design of an efficient blind equalizer for digital high...
TRANSCRIPT
DESIGN OF AN EFFICIENT BLIND EQUALIZER
FOR DIGITAL HIGH SPEED WIRELESS
COMMUNICATION SYSTEM
A THESIS
Submitted by
SUTHENDRAN K
(Reg.No.201008207)
In partial fulfillment for the award of the degree
Of
DOCTOR OF PHILOSOPHY
DEPARTMENT OF ELECTRONICS AND
COMMUNICATION ENGINEERING
KALASALINGAM UNIVERSITY
(Kalasalingam Academy of Research and Education)
Anand Nagar, Krishnankoil – 626 126
AUGUST 2015
CERTIFICATE
This is to certify that all corrections and suggestions pointed out by the Indian/Foreign
Examiner(s) are incorporated in the Thesis titled “DESIGN OF AN EFFICIENT BLIND
EQUALIZER FOR DIGITAL HIGH SPEED WIRELESS COMMUNICATION
SYSTEM” submitted by Mr.K.Suthendran, Reg.No.201008207.
SUPERVISOR
Place: Madurai
Date: 13.08.2015
Minutes of the Ph.D. Viva-Voce Examination of Mr.K.Suthendran
(Reg. No.201008207) held at 09.30 a.m. on 25th August in the Conference
hall of office of R & D, International Research Center, Kalasalingam
University, Anand Nagar, Krishnankoil-626126.
The Ph.D. Viva-Voce Examination of Mr.K.Suthendran (Reg. No.
201008207) on his/her Ph.D. thesis entitled “Design of an Efficient Blind
Equalizer for Digital High Speed Wireless Communication Systems” was
conducted on 25th August 2015 at 09.30 a.m. in the Conference Hall of office
of R & D, International Research Center at Kalasalingam University,
Anand Nagar, Krishnankoil-626126.
The following members of the Oral Examination Board were present:
1. Dr.T.Arivoli Supervisor & Convener Professor & Head / ECE,
Kalasalingam University,
Anand Nagar, Krishnankoil – 626 126
2. Dr.B.Priestly Shan Indian Examiner Principal,
Royal College of Engineering &
Technology, Akkikavu, Thrissur,
Kerala- 680604
3. Dr.M.Pallikonda Rajasekaran Chairman/DRC Prof & Head /ECE Kalasalingam University Anand Nagar, Krishnankoil – 626 126
The candidate, Mr.K.Suthendran, presented the salient features of his
Ph.D. work. This was followed by questions from the Board Members. The
queries and clarifications raised by the Foreign and Indian Examiners were
also put to the candidate. The candidate answered the questions to the full
satisfaction of the Board Members.
The corrections and suggestions pointed out by the Indian/Foreign
examiner have been carried out and duly incorporated in the thesis.
Based on the candidate's research work, his presentation and also the
clarifications and answers by the candidate to the questions raised by the
examiners, the Board recommends that K.Suthendran be awarded the Ph.D.
degree in the FACULTY OF ELECTRONICS AND
COMMUNICATION ENGINEERING.
Dr.T.Arivoli Dr.B.Priestly Shan
Supervisor & Convener Indian Examiner
Dr.M.Pallikonda Rajasekaran Chairman/DRC
KALASALINGAM UNIVERSITY
(Kalasalingam Academy of Research and Education)
KRISHNANKOIL 626 126
DECLARATION
I hereby declare that the thesis entitled “Design of an Efficient Blind Equalizer for
Digital High Speed Wireless Communication System” submitted by me for the Degree of
Doctor of Philosophy in Department of Electronics and Communication Engineering is
the result of my original and independent research work carried out under the guidance of
Dr.T.ARIVOLI and it has not been submitted for the award of any degree, diploma,
associateship, fellowship of any University or Institution.
SUTHENDRAN .K
i
KALASALINGAM UNIVERSITY
(Kalasalingam Academy of Research and Education)
Anand Nagar, Krishnankoil – 626 126
BONAFIDE CERTIFICATE
Certified that this thesis titled “DESIGN OF AN EFFICIENT BLIND
EQUALIZER FOR DIGITAL HIGH SPEED WIRELESS
COMMUNICATION SYSTEM” is the bonafide work of
Mr. SUTHENDRAN K, who carried out the research under our supervision.
Certified further, that to the best of our knowledge the work reported herein
does not form part of any other thesis or dissertation on the basis of which a
degree or award was conferred on an earlier occasion on this or any other
scholar.
Dr. T. ARIVOLI
SUPERVISOR
Professor & Head
Department of ECE
Vickram College of Engineering
Enathi, Madurai – 630 561
Tamilnadu, India.
ii
ABSTRACT
Information traveling through a channel undergoes various forms of
distortion. The most common is Inter-symbol-interference, which is known as
ISI. Inter symbol interference induced errors can cause the receiver to fail to
reconstruct the original data. Equalizers in the receivers, which are special
kind of filters, mitigate the linear distortion produced by the channel. If the
channel’s time varying characteristics are known a priori, then optimum
setting for equalizers can be worked out. But in practical systems, the
channel’s time varying characteristics are not known a priori, so adaptive
equalizers are used. Adaptive equalizers are adapt, or change the value of its
taps as time progresses. There are two main types of adaptive equalizers,
trained equalizers and blind equalizers. In trained equalizers there is a pseudo-
random pattern of bits called training sequence known both to receiver and
transmitter. But equalizers for which no such initial training is provided are
called BLIND EQUALIZERS. A Blind equalizer is able to compensate
amplitude and delay distortion of a communication channel using only the
channel output samples and knowledge of basic statistical properties of the
data symbols. One of the major disadvantages is that all blind equalizers
converge very slowly. However, variable step size can speed up the
convergence rate by balancing the steady state error. The novel idea projected
here is that instead of selecting an optimum step size as the starting value and
then decrementing it in iterations, as others did for the reconstruction of all
symbols, chose a small step size as the starting value and incrementing it in
iterations only for very first symbol by balancing maladjustment and
convergence. The updated maximum step size is treated as the starting point
and subsequently reducing the step size in iterations for the remaining symbols
and thus gained better performance.
iii
In this work, different equalization algorithms are simulated, which includes
1. Least Mean Square Algorithm
2. Sato’s Blind Equalization Algorithm
3. Godard’s Blind Equalization Algorithm (Constant Modulus Algorithm)
iv
ACKNOWLEDGEMENT
I thank my advisor Dr. T.Arivoli for his guidance and support. It has
been a true privilege to work with a well-reputed advisor at Kalasalingam
University. His sincere guidance has helped me to shape up my research and
career. I hope to collaborate with him in the future. Special thanks are the due
to Dr.S.SaravanaSankar, Vice Chancellor, Kalasalingam University for his
valuable support. I am grateful to all the committee members for their
suggestions and time. Their suggestions were helpful in improving the quality
of this dissertation. I am thankful to my mother and sister for their
unconditional love and support. I am grateful to my father and brothers for the
sacrifices they made to ensure a high quality education for me. I am grateful to
all my gurus and teachers for their guidance and wisdom. I thank my cousin
G.Balaji for his encouragement and support. I thank my friends (M.Satheesh
Kumar, R.Kalidoss, S.Sriram Sundar, M.Raja, A.Amutha kannan, Neelakanda
Bharathi Raja, Senthil, and Prabhakaran) for their support. I thank
K.Meenakshisundaram for helping me with administrative tasks. I express my
sincere thanks to my co-researchers Mr. K. Rajakumar, Mrs. A. Lakshmi and
Mrs. Anish Ponyamini and Mr.T.Ramu for their cooperation and help during
the research work.
SUTHENDRAN K
v
TABLE OF CONTENTS
CHAPTER TITLE PAGE NO. NO.
ABSTRACT ii
LIST OF TABLES ix
LIST OF FIGURES xi
LIST OF SYMBOLS AND ABBREVIATIONS xv
1 INTRODUCTION 1
1.1 THE COMMUNICATION SYSTEMS 1
1.2 ADAPTIVE FILTERS 7
1.3 FILTER DESIGN 11
1.4 CHANNEL ESTIMATION 14
1.5 ORGANIZATION OF THESIS 16
1.6 SUMMARY 17
2 LITERATURE REVIEW 18
2.1 INTRODUCTION 18
2.2 LEAST MEAN SQUARE ADAPTIVE ALGORITHMS
CONVERGENCE RATE, COMPLEXITIES
AND ITS APPLICATIONS 18
2.3 VARIABLE STEP SIZE TECHNIQUES BASED
ON LEAST MEAN SQUARE ALGORITHM 21
2.4 BLIND EQUALIZATION BASED ON SATO
ALGORITHM 27
2.5 BLIND EQUALIZATION BASED ON GODARD
ALGORITHM (CMA) 30
2.6 VARIABLE STEP SIZE TECHNIQUES FOR
BLIND EQUALIZATION ALGORITHMS 34
vi
CHAPTER TITLE PAGE NO. NO. 2.7 THE KNOWLEDGE GAP IDENTIFIED FROM
THE EARLIER INVESTIGATIONS 38
2.8 AIM OF THE RESEARCH WORK 39
2.9 OBJECTIVES OF THE RESEARCH WORK 39
2.10 SUMMARY 40
3 ADAPTIVE EQUALIZER 41
3.1 INTRODUCTION 41
3.2 EQUALIZER AND ITS OPERATING MODES 42
3.3 ADAPTIVE LEAST MEAN SQUARE EQUALIZER 44
3.3.1 Basic Concept 44
3.4 PSEUDOCODE OF VARIABLE STEP SIZE LEAST
MEAN SQUARE EQUALIZER 47
3.5 VARIABLE STEP SIZE LEAST MEAN SQUARE
EQUALIZER 48
3.5.1 The Channel Model 50
3.5.2 Simulation Results 50
3.6 SUMMARY 58
4 BLIND EQUALIZER 59
4.1 INTRODUCTION 59
4.2 IMPORTANCE OF BLIND EQUALIZER 60
4.3 EVOLUTION OF BLIND EQUALIZER 61
4.4 SATO’s BLIND ALGORITHM 62
4.5 SIMULATION RESULTS OF SATO’s BLIND
ALGORITHM 64
4.6 GODARD’s BLIND ALGORITHM (CMA) 70
4.7 SIMULATION RESULTS OF GODARD’s
BLIND ALGORITHM 71
vii
CHAPTER TITLE PAGE
NO. NO. 4.8 SUMMARY 76 5 VARIABLE STEP SIZE TECHNIQUES FOR SATO
BASED BLIND EQUALIZER 78
5.1 INTRODUCTION 78
5.2 VARIABLE STEP SIZE SATO’s BLIND
ALGORITHM 78
5.3 PSEUDOCODE OF VARIABLE STEP
SIZE SATO BASED BLIND EQUALIZER 80
5.4 SIMULATION RESULTS OF VARIABLE STEP
SIZE SATO BASED BLIND EQUALIZER 81
5.5 SUMMARY 87
6 VARIABLE STEP SIZE TECHNIQUES OF GODARD
BASED BLIND EQUALIZATION ALGORITHM (CMA) 88
6.1 INTRODUCTION 88
6.2 PSEUDOCODE OF VARIABLE STEP SIZE GODARD
BASED BLIND EQUALIZATION ALGORITHM 88
6.3 VARIABLE STEP SIZE GODARD BASED BLIND
EQUALIZATION ALGORITHM (CMA) 89
6.4 SIMULATION RESULTS OF VARIABLE STEP
SIZE BLIND EQUALIZER 92
6.5 SUMMARY 98
7 CONCLUSION 99
APPENDIX A : PROGRAMMING CODE FOR
EQUALIZER ALGORITHMS 104
viii
APPENDIX B: HILBERT- TRANSFORM PAIR 109
REFERENCES 110
LIST OF PUBLICATIONS 120
CURRICULUM VITAE 122
ix
LIST OF TABLES
TABLE
NO. TITLE
PAGE
NO.
3.1 Comparison of SNR vs. Iterations for LMS Adaptive
Equalizer with Step Size Parameter µ = 0.015.
56
3.2 Comparison of SNR vs. Iterations for LMS Adaptive
Equalizer with Step Size Parameter µ=0.25.
57
3.3 Comparison of SNR vs. Iterations for LMS Adaptive
Equalizer with Variable Step Size.
57
4.1 Comparison of SNR vs. Iterations for LMS Adaptive
Equalizer with Step Size Parameter µ = 0.015
67
4.2 Comparison of SNR vs. Iterations for SATO based
Blind Equalizer with Step Size Parameter α = .0006
68
4.3 Comparison of SNR vs. Iterations for SATO based
Blind Equalizer with Step Size Parameter α = 0.6
68
4.4 Comparison of SNR vs. Iterations for SATO based
Blind Equalizer with Step Size Parameter α = .0006
74
4.5 Comparison of SNR vs. Iterations for SATO based
Blind Equalizer with Step Size Parameter α = .06
74
4.6 Comparison of SNR vs. Iterations for Godard based
Blind Equalizer with Step Size Parameter µ = .06
75
5.1 Comparison of SNR vs. Iterations for Sato based
Blind Equalizer with Step Size α = .0006
85
5.2 Comparison of SNR vs. Iterations for Variable α
Blind approach (Linear)
86
5.3 Comparison of SNR vs. Iterations for Variable α
Blind approach (Non Linear)
86
x
TABLE
NO.
6.1
TITLE
Number of Iterations for Sato Blind approach with
variable step size
PAGE
NO.
96
6.2 Number of Iterations for Godard blind approach with
fixed step size 96
6.3 Number of iterations for proposed Godard blind
approach with variable step size 97
xi
LIST OF FIGURES
FIGURE
NO. TITLE
PAGE
NO.
1.1 Basic Elements of the Communication System. 2
1.2 Analog to Digital Converter. 3
1.3 Digital to Analog Converter. 4
1.4 Mathematical Notational View of Additive Noise 6
1.5 Symbols for Basic Building Block of Digital Filter
Design.
13
1.6 The block Diagram of the channel estimator. 15
2.1 The region of Di of variable step size for 16-QAM
signal
36
3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8
3.9
3.10
3.11
Generalized Block Diagram of Equalizer.
Generalized Diagram of Equalizer with N taps
Flowchart for Variable Step Size LMS Algorithm
The Channel Model
The PAM Symbol 3
The ISI model for PAM Symbol 3
The PAM Symbol 3 with ISI and AWGN Noise
The Equalizer output for PAM Symbol 3 after 1st
iteration
Reconstructed Symbol 3 using LMS algorithm and
SNR = 30dB (3708 iterations).
Reconstructed Symbol 3 using Variable Step Size
LMS algorithm and SNR = 30dB (19 iterations)
Mean Square Error comparison between LMS and
proposed VSS LMS approach.
41
44
48
49
50
50
51
51
52
52
53
xii
FIGURE
NO.
3.12
3.13
3.14
3.15
4.1
4.2
4.3
4.4
4.5
4.6
4.7
4.8
4.9
4.10
4.11
4.12
4.13
TITLE
The Equalizer output for PAM Symbol 3 after 1st
iteration
Reconstructed Symbol 3 using LMS algorithm and
SNR = 30dB (3708 iterations).
Reconstructed Symbol 3 using Variable Step Size
LMS algorithm and SNR = 30dB (19 iterations).
Mean Square Error comparison between LMS and
proposed VSS LMS approach.
General block diagram for Blind Equalizer
The Sato based Blind Equalizer with 5 taps
The PAM symbol1
The ISI model for PAM symbol1
The PAM symbol 1 with ISI and AWGN noise
The PAM symbol 1 output after 1st iteration
The reconstructed PAM symbol 1 using LMS
Algorithm with µ=0.015 and SNR = 20dB (84
iterations)
The reconstructed PAM symbol 1 using Blind
algorithm with α =0.6 and SNR = 20dB
(9 iterations)
The Mean Square Error comparison using LMS
and Blind algorithm
Godard scheme for Blind equalization and carrier
tracking
The PAM symbol 2
The ISI model for PAM symbol 2
The PAM symbol 2 with ISI and AWGN noise
PAGE
NO.
55
55
55
56
62
63
64
64
65
65
66
66
66
69
71
72
72
xiii
FIGURE
NO
4.14
4.15
4.16
4.17
5.1
5.2
5.3
5.4
5.5
5.6
5.7
5.8
6.1
6.2
6.3
6.4
TITLE
The equalizer output for PAM symbol 2 after 1st
iteration
The reconstructed PAM symbol 2 using Sato’s
blind algorithm with α =0.0006 and SNR = 30dB
(12950 iterations)
The reconstructed PAM symbol 2 using Godard’s
blind algorithm with α =0.06 and SNR = 30dB
(4640 iterations)
The Mean Square Error comparison of Sato and
Godard Blind Equalization algorithm (CMA)
The flowchart for VSS Sato based blind equalizer
The PAM symbol 4
The ISI model for PAM symbol 4
The PAM symbol 4 with ISI and AWGN noise
The equalizer output of received PAM symbol 4
after 1st iteration
The reconstructed PAM symbol 4 using Sato
algorithm with fixed step size (24483 iterations)
The reconstructed PAM symbol 4 using Sato
algorithm with variable step size (177 iterations)
Mean Square Error comparison between Sato’s
Blind and variable step size blind algorithm
The flowchart for variable step size Godard
algorithm
The PAM symbol 5
The ISI model for PAM symbol 5
The PAM symbol 5 with ISI and AWGN noise
PAGE
NO
72
73
73
73
78
82
83
83
83
84
84
84
90
93
93
xiv
FIGURE
NO
6.5
6.6
6.7
6.8
TITLE
The equalizer output for PAM symbol 5 after 1st
iteration
The reconstructed PAM symbol 5 using VSS Sato
algorithm with SNR=30dB (26 iterations)
Godard algorithm with SNR=30dB (1350
iterations)
Mean Square Error comparison between VSS Sato
algorithm and VSS Godard algorithm
PAGE
NO
94
94
94
95
xv
LIST OF SYMBOLS AND ABBREVIATIONS
ISI - Inter Symbol Interference
MSE - Mean Square Error
LMS - Least Mean Square
RLS - Recursive Least Squares
PAM - Pulse Amplitude Modulation
FIR - Filter Impulse Response
IIR - Infinite Impulse Response
SNR - Signal to Noise Ratio
VSS - Variable Step Size
AWGN - Additive White Gaussian Noise
QAM - Quadrature Amplitude Modulation
CMA - Constant Modulus Algorithm
MIMO - Multiple Input Multiple Output
α - Attenuation factor
µ - Step size parameter
λ - Step size parameter
s(t) - Original information
n (t) - Noise
r(t),y(t) - Received Signal
X (n) - Digitized input
X (z) - Filter Input
Y (z) - Filter Output
H (Z) - Transfer function of filter
fc - Cutoff frequency
1
CHAPTER 1
INTRODUCTION
1.1 THE COMMUNICATION SYSTEMS
Communication, generally, is the exchange of information between
transmitter and receiver. On the basis of electrical sense, it refers to sending,
receiving and processing of the information by electrical means. On other side,
telecommunication refers to the communication over a distance greater than
would normally be possible without artificial acts.
A communication system, in today’s world, is a dominating factor,
which is the most important criterion for human development. In fact, the
people in developed nations are enjoying the more communication systems.
Therefore, the role of communication system plays important factor for the
development of any society or their country. Communication, in general sense,
is the transmission of information from one point to another point. Therefore,
while we say there is a communication, there must be three basic elements.
They are Transmitter, Channel and Receiver.
For example, suppose as we are communicating to each other, the
person speaking is the transmitter, the other person who is listening is the
receiver, and the medium, that is air, is the channel. The basic element of the
communication systems are shown in figure 1.1. This gives the brief block
metric views of general communication.
2
Figure 1.1 Basic Elements of the Communication System.
The communication systems can be divided into two types based on
the signal to be transmitted. They are the basic of the types of the signal to be
transmitted, the communications system can be transmitted; the
communication system can be divided into two types. They are
Analog communication,
Digital communication.
