SIM UNIVERSITY
SCHOOL OF SCIENCE AND TECHNOLOGY
PERFORMANCE ANALYSIS AND COMPARISON
OF ADAPTIVE CHANNEL EQUALIZATION
TECHNIQUES FOR
DIGITAL COMMUNICATION SYSTEM
STUDENT
: PYI PHYO MAUNG (M0706331)
SUPERVISOR
: DR LIM BOON LUM
PROJECT CODE: JUL2009/ENG/003
A project report submitted to SIM University
in partial fulfillment of the requirements for the degree of
Bachelor of Electronics Engineering
May 2010
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ABSTRACT
The objective of this project is to study, analyze and compare on three different types of
adaptive channel equalization techniques such as Least Mean Square (LMS), Recursive
Least Square (RLS) and Gradient Adaptive Lattice Algorithm.
This report consists of four main chapters which are Chapter 2 LMS algorithm, Chapter 3
RLS algorithm, Chapter 4 Gradient Adaptive Lattice algorithm and Chapter 5
Comparison Study. Much of these chapters are concerned with a detailed mathematical
analysis of three algorithms. Mathematical analysis is a very important tool, since it
allows many properties of the adaptive filter to be determined without expending the time
or expense of computer simulation or building actual hardware.
In this project, simulation and analysis use the MATLAB software. Simulation is often
used as a method to verify the mathematical analysis. The goals of the analysis in these
chapters are to determine the rate of convergence, misadjustment, tracking, robustness,
computational complexity and structure of the algorithms.
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ACKNOWLEDGEMENTS
Firstly, I would like to express my sincere and heartfelt appreciation to my project
supervisor, Dr. Lim Boon Lum for his exceptional guidance, invaluable advice and
wholehearted support in matters of practical and theoretical nature throughout the project.
His constant time check and meet ups had certainly motivated me in the completion of
the project. Throughout my report writing period, he provided encouragement, sound
advice, good teaching, good company, and lots of good ideas. The completion of the
Final Year Project would not be possible without his excellence supervision.
Secondly, I am indebted to my employer, Institute of Microelectronics (A*Star) for
allowing me to further study towards Bachelor Degree. I owe a favor to my reporting
officer; Dr. Selin Teo and colleagues for allowing me to take time off and exam leave
from work during the course of study.
Lastly, I would like to thank all the lectures and friends in UniSIM and also to my parent,
my wife and my daughter who have given their fullest support me throughout this
endeavor.
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TABLE OF CONTENTS
ABSTRACT .................................................................................................................... - 2 ACKNOWLEDGEMENTS ............................................................................................ - 3 TABLE OF CONTENTS ................................................................................................ - 4 LIST OF FIGURES ........................................................................................................ - 6 LIST OF TABLES .......................................................................................................... - 8 CHAPTER 1 ................................................................................................................... - 9 Introduction ..................................................................................................................... - 9 1.1
Background ................................................................................................. - 9 1.2
Review of Literature ................................................................................... - 9 1.3
Objective and Scope ................................................................................. - 12 CHAPTER 2 ................................................................................................................. - 13 The Least Mean Squares (LMS) Algorithm ................................................................. - 13 2.1 Introduction ..................................................................................................... - 13 2.2 Derivation of the LMS Algorithm .................................................................. - 14 2.2.1 Basic Idea of Gradient Search Method .................................................... - 14 2.2.2 Derivation of the LMS Algorithm ........................................................... - 16 2.2.3 Definition of Gradient search by Steepest Descent Method .................... - 21 2.3 Simulation Results .......................................................................................... - 25 2.3.1 Simulation Model..................................................................................... - 25 2.3.2 Channel Model 1 ...................................................................................... - 26 2.3.3 Channel Model 2 ...................................................................................... - 28 2.3.4 Channel Model 3 ...................................................................................... - 30 2.3.5 Channel Model 4 ...................................................................................... - 32 2.3.6 Observation & Analysis ........................................................................... - 34 CHAPTER 3 ................................................................................................................. - 36 The Recursive Least Squares (RLS) Algorithm ........................................................... - 36 3.1 Introduction ..................................................................................................... - 36 3.2 Derivation of the RLS Algorithm ................................................................... - 37 3.3 Simulation Results .......................................................................................... - 46 3.3.1 Channel Model 1 ...................................................................................... - 46 3.3.2 Channel Model 2 ...................................................................................... - 48 3.3.3 Channel Model 3 ...................................................................................... - 50 3.3.4 Channel Model 4 ...................................................................................... - 52 3.3.5 Observation & Analysis ........................................................................... - 54 CHAPTER 4 ................................................................................................................. - 56 The Gradient Adaptive Lattice (GAL) Algorithm ........................................................ - 56 4.1 Introduction ..................................................................................................... - 56 4.2 Derivation of the Gradient Adaptive Lattice Algorithm ................................. - 57 4.3 Simulation Results .......................................................................................... - 74 -4-
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4.3.1 Channel Model 1 ...................................................................................... - 74 4.3.2 Channel Model 2 ...................................................................................... - 76 4.3.3 Channel Model 3 ...................................................................................... - 78 4.3.4 Channel Model 4 ...................................................................................... - 80 4.3.5 Observation & Analysis ........................................................................... - 82 CHAPTER 5 ................................................................................................................. - 84 Comparison Study......................................................................................................... - 84 5.1 Simulation Results .......................................................................................... - 84 5.2 Observation and Analysis ............................................................................... - 88 5.3 Summary ......................................................................................................... - 90 CHAPTER 6 ................................................................................................................. - 92 Project Management ..................................................................................................... - 92 6.1 Project Plan and Schedule ............................................................................... - 92 6.2 Project Tasks Breakdown and Gantt chart ...................................................... - 94 CHAPTER 7 ................................................................................................................. - 96 REVIEW & REFLECTIONS ....................................................................................... - 96 7.1 Skills Review .................................................................................................. - 96 7.2 Reflections ...................................................................................................... - 97 7.3 Conclusions ..................................................................................................... - 98 References ..................................................................................................................... - 99 Appendix – A .............................................................................................................. - 101 Signal-Flow Graphs .................................................................................................... - 101 Appendix – B .............................................................................................................. - 102 Abbreviations .............................................................................................................. - 102 Appendix – C .............................................................................................................. - 103 Principal Symbols use in LMS ................................................................................... - 103 Principal Symbols use in RLS .................................................................................... - 104 Principal Symbols use in GAL ................................................................................... - 105 -
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LIST OF FIGURES
Fig-2.1 Block diagram representation of the LMS algorithm....................................... - 13 Fig-2.2 The LMS Adaptive Filter ................................................................................. - 16 Fig-2.3 Block Diagram of Adaptive Equalizer Experiment ......................................... - 25 Fig-2.4 Learning Curve of Channel Model 1................................................................ - 27 Fig-2.5 Channel Model and Equalizer Response of LMS Algorithm (Channel Model 1) .. 27 Fig-2.6 Learning Curve of LMS Algorithm (Channel Model 2) .................................. - 29 Fig-2.7 Channel and Equalizer Response of LMS Algorithm (Channel Model 2) ...... - 29 Fig-2.8 Learning Curve of LMS Algorithm (Channel Model 3) .................................. - 31 Fig-2.9 Channel and Equalizer Response of LMS Algorithm (Channel Model 3) ...... - 31 Fig-2.10 Learning Curve of LMS Algorithm (Channel Model 4) ................................ - 33 Fig-2.11 Channel and Equalizer Response of LMS Algorithm (Channel Model 4) .... - 33 Fig 3.1 - Block diagram representation of the RLS algorithm ..................................... - 36 Fig 3.2 – Transversal filter with time-varying tap weights ........................................... - 38 Fig-3.3 Learning Curve of RLS Algorithm (Channel Model 1) ................................... - 47 Fig-3.4 Channel and Equalizer Response of RLS Algorithm (Channel Model 1) ....... - 47 Fig-3.5 Learning Curve of RLS Algorithm (Channel Model 2) ................................... - 49 Fig-3.6 Channel and Equalizer Response of RLS Algorithm (Channel Model 2) ....... - 49 Fig-3.7 Learning Curve of RLS Algorithm (Channel Model 3) ................................... - 51 Fig-3.8 Channel and Equalizer Response of RLS Algorithm (Channel Model 3) ....... - 51 Fig-3.9 Learning Curve of RLS Algorithm (Channel Model 4) ................................... - 53 Fig-3.10 Channel and Equalizer Response of RLS Algorithm (Channel Model 4) ..... - 53 Fig 4.1 - Block diagram representation of the Lattice Structure................................... - 56 Fig 4.2 - Forward Prediction Error Filter ...................................................................... - 58 Fig 4.3 - Backward Prediction Error Filter ................................................................... - 60 Fig 4.4 - Single Stage Lattice Predictor ........................................................................ - 63 Fig 4.5 - Multistage Lattice Predictor ........................................................................... - 64 Fig 4.6 - Desired-response estimator using a sequence of m backward prediction errors ... 71 -
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Fig-4.7 Learning Curve of Lattice Algorithm (Channel Model 1) ............................... - 75 Fig-4.8 Channel and Equalizer Response of Lattice Algorithm (Channel Model 1).... - 75 Fig-4.9 Learning Curve of Lattice Algorithm (Channel Model 2) ............................... - 77 Fig-4.10 Channel and Equalizer Response of Lattice Algorithm (Channel Model 2).. - 77 Fig-4.11 Learning Curve of Lattice Algorithm (Channel Model 3) ............................. - 79 Fig-4.12 Channel and Equalizer Response of Lattice Algorithm (Channel Model 3).. - 79 Fig-4.13 Learning Curve of Lattice Algorithm (Channel Model 4) ............................. - 81 Fig-4.14 Channel and Equalizer Response of Lattice Algorithm (Channel Model 4).. - 81 Fig-5.1 Comparison Learning Curve of Channel Model 1 at SNR 20dB ..................... - 84 Fig- 5.2 Equalizer and Channel Model 1 response at SNR 20dB ................................. - 84 Fig-5.3 Comparison Leaning Curve of Channel Model 2 at SNR 20dB ...................... - 85 Fig- 5.4 Equalizer and Channel Model 2 response at SNR 20dB ................................. - 85 Fig-5.5 Comparison Learning Curve of Channel Model 3 at SNR 20dB ..................... - 86 Fig- 5.6 Equalizer and Channel Model 3 response at SNR 20dB ................................. - 86 Fig-5.7 Comparison Learning Curve of Channel Model 4 at SNR 20dB ..................... - 87 Fig- 5.8 Equalizer and Channel Model 4 response at SNR 20dB ................................. - 87 -
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LIST OF TABLES
Table- 2.1 Mean Square Error of LMS Algorithm ....................................................... - 34 Table- 2.2 Rate of Convergence of LMS Algorithm .................................................... - 34 Table- 3.1 Mean Square Error of RLS Algorithm ........................................................ - 54 Table- 3.2 Rate of Convergence of RLS Algorithm ..................................................... - 54 Table- 4.1 Mean Square Error of GAL Algorithm ....................................................... - 82 Table- 4.2 Rate of Convergence of GAL Algorithm .................................................... - 82 Table - 5.1 Mean Square Error of LMS, GAL & RLS Algorithms .............................. - 88 Table - 5.2 Number of Iterations to Converge in LMS, GAL & RLS Algorithms ....... - 89 Table - 6.1 Project Tasks Breakdown ........................................................................... - 94 Table - 6.2 Gantt chart .................................................................................................. - 95 -
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CHAPTER 1
Introduction
1.1 Background
Many problems encountered in communications and signal processing every day. These
are removing noise and distortion due to physical processes that are time varying or
unknown or possibly both. These types of processes represent some of the most difficult
problems in transmitting and receiving information.
The area of adaptive signal processing techniques provides one approach for removing
distortion in communications, as well as extracting information about unknown physical
processes. A short consideration of some of these problems show that distortion is often
present regardless of whether the communication is conversation between people or data
between physical devices.
One of the first applications of adaptive filtering in telecommunications was the
equalization of frequency-dependent channel attenuation in data transmission. The
frequency response of most data channels does not vary significantly with time.
However, this response is frequently not known in advance. Hence, it is necessary for the
data transmission system designer to build in the means to adapt to the channel
characteristics, and perhaps also track time-varying characteristics.
1.2 Review of Literature
The purpose of adaptive channel equalization is to compensate for signal distortion in a
communication channel. Communication systems transmit a signal from one point to
another across a communication channel, such as an electrical wire, a fiber-optic cable, or
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a wireless radio link. During the transmission process, the signal that contains
information might become distorted. To compensate for this distortion, we can apply an
adaptive filter to the communication channel. The adaptive filter works as an adaptive
channel equalizer. It monitors channel conditions continuously and to readjust itself when
required so as to provide optimum equalization.
System
Input
x
Channel
System
Output
Adaptive
Filter
y
Error
e(n)
Σ
+
Desired
Response
d(n)
Delay
Version
Fig 1.1 - Block diagram representation of Adaptive Channel Equalization
In general, an adaptive channel equalizer has a finite number of parameters that are
adjusted by adaptive algorithms to optimize some performance criterion. An adaptive
equalization filter adjusts to minimize the mean square error between actual and desired
response at the filter output.
In a wireless environment, the channel model is highly dependent on the physical
location of the reflectors. As objects move, the channel model must change accordingly.
In particular, if either the transmitter or receiver is mobile, every channel coefficient will
gradually change, and the adaptive channel equalizer must change accordingly.
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The most common adaptive algorithms which used in channel equalizer are Least Mean
Square (LMS), Recursive Least Square (RLS) and Gradient Adaptive Lattice Structure.
Different algorithms have different performances and some disadvantages. In the final
analysis, the choice of one algorithm over another is determined by one or more of the
following factors:
1. Rate of convergence
2. Misadjustment
3. Tracking
4. Robustness
5. Computational Complexity
6. Structure
The rate at which an adaptive filter converges to the system is important, particularly in
situations where the signal statistics are varying in time. Unfortunately, speed of adaption
is typically inversely proportional to computational complexity.
Misadjustment measures the offset between the actual signal and the signal produced by
the adaptive filter. Misadjustment is usually inversely proportional to both speed of
adaption and computational complexity.
When an adaptive filtering algorithm operates in a non-stationary environment, the
algorithm is required to track statistical variations in the environment. The tracking
performance of the algorithm is influenced by rate of convergence and the steady-state
fluctuation due to algorithm noise.
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For an adaptive filter to be robust, the small disturbances can only result in small
estimation errors. The disturbances may arise from a variety of factors, internal or
external to the filter.
Computational complexity measures the amount of computation which must be expended
at each time step in order to implement an adaptive algorithm and is often the governing
factor in whether real-time implementation is feasible.
The structure of algorithm determines the manner of information flow which is
implemented in hardware form. An algorithm whose structure exhibits high modularity,
parallelism, or concurrency is well suited for implementation using very large-scale
integration (VLSI).
1.3 Objective and Scope
The objective of this project is to study, analysis and comparison on three different types
of adaptive channel equalization technique such as Least Mean Square (LMS), Recursive
Least Square (RLS) and Gradient Adaptive Lattice Structure. The main focus of this
project is to analyze and investigate the advantages and shortfall of the three techniques
and also to determine an improving solution to provide a better channel equalization.
The project will include the following main tasks:
1. Study and analyze on Least Mean Square (LMS) algorithm
2. Study and analyze on Recursive Least Square (RLS) algorithm
3. Study and analyze on Gradient Adaptive Lattice algorithm
4. Simulate on MATLAB software and comparison
5. Evaluate and Implement to get better performance based on simulation result
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CHAPTER 2
The Least Mean Squares (LMS) Algorithm
2.1 Introduction
The Least Mean Square (LMS) algorithm, introduced by Widrow and Hoff in 1960, is a
linear adaptive filtering algorithm, which uses a gradient-based method of steepest
decent. The LMS algorithm uses the estimates of the gradient vector from the available
data and incorporates an iterative procedure that makes successive corrections to the
weight vector in the direction of the negative of the gradient vector which eventually
leads to the minimum mean square error.
The LMS algorithm consists of two basic processes. These are filtering process and
adaptive process. A filtering process involves computing the output of a linear filter in
response to an input signal and generating an estimation error by comparing this output
with a desired response. An adaptive process involves the automatic adjustment of the
parameters of the filter in accordance with the estimation error. The combination of these
two processes working together constitutes a feedback loop.
Input vector
X(k)
Y(k)
Transversal Filter

