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The unified adaptive equalization algorithm

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The unified adaptive equalization algorithm
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Ho, Hai Thanh
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vii, 87 leaves : illustrations ; 29 cm

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Bibliography:
Includes bibliographical references (leaves 69-70).
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Submitted in partial fulfillment of the requirements for the degree, Master of Science, Department of Electrical Engineering.
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by Hai Thanh Ho.

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|University of Colorado Denver
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Full Text
THE UNIFIED ADAPTIVE EQUALIZATION ALGORITHM
by
Hai Thanh Ho
B.S.E.E., University of Colorado, 1988
A thesis submitted to the
Faculty of the Graduate School of the
University of Colorado in partial fulfilment
of the requirements for the degree of
Master of Science
Department of Electrical Engineering
1989


This thesis for the Master of Science degree by
Hai Thanh Ho
has been approved for the
Department of
Electrical Engineering
Marvin Anderson
Date 3 O AJo u ^7


Ho, Hai Thanh (M.S., Electrical Engineering)
The Unified Adaptive Equalization Algorithm
Thesis directed by Associate Professor Douglas A. Ross
This thesis introduces the Unified Adaptive Equalization
Algortihm which can be used to implement nine types of equalization
schemes. Also, a performance study is conducted with three
adaptation algorithms: Recursive Least Square, Recursive Minimum
Variance, and Recursive Stochastic Gradient.
The form and content of this abstract are approved. I recommend
its publication.
Signed
Douglas A. Ross


ACKNOWLEDGMENTS
The author wishes to express sincere appreciation to Dr.
Douglas A. Ross for his guidance and support during the course of this
thesis. Also, he would like to thank the committee members
professors Bill Wolfe and Marve Anderson.
Special thanks to the author's family for their never ending
support.


CONTENTS
PART I INTRODUCTION & THEORY DEVELOPMENT
1. Introduction..........................................1
1.1 Mathematical Development......................6
2. Estimation Algorithm Development......................9
2.1 Recursive Least Square (RLS)..................10
2.2 Recursive Minimum Variance (RMV)..............17
2.3 Recursive Stochastic Gradient (RSG)...........21
2.4 Generalized Recursive Algorithm (GRA) ........24
PART II ADAPTIVE EQUALIZATION & APPLICATIONS
3. Adaptive Equalization.................................26
3.1 Derivation of Equalization algorithm ..........28
3.2 Generalized & Unified Adaptive................32
Equalization Algorithms (GAEA) & (UAEA)
3.3 Simulation and Performance Study..............39
3.3.1 MR Band-pass channel.....................44
3.3.2 Coaxial transmission line..............59
4. Conclusion ...........................................66
BIBLIOGRAPHY.............................................69
APPENDIX Simulation software & programs..................71


Figures and Tables
Figures: PAGE
1.1. Adaptive equalization shceme. 3
1.2. Block diagram of system. 6
2.1. Estimation scheme. 9
2.2. Generalized Recursive Algorithm block diagram. 24
3.1. Communication network. 27
3.2. Adaptive equalization scheme. 27
3.3. System identification scheme. 28
3.4. Estimation of inverse channel scheme. 30
3.5 HR equalizer structure. 32
3.6 FIR equalizer structure. 33
3.7. DFE Equalizer structure. 34
3.8. Block diagram of UAEA. 37
3.9. Comminication network with adaptive equalization. 43
3.10. Magnitude response of channel. 45
3.10b. Phase response of channel. 46
3.11a. Transmitted signal u(k). 46
3.11b. Received signal y(k). 47
3.12. Equalized output. 47
3.13a. Equalized channel magnitude response. 48
3.13b. Equalized channel phase response. 48
3.14a. Eye diagram of transmitted signal u(k). 49
3.14b. Eye diagram pf received signal y(k). 50
3.14c. Eye diagram of equalized signal u(k). 50
3.15. Equalizer filter coefficients. 51
3.16a. Random transmitted signal u(k). 52
3.16b. Received signal y(k). 52
3.16c. Equalized signal u(k). 53
3.17. MSE responses with crv = .01. 54
3.18. MSE responses with ctv = .03. 54
3.19a. RMV MSE responses. 56
3.19b. RLS MSE responses. 56


vii
3.19c. RSG MSE responses. 57
3.20. Coaxial channel circuit model. 59
3.21a. Coaxial channel magnitude response. 61
3.21b. Coaxial channel phase response. 62
3.21c. Coaxial impulse response. 62
3.22. Transmitted signal u(k) and received signal y(k). 63
3.23. Equalized signal u(k). 64
3.24. MSE response of HR RLS. 65
Table:
3.1. Algorithm performance summary. 58


Chapter 1
INTRODUCTION
One of the very popular and important subject in the areas
of digital signal processing and system engineering is Adaptive
Estimation Algorithms. In myriads of applications, scientists
need to faithfully estimate a certain set of parameters
belonging to a stochastic signal or system. Some of the
applications are system identification, adaptive equalization,
adaptive control, noise cancelation, and adaptive array
beamforming. The invention of estimation algorithms dates
back to the early 1960's and more and more different
algorithms of different structures and performances have been
developed to satisfy the -need of today's technology. Two of the
widely used mathematical models in representing a signal or a
system are the State Space model and Input/Output (or ARMAX)
model In many applications the ARMAX model is used more
because it is simpler to implement and requires less
computation.
Data transmission over telephone and radio lines have been
made efficient in bandwidth and minimal error recovery of the
signal by the use of equalization to compensate for the
distortion caused by the channel. Many types of equalizers have
been developed over the years and each has their advantages
and disadvantages. The process of selecting the type of
equalizer involves examining the particular application and
factors such as computation power, performance, cost,
complexity and type of channel. One of the popular equalizers
is the FIR (Finite Impulse Response) transversal filter [2]


2
which is popular because of its unconditionaly stable
characteristic and simple implementation. The Weiner MR
(Infinite Impulse Response) equalizer was shown [2] to have
better performance than the FIR but requires the channel
characteristics to be known ahead. So in the case where the
channel is unknown, the adaptive Kalman equalizer [3] can be
considered but has a nonlinear form and therefore the
linearization causes the algorithm not to always converge [4].
The last two types of equalizers which are not as simple as the
FIR nor complex as the Kalman are the adaptive HR [2,5] and the
HR Decision Feedback Equalizer DFE [2]. The performance of
these equalizers [2] beats the FIR because they can model the
transfer function poles more readily and accurately. However,
the Kalman equalizer can outperform these when it converges
[3]. The HR equalizer has received little attention in recent
years due to the development of the DFE. However, although the
DFE has better MSE performance compared to the HR, it has the
potential of propagating decisions errors [6] which can degrade
the performance but fortunately, not catastrophically. In
summary, in many equalization problem where the channel
characteristics are not known in advanced, the use of adaptive
equalizer is very necessary. The popular types of adaptive
equalizer (AE) are the FIR, HR, KALMAN, and the DFE which have
been studied and implemented by many researchers and
engineers.
The three types of adaptive equalizers: FIR, HR and DFE all
need the adaptive algorithm to search for the optimal
coefficients of the filter. Although, in the literatures of AE,


3
the structures and design of AE were discussed along with the
popular algorithms such as the Recursive Least Square and
Gradient, many of the details such as comparision of
performances of various algorithms, and the generality of FIR,
MR and DFE with respect to the algorithm structure have not
yet been investigated.
The motivation of this thesis was to develop the unified
adaptive equalization algorithm that in which it can be used to
implement nine different equalization schemes. Each scheme
involves an equalizer filter and an adaptive coefficient update
algorithm as shown in figure 1.1. The three types of filter are
Finite Impulse Response (FIR), Infinite Impulse Response (HR),
and Decision Feedback Equalizer (DFE), and the three types of
coefficient update algorithm are Recursive Least Square (RLS),
Recursive Minimum Variance (RMV), and the Recursive
Stochastic Gradient (RSG).
Figure 1.1. Adaptive equalization scheme.