Analog communication is the first type of communication systems,
in which the base band information signal is analog. One prominent example
of analog communication is radio broadcasting.
In the digital communication, the base band information is digital.
The data communication is pure digital. In some cases, the analog is converted
to digital and then transmitted. In the receiver, the digital signal is extracted
and then converted back to analog. Though, it is complicated in its structure, it
has many advantages over the analog communication. Today’s
communication systems are dominated by digital communication systems.
The following is a brief explanation about the basic communication
elements.
Transmitter
Transmitter does the job of transmitting the information. Therefore,
transmitters are the devices that impress source information on to an electrical
Source of
Information Transmitter Channel Receiver
User of
Information
Communication Systems
3
wave (or carrier) appropriate to a particular transmission medium (for example
optical fiber, cable, free space etc.). In short, the purpose of transmitter is to
transform the message signal produced by the source of information in to a
form suitable for transmission over the channel.
Figure 1.2 Analog to Digital Converter.
Communication systems use the various sources of information
such as speech, still picture images as used by facsimile machines, moving
picture images as used in television, binary data as used in personal
computers. Thus the sources of information can be either of analog types such
as speech signal or binary types such as datas in PC.
As said earlier, the analog transmission systems just assign the
suitable carrier frequency to the available source of information and transmit
it. But in the case of digital communication, it has extra procedure to code the
information source in to the binary form. Each bit from the ADC converter or
source encoder is transmitted separately. The conceptual block diagram for
analog to digital converter is as shown in figure 1.2.
Channel
Communication channel is the physical medium that is used to send
the signal from the transmitter to the receiver. There are lots of
communication mediums available to propagate the information signals. Here
are some of them,
Wire-line channels
Fiber-optic channels
Wireless electromagnetic channels
Storage channels
Sampler Quantizer Encoder
4
The wire-line channels are the cheapest medium for signal
transmission and have been used for long time for low frequency signals.
Twisted pair wire-lines and coaxial cables are the examples of wire-line
channels. The optical fibers are being used recently and they provide
enormous bandwidth. The wireless electromagnetic channels make use of free
spaces. Based on the frequency applicability of the frequency spectrum, free
spaces are divided in to three divisions. They are,
Ground wave propagation
Sky-wave propagation
Line-of-sight propagation
Whatever be the best physical medium used for the transmission of
the information, the transmitted signal is being corrupted in a random manner.
The varieties of noises, which corrupt the signal, will be dealt in later in this
chapter.
Receiver
Receivers are sub systems that extract information from the
transmitted carriers. In fact, the receiver does the job of complementary
operations to the transmitter. Therefore, whatever is the type of receiver, its
most important function is demodulation (and decoding in the case of digital
receiver). Analog receiver has a simple mechanism for demodulation, where
as in the case of digital receiver, it has extra circuitry and it is shown in
figure 1.3.
Figure 1.3 Digital to Analog Converter
Decoder Dequantizer
Reconstruction
Filter
5
In noisy environment, it is very difficult to extract the information
transmitted. This is because these noises corrupt the information signal. To
cancel the effect of the noise and extract the real information signal, the
receiver needs some extra mechanism. This extra mechanism is nothing but
the filters, which is the core topic of this work.
Noises
As explained earlier, the receiver is unable to get a noise free signal.
Basically, there are three kinds of noises exist in the communication systems.
They are,
Quantization noise
Inter Symbol Interference induced noise
Channel noise
Quantization noise occurs when the analog signal is quantized.
During analog to digital conversion, the analog signals are discretised and the
amplitude of the discrete signal is coded in to a binary form that is either ‘1’ or
‘0’. But unfortunately the discrete form of signal representation cannot be able
to represent the analog signal completely. Therefore, there will be some
deviation to the original signal, the cause of which is called the quantization
noise.
Inter Symbol Interference is the kind of noise, which is created by
signal itself. The signal reaches the receiver by different paths due to
reflections. The signal with no reflection reaches first and the same signal will
be delayed if it is reflected. The delayed signal overlaps with the next
information signal causing distortion of the signal, which is known as Inter
Symbol Interference.
Additive White Gaussian Noise is a basic model used in
Information theory to mimic the effect of many random processes that occur in
nature. The modifiers denote specific characteristics: Additive because it is
6
added to any noise that might be intrinsic to the information system. White
refers to the idea that it has uniform power across the frequency band for the
information system. It is an analogy to the color white which has uniform
emissions at all frequencies in the visible spectrum. Gaussian because it has a
normal distribution in the time domain with an average time domain value of
zero.
There are lot of channel noises, such as thermal and shot noise,
generated by electronic devices; man-made noise, generated by human; and
the atmospheric noise, generated by like electrical lightening during the
thunderstorms.
Nyquist, the father of mathematical simulations for communications
systems, formulated the signals to a mathematical form. For example, if the
transmitted sequence is s (t), and the noise introduced at channel during the
propagation is n(t), then the received signals at the receiver can be represented
as
)t(n)t(s)t(y +α= (1.1)
Where,
α= is the attenuation factor
The block diagrammatic notation of the equation 1.1 is shown in figure 1.4.
Figure 1.4 Mathematical Notational View of Additive Noise
Information Signal s(t)
Noise n(t)
Received Signal
r(t)=α s(t)+n(t)
Channel
7
1.2 ADAPTIVE FILTERS
In the trendy electronic communication, plenty of effort has been
dedicated to utilize the accessible channel bandwidth expeditiously. Inter-
symbol Interference (ISI) and Thermal noise are the two main factors that are
limiting the performance of information transmission systems. In essence, the
ISI is generated by dispersion within the transmit filter, the transmission
medium, and receive filter. Yun Zhao et al [89], within the transmission
medium, which is a band-limited (frequency selective) time dispersive
channel, the ISI is caused by multipath propagation. The result is that the
modulated pulses spread in time into adjacent symbols, and distort the
transmitted signals inflicting information errors at the receiver Monika
Pinchas et al [50]. Thermal noise is generated at the face of the receiver. For
wireless channels, which are bandwidth-limited, the ISI has been recognized
as the major downside in high speed information transmission. The standard
band restricted filters fail to recover the information once the received symbol
contains ISI and in-band noise. The Inter-symbol Interference can be removed
by using equalization techniques Alban Goupil et al [4].
Generally, the term equalization is used to explain any signal
process operation that minimizes the ISI S.U.H.Qureshi et al [59]. Digital
signal processing based equalizer systems become more essential in various
applications including information, voice, and video communications. The
equalizer may be a digital filter, placed between sampler circuit and decision
algorithm within the band restricted communication model. An equalizer
inside the receiver compensates for average range of expected channel
amplitude and delay characteristics. Equalizer algorithmic program, equalizer
structure and the rate of amendment of the multipath radio channel are three
main factors that have an effect on the time spread over which an equalizer
converges. Two important issues in equalizer design and implementation are
8
its complexity and its training. For frequency selective channel, the equalizer
enhances the frequency parts with small amplitudes and attenuates the robust
frequencies within the received frequency response and for a time-varying
channel.
An equalizer corrects the channel frequency response variation and
cancels the multipath effects. They are specifically designed for multipath
correction and are therefore usually termed as echo-cancellers or deghosters
David Smalley et al [15]. For this effect, it will need considerably longer filter
length than that of easy spectral equalizers; however, the principles of
operation are basically the same. These filters have an equalized impulse
response having zero ISI and zero channel distortion. This implies that
convolution of the channel response and the equalizer impulse responses
should be equal, having one at the centre tap and nulls at the opposite sample
points inside the filter span Ye Li et al [87], David Smalley et al [93]and (A.
Benveniste et al [13].
Automatic synthesis and adaptation are the two strategies used to
estimate the filter coefficients. In automatic synthesis methodology, the
equalizer generally compares a received time-domain reference signal thereto
of an ingenuous training signal. This is often holding on within the receiver
and a time-domain error signal is decided. The calculated error signal is
employed to estimate the inverse filter coefficient. In an adaptation filter
synthesis methodology, the equalizer calculates the error signal supported by
the distinction between the outputs of the equalizer. The estimated transmitted
signal is generated by a decision device. The filter coefficient values are
changed at iterations corresponding to the error signal value and they are
optimized for zero error. The main disadvantage of this automatic synthesis
equalization methodology is related to the overhead of sending training signal,
9
which should at least have the length of the filter tap. This training of the filter
to converge at the startup could be part of the initialization overhead.
The mobile weakening channel may be random and time varying;
equalizers should track the time varying characteristics of the channel, and
therefore known as adaptive equalizers. Adaptive channel equalization is a
good tool in mitigating inter-symbol interference (ISI) caused by linear
distortions in unknown channels Rappaport Theodore et al [63] and
Giannakis.G.B. et al [22].
In general, the error estimation is computed with the aid of the
received vector and the desired response, and it is used to create the adjustable
filter coefficients values. Depending on the chosen filter structure, the
adjustable coefficients are also in style of tap weight reflection coefficients, or
rotation parameters. However, the elemental distinction between the assorted
applications of adaptive filtering arises within the manner during which the
required response is extracted.
Training and tracking are the two general operational modes of an
adaptive equalizer. First, a legendary training sequence pseudorandom binary
signal of fixed length is transmitted by the transmitter. With this, the equalizer
at the receiver could adapt to a correct weight for minimum bit error rate
(BER) detection. Following this training sequence, original information is
transmitted and adaptive equalizer utilizes the recursive formula to gauge the
channel, and therefore estimates the filter coefficients to compensate the
distortion created by multipath within the channel. Equalizers need periodic
preparation so as to keep up effective channel variation. In digital
communication systems, user information is generally segmented into short
time blocks or time slots. Time division multiple access (TDMA) wireless
systems are notably compatible for equalizers. Owing to time variable nature
10
of wireless channels, training signals should be sent often and this occupies
additional information measure.
Even though trained strategies have many disadvantages, they're
typically adequate. The throughput of the system drops owing to the time slots
occupied by the training signal. Another disadvantage is that the training
signal isn't always familiar at the receiver, e.g., in an exceedingly non
cooperative (surveillance) surroundings. Finally, the quicker time varying
channel needs training sequence more often to train the equalizer. This results
in more reduction within the throughput of the system.
The Blind algorithms are ready to exploit characteristics of the
transmitted signals and don't need training sequences. They’re called so
because they supply equalizer convergence without burdening the transmitter
with training overhead. These fashionable algorithms are able to acquire
equalization through property restoral techniques of the transmitted signal. In
general, even if the initial error rate is high, blind equalization technique
directs the coefficient adaptation method towards the optimum filter
parameters. A Blind Equalizer can compensate the amplitude and the delay
distortions of a communication channel by using solely the channel output
samples and the data of the basic statistical properties of the information
symbols. The key advantage of blind equalizers is that there are no training
sequences and thus no bandwidth is wasted by its transmission. Blind
equalization is effective for a high-speed digital radio, digital mobile
communication systems, multi-point networks, cable TV, and digital terrestrial
TV broadcasting Kil Nam Oh et al [38] and A. Benveniste et al [13].
The major downside is that the equalizer can usually take an
extended time to converge as compared to a trained equalizer. The necessity
for blind equalizers within the field of information communications is greatly
11
mentioned by Godard Dominique N. Godard et al [15], within the context of
multipoint networks. Blind joint equalization and carrier recovery might
realize application in digital communication system over multipath weakening
channels. Moreover, it's applied in extremely non-stationary digital mobile
communications, wherever it's impractical to use training sequences.
These techniques embrace algorithms like the SATO algorithm and
Constant modulus algorithm (CMA).The first self-recovering algorithm was
proposed by Sato Y.Sato et al [88] in 1975 for equalization of PAM signals.
The sole limitation of Sato algorithm is that it recovers only single carrier.
This limitation is overcome by Godard proposal.
To enlarge the performance of the blind equalization algorithm, a
standard way is to formulate the step size to be variable rather than fixed Raja
Uyyala et al [60]; that is, choose large step size values during the initial
convergence of the blind equalization algorithm, and use small step size values
when the system is close to its steady state. In other words, select a large step
size value in the transient phase and a small step size value in the steady state
noise level.
1.3 FILTER DESIGN
The Blind Equalizers, in practice, are designed using FIR digital
filter design technique. But, there are search to make use of IIR filter design
technique, which has considerable advantage over FIR. This project, though,
not considering the design part, the analysis itself is very near to the design.
This chapter concentrates only on the fundamentals of filter design. It first
deals with analog filter design and then deals with digital filter design.
12
Analog Filter Design
Analog filtering is performed on continuous-time signals and yield
continuous-time signals. The basic analog filter design makes use of
operational amplifiers, resistors, and capacitors. In reality, analog filters are
more difficult to design and analyze, then are their digital counterparts,
because the analog filters are based on differentiation. The general definitive
transfer function of the analog filter is given in equation below.
k
kM
kkk
kN
kk
dt
tydb
dt
txdaty
)()()(
10
(1.1)
Most of the analog filters are designed to meet the specifications in the
frequency response. Here we have some of the analog filters in brief.
Butterworth Filter
Butterworth filters are very popular in analog filter design because,
its pass-band and stop-band both are of ripple free. But, it achieves this at the
expense of relatively wide transition region.
Chebyshev Filter
Chebyshev filter has a smaller transition region than the same order
Butterworth. But it has ripple either on stop-band or pass-band.
Elliptic Filter
The elliptic filter has the shortest transition region with ripple on
both bands.
13
Digital Filter Design
Designing of digital filters are easier than analog filters, because it
uses the operations such as addition, multiplication, and data movements only.
Figure 1.5 Symbols for Basic Building Block of Digital Filter Design
The symbolic representations for basic building blocks of digital
filter are as shown in figure 1.5.
There are mainly two branches of digital filter design. They are,
Non-recursive,
Recursive.
Non-recursive Filter Design
A non-recursive filter generates its output by simply weighting the
inputs by constants and then summing the weighted inputs. Finite Impulse
Response (FIR) is the best example of non-recursive filter design. In FIR filter
design, if the system signal is in analog form, then it can be converted in to the
digital form by using the transformation procedure. The transfer function for
realizing the FIR filter is given in equation 1.2, below
Multiplier
X(n) a X(n) X(n) Z-1 X(n)-1
Unit delay
X1(n)
X2(n)
X1(n) + X2(n)
Addition
14
nN
n
zxhzH
1
0
)()( (1.2)
There are plenty of methods available to design FIR filter, such as direct form,
parallel form, cascaded form etc.
Recursive Filter Design
In recursive filter design, the output is not only a function of the
inputs, but is also depends upon the past outputs. Infinite Impulse Response
(IIR) is the best example of recursive filter design. The transfer function for
designing the IIR filter is given in equation 3.3, below.
N
k
kk
N
n
kk
za
zb
zX
zYzH
0
0
1)(
)()(
(1.3)
1.4 CHANNEL ESTIMATION
A channel is the medium, which is used to transfer the data or
information from transmitter to receiver. Channels include the physical
medium like free space, fiber, waveguides etc. The characteristics of any
physical medium is that, the transmitted signal is corrupted by various
ways such as frequency and phase distortion, inter symbol interference,
thermal noise etc. and the receiver receives only the corrupted signal .
Channel estimation is defined as the process of characterizing
the effect of the physical channel on the input sequence. It helps to
mitigate the channel effect and reproduce the input sequence from the received
signal. In order to provide reliability and high data rates at the receiver, the
system needs an accurate estimate of the time-varying channel. Furthermore,
mobile wireless systems are one of the main technologies which used to
provide services such as data communication, voice, and video with quality of
15
service (QoS) for both mobile users and nomadic. We can say a channel is
well estimated when its error minimization criteria is satisfied.
Figure 1.6 The block diagram of the channel estimator
The modulated corrupted signal from the channel has to undergo
channel estimation using LMS, MLSE, MMSE, RMS etc. before the
demodulation takes place at the receiver side. A typical channel estimator is
shown in Figure 1.6.
A signal detector needs to know the channel impulse response
characteristics to ensure successful equalization. Note that equalization
without separate channel estimation is also possible (e.g., with linear,
decision-feedback, blind equalizers Haykin.S et al [68].After detection, the
signal is channel decoded to extract the original message.
Source Channel
Encoder Modulato
r
Multipath
Channel
Receiver
Filter Detector Channel
Decoder
Channel
Estimator
Nois
16
Channel estimation is an important technique especially in mobile
wireless network systems where the wireless channel changes over time,
usually caused by transmitter and/or receiver being in motion at some speed.
1.5 ORGANIZATION OF THESIS
The thesis is organized in the form of seven chapters.
Chapter-1 This chapter of the thesis provides the required
introductory concepts on filters. It provides a short overview of the historical
background of the filters and the need for equalization algorithms.
Chapter-2 This chapter reviews the existing literatures on LMS
adaptive equalization algorithm, variable step size LMS algorithm, Sato based
blind equalization algorithm, Godard based blind equalization algorithm also
known as Constant Modulus Algorithm (CMA) and variable step size blind
equalization algorithm. It also discusses the motivation and objectives of the
thesis.
Chapter-3 This chapter explains about adaptive LMS equalization
algorithm. The objective of this chapter is to analyze the performance of this
algorithm in noisy environment with fixed step size and variable step size.
Chapter-4 This chapter explains about blind equalization algorithm
namely Sato based blind equalization and Godard based blind
equalization(CMA). The objective of this chapter is to analyze the
performance of these algorithms in noisy environment with fixed step size.
Chapter-5 This chapter explains about variable step size Sato based
blind equalization algorithm. The objective of this chapter is to analyze the
performance of this algorithm in noisy environment and to compare the results
of this algorithm with existing algorithm for PAM input symbol.
17
Chapter-6 This chapter explains about variable step size Godard
based blind equalization algorithm (CMA). The objective of this chapter is to
analyze the performance of this algorithm in noisy environment and to
compare the results of this algorithm with existing algorithm for PAM input
symbol.
Chapter-7 This chapter of the thesis concludes with a summary of
the outcomes of the research work, augmented with the future research
directions that arise from the investigations that have been carried out.
1.6 SUMMARY
This chapter has highlighted the fundamental concepts of the
communications systems, and the types of noises.
The fundamental concept of design considerations for a filter is
also discussed in this chapter.
The conventional filter, i.e. band limited filter is also discussed.
Outline of the thesis has been provided
The next chapter describes the literature review, knowledge gap
identified and objective of the research.
18
CHAPTER 2
LITERATURE REVIEW
2.1 INTRODUCTION
This thesis considers various aspects of equalization with special
reference to their variable parameter and also their characteristics to provide
the background studies on the issues and to highlight the relevance of the
current study.
This chapter includes reviews of available research reports on
Least Mean Square adaptive algorithms convergence rate,
complexities and its applications.
Variable step size techniques of Least Mean Square algorithm.
Sato based blind equalization algorithms convergence rate,
complexities and its applications
Godard based blind equalization algorithms (CMA) convergence
rate and complexities.
Variable step size techniques of blind equalization algorithms.
2.2 LEAST MEAN SQUARE ADAPTIVE ALGORITHMS
CONVERGENCE RATE, COMPLEXITIES AND ITS
APPLICATIONS
B. Widrow and M. E. Hoff Jr. et.al [81] proposed LMS algorithm,
to obtain optimum filter weights, by adjusting the filter weights in such
manner to converge to the optimum filter weight. The weights are initialized
to zero, and at iterations the weights are revised by estimating the mean square
error gradient.
19
Haykin.S et.al [68] have proposed the LMS algorithm that is
responsive to the scaling of its input, causes disadvantage and leads to trouble
in choosing a step size that guarantees stability of the algorithm. Normalized
Least Mean Squares filter (NLMS) is the modified version of LMS algorithm
which solves this problem by normalizing the power of the input.
