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 Title:
 Performance analysis of adaptive blind equalization algorithms for noisy FIR and IIR channels
 Creator:
 Althahab, Awwab Qasim ( author )
 Place of Publication:
 Denver, CO
 Publisher:
 University of Colorado Denver
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 2013
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 Adaptive filters ( lcsh )
Adaptive signal processing ( lcsh ) Signal processing  Digital techniques ( lcsh )
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 bibliography ( marcgt )
theses ( marcgt ) nonfiction ( marcgt )
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 Thesis:
 Thesis (N.S.)University of Colorado Denver. Electrical engineering
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 Includes bibliographic references.
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 Department of Electrical Engineering
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 by Awwab Qasim Althahab.
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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 InterSymbolInterference (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 Tworay 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 symboleye diagram, using RLS algorithm
with the raisedcosine channel.......................................51
4.5 Meansquare 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 Meansquare symbol error of RLS algorithm............................. 55
4.10 Eye diagram of equalizer output, using cyclic LMS
with the raisedcosine channel........................................56
4.11 Meansquare symbol error of cyclic LMS ............................... 56
4.12 Equalized eye diagram, using LMS algorithm with the raisedcosine channel
.............................................................................57
4.13 Meansquare 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 Meansquare symbol error of LMS algorithm..............................59
4.16 Meansquare symbol error, Comparison using raisedcosine channel......60
4.17 Meansquare symbol error, Comparison using second class of channel model
......................................................................61
4.18 Eye diagram of equalizer output, using LP method
with the braisedcosine channel.......................................62
4.19 Meansquare 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 Meansquare 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 Meansquare 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 Meansquare 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 intersymbolinterference
(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 selfrecovering
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 socalled '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
intersymbol 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 socalled
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) Continuoustime input information signal
w(n) Input discretetime sequences
w(n) Equalizer output
m Dirac delta function
Continuoustime impulse response of composite channel
h(n) Discretetime equivalent of hc(t)
/loOO.hiO) Finitelength impulse responses of the two subchannels
h Vector form of channel impulse response
h,h, Vector form of the two subchannels coefficients
vr(0 Continuoustime additive noise
v(n) Discretetime equivalent of vc(t)
xc(t) Continuoustime channel output
y,(0 Continuoustime 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
Discretetime 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 stepahead 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
Rolloff factor
P
rc(t,p)...................Raisedcosine 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 inputmultiple 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 socalled 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 16QAM (quadrature amplitude
modulation) since the symbols are usually complex with inphase 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 inputoutput communication system using MQAM
is that the transmitter generates a sequence of complexvalued random input
data (w(Z)}, each element of which belongs to a complex alphabet A (or
constellation) of MQAM 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 continuoustime information signal, S(t) is a pulse of duration rs
with an amplitude w(Z).
The unknown complexvalued 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 discretetime sequences,
respectively, it is convenient to rewrite (2.5) as an equivalent discretetime system
yOO = E^co w(l)h(n IP) + i
= x(n) + v(n). (2.6)
where h(n) and v(n) are the discretetime equivalents of hc(t) and vc(t),
respectively, x(n) is the discretetime 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 inputmultiple output model, an integer P > 1 of measurements
are performed for each transmitted symbol, provided the continuoustime 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(nl)+v1(ri). (2.9)
Note that h0(ri) and (n) are finitelength 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 finitelength 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 finitelength when it is used as a
vector form. Assume that the finitelength 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^Ln1)}.
(2.14)
then
y(n)
To 00' h0i 0) h0(l) o 1
.yiOO. MO) /id!) . KUbD)
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
1input/ 2output 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 finitelength 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(Lgl)]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 inputoutput
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 Kdhi) : ^(1)
0
0 .. hdd1).
(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) hddd fti(0)
0 /\ 1 o X2 \ ^1) 0 hddd ! ftiCl)
0 0
1 0 1 o X2 0 .. hdui).