4
Together, nine adaptive equalizers listed below can be formed
1) HR RLS
2) HR RMV
3) HR RSG
4) FIR RLS
5) FIR RMV
6) FIR RSG
7) DFE RLS
8) DFE RMV
9) DFE RSG
The Unified Adaptive Equalization Algorithm (UAEA) is to be
developed by integrating the Generalized Recursive Algorithm
(GRA) and the Generalized Adaptive Equalization Algorithm
(GAEA) where both of these are also to be developed in this
thesis. In
addition, an extensive evaluation and comparision of the
performance of the four estimation algorithms 1-4 in adaptive
equalization. The performance factors to be examined are:
stability, rate of convergence, steady state error, noise
immunity and computation complexity. Part of the results of
the study will be simulation results.
In chapter two, the concept of estimation and the
estimation algorithms will be discussed along with the
development of the three algorithms RLS, RMV and RSG. It will
also be shown that these three algorithms share a common


5
structure so that a Generalized Recursive Algorithm (GRA) can
be formed. In chapter three, the four estimation algorithms
are modified to become equalization algorithms. Similiar to the
GRA, the GAEA will be introduced and then combined with the
GRA to form the UAEA. Also In this chapter, a performance
study will be conducted using hyperthetical HR band-pass
channel and a Coaxial transmission line channel. Chapter four
ends the thesis with conclusions and final remarks.


6
1.1. Mathematical Development.
In this section, we will establish the mathematical
notations to be used later for developing the estimation
algorithms.
A given system or process can generally be represented by a
mathematical model based on physical laws of nature and/or
past hystory of its behavior. The model can take on various
forms. Here are some of the common models being used in real
applications
i) Ordinary, Partial Difference Equations
ii) Algebraic Equations
iii) State Space Equations
iv) Impulse Response
v) Transfer Function
We will be working with Regression models which can be of
type i, and v.
Consider the block diagram of a system shown in figure 1.2
below
Figure 1.2. Block diagram of system.


7
where
u: Input signal
y: Output of system
v: Output white noise with statistics
E{v) = 0; E{v2} = ov2
0: Parameter vector of system
The input output relationship of the system is described by the
difference equation as
Vk -a1 Yk-1 a2Vk-2 - anyk-n +
b0uk + b1 uk-1 + b2uk-2 + + bmuk-m
Defining the following vectors
0 = [-a., -a2 ... -an b0 b1 ... bm]' (1.1.2a)
and
[Vk-1 Vk-2 ^k-n uk uk-1 - uk-rJ (1-1-2b>
where denotes the transpose of a matrix. Now we combine
(1.1.1) and (1.1.2) to write y(k) in vector form as
yk = *'kB + vk
(1.1.3)
Next, if we take the Z-transform of the difference equation


8
(1.1.1) we will get
(1.1.4)
Y(z)[1 + a-jz-"1 + ... + anz'n] = U(z)[bQ + b^-1 + ... + bmz'm]
where z is the complex frequency variable. Thus, the transfer
function which represents the input output relationship in the
frequency domain is
Y(z) zn"m(bQZm + b-|Zm1 +... + bm)
H(z) ---------------------:----------------- (1.1.5)
U(z) zn + a-|Zn'1 +... + an
N
' 5 will use the notations established in the previous equations throughout
Ha next chapters.


Chapter 2
ESTIMATION ALGORITHM DEVELOPMENT
Estimation algorithm is the result of optimization of a
performance index that is a function the signals and
parameters we wish to estimate. Consider the estimation
scheme
Process
Figure 2.1. Estimation scheme.
Where
u(k) : Input signal
y(k) : Output signal
v(k) : Gaussian output noise
0 : System parameter vector


10
A'
0 : Estimated parameter vector
A
y(k) : Estimated output
e(k) : Output estimation error
The task of the estimation algorithm is to adjust the estimated
vector 0 so that the error signal e(k) converges to a minima.
The assumption is that u(k) and y(k) are available. Also, it
should be mentioned that the output y(k) can be thought as
being generated by an input u(k) through a real or fictitious
process with 0 representing the parameters of its transfer
function. This concept can be used to develop and distinguish
the system identification and adaptive equalization algorithms.
Optimization theory is applied to a wide range of models,
the two type models that are often used are the State Space
model and the Input/Output model discussed in section 1.1. Our
algorithms will be based on Input/Output model We will
develop three different algorithms RLS, RMV, and RSG. These
three algorithms will show to have similiar structure and
therefore a generalized recursive algorithm (GRA) will be
created.
2.1. Recursive Least Square (RLS) Algorithm
The RLS algorithm is a recursified version of the offline
Least Square algorithm. We therefore start with Least Square
Development.


11
Consider the system
y(k) = -a1 y(k-1) a2y(k-2) ... any(k-n) + (2.1.1)
bgu(k) + b.ju(k-1) + b2u(k-2) + ... + bmu(k-m)
or
y(k) = <(>'(k)0 + v(k) (2.1.2)
where 6, <|> and v are defined in section 1.1. Suppose we have
collected a set of data points (y(N) y(N+1) ... y(k)} and we would
like to estimate the unknown parameter vector 0 based on these
data points. Let us first define the augmented y as
y = [y(N) y(N+1) ... y(k)]' (2.1.3)
From the relationship in (2.1.2) we see that y can be written
as
(2.1.4)
y(N) y(N-1) y(N-2) ... y(N-n) u(N) ... u(N-m) -a^ v(N)
y(N+1) y(N) y(N-1) ... y(N+1-n) u(N+1) ... u(N+1-m) . -a2 v(N+1)
y(k) y(k-1) y(k-2) ... y(k-n) u(k) ... u(k-m) bm v(k)


We see that equation (2.1.4) can also be written
y = fo'(N) '(N+1) ... where the noise vector v is
v = [v(N) v(N+1) ... v(k)]' (2.1.6)
We next define
Thus, the output vector may now be written as
y = We now derive the Least Square algorithm. As mentioned
earlier that optimization involves minimizing a cost function
J(0) and usually a function of the error. Consider the error
squared cost function
J(0) (y -y)'(y-y) (2.1.9)
= e'e
= (y o0)'(y-<&0)
The optimization can be done by finding the minima of the
quadratic function J(0) which means taking the derivative of
J(0) and set it to zero to solve for 0. The derivative of (2.1.9) is
8J(0)
= -2 80
(2.1.10)


1
Setting it equal to zero we get
or
O' = A
0 = 0
e* = [0'0]"10'y
(2.1.11)
(2.1.12)
Thus, (2.1.12) is the minimal least square solution of 0 based
on the (k-N) data points. Lets put in the time subscript to
specified that the latest data point is at tk we get
0k ro'kk]-Vkyk (2.1.13)
In many cases where the LS solution is to be applied to real
time processes it needs to update its solution to include the
next incoming data point. Suppose we receive the next data
point at time tk+1 then LS solution in equation (2.1.13) with
the updated vectors
ek+1 = [k+1 k+ll1 fk+1yk+1 (2.1.14)
Notice that this form of a solution requires us to perform the
matrix inversion which is very computationally expensive. A
way to get around this and facilitate the computation is to
develop a recursive form of (2.1.14). The idea of recursion is
A
letting the current estimate 0(k+1) be a function of the
previous estimate 6(k). We would expect that the computation
will lessen because we use the results from the previous
iterations.