Plackett.R.L et.al [57] proposed RLS algorithm which uses the
input as deterministic, whereas for the LMS and other similar algorithm uses
the stochastic input. The RLS exhibits fast convergence compared to most of
its challenger. However, this results in high computational complexity.
After Gauss C.F et.al [21], reinvention of zero forcing equalizer was
done by Robert Lucky et.al [29].Zero Forcing Equalizer which utilizes the
inverse frequency response of the channel and it is employed in modern
communication systems. Veeraruna Kavitha et.al [74] proposed zero forcing
algorithms which are studied greatly for IEEE 802.11n (MIMO). The name
Zero Forcing represents the mitigation of the Inter Symbol Interference (ISI)
to zero in a noise free environment.
The zero-forcing equalizer will amplify the noise to the highest
degree at frequencies f where the channel response H(j2πf) has a very small
magnitude in the attempt to reverse the channel when the channel model is
noisy. A balanced linear equalizer is the minimum mean-square error
equalizer, which does not completely mitigate the ISI effect, but minimizes the
overall power of the noise and ISI components in the output.
Fengqi Yu and Alan N.Willson, Jr. et al [20] have proposed an
interleaved architecture to implement the adaptive lattice algorithm. It is a
technique to overcome the slow LMS convergence problem .But it has high
complexity. So it has not been widely implemented in silicon.
20
Kun-Chien Hung, David W.Lin and Chun-Nan Ke et.al [40]
proposed a variable-step-Size multi-modulus blind decision-feedback
equalization for high-order QAM based on Boundary MSE Estimation. In
their approach, variable-step-size method works in a multistage, gear shifting
fashion rather than employing a continuously varying step size as some other
researchers have considered. Ahmad Tariq Sheikh et.al [3], demonstrated that
SNR is used to switch to Decision Directed mode with little concern of
divergence afterwards.
Feng TONG, Bridget Benson, Ying Li and Ryan Kastner et.al [19]
have proposed channel equalization based on data reuse LMS algorithm for
shallow water acoustic communication. To mitigate the effect of Inter-symbol
Interference caused by multipath propagation, the Data Reuse-LMS algorithm
is integrated with Fractionally Spaced Equalizer-Decision Feedback Equalizer
structure to form an adaptive channel equalizer for the coherent acoustic
communication link.
The Data Reuse algorithm is given by the following equations
Step 1: Initialization: i=0
,0,0, , kkkk WWee
Step 2: Loop While i≤ N-1
ikTkkik WXde ,,
kikikik XeWW ..2 ,,1,
Step 3: Update
1,,1 kkWW Nkk Go to Step1
21
Obviously, N=1 reduces to the classic LMS update. The algorithm
is initialized with zero, and calculating tap weights at each iteration and
updating the same is done until the noise level becomes minimum.
Athar Qureshi, Triantafyllos Kanakis and Predrag Rapajic et al [9]
analyzed the adaptive combining of signals with unequal noise variances. And
it provides uneven step sizes at each branch of combiner by the inclusion of
multiplicative factor of respective channel inverse of noise variance in LMS
algorithm. Their study is very useful for combining the wireless
communication system adaptive signal with unequal noise variance and simple
in computational complexity.
2.3 VARIABLE STEP SIZE TECHNIQUES BASED ON LEAST
MEAN SQUARE ALGORITHMS
Raymond H. Kwong and Edward W. Johnston et al [43] proposed a
variable step size LMS algorithm where the step size is adjusted based on
square of the prediction error. The inspiration is that a large prediction error
will cause the step size to increase to provide faster tracking while a small
prediction error will result in a decrease in the step size to yield smaller
maladjustment. The adjustment equation is simple to implement, and its form
is such that a detailed analysis of the algorithm is possible under the standard
independent assumptions commonly. Here the step size µk is time varying with
its value determined by the number of sign changes of an error surface
gradient estimate. The step size value lies between µmin and µmax to guarantee
the stability of the algorithm.
µmax if µk+1>µmax
µk+1= µmin if µk-1<µmin
µ’k+1 Otherwise
22
where 0<µmin<µmax.
T.Aboulnasr and K.Mayyas et al [1] then proposed a new VSS LMS
algorithm, where the step size is varied based on the square of the time-
averaged estimate of the autocorrelation of e (n) and e (n-1). Therefore, the
algorithm can successfully regulate the step size as in algorithm proposed by
Raymond H. Kwong and Edward W. Johnston at al [43] and while
maintaining the exemption against independent noise trouble. Here the
adaptation step size is adjusted using the energy of the instantaneous error the
weight update equation is given by
W (n+1) =W (n) +µ (n) e (n) X (n)
And the step size update expression is
µ(n+1)=αµ(n)+ e2(n)
Where 0<α<1, >0and µ(n+1)is set to µmin or µmax when it falls below or
above these lower and upper bounds respectively.
Richard.W.Harris, Douglas.M.Chabries and F.Avery Bishop et al
[26] and [27] proposed another variable step size algorithm, where the step
size is adjusted based on the method of steepest descents but utilizes an
independent feedback constant µp for each filter weight in a transversal filter
implementation. The values of each of the feedback constants vary according
to an estimate of the distance to the mean-square-error minimum thereby
providing rapid convergence. The advantages of this algorithm are that only a
modest increase in computation (= 15 percent) over the LMS algorithm is
required while convergence time is reduced, in some instances, by a factor of
50.Although the VS algorithm is similar in many respects to older stochastic
algorithms, a variable step size offers good convergence characteristics with
non-stationary input signals. Here all the data symbols are reconstructed in the
same manner that is step size value adjustment towards the successful
23
reconstruction always starts with maximum value at initial iterations results
faster convergence and geared down to small step size value which results
minimum steady error.
John Mathews and Zhenhua Xie et al [94] and Kying Xiao et al [45]
proposed a stochastic gradient algorithm that overcomes the slow rate of filter
convergence. Here the step size is adjusted based on the negative of the
estimated gradient squared error with respect to the step size. Earlier, the
method was launched by Shin and Lee. Assumptions of their analysis
indicated that the initial choice of the step size value is very important.
However, the steady-state behavior of the adaptive filter depended on it.
Particularly, their analysis predicts that the step size value in steady-state is
always higher than the initial step size value and is a function of the initial step
size. This entails that the steady-state error will be large and will depend on
the initial step size. These statements are incongruous to what has been
observed in practice. Experimental results have shown that these algorithms
have very good convergence speeds as well as small maladjustments,
irrespective of the initial step sizes. Here the step size minimum and maximum
range is chosen to guarantee the stability of the algorithm and minimum steady
error.
0 < µ(n) < 2/3 tr{R}
Where tr{(.)} denotes the trace of the matrix (.) and R is the autocorrelation
matrix of the input vector given by
R= E{X(n) XT(n)}
If µ(n) falls outside between 0 and 2/3 tr{R}, we can bring it inside the range
by setting it to the closest of 0 and 2/3 tr{R}.
If µ(n) falls outside between 0 and 2/3 tr{R}, it should be brought
we can bring it inside the range by setting it to the closest of 0 and 2/3 tr{R}.
24
Boˇ zoKrstaji´ c, LJubiša Stankovi´ c and ZdravkoUskokovi ´ et al
[12] proposed another approach to variable step-size LMS algorithm. Here the
proposed algorithm is implemented for non stationary environments in a
system identification setup.
Thamer M.J. Al-anbaky et al [70] and [71] proposed modified
version of VSSLMS algorithm based on adaptive FIR equalizer. The proposed
algorithms used recursively adjusted adaptation step size based on the
performance surface gradient square.
Thamer M. Jamel et al [70] and [71] have proposed distributed step
size LMS algorithm for adaptive FIR equalizer based on rough estimate of the
performance surface gradient square. Xiong Z et al [83], the adjusted step size
is then distributed among the weights coefficients in exponential form to get
faster convergence and minimum error level in the steady state.
Sayed A. Hadei and Paeiz Azmi et al [65] have proposed a Novel
Adaptive Channel Equalization Method Using Variable Step-Size Partial Rank
Algorithm based on unified approach. The projected adaptive filter is
characterized by its fast convergence speed, and reduced steady state mean
square error in comparison with the ordinary PRA.
Wang Junfeng and Zhang Bo et al [79] proposed Adaptive
Equalizer Based on Variable Step LMS Algorithm. In their study, e (n)
gradually decreases and approaches zero value; µ value changes similar to e
(n).When e (n) = 0, µ = 0. Therefore, monotone and smooth cure of
mathematical function between e(n) and µ can be concluded. The curve is
through origin with µ changing by adjust e(n). It is studied that arc-Tangent
curve is consistent with the variation of step factor. Therefore, variable step
25
size LMS algorithm based on arc-tangent function is called atan-LMS
algorithm.
Shihab Jimaa et al [66] proposed Convergence Evaluation of a
Random Step-Size NLMS Adaptive Algorithm in System Identification and
Channel Equalization. Here the step size is adjusted using random step-size
approach in the adaptation process of the NLMS adaptive algorithm.
AjjaiahH.B.M , Prabhakar V Hunagund , Manoj Kumar Singh and
P.V.Rao et al [5] and [6] proposed Adaptive Variable Step Size in LMS
Algorithm using Evolutionary Programming. The algorithm runs iteratively
and convergence to the optimal step-size which minimizes the steady-state
error rate at each iterations. Vicente Zarzoso et al [75], Initialization for the
step-size value is not required and fittest step size in that generation taken
for that particular iteration. From previous generation a new generation is
created by mutation process, for next iteration and the process will keep
continue until all iteration is completed.
Hong Chae Woo et al (2012) proposed Variable Step Size LMS
Algorithm using Squared Error and Autocorrelation of Error. In this algorithm,
the squared estimation error and the autocorrelation of errors are used for the
step size adjustment to achieve the faster convergence and the robustness.
Emin TUĞCU, Fatih ÇAKIR and Ali OZEN et al [17] proposed A
New Step Size Control Technique for Blind and Non-Blind Equalization
Algorithms. Here, VSS-LMS non-blind equalizer based on cross correlation of
channel output and error signal has been proposed as a solution to the problem
of slow convergence of the fixed step size conventional CMA blind and LMS
non-blind equalizer. Thus, the conflict is removed between the convergence
26
rate and low steady state error of the fixed step-size conventional CMA and
LMS algorithm. Swathi et al [69].
Mashhoor AlTarayrah and Qasem Abu Al-Haija et al [47] proposed
Adaptive Channel Equalization for FBMC Based on Variable Step Size and
Mean-Squared Error. Filter Bank Multi Carrier system used to mitigate the
Inter Carrier Interference (ICI) and Inter Symbol Interference (ISI) and
converts the channel as frequency selective one. As a result complex equalizer
is not required thus simple adaptive LMS equalizer is enough to solve this
issue.
Adam.R et al [2] proposed techniques to identify and equalize the
MC-CDMA channel using the LMS Algorithm and Takagi-Sugeno Fuzzy
System.
U Irusta, S Ruiz de Gauna, J Ruiz, E Aramendi, A Lazkano, JJ
Gutierrez [34] proposed a variable step size LMS algorithm for the
suppression of the CPR Artefact from a VF Signal.
Ajjaiah H.B.M and V. Hunagund et al [5] and [6] have proposed
Variable Step Size of LMS Algorithm Using Partical Swarm Optimization.
Revati Joshi and Ashwinikumar Dhande et al [64] have evaluated
the performance of Least Mean Square beam forming algorithm in the form of
normalized array factor and mean square error by varying the number of
elements in the array and the placing between the sensor elements. Here the
initial weight value is assumed to be zero and the successive corrections of the
weight vector eventually leads to the minimum value of the mean squared
error. The step sizes varied between 0 and λmax. Where λmax is the largest Eigen
value of the correlation matrix.
27
All the above mentioned work has been done based on Least Mean
Square algorithm. The problem with this LMS adaptive equalizer is training
sequence which results in additional bandwidth requirement. This problem can
be overcome by blind equalizer.
2.4 BLIND EQUALIZATION BASED ON SATO ALGORITHM
A Method of Self-Recovering (Blind) Equalization for Multilevel
Amplitude-Modulation Systems was proposed by Yochi Sato et al [88] and it
is a development of LMS algorithm that can also be extended to other
modulation schemes. Application oriented blind equalization for QAM was
proposed by Dominique N. Godard et al [15]. In this, in addition to blind
equalization carrier recovery is also possible.
Vijitha Weerackody and Saleem A. Kassam et al [76] and [77]
proposed a method to accelerate the convergence speed of a blind equalization
algorithm using lattice filters. The incorporated lattice structures in Sato
algorithm results in an increase of convergence rate by an order of magnitude.
Vijitha Weerackody, Saleem A. Kassam and Kenneth R. Laker et al
[76] and [77] proposed a convergence model for the performance analysis of
some blind equalization algorithms. They derived an expression for the Mean
Square Error of the equalizer at iterations in terms of the first and second order
moments of the equalizer taps. Then, recursive relations are derived for the
first and second order moments of the taps with the statistics of the data
sequence, the channel impulse response and the step-size parameter, a, as the
variables.
Rodney A. Kennedy, Brian D.O. Anderson, Zhi Ding, and
C.Richard Johnson Jr et al [16] proposed method to find a local stable
28
minimum in Sato based blind equalization by using recursive identification
scheme.
Zhi Ding, Rodney A. Kennedy, Brian D. O. Anderson and C.
Richard Johnson, Jr. et al [16] analyzed the local convergence of the Sato
blind equalizer and generalizations under practical constraints. From the study,
it is observed that center tap initialization is not sufficient to avoid such slow-
convergence. They have also provided simulation proof for slow-convergence.
Wolfgang H. Gerstacker, Robert F.H. Fischer, and Johannes B.
Huber et al [82] proposed blind equalization for digital cable transmission
with Tomlinson-Harashima precoding and shaping. The study includes a
designed fixed precoder for a cable with typical characteristics. Since self-
recovering equalization is unfeasible for a system with THP, authors proposed
a new joint precoding/shaping scheme called dynamics shaping, which
restricts the dynamics of the effective data sequence while retaining power
efficiency, as long as the design parameters, such as maximum absolute value
of the effective data sequence, are properly selected.
Timoleon Vaidis and Charles L. Weber et al [73] proposed chunk
adaptive techniques for channel detection and data demodulation over band-
restricted channels. They have also projected a new way to implement the
Viterbi algorithm (VA) for maximum-likelihood data sequence estimation
(MLSE) in a known channel environment and utilize it to derive chunk
adaptive techniques for joint channel and data estimation, when the channel-
impulse response (CIR) is unknown.
Gi Hun Lee and Rae-Hong Park et al [23] projected blind
equalization scheme for QAM input based on shell partition Gi Hun Lee et al
[23]. Here shell boundaries are identified by maximum likelihood (ML)
29
evaluation. The shell partition-based Joint Blind Equalizations also are
constructed by replacing the DD algorithm by the proposed shell partition-
based algorithms. To improve the performance of the JBE, the proposed
SPCMA replaces the DD mode of the JBE, resulting in the JDE-SPCMA that
concatenates the CMA with a SPCMA. Also the shell partition-based hybrid
algorithm (SPHA) that combines the SPCMA and the SPGSA replaces the DD
algorithm to generate the JBE-SPHA.
Heinz Mathis and Scott C. Douglas et al [30] proposed blind
deconvolution of Impulsive signals using a modified Sato algorithm. Here,
authors have provided a theoretical explanation as to why Buss gang-type
algorithms fail to deconvolve impulsive signals from their filtered
measurements. Then, they projected a novel adaptation of the Sato algorithm
to enable it to deconvolve impulsive signals with the following advantages
that it has less computational complexity, avoid signal prewhitening, and
require only multipliers and adders to implement.
Vinod Sharma and V. Naveen Raj et al [78] analyzed the
Convergence and Performance of Godard Family and Multimodulus
Algorithms for Blind Equalization. Godard family includes Sato and Constant
Modulus algorithm.
Muhammad Lutfor Rahman Khcan, Mohammed H.
Wondimagegnehu and Tetsuya Shimamura et al [51] proposed amplitude
banded Sato algorithm for blind channel equalization. Here in the signal
reconstruction process, Sato equalizer uses a linear adaptive LMS algorithm.
In recent times, Shimamura et al derived a new Least Mean Square based non-
linear adaptive algorithm, called amplitude banded LMS algorithm, which
considers the amplitude information of the channel output in the coefficient
adaptation process of the equalizer. The ABLMS algorithm exhibits better
30
performance than the conventional LMS algorithm. Then later Amplitude
Banded Godard and Amplitude Banded Sato algorithms have been proposed
for blind channel equalization. They have demonstrated that the Amplitude
Band Godard and Amplitude Banded Sato algorithms which performs better
than the existing CMA and Sato based blind equalization algorithms,
correspondingly, for simple communication channel equalization. The
Amplitude Banded Sato algorithm works more precisely than the Amplitude
Banded Godard algorithm, and the increased division number and the use of
parallel structure improve the performance of the ABSato algorithm further.
2.5 BLIND EQUALIZATION BASED ON GODARD
ALGORITHM (CMA)
Langford B. White et al (1996) proposed blind equalization of
Constant Modulus signals using an adaptive observer approach. Here the
problem of blind equalization of constant modulus signals that are corrupted
by frequency selective multipath signal propagation and additive white
Gaussian noise. The method adaptive observer is utilized to adjust the tap
weights of an FIR equalizer with the aim of to reinstate the signal's constant
modulus property. In order to guarantee local stability, the nonlinear observer
gain is selected using fake algebraic Riccati methods.
Nikhil Deshpande et al [96] studied fast recovery equalization
techniques for DTV Signals. Here the authors considered Godard’s algorithm,
Sato’s algorithm, G-Pseudo Error algorithm utilizing Sato’s cost function and
G-Pseudo Error algorithm using Godard’s function. Their analysis includes
these algorithms for 8 VSB transmission format and recommends the most
suitable algorithm. The performance comparison parameters include MSE,
convergence characteristics, computational load and accuracy of estimating
the channel.
31
W. Pora, J.A. Chambers and A.G. Constantinides et al [56]
proposed variable step size equalization for fast-fading channels using the
combination of Kalman filter and constant modulus algorithm. The constant
modulus algorithm is not capable to track the time-variations accurately for
the reason that the magnitude of the received corrupted signal changes too
quickly. For the time-varying channels the Kalman filter suits well to track the
characteristics of the channel although needs training sequence for successful
reconstruction of information. As a result, a combination of CMA and KF
algorithm is proposed with the aim of to make use of the advantages of both
algorithms. The connected step sizes of the CMA and the KF algorithm are
also adjusted based on the magnitude of the output.
Kutluyıl Do˘ gan¸cay and Rodney A. Kennedy et al [16] proposed
least squares approach to blind channel equalization. He has done closed-form
derivation for the LS Solution and extracted the equalizer parameters. His
proposal results in high computational complexity if a long equalizer is used
because of pseudo inversion of a large matrix. The matrix inversion problem is
dealt by descent and recursive methods. The applications where only short
channel output observations are enough for those this algorithm may be better
choice. The Godard algorithm generally require long channel output
observations to converge to an open- or closed-eye parameter setting
depending on the filter tap coefficient initialization.
Thomas J. Endres, Samir N. Hulyalkar, Christopher H. Strolle, and
Troy A. Schaffer et al [72] proposed low-complexity and low-latency
implementation of the Godard/CMA update. Here the chip area and signal
latency are both significantly reduced by not using any multipliers. One
approach uses region-based quantization and the other uses decision-directed
32
CMA update term. The quantized error term is calculated using a look-up
table in place of costly multipliers and adders.
Guo Li Li Ning' Guo Yan Zhou Jiongpan et al [25] proposed the
special step size Constant Modulus algorithm Convergence Behavior. Here
they compared the performance of CMA1-2 and CMA2-1 algorithms
convergence behaviors based on special step size. By selecting certain step
size, which is decided by the singular value of the input data, certain signal
could be instantly detached from the output.
