and radio channels

typically distort the transmitted signal. During transmission, the essential features of

the transmitted signal are corrupted in a random manner.

Figure 2.2 Basic elements in a digital communication system.

At the receiver end of a digital communication system, the digital

demodulator processes the channel corrupted signal waveform to convert the

waveform into a sequence of number that represents the transmitted signal. Then, this

sequence is passed through a channel decoder, which attempts to recover the original

information sequence. How well the original signal is recovered depends on many

11

factors such as transmitted signal power, characteristics of the channel, the amount of

noise and nature of interference.

QAM 16 and higher is widely used in communication systems, and it

combines both phase shift keying (PSK) and amplitude shift keying (ASK)

modulations. A simple concept of input-output communication system using M-QAM

is that the transmitter generates a sequence of complex-valued random input

data (w(Z)}, each element of which belongs to a complex alphabet A (or

constellation) of M-QAM symbols. The data sequence (w(Z)} is sent through a

baseband equivalent unknown complex linear time invariant (LTI) channel whose

output x(t) is observed by the receiver. The function of the receiver is to estimate the

original data (w(Z)} from the received signal x(t) [7], QAM 16 will be used in this

thesis because its constellations allow for more bits per symbol and thus more bits per

hertz.

2.2 Channel Model

In a digital communication system, the transmitter modem takes R bits of

binary data at a time and encodes them into one of 2R analog symbols for

transmission (at the signaling rate) over an analog channel. At the receiver the analog

signal is sampled and decoded into the required digital format.

Assume that at the transmitter modem, the Ith set of R binary digits is mapped

into a pulse of duration Ts seconds and an amplitude w(Z). Thus the modulator output

signal, which is the input to the communication channel, is given as

12

Wc(t) = Zi w(z)5(t ITS).

(2.1)

where wc(t) is the continuous-time information signal, S(t) is a pulse of duration rs

with an amplitude w(Z).

The unknown complex-valued LTI communication channel with the impulse

response h(t) composited with the known transmit and receive filters is shown in

Figure 2.3. The overall composite channel hc(t) is finite impulse response (FIR).

Additive noise vc(t) that is assumed to be stationary and uncorrelated with the

information symbols w(n) can be included in this model.

vc(t)

wc(t)

p

y(n)

*-

Figure 2.3 Fractionally sampled (P = 2 or more) communication system.

The composite channel (whose impulse response is hc(t)) is non ideal, which

means hc(t) is nonzero for t A 0. An integer P in Figure 2.3 denotes the amount of

fractionally sampling (oversampling). The channel output can be modeled as the

convolution of the input signal and composite channel response. The signal at the

sampler is:

yc(t) = (wc(t) hc(t)) + vc(t),

13

(2.2)

XCO . .

-co wc(.T)hc(t t) dr + vc(t).

substituting (2.1) into (2.2), we get

yc(0 = Zr=-co f_l w(l)S(r lTs)hc{t r)dr + vc(t). (2.3)

yc(0 = Ii w(l)hc(t ITS) + vc(t). (2.4)

If yc(t) is sampled at t = s/p, the received discrete data signal (y(n) =

yc(t) fort = nrVp) is

y(n) = w(Z)/ic ^n7Vp Z7;^ + vc(nTs/p). (2.5)

Considering w(n) and y(n) are the input and output discrete-time sequences,

respectively, it is convenient to rewrite (2.5) as an equivalent discrete-time system

yOO = E^-co w(l)h(n IP) + i

= x(n) + v(n). (2.6)

where h(n) and v(n) are the discrete-time equivalents of hc(t) and vc(t),

respectively, x(n) is the discrete-time equivalent of the noise free received signal,

and index n can be any integer (n = , 3, 2, 1,0,1,2, 3, ) [11],

14

In the single input-multiple output model, an integer P > 1 of measurements

are performed for each transmitted symbol, provided the continuous-time observation

is oversampled compared with the transmitted symbol rate, leading to create a

fractionally spaced scenario. Throughout this thesis, it is assumed that P = 2 to make

our model in the oversampling scheme (see Figure 2.4).

v0(n)

w(n)

v^ri)

Figure 2.4 SIMO multichannel model (P = 2).

now, (2.6) can be rewritten in fractionally spaced system as follows

Vi(n) = w(/)/ij(n Z) + Vi(n) for i = 0,1

(2.7)

with

y0(n) = w(n) /i0(n) + v(n),

lT=-oo w(Z)/i0(n 0 + v0(n).

= CO

(2.8)

and

15

yi(n) = w(n) h^n) + 1^(71),

= l?=-o0w(l)h1(n-l)+v1(ri). (2.9)

Note that h0(ri) and (n) are finite-length channel impulse responses of the

first and second channels, respectively, at rate .

2.3 Vector Representation

A vector representation is very important and useful in our system analysis

and equalizer implementation because it makes our finite-length subchannels

hQ(n) and (n) easy to understand and modify. The input, noise and output such as

w(n), v(n) andy(n) can be also expressed in vector form. The index n in these

signals is usually infinite length. However, it will be finite-length when it is used as a

vector form. Assume that the finite-length channel h(n) is causal with length 2Lh

such that each subchannel has a length Lh. Considering the following vectors:

y(n) = [y0(n) yi(n)]T. (2.10)

v(n) = [v0(n) vt(ri)]T. (2.11)

pio(O) /i0(l) ... k( 0) ^(1) ... K(Lh 1) h\(Lh 1)1 (2.12)

ho = [/io(0) h0(l) ... h0(Lh~ 1)]. (2.13)

16

and

hi = [/i!(0) /ii(l) ... h^Ln-1)}.

(2.14)

then

y(n)

To 00' h0i 0) h0(l) o 1

.yiOO. MO) /id!) . KUb-D)

w(n)

w(n 1)

w(n (Lh 1))

+

v0(n)'

v1(n)_

(2.15)

or

r^-i

yin) =

y0d)'

J'l(w).

^ win l)h0(l) + v0(n)

1=0

^ win 1)^(1) + v^ri)

L 1=0

= 1 win ~ OhiO + v(n).

(2.16)

where h is a vector form of the two subchannels, h0 and hj represent the vector form

coefficients of the first and second channels, respectively. The noisy output vector of

1-input/ 2-output causal 2Lh FIR channel filters driven by the input sequence is yin).

Note that (Lh 1) is the subchannel polynomial order.

17

2.4 Overall System Model with Equalizer

A finite impulse response filter g(n) as an equalizer with vector parameters g

is applied to our channel model to make our system into a complete system form (see

Figure 2.5). The goal of the equalizer is to remove the distortion caused by channel

1ST

v0(ri)

w(n)

w(n)

Figure 2.5 Multichannel system models with equalizer g(n).

Assume that the finite-length equalizer vector g can be defined in a same

manner as h (as mentioned in previous section) with length 2Lg such that each

subequalizer has a length Lg. The following vectors are defined

g = [0o(O) t/,,0 ) .... g0(Ls 1)

Si(0) 0i(l) ...... l)]r. (2.17)

with

go = [0o(0) 0o(1) ..... g0(Lg-l)]T. (2.18)

18

and

gi = [0i(0) 0i(l) ...... gi{Lg 1)]T.

