Citation
Adaptive blind equalization with applications in communication sysytems [sic]

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Title:
Adaptive blind equalization with applications in communication sysytems [sic]
Creator:
Stuck, Gordon H
Place of Publication:
Denver, Colo.
Publisher:
University of Colorado Denver
Publication Date:
Language:
English
Physical Description:
182 leaves : illustrations ; 28 cm

Thesis/Dissertation Information

Degree:
Master's ( Master of Science)
Degree Grantor:
University of Colorado Denver
Degree Divisions:
Department of Electrical Engineering, CU Denver
Degree Disciplines:
Electrical Engineering
Committee Chair:
Radenkovic, Mike
Committee Members:
Bialasiewicz, Jan
Moore, Linda

Subjects

Subjects / Keywords:
Digital communications ( lcsh )
Signal processing -- Digital techniques ( lcsh )
Equalizers (Electronics) ( lcsh )
Algorithms ( lcsh )
Algorithms ( fast )
Digital communications ( fast )
Equalizers (Electronics) ( fast )
Signal processing -- Digital techniques ( fast )
Genre:
bibliography ( marcgt )
theses ( marcgt )
non-fiction ( marcgt )

Notes

Bibliography:
Includes bibliographical references (leaves 181-182).
General Note:
Department of Electrical Engineering
Statement of Responsibility:
by Gordon H. Stuck.

Record Information

Source Institution:
|University of Colorado Denver
Holding Location:
|Auraria Library
Rights Management:
All applicable rights reserved by the source institution and holding location.
Resource Identifier:
50726356 ( OCLC )
ocm50726356
Classification:
LD1190.E54 2002m .S78 ( lcc )

Full Text
ADAPTIVE BLIND EQUALIZATION
WITH APPLICATIONS IN COMMUNICATION SYSYTEMS
by
Gordon H. Stuck
B.S., Metropolitan State College of Denver, 1992
A thesis submitted to the
University of Colorado at Denver
in partial fulfillment
of the requirements for the degree of
Master of Science
Electrical Engineering
2002


This thesis for the Master of Science
degree by
Gordon H. Stuck
has been approved
by
Date
Linda Moore


Stuck, Gordon H. (M.S., Electrical Engineering)
Adaptive Blind Equalization with Applications in Communication Systems
Thesis directed by Associate Professor Mike Radenkovic
ABSTRACT
Adaptive blind identification and equalization with applications in
communication systems are presented. Algorithm development and simulation
results are given for direct and adaptive solutions of blind fractionally spaced
equalization in noisy FIR channels. Additionally, the Constant Modulus Algorithm
(CMA), in the context of fractionally spaced equalizers, is given with theoretical
background, robustness considerations, and simulation verification. The theory and
simulations are primarily verification of results in the recent literature, however
some intermediate steps and additional derivations are presented.
This abstract accurately represents the content of the candidate's thesis. I
recommend its publication.
Signed
Mike Radenkovic
in


ACKNOWLEDGEMENT
My thanks to my advisor, Dr. Mike Radenkovic, for his guidance,
knowledge, and especially patience during this thesis project and my graduate career
at UCD. In addition, I am indebted to Sid Henderson for his valuable discussions
that helped me understand some key concepts in this work. My appreciation to my
coworkers and friends Erwin Siegel, Mark Mauerhan, Richard Rasmussen, and
Grant Szabo for their advice, support, and help with editing. Also I am grateful to
the management at Agilent Technologies, especially Giampaolo Tardioli and Tom
Beckman, for arranging a temporary relocation to Denver to do work on this project.
Finally, my sincere and deepest thanks go to my family: my mother Lynn, stepfather
Luis, stepmother Pamela, father Harvey, sisters Amy and Nancy, and brother
Randy.


CONTENTS
Figures....................................................................x
Tables.....................................................................xii
Chapter
1. Introduction to Blind Equalization......................................1
2. Overview................................................................3
3. Notations used Throughout Text..........................................5
4. Problem Description and Mathematical Framework..........................7
4.1 The Composite Channel Model........................................... 9
4.2 Baud and Fractional Sampling...........................................11
4.3 Multichannel Model for Fractionally Spaced Channel.....................12
4.4 Alternate Vector Representation........................................14
4.5 The Equalizer..........................................................15
4.6 Overall Model Derived from Baud Spaced System..........................16
4.7 Pure Delay Equalizer for BSE System is Not
Achievable............................................................17
4.8 Overall Model Derived from Fractionally Spaced
System................................................................18
4.9 Noise Free Optimal Zero Forcing (ZF) Equalizer.........................21
4.10 Non Blind Optimal MMSE Weiner Equalizer in the
Presence of Noise.....................................................22
v


5. Direct Blind Equalization Using Second Order
Statistics (SOS).......................................................25
5.1 Cylic Statistical Property of Fractionally Sampled
y(n)...................................................................26
5.2 Zero Forcing Direct Blind Equalizer.....................................27
5.3 MMSE Criterion (Weiner) Direct Blind Equalizer..........................29
5.4 Estimation of C2y from Sample Statistics................................32
5.5 Direct Blind Equalizer Calculation Methods..............................32
5.5.1 Batch..................................................................33
5.5.2 Recursive Non-Adaptive.................................................33
5.5.3 Recursive Adaptive Using Cyclic RLS....................................34
5.5.4 Recursive Adaptive Using Cyclic LMS....................................35
6. Direct Blind Equalization using Higher Order
Statistics (HOS) with the Constant Modulus
Algorithm (CMA)........................................................38
6.1 The CMA Algorithm..................................................... 38
6.2 Development of CMA from Godard [2]......................................40
6.3 More General Criteria...................................................43
6.4 CMA Cost Surface........................................................44
6.5 Robustness of CMA and Effects on the Cost Surface.......................47
6.5.1 Robustness to Equalizer Length.........................................48
6.5.2 Robustness to Channel Disparity........................................48
6.5.3 Robustness to Noise....................................................49
6.5.4 Robustness to a Non Constant Modulus Source............................51
vi