Xi-Lin Li and Xian-Da Zhang [84] studied a family of generalized
Constant Modulus algorithms for blind equalization and proposed a family of
Generalized Constant Modulus algorithms. Fascinatingly, this class contains
not only the well-known CMA, but also the recently proposed Sign Godard
algorithm, Square Contour algorithm, Generalized Square Contour algorithm,
and Sign Square Contour algorithm as special examples. Additionally, the
novel generalization cost function estimates other new equalization
algorithms, such as the extended Constant Modulus algorithm here, which is
capable of execute ISI elimination and carrier-phase recovery at the same
time. From their simulations, the ECMA algorithm shows quick initial
convergence, and can keep away from the deteriorate diagonal solutions and
undesired rotated solutions simultaneously.
Naveed R. Butt and L. Cheded et al [52] studied an improved CMA-
based hybrid algorithm for blind channel equalization that overcomes the two
main limitations of the conventional CMA. The proposed hybrid algorithm is
capable of tracking dispersive channels with a faster convergence than any of
its CMA-based counterparts.
33
Ram Babu. T and P.Rajesh Kumar et al [62] studied blind channel
equalization using Constant Modulus algorithm. The proposed cost function is
relating the deformation of the Constant Modulus error surface. They have
predicted a linear channel model driven by a QAM, PAM, BPSK source and
become an adaptive filter using CMA based on BERGulator Simulator Tool.
They have characterized the frequency response with the help of BERgulator
tool, Impulse response analysis and channel options. BERgulator plots the
error histories with magnitude & directional coefficients.
Shafayat Abrar and Asoke K. Nandi et al [67] studied a multi
modulus approach for blind equalization of square-QAM Signals. From the
existing function new cost function derived by including some modifications.
Resulted two novel, generic and efficient multi modulus families of blind
equalization algorithms for use in higher-order quadrature amplitude
modulation based digital communication systems. Evaluation of equalizer gain
and dynamic convergence has been described in detail.
Athanasios Vgenis, Constantinos S. Petrou, Constantinos B.
Papadias, Ioannis Roudas, and Lambros Raptis et al [8] studied nonsingular
Constant Modulus equalizer for PDM-QPSK Coherent Optical Receivers. The
adaptive filters using the constant modulus algorithm frequently converge to a
singular coefficient matrix that produces the same signal at multiple outputs.
They have dealt this issue in the context of optical communications systems
with polarization-division multiplexing and coherent receivers.
Yangyang Fan, Xue Chen, Weiqin Zhou, Xian Zhou, Hai Zhu [86]
compared the Constant Modulus algorithm and LMS Equalization algorithms
for optical coherent receivers. They have considered that these algorithms are
studied to select the optimal adaptive algorithm for electrical dispersion
equalizer in optical coherent receivers at 100Gbps and concluded that CMA is
34
better than the LMS according to the comparison from equalization
performance and computation complexity.
A.B. Djebbar et al [17] proposed least square fitting-Constant
Modulus algorithm based blind equalization algorithms for multiuser multi-
carrier code division multiple access systems under Rayleigh multipath fading
channel. By combining a least square fitting and constant modulus algorithm
criteria the study was completed and exhibited the robustness and
effectiveness of this proposed algorithm.
Amin Mohamed Nassar and Eng. Waleed EI Nahal et al [7]
proposed adaptive blind equalization technique to enhance the constant
modulus algorithm performance. The Step-size is exponentially weighted for
Recursive Least Squares Constant Modulus Algorithm, based upon the
combination between the Exponentially Weighted Step-size Recursive
Least Squares algorithm and the Constant Modulus Algorithm, by
providing several assumptions to obtain faster convergence rate to an
optimal delay where the Mean Squared Error is minimum, and so this
selected algorithm can be implemented in digital system to improve the
receiver performance.
2.6 VARIABLE STEP SIZE TECHNIQUES FOR BLIND
EQUALIZATION ALGORITHMS
Khurram Shahzad, Muhammad Ashraf and Raja Iqbal [39] proposed
Variable Step Size Blind Equalization Scheme using Constant Modulus
Algorithm.
Doaa Ashmawy, Kevin Banovic, Esam Abdel-Raheem, Mohamed
Youssif, Hala Mansour and Mahmoud Mohanna et al [10] proposed a variable
35
step size equalization algorithm to speed up convergence based on
combination of Modified Constant Modulus algorithm and Decision Directed.
The same technique is used with joint CMA and DD algorithm and exhibits
improved performance.
Wei Xue, Xiaoniu Yang, and Zhaoyang Zhang et al [80] proposed a
variable step size modified constant modulus algorithm (VSS-MCMA)
Chahed et al [14]. The step size of the algorithm is adjusted according to the
region where the received signal lies in the constellation plane. The VSS-
MCMA can obtain fast convergence rate, small steady state MSE and the
recovery of the phase rotation and frequency offset. The simulation results for
16-QAM signals demonstrate the effectiveness of the VSS-MCMA in the
equalization performance enhancement.
In Figure 2.1 , when the output of the equalizer lies outside of the
region Di a larger step size is chosen, and when the output of the equalizer lies
in the region Di a smaller step size is chosen. Then the variable step size
scheme can be written as
µ= µ0 if y(k) Є UDi
µ1 if y(k) Є Di
Where µ0 > µ1
36
Figure 2.1 The region of Di of variable step size for 16-QAM signal
Baofeng ZHAO and Jia LIU et al [11] studied the influence of MSE
on tracking channel and anti-interference of a Variable Step-size CMA. Here
the convergence speed and convergence precision are contradictory for step-
size.
Roozbeh Hamzehyan, Reza Dianat, and Najmeh Cheraghi Shirazi et
al [67], proposed a variable step size blind equalization algorithm based on
MCMA that could automatically adapt the step size depending on whether the
current equalizer output is in decision circle or not. The effective adaption step
size and the radius of decision circle were made continuously variable and
decreasing with the decrease in the output error. Such characteristics are
beneficial to attain fast convergence speed and low steady-state mean-square
error in equalization process. Here step size value is chosen based on where
the symbols fall on the plot. If more than 90 percent of symbols fall within the
decision circle and step size parameter are adapted with Ri+1= µR Ri (0 < µR <
1) and λi+1= µλ λi (0 < µλ < 1) respectively. This procedure continues until Ri <
0.4Dmin. After this point algorithm switches to LMS algorithm.
37
Merve Abide Demir and Ali Ozen et al [29] proposed a variable
step size algorithm to solve the slow convergence issue. Here the step size is
adjusted based on error autocorrelation. It is applied to time domain channel
equalization of a single carrier.To improve the performance of the blind
equalization algorithm, an outstanding way is to formulate the step size
variable rather than fixed, That is, choose large step size values during the
initial convergence of the blind equalization algorithm, and use small step size
values when the system is close to its steady state or select a large step size
value in transient phase and a small step size value in steady state noise level.
Iorkyase, E.Tersoo and Michael.O et al [33] proposed blind
adaptive equalization algorithm to improve the rate of convergence for fast
time varying digital communication systems. Here the step size value is
adjusted between minimum and maximum range.
µmax, if µ(n)> µmax
µ(n)= µmin, if µ(n)< µmin
µ(n), Otherwise
Where condition of 0 < µmin < µmax must be satisfied. The initial
value of variable step size µ(0) is according to the upper bound constant µmax.
Radhakrishna.Y and T. Ravi Kumar Naidu [61] proposed extended
variable step size Constant Modulus algorithm to improve the convergence
rate for noise colorings with large Eigen value spreads.
Ying Xiao and Fuliang Yin et al [42] proposed variable step size
blind equalization based on sign gradient algorithm under impulse noise
environment. The signum operation on the iterative gradient can suppress the
impulse noise effectively, which ensures the blind equalization algorithm to
obtain robust convergence performance. Furthermore, a variable step size
algorithm is designed according to the iterative times and the reliability of the
output signal without man-made parameters setting to improve the
performance of sign gradient algorithm.
38
Thamer M. Jamel,Mohammed Abed Shabeeb and Abed AL–Abass
Muhseen Jassem et al [71] proposed a hybrid variable step-size MCMA Blind
Equalizer algorithm for QAM signals. Here the step size of the algorithm is
varied based on to the combined absolute difference error with iteration
number. The proposed algorithm can obtain both fast convergence rate, and a
small steady state MSE compared with traditional MCMA and other variable
step size MCMA. Here µ (k+1) bounded between two values µmax and µmin as
µmax, if µ(k+1)> µmax
µ(k+1)= µmin, if µ(k=1)< µmin
µ (k+1), Otherwise
2.7 THE KNOWLEDGE GAP IDENTIFIED FROM THE
EARLIER INVESTIGATIONS
The literature survey presented reveals the following knowledge
gap:
Much work has been done on Least square algorithm with fixed step
size and variable step size least algorithm. The sole limitation of the
trained adaptive equalizers is the occupancy of additional
bandwidth which is overcome by blind equalization.
Limited work has been done on Sato based blind equalization
because it recovers only single carrier.
Godard based blind equalization recovers dual carrier so much work
has been done on this algorithm.
Since blind equalizers do not require training sequence to track the
time varying characteristics of the channel it results in slow
convergence. To solve this issue some variable step size techniques
has been proposed.
39
2.8 AIM OF THE RESEARCH WORK
The Inter Symbol Interference is the major issue in wireless
communication systems and it is effectively mitigated by
equalizers. Trained equalizers may require additional bandwidth to
do the same compared with blind equalizer. However, blind
equalizers performance is limited by slow convergence rate.
Therefore the ultimate aim of this work is to design an efficient
blind equalizer with fast convergence rate for digital high speed
wireless communication systems.
2.9 OBJECTIVES OF THE RESEARCH WORK
The objectives of the research are:
To analyze the performance of adaptive LMS algorithm and speed
up the convergence rate by variable step size adaptive LMS
algorithm for PAM symbol.
To analyze and compare the performance of adaptive LMS
algorithm and Sato based blind equalizer for PAM symbol.
To analyze and compare the performance of Sato based blind
equalizer and Godard based blind equalizer for PAM symbol input.
To solve the slow convergence rate issue by proposing variable step
size techniques for Sato based blind equalizer for PAM symbol
input.
To solve the slow convergence rate issue by proposing variable step
size technique for Godard based blind equalizer (CMA) for PAM
symbol input.
40
2.10 SUMMARY
This chapter has provided
An exhaustive review of research works on trained adaptive least
mean square algorithm and blind equalization algorithms based on
Sato and Godard proposals reported by the previous investigators
The knowledge gap from the earlier investigations
The objectives of the present work
The next chapter explains the adaptive LMS equalization algorithm
and its performance in noisy environment with fixed step size and variable
step size.
41
CHAPTER 3
ADAPTIVE EQUALIZER
3.1 INTRODUCTION
Equalization is an important problem in digital high-speed
communication systems and it has received a great amount of attention. In
high-speed wireless communication systems, equalization process is needed to
suppress the inter symbol interference caused by multipath channels.
Conventional equalization techniques use training signals. When wireless
channels, especially in fast variant mobile channels, the training symbols must
be sent frequently. As such, a lot of band width will be occupied by the
training symbols.
The training sequence allows the adaptation of the equalizer
parameters to minimize some error measurement between the actual equalizer
output and the desired response. When a linear filter is used to implement the
equalizer, there are many adaptive algorithms that can be used to adapt the
filter weights, for example the well-known LMS.
Conventional equalization requires transmitting a training sequence
that is known at the receiver. A replica of this sequence is made available at
the receiver, which is synchronized with the transmitter, thereby making it
possible for adjustments to be made to the equalizer coefficients in accordance
with the adaptive filtering algorithm employed in the equalizer design. When
the training is completed, the equalizer is switched to its decision–directed
(DD) mode, and normal data transmission may then commence.
42
3.2 EQUALIZER AND ITS OPERATING MODES
In a simple sense, equalizer is the special kind of filter. The
conventional band limited filter fails, if it receives,
In-band Noise (Noise with same frequency as that of signal band)
ISI.
These complications at the receiving signal can be normalized by
the a special kind of filter, i.e. by the equalizer. In other words, equalizers are
also called compensator of ISI. The basic symbolic block diagram of
equalizing filter is shown in Figure 3.1,
Figure 3.1 Generalized Block Diagram of Equalizer.
The original information can be an image, data or video transmitted
towards the receiver and there may the possibility for noise occurrence when it
travels through the medium. So, the corrupted information is received at the
receiver side and it is passed through the equalizer where in feedback
Noise
Signal
Equalizer
Equalizing Algorithm
Equalized Signal
43
equalization algorithm is employed. At iterations it compares the original
information with the desired information and estimates the difference between
the same and slowly it removes that error. Finally the equalized signal comes
as an output of the filter.
The general operating modes of an adaptive equalizer are,
Training Mode:
A known, fixed length training sequence is sent by the transmitter
so that the receiver’s equalizer may adapt to a proper setting for minimum Bit
Error Rate (BER) detection.
Tracking Mode:
Following this training sequence, the user data is sent, and the
adaptive equalizer at the receiver utilizes a “recursive algorithm” to evaluate
the channel and estimate the filter coefficients to reduce the distortion.
Example: Least Mean Square Adaptive Equalizer.
Disadvantage:
For time varying channels or non cooperative environment, training
sequence must be sent frequently, which utilizes more Bandwidth.
On the basis of its functionality, equalizers are sub-divided into two
categories. They are,
Blind equalizer,
Adaptive equalizer.
Even if the training sequences are sent by the transmitter, all the
complicated noises can be cancelled by a special type of equalizer called the
blind equalizer. This is the core subject of this work, and is analyzed at
44
Chapter 4 and 5. If the reference input sequences are provided, then the
filtering job can be done by using adaptive equalizer. This chapter deals the
second option. It mainly concentrates on the Least Mean Square (LMS)
algorithm, which is the only dominating adaptive equalizer algorithm. LMS
algorithm is also the basis for all other algorithms in whole equalizer arena.
3.3 ADAPTIVE LEAST MEAN SQUARE EQUALIZER
The concept of adaptive equalizer was introduced in 1965 by the
researcher Lucky, on the topic “Peak Distortion Criterion”. Latter, it was
famed with “Zero Forcing Algorithm”. But in practice it didn’t give
satisfactory result. Thus, the most successful approach, i.e. the LMS algorithm
came forward. This, not only succeeded, but also dominated the whole field.
In this section, the standard Least Mean square (LMS) adaptive equalizer
algorithm have been discussed.Figure.3.2shows the structure of an adaptive
least mean square algorithm that encompasses the matched filtering action.
The principle of this algorithm is to regulate the equalizer tap coefficients with
the reference to the required response.
3.3.1 Basic Concept
In mathematics, if we differentiate any function and equate it to
zero, then this gives the minimum of the function. LMS algorithm utilizes this
philosophy of mathematics. It takes the mean square of the error function and
differentiates it at iterations. Every time we do the iteration, the error get
reduced, and at last to zero. The zero error is nothing but the output sequence
has become identical to that of the input reference sequence. Since the output
sequence has tried to adapt to the input reference sequence, this kind of
equalizer is called the adaptive equalizer. The incoming signals to the receiver
are always contaminated by the noises, especially channel noise and ISI.
Figure 3.2 Generalized
The output y[n] of the tapped delay line equalizer corresponding to the input
sequence{x[n]} is outlined by the discrete convolution,
Where wk is that the k
within the equalizer.
The tap weight represents the adaptive filter coefficients. At each
sampling instant, the error is determined by comparing the specified response
that is understood at the receiver and therefore the actual response. With the
assistance of obtained error, t
N
kkwny
0
][
Generalized diagram of Equalizer with N taps
The output y[n] of the tapped delay line equalizer corresponding to the input
sequence{x[n]} is outlined by the discrete convolution,
is that the kth tap weight, N+1 is that the total range of taps present
The tap weight represents the adaptive filter coefficients. At each
sampling instant, the error is determined by comparing the specified response
that is understood at the receiver and therefore the actual response. With the
assistance of obtained error, the filter will estimate the direction during which
knx ][
45
with N taps
The output y[n] of the tapped delay line equalizer corresponding to the input
(1)
tap weight, N+1 is that the total range of taps present
The tap weight represents the adaptive filter coefficients. At each
sampling instant, the error is determined by comparing the specified response
that is understood at the receiver and therefore the actual response. With the
he filter will estimate the direction during which
46
the tap weights ought to be modified and therefore the adaptation is also
achieved within the adaptive filter. Let a[n] denote the known response of nth
transmitted binary symbol. The distinction between the specified response a[n]
and therefore the actual response y[n] of the equalizer denote the error signal
e[n], and expressed as e[n] = a[n] – y[n]. In LMS algorithm, individual tap
weights change to each iteration are controlled by the obtained error value
e[n]. LMS algorithm is expressed as follows.
Let µ denote the step size parameter of the filter. From the figure.3.2, the input
signal given to the kth tap weight at time step n is x[n-k]. Hence, using
as the previous value of the kth tap weight at time step n, the updated value of
this tap weight at time step n+1 is calculated from the equation (2), defined
by,
(2)
Where,
k=0, 1,…..,N
(3)
The adaptive least mean square algorithm represented by equation (2) and (3).
signal
Error*
weight
tapk
toapplied
signalInput
*parameter
size Step
weight
tapk
of
valueOld
weight
tapk
of value
Updated
ththth
)(^
nw k
][][][]1[^^
neknxnwnw kk
N
k
k knxnwnane0
^
][][][][
47
3.4 PSEUDOCODE OF VARIABLE STEP SIZE LEAST MEAN
SQUARE EQUALIZER
X=PAM Symbol
ISI=X with five reflections (Different amplitude with
different delay)
Y=X+ISI
Snrvalue=30
Received Symbol=AWGN(Y, 25)
Received Symbol = PAM symbol 1, PAM symbol 2…, PAM symbol N
Tap weights are initialized with zero (C1, C2, C3, C4 and
C5=0)
Step size µ=0.015
Equalizer input (input1) = corrupted PAM symbol1
V=1;The iteration procedure for first symbol
reconstruction:
Loop 1: while (V<Snrvalue)
C1=C1+delta.*error.*(input1);
Estimating C2, C3, C4 and C5
Calculating the output difference values at three different
sampling points
at iterations
If (output difference < 0.001)
Reconstruction is stopped and updated step size and tap
weights are used as commencing value for subsequent symbol
reconstruction
Otherwise
Step size is adjusted in either linear or non linear
fashion
Count=Count+1;
V = 20*log10 (norm (newy1 (:)) /norm (newy1 (:)-out1
(:)));
End
The Iteration Procedure for subsequent symbol
reconstruction:
Updated step size and tap weights are used as commencing
value for subsequent symbol reconstruction
Go Loop1
48
3.5 VARIABLE STEPSIZE LMS EQUALIZER
The flowchart of the proposed algorithm is shown in Figure 3.3.In
adaptive LMS equalization algorithm the initial value of the tap parameter (µ)
is chosen between the minimum and maximum values and these values are
finite to confirm the convergence and stability of the algorithms. The fixed
µmax is chosen with respect to the stability condition of the algorithm, while
µmin is chosen to confirm desired steady-state performance. In the proposed
novel approach the tap parameter value starts with 0.015 (optimum value
identified for LMS algorithm by R.Perry et al [55] to reconstruct the very first
symbol and this value is incremented by small constant (s) for each iteration.
The output difference between successive iterations is calculated
and this value is used to stop the iteration process for reconstruction of very
first symbol. The updated tap parameter value is chosen as beginning value for
subsequent symbols and using this value the iteration for reconstruction of
very first symbol is stopped. The updated tap parameter value is chosen as
beginning value for subsequent symbols. In the reconstruction of the
subsequent symbols µ value is decremented by a small constant at iterations
and, the iteration is stopped when the µ value become minimum. For the first
symbol estimation, the specified SNR can be achieved by changing the output
difference value to stop the iteration and for subsequent symbols estimation;
by changing µmin value the specified SNR output can be obtained.