(2.19)

where g0 and g1 represent parameter vectors of the first and second equalizers,

respectively. Let w(n) be the output signal of equalizer, and then the input-output

relationship can be written in the following form

w(n) = w(n) [/i0(n) g0(ri)] + v0(ri)

+w(n) [h^n) g^n)] + vt(n),

(2.20)

w

(n) = w(n) * g0(l)h0(n /)] + v0(ri)

1=0

+w(n) \Z\Iq1 giWKtji /)] + Vi(n). (2.21)

To simplify (2.21), let's consider our system in the absence of noise and define

Hr that is associated with vector h to be a channel convolution matrix with the length

(Lh + Lg l) x PLg as below

HT

\ ho(0) 0

/Ii(0) ho(0)

/lo(l) h- (0) ho(0)

hi(Lh ~V h0(Lh 1)

6

o

o

o

(2.22)

ho(Lh 1)

hi(Lh 1)J

19

The convolution matrix Hr is also called Sylvester or block Toeplitz matrix

[15] of the two subchannels, and P is equivalent to linear filters that the input signal

w(n) passes through (P = 2). Alternative representation of (2.22) is found in some

books and is written in the following form

HT = [HT0 : HT1],

fto(0) 0 0

fto(D fto(0) h0(l) 0

1 C o fto(0)

0 1 e O : h0(i)

0

0 1 e o

fti(0) 0 0

hid) fti(0) hid) 0

K (Lh 1) fti(0)

0 Kdh-i) : ^(1)

0

0 .. hdd-1).

(2.23)

then (2.21) can be written as below

w(n) = (H7g}w(n)

[HT0

:HTl]

go

Si.

w(n).

(2.24)

O o -X2 0 0 0 0

h0W ho(0) dd) fti(0)

hoW 0 dd) 0

1 o -X2 ho(P) hdd-d fti(0)

0 /\ 1 o -X2 \ ^1) 0 hdd-d ! ftiCl)

0 0

1 0 1 o -X2 0 .. hdu-i).

' 50(0) '

5o(l)

1 o 1)

5i(0) >

5i(l)

-5l (Lg ~ 1)1

(2.25)

20

with

T

w(n) = [w(n) w(n 1) ... w(n Lh Lg + 1)] . (2.26)

where Hr0 and \ are the convolution matrices of the two subchannels. Note that

there is no need to know the channel order Lh precisely, only the equalizer length Lg

has to be chosen so that the following hypotheses hold in order to have sufficient

channel diversity

H-l) Lh+ Lg 1 < PLg (Hr has strictly more columns than rows).

H2) Hr has full row rank Lh + Lg 1 which means the matrix Hr is invertible.

The length of equalizer Lg must be chosen large enough (i.e. Lg> Lh 1) in

order to have sufficient equations to solve the equalizer. In this case, a perfect

equalization can be achieved. It has been shown as in [8], [10], [11], [18], and [24]

that the block Toeplitz structure of Hr implies that it will be full row rank if and only

if there are no common subchannel roots and Lg > Lh 1. It is also found that there

is a unique solution of the equalizer equations if Lg = Lh 1 .

21

CHAPTER III

BLIND ADAPTIVE CHANNEL EQUALIZATION METHODS

3.1 Adaptive Filter Theory

(The beginning of this section is inspired from [13].)

An adaptive filter is defined as a self-designing system that relies for its

operation on a recursive algorithm, which makes it possible for the filter to perform

satisfactorily in an environment where knowledge of the relevant statistics is not

available. Linear and non linear are main groups of adaptive filters. Linear adaptive

filters compute an estimate of a desired response by using a linear combination of the

available set of observables applied to the input of the filter. Otherwise, the adaptive

filter is said to be non linear. In this chapter, adaptive filters theories are described for

different kinds of algorithms.

Adaptive filters can be integrated in systems with different functionalities,

being predictive deconvolution, system identification, channel equalization, spectral

analysis, signal detection, beamforming, interference and noise cancellation examples

of such. Also, adaptive filters can adjust to unknown environment, and even track

signal or system characteristics varying over time.

22

3.2 Adaptive Equalization

In digital communication systems, one application of adaptive filters is

adaptive channel equalization that removes ISI generated from unknown channel. ISI

can be compensated or removed from received signals by using many approaches

such as optimum receiver which uses maximum likelihood estimation, suboptimum

receiver which uses a linear equalizer or a decision-feedback equalizer [20], The three

equalization methods assume impulse response of the channel characteristics or

frequency response of the channel characteristics were known a priori to the receiver.

However, in most practical digital communication systems that use equalizers the

channel characteristics are not known a priori. Therefore, training sequences that

provide different realizations of a desired reference signal can be used to estimate the

channel and find the necessary equalizer. The reliance of an adaptive channel

equalizer on a training sequence requires that the transmitter cooperates by resending

the training sequence, lowering the effective data rate of the communication link [7].

In high speed digital communication systems, the transmission of a training

sequence is either impractical or very costly in terms of data throughput. Another

drawback of using training sequences for channel equalization is that time and

bandwidth are consumed for the equalization process, causing most adaptive filters

whose desired reference signals are training sequences cannot be used. For these

reasons, blind adaptive channel equalization algorithms have been invented and

developed, and they have been the best approaches that satisfy our system models.

23

The purpose of this thesis is applying blind adaptive channel equalization with the

variety of algorithms.

3.3 Blind Adaptive Equalization

In some applications, such as multipoint communication networks, the input

signal to the channel is unknown to the receiver. Only statistical properties of the

input structure that blind equalization based on to recover the unknown input

sequence are known. Therefore, it is desirable for the receiver to synchronize the

received signal and to adjust the equalizer (self-adaptation) without having a known

training sequence available. Equalization techniques based on the initial adjustments

of the coefficients without benefit of the training sequence are said to be blind. Now,

we can define a blind adaptive filter is an equalizer that uses one of these techniques

to perform recursive adjustments of its parameters without the aid of training

sequences.

3.4 Blind SIMO Channel Equalization Methods

Using Second Order Statistics (SOS)

In the blind methods of equalization, some structure properties of the input

signal are used such as whiteness in conjunction with the receiver outputs in order to

estimate the equalizer. Also, most commonly used adaptive algorithms for blind

channel equalization do not require extra bandwidth for training. Therefore, blind

equalizers have gotten great research and practical interest, and many effective

algorithms have been proposed for last few years [19] and [23], In recent papers, ( see

for example [3], [18], [26] and [27]), it has been shown that the second order statistics

24

(SOS) of the channel output y(n) is sufficient to estimate most communication

channels when the received signals are fractionally sampled (oversampling), or

multiple antennas are used.

The methods introduced in [9], [10], [11] and [22] have been used SOS of the

channel output to directly estimate the equalizer without finding the channel impulse

response as a first step. These types of methods are called blind methods. In the

following section, the cyclic statistical property of the fractionally sampled

observation [y(n)] is described because some calculation algorithms that will be

discussed later in this thesis use this property.

3.5 Cyclic Statistics

In this section, we describe the covariance of yN(n) which will be useful for

finding the equalizers directly from the noise-free channel output. The following

vectors are defined to represent N vector observations of the SIMO fractionally

spaced model (see Figure 2.5).

TwOO = [yoO0 ToOi 1) ... y0(n N + 1) y^n) y^n 1)

... yt{n-N + l)]7)

xw(n) = [x0(n) x0(n 1) ... x0(n N + 1) x1(n) x1(n 1)

... x1(n N + 1)]T,

25

vN(n) = [v0(n) v0(n 1) ... v0(n N + 1) 1^(71) vx{n 1)

... 1^(71 N + 1)]T.

(3.1)

The correlation of the scalar output y(n) in equation (2.6) is defined

c2y(n; rri) = Â£'{y(n)y*(n + m)}.

(3.2)

-2y

(jim, rri) = ^ ^

K = CO ln= CO

c2w(^2 li)Kn 2/1)/i*(n 212)

+c2v(rri).