' 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
Hl) 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 selfdesigning 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 decisionfeedback 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 (selfadaptation) 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 noisefree 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{nN + 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 timevarying 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 meansquare 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 noisefree 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+HC:, 1)
(3.15)
The above equations (3.14) and (3.15) give to us the zerodelay 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 tapinput
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
Nl
c\y{N) = Â£ A'y4(0yr(0 + yt3mylm
1=0
= AC*y(Ni)+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 ii
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(Lgl) dg*1(Lgi)_
(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 steadystate 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 tapinput 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 SteepestDescentBased 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 meansquare 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 steepestdescent
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 timeaveraging 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
Hl) 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
mixedphase 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 noisefree prediction error signal
related to the input symbol sequence w(n).
Assume that the finitelength equalizer vector g be a 2Lg x 1 complexvalued
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(',)yjg(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 onestep 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 singleinput twooutput 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)
Aiq1)
w(i),
*2(0
C(q~1)F(q~1)
Aiq1)
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)
Ciq1) =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(q1) are of the same order, and assuming that V^q1) 1S a stable
operator. Also, 5(q_1) and C(q1) 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 predictionbased equalizer is shown in Figure 3.2, where y(t + 1) is one
stepahead 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)
Diq1)
Xi(t) +
Siq1)
Diq1)
x2(0
(3.37)
which is optimal in the meansquare 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)
Htq1)
[B(q~1)R(q~1) + C(q')S(q')]F(q')
A(q')D(qi)
q[B(q 1)F(q  A{q x)bQ]
A (q1)
(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£71)F(c71) 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
Riq1) =r0+ rxq~x + + rNiq~N\
Siq1) = s0 +s1cT1 + + sN2q~Nz. (3.44)
Then (3.42) has a solution with respect to F(q_1) and S(q_1) if JVl > iV0,
JV2 > iV0, where iV0 = max(L 1 ,nF 1). A unique solution exists if 5(q_1) and
C(q1) 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(<71)y(i + 1) = R{q~1)xi(.0 + 5(q1)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'nl 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(0p(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), JVl > 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 meansquareerror (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 l4/?2t2/Ts2'
(4.4)
with rolloff 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 tworay
multipath mobile radio environment. This channel is reported by [1] and [21], The
discretetime 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 Tworay 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. % *
,3t11
2 1.5 1 0.5 0 0.5 1 1.5 2
Re
Figure 4.4 Equalized symboleye diagram, using RLS algorithm with the raised
cosine channel.
51
MMSE
Figure 4.5 Meansquare 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
0.0685yO.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 meansquare 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 Meansquare 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
raisedcosine tworay multipath mobile radio) with Lg = Lh = 4. Figure 4.10 shows
the equalized eye diagram, whereas Figure 4.11 depicts the meansquare symbol error
for cyclic LMS algorithm across 3000 symbols. The additive noise is the same as last
experiment. Trading off speed convergence with steadystate 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 raisedcosine
channel.
MMSE
Figure 4.11 Meansquare 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 raisedcosine
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 meansquare
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 raisedcosine channel.
57
MMSE
Figure 4.13 Meansquare 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 meansquare
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 Meansquare 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 raisedcosine 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 Meansquare symbol error, Comparison using raisedcosine 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 Meansquare 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 tworay 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 raisedcosine
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 Meansquare 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 meansquare 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 Meansquare 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
Aiq1) = IO.8C71 + 0.41c?2. (4.9)
FOr1) = l + (0.50.60<7_1. (4.10)
Biq1) = 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 meansquare 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 =Aiq1)+ Vl + Sl
C(q 1)E(q x)
*2 =Aiq1)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 meansquare
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 Meansquare 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 intersymbolinterference (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 raisedcosine 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
timevarying transmission channel and see which one provides computationally
efficient implementations with and without presence of noise.
73
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Full Text 
PAGE 1
PERFORMANCE ANALYSIS OF ADAPTIVE BLIND EQUALIZATION ALGORITHMS FOR NOISY FIR AND IIR 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
PAGE 2
ii This thesis for the Master of Science d egree by Awwab Qasim Althahab h as been approved for the Electrical Engineering Degree by Miloje Radenkovic Chair Yiming Deng Tim Lei 02 / 21 / 2013
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iii Althahab, Awwab Qasim (M.S., Electrical Engineering) Performance Analysis of Adaptive Blind Equalization Algorithms for Noi sy FIR and IIR 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 (IIR) channels (their characteristics are unkn own) 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 O rder 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
PAGE 4
iv concept. These algorithms operate blindly in the practical situation, and they can achieve a complete adap tation 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 t o 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 bli nd adaptive equalization algorithms through computer simulations. The form and content of this abstract are approved. I recommend its publication. Approved: Miloje Radenkovic
PAGE 5
v 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.