49
Figure 3.3 Flowchart for Variable Step Size LMS Algorithm.
No
Yes
Start
Initialization Tap weight initialization (near to zero)
Previous iteration =0 Step Size (µ) =0.015
Compute equalizer output
Calculate output difference =Current iteration value-Previous iteration value
Previous iteration=Current iteration
Output difference<0.001
Or< -60dB
µk+
1 = µ
k + factor (S
) Reconstructed Symbol
1
No
Yes
Compute equalizer output
Calculate output difference =Current iteration value-Previous iteration value
Previous iteration=Current iteration
µ<=0.015
µk+
1 = µ
k ─ factor(S
)
To reconstruct subsequent symbol Use updated tap weight from previous symbol.
Optimized µvalue from first symbol is considered.
Reconstructed Subsequent
Symbols
50
3.5.1 The Channel Model
The channel model is shown in Figure 3.4.Consider a pulse
amplitude modulated symbol ak passes through a channel with memory C(z)
introducing inter symbol interference (ISI) and the resultant bk is corrupted by
Additive White Gaussian Noise (AWGN) nk resulting in the received sequence
rk. An equalizer D(z) is to be used to mitigate the ISI. The cascaded channel
and equalizer is denoted as G(z).The transmitted symbol can be estimated
from the equalizer output symbol yk using symbol by symbol decision device
and is given by
1
0
. k
l
iik rcy
Where Ci is the tap weight coefficient, l is the length of the equalizer and rk is
the noisy observation of the channel.
Figure 3.4 The Channel Model.
3.5.2 Simulation Results
Simulations results of trained LMS adaptive algorithm and variable
step size LMS algorithm are compared. First, the PAM symbol is generated
with the amplitude of 2Volt ( i.e. -1V to +1V) and the duration of 0.2 seconds
(approximately) as shown in Figure 3.5. It is mixed with five reflections of
yk rk
nk
bk ak
G(z)
C(z) D(z)
51
same symbol but with different amplitude and time delay (because, it is
considered as the worst case of ISI) is shown in Figure 3.6. Again, it is mixed
with the Additive White Gaussian Noise of 25dB SNR is shown in Figure 3.7.
The resultant waveform is chunked in to 0.0025 seconds symbols called as
received PAM symbol 1, PAM symbol 2, and so on is shown in Figure 3.8.
These PAM symbols are processed by equalizer and the original symbols are
reconstructed. The entire work carried out in similar fashion. Here PAM
symbol 3 is reconstructed in sequential manner (i.e after reconstruction of
PAM symbol 1 and PAM symbol2).
Figure 3.5 The original PAM symbol.
Figure 3.6 The ISI model for original PAM symbol.
52
Figure 3.7 Additive White Gaussian Noise with SNR =25dB
Figure 3.8 The original PAM symbol with noise
PAM Symbol 1
PAM Symbol 2
PAM Symbol 3
PAM Symbol 4
PAM Symbol 5
53
The PAM symbol 3 is shown in Figure 3.9, The five reflections with
relative amplitude (0.7, 0.6, 0.5, 0.3 and 0.1) are shown in Figure 3.10. PAM
symbol 3 with ISI and Additive White Gaussian Noise (AWGN) with 25dB
SNR is shown in Figure 3.11, which is taken as the input to the equalizer. The
equalizer has been implemented by a linear transversal filter with a five tap
complex circuitry. The sequence of PAM symbols are generated from the
source and symbol by symbol processing is done at the equalizer. Simulation
results are shown for PAM Symbol 3.
Experiment1: Constant µ (µconstant)
Small µ value results minimum steady state error but results in slow
convergence. High µ value will speed-up the convergence however lead to
maximum maladjustment. The variable tap parameter value is restricted to the
range [µmin=0.015, µmax=0.25] to guarantee stability of the algorithm.
Figure 3.9 The PAM Symbol 3.
Figure 3.10 The ISI model for PAM Symbol 3.
54
Figure 3.11 PAM Symbol 3 with ISI and AWGN Noise
The waveforms are shown in Figure 3.12, Figure 3.13 and, Figure
3.14 and Figure 3.15are the results of simulations for the simulated equalizer
output for PAM symbol 3 after first iteration, Reconstructed Symbol 3 after
completion using LMS algorithm with SNR 30dB, Reconstructed symbol 3 by
after completion using variable step size LMS approach with SNR 30dB
respectively. and The MSE comparison between LMS and variable step size
LMS approaches is shown in figure 3.11.respectively. In this Figure. 3.9 and
Figure 3.10 seems to be identical because both are reconstructed with same
SNR 30dB however number of iterations differs. Table 3.1 shows the number
of iterations taken by LMS algorithm, with different SNR values for the
reconstruction of symbols 1, 2 , 3, 4 and 5 using step size parameter µ= 0.015.
Experiment 2: µ with linear increment and decrement (µlinear)
For the above mentioned input, the parameter value chosen as 0.015
and tap weights are initialized with center tap (only center tap has value one
and others are near zero). The very first symbol is reconstructed by using
linear increment in µ. i.e., µ is incremented by constant factor (s=0.02) for
every iteration as
µk+1 = µk + s; where s=0.02
55
The output difference between the current iteration and the former
iteration is calculated at in three different sampling points. If all the three
output values are less than 0.001 (based on experimental analysis, output
difference value is chosen to be 0.001 ),the iteration for reconstruction of very
first symbol will be stopped. The updated µ(=0.51) is fixed as optimum or
starting value for subsequent symbol. In the reconstruction of the following
symbols the µ is decremented by a the same factor (say 0.02) at each iteration
as
µk+1 = µk – s; where s=0.02
When µ reaches 0.015, the iteration is stopped. On trial and error
basis, for constant µ input the optimum µ value is 0.015. But in the proposed
approach, it is found that µmax fixed at 0.51 and with updated tap weights,
consequent symbols were reconstructed with 30dB SNR in fewer iterations
with stability. While giving µ as 0.51 with center tap initialization for
experiment 1 ends up in maladjustment and hence subsequent symbols cannot
be reconstructed.
The results for the tap adjusting coefficient value (µ) equal to 0.015)
to reconstruct the PAM signal are shown in table 3.1. For a variable step size,
we get better convergence as shown in table 3.2. The Simulation results show
that proposed algorithm has quicker convergence rate compared to that of
LMS algorithm. That is, the number of iterations to obtain the same output
SNR for identical symbol is much lesser within the LMS approach with
µ= 0.015.
56
Figure 3.12 The Equalizer output for PAM Symbol 3 after 1st iteration
Figure 3.13 Reconstructed Symbol 3 using LMS algorithm and
SNR = 30dB (3708 iterations).
Figure 3.14 Reconstructed Symbol 3 using Variable Step Size LMS
algorithm and SNR = 30dB (19 iterations).
57
Figure 3.15 Mean Square Error comparison between LMS and proposed
VSS LMS approach.
Table 3.1. SNR vs. Iterations for LMS Adaptive Equalizer with Step Size
Parameter µ= 0.015
SNR
in dB
(Output)
Number of Iterations
Symbol 1 Symbol 2 Symbol 3
10 353 407 192
15 907 1445 514
20 1903 4029 1163
25 3995 7101 2389
30 6799 9728 3708
58
Table 3.2. SNR vs. Iterations for LMS Adaptive Equalizer with Step Size
Parameter µ=0.25
SNR
in dB
(Output)
Number of Iterations
Symbol 1 Symbol 2 Symbol 3
10 8 9 4
15 22 34 12
20 48 135 25
25 90 197 53
30 143 259 97
Table 3.3. SNR vs. Iterations for LMS Adaptive Equalizer with Variable
Step Size
SNR
in dB
(Output)
Number of Iterations
Symbol 1 Symbol 2 Symbol 3
10 9 8 3
15 17 24 6
20 28 37 11
25 40 74 15
30 54 95 19
3.6 SUMMARY
In this work, we compared the performance of LMS adaptive
equalizer with fixed step and variable step size LMS equalizer. Observations
from table 3.1, table 3.2 and table 3.3 show that, the specified SNR will be
obtained with less number of iterations in VSS LMS equalization by selecting
best µ value. The disadvantage of adaptive trained equalizer is additional
bandwidth occupancy for training sequence. In order to overcome this issue
blind equalization concept is suitable. The next chapter explains about blind
equalization concept and it is compared with trained adaptive LMS equalizer.
59
CHAPTER 4
BLIND EQUALIZER
4.1 INTRODUCTION
In the previous chapter, the equalizer, which requires a training
sequence, has been explained. There are many communication systems in
which transmitting a training sequence may not be possible. In such cases, a
blind equalizer will be employed. The simple definition of filter, “to limit the
spectrum of the signal to some band of frequency”, would not be sufficient to
define the equalizer. Because, it does more than that.
The throughput of the system drops due to the time slots occupied
by the training signal. Fast time varying channels need training sequence more
often to train the equalizer. This results in more reduction of the throughput of
the system. Another disadvantage is that the training signal is not known at the
receiver, e.g., in an exceedingly non cooperative (surveillance) surroundings.
Even though trained strategies have such disadvantages, they are typically
adequate.
The Blind algorithms exploit characteristics of the transmitted
signals and do not need training sequences. They are called so because they
provide equalizer convergence without burdening the transmitter with training
overhead. These fashionable algorithms are able to acquire equalization
through property restoral techniques of the transmitted signal. In general,
even if the initial error rate is high, the blind equalization technique directs the
coefficients towards the optimum filter parameters. A Blind Equalizer is in a
position to compensate amplitude and delay distortions of a communication
60
channel using the channel output samples and the data of the basic statistical
properties of the information symbols. The key advantage of blind equalizers
is that there's no training sequence to calculate the tap weight coefficients; thus
no bandwidth is wasted by its transmission. Blind equalization is effective for
a high-speed digital radio, digital mobile communication systems, multi-point
networks, cable TV, and digital terrestrial TV broadcasting.
The major downside is that the equalizer can usually take extended
time to converge as compared to a trained equalizer. The necessity for blind
equalizers in the field of information communications is mentioned by
Godard, within the context of multipoint networks. Blind joint equalization
and carrier recovery might realize application in digital communication system
over multipath weakening channels. Moreover, it's applied in extremely non-
stationary digital mobile communications, wherever it's impractical to use
training sequences. These techniques embrace algorithms like the SATO
algorithm and Constant modulus algorithm (CMA) Meng Zang et al [49].
4.2 IMPORTANCE OF BLIND EQUALIZER
Importance is the key catalyst for evolution procedure. Difficulties
faced during Second World War revolutionized the field of communications
systems. The necessity of filter was realized when the some important
messages were corrupted and thus were unable to deliver it. Another bigger
step forward in the field of communication is the high frequency multi-path
communication. The latest development in the field of communications under
the sea, which is extremely complicated, because the physical media, Both
these developments in the field of communications demanded automatic or
blind types of channel equalization. And, finally the most dynamic
development in the field of communication is the high speed computer
communication, in which blind types of equalization are needed. In fact, the
61
latest developments in the field of communications were not possible without
invention of the blind equalizer.
4.3 EVOLUTION OF BLIND EQUALIZER
Needs lead to evolution. During the end of 1960’s, lots of
communication researchers were engaged to fulfill the invention of blind
equalizer. Many of them claimed to have invented a blind equalizer. They
wanted the received output sequence to be same as that of input sequence
automatically. Unfortunately, no algorithm was successfully used in practical
applications, in 1975, the prolonged uneasiness in the field of communications
system was broken by the researcher from the country of rising sun.
After prolonged effort, in 1975 Sato Yoichi, a Japanese researcher,
introduced a totally new concept for getting the output sequence as same that
of input sequence automatically. He formulated a typical cost function and
named his proposal as a self-recovering equalization for a multi-level
amplitude modulation system. Sato’s cost function was accepted by almost all
communication pundits, because it was practically feasible. Instead of Sato’s
own word, “Self-recovering equalization”, researchers started mentioning it in
an easy manner, i.e. “Blind Equalization”. Therefore, in fact Mr. Sato is the
father of Blind Equalizer. Sato’s algorithm is analyzed in details on section
4.4.
Sato’s cost function was not able to fulfill all the flying demands of
communications. It was bounded with some limitations. To fulfill these
limitations, in 1980, a French researcher, named Dominique N. Godard et al
[15] introduced another cost function, which added extra feature of carrier
tracking. Godard blind equalization is explained detailed on section 4.5.
62
Again, in the year 1980, a couple of French researchers proposed
another cost function for blind equalization, in which same features as that of
Godard cost function were included. However, their algorithm was not able to
get global recognition. As the year progresses, the development in blind
equalizer is also progressed.
The Sato and the Godard Blind algorithms have been discussed in
the coming chapters.
4.4 SATO’s BLIND ALGORITHM
Sato was the one who first introduced the blind equalization for
multilevel pulse amplitude modulation, wherever there is no reference
sequence available. Godard combined Sato’s idea with a decision Directed
(DD) algorithm and acquired a replacement blind equalization scheme for
QAM data transmission. Blind equalization has attracted significant scientific
interest due to its potentials in terms of overhead reduction and simplification
of point to multipoint communication. Sato’s algorithm was designed just for
real valued signal and PAM. However, its advanced valued extension was
derived by Godard. The cost function proposed by SATO is given in (4.1),
)})(.{()( 2kk
sato
ysignyEAJ (4.1)
Where,
ky = output of the equalizer
01
01)(
k
k
ky
yysign
(4.2)
|)(|
)( 2
k
k
aE
aE
(4.3)
Sign denotes the standard signum function of a real scalar. γ referred as
scaling factor and ak denotes the input data sequence.
63
Figure 4.1 General block diagram for Blind Equalizer
Figure 4.1 shows the general block diagram of the Blind Equalizer.
It seems that Sato’s proposal has been developed over LMS algorithm that
uses steepest decent criteria for convergence process. Mathematically, if any
equation is differentiated and equate it to zero, then the minimum of the
function will be obtained. Substituting the minimum to the steepest-descent
criteria, the tap weight coefficients can be obtained for the equalizer. By
differentiating eqn (4.1) and substitute it to the steepest-descent criteria, (4.4)
will be obtained as shown. The algorithm of SATO’s blind equalization relies
on (4.4) that is employed for training the output sequences,
)](.[.ˆˆ
1 kkkkk ysignyrAA (4.4)
Where,
kA = Weight used for training
Α = Tap-adjusting coefficient
yk = Output sequence
rk = Input sequence
1.
k
iik xar
(4.5)
Since this algorithm works under the iteration principle, at each
iteration; it tries to adapt its output sequence to the self-realized input
nk
�� ak
Ak rk
Decision Circuit Blind Equalizer
64
sequence. Thus, it's conjointly referred to as self-learning equalizer. The
convergence rate and precision to output sequence are the two main design
concerns in SATO’s blind equalization. To induce the simplest result from
Sato’s algorithm, the design considerations ought to be optimized on the basis
of its parameters. So, it will converge in quickly with a high precision output
sequence. This could be more or less guided by tap-adjusting coefficient ‘α’,
as a result of the remaining parameters are fixed in equation (4.4). The Figure
4.2 depicts the structure of SATO based blind equalizer with five taps. In blind
approach, the retrieval of each symbol is done with centre tap initialization.
The tap coefficient values are calculable using equation (4.4).
Figure 4.2 The Sato based Blind Equalizer with 5 taps.
4.5 SIMULATION RESULTS
Simulations have been carried out to evaluate the performance of
LMS adaptive algorithm and SATO Blind algorithm. Here we aimed to
reconstruct PAM symbol 1. The PAM symbol 1 is shown in Figure 4.3, The
65
five reflections with relative amplitude (0.7, 0.6, 0.5, 0.3 and 0.1) are shown in
Figure 4.4. PAM symbol 1 with ISI and Additive White Gaussian Noise
(AWGN) with 25dB SNR is shown in Figure 4.5, which is taken as the input
to the equalizer. The equalizer has been implemented by a linear transversal
filter with a five tap complex circuitry. The sequence of PAM symbols are
generated from the source and symbol by symbol processing is done at the
equalizer. Simulation results are shown for PAM Symbol 1.
Figure 4.3 The PAM symbol1.
Figure 4.4 The ISI model for PAM symbol1.
66
Figure 4.5 The PAM symbol 1 with ISI and AWGN noise
Figure 4.6 The PAM symbol 1 output after 1st iteration
Figure 4.7 The reconstructed PAM symbol 1 using LMS algorithm with
µ=0.015 and SNR = 20dB (84 iterations)
67
Figure 4.8 The reconstructed PAM symbol 1 using Blind algorithm with
α =0.6 and SNR = 20dB (9 iterations)
Figure 4.9 The Mean Square Error comparison using LMS and
Blind algorithm
The waveforms shown in Figure 4.6, Figure 4.7, Figure 4.8 and
Figure 4.9 are the results of simulations for the PAM symbol 1 output after 1st
iteration, reconstructed symbol 1 by using LMS approach, reconstructed
symbol 1 by using Blind approach and MSE comparison between LMS and
Blind approaches respectively. In this figure 4.7 and figure 4.8 seems identical
68
because both are reconstructed with same SNR 20dB however number of
iterations differs. Table 4.1 shows the quantity of iterations taken by LMS
algorithm, with completely different SNR value for the reconstruction of
symbol 1, 2 , 3, 4 and 5 using step size parameter µ = 0.015. In this work, we
have used the tap adjusting coefficient value (α = 0.6x10-3), as projected by
SATO to reconstruct the PAM signal that is shown in table 4.2. For a value of
α=0.06, we get better convergence as shown in table 4.3. But, whereas further
increasing the value of α (> 0.06) ends up in unsuccessful reconstruction of
original PAM symbols. The Simulation results show that Sato’s Blind
algorithm with optimum α value has quicker convergence rate compared to
that of LMS algorithm
Table 4.1. SNR vs. Iterations for LMS Adaptive Equalizer with Step Size
Parameter µ = 0.015
Output
SNR
(in dB)
Number of Iterations
Symbol 1 Symbol 2 Symbol 3 Symbol 4 Symbol 5
10 15 20 10 13 5
12 22 36 14 44 12
14 31 62 20 131 33
16 44 112 30 301 58
18 58 158 39 496 87
20 84 205 56 752 205
69
Table 4.2 SNR vs. Iterations for SATO based Blind Equalizer with
Step Size Parameter α = .0006
Output
SNR
(in dB)
Number of Iterations
Symbol 1 Symbol 2 Symbol 3 Symbol 4 Symbol 5
10 88 233 156 251 4
12 170 542 322 1115 277
14 286 1091 507 2900 733
16 445 2353 804 7955 1262
18 620 3982 1091 10907 2531
20 802 5608 1456 14123 4011
Table 4.3. SNR vs. Iterations for SATO based Blind Equalizer with
Step Size Parameter α = 0.06
Output
SNR
(in dB)
Number of Iterations
Symbol 1 Symbol 2 Symbol 3 Symbol 4 Symbol 5
10 1 3 2 3 1
12 2 6 3 11 3
14 3 12 5 42 10
16 4 19 8 77 16
18 6 37 12 140 26
20 9 50 15 110 38
Limitations
The limitation of Sato’s algorithm is that it recovers only single
carrier, whereas in practice the most sophisticated communication systems
employ dual carrier modulations, such as quadrature amplitude modulation.
This limitation is overcome by Godard proposal.
70
4.6 GODARD’s BLIND ALGORITHM (CMA)
After Sato’s initiation, in 1975, the race on blind equalization
algorithm took positive impetus. Godard algorithm was one of the best among
them. His introduction for dual carrier blind equalization was a new milestone,
which is practically feasible and conceptually simple blind algorithm. It
received immediate global acceptance. Godard has developed Sato’s cost
function in such a fashion that Sato’s cost function became a special case of
Godard cost function. In fact, Godard has substantially generalized the cost
function. The cost function proposed by D.N.Godard in 1980 is given in
equation (4.6) below:
])[()2/1()( 2
ppGod RyEpCJ
(4.6)
Where,
][/][
2 p
k
p
kp aEaER
p=dispersion constant and
p=1, 2, 3, 4....