w (k )
Adaptive weightControl Mechanism
Output
Error
e(k)
-
Σ
+
Desired
Response
d(k)
Fig-2.1 Block diagram representation of the LMS algorithm
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The LMS algorithm is the simplest and is the most universally applicable adaptive
algorithm to be used. This algorithm uses a special estimate of the gradient that is valid
for the adaptive linear combiner. This algorithm is important because of its simplicity and
ease of computation.
It does not require explicit measurement of correlation functions, nor does it involve
matrix inversion. Accuracy is limited by statistical sample size, since the weight values
found are based on real-time measurements of input signals.
2.2 Derivation of the LMS Algorithm
2.2.1 Basic Idea of Gradient Search Method
   min   w  wopt 2
where  is eigenvalue equal to v00 in the univariable case
The first derivative,
d
 2 w  wopt  .....................................................................(2.1)
dw
The second derivative,
d 2
 2
dw 2
Simple Gradient Search Algorithm with only a single weight,
wk 1  wk     k  ..........................................................................(2.2)
From equation (2.1)
k 
d
dw
w wk
 2 wk  wopt 
Substitute in equation (2.2)
wk 1  wk  2  wk  wopt  ................................................................(2.3)
Rearranging equation (2.3)
wk 1  1  2 wk  2wopt .............................................................(2.4)
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The equation (2.4) is a linear, first order, constant-coefficient, ordinary difference
equation.
For few iterations, starting with initial guess w0 ,
The first three iterations,
w1  1  2 w0  2wopt
w2  1  2  w0  2wopt 1  2   1
2
w3  1  2  3 w0  2 wopt   1  2    1  2   1
2

From above result, generate for k th iterations:
k 1
wk  1  2  w0  2wopt  1  2 
k
n
n 0
 1  2  w0  2wopt
k
1  1  2 
1  1  2 
k
 1  2  w0  wopt  wopt 1  2 
k
k
wk  wopt  1  2  w0  wopt 
k
wk  wopt  1  2  w0  wopt  ................................................................(2.5)
k
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2.2.2 Derivation of the LMS Algorithm
Y(k)

Xk
Z-1
Xk
wN
w3
w2
w1
Z-1
X k2
Xk1
Xk(N1)
-

LMS ALGORITHM
ek
+
dk
Fig-2.2 The LMS Adaptive Filter
Weight Vector
 w0 
 w 
 1 
W   w2  ........................................................................................(2.6)


  
 wN 1 
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Input Signal Vector
 xk 
 x

 k 1 
X k   xk 2  ...................................................................................(2.7)


  
 xk ( N 1) 


Output Signal Vector
Yk  W T X k  X kT W
Yk  w0
w1
 xk 
 x

 k 1 
 wN 1   xk 2 


  
 xk ( N 1) 


w2
N 1
Yk   wi xk i ...................................................................................(2.8)
i 0
Error signal
ek  d k  Yk
ek  d k  W T X k
ek2   d k  W T X k
2
ek2  d k2  2d kW T X k  W T X k X kT W
  ek2
  
d k2
  d k2
  2 
d k X kT W  W T   X k X kT
W
  2 P T W  W T RW
 d k xk 
 d x

 k k 1 
where P    d k x k  2  ; cross correlation between d k and X k





d k x k ( N 1) 


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 xk xk
x x
 k 1 k
R    xk 2 xk

 
 

MSE
x k x k 1
x k 1 x k 1



xk xk 2

    ek2



 ; auto correlation matrix of X k


x k ( N 1) x k ( N 1) 


   d k2   2 P T W  W T RW ........................................................(2.9)
With stationary X k and d k , to minimize 


 
 d k2

2 P T W W T RW



W
W
W
W
0 
0 
     2 P  2 RW

 
0 
The gradient Vector of the performance function is
  2P  2RW ................................................................................(2.10)
To optimize, is to get Wiener Solution
Let   0 to get Wopt
 2 P  2 RWopt  0
2 RW opt  2 P
Wopt  R 1 P (Wiener Solution)..........................................................(2.11)
For  min , W  Wopt
 min   d k2   2PT Wopt  Wopt  T RWopt
 min   d k2   2 P T R 1 P  P T R T R R 1 P
 min   d k2   2 P T R 1 P  P T R T P
R is symmetric; R= R T
 
R T  R 1
T
 R 1
 min   d k2   2 P T R 1 P  P T R 1 P
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 min   d k2   P T R 1 P
 min   d k2   PT Wopt .....................................................................(2.12)
When W  Wopt ,
 W    Wopt   2PTW  W T RW  2PTWopt  Wopt T RWopt
  W T RW  Wopt  RWopt
T
  2P T W  Wopt 
  W  Wopt  RW  Wopt    2 P T R 1 RW  Wopt 
T
 W  Wopt  RW  Wopt   2Wopt  RW  Wopt 
T
T
 W  Wopt  RW  Wopt 
T
define V  W  Wopt ,
 V   V T RV   min ...........................................................................(2.13)

 (V )
 2 RV .............................................................................(2.14)
V
when   0 , V  Vopt , W  Wopt
  2 RVopt  0 ...................................................................................(2.15)
Finding eigenvalues / eigenvectors for a square matrix R, a vector q is called an
eigenvector of R.
If q  0 ,
R q   q ;  is called an eigenvalues of R
R q   q  0
R   I  q  0
So,  is an eigenvalues for non-zero q
If R  I  is singular
RQ   Q
Q is eigenvector matrix of R
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q00
q
Q =  10
q20

 
q01 q02 

q11
 ...................................................................(2.16)




0
0

 is eigenvalues matrix =  0


 0
0
0
1
2
0


 .......................................(2.17)



n 
R  Q 1 Q Normal form
The eigenvector matrix Q can be made orthogonal
QT Q  I
Q 1  QT ................................................................................(2.18)
From equation (2.13)
   min  V T RV
   min  V T QQ 1V
  min  V ' V '
T
where V '  Q 1V  QT V
Summary of 3 coordinate systems principal
Natural: W
   min  W  Wopt T RW  Wopt 
Translation: V  W  Wopt
   min  V T RV
Rotation: V '  QTV  Q 1V , V  QV '
   min  V ' T V '
- 20 -
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2.2.3 Definition of Gradient search by Steepest Descent Method
The weights are adjusted in the direction of the gradient at each step which the
function is guaranteed to be reduced in value. The direction in which function decreases
most rapidly is determined by the negative of the gradient   of function at the current
point.
Wk 1  Wk     k 
Wk 1  Wk    2 RV 
Wk 1  Wk  2RWopt  Wk 
Wk 1  Wk  2RWopt  2 RW k
Subtract both side Wopt ;
Wk 1  Wopt  Wk  Wopt  2RWopt  2RWk
Wk 1  Wopt  I  2R Wk  Wopt  .................................................................(2.19)
Transforming to the principal coordinate system,
Translation: V  W  Wopt
Equation (2.19) become
Vk 1  I  2R Vk ..........................................................................................(2.20)
Rotation: V  QV ' , Vk 1  QVk'1 , Vk  QV k'
QVk'1  I  2R QVk'
Multiply with Q 1 both sides,
Q 1QVk'1  Q 1 I  2 R QVk'


Vk'1  Q 1 IQ  2Q 1 RQ Vk' .........................................................................(2.21)
Recall equation (2.16) and (2.17)  R  QQ 1 , Q 1 RQ  
Substitute in equation (2.21) and Q 1 IQ  I
Vk'1  I  2  Vk'
After n th iteration, compare with equation (2.5)
Vk'  I  2 V0'
k
- 21 -
______________________________________________________________________________
1  2 0 k

'
Vk  



1  21 k
1  22 k


V '
 0


The steepest descent algorithm is stable and convergent
When Limk  I  2  0
k
  1  2 p   1 ;  p is eigenvalues where p = 0, 1, 2, 3,…, N-1
 1  1  2 p  1
 2   2  p
and
 2  p  0 ......................................................(2.22)
From equation (2.22),
2  2 p
 p  1
max is maximum eigenvalue

1
max
.............................................................................................(2.23)
From equation (2.22)
 2  p  0
 p  0
  0 ..................................................................................................(2.24)
Combine equation (2.23) and (2.24)
0  
1
max
........................................................................................(2.25)
Effects on the weight –vector solution
Referring to equation (2.10)
  2P  2RW
Let  k = the gradient estimation at k th iteration

Define  k   k  N k ....................................................................................(2.26)
- 22 -
______________________________________________________________________________
Where N k  the gradient estimation noise vector at kth iteration
( vector of size L+1)

 k  the gradient estimation at the kth iteration with the gradient
estimation noise
Steepest descent formula without noise,
Wk 1  Wk     k 
Steepest descent formula with gradient noise,
  
Wk 1  Wk      k  ....................................................................................(2.27)


Substitute equation (2.26) into equation (2.27)
Wk 1  Wk    k  N k  ................................................................................(2.28)
Wk 1  Wk   k  N k
Subtract Wopt both sides,
Wk 1  Wopt  Wk  Wopt   k  N k
Translate coordinate vector, V  W  Wopt ,
Vk 1  Vk   k  N k
Recall equation (2.14),   2RV
Vk 1  Vk  2RVk  N k
Equation (2.28) become,
Vk 1  I  2R  Vk  N k ..............................................................................(2.29)
Relate to principal coordinate using V  QV ' , N  QN ' and N '  Q 1 N
QVk'1  I  2 R  QVk'  N k
Multiply with Q 1 both sides,


Vk'1  Q 1 IQ  2Q 1 RQ Vk'  Q 1 N k
Vk'1  I  2  Vk'  N k' ..............................................................................(2.30)
- 23 -
______________________________________________________________________________
Consider the first three iterations,
V1'  I  2 V0'  N 0'
V2'  I  2 V1'  N1'
V2'  I  2 V0'   I  2N0'  N1'
2

V2'  I  2 V0'   I  2N0'  N1'
2

V3'  I  2 V2'  N 2'
V3'  I  2 V0'   I  2 N0'   I  2N1'  N 2'
3
2
V3'  I  2 V0'   I  2 N 0'  I  2N1'  N 2'
3
2

Equation (2.30) obtained at kth iteration,
k 1
Vk'  I  2  V0'    I  2  N k' i 1 ...................................................(2.31)
k
i
i 0
Recall equation (2.25),  is in the stability range 0   
1
max
,
Multiply equation (2.25) with 2max
0  2max   2 .................................................................................................(2.32)
Subtract equation (2.32) from 1,
1  1  2max   1  2
1  1  2max    1
For the worse case of equation (2.31), k  
Then the first term of equation (2.31) becomes negligible,
 the steady-state effects of gradient noise on the weight vector solution is