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vi 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. Miloj e 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 a nd 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
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vii T ABLE OF CONTENTS CHAPTER I. 1 1.1 Introduction and Background Information to Blind A daptive Equalizer 1.2 1.3 Mathematical Notations II. MODEL OF COMMUNICATION SYSTEM AND MATHEMATICAL 10 2.4 Overall System Model with Equalizer III. BLIND ADAPTIVE CHANNEL EQUALIZATION METHODS 22 3.1 Adaptive Filter The ory 3.4 Blind SIMO Channel Equalization Methods Using Second Order
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viii 3.5 Cyclic S tatistics 3.6 Direct Blind MMSE Equalizers 3.7 Methods for Blind Equalizer Calculation 3.7.1 Recursive Blind A daptive Using Recursive Least Square Algorithm (RLS) 3.7.2 Recursive Blind A daptive Using Cy clic Least Me an Square 3.7.3 Recursive Blind A daptive U sing Least Mean Square 3.7.4 Unbiased Blind A daptive Using Gradient Projection 3.7.5 Recursive Blind A daptive Using Recursive Extended Least IV. PERFORMANCE OF ADAPTIVE BLIND EQUALIZATION ALGORITHMS AND SIMULATIONS 4 5 4.1 Performance Evaluations 4.1 .1 Mean Square Error (MSE) 4.1.2 Symbols Constellation Plots 4.2 Simulations 4.2.1 Experiment 1a 4.2.2 Experiment 1b 4.2.3 Experiment 2 4.2.4 Experi ment 3a
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ix 4.2.5 Experiment 3b 4.2.6 Experiment 4a 4.2.7 Experiment 4b 4.2.8 Experiment 5a 4.2.9 Experiment 5b V. CONCLUSION 7 2 7 4
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x LIST OF FIGURES FIGURE 2.1 General model of a com 2.2 Basic elements in a digital 2.3 Fractionally sampled ( 13 2.4 SIMO multichannel model ( ... 15 2.5 Multichannel system models with equalizer 18 3.1 SIMO channel Model ..... 40 3.2 Predictor Bas .. 40 4.1 Two ray multipath channel, the magnitude of impulse response 4.2 The zeros of and the subchannels zeros 4.3 Constellation plot of the received se .. 51 4.4 Equalized symbol eye diagram, using RLS algorithm w ith the raised cosine channel 4.5 Mean square symbol er
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xi 4.6 T he magnitude and zeros of impulse response 4.7 Eye diagram of the received sig 54 4.8 Equalizer output symbols, using RLS algorithm with the second 4.9 Mean square symbol err 4.10 Eye diagram of equalizer output, using cyclic LMS with the raised 56 4.11 Mean square symbol er 56 4.12 Equalized eye diagram, using LMS algorithm with the raised cosine channel 4.13 Mean square symbol err or of LMS algorithm 4.14 Eye diagram of equalizer output, using LMS algorithm w i ... 59 4.15 Mean square symbol er .. 59 4.16 Mean square symbol e rror, Comparison using raised 4.17 Mean square symbol error, Comparison using second class of channel model 4.18 Eye diagram of equalizer output, using LP method with the b raised ....................................................... 62 4.19 Mean square symbol error of LP met
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xii 4.20 Eye diagram of equalizer output, using LP algorithm w ith the second class of c hannel model 4.21 Mean square symbol er .. 65 4.22 Eye diagram of received signa 4.23 Eye diagram of equalizer out put, using RELS algorithm with the third 7 4.24 Mean square symbol er 68 4.25 Scatter plot of equalized symbols, using RELS algorithm with the presence of modeling er ror ... 69 4.26 Sample mean square err 70 4.27 Mean square symbol error, Comparison for those five algorithms w ith the second class of channel model
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1 CHAPTER I INTRODUCTION 1.1 Introduction and Background Information to Blind Adaptive Equalizer C ommunication 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 ha ve 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 o f 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 interf erence (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 band limited 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 t he detection of those symbols since their spreading can corrupt the
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2 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) occurri ng 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 se nding 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 t hat does not need a training sequence is invented. Such approach is termed 'b lin d' 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
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3 unknown signals that are sent through a linear channel without any knowledge of its cha racteristics 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 eq ualization with no need of transmitter assistance (sending training signals). Consequently, a better channel bandwidth efficiency is obtained.