The block diagrammatic view of dual carrier communication channel using the
blind equalization filter is as shown in Figure 4.10.
Figure 4.10 Godard scheme for Blind equalization and carrier tracking
rk
nk
yk
Phase
Splitter
Adaptive
Equalizer
Decision
Device
Carrier
Tracking
Cos (2πfct)
Sin(2πfct)
71
As like Sato’s algorithm, Godard proposal was additionally the
development version of LMS algorithm. Besides the major algorithm, it also
provides another additional algorithm (by product) for carrier recovery, which
was not enclosed in Sato’s algorithm. If we differentiate (4.6) with respect to
some constant and applied to the steepest descent algorithm, it offers the most
prominent result as shown (4.7) and (4.8).
)]|(|||µr 2k1
^
pp
kp
kkkk RyyyWW
(4.7)
)]exp(Im[^
1
^
kkkkk HjzaHH
(4.8)
Where,
W=Weight used for training
rk=Input sequence
yk=Output sequence
Rp=Constant scalar
ak=Decision output
zk=Input to the decision circuit
4.7 SIMULATION RESULTS
Simulations have been carried out to evaluate the performance of
LMS adaptive algorithm and SATO Blind algorithm. For performance
analysis, we considered the transmission of PAM symbols having ISI with five
reflections and AWGN as noise being given as input to the equalizer.
In this approach, the input data sequence was assumed to be
independent and drawn from PAM signaling sources. The equalizers are
implemented by a linear transversal filter with a five complex tap circuitry.
Here we aimed to reconstruct PAM symbol 2 in sequential manner (i.e., after
the reconstruction symbol 1).
72
The waveforms shown in Figure 4.11, Figure 4.12, Figure 4.13 are the
results of simulations for original PAM symbol 2, ISI model for PAM symbol
2, PAM symbol 2 with ISI and AWGN noise are taken as input to the
equalizer. The waveforms shown in Figure 4.14, Figure 4.15, Figure 4.16 and
Figure 4.17 are results of the equalizer output for PAM symbol 2 after 1st
iteration, Reconstructed PAM symbol 2 by using Sato approach,
Reconstructed symbol 2 by using Godard based Blind approach and MSE
comparison between Sato and Godard Blind approaches respectively. In this
Figure 4.15 and Figure 4.16 seems identical because both are reconstructed
with same SNR 30dB however number of iterations differs. Table 4.4 shows
the quantity of iterations taken by Sato algorithm, with completely different
SNR value for the reconstruction of symbol 1, 2 , 3, 4 and 5 using step size
parameter α= 0.0006.
Figure 4.11 The PAM symbol 2
73
Figure 4.12 The ISI model for PAM symbol 2
Figure 4.13 The PAM symbol 2 with ISI and AWGN noise
Figure 4.14 The equalizer output for PAM symbol 2 after 1st iteration
74
Figure 4.15 The reconstructed PAM symbol 2 using Sato’s blind
algorithm with α =0.0006 and SNR = 30dB (12950 iterations)
Figure 4.16 The reconstructed PAM symbol 2 using Godard’s
blind algorithm with α =0.06 and SNR = 30dB (4640 iterations)
Figure 4.17 The Mean Square Error comparison of Sato and Godard
Blind Equalization algorithm (CMA).
0 2000 4000 6000 8000 100000
20
40
60
80
100
120
Number of Iterations
MS
E
Sato Algorithm Vs Godard Algorithm
Sato-MSE
Godard-MSE
75
Table 4.4 SNR vs. Iterations for SATO based Blind Equalizer with Step
Size Parameter α = .0006
SNR
in dB
(Output)
Number of Iterations
Symbol 1 Symbol 2 Symbol 3 Symbol 4 Symbol 5
10 86 232 157 244 5
15 342 1638 623 6265 1436
20 804 5146 1449 13554 4285
25 1460 8990 2888 27889 7555
30 3803 12950 6555 42365 9637
Table 4.5 SNR vs. Iterations for SATO based Blind Equalizer with
Step Size Parameter α = .06
SNR
in dB
(Output)
Number of Iterations
Symbol 1 Symbol 2 Symbol 3 Symbol 4 Symbol 5
10 1 3 2 3 1
15 4 16 6 41 12
20 8 49 14 166 36
25 15 94 28 220 58
30 43 117 71 318 100
76
Table 4.6 SNR vs. Iterations for Godard based Blind Equalizer
with Step Size Parameter µ = .06
SNR
in dB
(Output)
Number of Iterations
Symbol 1 Symbol 2 Symbol 3 Symbol 4 Symbol 5
10 2806 1216 1315 1914 1182
15 4838 2344 1380 2141 1478
20 8930 3334 2236 2153 1536
25 11593 4222 9088 2681 2214
30 14285 4640 41041 2820 4060
4.8 SUMMARY
In section 4.5, the performance of LMS adaptive equalizer and
SATO based Blind equalizer have been compared. Observations from table
4.2 and table 4.3 show that, the specified SNR is obtained with less number of
iterations in SATO based Blind equalizers by selecting best α value.
In section 4.7, the performance of Sato blind equalizer and Godard
based blind equalizer have been compared. Observations from table 4.5 and
table 4.6show that the specified SNR is obtained with less number of iterations
in SATO based mostly blind equalizers by selecting best α value. Increase in
the tap adjusting coefficient value of Sato algorithm (e.g., α=0.06) provides a
much quicker convergence. When α=0.07 some symbols have converged
quickly, but some symbols have not converged at all due to maladjustment.
Similarly for higher α values (α > 0.07), none of the PAM symbols got
converged.
Godard based blind algorithm (CMA) with same step size as
proposed by Sato for PAM symbols (0.6x10-3) takes more number of iterations
77
and increase in the step size provides a faster convergence. When µ=0.06 only
few symbols have converged quickly. Likewise for higher values (µ > 0.06),
none of the PAM symbols got converged.
The next chapter presents the variable step size techniques for Sato
based blind equalizer with increased convergence rate and with small
maladjustment.
78
CHAPTER 5
VARIABLE STEP SIZE TECHNIQUES FOR SATO BASED
BLIND EQUALIZER
5.1 INTRODUCTION
It is known that blind equalizers do not require training sequence to
track the time varying characteristics of the channel. But, it ends up in slow
convergence to realize a selected Signal to Noise Ratio (SNR) at the output.
However, variable tap parameter (α) will speed up the convergence rate and
also minimizes the maladjustment for a blind equalizer. In this work, two
variable tap parameter techniques were used for Sato based blind equalizer
algorithm. Simulation results for Pulse Amplitude Modulated (PAM) signal
show that the proposed approach has a higher convergence rate than the
existing Sato algorithm with a fixed α value.
5.2 VARIABLE STEP SIZE SATO’s BLIND ALGORITHM
The flowchart of the proposed algorithm has been shown in Figure
5.1.In Sato’s blind equalization algorithm the initial value of the tap parameter
(α) is chosen between the minimum and maximum values and this range of
values is finite to guarantee the convergence and stability of the algorithms.
The fixed αmax is chosen with respect to the stability condition of the
algorithm, while αmin is chosen to confirm desired steady-state performance
Zhao Shengkui et al [92], Xue Wei et al [85] and Yuan Gao et al [90]. In the
proposed approach, the tap parameter value starts with 0.0006 (optimum value
identified by Sato for PAM signal) to reconstruct the very first symbol and this
value is incremented by small constant (s) for each iteration. The output
difference between successive iterations is calculated and this value is used to
stop the iterations for the reconstruction of very first symbol.
79
Figure 5.1 The flowchart for VSS Sato based blind equalizer
No
Start
Initialization Tap weight=Center tap initialization
Previous iteration =0 Alpha(α)=0.0006
Compute equalizer output
Calculate output difference =Current iteration value-Previous iteration value
Previous iteration=Current iteration
Output difference<0.001 or < -60dB
αk+
1 = α
k + factor (S
) Reconstructed Symbol
1
No
Yes
Compute equalizer output
Calculate output difference =Current iteration value-Previous iteration value
Previous iteration=Current iteration
α<=0.0006
αk+
1 =
αk ─
factor(S)
To reconstruct subsequent symbol Use updated tap weight from previous symbol.
Optimized α value from first symbol is considered.
Reconstructed Subsequent
Symbols
80
The updated tap parameter values are chosen as beginning value for
subsequent symbols. In the reconstruction of the subsequent symbols α value
is decremented by same constant value at each iteration, and when α reaches
0.0006, the iteration is stopped. For first symbol estimation, the specified SNR
can be achieved by changing the output difference value to stop the iteration.
For subsequent symbols estimation αmin value decides the iterations to be
stopped.
5.3 PSEUDOCODE VARIABLE STEP SIZE SATO’s BLIND
ALGORITHM
X=PAM Symbol
ISI=X with five reflections (Different amplitude with
different delay)
Y=X+ISI
Snrvalue=30
Received Symbol=AWGN(Y, 25)
Received Symbol = PAM symbol 1,PAM symbol 2…,PAM symbol N
Tap weights are center tap initialized (C1,C2,C4 and
C5=0,C3=1)
Step size (alpha) =0.0006
Equalizer input (input1) = Corrupted PAM symbol1
V=1;
The iteration procedure for first symbol reconstruction:
Loop 1: while (V<Snrvalue)
Estimating C2, C3, C4 and C5
Calculating the output difference values at three different
sampling points
at iterations
If (output difference < 0.001 or < -60dB)
Reconstruction is stopped and updated step size and tap
weights are used as commencing value for subsequent symbol
reconstruction
Otherwise
Step size is adjusted in either linear or non linear
fashion
81
Count=Count+1;
V = 20*log10 (norm (yk (:)) /norm (yk (:)-out1 (:)));
End
The Iteration Procedure for subsequent symbol
reconstruction:
Updated step size and tap weights are used as commencing
value for subsequent symbol reconstruction
Go Loop1
5.4 SIMULATION RESULTS OF VARIABLE STEP SIZE SATO
BASED BLIND EQUALIZER
The performance of the improved blind algorithm has been studied
for PAM symbols as done by Sato. The PAM symbol is shown in 5.2.The ISI
with five reflections with relative amplitude (0.7, 0.6, 0.5, 0.3 and 0.1) is
shown in fig. 5.3.The symbol 4with ISI and Additive White Gaussian Noise
(AWGN) with 25dB SNRis shown in 5.4, which is taken as the input to the
equalizer. The equalizer has been implemented by a linear transversal filter
with a five tap complex circuitry as shown in Figure 4.2.
Experiment1: Constant α (αconstant) Small α value results minimum steady state error but results in slow
convergence. Higher α value will speed-up the convergence; however, lead to
more maladjustment Zhao Shengkui et al [92]. So, the variable tap parameter
value is restricted to the range [αmin=0.0006, αmax=0.15] to guarantee stability
of the algorithm.
Experiment 2: α with Linear increment and decrement (αlinear)
For the above mentioned input, the α parameter value is chosen as
0.0006 and tap weights are initialized - with center tap value to be ‘1’ and all
other values to be zero David Smalley et al [93] and A. Benveniste et al [13].
The very first symbol is reconstructed by using linear increment in α. i.e., α is
incremented by constant factor (s=0.02) for every iteration as
82
αk+1 = αk + s; where s=0.02
The output difference between the current iteration and the previous
iteration is calculated at three different sampling points (locations).If all the
three output difference values are less than 0.001, the iteration for
reconstruction of very first symbol is stopped., the output difference value
0.001 is chosen to stop the iteration (based on the experimental analysis).The
updated α (=0.2406) is fixed as the optimum or the starting value for the
subsequent symbol. In the reconstruction of the following symbols, α is
decremented by the same factor (say 0.02) at each iteration as:
αk+1 = αk – s; where s=0.02
When α reaches 0.0006, the iteration is stopped. On trial and error
basis, for constant α input the optimum α value is 0.15. But in proposed
approach, αmax is found be 0.2406 and with updated tap weights consequent
symbols reconstructed with 30dB SNR in few iterations with stability. While
giving the fixed α value as 0.2406 with center tap initialization for experiment
1, it ended up with maladjustment and hence subsequent symbols could not be
reconstructed.
Experiment 3: α with Nonlinear increment and decrement (αnonlinear)
For an equivalent input, step size parameter value chosen as 0.0006
and tap weights are initialized with center tap. The very first symbol
reconstructed by using nonlinear increment of step size parameter value. That
is, step size value has been calculated with the assistance of iteration count as
αk+1 = αk + count * 0.001
The output difference between current iteration and previous
iteration is calculated in three different sampling points and if all the three
output values are less than 0.001,the iteration for reconstruction of very first
symbol is stopped. The updated step size parameter (0.32) is fixed as optimum
83
or beginning value for subsequent symbol. In the reconstruction of the next
symbol the step size parameter value is decremented in nonlinear method as
αk+1 = αk - count * 0.001
When α is less than or equal to 0.0006, the iteration is stopped. By
varying the output difference value and step size parameter value the required
SNR output can be obtained.
The equalizer output waveforms for PAM symbol 4areshown in the
following figures. Figure 5.5 and 5.6 show the output of PAM symbol 4after
1st iteration and after full recovery by using Blind approach. The figures 5.7
shows the self realized output symbol 4 by using variable α Blind approach.
The figure 5.8 show the MSE comparison between fixed α Blind and variable
α Blind approaches.
Figure 5.2 PAM symbol 4
84
Figure 5.3 The ISI model for PAM symbol 4
Figure 5.4 The PAM symbol 4 with ISI and AWGN noise
Figure 5.5 The Equalizer Output of Received Pam Symbol 4 After 1st
Iteration
85
Figure 5.6 The Reconstructed Pam Symbol 4 Using Sato Algorithm With
Fixed Step Size (24483 Iterations)
Figure 5.7 The Reconstructed Pam Symbol 4 Using Sato Algorithm With
Variable Step Size (177 Iterations)
Figure 5.8 Mean Square Error comparison between Sato’s Blind and
variable step size blind algorithm
86
Table 5.1, Table 5.2 and Table 5.3 shows the number of iterations
taken by Sato’s blind algorithm , proposed variable α approach (Linear) and
(Non Linear), with different Signal to Noise ratio value for the reconstruction
of symbol 1, 2 and 3 respectively. In this work, the same tap adjusting
coefficient value (α = 0.6x10-3) is used as proposed by Sato to reconstruct the
PAM signal. For a variable α blind approach, better convergence is obtained as
shown in table 5.2. The Simulation results show that the proposed variable α
blind approaches has increased the convergence rate compared to that of
Sato’s fixed α blind algorithm. That is, the number of iterations to obtain the
same output SNR for identical symbol is much lesser in the variable α blind
approach.
Table 5.1 SNR vs. Iterations for Sato based Blind Equalizer with Step
Size α = .0006
Output
SNR
(in dB)
Number of Iterations for Sato Blind Approach with α = 0.0006
Symbol 1 Symbol 2 Symbol 3 Symbol 4 Symbol 5
10 31 81 20 15 1
15 199 592 292 956 301
20 639 2926 981 7120 2495
25 1182 6583 1792 17456 4504
30 2114 9950 4452 24483 6018
87
Table 5.2 SNR vs. Iterations for Variable α Blind approach (Linear)
Output
SNR
in Db
Number of Iterations for Variable α Blind approach
(Linear)
Symbol 1 Symbol 2 Symbol 3 Symbol 4 Symbol 5
10 2 1 1 1 1
15 4 5 2 7 2
20 7 11 5 41 11
25 9 468 10 122 30
30 12 1454 31 177 82
Table 5.3 SNR vs. Iterations for Variable α Blind approach
(Non Linear)
Output
SNR
in dB
Number of Iterations for Variable α Blind approach
(Non Linear)
Symbol 1 Symbol 2 Symbol 3 Symbol 4 Symbol 5
10 4 3 2 2 1
15 10 15 12 18 13
20 34 45 28 73 31
25 49 520 56 212 89
30 61 1973 83 364 104
5.5 SUMMARY
The convergence rate of proposed algorithm is very much
encouraging .The only disadvantage of Sato’s algorithm is that it recover only
single carrier, whereas in practice the most communication system employs
dual carrier modulation systems, like quadrature amplitude modulation. This
limitation is overcome by Godard proposal. By using variable tap parameter
technique convergence of Godard algorithm can also be improved. The next
chapter presents the variable step size Godard based blind equalization
algorithm for PAM symbol input.
88
CHAPTER 6
VARIABLE STEP SIZE TECHNIQUES OF GODARD BASED
BLIND EQUALIZATION ALGORITHM (CMA)
6.1 INTRODUCTION
The modern digital high speed wireless communication system
demands quick convergence rate and low steady state error for the equalizers.
The balancing between the demands can be achieved by selecting correct step
size. Thus, it is essential to define new algorithms or optimize the available
algorithms to equalize channels and mitigate noise in communications. It is
known that time varying step size blind equalization technique can speed up
the convergence rate and minimize the maladjustment. This work presents a
variable step size (VSS) approach for Godard blind equalization algorithm to
resolve the conflict between the convergence rate and precision of the fixed
step-size Godard algorithm. The results of this proposed approach is compared
with the existing variable step size Sato algorithm for a pulse amplitude
modulated (PAM) input symbol.
6.2 PSEUDOCODE OF VARIABLE STEP SIZE GODARD
BASED BLIND EQUALIZATION ALGORITHM (CMA)
X=PAM Symbol
ISI=X with five reflections (Different amplitude with
different delay)
Y=X+ISI
Snrvalue=30
Received Symbol=AWGN(Y, 25)
Received Symbol = PAM symbol 1, PAM symbol 2…, PAM symbol N
Tap weights are center tap initialized (C11, C12, C21, C22…
and C56=0)
Step size (µ) =0.06, ?=115
89
Equalizer input (input1) = Corrupted PAM symbol1
V=1;
The iteration procedure for first symbol reconstruction:
Loop 1: while (V < Snrvalue)
Estimating C11, C12, C22, C23, C33, C34, C44, C45, C55 and
C56
Calculating the output difference values at three different
sampling points
at iterations
If (output difference < 0.001)
Reconstruction is stopped and updated step size and tap
weights are used as commencing value for subsequent symbol
reconstruction
Otherwise
Step size is adjusted in either linear or non linear
fashion
Count=Count+1;
V = 20*log10 (norm (y11 (:)) /norm (y11 (:)-out1 (:)));
End
The Iteration Procedure for subsequent symbol
reconstruction:
Updated step size and tap weights are used as commencing
value for subsequent symbol reconstruction
Go Loop1
6.3 VARIABLE STEP SIZE GODARD BASED BLIND
EQUALIZATION ALGORITHM (CMA)
The flowchart of the proposed algorithm is shown in Figure 6.1.The
only limitation of Godard blind algorithm is slow rate of convergence and this
problem can be compensated by VSS Godard blind algorithm. Instead of
single step size as like Sato algorithm, here µ and λ are present. The selection
of step size must be balanced between the following two scenarios.
A small step size µmin will guarantee small noise level in steady
state, but the algorithm will converge slowly .On the other hand, a large step
90
size µmax will commonly provide faster convergence and enhanced tracking
capabilities at the cost of higher noise level. The novel idea projected here is
that instead of selecting an optimum step size as the starting value and then
decrementing it in iterations, as others did, for the reconstruction of all
symbols, chose a small step size as the starting value and incrementing it in
iterations only for very first symbol by balancing maladjustment and
convergence. The updated maximum step size is treated as a starting point and
subsequently reducing the step size in iterations for the remaining symbols and
thus gained better performance. In this approach the step size value starts with
0.06 to reconstruct the very first symbol and this value is Incremented by a
small constant (s) for each iteration. The Sato identified optimum value
(0.0006) takes large number of iterations for reconstruction. And the other λ is
chosen as fixed value of 115 based on compromise between convergence rate
and stability of the algorithm.