14
The complete derivation of the RLS algorithm can be found
in many of the adaptive filter textbooks [7]. Therefore, we will
state the key ideas and important steps here. First, the vectors
that includes the next date point at tk +-j can be written in the
augmented form
Yk
Yk+1 =
vk+1 =
Vk+1
Vk
Vk+1
(2.1.15)
(2.1.16)
^k+1 =
Ok
(2.1.17)
Also, defining the matrix P as
pk+1 = ['k+1 k+111
= [0'k Ok + k+i 'k+i]
-1
(2.1.18)
Subtituting these equations into the solution equation (2.1.13)
we get
ek+1 = pk+1'k+iyk+1 (2.1.19)
Let next us state the matrix Inversion Lemma
Lemma: Let A, C and A, BCD be nonsingular square matrices,
the following matrix idenity holds
(A + BCD)-1 = A'1 A'1B(C1 + DA1B)_1DA1
(L1)


15
And applying this to the equation of Pk+1 (2.1.18) where we let
A = 0'k<|)k> B = k+1, C = I, D = 'k+1
we get the recursive form of P as
pk+1 = pk fVk+lC1 +^+1 Pk^k+I^'^'k+I pk (2.1.20)
Thus this form of P involves computing an inverse of a scalar
instead of a matrix. We can now subtitute (2.1.20) into the
solution (2.1.19) and get
k+1 = [pk Pk^k+l^+^k+lPk^k+ir^'k+lPk^^kyk ^k+l Vk+1^
(2.1.21)
It can be shown, after some algebraic manipulation, that 0 can
be put in recursive form as
ek+1 = ek + Pk^k+l^^'k+lPk^k+^^yk+l ^'k+1 ek^ (2-1-22)
For compactness, we define
*-k+1 = pk and subtituting (2.1.23) into (2.1.22) and (2.1.20) yields


1 6
ek+1 = ek + Lk+1 tVk+1 -4>,k+1elJ (2.1.24)
and
pk+1 = pk Lk+1 ^k+1 pk (2-1 -25)
Finally, using (2.1.23), (2.1.24) and (2.1.25) the RLS algorithm
can be summarized as:
process:
yk = k0+uk (2.1.26)
^'k = frk-1 yk-2 yk-n uk uk-m^
0 = [-a1 -a2 . -an bQ . bm]
Vk = white Gaussian noise N(0,a2)
Algorithm:
Lk+1 = pk4*k+1 + pk+1 = pk" Lk+1^ k+1 pk
ek+1 =ek + Lk+1 (yk+1 ^'k+1 ek>
(2.1.27)


17
2.2 The Recursive Minimum Variance (RMV) Algorithm
The RMV algorithm is a result of the optimization of a
somewhat different cost function from that of the RLS. The
RLS minimizes the error between the actual output yk and the
estimated output yk. Whereas the RMV, is based on the
minimization of the error between the actual parameter vector
A
0 and the estimated parameter 0. Let us state the properties of
the RMV
i) Linear estimator
0k+1 f(0k) f 's ,'near functional
ii) Unbiased estimator
E(0k) = E(0)
iii) Minimum variance
Jmin(e)-Emin([9k-V[k-8kl)
We will use these three properties to develop our RMV
algorithm. Recall the system model
*k **'k+10k + vk+1
(2.2.1)
where vk is a zero mean white noise sequence
E{vk} 0
E{vk2) av


18
Suppose that the parameters are slow varying with respect to
our sample period so that we can assume:
ek+1 = ek (2.2.3)
To satisfy the first property i) we choose the model of the
estimator as
ek+1 88 Ak+1 ek + Lk+1 Vk+1 (2.2.4)
where the matrix A and the vector L are to be determined so
that the three stated properties are satisfied. Subtituting
(2.2.1) into (2.2.4) for yk+1 we get
a A
ek+1 =Ak+1ek + Lk+1 U'k+^k+l +vk+1) (2.2.5)
Taking the expectation on both sides of (4) we get
E{ 0k+1} = Ak+1E{0k} + Lk+1 <|),k+1E{ek +1} + Lk+1E{vk}
88 Ak+1 E{0kJ + Lk+1 ^'k+1 E^ek +1 > (2.2.6)
Combining the unbiased property ii) and equation (2.2.6) we see
that
1 = Ak+1 + Lk+1 ^'k+1
or
Ak+1 88 1 Lk+1 ^'k+1
(2.2.7)
and I is the identity matrix with appropriate dimension.


19
Subtituting (2.2.7) back into (2.2.4) we will get
A
ek+1 * " Lk+1 ^'k+1 >ek + Lk+1 ^k+1 (2.2.8)
= ek+ Lk+i(yk+1 ^'k+1 ek)
It is now left to determine the vector Lk+1 such that the
optimal estimate can be obtained. The remaining property to be
satisfied is property iii) which states that the estimate 0
should minimize the MSE cost function
j(e) = E{(0k+1 8k+1 ) (8k+1 ek+1 )} (2.2.9)
- E{8k+1 k+T l
Defining P the Covariance matrix:
P = E{00'} (2.2.10)
The minimization of the quadratic cost function (2.2.9) would
mean that find the parameter vector 0 such that Trace[P(k)] is
minimized.
But let us first find a useful form for P. Notice that by using
equation (2.2.3) and (2.2.8) the error vector 0 can be written in
recursive form as
ek+1 = ek+1 ek+1 (2.2.11)
ek+1 _{0k+ Lk+l(Vk+1 ^'k+1 ek>>
= ek" Lk+l(Vk+1 ^k+1 ek>
Using equation (2.2.11), it can be shown by algebraic
manipulation that the covariance matrix P can be written in
recursive form


20
Pk+1 = pk pk ^k+1 k+1 + *~k+1 ^ k+1 pk + ^ k+1 *3k - L,k+1k+1Pk (2-2.12)
Now, performing the differentiation
^r pk+1
------- = Pk^k+I + Lk+11 ^ k+1 pk ^k+1 + avl"1
0Lk+1 (2.2.13)
Solving for Lk+1 we get
Lk+1 = Pk ^k+1 I ^'k+1 Pk ^k+1 +v]'1 (2.2.14)
Subtituting (2.214) into (2.2.11) to get the compact recursive
form
pk+1 = pk pk ^k+1 L'k+1 + pk <>k+1 Lk+1 pk ^'k+1
= pk 4<+1 Pk^k+1 (2.2.15)
Finally, the RMV algorithm can be summarized by (2.2.8),
(2.2.14) and (2.2.15)
Lk+1 ~ Pk ^k+1 (^ k+1 pk ^k+1 + CTv^ ^
pk+1 = pk" Lk+1 pk ^'k+1
ek+1 = k +Lk+ltyk+1 -^k+Al
(2.2.16)


21
Comparing this algorithm with the RLS algorithm, we recognize
that they have very similiar structure. More about this in
section 2.4.
2.3 Recursive Stochastic Gradient Algorithm (RSG)
The idea of the RSG algorithm is that one optimizes the
approximation of the cost function. That is, using the Taylor's
series expansion to approximate the cost function at tk+1 about
xk
J(0k+1) = J(0k) + aj(9k+i)' (0k+1 ek) (2.3.1)
90k
The optimization is performed by maintaining the condition
J(9k+-|) < J(9k)
(2.3.2)
throughout the iterative search and J will converge to its
minima. The condition (2.3.2) can be satisfied if at each
iteration of equation (2.3.1) this condition is true
ek+1 ek
n(k)3J(0k+-|)
3(ek)
(2.3.3)
where p(k) > 0 is the search step size. Let's subtitute (2.3.3)
into (2.3.1)
J(0k+1) = J(0k) n{dJ'(9k+l)dJ(9k+l) i
30k 30k
(2.3.4)


22
We can see that the second term on the right hand side of
(2.3.4) is always positive and with the negative sign in front
makes
J(ek+-|) < J(9|<)
which satisfies our optimization criteria. Therefore, equation
(2.3.3) motivates the stochastic Gradient Algorithm
^k+1 = ek M^J(0) (2.3.5)
30 0 = 0k
We next proceed to find the gradient of J. Recall the system
model
Vk+1 *'k+1 9 + vk+1 <2-3-5)
and the estimated output is
yk+i = 4>'k+1 0 k+1 (2.3.7)
Defining the error square cost function as
J(0) = (1/2){yk+1
= <1/2)[e2k+1] (2.3.8)
Subtituting (2.3.7) into (2.3.8) we get
J(0) = 1/2[yk+1 -0'k+^k)2 (2.3.9)
Next we solve for the gradient of J by