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4 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 perform ance, 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 equalizatio n 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 a chieve 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 rec eived 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
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5 Squ are (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 (M SE) that can be used to compare the equalization algorithms. The channel model cla sses 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.
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6 1.3 Mathematical Notations Continuous time input information signal time sequences time impulse response of composite channel time equivalent of length impulse responses of the two subchannels time additive noise time equivalent of time channel output time channel output with additive noise input to equalizer
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7 time impulse response of linear equalizer bequalizers the conjugate transpose matrix .. Correlations of the stationary input, noise, and channel noisy output function ................................. Prediction error variance
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8 .. Polynomials of IIR channels ahead prediction of .. Polynomials of IIR equalizer
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9 off factor ed cosine channel error multiple o utput
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10 CHAPTER II MODEL OF COMMUNICATION SYSTEM AND MATHEMATICAL FRAMEWORK 2.1 Introduction A g eneral and basic model for a communication system is comprised of digital signal that is transmitted by a transmitter through an analog channel to a r eceiver ( see Figure 2.1 ) Source Info. Received Info. Figure 2.1 General model of a communication system Source information may be either a digital or an analog signal. Th e 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 cal led 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 Transmitter Channel Receiver
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11 communication channel s. 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 modu lation ) 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 sign al are corrupted in a random manner. Source Info. Received Info. 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 sequen ce is passed through a channel decoder, which attempts to recover the original information sequence. How well the original signal is recovered depends on many Digital Modulator Channel E ncoder Source E ncoder Channel Digital D emodulator Channel Decoder Source Decoder
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12 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 t ransmitter generates a sequence of complex valued random input data each element of which belongs to a complex alphabet A (or constellation) of M QAM symbols. The data sequence is sent through a baseband equivalent unknown complex linear ti me invariant ( LTI ) channel whose output is observed by the receiver. The function of the receiver is to estimate the original data from the received signal [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 bits of binary data at a time and encodes them into one of 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 set of binary digits is mapped into a pulse of duration seconds and an amplitude Thus the modulator output signal, which is the input to the communication channel, is given as
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13 (2.1) where is the continuous time information signal, is a pulse of duration with an amplitude The unknown complex valued LTI communication channel with the impulse response c omposited with the know n transmit and receive filters is shown in Figure 2.3. The overall composite channel is finite impulse response (FIR). Additive noise that is assumed to be stationary and uncorrelated with the information symbols can be included in this model. Figure 2.3 Fractionally sampled ( communication system The composite channel (whose impulse response is ) is non ideal, which means is nonzero for An integer 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:
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14 (2.2) substituting (2.1) into (2.2), we get (2.3) (2.4) If is sampled at the received discrete data signal ( for ) is ( 2.5) Considering and are the input and output discrete time sequences, respectively, it is convenient to rewrite (2.5) as an equivalent discrete time system (2.6) where and are the discrete time equivalents of and respectively, is the d iscrete time equivalent of the noise free received signal, and index can be any integer ( [11].
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15 In the single input multiple output model, an integer 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 to make our model in the oversampling scheme (see Figure 2.4). Figure 2.4 SIMO multichannel model ( now, (2.6) can be rewritten in fractionally spaced system as follows for (2.7) with (2.8) and
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16 (2.9) Note that and 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 easy to understand and modify The input, noise and output such as can be also expressed in vector form. The index in these signals is usually in finite length. However, it will be finite length when it is used as a vector form. Assume that the finite length channel is causal with length such that each subchannel has a length Considering the following vectors: (2.10) (2.11) (2.12) with (2.13)
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17 and (2.14) then (2.15) or (2.16) where is a vector form of the two subchannels, represent the vector form coefficients of the first and second channels, respectively. The noisy output vector of 1 input/ 2 output causal FIR channel filters driven by the input sequence is Note that is the subchannel polynomial order.