(3.3)

where m = 0,1, ...,Lg 1, c2w(m) = E{w(n)w*(n + m)} and c2v(v) =

E{v(n)v*(n + m)} are the correlations of the stationary input and noise, respectively.

It can be proven from (3.3) that the correlation is periodically time-varying in n with

period 2, see for proving [25],

c2y(n; m) = c2y(n + 21; m) V l integer. (3.4)

Due to the periodicity of the correlation matrix, we can use yN (n) in similar

manner instead of using y(n) to find the correlation matrix C2y as below

c2y = E{yN(r0yN(n)H}.

(3.5)

26

3.6 Direct Blind MMSE Equalizers

In this section, we address noise suppression and consider FIR Weiner filters

to determine the minimum mean-square sense estimate of equalizer output w(n)

using only y(n). The objective is to compute the equalizer g(n) so that the following

cost function / is minimized.

/ = E{\w(ri) w(n d)\2}. (3.6)

where d is the integer delay between input w(n) and noise-free channel outputs

Xi(ri), i = 0,1. The output of equalizer w(n) can be written as below

w(n) = zffo^oCOyoCn 0 + 1^ V(0yi(n H

W(n) = T,i=oT,Lli01 gi(.Qyi(n l). (3.7)

We substitute (3.7) into (3.6) and take the first complex partial derivative with

respect to the unknown equalizer coefficients and set it equal zero to minimize /

d

dg*k (pi)

E

I.i=o ij'f o19i(0yi(n -I)- w(n d)

= 0.

(3.8)

where k = 0,1 and m = 0,1, ...,Lg 1. After simplification (see [25]), it yields to

orthogonality condition

gi(.0E{yi(n l)yk*(n m)}

i=0 1=0

E{w(n d)yk*(ji m)} = 0.

(3.9)

27

Since the input symbols are assumed to be i.i.d., and uncorrelated with the

noise, the second term of (3.9) is exist if and only if l = n d. Therefore, the second

term can be written as follow

Lg 1

E{w(n d)yk*(ji m)} = E \ w(n d) 'S' w*(V)hk(n m V)

= E{w(n d)w*(n d)

x h*k(n m (n d))},

E{w(n d)w*(n d)} h*k(d m),

- alh*k(d -m).

substituting (3.10) into (3.9), yields

i Lg i

^ ^ gi(l)E{yi(n l)yk{n m)} a^h*k(d m) = 0,

i=0 1=0

ZUoZilo1 di(0E{yi(n l)yk*(n m)} = o*h*k(d m). (3.11)

the above equation can be written in a vector form as

E {y*Lg(.n)y[g(n)}gd = cr^H\:,d + 1).

C$gd = a*H\:,d + l).

go=^[C2*y] H*(:,l).

(3.10)

(3.12)

(3.13)

(3.14)

28

gd = oi[C$Cg* KJl+H-C:, 1)

(3.15)

The above equations (3.14) and (3.15) give to us the zero-delay MMSE

equalizer by substituting d = 0 and the arbitrary nonzero delay (d =Â£ 0) MMSE

equalizer.

3.7 Methods for Blind Equalizer Calculation

This section introduces and describes five methods for blind equalizer

calculation with their derivations. These blind algorithms are procedures for self

adjusting the parameters of an adaptive filter to minimize the cost function. We

describe the general form of adaptive FIR and HR filtering concept.

3.7.1 Recursive Blind Adaptive Using Recursive Least

Square Algorithm (RLS)

This algorithm addresses the problem of blind adaptive equalization of finite

impulse response channels (FIR) with the exploiting the diversity induced by sensors

arrays or fractionally sampled. It analyses and solves the case when we have parallel

subchannels in the presence or absence of noise. It is shown later that the resulting a

priori error converges towards a scalar version of the input symbol sequence and the

equalizer parameters are estimated by using Recursive Least Square (RLS) algorithm.

This algorithm is very useful for tracking time varying channel. The RLS algorithm

uses information from all past input samples (and not only from the current tap-input

samples) to estimate the static correlation matrix of the input vector C^y. To start

29

driving this algorithm, it is assumed that the additive noise is to be uncorrelated with

input samples and a forgetting factor A (0 < A < 1) is included to reduce the effect of

past observations on the statistics correlation matrix estimate C^y.

ClyiN) = EO^My^Oi)},

N

1=0

At Z = JV, the correlation matrix is simplified as

N-l

c\y{N) = Â£ A-'y4(0yr(0 + yt3mylm

1=0

= AC*y(N-i)+y[g(N)yZg(N).

(3.16)

The result of (3.16) is a recursive form, and it can be applied to (3.14) to find

MMSE equalizer g0 with zero delay. However, using the matrix inversion lemma [13,

p.480], a direct method to find g0 is developed without requiring a matrix inverse or

a -i-i

pseudoinversion of C^y. This can be done by defining P(1V) =

c*2ym

and

applying the matrix inversion lemma with (3.16)

P(iV) = A^PQV 1)

A~2P(N 1 )y*lg{N l)yTlg{N 1 )P(JV 1)

1 + A-'y[g(N 1)P(JV Y)ylg{N 1)

(3.17)

30

The above equation can be applied to (3.14) to estimate the MMSE zero delay

equalizer g0(iV)

A i_3^V + l

g(iV) = P(W)i^I-
The equalizer estimate form in (3.18) does not require a matrix inverse and,

hence, is computationally feasible for adaptive implementation. If A = 1 is chosen,

(3.18) provides a method to recursively compute the time invariant equalizer taps,

thereby reducing the memory requirements for long data records [11], It is very

A

important to mention that the convergence of C2y(N) is affected by the forgetting

factor A and, thereby, it affects the accuracy of the estimator. The forgetting factor A

A

is usually chosen between [0.98, 1], The delay MMSE equalizer gd(JV) can be driven

in a similar way using the same above steps in this method.

3.7.2 Recursive Blind Adaptive Using Cyclic Least

Mean Square Algorithm (cyclic LMS)

This algorithm also considers the problem of blind adaptive equalization of

finite impulse response channels (FIR) with the exploiting the diversity induced by

sensors arrays or fractionally sampled. It analyses and solves the case when we have

parallel subchannels in the presence or absence of noise. The cyclic LMS algorithm is

very popular, and it uses the Stochastic Gradient Descent approach for updating the

equalizer coefficients at each symbol. It provides an instantaneous approximation

estimate to the gradient vector of the cost function. The goal of the cyclic LMS

method is to estimate the equalizer coefficients to achieve the least mean squared

31

error. It is shown later that the resulting error converges towards a scalar version of

the input symbol sequence. The update of the coefficients is performed in the

following gradient descent equation

goOV) = g0QV 1) ^V/o(JV). (3.19)

where p is the step size parameter which controls the moving distance along the error

A

surface, and VJ0(N) is the instantaneous approximation to the gradient of the cost

function Jo = E{|w(n) w(n)|2}.

A

V/0

dJo dJo dJo dJo

dJo dJo

-iT

3go(0) dg*(0) 3g5(l) dg*(l) dg*0(Lg-l) dg*1(Lg-i)_

(3.20)

The equation (3.20) can be simplified in a same manner as in the derivation of the

MMSE equalizer (see section 3.6), yields

m = E{yl1 yr}go-^H"(l,:).

(3.21)

as a result, the instantaneous approximation at time JV is obtained by

V/oQV) = y*Lg(N)y[g(N)g0(N 1) a2H"(l,:).