7. Performance Measures of Equalization Algorithms.......................53
7.1.1 MSE..................................................................54
7.2 ISI...................................................................56
7.3 Constellation Plots...................................................57
8. Channel Models Used in Simulation.....................................59
8.1 Channel Class RC: Raised Cosine......................................59
8.2 Channel Class AST: AppSigTec.........................................60
8.3 Channel Class NRR: Nearly Reflected Roots (i.e.
Loss of Disparity)....................................................61
8.4 Channel Class NUC: Near Unit Circle..................................62
8.5 Channel Class B: Benign............................................ 62
9. Simulations...........................................................64
9.1 Experiment 1: Performance in Noise with RC
Channel...............................................................65
9.1.1 Batch Method....................................................... 65
9.1.2 Recursive Non Adaptive Using Equations (5.32) and
(5.29)................................................................68
9.1.3 Optimal MMSE Weiner Equalizer from Equation
(4.32)................................................................68
9.2 Experiment 2: Channel Class AST Empirical Channel.....................69
9.2.1 Batch Method.........................................................69
9.2.2 Cyclic RLS Using Equation (5.38).....................................72
9.2.3 Cyclic LMS Using Equation (5.39).....................................72
9.2.4 CMA Using Equation (6.4).............................................75
vii


9.3 Experiment 3: Channel Class NRR......................................78
9.3.1 Batch Method........................................................78
9.3.2 Cyclic RLS Using Equation (5.38)....................................80
9.3.3 Cyclic LMS Using Equation (5.39)....................................80
9.3.4 CMA Using Equation (6.4)............................................83
9.3.5 CMA with Output Rotation Tweak......................................85
9.4 Experiment 4: Time Varying Channels..................................88
9.4.1 Cyclic LMS Using Equation (5.39)....................................88
9.4.2 Cyclic RLS Using Equation (5.38).................................. 91
9.4.3 Recursive Non Adaptive using Equations (5.34) and
(5.35)............................................................. 93
10. Matlab Code........................................................ 95
10.1 Filename: ell.m......................................................95
10.2 Filename: el 11.m...................................................100
10.3 Filename: el2.m.....................................................105
10.4 Filename: e211.m....................................................110
10.5 Filename: e221.m....................................................115
10.6 Filename: e213.m....................................................121
10.7 Filename: cmal2.m...................................................127
10.8 Filename: e311.m....................................................134
10.9 Filename: e321.m....................................................138
10.10 Filename: e312.m..................................................144
viii
10.11 Filename: cma31.m
150


10.12 Filename: cma31r.m
10.13 Filename: e41.m...
10.14 Filename: e42.m...
10.15 Filename: e421.m..
References...............
157
164
170
176
181
IX


FIGURES
Figure
4.1 Very general communications system model...................................7
4.2 More detailed digital communications system................................8
4.3 Correctly designed pulses have no interference at the
T-spaced sampling intervals............................................. 9
4.4 Baud spaced (P=l) and fractionally spaced (P=2 or
more) composite channel................................................ 10
4.5 SIMO multichannel model for FS channel (P=2)............................ 13
4.6 Baud spaced system model with linear equalizer
g(n).....................................................................15
4.7 Fractionally spaced system model with linear
equalizer g(n)...........................................................16
4.8 Fractionally spaced multichannel system model with
linear equalizer g(n)....................................................16
4.9 Overall baud-spaced system................................................16
6.1 JCmibpsk fr a well behaved noiseless real-valued
channel..................................................................45
6.2 JCM contours (solid) and JMSE overlay (dashed) for a
well behaved noiseless channel...........................................50
6.3 Effect of Non-CM 32PAM source (Kro = 1.8) on
JCM for a well behaved noiseless channel.................................51
7.1 Example constellation plots...............................................58
x


8.1 Two ray multi-path channel with raised cosine pulse
shape............................................................60
8.2 NRR channel......................................................62
9.1 Experiment 1 blind MMSE batch method.............................66
9.2 Blind ZF-MMSE FSE batch method...................................71
9.3 Blind ZF-MMSE FSE cyclic LMS method..............................74
9.4 Blind CMA FSE with three MSE calculation methods.................77
9.5 Blind ZF-MMSE FSE batch method...................................79
9.6 Blind ZF-MMSE FSE cyclic LMS method..............................82
9.7 Blind CMA FSE with three MSE calculation methods.................84
9.8 CMA constellation rotation.......................................85
9.9 Blind CMA FSE with three MSE calculation methods.................87
9.10 CMA constellation with manual rotation to match
input QAM 16 phase...............................................88
9.11 Blind ZF-MMSE FSE cyclic LMS method..............................90
9.12 Blind ZF-MMSE FSE cyclic RLS method..............................92
9.13 Blind ZF-MMSE FSE recursive non adaptive method..................94


TABLES
Table
8.1 Nearly common subchannel roots....................................61
9.1 Optimal Weiner MMSE equalizer comparison..........................69
s
I
!
xii