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1 8 2.4 Overall System Model with Equalizer A finite impulse response filter as an equalizer with vector parameters is applied to our channel model to make our system into a complete syst em form (see Figure 2.5). The goal of the equalizer is to remove the distortion caused by channel ISI. Figure 2.5 Multichannel system models with equalizer Assume that the finite length equalizer vector can be defined in a same manner as (as mentioned in previous section) with length such that each subequalizer has a length The following vectors are defined (2.17) with (2.18)
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19 and (2.19) where and represent parameter vectors of the first and second equalizers, respectively. Let be the output signal of equalizer, and then the input output relationship can be written in the following form (2.20) (2.21) To simplify (2.21), let's consider our system in the absence of noise and define that is associated with vector to be a channel convolution matrix with the length as below (2.22)
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20 The convolution matrix is also called Sylvester or block Toeplitz matrix [15] of the two subchannels, and is equivalent to linear filters that the input signal passes through ( Alternative representation of (2.22) is found in some books and is written in the following form (2.23) then (2.21) can be written as below (2.24) (2.25)
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21 with (2.26) where and are the convolution matrices of the two subchannels. Note that there is no need to know the channel order precisely, only the equalizer length has to be chosen so that the following hypothes es hold in order to have sufficient channel diversity ) ( has strictly more columns than rows). ) has full row rank which means the matrix is invertible. The length of equalizer must be chosen large enough (i.e. ) 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 implies that it will be full row rank if and only if there are no common subchannel roots and It is also found that there is a unique solution of the equalizer equations if
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22 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 sat isfactorily 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 ava ilable 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 system s with different f unctionalities, 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 unkn own environment, and even track signal or system characteristics varying over time.
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23 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 receive r which use s maximum likelihood estimation suboptimum receiver which use s 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 communicat ion systems that use equalizers the channel characteristics are not known a priori. Therefore, training sequences that provide different realizations of a des ired 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 effec tive 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 e qualization 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 be en invented and developed, and they have been the best approaches that satisfy our system models.
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24 The purpose of this thesis is applying blind adaptive channel equalization with the variety of algorithms. 3.3 Blind Adaptive Equalization In some application s, 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 Equali zation Methods Using Second Order Statistics (SOS) In the blind methods of equalization, so me 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 g otten great research and practical interest, and ma n y 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
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25 (SOS) of the c hannel output 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 chan nel 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 is described because some calculation algorithms that will be discussed later in this thesis use this property. 3.5 Cyclic S tatistics In this section, we describe the covariance of which will be useful for finding the equalizers directly from the noise free channel output The f ollowing vectors are defined to represent vector observations of the SIMO fractionally spaced model (see Figure 2.5). ,
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26 (3.1) The correlation of the scalar output in equation (2.6 ) is defined (3.2) (3.3) w here and 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 with period 2, see for proving [25] integer. (3.4) Due to the periodicity of the correlation matrix, we can use in similar manner instead of using to find the correlation matrix as below (3.5)
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27 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 using only The objective is to compute the equalizer so that the following cost function is minimized. (3.6) where is the integer delay between input and nois e free channel outputs The output of equalizer can be written as below (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 (3.8) where and After simplification (see [25]), it yields to orthogonality condition (3.9)
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28 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 Therefore, the second term can be written as follow } (3.10) substituting (3.10) into (3.9), yields (3.11) the above equation can be written in a vector form as (3.12) (3.13) (3.14)
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29 (3.15) The above equations (3.14) and (3.15) give to us the zero delay MMSE equalizer by substituting and the arbitrary nonzero delay 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 ad justing the parameters of an adaptive filter to minimize the cost function. We describe the general form of adaptive FIR and IIR filtering concept. 3.7 .1 Recursive Blind A daptive 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 subchan nels 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 a lgorithm 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 To start
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30 dri ving this algorithm, it is assumed that the additive noise is to be uncorrelated with input samples and a forgetting factor is included to reduce the effect of past observations on the statistics correlation matrix estimate At the correlation matrix is simplified as (3.16) The result of (3.16) is a recursive form, and it can be applied to (3.14) to find MMSE equalizer with zero delay. However, using the matrix inversion lemma [13, p.480], a direct method to find is developed without requiring a matrix inverse or pseu doinversion of This can be done by defining and applying the matrix inversion lemma with (3.16)