(3.22)

substituting (3.22) into (3.19), yields to the following equation that the equalizer

coefficients can be estimated

goQV) = g0QV 1) ^[y*Lg{N)ylg(iV)go(N 1) a2H"(l, :)](3.23)

32

The speed of convergence and steady-state performance rely on choosing of

the step size /i. The convergence analysis issues are not addressed in this thesis, and

they have been studied in many papers and books such as [13],

The cyclic LMS algorithm, as we can see through derivation, does not rely on

an explicit matrix inverse. Therefore, it is not sensitive to nearly common subchannel

roots. In addition, it has extremely low computational complexity, but it has slow

convergence [13, pp. 334- 335],

3.7.3 Recursive Blind Adaptive Using Least Mean

Square Algorithm (LMS)

The LMS algorithm is widely used in various applications of adaptive filtering

due to its computational simplicity. This algorithm is similar to previous section, the

step size parameter /i, which affects the convergence speed of the LMS, is used to

control the moving distance along the error surface. They use information only from

the current tap-input symbols. Also, they both depend on the statistics of the input

w(n) and the output y(n) signals since their updating equations consist of first

complex partial derivative to the cost function that has expectation to those two

signals. However, this algorithm uses a Steepest-Descent-Based algorithm for

updating the equalizer coefficients at each symbol [9].

In section 3.6, the optimal solution for parameters of the adaptive filter is

driven. The objective of this optimal (Wiener) solution is to compute the equalizer

g(n) by minimizing the cost function / relying only on using the observation y(n). In

33

other word, this solution leads to the minimum mean-square error in estimating the

reference signal w(n). The optimal (Wiener) solution is given by

go = [C2*"]+Â£{w(n)y*(n)},

go=^[C2*y]+H*M). (3.24)

where C^y = E{y^(ri)y^(n)} and let p = E{w(ji)y*N(n)} which both are unknown.

Now, the LMS method updates the equalizer coefficients using a steepest-descent-

based algorithm which in turn can be used to search the Wiener solution of equation

(3.24) as follows

A

g(n + 1) = g(n) -p/(n),

g(n + 1) = g(n) + 2p[p(n) - (n)g(n)]. (3.25)

A

where /(n) is an instantaneous estimate to the gradient vector of the cost function

with respect to the filter coefficients. Employing instantaneous approximation

A

estimates C2y(n) and p(n) for C^y and p is one possible solution to estimate the

gradient vector as follow

A

Czy(n) = y*N(n)yl(n). (3.26)

p(n) = w(n)y(n). (3.27)

34

The gradient estimate is given by

/(n) = -2[p(n) - (n)g(n)]

= 2 w(n)y*N(n) + 2y(n)y^(n)g(n)

= -2y*(n)[w(n) -yÂ£(n)g(n)]

= 2 y^(n)err(n). (3.28)

This can be applied back to (3.25), the resulting is the updating equalizer parameters

estimate of the least mean square LMS algorithm. In summary, the updating

equations for the LMS algorithm are described by

err{n) = w(n) yN(n)g(n). (3.29)

g(n + 1) = g(n) + 2 fiy^in)err(n). (3.30)

The LMS coefficients update which is illustrated in the equation (3.30) is a

form of time-averaging that smooth the errors in the instantaneous gradient

calculation to obtain a more reasonable estimate of the true gradient [9],

35

3.7.4 Unbiased Blind Adaptive Using Gradient Projection Technique

This algorithm considers the problem of blind adaptive equalization of finite

impulse response channels (FIR) with the exploiting the diversity induced by sensors

arrays or fractionally sampled. It analyses and solves the case when we have parallel

subchannels in a noisy environment. An adaptive equalizer is developed depending

on the FIR quadratically constrained filter which uses the Gradient Projection (GP)

technique as a solution for its implementation at the receiver. It has been shown that

the (GP) technique as described for example in the work by [6] is the most direct

solution to the adaptive implementation of quadratically constrained filters. Also, the

Gradient Projection algorithm can be used in adaptive beamforming problems at low

cost sense [10], It is shown later that the resulting prediction error converges towards

a scalar version of the input symbol sequence and the adaptive filter parameters are

estimated by using (GP) algorithm. In addition, the additive noise is assumed to be

uncorrelated with input samples.

In order to start driving this method, recall form chapter two that there is no

need to know the channel order Lh precisely, only the equalizer length Lg has to be

chosen such that the following hypotheses hold in order to have sufficient channel

diversity

H-l) Lh+ Lg 1 < PLg (Hr has strictly more columns than rows).

H2) Hr has full row rank Lh + Lg 1 which means the matrix Hr is invertible.

36

The notion of linear prediction method (LP) is whitening the observation to

find a channel inverse filter, using the following a priori constraint:

H3) The input sequence w(n) is white, E{w(ri)w*(n k)} = 8k.

It is a good point to mention that there is limitation in the SISO case because

the whiteness constraint is too weak to allow phase and amplitude equalization of a

mixed-phase channel. However, in the SIMO model, this limitation vanishes due to

hypotheses and H2, so the exact FIR inverse of the nonideal channel exists. It has

been shown in [10] and [24] that the resulting noise-free prediction error signal

related to the input symbol sequence w(n).

Assume that the finite-length equalizer vector g be a 2Lg x 1 complex-valued

vector of prediction coefficients, and errk(n) be the prediction error defined for k =

0,1. The prediction error achieves its optimality in the noise free case if and only

if errk(ji) = hk(0)w(n), and thus the prediction error variance /(g) = E\errk(n)\2

is minimized. The performance versus SNR of this theoretical equalization method

would critically depend on the realization of the particular coefficient hk(0), so it is

more appropriate to exploit the predictor as a tool to identify the channel [10], A

major drawback of the prediction error filtering is sensitivity to channel additive

noise. For this reason, a modified prediction scheme will be introduced to allow the

adaptive computation of an unbiased predictor in a noisy environment. Adaptive filter

parameters are estimated by using the Gradient Projection technique. The algorithm

which is first introduced by [10] goes as follows:

37

errk(ri) = yk(ri) g(n l)HyN{n 1).

(3.31)

uk(ri) =yN(n- 1 )errfc(n)*. (3.32)

g(n) = g(n 1) + auk(n). (3.33)

g(n)=S(',)yj|g(Jl)|. (3.34)

where a is a small step size and y is a constraint value. The performance behavior of

the mean square error is influenced by the choice of y in this particular algorithm. It

is not difficult to see that (3.33) is a standard LMS update of equalizer parameters.

This algorithm has a very low computational cost, and it shows desirable robustness

properties [10],

3.7.5 Recursive Blind Adaptive Using Recursive Extended

Least Square Algorithm (RELS)

This algorithm considers the problem of blind adaptive equalization of infinite

impulse response (HR) channels without requiring the channel diversity condition,

and it analyses and solves the case when we have parallel subchannels having

common zeros. This common factor is assumed as a minimum phase filter, while

overall subchannels can be a non minimum phase systems. An equalizer is developed

based on the optimal HR filter as a predictor of the received signal. The main criterion

of this method is a one-step ahead prediction of one of the subchannel outputs [1],

[2], [10] and [22], It is shown that the resulting prediction error converges towards a

scalar version of the input symbol sequence and the adaptive predictor parameters are

38

estimated by using Recursive Extended Least Squares (RELS) Algorithm. In addition,

it is assumed that the input samples are mutually uncorrelated, and the parameter

estimates are updated when each single signal is received [22],

Prediction -Based Equalizer

Consider the case of single-input two-output system model which means the

receiver performs two measurements for each transmitted symbol. An equivalent

representation to our channel model is shown in Figure 3.1.