1. Introduction to Blind Equalization
Linear equalizers are used in digital communication systems at the receiver
to help maximize the information throughput. In general, digital communication
involves the transmission of analog pulses, also known as symbols, over a channel.
Typically the channel is a dispersive medium that introduces memory and spreads
the signal over time. This spreading of the symbols can corrupt the precise time
spacing and make the symbols spill over and corrupt adjacent symbols. This
phenomenon is referred to as inter-symbol interference (ISI). As ISI grows, the
probability of detecting a bit or symbol error at the receiver also grows. At high
enough data rates all physical channels, such as coax, fiber optic, microwave, and
twisted pair, exhibit ISI.
Equalization methods estimate the linear filtering dispersion characteristics
of the channel and insert an inverse filter at the receiver. The combined channel-
equalizer linear filter pair then acts as an ideal channel. In the time domain this
would be a Dirac delta impulse response. In the frequency domain this would be a
flat line passing all frequencies (i.e. information) without distortion.
Usually the non-ideal characteristics of the communications channel are not
known a priori, especially in high speed, high capacity channels. The term a prioi
1


is used because some equalizations methods do eventually figure out (and use) the
channel characteristics.
Prior to the 1980s, most linear equalization methods involved using a
training signal that was sent by the transmitter and was already known at the
receiver. This consumes valuable channel capacity:
A pre-arranged training sequence which is known a priori at the transmitter
and receiver can be used to estimate the sequence, though periodic re-
training my be necessary. Certain applications where a training sequence is
either too costly in usable bandwidth or where a training sequence is
impractical require a receiver design which operates on the received signal
and possibly some statistics of the source, but not on the actual source
sequence itself. Such an approach is termed blind. [6]
2


2. Overview
This thesis begins by presenting some introduction to the general problem of
blind equalization in communication systems and gives theory and simulation
verification of the equalization techniques of Giannakis and Halford [1], It presents
the theory and simulations related to the Constant Modulus Algorithm (CMA)
technique and explains channel models and performance measures. Chapter 3 is a
list of mathematical notations used throughout the text.
Chapter 4 introduces a description of the problem of blind equalization in
communication systems including channel models, baud and fractional sampling,
and system representations via vectors and matrices, and describes the non-blind
optimal Weiner Minimum Mean Squared Error (MMSE) equalizer. This is useful
for performance comparison of methods in the following chapters.
Chapter 5 presents the theory behind second-order-statistics direct blind
equalization methods including Zero Forcing (ZF) and ZF-MMSE and shows the
conditions needed for perfect equalization (PE). It also gives the equalizer
calculation methods of batch, recursive non-adaptive, cyclic Least Mean Squares
(LMS), and cyclic Recursive Least Squares (RLS).
Chapter 6 develops the theory behind the Constant Modulus Algorithm
(CMA), a popular higher-order-statistics blind equalization method and gives a
3


more general criteria related to the CMA. In addition it presents the CM criteria
cost surface and studies its robustness under violations of the conditions of PE.
Chapter 7 gives a description of the performance measures, such as mean
squared error (MSE) and inter-symbol interference (ISI) by which the equalization
techniques are compared. Though MSE can be found using input/output data, it
gives a derivation using only system parameters.
Chapter 8 introduces the channel models used in simulation.
Chapter 9 gives a description of the simulations, including graphs and results
with comparisons to [1].
Chapter 10 is a listing of the Matlab code used in the simulations.
4


3. Notations used Throughout Text
co().............. Information (source) symbols, discrete time
to................. Information (source) vector, discrete time
hc(t) .............Composite channel, continuous time
h ................. Channel impulse response vector, discrete time
vc(t)..............Additive noise, continuous time. Assumed to be stationary as
well as uncorrelated to co(n)
v(n), v..............Additive noise (function and vector form), discrete time
g(n), g..............Linear equalizer (function and vector form), discrete time
f (n),f.............. Overall system impluse response (function and vetor form),
discrete time
(n),o>..............Equalizer output. An estimate of the source symbols (fucntion
and vector form), discrete time
y(n), y..............Input to equalizer (function and vector form), discrete time
y0(n),y1(n),y0,yi.... Even and odd subchannel inputs to equalizers (function and
vector form), discrete time
y(n) = [y0(n)y1(n)]r
0T.................
. Combined even and odd subchannel inputs
Transpose
5


Conjugation
O*........
( )H or( )*T.......Conjugation and Transpose (Hermitian)
r\t
w ................. Pseudoinverse
* ................. Convolution
w.r.t.............. with respect to
6


4. Problem Description and Mathematical Framework
A simple and general model for a communications system is comprised of a
transmitter, channel, and receiver. See Figure 4.1.
Figure 4.1 Very general communications system model
The transmitter is fed from an information source, which for the purposes of
this thesis we can assume to be binary. The transmitter typically takes groups of n
bits from the binary stream input to encode a symbol a, from a finite alphabet, A.
An simple example alphabet for four-level PAM would be A = {-3, -1, 1, 3}
corresponding to the bit pair set {00, 01, 10, 11}. In this case n=2. Therefore the
total number of symbols is 2n = 4.
Often the symbols are complex with in-phase and quadrature components
such as in the QAM 16 modulation scheme, which will be the primary scheme used
in the analysis and simulations in this thesis.
QAM 16 has a complex-valued alphabet with n=4 allowing for 16 symbols.
This is representative of the dense and complex alphabets used in practice [6].
Compared to simpler modulation schemes, QAM16s dense constellation allows for
more bits-per-symbol and thus more bits per hertz. This results in a higher
7


bandwidth efficiency. QAM (16 and higher) is carried by many modem systems
[4].
-tb(n)
Symbol Estimate
hc(t) + vc(t)
Primary focus of this paper
Figure 4.2 More detailed digital communications system
In the transmitter, the complex symbols are modulated onto the amplitude,
frequency, and/or phase of analog pulses. These analog pulses are usually designed
so that adjacent symbols dont interfere with each other. Ideally one could transmit
infinite-frequency impulses which would not require pulse shapes that do not
interfere with each other. But bandwidth is limited. Therefore one popular type of
pulse is a raised-cosine or root-raised-cosine [7, pg. 546]. Both have time-domain
nulls at the symbols intervals and conserve spectral bandwidth. See Figure 4.3 for
an example.
8