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31 The above equation can be applied to (3.14) to estimate the MMSE zero delay equalizer (3.18) The equalizer estimate form in (3.18) does not require a matrix inverse and, hence, is computationally feasible for adaptive implementation. If is chosen, (3.18) provides a method to recursively compute the time invariant equalizer taps, thereby reduc ing the memory requirements for long data records [11]. It is very important to mention that the convergence of is affected by the forgetting factor and, thereby, it affects the accuracy of the estimator. The forgetting factor is usually c hosen between [0.98, 1]. The delay MMSE equalizer can be driven in a similar way using the same above steps in this method. 3.7.2 Recursive Blind A daptive 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 su bchannels 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
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32 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 (3.19) where is the step size parameter which controls the moving distance along the error surface, and is the instantaneous approximation to the gradient of the cost function . (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 (3.21) as a result, the instantaneous approximation at time is obtained by (3.22) substituting (3.22) into (3.19), yields to the following equation that the equalizer coefficients can be estimated (3.23)
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33 The speed of convergence and steady state performance rely on choosing of the step size 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 Re cursive Blind A daptive 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 paramet er 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 and the output 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 by minimizing the cost function relying only on using the observation In
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34 other word, t his solution leads to the minimum mean square error in estimating the reference signal The optimal (Wiener) solution is given by (3.24) where and let 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 (3.25) where is an instantaneous estimate to the gradient vector of the cost function with respect to the filter coefficients Employing instantaneous approximation estimates and for and is one possible solution to estimate t he gradient vector as follow (3.26) (3.27)
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35 The gradient estimate is given by (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, t he up dating equations for the LMS algorithm are described by (3.29) (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 c alculation to obtain a more reasonable estimate of the true gradient [9].
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36 3.7.4 Unbiased Blind A daptive 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 subchan nels 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) te chnique 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 precisely, only the equalizer length has to be chosen such that the following hypotheses hold in order to have suff icient channel diversity ) ( has strictly more columns than rows). ) has full row rank which means the matrix is invertible.
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37 The notion of linear prediction method (LP) is whitening the observation to find a channel inverse filter, using the following a priori constraint: ) The input sequence is white, It is a good point to mention that there is limitation in the SISO case because the whiteness constraint is too weak to a llow phase and amplitude equalization of a mixed phase channel. However, in the SIMO model, this limitation vanishes due to hypotheses and so the exact FIR inverse of the nonideal channel exists. It has been shown in [10] and [24] that the resulti ng noise free prediction error signal related to the input symbol sequence Assume that the finite length equalizer vector be a complex valued vector of prediction coefficients, and be the prediction error defined for The prediction error achieves its optimality in the noise free case if and only if and thus the prediction error variance is minimized. The performance versus SNR of this theoretical equalization method would critically depend on the realization of the particular coefficient 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 nois e. 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:
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38 (3.31) (3.32) (3.33) (3.34) where is a small step size and is a constraint value. The performance behavior of the mean square error is influenced by the choice of 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 A daptive Using Recursive Extended Least Square Algorithm (RELS) This algorithm considers the problem of blind adaptive equalization of i nfinite impulse response (IIR) 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 IIR 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
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39 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 mea surements for each transmitted symbol. An equivalent representation to our channel model is shown in Figure 3.1. where is a unit delay operator, is the transmitted symbol and =1+ = + (3.36) = + =1+ where L is the pol ynomial channel order. Assuming that polynomials , and are of the same order, and assuming that is a stable operator. Also, and are coprime polynomials as well as is a
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40 minimum phase polynomial. In genera l, all quantities in (3.35) and (3.36) can be complex numbers. Figure 3.1 SIMO c hannel m odel (IIR). Figure 3.2 Predictor based e qualizer
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41 The prediction based equalizer is shown in Figure 3.2, where is one step ahead prediction of Let and be the fil t er operators and the polynomials in so is which is optimal in the mean square sense. The polynomials and are computed by minimizing the following cost function : (3.38) Note that we can use instead of as a reference signal and find the predictor by minimizing From (3.35) and (3.37), can be written in the following form where and
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42 since is not a factor of from (3.41) it follows that =0 if and only if whereas and are solutions of the following polynomial equation: = (3.42) notice also that gives and from (3.39), we get the prediction error (3.43) Let the predictor operators and are defined as (3.44) Then (3.42) has a solution with respect to and if , where A unique solution exists if and are coprime, and at least one of the following holds: In practice, we cannot use (3.42) to calculate and since polynomials and are unkn own. 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).