Vk'     I  2   N k' i 1 ...........................................................................(2.33)
i
i 0
- 24 -
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2.3 Simulation Results
2.3.1 Simulation Model
This simulation studies the use of the LMS algorithm for adaptive equalization of a linear
dispersive channel that produces distortion. This simulation uses 4 different channel
models which have 13th order complex signal, 10th order real and 3rd order real signal.
Fig-2.3 shows the block diagram of the system model used to carry out the simulation in
MATLAB software. Random-number generator (1) provides the test signal x(n), used for
probing the channel. Random-number generator (2) serves as the source of additive white
noise v(n) that corrupts the channel output. These two random-number generators are
independent of each other.
Random-number
Generator (2)
v(n)
X(n)
Random-number
Generator (1)
Σ
Channel
Adaptive
Equalizer
y
Error
e(n)
Delay
Version
Σ
+
Desired
Response
d(n)
Fig-2.3 Block Diagram of Adaptive Equalizer Experiment
The adaptive equalizer has the task of correcting for the distortion produced by the
channel in the presence of the additive white noise. By using suitable delay, randomnumber generator (1) also supplies the desired response to the adaptive equalizer in the
form of a training sequence.
- 25 -
______________________________________________________________________________
2.3.2 Channel Model 1
Simulation Condition
Channel transfer function:
H(z) = [ (0.10-0.03j) + (-0.20+0.15j) z 1 + (0.34+0.27j) z 2 + (0.33+0.10j) z 3 +
(0.40-0.12j) z 4 + (0.20+0.21j) z 5 + (1.00+0.40j) z 6 + (0.50-0.12j) z 7 +
(0.32-0.43j) z 8 + (-0.21+0.31j) z 9 + (-0.13+0.05j) z 10 + (0.24+0.11j) z 11 +
(0.07-00.06j) z 12 ]
Number of tap of LMS adaptive filter: 25
Delay of desired response: 18
Misadjustment of output: 10%
Signal-to-noise ratio SNR: 3dB, 10dB & 20dB
Number of run for averaging: 100
Number of iteration: 2000
- 26 -
______________________________________________________________________________
Learning Curve of LMS Algorithm
2
Channel Model 1
SNR 3
SNR 10
SNR 20
1.8
1.6
Mean Square Error
1.4
1.2
1
0.8
0.6
0.4
0.2
0
0
200
400
600
1000 1200
800
Number of Iteration
1400
1600
1800
2000
Fig-2.4 Learning Curve of Channel Model 1
Channel Response & Equalizer Response
3.5
3
Channel Model 1
SNR 3
SNR 10
SNR 20
LMS Algorithm
Channel Model 1
Frequency Response
2.5
2
1.5
1
0.5
0
0
0.1
0.2
0.3
0.4
0.5
0.6
Normalized Frequency
0.7
0.8
0.9
1
Fig-2.5 Channel Model and Equalizer Response of LMS Algorithm (Channel Model 1)
- 27 -
______________________________________________________________________________
2.3.3 Channel Model 2
Simulation Condition
Channel transfer function:
H(z) = [ (0.6) + (-0.17) z 1 + (0.1) z 2 + (0.5) z 3 + (-0.19) z 4 + (0.01) z 5 + (-0.03) z 6
+ (0.2) z 7 + (0.05) z 8 + (0.1) z 9 ]
Number of tap of LMS adaptive filter: 19
Delay of desired response: 14
Misadjustment of output: 10%
Signal-to-noise ratio SNR: 3dB, 10dB & 20dB
Number of run for averaging: 100
Number of iteration: 2000
- 28 -
______________________________________________________________________________
Learning Curve of LMS Algorithm
2
Channel Model 2
SNR 3
SNR 10
SNR 20
1.8
1.6
Mean Square Error
1.4
1.2
1
0.8
0.6
0.4
0.2
0
0
200
400
600
800
1000 1200
Number of Iteration
1400
1600
1800
2000
Fig-2.6 Learning Curve of LMS Algorithm (Channel Model 2)
Channel Response & Equalizer Response
2.5
2
Channel Model 2
SNR 3
SNR 10
SNR 20
LMS Algorithm
Frequency Response
Channel Model 2
1.5
1
0.5
0
0
0.1
0.2
0.3
0.4
0.5
0.6
Normalized Frequency
0.7
0.8
0.9
1
Fig-2.7 Channel and Equalizer Response of LMS Algorithm (Channel Model 2)
- 29 -
______________________________________________________________________________
2.3.4 Channel Model 3
Simulation Condition
Channel transfer function:
H(z) = [ (0.6) + (-0.17) z 1 + (0.1) z 2 + (0.5) z 3 + (-0.5) z 4 + (-0.01) z 5 + (-0.03) z 6 +
(0.2) z 7 + (-0.05) z 8 + (0.1) z 9 ]
Number of tap of LMS adaptive filter: 19
Delay of desired response: 14
Misadjustment of output: 10%
Signal-to-noise ratio SNR: 3dB, 10dB & 20dB
Number of run for averaging: 100
Number of iteration: 2000
- 30 -
______________________________________________________________________________
Learning Curve of LMS Algorithm
2
Channel Model 3
SNR 3
SNR 10
SNR 20
1.8
1.6
Mean Square Error
1.4
1.2
1
0.8
0.6
0.4
0.2
0
0
200
400
600
800
1000 1200
Number of Iteration
1400
1600
1800
2000
Fig-2.8 Learning Curve of LMS Algorithm (Channel Model 3)
Channel Response & Equalizer Response
2.5
2
Channel Model 3
SNR 3
SNR 10
SNR 20
LMS Algorithm
Frequency Response
Channel Model 3
1.5
1
0.5
0
0
0.1
0.2
0.3
0.4
0.5
0.6
Normalized Frequency
0.7
0.8
0.9
1
Fig-2.9 Channel and Equalizer Response of LMS Algorithm (Channel Model 3)
- 31 -
______________________________________________________________________________
2.3.5 Channel Model 4
Simulation Condition
Channel transfer function: H(z) = 1+2.2 z 1 +0.4 z 2
Number of tap of LMS adaptive filter: 5
Delay of desired response: 3
Misadjustment of output: 10%
Signal-to-noise ratio SNR: 3dB, 10dB & 20dB
Number of run for averaging: 100
Number of iteration: 2000
- 32 -
______________________________________________________________________________
Learning Curve of LMS Algorithm
2
Channel Model 4
SNR 3
SNR 10
SNR 20
1.8
1.6
Mean Square Error
1.4
1.2
1
0.8
0.6
0.4
0.2
0
0
200
400
600
800
1000 1200
Number of Iteration
1400
1600
1800
2000
Fig-2.10 Learning Curve of LMS Algorithm (Channel Model 4)
Channel Response & Equalizer Response
4
Channel Model 4
SNR 3
SNR 10
SNR 20
3.5
Frequency Response
3
LMS Algorithm
Channel Model 4
2.5
2
1.5
1
0.5
0
0
0.1
0.2
0.3
0.4
0.5
0.6
Normalized Frequency
0.7
0.8
0.9
1
Fig-2.11 Channel and Equalizer Response of LMS Algorithm (Channel Model 4)
- 33 -
______________________________________________________________________________
2.3.6 Observation & Analysis
The Fig- 2.4, 2.6, 2.8 and 2.10 are learning curves of four different channel models with
three different SNR values. By observing the learning curves, all channel models have
lower mean square error at 20dB and better rate of convergence at 3dB. But the mean
square error is very high at 3dB. It means better SNR value gives smaller mean square
error.
Mean Square Error of LMS Algorithm
Channel Model
3dB
10dB
20dB
1
1.1
0.5
0.18
2
1
0.45
0.25
3
0.85
0.41
0.18
4
1
0.38
0.09
Table- 2.1 Mean Square Error of LMS Algorithm
Estimate Number of Iterations ( LMS Algorithm )
Channel Model
3dB
10dB
20dB
1
200
500
1000
2
180
250
400
3
200
300
500
4
80
160
200
Table- 2.2 Rate of Convergence of LMS Algorithm
By observing rate of convergence at 20dB, channel model 1 has slower rate of
convergence than channel model 4. Because channel model 1 is complex signal and it
- 34 -
______________________________________________________________________________
takes longer time to converge. Channel model 4 is real value signal with 3rd order.
Therefore, channel model 4 converges very fast.
The results of learning curves confirm that the rate of convergence and mean squared
error of the adaptive equalizer is highly dependent on the signal-to-noise ratio (SNR) and
type of channel model.
The Fig- 2.5, 2.7, 2.9 and 2.11 show the comparison between channel model and
equalizer response. The mean square error of 10dB is about twice of 20dB but the
equalizer response at 10dB is comparable to the equalizer response at 20dB.
The equalizer response at 20dB is the best tracking and gives an approximate inversion of
the channel response because of its lower mean square error. However, there are some
differences due to the existence of the 10% misadjustment.
Based on the learning curves and channel response Vs equalizer response simulation
results, the equalizer perform very well at better SNR values.
- 35 -
______________________________________________________________________________
CHAPTER 3
The Recursive Least Squares (RLS) Algorithm
3.1 Introduction
The RLS algorithm as a natural extension of the method of least squares to develop and
design of adaptive transversal filters such that, given the least squares estimate of the tapweight vector of the filter at iteration n  1 . Therefore, we may compute the updated
estimate of the vector at iteration n upon the arrival of new data. The derivation based on
a lemma in matrix algebra known as the matrix inversion lemma.
An important feature of this algorithm is that its rate of convergence is typically an order
of magnitude faster than that of the simple LMS algorithm. However, this improvement
of performance is achieved at the expense of an increase in computational complexity of
the RLS algorithm.
Input vector
x(i)

y(i)  wH (n 1)x(i)
Transversal Filter

w ( n  1)
Output
Adaptive weightControl Mechanism
Σ
Error
e (i)
+
Desired
Response
d(i)
Fig 3.1 - Block diagram representation of the RLS algorithm
- 36 -
______________________________________________________________________________
3.2 Derivation of the RLS Algorithm
The Matrix Inversion Lemma
Let A and B be two positive-definite M-by-M matrices
C is positive-definite M-by-N matrix
D is positive-definite N-by-M matrix


Define A  B 1  CD 1C H .....................................................................................(3.1)
According to matrix inversion lemma, inverse of matrix A as

A1  B  BC D  C H BC

1
C H B ...................................................................(3.2)
To prove the matrix inversion lemma, multiply eq(3.1) by eq(3.2)

AA1  B 1  CD 1C H
 B  BC D  C
H
BC

1
CH B

AA 1  B 1 B  B 1 BC D  C H BC  C H B  CD 1C H B
1
 CD C BC D  C BC  C B
1
H
1
H


To show that AA1  I . Since D  C H BC D  C H BC
....................................(3.3)
H

1
 I and B 1 B  I
Rewrite eq(3.3)
AA1  I  C D  C H BC  C H B  CD 1 D  C H BC D  C H BC  C H B
1
1
 CD 1C H BC D  C H BC  C H B
1


AA1  I   C  CD 1 D  C H BC  CD 1C H BC
D  C H BC 1 C H B
AA1  I   C  CD 1 D  CD 1C H BC  CD 1C H BC
AA1  I   C  CD 1 D
D  C H BC 1 C H B
D  C H BC 1 C H B ...............................................................(3.4)
Since D 1 D  I , the second term from right hand side of eq(3.4),
Therefore AA1  I
- 37 -
 C  CD 1D  0
______________________________________________________________________________
RLS Algorithm
Z-1
Z-1
Z-1
w1 ( i )
w 0 (i )
x(i M 1)
x(i M  2)
x(i 1)
x (i)
wM 1 (i )
wM  2 ( i )



dˆ ( i X i )
e( i )

+
d( i )
Fig 3.2 – Transversal filter with time-varying tap weights
Input vector
 xi 
 x

 i 1 
X i   xi  2 


  
 xi ( M 1) 


Weight vector
 w0 
 w 
 1 
W   w2 


  
 wM 1 
 n i  exponential weighting factor, i = 1,2,3,…., n
- 38 -
______________________________________________________________________________
e(i )  d (i )  y (i )
 d (i)  w H (n  1) x(i) ...........................................................................(3.5)
Define Cost function
n
 n     n i ei   n wn 
2
2
,n 1
i 1
where  is regulation parameter,   0
n
 n     ni e 2 i   n wT n wn  ...............................................................(3.6)
i 1
Substitute eq (3.5) into eq (3.6)
 n     n i d i   w H n xi    n w H n wn 
n
2
i 1

   w
   n i d 2 i   2d i w H n xi   w H n xi 
n
2
n
H
n wn 
i 1




 n
  n

    n i d 2 i   2d i w H n xi       n i w H n xi  x T i wn   n w H n wn 
 i 1
  i 1

n
 n

n

    n i d 2 i   2  n i d i w H n xi   w H n   n i xi x T i    n  wn  ...(3.7)
i 1
 i 1

 i 1



The expression for  n can be made more concise by introducing Rn (auto-correlation
matrix) and z n  (cross-correlation matrix)
Rn     n i xi  x T i    n .......................................................................(3.8)

n
i 1
- 39 -
______________________________________________________________________________

n
z n     n i d i  xi  ......................................................................................(3.9)
i 1
eq (3.8) and (3.9) substitute in eq (3.7)


n
 n      ni d 2 i   2w H n z n   w H n  Rn  wn  .............................................(3.10)
 i 1

partial derivative of  n with respect to w
 n  
 n 
 2 z n   2 Rn wn 
w
setting  n  0, to determine optimal weight
0  2Rnwn  zn
optimal weight is

wn   R 1 n z n  ..........................................................................................(3.11)
Reformulate the required computations to make easy, the solution will start at iteration
(n-1)
 eq (3.8) become
Rn     n i xi  x T i    n
n
i 1
Rn     n i xi  x T i   xi  x T i    n
n 1
i 1
- 40 -
______________________________________________________________________________
Rn      n i 1 xi  x T i   xi  x T i    n
n 1
i 1
 n 1