Symbol n
Figure 4.3 Correctly designed pulses have no interference at the T-spaced sampling intervals
Next, the analog pulses are up-converted to RF. Sometimes its done in
multiple IF stages to increase dynamic range (not shown in Figure 4.2). The RF is
then bandpass filtered and transmitted through the channel. At the receiver, the
signal is filtered and down converted into baseband. Most likely this
downconversion is done in quadrature, similarly to the tranmitter.
4.1 The Composite Channel Model
The composite channel hc(t) includes the known transmit and receive filters
as well as the unknown channel (see Figure 4.2). This may include transmitter pulse
shaping (e.g. raised cosine). The model can also include additive noise vc(t) that is
assumed to be stationary and uncorrelated with the baseband information symbols
Cfl(n).
9


vc(t)
c(t)
J0
xc(t)VU yc(t) t = nT
P
X*___
y(n)
Figure 4.4 Baud spaced (P=l) and fractionally spaced (P=2 or more) composite channel
The continuous time information signal coc (t), for the purposes of this
thesis, is modeled as an impulse train of information symbols co(n):
coc(t)= £(0(f)5(t-fT).
t=-oo
The signal at the output of the sampler is
(4-1)
y c (t) = k (t) h c (t)] + vc (t) = J c (x)h c (t x)dx + vc (t). (4.2)
To derive an expression for yc (t) in terms of the discrete information
symbols co(n), combine (4.1) and (4.2),
yc(t)= X Jco(f)5(t-fT)hc(t-x)dx + vc(t). (4.3)
= £co(f)hc(t-fT) + vc(t).
(4.4)
10


Strictly speaking, the order of summation and integration in (4.3) cant be
interchanged. Practically speaking, the number of source symbols is large but finite
and the channel is FIR. Therefore (4.3) is absolutely integratable. Hence (4.4) is
justified [6].
4.2 Baud and Fractional Sampling
If yc(t) is sampled, as shown in Figure 4.4, at t = nT/P, the received
discrete signal y(n) = yc(t) for t = nT/P is
With P=l, (4.5) can be written as an equivalent baud-spaced discrete time
system
The n index in (4.5) and (4.6) above (n=...,-2,-1,0,1,2,...) corresponds to T-
spaced (baud spaced) time intervals.
Using P=2 (or greater) is the basis for creating a fractionally spaced system.
P=2 is the most popular choice for oversampling and is used throughout this thesis.
With this, (4.5) can be written as an equivalent fractionally spaced discrete time
nT
nT
(4.5)
yBS(n)= £co(^)h(n-.o+v(n).
(4.6)
system
11


(4.7)
Yfs (n) = X^)h(n -1) + v(n) = x(n) + v(n).
t=-~
Here note that the n index in (4.7), (n = ...,-2,-1,0,1,2,...) corresponds to T/2-spaced
(fractionally spaced) time intervals. However, the l index corresponds to baud
spaced intervals because the information symbols are always baud spaced.
The n index may be confusing. The reader should always look at the current
context because sometimes n corresponds to baud (T) spaced time intervals; and
other times n corresponds to fractionally (T/2) spaced time intervals. The primary
focus of this thesis is the latter, and therefore the FS subscript will be dropped
from yFS(n).
4.3 Multichannel Model for F ractionally Spaced
Channel
It can be seen from (4.7) that the output y(n) with n odd is a function of only
the odd coefficients of the channel model h(n) and noise samples v(n). The same
can be said for n even. Therefore upon defining y0(n) = y(2n) and
y,(n) = y(2n +1), the single-input, single-output (SISO) model of (4.7) becomes an
equivalent single-input, multiple-output (SIMO) multichannel model as shown in
Figure 4.5
12


v0(n)
with
y0(n)= ^co(i)h0(n -£) + v0(n)
e=-~
yi(n)= X(B(^)hi(n-^) + vi(n)
h0(n) = h(2n);h1(n) = h(2n+l)
x0(n) = x(2n);x1(n) = x(2n+l)
v0(n) = v(2n);Vj(n) = v(2n+l)
y0(n) = y(2n);y1(n) = y(2n+l).
Note that in (4.8) there is a change in the symbol spacing. The n in the
equations on the left corresponds to baud spacing within the subchannel signals h, x,
v, and y. The u' on the right hand side corresponds to fractional (T/2) spacing.
(4.8)
(4.9)
(4.10)
13


4.4 Alternate Vector Representation
A vector representation of the finite-length T/2-spaced channel h(n) and T-
spaced subchannels h0(n), h,(n)can be useful in equalizer and system analysis to
follow. The following vectors are defined
where Lh is the length of the T/2-spaced channel (note that the channel polynomial
first vector h has T/2-spaced elements (h0,h,, ...). However when used alone, h
can be considered to have T-spaced elements for analysis of baud spaced systems.
Vectors can also be used for the finite-length equalizer g and overall system
response f (these items will be developed later) in a similar manner to (4.11). In
addition the input, intermediate, and output signals such as co(), x(n), v(n), y(n),
and (b(n) can be expressed in vector form. These signals are generally considered
to have an infinite length index n, but when used in vector form they usually span a
finite-length recursive subset as a function of n. When used in the following
chapters, they will be clearly defined.
(4.11)
order would be Lh -1). In general (4.11) is for fractionally spaced analysis and the
14


4.5 The Equalizer
Consider now a more complete system model that includes a linear FIR
equalizer, g(n), for both the baud spaced and fractionally spaced systems (see Figure
4.6, Figure 4.7, and Figure 4.8) The equalizers g(n), g(n), g0(n),and gj(n)are
defined in a similar manner to h in (4.10). Also the equivalent finite vector T/2-
spaced representations are defined similar to h,
g = Lg 0 §1 §Lg-lJ
CfQ o II go Si -El.-2 h k r
8i = Si S3 * E Lg -1. rj
II O HO So Si -gLg-, r' r *
8i = Si §3 '§Lg-2 .
VLg even
(4.12)
In general (4.12) is for fractionally spaced analysis and the first vector g has T/2-
spaced elements (g0,g,, ...). However, when used alone g can be considered to
have T-spaced elements for analysis of baud spaced systems.
v(n)
to(n)
>d)(n)
Figure 4.6 Baud spaced system model with linear equalizer g(n). Note that n corresponds to baud
spacing.
15


v(n)
co(n)