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43 Taking advantage to our assumption that the nona daptive predictor (3.37) can be written in the form (3.45) Let (3 .46) and (3.47) Then (3.45) becomes (3.48) The number of inserted zeros in (3 .47) relies on the value of and the unknown degree Equation (3.48) is called the optimal nonadaptive prediction because is unknown. Therefore, the following adaptive predictor can be used (3.49) where (i) is an estimate of unknown and (3.50)
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44 Equation (3.49) is called a posteriori prediction, whereas represents a priori prediction. A priori prediction error is used to run the following extended least squares algorithm (first presented by [22]): (3.51) (3.52) , where I is the identity matrix, and is an arbitrary finite positive scalar. Initial values of in (3.51) and 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 precisely as long as one overfit, i.e., in (3.50), and [22].
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45 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 forgetting factor nu mber 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 p robability 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 b rief explanation to the MSE and symbol constellations as performance measures in the following section.
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46 4.1 Performance Evaluations 4.1 .1 Mean Square Error (MSE) In general, t he 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 ac curately 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. Howe ver, most real systems cannot achieve the ideality. The MSE is defined in a similar form as a cost function in (3.6) (4.1) This equation can be simplified and used in pract ice by a consistent sample estimate based on observations (4.2) where 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.
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47 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.
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48 4.2 S imulations In this section, the simulation result s 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 independen t 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 resu lts related to each single calculation method. 4.2.1 Experiment 1 a 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 fo ur symbols and describes for (4.3) where is the raised cosine given in [20, pp.546] as (4.4) with roll off factor while is the symbol duration. As in [11], is chosen.
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49 The above represents the composite causal approximation of a two ray multipath mobile radi o environment. This channel is reported by [1] and [21]. The discrete time equivalent channel is obtained by oversampling ( ) at a rate of (fractionally sampled) or for Figure 4.1 shows the magnitude of the impulse response while Figure 4.2 shows the zeros of and the zeros of the subchannels The subchannels parameters are (4.5) (4.6) Figure 4.1 Two ray multipath channel, the magnitude of impulse response
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50 Figure 4.2 The zeros of and the subchannels zeros Both Figures 4.1 and 4.2 are first presented in [11]. Using this type of channel to im plement the first equalizer calculation method that is Recursive Least Square (RLS) with zero delay ( ). It is assumed that the subchannel length and subequalizer length 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 that is assumed to be white with zero mean and 0.14 variance The performance of RLS algorithm depends on the forgetting factor which in this experiment is chosen to be 0.99. For simulation, use equations (3.17) and (3.18).
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51 Figure 4.3 Constellation plot of the received sequence for 16 QAM modulation Figure 4.4 Equalized symbol eye diagram, using RLS algorithm with the raised cosine channel
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52 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 4.2.2 Experiment 1 b In this section, an uncorrelated symbol sequence generated from a 16 QAM constellation is used. The channel mode l (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 The subchannels parameters are described as fol low:
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53 (4.7) (4.8) Figure 4.6 The magnitude and zeros of impulse response second class of channel model The RLS algorithm is implemented again using the second class of channel model with zero delay ( ). It is assumed that the subchannel length and subequalizer length The received symbols are presented in Figure 4.7, while F igure 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 that is assumed to be white with z ero mean and 0.14 variance. The forgetting factor is also 0.99.