Rn      n i 1 xi  x T i    n 1   xn  x T n 
 i 1



 n 1

Rn     n i 1 xi x T i   n 1   xn x T n  ...........................................(3.12)
 i 1

the expression inside the bracket on the right hand side of eq: (3.12) equals Rn  1
Rn   Rn  1  xn x T n  ..........................................................................(3.13)
Similarly for eq(3.9), Reformulate the required computations to make easy, the solution
will start at iteration (n-1)
n
z n     n i d i  xi 
i 1
z n     n i d i  xi   d n  xn 
n 1
i 1
 n 1

z n      n i 1 d i  xi    d n  xn  .........................................................(3.14)
 i 1



the expression inside the bracket on the right hand side of eq: (3.14) equals z n  1
zn  zn  1  d nxn .............................................................................(3.15)
- 41 -
______________________________________________________________________________

To compute wn  optimal weight vector in accordance with eq (3.11), we have to
determine R 1 n by using matrix inversion lemma.
Assumed auto-correlation matrix R(n) is nonsingular and invertible.
Eq(3.13): Rn   Rn  1  xn x T n 

Eq(3.1) : A  B 1  CD 1C H

Make following identifications:
A  Rn
B 1  Rn  1
C  xn
D 1  1
Apply matrix inversion lemma to eq (3.13)

Referring to eq(3.2): A1  B  BC D  C H BC


1
C H B , inverse of Rn is

R 1 n   1 R 1 n  1  1 R 1 n  1 xn  1  x T n  1R 1 n  1xn 
 1 R 1 n  1 
1
x T n 1 R 1 n  1
1 R 1 n  1xn x T n 1 R 1 n  1
...................................(3.16)
1  1 x T n R 1 n  1xn 
For convenience of computation,
Let
Pn   R 1 n  .................................................................................................(3.17)
- 42 -
______________________________________________________________________________
k n  
1 Pn  1 xn 
.....................................................................(3.18)
1  1 x T n Pn  1xn 
Substitute eq(3.17) and eq(3.18) into eq(3.16)
R 1 n   1 R 1 n  1 
Pn   1 Pn  1 
1 R 1 n  1xn x T n 1 R 1 n  1
1  1 x T n R 1 n  1xn 
1 Pn  1xn x T n 1 Pn  1
1  1 x T n Pn  1xn 
Pn  1 Pn  1  k n x T n1 Pn  1 ......................................................(3.19)
Rearranging eq(3.18)
k n  

1 Pn  1 xn 
1  1 x T n Pn  1xn 

k n  1  1 x T n Pn  1xn  1 Pn  1xn 
k n  1k n x T nPn  1xn   1 Pn  1xn
k n  1 Pn  1xn  1k nx T n Pn  1xn
k n   1 Pn  1  1k nx T nPn  1
 xn ...................................(3.20)
the expression inside the bracket on the right-hand side of eq(3.20) is equal to eq(3.19),
Pn
 eq(3.20) become k n  P n x n ......................................................................(3.21)
- 43 -
______________________________________________________________________________
substitute eq(3.17) into eq(3.21)
Gain vector k n   R 1 n  x n  .................................................................................(3.22)
Time Update for the tap-weight vector
From eq(3.11)

wn   R 1 n z n 
substitute eq(3.17) and eq: (3.15) into eq(3.11)

wn   Pnz n
 Pn zn  1  d nxn
 Pnzn  1  Pn d nxn ............................................................(3.23)
Substitute eq(3.19) into eq(3.23); only first term Pn on right-hand side



wn    1 Pn  1  k n  x T n 1 Pn  1 z n  1  Pn d n xn 
 Pn  1z n  1  k nx T nPn  1z n  1  Pnd nxn
Since Pn   R 1 n  and Pn  1  R 1 n  1 ,

wn   R 1 n  1z n  1  k n x T n R 1 n  1z n  1  R 1 n d n xn  ...................(3.24)
substitute eq(3.22) and eq(3.11) into eq(3.24)

k n   R 1 n  x n  and wn   R 1 n z n 

wn   R 1 n  1z n  1  k n x T n R 1 n  1z n  1  R 1 n d n xn 
- 44 -
______________________________________________________________________________




 wn  1  k n x T n  wn  1  R 1 n d n xn 
 wn  1  k n x T n  wn  1  k n d n 




 wn  1  k n d n   x T n  wn  1 .............................................................(3.25)



substitute eq(3.5) into eq(3.25), en   d n   x T n  wn  1


wn   wn  1  k n en  ..........................................................................................(3.26)
eq(3.26) for the adjustment of the tap-weight vector
eq(3.5) for the a priorio estimation error
Summary of RLS
Initialize the algorithm by setting

w0  0
P0   1 I , where  is regulation parameter,   0
For each instant time, n = 1,2,3,…, compute
 n  Pn  1 x n
k n  
 n 
  x T n  n 

en   d n   x T n  wn  1


wn   wn  1  k n en 
Pn  1 Pn  1  k n x T n1 Pn  1
- 45 -
______________________________________________________________________________
3.3 Simulation Results
This simulation studies the use of RLS algorithm with the exponential weighting
factor   0.999 , for the adaptive equalization of a linear dispersive communication
channel. The simulation model is the same model as LMS algorithm simulation model in
Section 2.3. (Fig-2.3)
3.3.1 Channel Model 1
Simulation Condition
Channel transfer function:
H(z) = [ (0.10-0.03j) + (-0.20+0.15j) z 1 + (0.34+0.27j) z 2 + (0.33+0.10j) z 3 +
(0.40-0.12j) z 4 + (0.20+0.21j) z 5 + (1.00+0.40j) z 6 + (0.50-0.12j) z 7 +
(0.32-0.43j) z 8 + (-0.21+0.31j) z 9 + (-0.13+0.05j) z 10 + (0.24+0.11j) z 11 +
(0.07-00.06j) z 12 ]
Number of tap of RLS adaptive filter: 25
Delay of desired response: 18
Misadjustment of output: 10%
Signal-to-noise ratio SNR: 3dB, 10dB & 20dB
Number of run for averaging: 100
Number of iteration: 2000
Exponential weighting factor   0.999
Regulation parameter  = 0.01
- 46 -
______________________________________________________________________________
Learning Curve of RLS Algorithm
2
SNR 3
SNR 10
SNR 20
Channel Model 1
1.8
Mean Square Error
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0
0
200
400
600
800
1000 1200
Number of Iteration
1400
1600
1800
Fig-3.3 Learning Curve of RLS Algorithm (Channel Model 1)
Channel Response & Equalizer Response
3.5
3
Channel Model 1
SNR 3
SNR 10
SNR 20
RLS Algorithm
Channel Model 1
Frequency Response
2.5
2
1.5
1
0.5
0
0
0.1
0.2
0.3
0.4
0.5
0.6
Normalized Frequency
0.7
0.8
0.9
1
Fig-3.4 Channel and Equalizer Response of RLS Algorithm (Channel Model 1)
- 47 -
______________________________________________________________________________
3.3.2 Channel Model 2
Simulation Condition
Channel transfer function:
H(z) = [ (0.6) + (-0.17) z 1 + (0.1) z 2 + (0.5) z 3 + (-0.19) z 4 + (0.01) z 5 + (-0.03) z 6
+ (0.2) z 7 + (0.05) z 8 + (0.1) z 9 ]
Number of tap of RLS adaptive filter: 19
Delay of desired response: 14
Misadjustment of output: 10%
Signal-to-noise ratio SNR: 3dB, 10dB & 20dB
Number of run for averaging: 100
Number of iteration: 2000
Exponential weighting factor   0.999
Regulation parameter  = 0.01
- 48 -
______________________________________________________________________________
Learning Curve of RLS Algorithm
Mean Square Error
2
SNR 3
SNR 10
SNR 20
Channel Model 2
1.5
1
0.75
0.5
0.25
0
0
200
400
600
800 1000 1200
Number of Iteration
1400
1600
1800
Fig-3.5 Learning Curve of RLS Algorithm (Channel Model 2)
Channel Response & Equalizer Response
2.5
2
Channel Model 2
SNR 3
SNR 10
SNR 20
RLS Algorithm
Frequency Response
Channel Model 2
1.5
1
0.5
0
0
0.1
0.2
0.3
0.4
0.5
0.6
Normalized Frequency
0.7
0.8
0.9
1
Fig-3.6 Channel and Equalizer Response of RLS Algorithm (Channel Model 2)
- 49 -
______________________________________________________________________________
3.3.3 Channel Model 3
Simulation Condition
Channel transfer function:
H(z) = [ (0.6) + (-0.17) z 1 + (0.1) z 2 + (0.5) z 3 + (-0.5) z 4 + (-0.01) z 5 + (-0.03) z 6 +
(0.2) z 7 + (-0.05) z 8 + (0.1) z 9 ]
Number of tap of RLS adaptive filter: 19
Delay of desired response: 14
Misadjustment of output: 10%
Signal-to-noise ratio SNR: 3dB, 10dB & 20dB
Number of run for averaging: 100
Number of iteration: 2000
Exponential weighting factor   0.999
Regulation parameter  = 0.01
- 50 -
______________________________________________________________________________
Learning Curve of RLS Algorithm
SNR 3
SNR 10
SNR 20
2.5
Mean Square Error
2.25
2
Channel Model 3
1.75
1.5
1.25
1
0.75
0.5
0.25
0
0
200
400
600
800 1000 1200
Number of Iteration
1400
1600
1800
Fig-3.7 Learning Curve of RLS Algorithm (Channel Model 3)
Channel Response & Equalizer Response
2.5
2
Channel Model 3
SNR 3
SNR 10
SNR 20
RLS Algorithm
Frequency Response
Channel Model 3
1.5
1
0.5
0
0
0.1
0.2
0.3
0.4
0.5
0.6
Normalized Frequency
0.7
0.8
0.9
1
Fig-3.8 Channel and Equalizer Response of RLS Algorithm (Channel Model 3)
- 51 -
______________________________________________________________________________
3.3.4 Channel Model 4
Simulation Condition
Channel transfer function: H(z) = 1+2.2 z 1 +0.4 z 2
Number of tap of RLS adaptive filter: 5
Delay of desired response: 3
Misadjustment of output: 10%
Signal-to-noise ratio SNR: 3dB, 10dB & 20dB
Number of run for averaging: 100
Number of iteration: 2000
Exponential weighting factor   0.999
Regulation parameter  = 0.01
- 52 -
______________________________________________________________________________
Learning Curve of RLS Algorithm
Mean Square Error
2
SNR 3
SNR 10
SNR 20
Channel Model 4
1.5
1
0.5
0
0
200
400
600
800
1000 1200
Number of Iteration
1400
1600
1800
Fig-3.9 Learning Curve of RLS Algorithm (Channel Model 4)
Channel Response & Equalizer Response
4
Channel Model 4
SNR 3
SNR 10
SNR 20
3.5
Frequency Response
3
RLS Algorithm
Channel Model 4
2.5
2
1.5
1
0.5
0
0
0.1
0.2
0.3
0.4
0.5
0.6
Normalized Frequency
0.7
0.8
0.9
1
Fig-3.10 Channel and Equalizer Response of RLS Algorithm (Channel Model 4)
- 53 -
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3.3.5 Observation & Analysis
The Fig- 3.3, 3.5, 3.7 and 3.9 are learning curves of four different channel models with
three different SNR values. By observing the learning curves, all channel models have
lower mean square error and faster rate of convergence at 20dB. It means better SNR
value gives smaller mean square error and faster rate of convergence in RLS algorithm.
The steady-state values of mean square error is very high in the first few iterations and
jump down immediately after this. The mean square error is high at 3dB but the rate of
convergence is comparable to the rate of convergence at 10dB.
Mean Square Error of RLS Algorithm
Channel Model
3dB
10dB
20dB
1
0.95
0.43
0.1
2
0.82
0.38
0.25
3
0.85
0.35
0.13
4
0.8
0.35
0.09
Table- 3.1 Mean Square Error of RLS Algorithm
Estimate Number of Iterations ( RLS Algorithm )
Channel Model
3dB
10dB
20dB
1
150
120
100
2
100
100
80
3
100
100
60
4
80
60
20
Table- 3.2 Rate of Convergence of RLS Algorithm
- 54 -
______________________________________________________________________________
At SNR 20dB, the mean square error of channel model 1 (complex) and channel model 4
(real) are 0.1 and 0.09 respectively. The RLS algorithm gives smaller mean square error
and it does not depend on type of channel. However, the rate of convergence of complex
channel is slower than the real value channel.
The results of learning curves confirm that the rate of convergence is very fast in RLS
algorithm and mean squared error of the adaptive equalizer is highly dependent on the
signal-to-noise ratio (SNR) and independent on the type of channel.
The Fig- 3.4, 3.6, 3.8 and 3.10 show the comparison between channel model and
equalizer response. The equalizer response of channel model 2, 3, and 4 (real) is better
tracking and gives an approximate inversion of the channel response compare to the
equalizer response of channel model 1 (complex). The misadjustment (10%) also does
not significantly differences in channel model 2, 3 and 4 (real).
- 55 -
______________________________________________________________________________
CHAPTER 4
The Gradient Adaptive Lattice (GAL) Algorithm
4.1 Introduction
The gradient adaptive lattice algorithm is a natural extension of the least mean square
filter in that both types of filter rely on a stochastic gradient approach for their
algorithmic implementations. The Fig- 4.1 shows the basic structure for the estimation of
a desired response which is based on a multistage lattice structure that performs both
forward f(n) and backward b(n) predictions. Firstly, we may derive the recursive order
updates for the multistage lattice predictor. Then, we may derive the corresponding
updates for the desired-response estimator.
Stage 1
f 0 ( n)
Stage M
Σ
f M 1 (n)
f1 ( n)
1
M
 1*
 M*
fM (n)
Σ
Input
Signal
x(n)
b0 (n)
Z-1
h 0*
Σ
b1 ( n )
b M  1 ( n)
Z-1
Σ
h 1*
hM* 1
h M*
Σ
Σ
Σ
bM (n)
YM(n)=Estimate of
desired response
d(n)
Fig 4.1 - Block diagram representation of the Lattice Structure
The order-recursive adaptive filters derived from the stochastic gradient approach are
simple to design, but approximate in nature. The simplicity of design results from the fact
that each stage of the lattice predictor is characterized by a single reflection coefficient.
- 56 -
______________________________________________________________________________
The lattice filters can be viewed as stage-by-stage orthogonalization of input data. This
property results in faster convergence than the tapped-delay-line implementation of LMS
filter. When a lattice filter is used as a linear predictor, the predictor order can be
increased simply by adding another lattice section without changing any of the previous
section. The tapped-delay-line implementation of the linear predictor does not have this
useful modularity property. The lattice filters have become popular in the adaptive signal
processing because of their fast convergence and modularity property.
4.2 Derivation of the Gradient Adaptive Lattice Algorithm
Levinson-Durbin Algorithm
Let
a m = the tap weight (m+1) x 1 vector of a forward prediction error filter of order m
B*
a m = the tap weight (m+1) x 1 vector of a backward prediction error filter of order m and
their complex conjugate
a m1 = the tap weight m x 1 vector of a forward prediction error filter of order m-1
B*
a m 1 = the tap weight m x 1 vector of a backward prediction error filter of order m and
their complex conjugate
 m = constant (reflection coefficient)
f m (n) = forward prediction error
bm (n) = backward prediction error
- 57 -
______________________________________________________________________________
Z-1
am,0 (n)
x(n  2)
x(n  1)
x(n )
am,1 (n)