Figure 4.7 Fractionally spaced system model with linear equalizer g(n). n corresponds to baud
spacing, n corresponds to T/2 spacing.
vo(n)
co(n)
(n)
v,(n)
Figure 4.8 Fractionally spaced multichannel system model with linear equalizer g(n). n corresponds
to baud spacing.
4.6 Overall Model Derived f rom Baud Spaced System
In the absence of noise, consider the overall baud-spaced system model f
co(n)
"Q)(n)
Figure 4.9 Overall baud-spaced system
which can be expressed as
L.-l
f (n) = £g(^)h(n -£) = g(n) h(n).
(4.13)
e=o
16


This convolution can be expressed as the inner product of a channel
convolution matrix H, associated with vector h, and the equalizer vector g as
f = Hg
(4.14)
with f = [f0f,...fLg+Lh_1Jr
The H channel convolution matrix is defined as
(4.15)
The combined channel-equalizer vector f is length Lh +Lg -1 and the
channel convolution matrix H is length (Lh +Lg -l)xLf, and is Toeplitz [18].
4.7 Pure Delay Equalizer for BSE System is Not
Achievable
The desired overall system response f in most applications is a pure delay
z5for some integer 5. However, the system of equations in (4.13) is always over
determined, so a finite length equalizer g which results in a zero-ISI pure delay
system is not achievable. In general, in the absence of noise an infinite length
equalizer would be required [6],
17


4.8 Overall Model Derived from Fractionally Spaced
System
In the absence of noise the overall baud-spaced system model of Figure 4.9
can be derived from the fractionally spaced models of Figure 4.7 and Figure 4.8.
The overall system transfer function, f, is always defined as baud spaced, regardless
of whether it was originally derived from a baud-spaced or fractionally-spaced
channel and equalizer. Here we will look at f(n) and f derived from the T/2-spaced
channel and equalizer:
Lg-1 Lg"l
f (n) = £g0 Wh0 (n -1)+ £g, (f)h, (n i); (4.16)
(=0 1=0
or in vector form:
f = h0*g0+h1*g1; (4.17)
and equivalently in the z (frequency) domain:
F(z_l) = H0 (z-1 )G0 (z-1) + H, (z-1 )G, (z~'). (4.18)
The form in (4.17) is equivalent to the integer-valued Diophantine equation.
This equation is attributed to Bezout and leads to the Bezout relationship [19] when
we let F(z_1) = z~5, which is our desired system response of a pure delay. In the
baud-spaced equalizer (BSE) case, the equalizer approximates the inverse of the
channel. But here in the fractionally spaced equalizer (FSE) case, g0 and gj and not
generally the inverse of the T/2-spaced channel.
18


Similar to equation (4.14), the convolution in (4.16) and (4.17) can be
expressed as the inner product of a channel convolution matrix and the equalizer
vector. However, since the symbols are transmitted at baud rate and not T/2, the
channel convolution matrix H has every other row removed. The result is a row
decimated version of (4.14)

(4.19)
" ho
fo u h2 hLh-2 hl h3 h0 h2 h, ho So gl
f |_^rh +Lg -l J hL-2 h3 h2
hLh-> 1 (4.20)
The number of equations in (4.20) is (Lh + Lg -1)/ 2.
Hi is a generalized Sylvester matrix [18] of the two subchannels. In some
texts (4.20) is written in an equally valid, alternate form
go r
fo g2
f2 = [HIH,] gL,-2 g. =
f g3
_gL,-,_
h0
h2
hL.
hL
hi
K-
h,
h3
hu-,
go
g2
81,-2
8,
g3
Sl.-,
(4.21)
19


where = [H0 | H,] is block Toeplitz and H0, H, are the convolution matrices of
the two subchannels. H0, Hj have a form similar to H in (4.15).
Due to fractional sampling and the corresponding row decimation of H in
Hx, the number of equations to be satisfied in (4.20) and (4.21) allows the system
of equations to become exactly determined. If the equalizer length Lg is chosen so
is square, the system of equations can be exactly solved if Hx is invertible (i.e.
full column rank). Therefore, with sufficient equalizer length Lg > Lh -1 the FSE g
can be found to achieve a pure delay, perfect equalization (PE)
system F(z_1) = z-8 (for some integer 5). It can be shown that if Lg = Lh -1 the
equalizer solution is unique.
Due to its block Toeplitz structure, Hx will be full row rank and invertible if
and only if there are no common subchannel roots in the H0(z_I) and
H^z-1) polynomials. When this condition is met, the system is considered to have
subchannel disparity.
In summary, PE (and hence zero ISI) is possible with a finite length
equalizer if these conditions are met:
Length: The equalizer is of sufficient length Lg > Lh -1
Noise: There is no additive channel noise
Disparity: There are no common subchannel roots in H0(z_l) and H,(z_1)
20


IID Source: The source co(n) is zero mean, iid, and has equiprobable symbols
Our analysis of equalization methods (e.g. optimal, blind, and adaptive) will
use FSEs only due to their ability to achieve PE with a finite length equalizer, their
use in literature, and their frequent application. Any mention of BSEs will be for
comparison only.
4.9 Noise Free Optimal Zero Forcing (ZF) Equalizer
When the conditions in the previous section are satisfied, and Lg = Lh -1 to
result in a square Hx matrix is used, the optimal ZF equalizer can be found by
solving (4.19)
However, for clarity recall that the goal is to create an overall delay function of
z8 for the system response So we replace f, above with
su = [Hj]%
(4.22)
ed+1 = [0 0-010 0]
(4.23)
d zeros
which has a 1 in the (d+l)th position. Also gd is defined as the equalizer
corresponding to the delay-d equalizer [1], So (4.23) can be rewritten as
& = [hJ'
(4.24)
21