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54 Figure 4.7 Eye diagram of the received signal for 16 QAM modulation Figure 4.8 Equalizer output symbols, using RLS algorithm with the second class of channel model
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55 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 4.2.3 Experiment 2 In this experiment, the cyclic LMS algorithm, which is first presented in [11], is simulated with zero delay ( ) using the first class of channel model (the raised cosine two ray multipath mobile radio) with Figure 4.10 shows the equalized eye diagr am, 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
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56 of cyclic LMS relies on the step size which throughout this thesis is chosen to be 0.0025. For simulation, use equation (3.23). Figure 4.10 Eye diagram of equalizer output, using cyclic LMS with the raised cosine channel Figure 4.11 Mean square symbol error of cyclic LMS
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57 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 However, the cyclic LMS doesn't work with the second class of channel model. To make it work, the conjugate calculation 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 ( ). 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). Figure 4.12 Equalized eye diagram, using LMS algorithm with the raised cosine channel
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58 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 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 t he equalized eye diagram, whereas Figure 4.15 depicts the mean square symbol error for LMS algorithm across 3000 symbols.
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59 Figure 4.14 Eye diagram of equalizer output, using LMS algorithm with the second class of channel model Figure 4.15 Mean square symbol error of LMS algorithm
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60 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 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 multip ath mobile radio ( ). 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 ( ). Figure 4.16 Mean square symbol error, Comparison using raised cosine channel
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61 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
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62 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 chan nel used is the raised cosine two ray multipath mobile radio with The small step size is chosen to be 0.002. The performance of linear prediction relies on the constraint value due to its impact on the MSE behavior. Throughout this thesis 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). Figure 4.18 Eye diagram of equalizer output, using LP method with the raised cosine channel
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63 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 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 which is the leading coefficient in The angle of rotation is and the magnification is
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64 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. Figure 4.20 Eye diagram of equalizer output, using LP algorithm with the second class of channel model
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65 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 The angle of rotation is and the magnification is
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66 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) IIR channel model (4.9) (4.10) (4.11) (4.12) Our assumptions with respect to additive noise are still the same. Figure 4.22 shows the received symbols while Figure 4.23 shows the equalized symbols eye diagram. As perf ormance measures, the mean square symbol error for RELS algorithm across 3000 symbols is estimated as shown in Figure 4.24 using the following equation (4.13) For simulation, use equation (3. 49) (3.53).
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67 Figure 4.22 Eye diagram of received signal for 16 QAM modulation Figure 4.23 Eye diagram of equalizer output, using RELS algorithm with the third class of channel model (IIR).
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68 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 modelin g error in the channel dynamics, and we use the same polynomials as in Experiment 5a,
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69 where and are white noise with zero mean and variance=0.14. It is important to mention that are independent seq uences. Let are modeling errors given by where 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 in (4.15 ). Figure 4.25 Scatter plot of equalized symbols, using RELS algorithm with the presence of modeling error using (IIR ) channel.
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70 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 in experiment 5a, whereas the MSE of ex periment 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 t he 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 ( ).
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71 Figure 4.27 Mean square symbol error, Comparison for those five algorithms with the second class of channel model
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72 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 i nter 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 channel s. 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 occur red 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 statis tical 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 realiz ed 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) IIR filter. The latest channel model was performed with and without modeling error.
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73 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, i t 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 performan ce. 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.
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74 R EFERENCES [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 domain blind Proc. Int. Conf. Acoust., Speech, Signal Processing vol. 3, Detroit, MI, 1995, pp. 1968 1971. [3] E. W. Bai and M.Fu 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. determined mixtures Signal Proc ., vol. 86, pp. 2271 2281, Septembe r 2006. 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, order cyclic IEEE Trans. Signal Processing vol. 42, pp. 1260 1264, May 1994. [9] P. S. R. Diniz, Adaptive Filtering Algorithms and Practical Implementation 3 rd ed. New York, NY: Springer Science & Business Media, LLC, 2008.
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