Xi(t)

Aiq-1)

w(i),

*2(0

C(q~1)F(q~1)

Aiq-1)

w(i).

(3.35)

where q 1 is a unit delay operator, w(i) is the transmitted symbol and

A(q_1) =1+a1q~1 + I- aLq~L ,

B(q 1 + I- bLq L, (3.36)

Ciq-1) =c0+c1q1 + + cLq~h ,

F(q 1) =l+f1q 1 + + fnFq np.

where L is the polynomial channel order. Assuming that polynomials A(q_1),

5(q_1), and C(q-1) are of the same order, and assuming that V^q-1) 1S a stable

operator. Also, 5(q_1) and C(q-1) are coprime polynomials as well as F(q_1) is a

39

minimum phase polynomial. In general, all quantities in (3.35) and (3.36) can be

complex numbers.

w(i)

Xi

A(q~i)

C(q~1)F(q~1) *2

w A(q~i) W

Figure 3.1 SIMO channel model (HR).

Figure 3.2 Predictor based equalizer.

40

The prediction-based equalizer is shown in Figure 3.2, where y(t + 1) is one

step-ahead prediction of x1(t + 1). Let S(q_1) and D(q~1) be the filter

operators and the polynomials in q_1, so y(t + 1) is

y(i + 1)

K&T1)

Diq-1)

Xi(t) +

Siq-1)

Diq-1)

x2(0-

(3.37)

which is optimal in the mean-square sense. The polynomials R(q x), S(q x) and

D(q~1') are computed by minimizing the following cost function / :

/ = E^x^i + 1) y(t + 1)|2).

(3.38)

Note that we can use x2(t) instead of x1(t) as a reference signal and find the

predictor by minimizing Â£'(|x2(t + 1) y(t + 1)|2). From (3.35) and (3.37),

x1(t + 1) can be written in the following form

x1(t + 1) y(t + 1) = x(t) + b0w(i + 1). (3.39)

where

x(t) = H(q 1)w(t)-

and

(3.40)

Htq-1)

[B(q~1)R(q~1) + C(q-')S(q-')]F(q-')

A(q-')D(q-i)

q[B(q 1)F(q - A{q x)bQ]

A (q-1)

(3.41)

41

since F(q_1) is not a factor of from (3.41) it follows that Fl(q~1)=0 if and

only if D(q_1) = F(q_1) whereas F(q_1) and S(q_1) are solutions of the following

polynomial equation:

= Â£7[jB(Â£7-1)F(c7-1) Aiq-^bo] (3.42)

notice also that H(q_1) = 0 gives x(i) = 0 and from (3.39), we get the prediction

error

x1(t + 1) y(i + 1) = h0w(i + 1). (3.43)

Let the predictor operators F(q_1) and 5(q_1) are defined as

Riq-1) =r0+ rxq~x + + rNiq~N\

Siq-1) = s0 +s1cT1 + + sN2q~Nz. (3.44)

Then (3.42) has a solution with respect to F(q_1) and S(q_1) if JV-l > iV0,

JV2 > iV0, where iV0 = max(L 1 ,nF 1). A unique solution exists if 5(q_1) and

C(q-1) are coprime, L > nF and at least one of the following holds: ^ =

JV0 or N2 = N0.

In practice, we cannot use (3.42) to calculate F(q_1), S(q_1) and F(q_1)

since polynomials 5(q_1), C(q~r) and F(q_1) are unknown. Therefore, we cannot

use a nonadaptive predictor (3.37) to calculate prediction error (3.43). A recursive

algorithm is proposed for directly estimating the unknown parameters in (3.37).

42

Taking advantage to our assumption that D(q 1) = F(q 1), the nonadaptive

predictor (3.37) can be written in the form

f(<7-1)y(i + 1) = R{q~1)xi(.0 + 5(q-1)x2(0. (3.45)

Let

0o(Or = [*i(0< ~ iV1),x2(i),

...,x2(i- N2),-y(i),...,-y(i- N3)], N3 > nF 1, (3.46)

and

#0 fo, > T'n-l S0, > SNz, fit i fnF> < oj.

Then (3.45) becomes

y(i + 1) = 0" 0o(O-

(3.47)

(3.48)

The number of inserted zeros in (3.47) relies on the value of JV3 and the

unknown degree nF. Equation (3.48) is called the optimal nonadaptive prediction

because 0O is unknown. Therefore, the following adaptive predictor can be used

y(i + 1) = 0(i + 1)H0(O- (3.49)

where 6(1) is an estimate of unknown 0O, and

0(i)r = [x1(i), , Xi(t iVi),x2(i),

...,x2(i JV2), -y(i), ...,-y(i JV3)]. (3.50)

43

Equation (3.49) is called a posteriori prediction, whereas y(i + 1) = 0(t)H0(t)

represents a priori prediction. A priori prediction error x-,(i + 1) y(i + 1) is used

to run the following extended least squares algorithm (first presented by [22]):

6{i + 1) = 0(0 + p(O0(Oe(i + 1)*, (3.51)

e(i + 1) = x1(t + 1) 0(OH0(O, (3.52)

p(i l)0(O0(OHp(i 1) p(0-p(t 1) i + 0(O"p(i 1)0(0 ' (3.53)

p(0) = p0I, p0 > 0,

where I is the identity matrix, and p0 is an arbitrary finite positive scalar. Initial

values of 0(0) in (3.51) and y(/c), k < 0 in (3.50) are chosen arbitrary. Predictor

coefficients are directly computed using the RELS algorithm (3.49)-(3.53). There is

no need to know the channel order L and nF precisely, as long as one overfit, i.e., in

(3.50), JV-l > N0, N2 > N0, N0 = max(L 1 ,nF 1), and JV3 > nF [22],

44

CHAPTER IV

PERFORMANCE OF ADAPTIVE BLIND EQUALIZATION

ALGORITHMS AND SIMULATIONS

This section presents the results of simulations using MATLAB to examine

the performance behaviors of various adaptive /recursive algorithms described in

chapter 3. The performance of adaptive blind algorithms is assessed through this

thesis by calculating the mean-square-error (MSE), and showing constellation plots of

each single method. The major mean of comparison is the error cancellation

capability using algorithms that rely on the parameters such as step size /i, forgetting

factor A, number of iterations and their MSE performance. The principle advantage of

these algorithms is to remove ISI, which is generated during transmission, by

equalizing the transmission channels, or we can say in a more fashionable way it is to

reduce the error probability in the decision at the receiver. The calculation methods

that will be simulated in this chapter are cyclic LMS, LMS, RLS, Linear Prediction

(using GP technique) and RELS. Before the simulation results are presented, it is

useful to look at the brief explanation to the MSE and symbol constellations as

performance measures in the following section.

45

4.1 Performance Evaluations

4.1.1 Mean Square Error (MSE)

In general, the minimum mean square error (MSE) is a metric indicating how

well a system can adapt to a given solution. A small minimum MSE is an indication

that the adaptive system has accurately modeled, predicted, adapted, and/or

converged to a solution for the system. In other words, the algorithms will achieve

better performance. A very large MSE usually indicates that the adaptive filter cannot

accurately model the given system, or the initial state of the adaptive filter is an

inadequate starting point to cause the adaptive filter to converge. The ideal MSE is

when it reaches zero. However, most real systems cannot achieve the ideality. The

MSE is defined in a similar form as a cost function in (3.6)

MSE = E{\w(n d) w(ri)\2}. (4.1)

This equation can be simplified and used in practice by a consistent sample

estimate based on N observations

MSE(N) = ^En=olw(n d) w(n)|2. (4.2)

where d is the desired delay. Most of the time, the graphs illustrate the MSE versus

time index samples. The MSE achieves its asymptotic steady state level after a

sufficient number of symbols.