23
5J(9k)
aek
A
(Yk+1 ^'k+10k) ^k+1
A
k+lfrk+l ^'k+1 ek)
Subtituting (2.3.10) into (2.3.5) we get
ek+1 = ek -M-^k+l^k+l ^'k+1 ek)
Next we define L as
Lk+1 =
Finally, using (2.3.11) and (2.3.12) we
Recursive Stochastic Gradient Algorithm
Lk+1 = ^k+1
ek+1 = ek Lk+1 (2.3.10)
(2.3.11)
(2.3.12)
summarize the
(2.3.13)
1 > |i > 0


24
2.4. Generalized Recursive Algorithm (GRA)
It is recognized that the three algorithms RLS, RMV, annd
RSG share a same structure for the estimator 0 which has the
form
A A A
ek+1 = ek -Lk+l(yk+1 ^k-t-1 ek) (2.4.1)
where each algorithm is different from the other in the
computation of the error weight vector L. This is shown in
figure 2.2.
RLS
Figure 2.2. Generalized Recursive Algorithm block
diagram.


25
In hardware implementation, switching from one algorithm to
the other can be conveniently done by only changing the module
that computes L. Thus, the Generalized Recursive Algorithm is
implemented as shown in fig. 2.2 and can be stated by (2.1.27),
(2.2.16) and (2.3.13) as
ek+1 = ek Lk+1 (Vk+1 RLS: Lk+1 Pk^k+lC + ^'k+1 Pk^k+l)"1 (2.4.3)
pk+1 = pk-Lk+lfk+1pk
RMV:L|^+-j = Pk^k+1 t ^ k+1 Pk ^k+1 + av^ (2.4.4)
pk+1 = pk Lk+1 Pk^'k+I
RSG: Lk+1 = P<|)k+i (2.4.5)
1> p > 0
a


Chapter 3
ADAPTIVE EQUALIZATION
Data transmission over the telephone and radio lines have
been made efficient in bandwidth and minimal error recorvery
of the signal by the use of adaptive equalization to compensate
for the distortion caused by the channel. A communication
network is shown in figure 3.1. In this figure, the "channel"
would include the effects of the transmitter, the modulator,
the channel distortion, and the demodulator. The equalizer is to
try to reproduce the signal at the input of the channel.
Equalization has been researched and investigated for the last
two decades, some of the early work done in this area were by
Lucky [8] and Price [9], and the recent survey of the literature
and complete bibliorgraphy can be found in the publication of
Cowan [2] Qureshi [10], Proakis [5] and Gordano [11].
In the application of the estimation algorithms RLS, RSG,
and RMV developed in chapter two, the FIR, MR equalizer and the
DEF all require an adaptive algorithm to compute the filter
coefficients. The adaptive scheme is shown in figure 3.2. It is
the goal of this chapter to develope the Unified Adaptive
Equalization Algorithm (UAEA) which encompasses all nine fo
the following algorithms: FIR RLS, FIR RMV, FIR RSG, HR RLS,
HR RMV, HR RSG, DFE RLS, DFE RMV, and DFE RSG. Also, we will
show the applicability of the algorithms and evaluate the
performance of the three HR algorithms. We will study two
types of channels, they are: 1) HR band-pass channel and 2)
Coaxial transmission line channel.


27
v(k) noise
u(k) A y(kV Adaptive u(k)_
Input cnonric i Equalizer
Figure 3.1 Communication network
Figure 3.2 Adaptive Equalization scheme.


28
3.1 Derivation of the Equalization Algorithm
The algorithms developed in chapter 2 were implicitly
based on the idea of system identification. In other words, the
desired signal y(k) can be assumed to have been generated from
an input u(k) through the system with vector 0 containing the
coefficients of the transfer function.
Process
Figure 3.3 System identification scheme.


29
where
u(k) : Given input
y(k) : Given output
0 : Parameter vector of process transfer function
A
y(k) : Estimated output
0s : Estimated transfer function parameter vector
v(k) : White Gaussian noise N(o,c2)
and
yk = 'k0 + vk (3.1.1)
k = [uk_*|Uk_2-- u k-n Vk Vk-1 Vk-J
0 = [-ara2 . -an bQb1 . bn]'
vk = White Gaussian noise N(0, a2)
and the algorithm that optimizes the error e(k) to produce y(k)
has the form
ek+1 = ek + Lk+l(Vk Vk) (3.1.2)
= 0k+ Lk+1 (yk" ^'k+l0^
Now, consider the scheme with the process being the inverse
model of the channel


30
channel
noise
v(k)
Process
Figure 3.4. Estimation of inverse channel scheme.
Here, the process represents the inverse of the channel which
is implicit. In this case, the algorithm tries to produce the
estimate of u(k) and its input is y(k) whereas the reverse in the
case of system Identification. If there exists an inverse model
then it can be described by the process
uk 4>'k9 + uk
'k luk-1uk-2 u k-n Vk Vk-1 Vk-nl <3-1 -3>
6 = [-ara2 -an b0b., ... bn]'
= white Gaussian noise N(o, So one can see that the difference is only in the vector 0 where
the input and output samples are interchanged. Also, the order
of the numerator and denomerator is to be selected large


31
enough to model the inverse of the channel. Similiarly, in
modifying the algorithm so that it estimates u(k) instead of
y(k) we perform the similiar modification of interchanging the
input and output samples. Hence, we get the form
ek+1 = ek + Lk+1 (ukuk) (3.1.4)
" ek + Lk+1 H-^'k+l ek)
Notice that the algorithm has a general form, the computing of
the error weight vector L(k+1) depends on which algorithm we
choose, the RLS, RMV or the RSG. Other methods that were used
in the literatures in deriving FIR RLS [11] and the MR gradient
[5] adaptive equalization algorithms were based on different
approaches but yielded equivalent results.
In summary, the AE algorithm can be summarized by (3.1.3)
and (3.1.4) as follows:
k = K-1 uk-2 u k-n Vk Vk-1 Vk-nl <3-1 -5)
0 = [-ara2 . -an bQb1 ... bn]
uk = 'kek
a a /s
ek+1 ek + Lk+1 (uk' ^k+1 ek)
Lk+1 =Pk'k+iPkk+l)'1
Pk+1 = Pk Lk+1 ^ k+l Pk
Having the equalization algorithm developed, we next
investigate how it can be applied to various equalizer
structures.


32
3.2 Generalized & Unified Adaptive Equalization
Algorithms (GAEA) & (GAEA)
In the previous section, the AE algorithm has the structure
of an HR equalizer shown in figure 3.5. This can be extended to
accomodate other equalizer structures such as FIR and DFE.
Although, papers have been written on the performances of each
equalizer and their advantages and disadvantages [2,6], but no
general algorithm has been introduced to satisfy all three kinds
of equalizer structures. In this section, we will point out the
similiarities of the three equalizers and introduce the
generalized adaptive equalizer algorithm (GAEA). In figure 3.5,
3.6 and 3.7 the structures of the equalizers are shown.
Figure 3.5. MR equalizer strucure.