x (n  M )
x(nM1)
Z-1
Z-1
am,2 (n)
am,M1(n)


am,M (n)
fm-1(n)

Fig 4.2 - Forward Prediction Error Filter
A forward predictor may modify into a backward predictor by reversing the sequence in
which its tap weights are positioned and also taking the complex conjugates of them.
Order updated equation of a forward prediction-error filter
 0 
a 
a m   m1    m  B*  .....................................................................(4.1)
 0 
a m1 
In expended form
 am,0   am 1,0 
 0 
 a  
a*


 m,1   am 1,1 
 m 1, m 1 
        m   

 
 *


am, m 1  am 1, m 1 
 am 1,1 
 am, m   0 
 am* 1, 0 




Equivalently
am,0  am1,0   m am* 1,m
, then am* 1,m  0
am,m  am1,m   m am* 1,0
, then am1,m  0
am,k  am1,k   m am* 1,mk
, k  0, 1, 2, , m ....................................(4.2)
- 58 -
fm(n)
______________________________________________________________________________
Let m  k  l , m  l  k , then substitute in eq (4.2), am* 1,m  0 and am1,m  0
am,ml  am1,ml   m am* 1,l
, l  0, 1, 2, , m ...................................(4.3)
Complex conjugate both side of eq(4.3)
am* ,ml  am* 1,ml   m* am1,l
, l  0.1, 2, , m .....................................(4.4)
In expended form
 am* , m   0 
 am 1, 0 
 *   *

 a

am , m 1  am 1, m 1 
 m 1,1 
         m*   
 *   *



 am ,1   am 1,1 
am 1, m 1 
 a*   am* 1, 0 
 0 

 m,0  
am* ,m  am* 1,m   m* am1,0
, then am* 1,m  0
am* ,0  am* 1,0   m* am1,m
, then am1,m  0
Order updated equation of a backward prediction-error filter
 0 
a 
B*
a m   B*    m*  m1  ....................................................................(4.5)
 0 
a m1 
- 59 -
______________________________________________________________________________
x(n  2)
x(n  1)
x(n )
Z-1
am* ,M (n)
am* ,M1(n)
x (n  M )
x(nM1)
Z-1
Z-1
am*,M2(n)
am* ,1 (n)



am* ,0 (n)
bm-1(n)

Fig 4.3 - Backward Prediction Error Filter
Order-update Recursions for the prediction errors
Let
f m (n) = the forward prediction error produced at the output of the forward predictionerror filter of order m
f m1 (n) = the forward prediction error produced at the output of the forward predictionerror filter of order m-1
bm (n) = the backward prediction error produced at the output of the backward prediction-
error filter of order m
bm 1 (n  1) = the delayed backward prediction error produced at the output of the
backward prediction-error filter of order m-1
M = stage of lattice predictor
 m = reflection coefficient
x(n) = sample value of tap input in transversal filter at time n
- 60 -
bm(n)
______________________________________________________________________________
x m1 (n) = (m+1) x 1 tap input vector
 x m ( n) 
x m1 (n)     .....................................................................................(4.6)
 x(n  m)
Equivalent form
 x ( n) 
x m1 (n)     .....................................................................................(4.7)
 x m (n  1)
Consider the forward prediction-error filter of order m
Form the inner product of the (m+1) x 1 vector a m and x m1 (n) by pre-multiplying
x m1 (n) by Hermitian transpose of a m
(1) Multiply the left hand side of eq(4.1) and the left hand side of eq(4.6),
f m (n)  a m x m1 (n) ........................................................................................(4.8)
H
(2) Multiply the first term on the right hand side of eq(4.1) and the right hand side of
eq(4.6),
a
H
m1


 0 x m1 (n)  a
H
m1
 a m1 x m (n)
 x m ( n) 
 0   
 x(n  m)

H
 f m1 (n)
..........................................................................(4.9)
(3) Multiply the second term on the right hand side of eq(4.1) and the right hand side of
eq(4.7),
- 61 -
______________________________________________________________________________
0
 a
BT
m1
x
m1

( n)  0  a
BT
m1
 a m1 x m (n  1)

 x ( n) 
  


 x m (n  1)
BT
.....................................................................(4.10)
 bm1 (n  1)
Combine the results of multiplication eq(4.8), eq(4.9) and eq(4.10),
f m (n)  f m 1 (n)   m* bm 1 (n  1) .....................................................................(4.11)
Consider the backward prediction-error filter of order m
B*
Form the inner product of the (m+1) x 1 vector a m 1 and x m1 (n) by pre-multiplying
B*
x m1 (n) by Hermitian transpose of a m 1
(1) Multiply the left hand side of eq(4.5) and the left hand side of eq(4.6),
bm (n)  a m x m1 (n) .......................................................................................(4.12)
BT
(2) Multiply the first term on the right hand side of eq(4.5) and the right hand side of
eq(4.7),
0
 a
BT
m1
x
m1

( n)  0  a
 a m1 x m (n  1)
BT
m1

 x ( n) 
  


 x m (n  1)
BT
 bm1 (n  1)
.....................................................................(4.13)
(3) Multiply the second term on the right hand side of eq(4.5) and the right hand side of
eq(4.6),
- 62 -
______________________________________________________________________________
a
H
m1


 0 x m1 (n)  a
 x m ( n) 
 0   
 x(n  m)

H
m1
 a m1 x m (n)
H
 f m1 (n)
..........................................................................(4.14)
Combine the results of multiplication eq(4.12), eq(4.13) and eq(4.14),
bm (n)  bm1 (n  1)   m f m1 (n) ....................................................................(4.15)
Matrix form of combined eq(4.11) & eq(4.15)
 f m (n)  1  m*   f m 1 (n) 

 b ( n)   
 , m  1, 2,, M .......................................(4.16)
 m   m 1  bm 1 (n  1)
bm1 (n  1)  z 1 bm1 (n)
f m1 (n)
Σ
fm(n)
Σ
bm(n)

*
bm1(n)
Z-1
Fig 4.4 - Single Stage Lattice Predictor
- 63 -
______________________________________________________________________________
Multistage Lattice Predictor
Stage 1
f0 (n)
Stage M
f M1 (n)
f1(n)
Σ
1
m
 1*
 m*
Σ
f M (n)
Input
Signal
x(n)
b0 ( n)
Σ
Z-1
b1 ( n )
bM1 (n)
Z-1
Σ
bM (n)
Fig 4.5 - Multistage Lattice Predictor
Assume that input data are wide-sense stationary and  m is complex value
 m*   m
Define Cost function is
Jm 


1
2
2
 f m (n)  bm (n) .........................................................................(4.17)
2
Substitute eq(4.11) & eq(4.15) into eq(4.17),  m* bm1 (n  1)   m bm* 1 (n  1) and
 m f m1 (n)   m* f m*1 (n)
Jm 
2
1 
2
 f m 1 (n)   m* bm 1 (n  1)  bm 1 (n  1)   m f m 1 (n) 


2 
Jm 
2
1
2
2
  f m 1 (n)  2 f m 1 (n) m* bm 1 (n  1)   m* bm 1 (n  1)  bm 1 (n  1)
2
 2 bm 1 (n  1) m f m 1 (n)   m f m 1 (n)
- 64 -
2

______________________________________________________________________________
Re-arrange above equation as:
Jm 
2
1 
2
2
2
 f m 1 (n)   m* bm 1 (n  1)  bm 1 (n  1)   m f m 1 (n)  


2 
 f m 1 (n) m* bm 1 (n  1)    bm 1 (n  1) m f m 1 (n)

...........(4.18)
Differentiating eq(4.18), J m with respect to the reflection coefficient  m ,
f m1 (n)bm* 1 (n  1)  bm1 (n  1) f m*1 (n)





J m 1  

2
2 
 
E  m f m 1 (n) 
E  m bm 1 (n  1) 
 m 2   m
 m




 m  f m 1 (n)bm* 1 (n  1) 
 m*  bm 1 (n  1) f m*1 (n)
 m
 m




 
 


 
 




J m

2
2
  m E f m1 (n)  E bm1 (n  1)   f m1 (n)bm* 1 (n  1) 
 m  bm* 1 (n  1) f m1 (n)
 m
 m

J m
2
2
  m E f m1 (n)  E bm1 (n  1)  2 E bm1 (n  1) f m*1 (n) ............................(4.19)
 m
To optimize the reflection coefficient,   0 , eq(4.19) become
 m ,o  
2 E bm1 (n  1) f m*1 (n)
E f m1 (n)
2
  E

bm1 (n  1)
2

..................................................................(4.20)
Eq(4.20) for the optimum reflection coefficient  m,o is also known as Burg Formula.
Replacing the expectation operator E with time average operator
1 n
 , thus we get the
n i 1
Burg estimate for the reflection coefficient  m,o for stage m in the lattice filter. Eq (4.20)
become
- 65 -

______________________________________________________________________________
2
̂ m (n)  
1 n
bm1 (i  1) f m*1 (i )

n i 1

1 n
2
2
f m1 (i )  bm1 (i  1)

n i 1

n
̂ m (n)  
2 bm1 (i  1) f m*1 (i )
i 1
 f
n
i 1
(i )  bm1 (i  1)
2
m 1
2

...........................................................................(4.21)
Reformulate this estimator into an equivalent recursive structure
For denominator of eq(4.21)
Define total energy of both forward and backward prediction errors
n

 m1 (n)   f m1 (i)  bm1 (i  1)
i 1
n 1

2
2
 .......................................................................(4.22)

 m1 (n)   f m1 (i)  bm1 (i  1)  f m1 (n)  bm1 (n  1) ................................(4.23)
i 1
2
2
n 1

2
2

Since  m1 (n  1)   f m1 (i )  bm1 (i  1) , eq(4.23) become
i 1
2
2
 m1 (n)   m1 (n  1)  f m1 (n)  bm1 (n  1) ..........................................................(4.24)
2
2
For numerator of eq(4.21)
n
n 1
i 1
i 1
 bm1 (i  1) f m*1 (i)    bm1 (i  1) f m*1 (i)   bm1 (n  1) f m*1 (n) ...........................(4.25)
Substituting eq(4.24) and (4.25) into eq(4.21)
- 66 -
______________________________________________________________________________
n 1
ˆ m (n)  