46

4.1.2 Symbols Constellation Plots

It is also called scatter or eye diagram. This type of performance measures is

usually useful to see if the algorithm used is working or not. It is a constellation

diagram of the observed signals and equalizers outputs.

47

4.2 Simulations

In this section, the simulation results are presented for all adaptive /recursive

methods that were discussed in the previous chapter. The calculation methods that

include cyclic LMS, LMS, RLS, LP and RELS are simulated using MATLAB. Some

experiments assumed that the input signal is independent and identically distributed

(i.i.d.) sequence with the zero mean and unit variance; however, other experiments do

not exploit this identity. Nine experiments will be performed in the following nine

sections such that each section includes graphs and results related to each single

calculation method.

4.2.1 Experiment la

In this experiment, an i.i.d. symbol sequence that is generated from a 16 QAM

constellation is used. The symbol levels along both axes are -1.5, -0.5, 0.5 and 1.5.

The continuous time channel that will be the first class of channel model used in the

simulation of this thesis spans four symbols and describes for t G [0,4Ts)

hc{t) = e-/2
where rc(t,(3) is the raised cosine given in [20, pp.546] as

rc(t,(3) = sinc(-) -

l.c 1

nK cos (17)

s l-4/?2t2/Ts2'

(4.4)

with roll-off factor /?, while Ts is the symbol duration. As in [11], /? = 0.35 is

chosen.

48

The above hc(t) represents the composite causal approximation of a two-ray

multipath mobile radio environment. This channel is reported by [1] and [21], The

discrete-time equivalent channel h(ri) is obtained by oversampling (P = 2) hc(t) at a

rate of Ts/2 (fractionally sampled) or h(ri) = hc(nTs/2) for n = 0,1, Figure

4.1 shows the magnitude of the impulse response h(ri) while Figure 4.2 shows the

zeros of h(n), and the zeros of the subchannels h0(n)and (n). The subchannels

parameters are

h0 = [0.52 j0.72 - 0.48 + )0.24 - 0.05 + j0.07 0.01 j0.02]T.

(4.5)

hx = [0.12 y'0.43 0.48 + y'0.41 0.13 j0.ll 0.04 + j0.03]T.

(4.6)

Two Ray Multipath: Raised Cosine Pulse Shape

Figure 4.1 Two-ray multipath channel, the magnitude of impulse response h(n).

49

Subchannel Zeros

Channel Zeros

Figure 4.2 The zeros of h(n), and the subchannels zeros h0(n)and h1(n).

Both Figures 4.1 and 4.2 are first presented in [11], Using this type of channel

to implement the first equalizer calculation method that is Recursive Least Square

(RLS) with zero delay (cl = 0). It is assumed that the subchannel length Lh = 4, and

subequalizer length Lg = 4. The received constellation is presented in Figure 4.3,

while Figure 4.4 shows the equalized eye diagram. Also, Figure 4.5 depicts the mean-

square symbol error for RLS algorithm across 3000 symbols. All of these figures in

the presence of the receiver noise v(ri) that is assumed to be white with zero mean

and 0.14 variance. The performance of RLS algorithm depends on the forgetting

factor A which in this experiment is chosen to be 0.99. For simulation, use equations

(3.17) and (3.18).

50

4

eye diagram of y1 channel output

Figure 4.3 Constellation plot of the received sequence for 16 QAM

modulation.

eye diagram of equalizer output

A * % u| ft KP 'A _% p r *

* X 1 l |w

m rm 4 * 4 *

*; Ji Wf 9 4 as*. si % b. % *

,3t-----------------------------------------------------------------1--------1

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

Re

Figure 4.4 Equalized symbol-eye diagram, using RLS algorithm with the raised-

cosine channel.

51

MMSE

Figure 4.5 Mean-square symbol error of RLS algorithm.

We can see from Figure 4.5 that this algorithm does a very good job of

equalization after as few as 1000 symbols. The MSE converges to 0.06665 at index

sample n = 3000.

4.2.2 Experiment lb

In this section, an uncorrelated symbol sequence generated from a 16- QAM

constellation is used. The channel model (second class), which is used in this

experiment, is reported by [22], and it has different coefficients than the previous

experiment. Figure 4.6 shows the magnitude and zeros of the impulse response h(n).

The subchannels parameters are described as follow:

52

Magnitude of Impulse Response

h0 = [0.5498 y'0.3174 0.4342 -y0.4596 0.0131 -)0.0919

-0.0275+)0.1592]7. (4.7)

^ = [0.0414 )0.0717 0.7168 y'0.3407 0.0745 j0.3636

0.0685-yO.1861]7. (4.8)

Second class channel model

Channel Zeros

Figure 4.6 The magnitude and zeros of impulse response h(ri), second class of

channel model.

The RLS algorithm is implemented again using the second class of channel

model with zero delay (d = 0). It is assumed that the subchannel length Lh = 4, and

subequalizer length Lg = 4. The received symbols are presented in Figure 4.7, while

Figure 4.8 shows the equalized constellation. Also, the mean-square symbol error is

estimated for this algorithm across 3000 symbols as shown in Figure 4.9. All of these

figures in the presence of the additive noise v(n) that is assumed to be white with

zero mean and 0.14 variance. The forgetting factor A is also 0.99.

53

eye diagram of y1 channel output

Re

Figure 4.7 Eye diagram of the received signal for 16 QAM modulation.

Re

Figure 4.8 Equalizer output symbols, using RLS algorithm with the second class of

channel model.

54

6

MMSE

Figure 4.9 Mean-square symbol error of RLS algorithm.

From Figure 4.9, we can see that this algorithm works well when the second

class of channel model is used. The MSE converges to 0.0709 at index sample

n = 3000.

4.2.3 Experiment 2

In this experiment, the cyclic LMS algorithm, which is first presented in [11],

is simulated with zero delay (d = 0) using the first class of channel model (the

raised-cosine two-ray multipath mobile radio) with Lg = Lh = 4. Figure 4.10 shows

the equalized eye diagram, whereas Figure 4.11 depicts the mean-square symbol error

for cyclic LMS algorithm across 3000 symbols. The additive noise is the same as last

experiment. Trading off speed convergence with steady-state error, the performance

55

of cyclic LMS relies on the step size /i which throughout this thesis is chosen to be

0.0025. For simulation, use equation (3.23).

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

Re

Figure 4.10 Eye diagram of equalizer output, using cyclic LMS with the raised-cosine

channel.

MMSE

Figure 4.11 Mean-square symbol error of cyclic LMS.

56

From Figure 4.11, the performance of cyclic LMS is better than RLS

algorithm since cyclic LMS has lower MSE and faster convergence. The MSE

converges to 0.02526 at index sample n = 3000. However, the cyclic LMS doesn't

work with the second class of channel model. To make it work, the conjugate

calculation yj* QV) in (3.23) has to be removed. Otherwise, it won't work.

4.2.4 Experiment 3a

In this experiment, The LMS algorithm is performed using the raised-cosine

channel with zero delay (d = 0). We still suppose the same assumption as in the

experiment 2. Figure 4.12 shows the equalized constellation, while the mean-square

symbol error is estimated for this algorithm across 3000 symbols as shown in Figure

4.13. For simulation, use equations (3.29) and (3.30).

Re

Figure 4.12 Equalized eye diagram, using LMS algorithm with the raised-cosine channel.