33
Figure 3.6. FIR equalizer structure.


34
y(k)' 7-1
Input z

Figure 3.7. DFE equalizer structure.
Recall the general HR AE algorithm in equation (3.1.8)
ek+1 = ek + Lk+1 (uk ^'k+1 ek) (3.2.1)
^ k = tuk-1 uk-2 u k-n ^k ^k-1 ^k-n^
A a A /\A/\ A /
6 = [-ara2 . -an bQb1 ... bn]
Examining the FIR equalizer in figure 3.6 we see that there is
no feedback structure, so therefore, the AE algorthim can be


35
modified appropriately by the vector <)> as
4>k = [0 . 0 yk . yk.n]' (3.2.2)
and so replacing the vector in (3.2.1) with (3.2.2) we'll get the
algorithm for FIR equalizer.
Recall that in digital communication, the message sent
u(k) belongs to a set of possible messages (u(k)}. The idea of
DFE is that it matches the output of the estimator u(k) to
closest possible element in the set (u(k)}. There are various
criteria on how a decision is made [ ] to minimize the error but
that is not the issue of this section. Once a decision has been

made, and giving the output of the decision device u (k), we can
see clearly that the AE algorithm can be modified by changing
the vector § as
k Iu*k-1u*k-2 u*k-n Vk Vk-nl' <3-2-3)
Again, replacing the vector <]> in (3.2.1) with (3.2.3) we'll get the
algorithm for the DFE equalizer. Thus, we see that the AE
algorithms for all three types of equalizers HR, FIR and DFE
share a common form and differ in only the vector . Therefore
the Generalized Adaptive Equalization Algotighm (GAEA) can be
formed using (3.2.1), (3.2.2) and (3.2.3)
ek+1 =ek + Lk+l(uk_ ek [-ai (k) -a2(k) . an(k) b0(k) . bn(k)]
(3.2.4)


36
>k-1uk-2---- k-nVkVk-l Vk-nl IIR
[0 ... 0 yk ... yk.n] FIR
Ju*k-1u*k-2 u*k-n Vk ^k-nl
where the vector L is determined upon which recursive
algorithm (RLS, RMV, RSG) was selected. If we combine the GRA
and the GAEA we can form a Unified Adaptive Equalization
Algorithm (UAEA) in which any of the following equalizers can
be implemented:
1) IIR RLS
2) IIR RMV
3) IIR RSG
4) FIR RLS
5) FIR RMV
6) FIR RSG
7) DFE RLS
8) DFE RMV
9) DFE RSG
The block diagram of the UAEA in fig. 3.8 shows that any one of
the nine equlaizers can be implemented by switching to the
appropriate module.


37
Figure 3.8. Block diagram of UAEA.
And the UAEA can be summarized as
ek+1 = ek + Lk+1 (uk ^'k+1 ek) (3.2.5)
0k = [-^(k) -a2(k) . an(k) bQ(k) . bn(k)]y
imi k = [Uk.! uk_2 .... u k.n yk yk_! .. yk.n] (3.2.6)
FIR: ^'k = [0 . 0 yk . yk.n] (3.2.7)
DFEifk [uk-i u*k_2 u*k-n Vk Vk-J
(3.2.8)


38
B-USi M<+1 ^k^k+1 ["* + ^k+1 ^k^k+1^ ^
pk+1 Pk'Lk+l^'k+lPk
P^-*' ^k+1 = pk ^k+1 t ^k+1 pk ^k+1 + av^
pk+1 = pk Lk+1 Pk^'k+1
BSSl M<+1 = M-^k+l
1> p > 0
(3.2.9)
(3.2.10)
(3.2.11)
In the next sections, we will apply the HR equalizers to the
various channels : HR band-pass and Coaxial line.


42
3.3 Simulation and Performance Study
In this section, we will apply the HR RLS, HR RMV and HR
RSG equalization algorithms to two types of channels: bandpass
and coaxial line. The performance study will involve examining
the following factors and outputs:
- Convergence rate
- Mean Square Error MSE
Stability
- Computation
- Noise immunity
- Time domain signals
- Frequency responses of channel, equalizer and the
composite
Eye diagrams the transmitted, received and equalized
signals
- Noisy and noise free enviroment.
Before we present the simulation let us first describe the
routines and assumptions involved in the equalization process.
Consider the system mode shown in Figure 3.9


43
Data
Source
u(k)
Desired Probe
channel 4- signal
= 1 generator
Figure 3.9. Communication network with adaptive
equalization
The cascade of the channel and the AE is to represent a
reliable communication channel. To do this, at the initial
training period, the transmitted data sequence u(k) is generated
from the probe signal generators at the transmitter side and
the receiver side in synchronism. During this period, the AE
performs the search for the optimum filter coefficients to
minimize the error e(k). Once a converged steady state
response of the algorithm has been reached, the training period
is completed and the adaptation stops. Next, the steady state
filter coefficients 0 will be used to perform the actual
equalization of random input sequence. At this time, the input
to the channel will be from the data source, and the output of
the equalizer will be open-looped detected. The assumption is
that the channel characteristics have not changed since the
training. Often, a time-variant channel have time variation
that are much slower than the signaling interval so therefore


44
can esentially be considered as time-invariant channel. In the
next two sections, we will apply our three algorithms to the
band-pass channel and the coaxial channel. The simulation
results will be uses to study the performance of the
algorithms.
3.3.1 MR bandpass channel
A communication channel usually have a bandpass
characteristics and if not equalized than bandwidth of
transmitted signal will be greatly limited. Therefore, an
effective equalizer cascaded with the channel should produce a
low pass frequency response. Consider the simulation scheme
shown in Figure 3.8 with the channel represented by H(z) as
.7z2-.4z+.1 (3.3.1.1)
H(z) = ------------
z2 -.1z + .8
and the output noise v(k) with statistics
E{vk} =0 (3.3.1.2)
E{v2k) = n order to model the inverse of the channel poles and zeros, the order
jf the equalizer has to be at least equal to number of poles of the
channel. In this example, the order of the channel is two, hence, the


45
equalizer will have the form:
H'1 (r) =
a-j + a2Z + 83
z2 + b1 z + bj
(3.3.1.3)
with the normalized sampling period T=1 the frequency response
of the channel is given in figure 3.10
K1 pr
1(
Freq. response of Channel H(z)
If
freq. w
Figure 3.10a. Magnitude response of channel


46
-1
Phase response of Channel H(z)
Figure 3.10b. Phase response of channel
This channel has a nonlinear amplitude supression and distortion
at frequencies below the bandpass frequency. Also, notice the
phase delay associated at the low frequency band. Suppose we
would like to design a reliable communication network that
operates in the bandpass region and also in the low frequency
band. Shown below in figure 3.11 are the transmitted signal u(k)
and the distorded received signal y(k).
Time k
Figure 3.11a. Transmitted signal u(k)


47
Figure 3.11b. Received signal y(k).
The received signal y(k) exhibits the effect of channel distortion
and enviroment noise. Applying HR RMV adaptive equalization to
this communication network we get the equalized signal u(k)
shown below in figure 3.12
Figure 3.12 Equalized output.


48
Furthermore, it is expected that the frequency response of the
channel and equalizer should have improved. Indeed it has, figure
3.12 shows the cascade frequency response of the equalized
channel
K1 -
Freq. response of Channel & Equalizer
Figure 3.13a Equalized channel magnitude response
DO
PO
0
Phase response of Channel & Equalizer
- .DO
-2D0
3 4
freq. w
Figure 3.13b Equalized channel phase response


50
Figure 3.14b Eye diagram of received signal y(k)
Figure 3.14c Eye diagram of equalized signal u(k)


51
Again, as one can see the effect of the channel and noise which
resulted in a very distorted eye diagram of the received signal.
In practice, such condition is absolutely unacceptable. In figure
3.13c, the equalized signal shows much improvement in its eye
diagram. And figure 3.15 shows the evolution and convergence of
the equalizer filter coefficients during the training period.
Figure 3.15 Equalizer filter coefficients
After the training period is over, the equalizer steady state
filter coefficients will be used to equalize random input data.
We next study how the equalizer perform open loop. Through
simulation, below are the signals that were recorded.