91
Figure 6.1 The flowchart for variable step size Godard algorithm
Reconstructed Subsequent
Symbols
No
Yes
Start
Initialization Tap weight initialization (near to zero)
Previous iteration =0 Step Size (µ) =0.06 and λ=115
Compute equalizer output
Calculate output difference =Current iteration value-Previous iteration value
Previous iteration=Current iteration
Output difference<0.001
Or< -60dB
µk+
1 = µ
k + factor (S
) Reconstructed Symbol 1
No
Yes
Compute equalizer output
Calculate output difference =Current iteration value-Previous iteration value
Previous iteration=Current iteration
µ<=0.0006
µk+
1 = µ
k ─ factor(S
)
To reconstruct subsequent symbol Use updated tap weight from previous symbol.
Optimized µvalue from first symbol is considered.
Reconstructed Subsequent Symbols
92
The output difference between successive iterations is calculated to
stop the iteration for reconstruction of very first symbol. The updated step size
value is chosen as commencing value for subsequent symbols. In the
reconstruction of the subsequent symbols µ value is decremented by small
constant at iterations and the iteration is stopped when µ reaches the value
0.06. For the first symbol assessment, the specified SNR can be accomplished
by varying the output difference value to stop the iteration and for subsequent
symbols assessment; by changing µmin value the specified SNR output can be
obtained.
6.4 SIMULATION RESULTS
The performance of the improved Godard blind algorithm has been
studied for PAM symbols as done in section 5.3. The PAM symbols as shown
in Figure 6.2, and the ISI with five reflections with relative amplitude (0.7,
0.6, 0.5, 0.3 and 0.1) is shown in Figure 6.3 for symbol 5 with the ISI and
Additive White Gaussian Noise (AWGN) with 25dB SNR as shown in 6.4; are
taken as the input to the equalizer. The equalizer has been implemented by a
linear transversal filter with a five tap complex circuitry as shown in Figure
4.2.
Experiment1: Constant µ (µconstant)
Small step size (µ) value results minimum steady state error but
results in slow convergence. High step size (µ) value will speed-up the
convergence however lead to large maladjustment. The step size value is
restricted to the range [µmin=0.06, µmax=1.6] to guarantee stability of the
algorithm.
93
Experiment 2: µ with linear increment and decrement (µlinear)
For the above mentioned input, the step size value chosen as 0.06
and tap weights are initialized with center tap (only center tap has ‘one’ and
others are near zero) [23] and [24]. The very first symbol is reconstructed by
using linear increment in µ. i.e., µ is incremented by constant factor (s=0.001)
for every iteration as
µk+1 = µk + s; where s=0.001
The output distinction between current iteration and previous
iteration is calculated at in three different sampling points and if all the three
output values are less than 0.001 (based on experimental analysis, the output
difference value 0.001 is chosen to stop the iteration. The updated µ(=0.214) is
fixed as the optimum or the starting value for the subsequent symbol. In the
reconstruction of the following symbols, α is decremented by the same factor
(say 0.02) at each iteration as:
µk+1 = µk – s; where s=0.001
When µ reaches 0.0006, the iteration is stopped. On trial and error
basis, for constant µ input the optimum µ value is 1.6. But in proposed
approach it is found 2.14 as µmax and with updated tap weights consequent
symbols reconstructed with 30dB SNR in few iterations with stability. While
giving the fixed µ value as 2.14 with center tap initialization for experiment 1,
it ended up with maladjustment and hence subsequent symbols could not be
reconstructed.
94
Figure 6.2 PAM symbol 5
Figure 6.3 The ISI model for PAM symbol 5
Figure 6.4 The PAM symbol 5 with ISI and AWGN noise
95
Figure 6.5 The equalizer output for PAM symbol 5 after 1st iteration
Figure 6.6 The reconstructed PAM symbol 5 using VSS Sato algorithm
with SNR=30dB (26 iterations)
Figure 6.7 The reconstructed PAM symbol 5 using VSS Godard
algorithm with SNR=30dB (1350 iterations)
96
Figure 6.8 Mean Square Error comparison between VSS Sato algorithm
and VSS Godard algorithm
The waveforms are shown in Figure 6.5, Figure 6.6, Figure 6.7 and
are the results of simulations for the equalizer output for PAM symbol 5 after
1st iteration, self realized output symbol 5 by using variable step size Sato
blind approach, and self realized output symbol 5 by using variable step size
proposed blind approaches respectively Figure 6.8 show that the MSE
comparison between variable step size Sato blind approach and variable step
size proposed Godard blind approaches.
Table 6.1, Table 6.2 and Table 6.3 shows the number of iterations
taken by variable step size Sato blind algorithm, Godard blind algorithm with
fixed step size and proposed VSS Godard algorithm (Linear), with different
Signal to Noise ratio value for the reconstruction of symbol 1, 2, 3, 4 and 5
respectively. The Simulation results show that the proposed VSS Godard blind
approach has comparable convergence rate to that of existing VSS blind
algorithm.
97
Table 6.2 Number of Iterations for Sato Blind approach with
Variable step size
Output SNR
in dB
Number of Iterations for Sato Blind Approach with
variable step size
Symbol 1 Symbol 2 Symbol 3 Symbol 4 Symbol 5
10 2 1 1 1 1
15 4 4 2 8 1
20 7 10 5 34 4
25 9 421 8 59 8
30 12 1425 30 219 26
Table 6.2 Number of Iterations for Godard blind approach with
fixed step size
Output
SNR
in dB
Number of Iterations for Godard Blind approach
with fixed step size 0.06
Symbol 1 Symbol 2 Symbol 3 Symbol 4 Symbol 5
10 2806 1216 1243 1405 2148
15 2928 1390 1350 1811 2322
20 3214 1927 2082 1844 2869
25 3290 3848 8351 2295 5613
30 3576 4338 36181 2540 29084
98
Table 6.3 Number of iterations for proposed Godard blind approach with
variable step size
Output
SNR
in dB
Number of Iterations for Godard Blind approach
with variable step size
Symbol 1 Symbol 2 Symbol 3 Symbol 4 Symbol 5
10 1269 795 815 515 797
15 1300 827 901 563 831
20 1356 1120 973 681 892
25 1373 1570 1074 727 935
30 1373 1570 1076 1119 1350
6.5 SUMMARY
In this work, a variable step size technique has been proposed for
Godard based blind equalizer to resolve the conflict between the convergence
rate and accuracy of the fixed step-size Godard algorithm (CMA). The step
size of the algorithm is updated with respect to the differences in successive
outputs. From the simulation results, it is observed that proposed variable step
size Godard algorithm offers quicker convergence than fixed step size
(µ=0.06) Godard algorithm. But it is slower than Sato’s variable step size
blind approach.
99
CHAPTER 7
CONCLUSION
In this work the performance of LMS adaptive equalizer and
variable step size LMS equalizer has been compared. Observations made by
choosing the fixed step size value to be 0.015 which is identified as optimum
step value for trained adaptive LMS equalizer. Increase in the tap adjusting
coefficient value (e.g., µ=0.015) provides a much quicker convergence. When
µ=0.25, some symbols have converged quickly, but some symbols do not
converge (due to maladjustment). Similarly for higher values (µ> 0.25),
convergence does not occur for all PAM symbols. So, if the optimum value
for µ is calculated, the convergences are will be quicker. Rather than a fixed µ
value, variable µ value for iteration basis will be used to speed up the
convergence and minimize the maladjustment. The disadvantage of trained
equalizers is that they use additional bandwidth. This problem can be avoided
by using blind equalization algorithms proposed by Sato and Godard.
In non-cooperative environment placing known reference sequence
at the receiver is not possible. In this case, blind equalizers are needed to
reconstruct the original symbol. So, the performance of Sato based blind
equalizer has been analyzed and compared the results of trained adaptive LMS
equalizer. Observations from table 4.2 and table 4.3 show that, the specified
SNR will be obtained with less number of iterations in SATO based blind
equalizers by selecting best α value. Increase in the tap adjusting coefficient
value of Sato based blind equalizer (e.g., α=0.06) provides a much quicker
convergence. When α=0.07 some symbols have converged quickly, but some
symbols have not converged (due to maladjustment). Similarly, for higher
values (α > 0.07), convergence does not take place for all PAM symbols. So, if
the optimum value for α may be calculable, the convergence will be quicker.
100
Rather than a fixed step size value, variable step size value can be used to
speed up the convergence rate and minimize the maladjustment in iteration
basis. Observations from table 4.5 and table 4.6 show that, the specified SNR
will be obtained with less number of iterations in SATO based mostly blind
equalizers by selecting best α value. Increase in the tap adjusting coefficient
value of Sato algorithm (e.g., α=0.06) provides a much quicker convergence.
When α=0.07 some symbols have converged quickly, but some symbols do
not converge (due to maladjustment). Similarly for higher values (α > 0.07),
converge for all PAM symbols does not take place.
Godard based blind algorithm (CMA) with same step size as
proposed by Sato for PAM symbols (0.6x10-3) taking more number of
iterations and increase in the step size provides a faster convergence. When
µ=0.06 only few symbols have converged quickly. Likewise for higher values
(µ > 0.06), convergence for all PAM symbols does not takes place.
So, if the optimum value for α and µ may be calculable, the
convergence are going to be quick. Rather than a fixed α and µ value, variable
α and µ value can be used to speed up the convergence and minimize the
maladjustment in iteration basis. The sole limitation of Sato’s formula is that it
recovers only single carrier, whereas in practice the most sophisticated
communication system employs dual carrier modulation systems, like
quadrature amplitude modulation. This limitation is overcome by Godard
proposal. By using variable step size value the convergence of Godard can
also be improved.
So, the performance of Godard based blind equalization algorithm
has been analyzed and compared the results with Sato based blind equalization
algorithm for fixed step size value. Blind equalizers do not require known
sequence at the receiver side and that saves the bandwidth. However, it takes
101
more number of iterations to reconstruct the original symbol. The ill
convergence or slow convergence can be solved my using variable step size
technique instead of fixed step size value.SNR and MSE are used to estimate
the quality of the reconstructed symbols, which can also be used to stop the
tap weight calculation or stop the iterations for blind equalizers. Increase in α
value gives a convergence with increased maladjustment i.e., very high α
value does not converge. Using variable α values, the quicker convergence can
be obtained with minimum maladjustment. In this work, variable tap
parameter (α) techniques have been proposed for Sato based blind equalizer;
and also two different methods have been proposed to stop the iterations. First
method uses the differences in successive outputs and second method uses
particular tap parameter (α) value. The simulation results show that using the
proposed techniques, the desired SNR has been obtained with less number of
iterations and also with minimum steady state error compared with Sato’s
blind algorithm. From the results, it is also observed that variable αLinear offers
quicker convergence than variable αNonlinear.
Further, a variable step size technique is proposed for Godard based
blind equalizer to resolve the conflict between the convergence rate and
accuracy of the fixed step-size Godard algorithm. The step size of the
algorithm is updated with respect to the differences in successive outputs.
From the simulation results, it is observed that proposed variable step size
Godard algorithm offers quicker convergence than fixed step size (µ=0.06)
Godard algorithm. But it is slower than Sato’s variable step size blind
approach. The only disadvantage of Sato’s algorithm is that it recovers only
single carrier, whereas in practice the most sophisticated communication
system employs dual carrier modulation systems, like quadrature amplitude
modulation. But the computational complexity of proposed technique is
slightly higher than VSS Sato’s blind algorithm. The computational time taken
102
for reconstructing three symbols sequentially by VSS Sato algorithm is
3.869036 seconds and by proposed algorithm is 5.176232 seconds.
LIMITATIONS AND SCOPE FOR THE FUTURE WORK
Hardware design complexity is one of the important limitations.
However, by proposing a new method which could reduce the
calculations using number of adders and multiplexers may
overcome this problem and the same here is left as open issue.
The proposed blind equalization algorithms could be extended to
higher order modulations such as QAM.
Further study can be done on Blind Equalization algorithms for
QAM input and Medical Image input (CT, CR, MRI and MG).
Because, quick convergence equalization algorithms are much
needed in the medical imaging domain at the receiver side to
reconstruct original information.
Application-I: A doctor in the United States generates on an average close
to 70 Terabytes of data every year. If the Doctors happen to attend to
critical image of patient and need medical opinion for the patient from the
specialist, who is at distant place, then they will send their medical opinion
through internet. The memory size of the data generated by the
Doctor/Hospital is more than the size which could not be sent through e-
mail. However, it is possible by using some file transfer protocol
applications.
Application-II: In other real time scenario, assume the cine labs using
client server application, where server is located at a distant place, say,
Singapore and it is accessed by number of clients viz. Chennai, Mumbai
103
and Los Angles from different locations. That is all clients are
uploading/downloading the data (frames) from server.
In the above cases, there may be chances for noise occurrence when
the data travels through channel causing the data is corrupted by ISI means,
and then the equalizer plays a vital role to reconstruct the same. In order to
speed up this process variable step size techniques are much needed.
104
APPENDIX – A
Program for Least Mean Square Algorithm
%PAM Signal generation
%ISI Generation
offset_points = 4;
firstref(1:offset_points*1)=0;
firstref=[firstref 0.7*y1];
firstref(length(y1)+1:length(y1)+offset_points*1)=[];
secref(1:offset_points*2)=0;
secref=[secref 0.6*y1];
secref(length(y1)+1:length(y1)+offset_points*2)=[];
thirdref(1:offset_points*3)=0;
thirdref=[thirdref 0.5*y1];
thirdref(length(y1)+1:length(y1)+offset_points*3)=[];
fourthref(1:offset_points*4)=0;
fourthref=[fourthref 0.4*y1];
fourthref(length(y1)+1:length(y1)+offset_points*4)=[];
fifthref(1:offset_points*5)=0;
fifthref=[fifthref 0.3*y1];
fifthref(length(y1)+1:length(y1)+offset_points*5)=[];
figure(7);
plot(t1,y1,t1,firstref,t1,secref,t1,thirdref,t1,fourthref,t1,fifthref)
legend('original','firstref','secref','thirdref','fourthref','fifthref'),
title('The Effect of Intersymbol Interference.')
ylabel('Amplitude');
xlabel( ' Time' );
%The Initialization of Weights.
for i=1:100
C1(i)=0.0000000000001;
C2(i)=0.0000000000001;
C3(i)=0.0000000000001;
C4(i)=0.0000000000001;
C5(i)=0.0000000000001;
i=i+1;
105
end
%The received signal before applying to the equalizer
in1=y1;
in2=firstref;
in3=secref;
in4=thirdref;
in5=fourthref;
in6=fifthref;
in8=(in1+in2+in3+in4+in5+in6);
figure(8);
plot(t1,in8),
title(['The Received siganl before applying to the equalizer '])
ylabel('Amplitude');
xlabel( ' Time');
%AWGN with signal
input=awgn(in8,25);
figure(9);
plot(t1,input),
title('The PAM Symbol with Noise.')
ylabel('Amplitude');
xlabel( ' Time ');
% Received signal divided in to number of symbols
%The Output after every Weight Treatment.
out11=C1.*input1;
out21=C2.*input1;
out31=C3.*input1;
out41=C4.*input1;
out51=C5.*input1;
out1=(out11+out21+out31+out41+out51)./5;
delta=0.015;
%The Iteration Procedure.
while(v<snrvalue)
106
error=newy1-out1;
C1=C1+delta.*error.*(input1);
C2=C2+delta.*error.*(input1);
C3=C3+delta.*error.*(input1);
C4=C4+delta.*error.*(input1);
C5=C5+delta.*error.*(input1);
out1=C1.*input1;
out2=C2.*input1;
out3=C3.*input1;
out4=C4.*input1;
out5=C5.*input1;
out=(out1+out2+out3+out4+out5)./5;
count=count+1;
v = 20*log10(norm(newy1(:))/norm(newy1(:)-out1(:)));
disp(v)
end
figure(20);
disp(count)
disp(v)
plot(t,out1),title(['The Resultant Waveform-LMS Algorithm']),
ylabel('Amplitude');
xlabel( ' Time');
clear i;
Program for Sato based blind algorithm
while(V<snrvalue)
C1=C1-alpha.*input1.*(out1-abs(newy1).*out111);
C2=C2-alpha.*input1.*(out1-abs(newy1).*out111);
C3=C3-alpha.*input1.*(out1-abs(newy1).*out111);
C4=C4-alpha.*input1.*(out1-abs(newy1).*out111);
C5=C5-alpha.*input1.*(out1-abs(newy1).*out111);
out11=C1.*input1;
out21=C2.*input1;
out31=C3.*input1;
107
out41=C4.*input1;
out51=C5.*input1;
out1=(out11+out21+out31+out41+out51)./5;
V= 20*log10(norm(newy1(:))/norm(newy1(:)-out1(:)));
end
figure(21);
plot(t,out1),
title(['The Self Realized First symbol Output -Blind Approach'])
ylabel('Amplitude');
xlabel( 'Time');
clear j;
Program for Godard based blind algorithm (CMA)
while(v1<=snrvalue )
C11=C11+delta_p.*(conj(Y11)).*outC1.*(R21-(outC.^2));
C22=C22+delta_p.*(conj(Y22)).*outC1.*(R21-(outC.^2));
C33=C33+delta_p.*(conj(Y33)).*outC1.*(R21-(outC.^2));
C44=C44+delta_p.*(conj(Y44)).*outC1.*(R21-(outC.^2));
C55=C55+delta_p.*(conj(Y55)).*outC1.*(R21-(outC.^2));
C12=C12+delta_p.*(conj(Y12)).*outD2.*(R22-(outD.^2));
C23=C23+delta_p.*(conj(Y23)).*outD2.*(R22-(outD.^2));
C34=C34+delta_p.*(conj(Y34)).*outD2.*(R22-(outD.^2));
C45=C45+delta_p.*(conj(Y45)).*outD2.*(R22-(outD.^2));
C56=C56+delta_p.*(conj(Y56)).*outD2.*(R22-(outD.^2));
out11=Y11.*C11;
out22=Y22.*C22;
out33=Y33.*C33;
out44=Y44.*C44;
out55=Y55.*C55;
out12=Y12.*C12;
out23=Y23.*C23;
out34=Y34.*C34;
out45=Y45.*C45;
108
out56=Y56.*C56;
outC1=(out11+out22+out33+out44+out55)./5;
outD2=(out12+out23+out34+out45+out56)./5;
out1=outC1./(exp(-j.*Q1));
out2=outD2./(exp(-j.*Q2));
v1=20*log10(norm(y11(:))/norm(y11(:)-out1(:)));
count=count+1;
end
figure(22);
plot((t),real(out1)),hold on
title(['The resultant waveform-Godard algorithm']),
ylabel('Amplitude');
xlabel( ' Time');
109
APPENDIX B
HILBERT- TRANSFORM PAIR
Time Function Hilbert Transform
)2cos()( tftm c )2cos()( tftm c
)2sin()( tftm c )2cos()( tftm c
)2cos( tf c )2sin( tf c
)2sin( tf c )2cos( tf c
tt /sin tt /cos1
)(trec
2/1
2/1ln/1
t
t
)(t
t/1
)1/(1 2t
)1/( 2tt
t/1
)(t
110
REFERENCES
1. Aboulnasr, T and Mayyas, K, A roubust variable step-size lms-type algorithm: Analysis and simulations, IEEE Trans. Signal Processing, 45, 1997, 631-639.
2. Adam, R, Margetts and Philip Schniter, Adaptive chip-rate equalization of downlink multirate wideband CDMA, IEEE Transactions On Signal Processing, 53(6), 2005, 2205-2215.