57

MMSE

Figure 4.13 Mean-square symbol error of LMS algorithm.

As we see from Figure 4.13, the LMS method works well. The MSE

converges to 0.05596 at index sample n = 3000.

4.2.5 Experiment 3b

In this section, the second class of channel model is used to implement the

LMS algorithm. Our assumptions are still the same as in previous section. Figure 4.14

shows the equalized eye diagram, whereas Figure 4.15 depicts the mean-square

symbol error for LMS algorithm across 3000 symbols.

58

Re

Figure 4.14 Eye diagram of equalizer output, using LMS algorithm with the second

class of channel model.

MMSE

4.5 r

4 -

3.5 -

3 -

e 2.5

Â¥

E 2 -

1.5 -

1 -

0.5 ^

0 -

0

X 3000

Y: 0.04817

500 1000 1500 2000 2500 3000

Time Sample

Figure 4.15 Mean-square symbol error of LMS algorithm.

59

From Figure 4.15, the performance of this experiment is better than

experiment 3a since the MSE of LMS using the second class of channel model has

lower value at 3000 (index sample). For more clarity, the MSE converges to 0.04817

at index sample n = 3000.

The comparison between RLS, cyclic LMS and LMS is accomplished through

Figure 4.16 that shows the MMSE for the cyclic LMS, LMS and RLS. The channel

used for this comparison is the raised-cosine two ray multipath mobile radio (P = 2).

The plot demonstrates that the cyclic LMS has better performance since it has the

lowest MSE. The MSE of LMS is lower than the MSE line of the RLS at (n = 3000).

Figure 4.16 Mean-square symbol error, Comparison using raised-cosine channel.

60

The second comparison is also between cyclic LMS, LMS and RLS, but here

the second class of channel model is used. We can see from figure (4.17) that the RLS

has the best performance due to its MSE, which is lower than the MSE of both cyclic

LMS and LMS. As I mentioned earlier that the cyclic LMS doesn't work with this

channel class and it will not converge.

Figure 4.17 Mean-square symbol error, Comparison using second class of channel

model.

61

4.2.6 Experiment 4a

In this experiment, the unbiased blind adaptive using Gradient Projection (GP)

technique is implemented. It is first presented in [10], The channel used is the raised-

cosine two-ray multipath mobile radio with Lg = Lh = 4. The small step size a is

chosen to be 0.002. The performance of linear prediction relies on the constraint

value y due to its impact on the MSE behavior. Throughout this thesis, y is chosen to

be 1.5. Figures 4.18 and 4.19 illustrate the behavior of this algorithm in a white noise

situation. For simulation, use equations (3.31)(3.34).

2.5

2

1.5

1

0.5

I 0

-0.5

-1

-1.5

-2

Re

Figure 4.18 Eye diagram of equalizer output, using LP method with the raised-cosine

channel.

Eye diagram of equalizer output

'Jti

9% >

An.

Bf- Tr . 9

V -y S& {A#

: 1 j . & *

'4* 3$

*

62

MMSE

Figure 4.19 Mean-square symbol error of LP method.

From Figure 4.18 and 4.19, we see that this algorithm works well to equalize

the received noisy signal. The MSE converges to 0.06786 at index sample n =

50000. The problem with this algorithm is it has slow convergence. The amount of

rotation and magnification in the eye diagram (see Figure 4.18) is a function of

h0(0) = (0.52 j0.72) which is the leading coefficient in h0. The angle of rotation

is 54.16, and the magnification is |/io(0)| = 0.88.

63

4.2.7 Experiment 4b

The second class of channel model is simulated in this experiment with using

the same method as in experiment 4a. Our assumptions are still the same. Figure 4.20

shows the equalized eye diagram, while Figure 4.21 depicts the mean-square symbol

error across 50000 symbols.

Eye diagram of equalizer output

Figure 4.20 Eye diagram of equalizer output, using LP algorithm with the second

class of channel model.

64

MMSE

Figure 4.21 Mean-square symbol error of LP algorithm.

The algorithm works very well as we see from Figure 4.21 and the MSE

converges to 0.0608 at time sample (50000). The performance is slightly better than

that in experiment 4a. However, we are still facing the slow convergence problem.

The leading parameter h0(0) = (0.5498 jO.3174). The angle of rotation is 30,

and the magnification is |/io(0)| = 0.635.

65

4.2.8 Experiment 5a

In this section, we use a symbol sequence that is assumed to be uncorrelated

and generated from a 16- QAM constellation with -1.5, -0.5, 0.5 and 1.5 levels. The

RELS algorithm, which is first presented in [22], is simulated using the following

(third class) HR channel model

Aiq-1) = I-O.8C7-1 + 0.41c?-2. (4.9)

FOr1) = l + (0.5-0.60<7_1. (4.10)

Biq-1) = 0.5498 0.3174? + (0.4342 0.45960c?-1 + (0.0131 -

0.09190c?-2 + (-0.0275 + 0.1592QC?-3. (4.11)

CO?-1) = 0.0414 0.0717? + (0.7168 0.3407?)c?-1 + (0.0745 -

0.3636?)c?-2 + (0.0685 0.1861?)c?-3. (4.12)

Our assumptions with respect to additive noise are still the same. Figure 4.22

shows the received symbols x1(i), while Figure 4.23 shows the equalized symbols

eye diagram. As performance measures, the mean-square symbol error for RELS

algorithm across 3000 symbols is estimated as shown in Figure 4.24 using the

following equation

mse(n) = olxi(i + 1) y(? + 1) b0w(i + 1)|2. (4.13)

For simulation, use equation (3.49)-(3.53).

66

10

eye diagram of x1 channel output

Figure 4.22 Eye diagram of received signal for 16 QAM modulation.

E

Eye diagram of equalizer output

Re

Figure 4.23 Eye diagram of equalizer output, using RELS algorithm with the third

class of channel model (HR).

67

Time Sample

Figure 4.24 Mean square symbol error of RELS algorithm.

4.2.9 Experiment 5b

In this section, we run the algorithm with the presence of the modeling error

in the channel dynamics, and we use the same polynomials as in Experiment 5a,

*i(0 =----Aiq-1)----+ Vl + Sl

C(q 1)E(q x)

*2 =------Aiq-1)---W 1 + V2 1 + 52 1 '

(4.14)

68

where (i) and v2(0 are white noise with zero mean and variance=0.14. It is

important to mention that i;k(i), k = 1,2, and w(t) are independent sequences. Let

8k(i), k = 1,2, are modeling errors given by

8 8

where 5 is a parameter defining the size of the channel modeling error. Figures 4.25

and 4.26 illustrate the scatter plot of the equalized symbols and sample mean-square

error, respectively, with setting 6 = 0.1 in (4.15).

Eye diagram of equalizer output

Figure 4.25 Scatter plot of equalized symbols, using RELS algorithm with the

presence of modeling error using (HR) channel.

69

MMSE

Figure 4.26 Sample mean square error of RELS algorithm.

This algorithm shows a very good performance of equalizing the received

signal with and without presence of noise and modeling error as illustrated in both

Figures 4.24 and 4.26. The MSE converges to 0.04598 at index sample n = 3000 in

experiment 5a, whereas the MSE of experiment 5b converges to 0.06469 at the same

index sample. Therefore, it is clear that the algorithm provides a degree of robustness

with respect to receiver noise and modeling error [22],

The comparison between all of these algorithms is done with using the second

class of channel model. Figure 4.27 demonstrates that the performance of RELS

algorithm is the best since its MSE has the lowest value at the time sample (n =

3000).