52
Transmitted signal u(k)
1
0
[-1
-2
50 100 150 200 250 300 350 400
Time k
Figure 3.16a Random transmitted signal u(k)
Received signal y(k)
150 200 250
Time k
300
350
400
Figure 3.16b Received signal y(k)


53
2
1
0
-1
-2
0 50 100 150 200 250 300 350 400
Time k
Equalized signal u(k)

...Ajr-

.4%:-AA^\h.......-jf\.j
!.K/
*J\^
S2V
Figure 3.16c Equalized signal u(k)
As shown by the figures, the equalizer indeed continues its
ability to equalize after the training period.
The above equalization was performed by the MR RMV
algorithm. We next apply the RLS and RSG to same problem and
compare the performances. The time and frequency domain
signals will not exhibit significant noticeable differences.
Therefore, we will examine the performances in terms of
convergence rate, stability, mean square error (MSE), noise
immunity, and computation complexity. The mean square error
response of the three algorithms applied to the band-pass
channel are shown in figure 3.17 Also, in figure 3.18 shows how
the mse reponses of the three algorithms degrade when
significant noise was added.


54
Figure 3.17. MSE responses with ov = .01.
Figure 3.18. MSE responses with av = .03.
Based on the theoretical and simulation results we
summarize the performance analysis of the algorithms next.


55
Stability: In simulation, all three algorithms were found to be
always stable and converge in every simulation run under
reasonable noise enviroment. In theory, very few results are
available with which predict the MSE convergence of the
algorithms.
Convergence Rate: As expected, due to the simplicity of the
RSG algorithm it converged the slowest. The RMV converges
faster than the RLS because of the fact that the input to the
equalizer is contaminated with the noise v^ in which the RMV
compensates by weighting the performance index with av'1.
Mean Square Error: In noisy enviroment, fig 3.18 shows the
steady state mse of all three algorithms seem to be almost the
same. However, in low noise enviroment, the RLS and RMV have
lower mse than the RSG. This indicates that there is more
degredation in either the RSG or the RLS and RMV when noise is
present. See noise immunity.
Computation: The RSG requires the least amount of
computation. The RLS and RMV require about the same number of
operations, however, the RMV requires that the variance av of the
output noise v(k) is known.
Noise Immunity: Shown below are the MSE response of each
algorithm in the case with noise and without noise.


56
RMV MSE responses
" I 1 1 with noise sigv = .04 i
^ without noise
.5
50
100
150
Time k
200
250
300
Figure 3.19a RMV MSE responses
Figure 3.19b RLS MSE responses


57
RSG MSE responses
._______________i--------.-------------------------------------1---------------
, 50 100 150 200 250 300
Time k
Figure 3.19c RSG MSE responses
From fig. 3.19, It appears that the RMV has the most degradation
when the noise is present, hence, it is least immuned to noise.
The RLS shows a little more immunity to noise than the RMV. The
most immuned algorithm indicated by fig. 3.19c is the RSG.
The performance factors of the algorithms are categorized in the
table 3.1 shown next page:


58
Convergence rate MSE Computation Multiplies Additions Stability Noise imunity
RMV Fastest smallest 10n4+ 4n2 4 2 8n + 8n Stable Least
RLS fast smaller 10n4+ 4n 4 2 8n + 8n Stable Average
RSG slow small 6n 4n Stable Most
Table 3.1. Algorithms performance summary.


59
3.3.2. Coaxial Transmission Line.
In the previous section we showed the the performance of
the adaptive equalizer when applied to a hyperthetical
band-pass channel. The channel was represented by a
polynomial transfer function. In real life application, the
communication channels are not usually representable by a
polynomial function, therefore making the estimation of the
inverse model more difficult. Some of such channels are the
Coaxial transmission line [12], Telephone channel [11],
Multipath Radio channel [11], and the Troposcatter channel [11].
In this section, we investigate how the RLS RMV equalizer
performs with the Coaxial line channel.
The coaxial cable can be modeled based on the equivalent
circuit parameters: resistance, inductance, capacitance, and
conductance. These parameters contribute to the frequency
characteristics of the cable. A complete derivation of the cable
transfer function can be found in Ross [12], we will summarize
the results here.
A short length Ax of the coaxial cable can be modeled by an
equivalent circuit shown in fig. 3.20
i(x,t) LAx
v(x,t)
"o------------------
RAx
^\A
i(x+Ax,y
O
+
v(x+Ax,t)
O
^................. Ax ----------------------
Figure 3.20. Equivalent circuit of coaxial line.


60
where the series resistance and inductance, and parallel
capacitance and conductance are modeled by the lumped
parameters R,L,G, and C. In practice, these parameters may vary
with frequency. From this circuit, it can be shown that the
transfer function has the form
H(> Vout(yVin() (3.3.2.1)
= exp(-ax)exp(-jpx)
= exp(-yx)
where x is the length of the cable and
y = [(R + j = a + jp
is the propagation constant. To get the explicit form of the
transfer function we subtitute (3.3.2.2) into (3.3.2.1) and get
H(co) = exp{[(R + j With this transfer function, once the cabel parameters R,L,C,G
and the length x are determined, we can sample the function
H(co) at a rate fg and truncate at the end frequency fmax. The
effect is we have approximated this analog channel H(co) by a
discrete channel H(z). Then, taking the inverse Fourrier
transform of the sampled channel will give us the discrete
impulse response h(k) of the channel. After the impulse


61
response is obtained, we can convolve it with the input signal
u(k) to get the output y(k). This is part of the simulation task
when generating the output of the channel. To proceed with the
simulation we use the following set of values of the
parameters for the copper conductor coaxial line [13 ]


62
Figure 3.21b. Coaxial channel phase response .
And the corresponding time impulse response is obtained and
shown in figure 3.21c
Coaxial Channel impulse response
-0
Figure 3.1c. Coaxial channel impulse response.


63
As one can see, with this channel characteristic, there is not
as much phase and amplitude distortion as compared to the
band-pass channel used in section 3.2.1. Hence, using the binary
transmitted signal for trainning will not be effective and,
therefore, we select a more phase and amplitude sensitive
signal. And such signal has the following form
u(k) = I cos(27iAfnt + 6n ) (3.3.2.5)
where
n = 0..10
Af = sampling frequency = 10,000 Hz
-7t < 0n < 7c random phase
This signal, shown in figure 3.22, represents a random low pass
signal that covers the frequency spectrum from f=0 to f=10Af.
Figure 3.22. Transmitted signal u(k) and received signal
Y(k).


64
At the receiving end of the channel, the distorted received
signal exhibits both phase and amplitude distortion. This signal
is shown together with u(k) for easier comparision in figure
3.22.
Next, we simulate the HR RLS adaptive equalizer to see how
well it can equalize the distortion. Since the channel has a
somewhat complex model, we select the order of the equalizer
to be five. In figure 3.23, the output of the equalizer is shown.
Figure 3.23. Equalized signal u(k).
Again, the equalized signal is shown with the transmitted
signal for comparision. The plot in fig. 3.23 shows that the HR


65
RLS ability to perform satisfactorily with a real
communication channel. Futher justification can be seen with
the MSE response of the algorithm shown in figure 3.24
Time k xlO-4
Figure 3.24. MSE of RLS algorithm.
In this chapter, we have shown how an adaptive
equalization algorithm is derived based on the general
estimation algorithm established in chapter 2. We also
developed the Generalized Adaptive Equalization Algorithm
Equalization (GAEA) and the Unified Adaptive Equalization
Algorithm (UAEA). A performance study of the three HR
algorithms was conducted using computer software MATLAB.
Lastly, the HR RLS AE was applied to a real communication
channel and was found to have maintained its abilitity to
equalize.