3. Ahmad Tariq Sheikh and Shahzad Amin Sheikh, Efficient variants of square contour algorithm for blind equalization of QAM signals, World Academy of Science, Engineering and Technology,3, 2009, 180-188.
4. Alban Goupil and Jacques Palicot, An efficient blind decision feedback equalizer, IEEE Communications Letters,14(5), 2010,432-434.
5. Ajjaiah HBM, Prabhakar, V, Hunagund , Manoj Kumar Singh and Rao, P,V, Adaptive variable step size in LMS algorithm using evolutionary programming: VSSLMSEV, Signal Processing: An International Journal (SPIJ), 6 (2) , 2012, 78-85.
6. Ajjaiah HBM Prabhakar V, and Hunagundhamer, Variable step size of LMS algorithem using partical swarm optimization, International Journal of Research in Engineering and Technology,3,2014,324-327.
7. Amin Mohamed Nassar and Eng. Waleed EI Nahal, Blind equalization technique for cross correlation constant modulus algorithm (CC-CMA), WSEAS Transactions on Signal Processing, 6(2),2010, 23-32.
8. Athanasios Vgenis, Constantinos S, Petrou, Constantinos B, Papadias, Ioannis Roudas, and Lambros Raptis, Nonsingular constant modulus equalizer for PDM-QPSK coherent optical receivers, IEEE Photonics Technology Letters, 22(1), 2010,45 - 47.
9. Athar Qureshi, Triantafyllos Kanakis and Predrag Rapajic, Adaptive signal combining with unequal channel noise variances, IJSSST, 11(6), 2010,14-21.
10. Banovic, K, Esam, A , A novel radius adjusted approach for blind
adaptive equalizer, IEEE Signal Processing Letters, 13 (1), 2006,37-40.
111
11. Baofeng ZHAO and Jia LIU, Influence of MSE on tracking channel
and anti-interference of a variable step-size CMA, Journal of Computational Information Systems, 7 (10), 2011, 3646-3650.
12. Boˇ zo Krstaji ´ c, LJubiša Stankovi´ c and Zdravko Uskokovi ´ An approach to transform domain variable step-size LMS adaptive filter, EUSIPCO, 2004,1813-1816.
13. Benveniste,A, and Goursat, M , Blind equalizer, IEEE Transactions on Communications, 32, 1984,871-883.
14. Chahed, I, Belzile, J, and Kouki, AB, Blind decision feedback equalizer based on high order MCMA. In Canadian Conference on Electrical and Computer Engineering 4.Niagara Falls (Canada),4, 2004, 2111-2114.
15. David Smalley, Equalization concepts: a tutorial, Application Report, Atlanta Regional Technology Centre, SPRA140. 1994
16. Dominique N, and Godard, Self-recovering equalization and carrier tracking in two dimensional data communication systems, IEEE Transactions on Communications, 28(11), 1980,1867-1875.
17. Dogancay, K, Rodney, A, and Kennedy, Least squares approach to blind channel equalization, IEEE Transactions on Communications, 47 (11), 1999,1678 - 1687.
18. Djebbar AB , LSF-CMA based blind equalization algorithms for MC-CDMA systems, 7th International Workshop on Systems, Signal Processing and their Applications, 2011, 203 -206.
19. Emin TUĞCU, Fatih ÇAKIR and Ali OZEN, A new step size control technique for blind and non-blind equalization algorithms, Radioengineering,22(1), 2013,44-51.
20. Feng Tong, Bridget Benson, Ying Li and Ryan Kastner , Channel equalization based on data reuse LMS algorithm for shallow water acoustic communication, 2010 IEEE International Conference on Sensor Networks, Ubiquitous, and Trustworthy Computing, 2000,95-98.
112
21. Fengqi Yu Alan N, and Willson, Jr, An interleaved/pipelined architecture for adaptive lattice equalizer, Proc 43rd IEEE Midwest Symp on Circuits and Systems, 2000,856-859.
22. Gauss C,F, Theoria combinationis observationum erroribus minimis obnoxiae 1821, Werke, 4. Gottinge.
23. Giannakis GB, Hua, Y, Stoica, P, and Tong, L, Signal processing advances in wireless and mobile communications, Trends in Channel estimation and equalization. 1, 2000,Prentice Hall, Upper Saddle River, NJ.
24. Gi Hun Lee & Rae-Hong Park, Shell partition-based joint blind equalization for QAM systems, IEEE Transactions on Consumer Electronics, 46(2), 2000, 295 - 305.
25. Guo Yecai, He Longqing and Zhang Yanping, Design and implementation of adaptive equalizer based on FPGA, The Eighth International Conference on Electronic Measurement and Instruments (ICEMI), 2007,790-794.
26. Guo Li , Li Ning, Guo Yan and Zhou Jiongpan, Convergence behavior of the constant modulus algorithm controlled by special step size, 6th International Conference on Signal Processing, 1, 2002, 390 -392.
27. Harris, R, Chabries, D, and Bishop, FA, A variable step (VS) adaptive
filter algorithm, IEEE Trans. Acoust, Speech, Signal Processing, 34,
1986, 309-316.
28. Harris, R, Chabries, D, and Bishop, FA, A variable step(VS) adaptive
filter algorithm, IEEE Trans. Acoust, Speech, Signal Processing, 41, 1993,2075-2087.
29. Hao, Miin-Jong, Tsai, and Yu-Chi, ‘Channel estimation for nonlinear MIMO receiver with square envelope detection,’ International Journal Communication Systems, 26( 8), 2013,947-963.
30. http//en.wikipedia.org/wiki/Zero Forcing Equalizer
31. Heinz Mathis and Scott C, and Douglas, Bussgang blind deconvolution for impulsive signals, IEEE Transactions on Signal Processing, 51(7), 2003,1905 - 1915.
113
32. Hodges M, The GSM radio interface, British Telecom Technological Journal, 8(1), 1990, 31-43.
33. Hong Chae Woo, Variable step size LMS algorithm using squared error and autocorrelation of error, Procedia Engineering,41, 2012,47 - 52.
34. Iorkyase, E, Tersoo and Michael O, Improving the Rate of Convergence of Blind Adaptive Equalization for Fast Varying Digital Communication Systems, International Journal of Advanced Computer Science and Applications, 3(7), 2012,133-137.
35. Irusta, S Ruiz de Gauna, J Ruiz, E Aramendi, A Lazkano and Gutierrez, JJ, A variable step size LMS algorithm for the suppression of the CPR artefact from a VF signal, IEEE International journal of Computers in Cardiology , 32, 2005,179-182.
36. Jones DL, A normalized constant modulus algorithm, In IEEE Conference Record of the Twenty-Ninth Asilomar Conference on Signals, Systems and Computers Pacific Grove (USA),1, 1995,694-697.
37. Karami Ebrahim, Performance analysis of decision directed maximum likelihood MIMO channel tracking algorithm, International Journal of Communication Systems,26(12), 2013,1562-1578.
38. Kevin Banovic, Raymond Lee, Esam Abdel-Raheem, Mohammed AS, and Khalid, Computationally-efficient methods for blind adaptive equalization, in Proc.IEEE, 2005, 341-344.
39. Kil Nam Oh and Jae Hong Park, Property restoral approach to blind equalization of digital transmission channels, IEEE Transactions on Consumer Electronics, 43, (3), 1997.
40. Khurram Shahzad, Muhammad Ashraf and Raja Iqbal, Improved blind equalization scheme using variable step size constant modulus algorithm, Proceedings of the 7th WSEAS Int. Conf. on Signal Processing, Computational Geometry & Artificial Vision, 2007,86-90.
41. Kun-Chien Hung, David W, Lin and Chun-Nan Ke, Variable-step-size multimodulus blind decision-feedback equalization for high-order qam based on boundary MSE estimation, in Proc. IEEE-(ICASSP 2004), 2004, 881-884.
114
42. Kundu, A and Chakrabarty A, Fractionally spaced constant modulus algorithm for wireless channel equalization, Progress In Electromagnetics Research B, 4, 2008,237-248.
43. Kying Xiao and Fuliang Yin, Variable Step Size Blind Equalization Based on Sign Gradient Algorithm, TELKOMNIKA Indonesian Journal of Electrical Engineering,12(7),2014, 5491-5498.
44. Kwong, R, and Johnston, EW, A variable step size LMS algorithm, IEEE Trans.Signal Processing, 40,1992,1633-1642.
45. Langford B White, Blind equalization of constant modulus signal using an adaptive observer approach, IEEE Trans. On communication, 44(2), 1996,134-136.
46. Leou, Maw-Lin Wu, Chien-Min Liaw, Yi-Ching Su, and Hui-Kai ‘An orthogonalized blind algorithm for hybrid of adaptive array and equalizer,’ International Journal of Communication Systems,27(2), 2014,201-215.
47. Lilyi, Z Lei, Yunshan S, Variable step size CMA blind equalization based on nonlinear function of error signal, In International Conference on Communications and Mobile Computing. Kunming (China),1, 2009,396-399.
48. Mangoud MA, Effect of fading correlation on the performance of spatial multiplexed MIMO systems with circular antennas, Journal of Emerging Trends in Computing and Information Sciences,3(11) 2012,1453-1458.
49. Mashhoor AlTarayrah and Qasem Abu Al-Haija, Adaptive channel equalization for Fbmc based on variable length step size and mean-squared error, 3(4), 2013,66-76.
50. Mathews VJ and Xie Z , A stochastic gradient apadtive filter with gradient adaptive step size, IEEE Trans. Signal Processing, 41, 1993,2075-2087.
51. Meng Zhang, A novel blind equalization algorithm based on affine
projection and quantization estimation errors, IEEE 978-1-4244-5668. 2009
115
52. Monika Pinchas, Two blind adaptive equalizers connected in series for equalization performance improvement, Journal of Signal and Information Processing, 4, 2013, 64-71.
53. Muhammad Lutfor Rahman Khcan, Mohammed, H, Wondimagegnehu and Tetsuya Shimamura, Blind channel equalization with amplitude banded godard and sato algorithms, Journal of Communications, 4(6), 2009, 388-395.
54. Naveed, R, Butt and Cheded, L, An improved CMA-based hybrid algorithm for blind channel equalization, 9th International Conference on Signal Processing, 2008,1726 - 1730.
55. Nissim Panizel and Monika Pinchas, An approximated expression for the residual ISI obtained by blind adaptive equalizer and biased input signals, Journal of Signal and Information Processing,5,2014,155-178.
56. Nikhil Deshpande, Fast recovery equalization techniques for DTV signals, IEEE Transactions on Broadcasting, 43(4), 1997,370 -377.
57. Perry R, David R Bull, and Nix A, Pipelined DFE architectures using delayed coefficient adaptation, IEEE Transactions on Circuits and Systems-II: Analog and Digital Signal Processing, 45(7),1998,868-873.
58. Pora W, Chambers JA and Constantinides AG, Combination of kalman filter and constant modulus algorithm with variable stepsize for the equalisation of fast-fading channels, Electronics Letters, 34(18), 1998, 1718 -1719.
59. Plackett, RL, Some Theorems in Least Squares, Biometrika,37, 1950,149-157, ISSN 00063444.
60. Prandini M, Campi MC, and Leonardi, R, Optimal delay estimation and performance evaluation in blind equalization, International Journal of Adaptive Control and Signal Processing, 11, 1997,621-640.
61. Qureshi SUH, Adaptive equalization, in proc.IEEE , 73, 1985,1349-1387.
62. Raja Uyyala P and Satyanarayana, Improving the steady state error and convergence based on variable step size constant modulus blind equalization algorithm, International Journal of Computer Science and Network Security,14(10), 2014,107-110.
116
63. Radhakrishna Y, and T Ravi Kumar Naidu, Improving the convergence
rate based on extend variable step size CMA, International Journal of Engineering Trends and Technology (IJETT), 4, 2013, 1878-1882.
64. Ram Babu T and P Rajesh Kumar, Blind channel equalization using CMA algorithm, International Conference on Advances in Recent Technologies in Communication and Computing, 2009, 680 - 683.
65. Rappaport Theodore S, Wireless communications: principles and practice,2nd ed, Pearson Education, India, 2010.
66. Revati Joshi, and Ashwinikumar Dhande, Adaptive beamforming using Lms algorithm,International Journal of Research in Engineering and Technology, 3,2014,589-593.
67. Roozbeh Hamzehyan, Reza Dianat and Najmeh Cheraghi Shirazi, ‘New variable step-size blind equalization based on modified constant modulus algorithm, International Journal of Machine Learning and Computing, 2(1), 2012.
68. Sayed, A, Hadei and Paeiz Azmi, A novel adaptive channel equalization method using variable step-size partial rank algorithm, in Proc. IEEE AICT, 2010,201-206.
69. Shihab Jimaa, Convergence evaluation of a random step-size NLMS adaptive algorithm in systemidentification and channel equalization, adaptive filtering, Dr Lino Garcia (Ed.), ISBN: 978-953-307-158-9,InTech. 2011
70. Shafayat Abrar and Asoke K Nandi, Blind equalization of square-QAM signals: a multimodulus approach, IEEE Transactions on Communications, 58(6), 2010,1674 -1685.
71. Simon Haykin: Adaptive Filter Theory, Prentice Hall 2002, ISBN 0-13-048434-2
72. Swathi, V, Rajani Katiyar, and Padmaja K, ‘Blind equalization based on modified constant modulus algorithm’, International Journal of Advanced Research in Computer and Communication Engineering,3 (5), 2014,6523-6525.
117
73. Thamer MJ, Al-anbaky, A performance improvements of adaptive FIR equalizer using modiefed version of VSSLMS algorithm stochastic gradient adaptive filter with gradient adaptive step size, Journal of Engineering,13, 2007,1889-1900.
74. Thamer, M, Jamel, Mohammed Abed Shabeeb Abed AL–Abass Muhseen Jassem, A hybrid variable step-size MCMA blind equalizer algorithm for QAM signals, IJCCCE, 14(1), 2014,35-43.
75. Thomas J, Endres Samir N, Hulyalkar Christopher, H Strolle, and Troy A. Schaffer, Low-complexity and low-latency implementation of the godard/CMA update, IEEE Transactions on Communications,49(2). 2002.
76. Timoleon Vaidis and Charles L, Weber, Block adaptive techniques for channel identification and data demodulation over band-limited channels, IEEE Transactions on Communications, 46(2),1998,232- 243.
77. Veeraruna Kavitha and Vinod Sharma, Comparision of training, blind and semi blind equalizers in MIMO fading systems using capacity as measure , in ICASSP, IEEE Conference, 2005,589-592.
78. Vicente Zarzoso and Pierre Comon, Blind channel equalization with algebraic optimal step size, EUSIPCO-2005, Antalya, Turkey, Sept. 2005,4-8.
79. Vijitha Weerackody and Saleem A, Kassam , A blind equalization algorithm using lattice filters, IEEE International Conference on Communications, 1, 1988,376 -379.
80. Vijitha Weerackody, Saleem A, Kassam Kenneth. and Laker R, A convergence model for the analysis of some blind equalization algorithms, IEEE International Symposium on Circuits and Systems, 3, 1989, 2136 -2139.
81. Vinod Sharma and V. Naveen Raj, Convergence and performance analysis of godard family and multimodulus algorithms for blind equalization, IEEE Transactions on Signal Processing, 53(4), 2005,1520 -1533.
82. Wang Junfeng and Zhang Bo, Design of adaptive equalizer based on variable step LMS algorithm, Proceedings of the Third International
118
Symposium on Computer Science and Computational Technology (ISCSCT ’10), 2010,256-258.
83. Wei Xue, Xiaoniu Yang and Zhaoyang Zhang, A variable step size algorithm for blind equalization of QAM signals, Progress In Electromagnetic Research Symposium Proceedings, Cambridge, USA, July 5-8, 2010.
84. Widrow B and Hoff Jr ME, Adaptive switching circuits, Wescon Conv. Rec, IRE, 1960,96 -104.
85. Wolfgang H, Gerstacker Robert FH Fischer Johannes and B Huber, Blind equalization for digital cable transmission with Tomlinson-Harashima precoding and shaping, IEEE International Conference on Communications,1, 1995, 493 - 497.
86. Xiong Z, Linsheng L, and Dongfeng Z, , A new adaptive step-size blind equalization algorithm based on autocorrelation of error signal,. In 7th International Conference on Signal Processing. Bejing (China),2, 2004,1719-722.
87. Xi-Lin Li and Xian-Da Zhang, A family of generalized constant modulus algorithms for blind equalization, IEEE Transactions on Communications, 54(11), 2006, 1913 - 1917.
88. Xue Wei, Yang Xiaoniu and Zhang Zhaoyang, A variable step size blind equalization algorithm for QAM signals’, in Proc. IEEE ICMMT, 2010,1801-1804.
89. Yangyang Fan, Xue Chen, Weiqin Zhou, Xian Zhou, and Hai Zhu, The comparison of CMA and LMS equalization algorithms in optical coherent receivers, 6th International Conference on Wireless Communications Networking and Mobile Computing , 2010, 1 - 4.
90. Ye Li and Zhi Ding, Convergence analysis of finite length blind adaptive equalizers, IEEE Transactions on Signal Processing, 43(9), 1995, 2120-2129.
91. Yoichi Sato, A method of self-recovering equalization for multilevel amplitude-modulation systems, IEEE Transactions on Communications, 1975, 679-682.
119
92. Yun Zhao , Xiaonan Xue and Tingfei Zhang, ‘Receiver-channel based adaptive blind equalization approach for GPS dynamic multipath mitigation,’ Chinese journal of Aeronautics , 26(2), 2013,378-384,
93. Yuan Gao and Xinyun Qiu, A new variable step size CMA blind equalization algorithm, 24th IEEE Chinese control and decision conference, 2012,315-317.
94. Zhao B, Zhao J, Zhang L, A variable step size constant modulus blind equalization algorithm based on the variation of MSE. Journal of Taiyuan University of Technology, 36(4), 2005.
95. Zhao Shengkui, Man Zhihong and Khoo Suiyang , A Fast Variable Step-Size LMS Algorithm with System Identification, 2nd IEEE conference on Industrial Electronics and Applications 2007,2007, 2340-2345.
120
LIST OF PUBLICATIONS
International Journals
1. K.Suthendran and T.Arivoli, Performance comparison of adaptive and
blind equalization algorithms for wireless communication, Bonfring
International journal of research in communication engineering, 3,
2013, 1- 6.
2. K.Suthendran and T.Arivoli, Performance comparison of blind
equalization algorithms for wireless communication, International
Journal of Computer Applications, 85(13), 2014, 1-6.
3. K.Suthendran and T.Arivoli, Variable tap parameter techniques for
sato based blind equalizer, International Journal Earth Sciences and
Engineering, 7(3), 2014, 1192-1198.
4. K.Suthendran and T.Arivoli, Performance comparison of variable step
size techniques of sato and godard based blind equalizer, Fluctations
and Noise Letters, 14(3), 2015,1550024-1 to 1550024-14.
121
International Conference
1. K.Suthendran and T.Arivoli, ‘Performance comparison of adaptive
and blind equalization algorithms for wireless communication,’
Proceedings. of International conference ICECI-12, 2012.
2. K.Suthendran and T.Arivoli, ‘Performance comparison of adaptive
equalization algorithms,’ Proceedings of International conference
ICIESMS-15, 2015.
122
CURRICULAM VITAE
The author, Suthendran. K, born on 15-08-1980, has graduated in
Electronics and Communication Engineering from Madurai Kamaraj
University in the year 2002. He did his post graduation study in
Communication Systems from Anna University, Chennai in the year 2006. He
has totally 13 years of professional experience which includes both industry
and teaching. He served as Test Engineer in the Research and Development
section of Matrixview Technologies, Chennai for a couple of years. Then he
joined in Kalasalingam University, Krishnankoil as a Faculty in the
Department of Information Technology. His research area includes Wireless
Communication, Signal and Image Processing.