2 bm1 (i  1) f m*1 (i )  2bm1 (n  1) f m*1 (n)
i 1
 m1 (n  1)  f m1 (n)  bm1 (n  1)
2
..................................................(4.26)
2
Form the recursively computing the estimate ˆ m (n) , use the time-varying estimate
ˆ m (n  1) in place of  m to rewrite eq(4.11) and (4.15)
f m (n)  f m 1 (n)  ˆm* (n  1) bm 1 (n  1)
f m1 (n)  f m (n)  ˆ m* (n  1) bm1 (n  1) ......................................................................(4.27)
bm (n)  bm 1 (n  1)  ˆm (n  1) f m 1 (n)
bm1 (n  1)  bm (n)  ˆ m (n  1) f m1 (n) .......................................................................(4.28)
Rearrange 2nd numerator term of eq(4.26), bm1 (n  1) f m*1 (n)  f m*1 (n)bm1 (n  1)
2bm 1 (n  1) f m*1 (n)  bm 1 (n  1) f m*1 (n)  f m*1 (n)bm 1 (n  1) ......................................(4.29)
Substitute eq(4.27) and (4.28) into eq(4.29)


2bm1 (n  1) f m*1 (n)  bm1 (n  1) f m (n)  ˆ m* (n  1) bm1 (n  1)  f m*1 (n)bm (n)  ˆ m (n  1) f m1 (n) 
*


 bm1 (n  1) f m* (n)  ˆ m (n  1) bm* 1 (n  1)  f m*1 (n)bm (n)  ˆ m (n  1) f m1 (n) 
 bm1 (n  1) f m* (n)  ˆ m (n  1) bm1 (n  1)  f m*1 (n)bm (n)  ˆ m (n  1) f m1 (n)
2
 ˆ m (n  1) f m1 (n)  bm1 (n  1)
2
2
2
   f m*1 (n)bm (n)  bm1 (n  1) f m* (n) ..(4.30)
- 67 -
______________________________________________________________________________
Add the extra term ˆm (n  1) m1 (n  1) in eq(4.30) and substitute eq(4.24)
 ˆ m (n  1)  m1 (n  1)  f m1 (n)  bm1 (n  1)
2
2
  ˆ m (n  1) m1 (n  1)
  f m*1 (n)bm (n)  bm1 (n  1) f m* (n) 


 ˆm (n  1) m 1 (n)  ˆm (n  1) m 1 (n  1)  f m*1 (n)bm (n)  bm 1 (n  1) f m* (n) ..........(4.31)
Rearranging eq(4.21) to become recursively and substitute eq(4.22),
n
Eq(4.21): ̂ m (n)  
2 bm1 (i  1) f m*1 (i )
i 1
 f
n
i 1
(i )  bm1 (i  1)
2
m 1
2

n
ˆ m (n)  
2 bm1 (i  1) f m*1 (i )
i 1
 m1 (n)
n
2 bm1 (i  1) f m*1 (i )  ˆ m (n) m1 (n)
i 1
n 1
2 bm1 (i  1) f m*1 (i )  ˆ m (n  1) m1 (n  1) .................................................................(4.32)
i 1
Substitute eq(4.31) and (4.32) into numerator of eq(4.26)
n 1


2 bm1 (i  1) f m*1 (i )  2bm1 (n  1) f m*1 (n)
i 1

 ˆ m (n  1) m1 (n  1)  ˆ m (n  1) m1 (n)  ˆ m (n  1) m1 (n  1)  f m*1 (n)bm (n)  bm1 (n  1) f m* (n)


 ˆ m (n  1) m1 (n)  f m*1 (n)bm (n)  bm1 (n  1) f m* (n) ............................................(4.33)
- 68 -

______________________________________________________________________________
Combine numerator and denominator of eq(4.26) using eq(4.33) and eq(4.24)
n 1
Eq(4.26) : ˆ m (n)  
ˆ m (n)  
ˆ m (n) 
2 bm1 (i  1) f m*1 (i )  2bm1 (n  1) f m*1 (n)
i 1
 m1 (n  1)  f m1 (n)  bm1 (n  1)
2
2

 ˆ m (n  1) m1 (n)  f m*1 (n)bm (n)  bm1 (n  1) f m* (n)
 m1 (n)

ˆ m (n  1) m1 (n)  f m*1 (n)bm (n)  bm1 (n  1) f m* (n) 

 m1 (n)
 m1 (n)
ˆ m (n)  ˆ m (n  1) 
f
*
m 1

(n)bm (n)  bm 1 (n  1) f m* (n)
, m  1, 2,  , M . .....................(4.34)
 m 1 (n)
Modification to recursive formulas of Gradient Lattice Filter
Let
 = step-size parameter
 = constant (0    1)
 m = the sum of squared forward and backward prediction errors
 m   bm (n)
2  
f m (n)
2 .......................................................................(4.35)
 m1 (n) = total energy of both forward and backward prediction errors (define Eq
4.22)
Recall the eq(2.2), Simple Gradient Search method used in LMS algorithm with only a
single weight
wk 1  wk     k  ..........................................................................(2.2)
where  k 


w
- 69 -
______________________________________________________________________________
Apply Gradient- based method for updating the reflection coefficients
ˆ m (n)  ˆ m (n  1)     m  ..............................................................(4.36)
where  m 

 m
 m .........................................................................(4.37)
Substitute eq(4.35) into eq(4.37),
m 

 m
 bm (n) 2 

 m

2
f m ( n)
Apply chain rule for partial derivatives of backward and forward prediction errors
 m  2bm (n)

 m
 bm (n)   2 f m (n)

 m

f m (n)  ................................................(4.38)
Substitute eq(4.11) & eq(4.15) into eq(4.38)
 m  2bm (n)

 m
 bm1 (n  1)   m
f m 1 (n)   2 f m (n)

 m

f m 1 (n)   m* bm 1 (n  1)

 m  2bm (n) f m*1 (n)  2 f m* (n) bm 1 (n  1)
 m  2  f m*1 (n)bm (n)  bm 1 (n  1) f m* (n)  .............................................................(4.39)
Substitute eq(4.39) into eq(4.36)
ˆ m (n)  ˆ m (n  1)  2

f m*1 (n)bm (n)  bm 1 (n  1) f m* (n)  .......................................(4.40)
Eq(4.40) introduce 2  (step-size parameter) with a single weight.
For forward and backward prediction, 2  (step-size parameter) need to divide by total
energy  m1 (n) , that control the adjustment applied each reflection coefficient in
progressing from current iteration to the next iteration.
- 70 -
______________________________________________________________________________
Eq(4.40) become,
ˆ m (n)  ˆ m (n  1) 
2
 f m*1 (n)bm (n)  bm1 (n  1) f m* (n), m  1, 2, , M . ..........(4.41)
 m 1 (n)
Re-write Eq(4.24) in the form of a single-pole averaging filter that operates on the square
prediction errors
 m1 (n)    m 1 (n  1)  1    f m 1 (n)  bm1 (n  1)
2
2
 .....................................(4.42)
Desired-Response Estimator
Backward Prediction Errors
bm1(n)
bm (n)
hˆ 1*
hˆm* 1
hˆm*
Σ
Σ
Σ
b0 (n)
b1 (n)
ĥ0*
Ym(n)=Estimate of
desired response
d(n)
Fig 4.6 - Desired-response estimator using a sequence of m backward prediction errors
Let
b m (n) = backward prediction error vector
*
h m ( n) = weight vector
h m (n  1) = updated weight vector
*
- 71 -
______________________________________________________________________________
d * (n)  b m (n)h m (n  1) = desired response vector
H
*
h m (n  1) = adjustment weight vector
*
ym (n) = output signal, estimate of desired response d(n)
em* (n) = estimation error
2
b m (n) = squared Euclidean norm of b m (n)
m
y m ( n)   h k ( n) b k ( n)
*
k 0
y m ( n)  
m 1
h
*
k
k 0
(n) b k (n)   hm* (n) bm (n)
ym (n)  ym (n  1)  hm* (n) bm (n) ................................................................................(4.43)
Estimation error is defined by
em* (n)  d * (n)  y m (n)
em* (n)  d * (n)  b m (n)h m (n) .....................................................................................(4.44)
H
*
Squared Euclidean Norm
m
b m (n)   bk (n)
2
2
k 0
b m ( n)  
2
m 1
 b ( n)   b
k 0
2
k
m
( n)
2
b m (n)  b m1 (n)  bm (n) ...................................................................................(4.45)
2
2
2
Re-write Eq(4.45) in the form of a single-pole averaging filter
- 72 -
______________________________________________________________________________
b m ( n)
2
  b m 1 (n)  1    bm (n) ...................................................................(4.46)
2
2
Time update for the mth regression coefficient
h m (n  1)  h m (n  1)  h m (n) .................................................................................(4.47)
*
*
1

*
b m (n)b m (n) h m (n  1)  h m (n)
H
b m ( n)
2
where
*
1
1
b m ( n)
2
 b m (n)b m (n)
b m (n) b m (n)h m (n  1)  b m (n)h m (n)  ...................................................(4.48)
H
2

H
b m ( n)

*
*
H
*
Substitute d * (n)  b m (n)h m (n  1) ,
H
1

b m ( n)
*
b m (n) d * (n)  b m (n)h m (n)
H
2
*
 ..................................................................(4.49)
Apply eq(4.44), em* (n)  d * (n)  b m (n)h m (n)
H
1