4.10 Non Blind Optimal MMSE Weiner Equalizer in
the Presence of Noise
With channel noise v(n) present (note that here v(n) is fractionally 172
spaced from Figure 4.7 and the n notation has been dropped), which is a more real-
world scenario, PE is not possible, even for an FSE. So we desire to strike a
compromise between ISI reduction and noise amplification, where the latter is
present if a ZF equalizer is applied. So one approach is to minimize the data symbol
error e(n) in a mean-squared sense.
e(n) = d)n -ton_d (4.25)
where d is the choice of system delay. Vector to(n) has not been defined, but here
we want to make it a finite length Lg + Lh -1 (or Lf) regressor vector as a function
of baud spaced n
coLf (n) = [co(n) co(n-l) co(n-(Lg + Lh -2))] (4.26)
and also introduce the noise regressor vector vn using the alternate subchannel form
similar to g in (4.21)
VL (n)s[v(n-l) v(n-3) v(n-L -1) v(n) v(n-2)
6
v(n -Lg)]
(4.27)
It can be shown that the output can be expressed in the form
w(n) = to(n)Tf^ + v(n)Tg = to(n)THig + v(n)Tg . (4.28)
22


Then using ed+1 as defined in (4.23) the desired output o)(n d) is equal
to e(n) = (n)7^ + v(n)Tg = co(n)THig + v(n)Tg. (4.29)
Since the symbol sequence co(n) is assumed here to be iid with variance aj
and uncorrelated to the noise v(n) (with variance ov2), the mean square value of
the error e(n) is
MSE(g,d) = E{l e(n) I2} = a2 (Hig-ed+1)H(Hig-ed+1) + a2gHg . (4.30)
This expression has two minimizing parameters g and d. For the optimal MMSE
equalizer g, the solution is (see [6 pg 22], [8 pg 713], [5 pg 1931])
8opt-mmse ~ (Hj.H| H f-IL ) H^ed+1 (4.31)
and this is the classic Weiner filter. The corresponding MSE is found by using
(4.30) and (4.31) to obtain (see [6 pg 22])
min MSE(g,d) = MSE(d) = ed+1
SOPT-MMSE
i-H,(H"H,+-f-i,r,H;
'd+1
(4.32)
and the optimal value of delay d is found to be the minimum diagonal element of the
matrix in square brackets in (4.32)
23


^OPT-MMSE arS mjn
a

(4.33)
Jd,d ,
The reader should note that the optimal MMSE Weiner equalizer in (4.31) is
not blind, but requires exact knowledge of the channel parameters. In the blind and
adaptive equalization methods and simulations to follow, the optimal MMSE
equalizers performance is used as a comparison basis and creates theoretical limit to
how well the blind algorithms can perform.
24


5. Direct Blind Equalization Using Second Order
Statistics (SOS)
It has been shown in a variety of recent papers [12], [13], [14] that the
second order statistics (SOS) of the channel output y(n) contains enough
information to estimate most communication channels when the outputs are
fractionally sampled. Because of this, many effective blind methods have been
developed for estimating the channel from the output only SOS.
The SOS methods presented in [1] allow for finding the linear equalizer g(n)
directly from the data without first having to estimate the channel h(n). In this sense
it is a direct method.
Two of the methods in [1] will be presented in this thesis with some
comments and intermediate derivation steps for clarity and better understanding.
The first method is based upon (4.22) to find a ZF (i.e. ISI removal) direct blind
equalizer, however this method is not recommended with noise present. The second
method, which has similarities to the optimal MMSE equalizer in (4.31), finds a
MMSE direct blind equalizer that strikes a compromise between reducing ISI and
amplifying the noise. This method is also sometimes referred to by the criterion it is
minimizing: direct blind equalization using the MSE criterion.
i
25


Before these two direct blind equalization methods are presented, the cyclic
statistical property of fractionally sampled y(n) is shown because both methods use
this property.
5.1 Cylic Statistical Property of Fractionally Sampled y(n)
Similar to (4.26) and (4.27), define the following regressor vectors to
represent N vector observations of intermediate system signals x, v, and y of the
fractionally spaced, SIMO model of Figure 4.7.
xN(n) = [x0(n) x,(n) x0(n-l) x,(n-l) x0(n-N + l) x,(n-N + l)]T
vN(n) = [v0(n) v,(n) v0(n-l) v,(n-l) v0(n-N + l) v,(n-N + l)]T 5J
yN(n) = [y0(n) yi(n) y0(n-l) yi(n-l) y0(n-N + l) y,(n-N + l)]T
The correlation of fractionally spaced y(n) of Figure 4.4 and equation (4.7) is
c2y(n;m) = E{y(n)y*(n + m)} (5.2)
which expanded is
c2y(n;m) = jT ^c2(a(t2-^)h(n-2^)h*(n-2^2) + c2v(m) (5.3)
where c2(0(m) = E{oXn)*(n + m)} andc2v(m) = E{v(n)v(n + m)} From (5.3) it
can be shown that the correlation c2y(n;m)is periodically time varying in n with
period 2
c2y(n;m) = c2y(n + 2^;m)V ^integer (5.4)
To prove this, expand the r.h.s. of (5.4)
26


c2y(n + 2^;m)
X Xc2B(/2-/1)h(n-2(^I-£))h\n-2(£2-£))
el=^(2=^
+ c2v(m)
(5.5)
then, letting t\ = lx -l and/2 =l2-l, (5.5) becomes
c2y(n + 2^;m) = £ ^c2(0(^2,-^1,)h(n-2^1,)h(n-2^2,) + c2v(m) (5.6)
which has the same form of the r.h.s. of (5.3) and completes the proof that (5.4) is
valid.
The correlation matrix C 2y that is similar to (5.2), but uses regressor vector
yN(n) instead of y(n), is useful in upcoming derivations:
C2Ny=E{yN(n)yN(n)H}. (5.7)
5.2 Zero Forcing Direct Blind Equalizer
To develop an expression for a ZF direct blind equalizer, start by
considering the noise-free correlation matrix of x
C2x = HJC2nX (5.8)
and, since the input oo(n)is iid, C2(0 = o^I and therefore
C^ojHlH;. (5.9)
27