Chapter 4
Conclusion
The accomplishment of this thesis was developing the
Unified Adaptive Equalization Algorthim (UAEA) which is stated
in (3.2.5)-(3.2.11) and the block diagram is shown in figure 4.1.
The UAEA can implement any one of the three filter structures :
HR, FIR, and DFE and using any one of the three adaptation
algorithm: RLS, RMV, and RSG. It is the composition of the
Generalized Recursive Algorithm (GRA) (2.4.2)-(2.4.5) and the
Generalized Adaptive Equalization Algorithm (GAEA) (3.2.4)
which were also developed in this thesis.
Figure 4.1. Block diagram of UAEA.


67
In real life implementation of adaptive equalizers, the
initial task is to select the type of equalizer filter and
coefficient update algorithm. Knowing each type of filter and
algorithm offer different performance and often times it is
difficult to determine which types is most suitable. With the
UAEA, it offers the versatility of interchanging the various AE
schemes with minimum modification in hardware and software.
Therefore, it allows the user to build and test nine schemes of
adaptive equalization conveniently.
A performance study of the three MR equalization
algorithms was conducted by using computer simulation. The
results are summarized in the table shown below
Convergence rate MSE Computation Multiolies Additions Stability Noise imunity
RMV Fastest smallest ,n 4 2 10n +4n 4 2 8n +8n Stable Least
RLS fast smaller 10n4+ 4n 4 2 8n +8n Stable Average
RSG slow small 6n 4n Stable Most
Table 3.1. Algorithms performance summary.


68
The last part of the thesis was to apply the HR RLS
adaptive equalizer to a real communication channel. The
channel used was the Coaxial transmission line which has the
frequency characteristic response shown in figure 3.21. With
the results shown in figures 3.23 and 3.24, the equalizer
showed to maintain its ability to equalize the distortion.


69
BIBLIOGRAPHY
1. L. Ljung and T. Soderstrom, Theory and Practice of Recursive
Identification. The MIT Press, Cambridge, (1983).
2. B. Mulgreq and C. F. N Cowan, Adaptive Filters and Equalisers.
Kluwer Academic Publisher, Mass., (1988).
3. R. E. Lawrence and H. Kaufman, "The Kalman Filter for the
Equalisation of Digital Communications Channel," IEEE Trans.
Communications Technology. Vol. 19, (6), pp. 1137-1141, (Dec.
1971).
4. L. Ljung, "Assymptotic Behavior of the Extended Kalman Filter as a
Parameter Estimator for Linear Systems," IEEE Trans. Automatic
Control. Vol. AC-24, (1), pp. 36-58, (Feb. 1979).
5. J. G. Proakis, Digital Communications. McGraw Hill, New York,
(1983).
6. J. J. O'Reilly and A. M. de Oliveira Duarte, "Error Propagation in
Decision Feedback Receivers," IEEE Proceedings part F, Vol. 132,
(7), pp. 561-566, (Dec. 1985).
7. B. Widrow and S. D. Stearns, Adaptive Signal Processing. Prentice
Hall Inc., New Jersey, (1985).


70
8. R. W. Lucky and H. R. Rudin, "A Survey of the Communication Theory
Literature: 1968-1973," IEEE Trans. Inform. Theory. Vol. IT-19, pp.
725-739, (Nov. 1973).
9. R. Price, "Nonlinear Feedback Equalized PAM vs. Capacity for Noisy
Filter Channels, Proceedings 1972 IEEE Int. Conf.
Communications, pp. 22-12 to 22-17, (June 1972).
10. S. U. H. Qureshi, "Adaptive Equalisation," Proceedings IEEE Vol. 73,
(9), pp. 1349-1387, (Sept. 1985).
11. A. A. Giordano and F. M. Hsu, Least Souare Estimation with
Applications to Digital Signal Processing. John Willey & Sons, New
York, (1985).
12. D. A. Ross, "Distributed communications Architecture," University
of Colorado class report. (1983).
13. S. Ramo and J. Whinnery, Fields and Waves in Modern Radio. John
Wiley & Sons, New York, (1962).


APPENDIX: SIMULATION SOFTWARE & PROGRAMS
The software used for simulation in this thesis is MATLAB. Given
here are the written subroutines and programs to perform the
adaptive equalization simulation. The MATLAB version used was the
PC version, a two hundred-iteration simulation took about four
minutes on a 386 20 MHz system.


%
%clear
%io
%clc
clear
load data
clg
disp (' >')
M =size(Input)
N = 3;
disp (M)
INIT
~= ADAPTIVE EQUALIZATION=============
% number of iterations
% Select Algorithm
% Initialize variables & parameters
for k=l:M(2)
INPUTS
OUTPUT % Genearte plant output sequence
PARA
if k>l
if N>
if N=
if N=
end;
ERROR
SAVEDATA
%DISPLAY
end
% ALGORITHMS
1 RSG, end % Recursive Stochastic Algorithm
2 RMV, end % Recursive Minimum Variance
3 RLS, end % Recursive Least Square
% Save variables
% Display data
%PLOTTER


%==
========== Initialize Variables & Parameters=================
%=== ARMA model:
% y(k)= -al*y(k-1)-...-an*y(k-n) + bO*u(k)+...+bm*u(k-m)+v(k)
% THETA = [-al -a2 ... -an bO bl ... bm]' ;
% PHI = [y(k-1) y (k-2) ... y(k-n) u(k) u(k-l) ... u(k-m)]';
%=== i/o model: Y(Z) [Z'n+al*Z/'n-l + .. .+an]=U(Z) [bO*Z/Nm+bl*ZAm-l + .. .+bm]
n=5; % Order of Denomerator
m=5; % Order of Numerator
yl = 0;
y2=0;
ul = 0;
u2=0;
THETA = [.1 -.8 .35 -.2 .05)'; % [.6 .2 .7 -.4 .l]'test T = 20;
% enter plant parameters here
PHI1 = zeros(n+m+1,1);
PHI2 = zeros (n+n+1,1);
PHIB1 = zeros(n+n+1,1);
PHIB2 = zeros(n+n+1,1);
% ssss Generalized Algorithm:
% dim(THETAEST) = (n+n)xl
% dim(L) = (n+n)xl
% dim(P) = (n+n)x(n+n)
% dim(mu)=1x1
THETAEST = zeros(n+n+1,1);
L = zeros(n+n+1,1);
P = l*eye(n+n+1,n+n+1);
%==== Error Variables
% dim(yest) = lxl
% dim(es) = lxl
% dim(mse) = lxl
% dim(ES) = (m+n+l)x(m+n+l)
% dim(COV) = (m+n+1)x uestl = 0;
uest2 = 0;
esl = 0;
es2 =0;
msel = 0;
mse2=0;


dp dp dp
%==================== 10: Input/Output ===============
% This program is to compute the output of the coaxial
% transmission line by means of using IFFT and convolution.
clear
% Initializing the parameters of the line
x=30; % Km
1 = 2.357e-7;
G = .93e-6;
c = 9.428e-ll;
r = 51.8;
j = sqrt(-l);
format short e
df = 5000; % sampling frequency :small => T = small
fend = 1000000; % maximum frequency
Nf = fend/df;
% Loop to sample the channel frequency function exp(-x*gamma)
for f=df:df:fend
w = 2*pi*f;
indext = f/df;
gama = sqrt<(r+j*w*l)*(G+j*w*c));
Gama(indext) = gama;
temp(indext) = exp<-x*gama);
niag (indext) = abs (temp (indext)) ;
phase(indext) = angle(temp(indext))*180/pi;
% freq(indext) = f;
disp('f r gama temp mag phase')
disp([w r gama temp(indext) mag(indext) phase(indext)]')
pause
end
freq = df*[l:Nf);
TIMESIG