b m ( n)
2
*
b m (n)e* (n) ...............................................................................................(4.50)
Equivalent right hand side of eq(4.47) and eq(4.50),
h m (n  1)  h m (n) 
*
*
1
b m ( n)
2
bm (n) em* (n) .................................................................(4.51)
eq(4.51) introduce 2  (step-size parameter) that control the adjustment applied each
iteration
h m (n  1)  h m (n) 
*
*
2
b m ( n)
2
bm (n) em* (n) .................................................................(4.52)
- 73 -
______________________________________________________________________________
4.3 Simulation Results
This simulation studies the use of Gradient Adaptive Lattice algorithm with the step-size
parameter   0.001 and the constant   0.9 , for the adaptive equalization of a linear
dispersive communication channel. The simulation model is the same model as LMS
algorithm simulation model in Section 2.3. (Fig-2.3)
4.3.1 Channel Model 1
Simulation Condition
Channel transfer function:
H(z) = [ (0.10-0.03j) + (-0.20+0.15j) z 1 + (0.34+0.27j) z 2 + (0.33+0.10j) z 3 +
(0.40-0.12j) z 4 + (0.20+0.21j) z 5 + (1.00+0.40j) z 6 + (0.50-0.12j) z 7 +
(0.32-0.43j) z 8 + (-0.21+0.31j) z 9 + (-0.13+0.05j) z 10 + (0.24+0.11j) z 11 +
(0.07-00.06j) z 12 ]
Number of tap of Lattice adaptive filter: 25
Delay of desired response: 18
Misadjustment of output: 10%
Signal-to-noise ratio SNR: 3dB, 10dB & 20dB
Number of run for averaging: 100
Number of iteration: 2000
Step-size parameter,  : 0.001
Constant, lying in the range (0,1)  : 0.9
- 74 -
______________________________________________________________________________
Learning Curve of Lattice Algorithm
2
Channel Model 1
SNR 3
SNR 10
SNR 20
1.8
1.6
Mean Square Error
1.4
1.2
1
0.8
0.6
0.4
0.2
0
0
200
400
600
1000 1200
800
Number of Iteration
1400
1600
1800
2000
Fig-4.7 Learning Curve of Lattice Algorithm (Channel Model 1)
Channel Response & Equalizer Response
3.5
Channel Model 1
SNR 3
SNR 10
SNR 20
3
Frequency Response
2.5
Lattice Algorithm
Channel Model 1
2
1.5
1
0.5
0
0
0.1
0.2
0.3
0.4
0.5
0.6
Normalized Frequency
0.7
0.8
0.9
1
Fig-4.8 Channel and Equalizer Response of Lattice Algorithm (Channel Model 1)
- 75 -
______________________________________________________________________________
4.3.2 Channel Model 2
Simulation Condition
Channel transfer function:
H(z) = [ (0.6) + (-0.17) z 1 + (0.1) z 2 + (0.5) z 3 + (-0.19) z 4 + (0.01) z 5 + (-0.03) z 6
+ (0.2) z 7 + (0.05) z 8 + (0.1) z 9 ]
Number of tap of Lattice adaptive filter: 19
Delay of desired response: 14
Misadjustment of output: 10%
Signal-to-noise ratio SNR: 3dB, 10dB & 20dB
Number of run for averaging: 100
Number of iteration: 2000
Step-size parameter,  : 0.001
Constant, lying in the range (0,1)  : 0.9
- 76 -
______________________________________________________________________________
Learning Curve of Lattice Algorithm
2
SNR 3
SNR 10
SNR 20
1.8
Channel Model 2
1.6
Mean Square Error
1.4
1.2
1
0.8
0.6
0.4
0.2
0
0
200
400
600
1000 1200
800
Number of Iteration
1400
1600
1800
2000
Fig-4.9 Learning Curve of Lattice Algorithm (Channel Model 2)
Channel Response & Equalizer Response
2.5
2
Channel Model 2
SNR 3
SNR 10
SNR 20
Lattice Algorithm
Frequency Response
Channel Model 2
1.5
1
0.5
0
0
0.1
0.2
0.3
0.4
0.5
0.6
Normalized Frequency
0.7
0.8
0.9
1
Fig-4.10 Channel and Equalizer Response of Lattice Algorithm (Channel Model 2)
- 77 -
______________________________________________________________________________
4.3.3 Channel Model 3
Simulation Condition
Channel transfer function:
H(z) = [ (0.6) + (-0.17) z 1 + (0.1) z 2 + (0.5) z 3 + (-0.5) z 4 + (-0.01) z 5 + (-0.03) z 6 +
(0.2) z 7 + (-0.05) z 8 + (0.1) z 9 ]
Number of tap of Lattice adaptive filter: 19
Delay of desired response: 14
Misadjustment of output: 10%
Signal-to-noise ratio SNR: 3dB, 10dB & 20dB
Number of run for averaging: 100
Number of iteration: 2000
Step-size parameter,  : 0.001
Constant, lying in the range (0,1)  : 0.9
- 78 -
______________________________________________________________________________
Learning Curve of Lattice Algorithm
2
SNR 3
SNR 10
SNR 20
1.8
Channel Model 3
1.6
Mean Square Error
1.4
1.2
1
0.8
0.6
0.4
0.2
0
0
200
400
600
1000 1200
800
Number of Iteration
1400
1600
1800
2000
Fig-4.11 Learning Curve of Lattice Algorithm (Channel Model 3)
Channel Response & Equalizer Response
2.2
2
Lattice Algorithm
1.8
Channel Model 3
Channel Model 3
SNR 3
SNR 10
SNR 20
Frequency Response
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0
0.1
0.2
0.3
0.4
0.5
0.6
Normalized Frequency
0.7
0.8
0.9
1
Fig-4.12 Channel and Equalizer Response of Lattice Algorithm (Channel Model 3)
- 79 -
______________________________________________________________________________
4.3.4 Channel Model 4
Simulation Condition
Channel transfer function: H(z) = 1+2.2 z 1 +0.4 z 2
Number of tap of Lattice adaptive filter: 5
Delay of desired response: 3
Misadjustment of output: 10%
Signal-to-noise ratio SNR: 3dB, 10dB & 20dB
Number of run for averaging: 100
Number of iteration: 2000
Step-size parameter,  : 0.001
Constant, lying in the range (0,1)  : 0.9
- 80 -
______________________________________________________________________________
Learning Curve of Lattice Algorithm
2
SNR 3
SNR 10
SNR 20
1.8
1.6
Channel Model 4
Mean Square Error
1.4
1.2
1
0.8
0.6
0.4
0.2
0
0
200
400
600
800
1000 1200
Number of Iteration
1400
1600
1800
2000
Fig-4.13 Learning Curve of Lattice Algorithm (Channel Model 4)
Channel Response & Equalizer Response
4
Channel Model 4
SNR 3
SNR 10
SNR 20
3.5
Frequency Response
3
2.5
Lattice Algorithm
Channel Model 4
2
1.5
1
0.5
0
0
0.1
0.2
0.3
0.4
0.5
0.6
Normalized Frequency
0.7
0.8
0.9
1
Fig-4.14 Channel and Equalizer Response of Lattice Algorithm (Channel Model 4)
- 81 -
______________________________________________________________________________
4.3.5 Observation & Analysis
The Fig- 4.7, 4.9, 4.11 and 4.13 are learning curves of four different channel models with
three different SNR values. By observing the learning curves, all channel models have
almost same mean square error at each SNR value. For example, channel model 1 and
channel model 2 have 1.05 (MSE) at 3dB SNR, 0.45 and 0.43 (MSE) at 10dB. Therefore,
GAL algorithm gives almost same mean square error and does not depend on the type of
channel model. The better SNR value gives smaller mean square error.
Mean Square Error of Lattice Algorithm
Channel Model
3dB
10dB
20dB
1
1.05
0.45
0.18
2
1.05
0.43
0.22
3
0.98
0.4
0.15
4
0.88
0.42
0.08
Table- 4.1 Mean Square Error of GAL Algorithm
Estimate Number of Iterations ( Lattice Algorithm )
Channel Model
3dB
10dB
20dB
1
200
400
600
2
160
220
380
3
180
220
380
4
60
100
180
Table- 4.2 Rate of Convergence of GAL Algorithm
- 82 -
______________________________________________________________________________
The rate of convergence depends on both SNR values and type of channel model. By
observing different channel models, channel model 4 (real) is faster converging than
channel model 1 (complex). By observing different SNR values, the rate of convergence
at 20dB is slower than the rate of convergence at 3dB.
The results of learning curves confirm that the rate of convergence is fast in channel
model 4 (real) at 3dB SNR value. But the mean squared error of the adaptive equalizer is
highly dependent on the signal-to-noise ratio (SNR) and independent on the type of
channel.
The Fig- 4.8, 4.10, 4.12 and 4.14 show the comparison between channel model and
equalizer response. The equalizer response of all channel models has better tracking at
10dB and 20dB. The equalizer response at 3dB can not give approximate inversion of the
channel response. Between channel response and equalizer response have some
differences due to the existence of the 10% misadjustment.
Based on the learning curves and channel response Vs equalizer response simulation
results, the equalizer perform very well at real channel model with better SNR values.
- 83 -
______________________________________________________________________________
CHAPTER 5
Comparison Study
5.1 Simulation Results
Learning Curve of SNR 20dB
LMS
RLS
Lattice
1.8
1.6
1
0.8
1.2
Channel Model 1
0.7
0.6
1
0.5
0.4
0.8
0.3
0.2
0.6
0.1
50
0.4
100
150
200
250
300
350
400
450
0.2
0
0
100
200
300
400
500
600
Number of Iteration
700
800
900
Fig-5.1 Comparison Learning Curve of Channel Model 1 at SNR 20dB
Channel Response & Equalizer Response
3.5
Channel Model 1
LMS
RLS
Lattice
3
2.5
Frequency Response
Mean Square Error
1.4
0.9
SNR 20dB
Channel Model 1
2
1.5
1
0.5
0
0
0.1
0.2
0.3
0.4
0.5
0.6
Normalized Frequency
0.7
0.8
0.9
1
Fig- 5.2 Equalizer and Channel Model 1 response at SNR 20dB
- 84 -
______________________________________________________________________________
Learning Curve of SNR 20dB
1.4
1.2
LMS
RLS
Lattice
Channel Model 2
0.6
0.5
0.4
0.8
0.3
0.2
0.6
0.1
100
200
300
400
500
600
0.4
0.2
0
100
200
300
400
500
600
Number of Iteration
700
800
900
Fig-5.3 Comparison Leaning Curve of Channel Model 2 at SNR 20dB
Channel Response & Equalizer Response
2.5
2
Channel Model 2
LMS
RLS
Lattice
SNR 20dB
Channel Model 2
Frequency Response
Mean Square Error
1
1.5
1
0.5
0
0
0.1
0.2
0.3
0.4
0.5
0.6
Normalized Frequency
0.7
0.8
0.9
Fig- 5.4 Equalizer and Channel Model 2 response at SNR 20dB
- 85 -
1
______________________________________________________________________________
Learning Curve of SNR 20dB
1.8
LMS
RLS
Lattice
1.6
0.3
1.4
Channel Model 3
0.2
1
0.15
0.8
0.1
0.6
0.05
0.4
100
200
300
400
500
600
700
0.2
0
0
100
200
300
400
500
600
Number of Iteration
700
800
900
Fig-5.5 Comparison Learning Curve of Channel Model 3 at SNR 20dB
Channel Response & Equalizer Response
2.5
2
Channel Model 3
LMS
RLS
Lattice
SNR 20dB
Channel Model 3
Frequency Response
Mean Square Error
0.25
1.2
1.5
1
0.5
0
0
0.1
0.2
0.3
0.4
0.5
0.6
Normalized Frequency
0.7
0.8
0.9
Fig- 5.6 Equalizer and Channel Model 3 response at SNR 20dB
- 86 -
1
______________________________________________________________________________
Learning Curve of SNR 20dB
0.6
LMS
RLS
Lattice
Channel Model 4
0.22
0.2
0.18
0.16
0.4
0.14
0.12
0.1
0.3
0.08
0.06
0.04
0.2
0.02
0
50
100
150
200
250
0.1
0
100
200
300
400
500
600
Number of Iteration
700
800
900
Fig-5.7 Comparison Learning Curve of Channel Model 4 at SNR 20dB
Channel Response & Equalizer Response
4
Channel Model 4
LMS
RLS
Lattice
3.5
3
Frequency Response
Mean Square Error
0.5
2.5
2
SNR 20dB
Channel Model 4
1.5
1
0.5
0
0
0.1
0.2
0.3
0.4
0.5
0.6
Normalized Frequency
0.7
0.8
0.9
Fig- 5.8 Equalizer and Channel Model 4 response at SNR 20dB
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1
______________________________________________________________________________
5.2 Observation and Analysis
The simulation for comparison study uses the 20dB SNR value. The Fig 5.1, 5.3, 5.5 and
5.7 are the learning curve of channel model 1, 2, 3 and 4 respectively. Based on section
5.1 simulation results, two tables (Table 5.1 Mean Square Error and Table 5.2 Number of
Iterations to Converge) are generated.
For the Mean Square Error, RLS algorithm gives the smallest error among the three
algorithms. RLS algorithm can perform very well in the complex channel (channel model
1) compare to LMS and GAL. When RLS gives error 0.11, LMS and GAL give error 0.3
and 0.26 respectively. It means the error of LMS is 2 times and the error of GAL is 1.5
times higher than RLS algorithm. In the real channel (channel model 2, 3 and 4), the
difference of mean square errors are not significant. The performance of all algorithms is
same in the real channel model.
Mean Square Error
Channel Model 1
Channel Model 2
Channel Model 3
Channel Model 4
LMS
0.3
0.27
0.17
0.082
GAL
0.26
0.26
0.15
0.075
RLS
0.11
0.23
0.12
0.062
Table - 5.1 Mean Square Error of LMS, GAL & RLS Algorithms
For the rate of convergence, RLS algorithm is the fastest among the three algorithms. In
channel model 1 (complex) and channel model 2 & 3 (real), the rate of convergence of
RLS algorithm is about the same. Therefore, RLS algorithm supports significant rate of
convergence for both complex and real channel model. The channel model 4 converges
faster than other three channels because of its simplicity.
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______________________________________________________________________________
Number of Iterations to Converge
Channel Model 1
Channel Model 2
Channel Model 3
Channel Model 4
LMS
600
350
470
250
GAL
550
350
450
220
RLS
75
80
90
25
Table - 5.2 Number of Iterations to Converge in LMS, GAL & RLS Algorithms
The Fig 5.2, 5.4, 5.6 and 5.8 show the comparison between channel response and
equalizer response of different algorithm. The equalizer response of channel model 3 & 4
is better tracking and gives an approximate inversion of the channel response because of
its small error. In the all real channel models, the tracking of RLS and LMS are almost
same. In the channel model 1, the equalizer response of RLS gives the best tracking and
approximate inversion of the channel model because of the smallest error.
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______________________________________________________________________________
5.3 Summary
The LMS algorithm has the slowest rate of convergence and high MSE. However, the
computational complexity is simple and easily understandable. It uses transversal
structure. The LMS algorithm is an important member of the family of stochastic
gradient algorithms. The stochastic gradient means it is intended to distinguish the LMS
algorithm from the method of steepest descent, which uses a deterministic gradient in a
recursive computation.
The LMS algorithm does not require measurements of the pertinent correlation functions
and matrix inversion. Therefore, this algorithm is simple and standard benchmarked
against other algorithms.
The GAL algorithm has slightly faster rate of convergence and lower MSE than LMS
algorithm. However, the computation is more complex than LMS algorithm. This
algorithm uses multistage lattice structure and also a member of the family of stochastic
gradient algorithms.
The derivation of GAL algorithm involves the investigating Levinson-Durbin algorithm,
which computes a set of reflection coefficient and use it directly in the lattice structure.
The reflection coefficients provide a very convenient parameterization of many naturally
produced information signals.
The main advantages of lattice structure are low round off noise in fixed word length
implementation and relative insensitivity to quantization noise.
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______________________________________________________________________________
The RLS algorithm has the fastest rate of convergence and lowest MSE among the three
algorithms. The computation is very complex compare to others. The derivation is based
on a lemma in matrix algebra known as the matrix inversion lemma. This algorithm uses
transversal structure.
The fundamental difference between the RLS algorithm and LMS algorithm is the stepsize parameter in the LMS algorithm is replaced by the inverse of the correlation matrix
of the input vector in the RLS algorithm. The RLS algorithm has two limitations which
are lack of numerical robustness and excessive computational complexity.
The selection of algorithm depends on the application of usage. The selection for one
application may not be suitable for another. The application require to use better
performance and lower error, then it has to pay for high cost such as computational cost,
performance and robustness.
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______________________________________________________________________________
CHAPTER 6
Project Management
6.1 Project Plan and Schedule
At the beginning of the project, the proper planning is very important as it contributes the
success of the project. Time management factor is the “rule” of having a good graded
project. I have to juggle between my project and my full-time job. The proper plan of
this project is not only my contribution but also my project supervisor, Dr. Lim, did his
part to meet up regularly with me and constantly reminding me as not to have lapsed on
the planned schedule.
Project tasks are divided into nine sections:
1. Project Selection & Commencement Briefing
2. Literature search
3. Preparation of Capstone Project Proposal
4. Study and Analyze on LMS, RLS and Gradient Adaptive Lattice
5. Preparation of Interim Report
6. Simulation on MATLAB software and Comparison Study
7. Evaluation and Implementation on current disadvantages
8. Preparation for final report
9. Preparation for oral presentation
Task 1, schedule for 2 weeks to complete. This task is selecting the project that offer
from University. After getting the approval, the school gives briefing on the project
relating on all submission dates and rules.
Task 2, schedule for 3 weeks to complete. This task is doing literature search of the
project after 1st meeting with project supervisor. The literature searches mainly focus on
library reference books and online IEEE journals.
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______________________________________________________________________________
Task 3, schedule for 2 weeks to complete. This task is preparing the proposal that shows
how much understand on the project based on literature search. This task is very
important because of the judgment on continue the project.
Task 4, schedule for 6 weeks to complete. In this task, focus on study and analyze on
LMS, RLS and Gradient Adaptive Lattice algorithm that I studied in ENG313 Adaptive
Signal Processing module.
Task 5, schedule for 3 weeks to complete. In this task, prepare Interim Report and present
the progressive of study and analyze process of task 4.
Task 6 & 7, schedule for 13 weeks to complete. These two tasks are very important and
main tasks for this entire project. Fortunately, these 13 weeks are fall in school holiday.
So, I will give more time on simulation and implementation parts.
Task 8 & 9, schedule for 14 weeks to complete. In these tasks, prepare final report and
oral presentation based on the analysis, simulation and implementation of task 4, 6 & 7.
These two tasks also can consider as important task because final report is the show case
of the entire project and judge how much effort to put in the project.
The project commenced on 19th July 2009 with a submission date on the 17th May 2010.
Below Table 6-1 is the Project Task breakdown throughout the few months.
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______________________________________________________________________________
6.2 Project Tasks Breakdown and Gantt chart
Performance Analysis and Comparison of Adaptive Channel Equalization Technique
for Digital Communication System
Tasks Description
1. Project Selection & Commencement Briefing
2. Literature Search
2.1 Research on IEEE online journals, relevant reference
books
2.2 Analyze and study relevant books and journals
3. Preparation of Capstone Project Proposal
4. Study and Analyze on LMS, RLS & Gradient Lattice
5. Preparation of Interim Report
6. Simulation on MATLAB and Comparison
7. Evaluation and Implementation on current
disadvantages
8. Preparation of Final Report
8.1 Writing skeleton of final report
8.2 Writing literature review
8.3 Writing introduction of report
8.4 Writing main body of report
8.5 Writing conclusion of further study
8.6 Finalizing and amendments of report
9. Preparation of oral presentation
9.1 Review and extract important notes for presentation
9.2 Create poster and prepare for presentation
Start
Date
End
Date
19-Jul09
2-Aug09
2-Aug09
9-Aug09
23Aug-09
6-Sep09
18-Oct09
8-Nov09
20Dec-09
7-Feb10
7-Feb10
14-Feb10
21-Feb10
28-Feb10
21-Mar10
4-Apr10
18-Apr10
18-Apr10
2-May10
1-Aug09
22Aug-09
8-Aug09
22Aug-09
5-Sep09
17-Oct09
7-Nov09
19Dec-09
6-Feb10
17-Apr10
13-Feb10
20-Feb10
27-Feb10
20-Mar10
3-Apr10
17-Apr10
15May-10
1-May10
15May-10
Table - 6.1 Project Tasks Breakdown
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Duration
(weeks)
Resources
2
3
1
Library
resources
(reference
books)
2
Web
resources
(IEEE
journals,
source codes
&
related
reference
books)
2
6
3
6
7
10
1
Personal
Computer
1
1
3
2
2
4
2
2
MATLAB
Software
______________________________________________________________________________
Table - 6.2 Gantt chart
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______________________________________________________________________________
CHAPTER 7
REVIEW & REFLECTIONS
7.1 Skills Review
For this project, a great deal of knowledge of both mathematical derivations and
simulations are required to make it possible. Skills like project management and
information researches are greatly enhanced over the period of the entire project.
I have to say Project Management skill is the most important in making the project a
successful one. I studied PMJ300 Project Management module in the course of study that
gives me understanding of the theory. When I am planning and scheduling this project, it
gives me full picture of what is the Project Management. The tight schedule of juggling
between part time studies and a full time job makes it difficult. These were overcome as
proper planning and scheduling were carried out.
At the beginning of project, a lot of researches were done over the reference books from
Singapore Polytechnic Library and IEEE journals to know more about adaptive channel
equalization. A lot of time was spending on Project Overview and Literature Review as
of knowing what was already available in the area of study.
Mathematical derivation and modeling skills are required for this project which I have
studied the skills from the HESZ2001 module. MATLAB programming skill also
requires using in simulation of the model to analyze and compare with other model.
At the completion of this project, my Project Management skill, Research skill,
Mathematical derivation and modeling skill and MATLAB programming skill are
significantly improved.
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______________________________________________________________________________
7.2 Reflections
In this Final Year Project (FYP), SIM University had given me the opportunity to realize
my strengths and weakness. There were several problems that I have faced throughout the
entire project.
My weakness would be the understanding of adaptive signal processing theory at the
beginning stage and explanation on simulation results at the end of the project. I believe
that my understanding of adaptive signal processing in the course of study was not in
depth enough. However, after doing some researches and read up the reference books, I
manage to figure out and complete the task.
For MATLAB programming, I started by running simple programs like without noise
(SNR) and using real value channel model. Then, I edit slowly and research in further
simulation, analyzing and comparing what was needed to fulfill the project.
My strengths would be the time management and able to source for solutions once I have
encounter problems in the project. The time management is actually not easy for the
students especially like me having full time job and family.
During literature research and preparation proposal period, it has been hard to understand
the principles even though various resources are available. However, it was solved step
by step consulting with my project supervisor and friends. It proved that I have the
strength to solve the issue arose. Upon completion of the project, I am having better
understand on my strengths and weakness.
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______________________________________________________________________________
7.3 Conclusions
The use of an adaptive filter offers a highly attractive solution to the problem as it
provides a significant improvement in performance over the use of a fixed filter designed
by conventional methods. And the use of adaptive filters also provides new signal
processing capabilities. Thus, the adaptive filters have been successfully applied in such
diverse fields as communication, control, radar, sonar, seismology and biomedical
engineering.
The subject of adaptive filters constitutes an important part of signal processing and the
applications are used in wide range of area. Therefore, a lot of researchers are still doing
further research and implementing on the different algorithms and techniques.
I would consider this project is success as I had met the objectives of the project and
completed all the tasks that I set in the project proposal. Three different algorithms are
studied and analyzed by using mathematical derivation and simulation on MATLAB
software.
Throughout the duration of the entire project, I had picked up many valuable skills,
knowledge and exposure in the field of adaptive signal processing. In the past three and
half years in SIM University, my skills and knowledge had been tested as when the
problems arise. I have learnt to solve them from various researches and understanding the
route of error. Lastly, the project management skill, report writing skill and oral
presentation skill would benefit me and enhance me for my future and my career.
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______________________________________________________________________________
References
[1] Simon Haykin, “Adaptive Filter Theory”, Fourth Edition, Prentice-Hall, 2002.
[2] Widrow, B., and S.D.Stearns, “Adaptive Signal Processing”, Prentice Hall, 1985.
[3] Bellanger, M., “Adpative Digital Filters and Signal Analysis”, Marcel Dekker, 1987.
[4] Leon H. Sibul, “Adaptive Signal Processing”, IEEE press, 1987.
[5] Odile Macchi, “Adaptive Processing The Least Mean Squares Approach with
Applications in Transmission”, John Wiley & Sons, 1995.
[6] Michael L.Honig, “Adaptive Filters, Structures, Algorithms and Applications”,
Kluwer, 1984.
[7] Boca Raton, “Adaptive Signal Processing in Wireless Communications”, CRC, 2009.
[8] Jinho Choi, “Adaptive and Iterative Signal Processing in Communications”,
Cambridge University Press, 2006.
[9] S.Thomas Alexander, “Adaptive Signal Processing, Theory and Applications”,
Springer-Verlag, 1986.
[10] Patrik Wahlberg, Thomas Magesacher, “LAB2 Adaptive Channel Equalization”,
Department of Electrical and Information Technology, Lund University, Sweden, 2007.
[11] Janos Levendovszky, Andras Olah, “Novel Adaptive Channel Equalization
Algorithms by Statistical Sampling”, World Academy of Science, Engineering and
Technology, 2006.
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______________________________________________________________________________
[12] David Gesbert, Pierre Duhamel, “Unimodal Blind Adaptive Channel Equalization:
An RLS Implementation of the Mutually Referenced Equalizers”, IEEE, 1997.
[13] Georgi Iliev and Nikola Kasabov, “Channel Equalization Using Adaptive Filtering
with Averaging”, Department of Information Science, University of Otago
[14] Bernard Widrow and Michael A.Lehr, “Noise Canceling and Channel Equalization”,
The MIT PRESS, Cambridge, Massachusetts, London, England
[15] Jusak Irawan, “Adaptive Blind Channel Equalization for Mobile Multimedia
Communication”, Journal Teknik Elektro Vol. 6, No.2, September 2006: 73 – 78
[16] Muhammad Lutfor Rahman Khan, Mohamed H.Wondimagegnehu and Tetsuya
Shimamura, “Blind Channel Equalization with Amplitude Banded Godard and Sato
Algorithms”, Journal of Communications, Vol 4, No.6, July 2009.
[17] Otaru, M.U.; Zerguine, A.; Cheded, L., “Adaptive channel equalization: A
simplified approach using the quantized-LMF algorithm”, IEEE International
Symposium
on
Volume
18,
Issue
21,
May
2008
Page(s):1136
–
1139,10.1109/ISCAS.2008.4541623
[18] Pedro Inacio Hubscher and Jose Carlos M.Bermudez, “An Improved Statistical
Analysis of the Least Mean Fourth (LMF) Adaptive Algorithm”, IEEE Transactions on
Signal Processing, Vol 51, No. 3, March 2003.
[19] H.C.So and Y.T.Chan, “Analysis of an LMS Algorithm for Unbiased Impulse
Response Estimation”, IEEE Transcations on Signal Processing, Vol 51, No. 7, July
2003.
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______________________________________________________________________________
Appendix – A
Signal-Flow Graphs
In constructing block diagrams involving matrix quantities, the following symbols are
used:
a
c
Σ
b
The symbol denotes an adder with c = a + b.
a
+
c
Σ
-
b
The symbol denotes a subtractor with c = a - b.
x
h
y
The symbol denotes a multiplier with y = x h.
x(n)
Z -1
The symbol denotes the input sample delay operator.
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x(n-1)
______________________________________________________________________________
Appendix – B
Abbreviations
GAL
Gradient-adaptive lattice
ISI
Inter Symbol Interference
LMS
Least Mean Square
MSE
Mean Square Error
RLS
Recursive Least Square
SNR
Signal-to-noise ratio
VLSI
Very large-scale integration
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______________________________________________________________________________
Appendix – C
Principal Symbols use in LMS
X(k)
Input signal vector
Y(k)
Output signal vector from transversal filter
W(k)
Weight vector
Wopt(k)
Optimal weight vector
N(k)
Noise vector
e(k)
Estimated Error
d(k)
Desired response