Now, taking the complex conjugate and multiplying by the zero delay (d=0) zero
forcing equalizer of (4.24) results in
c;*go=Me>= where the last term is expressed in Matlab format (the first colon means all rows
and the 1 means the first column). Solving for g0 and expressing the last term as a
vector yields
g.=^[c;"]'[h(0) 0 ... Of (5.11)
and therefore the d=0 ZF equalizer can be found directly from the correlation matrix
of the data to within a scale ambiguity of a^h*(0). When the equalizer length is
chosen as Lg = Lh -1, C2x is square and the matrix pseudoinverse + in (5.11) can
be replaced by a matrix inverse -1 and the equalizer g0 found by solving (5.11) is
unique.
It is possible to find the delay-d ZF equalizer directly from (5.11). For
details on the derivation, see [1]. The result is
Cgd=Cg0=aX(:,d+l) (5.12)
g =<[C*2r note;* ]%(:,1). (5.13)
The delay zero (5.11) and delay-d (5.13) ZF direct blind equalizers remove
all the ISI from the system output in an ideal noise free environment. However,
with noise present it doesnt do a very good job. Instead the noise is colored or
28


enhanced by the equalizers g0or gd [1]. Thus we will explore a new criterion,
minimum mean squared error (MMSE) in the next section.
5.3 MMSE Criterion (Weine r) Direct Blind Equalizer
The ZF equalizer of the prior section does not address noise suppression.
However, real world signals usually include noise. Therefore we are motivated to
find alternate criteria. Similar to (4.24) we consider an FIR Weiner filter to find the
minimum mean squared error estimate of cb(n) using only the output y(n). So this
method finds the equalizer g(n) so that the MSE-criteria cost function
Jd,MSE = e{| cb(n) cofn d) I2} (5.14)
is minimized.
To begin the derivation, the output cb(n)is defined as
L.-l
L.-l
6(n) = £g0 (£)y0 (n £) + £ g, (f)h, (n -1)
1=0
e=o
1 Lg-l
(5.15)
=XXsi(£)yi(n-^-
i=0 (=0
To minimize Jd MSE we take its complex partial derivative w.r.t. the unknown
equalizer coefficients and set it equal to zero
2'
a
(5.16)
29


Swapping the expectation operator with the derivative, and recalling that
f(x)2=2f(x)f(x) (presumably due to the chain rule), and using a as defined
dx dx
in (5.16) temporarily,
= E dgk(m)
1 Lg->
X Xg.Wyi(n -1) (n -d)
i=o e=o
= 0
(5.17)
the 2 drops out due to equality with zero and the co(n d) drops out of the partial
derivative; thus
= E< I a I
dgl(m)
l H
L-l

i=0 (=0
> = 0.
(5.18)
The next simplification requires some inspection of the indices k, m, i, and 1. Doing
so it is easy to see that the partial derivative filters out all of the terms except
yk(n m) resulting in
= e{| a I |yk(n m)|}= 0.
(5.19)
Re-expanding a,
= E
1 L,-
X X & (^)yi (n ~ Xn d> |y k (n m)|
i=o e=o
) = 0.
(5.20)
Next apply the rule I a II b 1= ab*,
30


= E
1 Lg_1
2 2 Si Wyk (n m)yi (n-£)-(o(n- d)y* (n m)
i=o e=o
= 0
(5.21)
which yields the orthogonality condition
i Lg_1
= X X bi WE{yI(n m)Yi (n ^)}]- E{(n d)yk (n m)}= 0. (5.22)
i=0 t=0
Focusing on the second expectation,
E{co(n-d)y*(n-m)}, (5.23)
expand the y k (n m) term to yield
E jco(n d) (^)hk(n m f)|. (5.24)
Since the noise in uncorrelated with the input and the input is iid, the only
correlation above occurs when £ in (5.24) is equal to (n-d). The rest of the time the
correlation is zero. Therefore (5.24) becomes
E{co(n d)co* (n d)hk (n m (n d))}
= and applying this back to equation (5.22),
= X X ti (*)EK (n m)yj (n £)}] = a^h*k (d m).
i=o e=o
This equation can be written in matrix/vector form as
(5.25)
(5.26)
31


E{y*Lg/2(n)yL/2(n)}gd =<^HjI(:,d + l) or
(5.27)
(5.28)
and this is equivalent to (5.12) with the replaced with The final zero delay
and arbitrary delay MMSE equalizers are
To solve for C^y in (5.7) so that the blind equalizers can be computed,
consistent sample estimates must be used because in practice ensemble values are
not available; therefore
where N is the number of vector observations as introduced in (5.1).
It is known that the normalized sample estimate in (5.31) converges in a
mean square sense to the true value [1].
5.5 Direct Blind Equalizer Calculation Methods
This section presents four methods for equalizer calculation:
(5.29)
(5.30)
5.4 Estimation of C2yfrom Sample Statistics
(5.31)
32