%
TIMESIG
% Compute the impulse response of the channel.
% Generate two square wave inputs to channel.
% Compute channel output by means of convolution
y = real(ifft(temp)); % Channel imputlse response
dim = size(y);
Nt = dim (2) ;
fsquarel = 3000; % frequency of first input signal
fsquare2 = 100000; % frequency of second input signal
dt = l/(df*Nt); % The computed sampling period
T1 = 1/(fsquarel*dt);
T2 = 1/(fsquare2*dt);
% Loop to generate input signals
rand('normal')
harmo = 10;
for i=l:harmo
theta(i) = rand(l)*pi;
end
for k = l:dim(2)
time(k) = k*dt;
Input(k) = 0;
for i=l:harmo
Input(k) = Input (k) cos (2*pi*i*df'time(k) + theta(i))
end
T=T2;
square2;
Input2(k) = u2;
end
% Truncate the impulse response
% for i = round(dim(2)*3/4): dim(2)
% y (i) =0;
% end
% Computing output signals by convolution
Output = conv(Input,y);
Output2 = conv(Input 2, y);
IOPLOT % Plot results
save data freq time Input Output Input2 Output2


%
IOPLOT
% Plot out tiime and frequency signals of channel
cig
subplot(211)
%plot (f, R)
%grid
%title('Channel resistance: R(f)')
%pause
plot(freq,mag);
grid
title('Coaxial Channel freq. response')
xlabel('Frequency' )
grid
plot(freq,phase)
grid
title('Coaxial Channel phase response')
xlabel('Frequency')
delete cfl.met
meta cfl
pause
plot(time,y)
grid
title ('Coaxial Channel impulse response')
xlabel('Time k')
delete ct3.met
meta ct3


%============ Generate Input Sequence ==============
% Input u(k) = white noise with moment n(meanu,sigu)
udelayl uestl;
udelay2 = uest2;
ul = Input(k); % test signal
u2 = Input2(k); % information signal


78
%=============== Generate Output of Plant ==================
%
% v(k) = white noise sequence with moment n(meanv,sigv)
sigv = .005;
meanv = 0;
rand('normal'); % computing v(k+l)
v = rand(1,1) *sqrt(sigv) + meanv;
yl = Output(k) +v; % computing yl(k+l)
y2 = Output2(k) +v; % computing y2(k+l)



%=============== PARA: Parameterization ==================
%
for i= (n+n):-1:1; % updating PHI(k+l)
PHIB1(i+1)= PHIB1(i);
PHIB2(i+1)= PHXB2(i);
end
PHIBK1) = udelayl;
PHIB1 PHIB2(1) = udelay2;
PHIB2(n+1) = y2;
7 S


%=========== RSG (Recursive Stochastic Gradient) ==================
%
% J (THETA00) = ly(k)-yest(k)]A2
% mu(k+1) = alpha/(k+1) alpha>0
% L(k+1) = -mu(k+1)*PHI % THETAEST(k+1) = THETAEST(k) + L(k+1)[y(k+1)-PHI'(k+1)THETAEST(k)]
mu = .1; % updating mu (k+1)'
L = -mu*PHIBl; % computing L(k+1)
THETAEST = THETAEST L* (u -(PHIB1'"THETAEST));


%============ RMV (Recursive Minimum Variance) ==================
%
% J(THETA) = E((THEATA-THETAEST)*(THETA-THETAEST)') ,MSE
% L(k+1) = P(k)*PHI(k+1)[PHI'(k+1)*P(k)*PHI(k+1) + sigv(k))"-1
% P (k+1) = P (k) L()c+1)*PHI' (k+l)*P(k)
% THETAEST(k+1) = THETAEST(k) + L (k+1) [u(k+1)-PHI' (k+1)*THETAEST(k)]
L = P*PHIBl*inv(PHIBl'*P*PHIB1 + sigv);
P = P LPHIB1'*P;
THETAEST = THETAEST + L*(ul (PHIB1*THETAEST));


%=================== RLS (Recursive Least Square) ==================
%
% J(THETA) = [ul(k)-uestl (k)]A2
% L(k+1) = P(k)*PHIB1(k+1)[1 + PHIB1' (k+1)*P(k)*PHIB1 (k+1)]A-1
% P(k+1) = P(k) L(k+1)*PHIB1'(k+1)*P(k)
% THETAEST(k+1) = THETAEST(k) + L(k+1) [ul (k+1) PHIB1' (k+1)THETAEST(k)]
L = P*PHIB1/(1+(PHIB1'*P*PHIB1));
P = P (L*PHIB1'*P);
THETAEST = THETAEST + L*(ul (PHIB1'THETAEST));


dP dP dP dp dP
83
================ ERROR Simuation ==;
uest(k) = PHIB' (k) *THETA£ST (JO
es (k) = [uOO-uest (k) P2
mse(k) = E{es) ? mean square
uestl = PHIB1'*THETAEST;
uest2 = PHIB2'THETAEST;
esl = (ul-uestl)A2;
msel = msel +l/k* (esl -msel);
estimated input
; square error
error
es2 = (u2-uest2)*2;
mse2 = mse2 + (es2-mse2)/k;


84
% =================== Display desired variables
disp(' dispm = [k disp(dispm) k ul * ul (k) uestl v yl uestl(k) esl ] ; v (k) yl (k) esl (k) t
disp (' k u2 (k) uest2(k) v (k) y2 (k) es2 (k) t
disp([k u2 uest2 v y2 es2]);
dispt'PHIl')
disp(PHI1');
disp('PHIB1'), disp(PHIBl')
disp('PHI2'), disp(PHI2')
disp('PHIB2'), disp(PHIB2')
pause
clc;


85
%================ Save Data Sequence =====================
%
% Data are saved in Data Matrices DM1, DM2, ...
%
% DM1(k) = [ul 00 uestl(k)]
% DM3 DM1(k, 1) = ul;
DM1(k,2) = uestl;
DM1(k,3) = yl;
V (k) = v;
DM2 (k, 1) = u2;
DM2(k,2) = uest2;
DM2(k,3) = y2;
for i=l:n+n+l
DM3(k, i)=THETAEST (i);
end
DM4(k,1) = msel;
DM4

86
%=============== Plot Data ===================
%
plot(DM1) % Plot u(k) and uest(k)
title('ul(k) & uestl (k)');
grid
pause
plot(DM2)
title('u2(k) & uest2(k)');
grid
pause
plot(DM3)
title('THETAEST');
grid
pause
plot(DM4)
title('msel(k) & mse2 (k)');
grid


87
%============ Initialize Variables 4 Parameters=================
%=== ARMA model:
% y(k) = -al*y (k-1). .-an*y (k-n) + bO*u < k.) + . +bm*u (k-m) +v (k)
% THETA = [-al -a2 ... -an bO bl ... bm]' ;
% PHI = [y(k-l) y(k-2) ... y(k-n) u(k) u(k-l) ... u(k-m)]';
%=== I/O model: V (Z) [Z*n+al*Z*n-l+ . +an] =U (Z) [bOZAm+bl*Z/'m-l +. .
% Order of Denomeratcr
% Order of Numerator
.1 -.8 .35 -.2 .05]'; % [.6 .2 .1 -.4 .ij'test T = 2C;
% enter plant parameters here
PHI1 = zeros(n+m+1,1);
PHI2 = zeros(n+n+1,1);
PKIB1 = zeros(n+n+1,1);
PKIB2 = zeros (n+n+1,1);
% ===== Generalized Aloorithm:
% dim (THETAEST) = (r.+n)xl
% dim(L) = (n+n)xl
% aim(P) = (n+n)x(n-r.)
% dim(mu)=1x1
THETAEST = zeros (n+n-1,1);
L = zeros (.n+n + 1, 1) ;
? = l*eye(n+n+1,n+n-1);
%===f=== Error Variables
% dim(yest) = lxl
% dim(es) = lxl
% dim(mse) = lxl
% dim(ES) = (m+n+1)x(m+n+1)
.% dim(COV) = (m+n+1) x (m+n+1)
uestl = 0;
uest2 = 0;
esl = 0;
es2 = 0;
msel = 0;
mse2=0;
n=5;
C
yl"= 0;
y2=0 ;
ul = C;
u2=0 ;
THETA = [