Mean square error
 min
Minimum mean square error

Eigenvalue
max
Maximum eigenvalue
R
Auto correlation matrix of X(k)
P
Cross correlation matrix between d(k) and X(k)
Q
Eigenvector matrix of R

Eigenvalues matrix
k
Gradient vector

Step-size parameter
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______________________________________________________________________________
Principal Symbols use in RLS
X(i)
Input signal vector
Y(i)
Output signal vector from transversal filter
W(i)
Weight vector

wn 
Optimal weight vector
e(i)
Estimated Error
d(i)
Desired response

Exponential weighting factor
 n
Cost function

Regulation parameter,   0
R(n)
Auto correlation matrix
z(n)
Cross correlation matrix
k(n)
Gain vector
P(n)
Inverse of auto correlation matrix, R 1 n
 n
Vector
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______________________________________________________________________________
Principal Symbols use in GAL
am
B*
am
Tap weight (m+1) x 1 vector of a forward prediction error filter of order m
Tap weight (m+1) x 1 vector of a backward prediction error filter of order
m and their complex conjugate
a m1
Tap weight m x 1 vector of a forward prediction error filter of order m-1
B*
Tap weight m x 1 vector of a backward prediction error filter of order m
a m 1
and their complex conjugate
m
Constant (reflection coefficient)
 m,o
Optimum reflection coefficient
ˆ m (n)
Reflection coefficient after replacing the expectation operator with time
average operator
f m (n)
1 n
 ,
n i 1
Forward prediction error produced at the output of the forward predictionerror filter of order m
f m1 (n)
Forward prediction error produced at the output of the forward predictionerror filter of order m-1
bm (n)
Backward prediction error produced at the output of the backward
prediction-error filter of order m
bm 1 (n  1)
Delayed backward prediction error produced at the output of the backward
prediction-error filter of order m-1
M
Stage of lattice predictor
x(n)
Sample value of tap input in transversal filter at time n
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______________________________________________________________________________
x m1 (n)
*
(m+1) x 1 tap input vector
h m (n)
Weight vector
h m (n  1)
Updated weight vector
d * (n)
Desired response vector
h m (n  1)
Adjustment weight vector
ym (n)
Output signal, estimate of desired response d(n)
em* (n)
Estimation error

Step-size parameter

Constant (0    1)
m
The sum of squared forward and backward prediction errors
 m1 (n)
Total energy of both forward and backward prediction errors
m
Gradient vector
Jm
Cost function
*
*
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