70

mse(n)

Figure 4.27 Mean-square symbol error, Comparison for those five algorithms with the

second class of channel model.

71

CHAPTER V

CONCLUSION

This thesis has proved how adaptive blind algorithms (RLS, cyclic LMS,

LMS, LP and RELS) can equalize the wireless communication channels through

removing inter-symbol-interference (ISI). These techniques have proven to be quite

effective and powerful to combat ISI effect. ISI is mainly generated in dispersive

channels such as Radio and Mobile wireless channels. In a particular case, the mobile

cellular communication (multipath propagation of the transmitted signal) suffers from

severe ISI. Therefore, an adaptive filter as an equalizer can be placed at the receiver

to compensate or equalize the dispersion occurred during transmission. This filter or

the equalizer is a device positioned at the receiver to alleviate the effect of ISI and

thus the transmitted symbol sequence can be recovered.

Channel equalization methods used in this thesis rely only on the statistical

behavior of the received signals (Second order Statistics) in order to estimate the

transmitted sequence without requiring the knowledge of the channel characteristics.

The transmission environments used were a raised-cosine SIMO channel model

realized by fractionally sampled (FS) FIR filter, SIMO channel model with constant

coefficients over period realized by (FS) FIR filter and SIMO channel model realized

by (FS) HR filter. The latest channel model was performed with and without

modeling error.

72

It has been shown that the algorithms used in this thesis perform very well,

and they are very efficient and robust with respect to the channel distortions through

examining their mean square symbol error (MSE) performance via computer

simulations. Also, it is shown that the RELS algorithm has an advantage over the rest

of algorithms since it has the fastest MSE convergence.

For future research, our system might be extended to MIMO channel model

case, and we should study and examine our techniques performance. Also, we may

investigate the capability of adaptive blind channel equalization algorithms with using

time-varying transmission channel and see which one provides computationally

efficient implementations with and without presence of noise.

73

REFERENCES

[1] K. Abed Meraim, E. Moulines and P. Loubaton, Prediction error methods for

second-order blind identification, IEEE Trans. Signal Processing, vol. 45, pp.

694-705, Mar. 1997.

[2] K. Abed Meraim et al., Prediction error methods for time-domain blind

identification of multichannel FIR filters, in Proc. Int. Conf. Acoust., Speech,

Signal Processing, vol. 3, Detroit, MI, 1995, pp. 1968-1971.

[3] E. W. Bai and M.Fu, Blind system identification and channel equalization of

HR system without statistical information, IEEE Trans. Signal Processing, vol.

47, pp. 1910-1920, July 1999.

[4] A. Cichocki and S. Amari, Adaptive Blind Signal and Image Processing:

Learning Algorithms and Applications, Chichester: John Wiley & Sons, 2002.

[5] P. Comon and M. Rajih, Blind identification of under-determined mixtures

based on the characteristics function, Signal Proc., vol. 86, pp. 2271-2281,

September 2006.

[6] H. Cox, R. Zeskind and M. Own, Robust adaptive beamforming, IEEE Trans.

Acoust., Speech, Signal Processing, vol. ASSP-35, Oct. 1987.

[7] Z. Ding, Adaptive Filters for Blind Equalization, CRC Press LLC, 1999.

[8] Z. Ding and Y. Li, On channel identification based on second-order cyclic

spectra, IEEE Trans. Signal Processing, vol. 42, pp. 1260-1264, May 1994.

[9] P. S. R. Diniz, Adaptive Filtering Algorithms and Practical Implementation, 3rd

ed. New York, NY: Springer Science & Business Media, LLC, 2008.

74

[10]

D. Gesbert and P. Duhamel, Unbiased blind adaptive channel identification

and equalization, IEEE Trans. Signal Processing, vol. 48, pp. 148-158, Jan.

2000.

[11] G. Giannakis and S. Halford, Blind fractionally spaced equalization of noisy

FIR Channels: Direct and adaptive solutions, IEEE Trans. Signal Processing,

vol. 45, pp. 2277-2292, Sept. 1997.

[12] O. V. Goryachkin and E. I. Erina, Given correlation manifolds and their

application in blind channel identification, The open Statistics and Probability

Journal, 2009, 1, 55-64.

[13] S. Haykin, Adaptive Filter Theory, 2nd ed. Englewood Cliffs, NJ: Prentice-Hall,

1991.

[14] S. Haykin, Blind Deconvolution, S. Haykin Ed. Adaptive Filter Theory, NJ,

Prentice-Hall, Englewood Cliffs, 1991.

[15] R. A. Horn and C. R. Johnson, Matrix Analysis, New York, NY: Cambridge

Press, 1985.

[16] C. Johnson, P. Schniter, T. Endres, J. Behm, D. Brown and R. Cases, Blind

equalization using the constant modulus criterion: A Review, Processing of

the IEEE, vol. 86, pp. 1927-1950, Oct. 1998.

[17] J. Lebrun and P. Comon, Blind algebraic identification of communication

channels: symbolic solution algorithms, Appl. Ageber. Eng. Commun.

Comput., vol. 17, pp. 471-485, November 2006.

[18] E. Moulines, P. Duhamel, J. F. Cardoso and S. Mayrargue, Subspace methods

for the blind identification of multichannel FIR filters, IEEE Trans. Signal

Processing, vol. 43, pp. 516-525, Feb. 1995.

75

[19] B. Porat and B. Friedlander, Blind equalization of digital communication

channels using higher-order moments, IEEE Trans. Signal Processing, vol.

39, pp. 522-526, Feb. 1991.

[20] J. Proakis and M. Salehi, Digital Communications, New York: McGraw-Hill,

5th ed., 2008.

[21] M. Radenkovic, T. Bose and Z. Zhang, Self-tuning blind identification and

equalization of HR channels, EURASIP Journal on Applied Signal

Processing 2003:9, 930-937.

[22] M. Radenkovic, T. Bose, A recursive blind adaptive equalizer for HR channels

with common zeros, Circuit Syst. Signal Process, 28, 467-486, 2009.

[23] O. Shalvi and E. Weinstein, New criteria for blind deconvolution of

nonminimum phase systems (channels), IEEE Trans. Inform. Theory, vol. 36,

pp. 312-321, Mar. 1990.

[24] D. T. M. Slock, Blind fractionally-spaced equalization, perfect-reconstruction

filter banks and multichannel linear prediction, in Proc. Int. Conf. Acoust.,

Speech, Signal Processing, vol. IV, Adelaide, Australia, 1994, pp. 585-588.

[25] G. H. Stuck, Adaptive blind equalization with applications in communication

systems, MS Thesis, University of Colorado at Denver, April 2002.

[26] L. Tong, G. Xu and T. Kailath, Blind identification and equalization based on

second-order statistics: A time domain approach, IEEE Trans. Inform. Theory,

vol. 40, pp. 340-349, Mar. 1994.

[27] L. Tong, G. Xu, B. Hassibi and T. Kailath, Blind channel identification based

on second-order statistics: A frequency domain approach, IEEE Trans. Inform.

Theory, vol. 41, pp. 329-334, Jan. 1995.

[28] L. Tong and S. Perreau, Multichannel blind identification: From subspace to

maximum likelihood methods, Proc. IEEE, vol. 86, pp. 1951-1968, 1998.

76

[29] J. T. Tugnait, L. Tong and Z. Ding, Single-user channel estimation and

equalization, IEEE SignalProc. Mag., vol. 12, pp. 17-28, 2000.

77