5.5.1 Batch
The previous section gives a batch method of finding the ZF and MMSE
equalizers for both zero delay and any delay. The equalizer is found directly
without having to estimate the channel first.
Implementation by using the batch method is fairly straightforward. For
example, the sample estimator in (5.31) is used to calculate for a pre-determined
number of samples (e.g. N=100). Then the MMSE zero-delay equalizer g0 is found
according to (5.29). This equalizer can then be used in the system. Continuing this
process, a new equalizer is calculated every N samples.
However, it is also possible to find the estimate in an equivalent
recursive manner that avoids having to perform long and computationally wasteful
summations starting with the first symbol. Also, adaptive recursive methods that
include a forgetting factor, A, are able to perform in situations where the channel
parameters a time varying.
5.5.2 Recursive Non-Adaptive
The following method gives results that are equivalent to the simple batch
method, but does not require redundant wasteful summations starting at the first
symbol. However, the method here is non-adaptive. Similar to the batch method, it
does not perform well on channels with time varying parameters.
To derive this method, start with (5.31) and expand to
33


1
e^(N)=ir-r25,v!(<)yv!(<)+yv!(N-1)yv!(N"1)
(=0
N -1*. ^ 1 * ,XT T ,XT
C2y(N)+ ^ yLg/2(N l)yLg/2 CN 1)
(5.32)
which results is a recursive form that can be applied directly to (5.29) to find the
MMSE zero-delay equalizer g0. In addition, by using the matrix inversion lemma
[19, pg 480], a method for finding g0 is available that does not require the inversion
or pseudoinversion of C2y. To use it, letP(N) = [C2 (N)]-1 and apply the matrix
inversion lemma to (5.32)
p(Nr,=^p(N-l)-1+-U* /2(N-l)y* /2(N-1)
N
N g
(5.33)
P(N) = P(N-l)-
N-l
N yL /2(N-l)y7 /2(N-1) N
P(N-l)^--------------77; P(N -1) (5.34)
N-l
N
N-l
l + ^y^^N-D^W-Dy^^N-l)
This can be applied to (5.29) to find the zero delay MMSE equalizer
g0(N) = ^P(N)H(:,l). (5.35)
5.5.3 Recursive Adaptive Usin g Cyclic RLS
Using a method similar to the previous section, a forgetting factor
X (0 < X < 1) is introduced to lower the influence of past observations on the
34


correlation estimate C2y thus allowing it to track time variations of the channel
parameters
C;,(N) = £X."-'y- 2(/)y7 ,2) + y: ,2(N)y^2(N)
*=o (5.36)
= XC;y(N-l) + y*Lg/2(N)y[?/2(N).
This can be applied directly to (5.29) to find g0. A direct method for finding g0
without inversion or pseudoinversion of C2y is developed in a similar manner to
(5.33)-(5.35). Again let P(N) = [C2y(N)]_I and apply the matrix inversion lemma
P(N) = Ar'P(N-l)-
)t-2P(N-l)y*g/2(N-l)yJg/2(N-l)P(N-l)
\ + XyLg/2(N-mN-l)ylgl2(N-\)
(5.37)
This can be applied to (5.29) to find the zero delay MMSE equalizer
i iN+l
io(N)=irrP(N)HiH(:,1)' (538)
This method can be used in a similar manner to derive the delay d ZF or
delay d MMSE equalizer.
5.5.4 Recursive Adaptive Usin g Cyclic LMS
A popular method of recursive direct equalizer calculation uses the
stochastic gradient descent (SGD) approach in updating the equalizer coefficients.
SGD is used in the cyclic LMS method of [1] and the CMA method of [2] and [6].
The CMA is a higher order statistics method that will be covered in a later chapter.
35


The cyclic LMS method updates the equalizer estimate using
g0(N) = g0(N 1)i|iVJ0(N)
(5.39)
where pis the step size and VJ0(N)is the instantaneous approximation to the
MMSE-criterion cost function presented in (5.14) as J0 = e| cb(n)-co(n) I2}. Using
VJ0 =
dJn 3Jn dJn
d J
3-L
_3go(0) 3g,(0) ag0(l) dgl(l) dg0(Lg-2) dg,(Lg-l)
and the results of section 5.3,
(5.40)
VJ0 = E{yLg/2yL/2 teo - (;>!). (5.4i)
allows for the instantaneous approximation at time N,
VJ0(N) = yLg/2(N)y[g/2(N)g0(N-l)-o2Hji(:,l) (5.42)
which can be used in conjunction with (5.39) to compute the equalizer estimate
g0(N)as
go(N) = g0(N-l)-|p[y*Lg/2(N)y^g/2(N)g0(N-l)-a2H(:>1)]- (5-43)
The choice of p is important in speed of convergence and steady state
performance. In simulations to follow, p = 2.5xl03 and p = 1 x 10 3are typical
values.
The cyclic LMS method presented here and used later in simulations does
not rely on an explicit matrix inverse and therefore is not as sensitive to nearly
36


common subchannel roots. It is also somewhat low in computational complexity,
but slow to converge [19, p 334-335]
37


6. Direct Blind Equalization using Higher Order
Statistics (HOS) with the Constant Modulus
Algorithm (CMA)
The constant modulus algorithm (CMA) is arguably the most popular blind
algorithm for cold startup of a tapped delay line equalizer. It was originally
proposed by Godard [2] and developed by Treichler and Agee in [15] and has been
the most studied blind adaptation algorithm of the 1990s. It has seen recent
application in VLSI chips for HDTV demodulators and also an explosion of use in
emerging wireless communications technologies [5].
6.1 The CMA Algorithm
The CMA algorithm is considered a higher order statistics (HOS) method
because it accumulates the fourth order moment of the received signal [3]. It is
considered constant modulus because the criterion it tries to minimize, JCM,
operates purely on the magnitude of the equalizer output. It penalizes deviations in
the modulus (i.e. magnitude) of the equalized signal away from a fixed value. It
does not operate on phase; and therefore the phase shift of this adaptive equalizer is
not determined. This can be seen in the output constellation rotation in the
simulations later in this thesis. However, this is not a problem because most modem
communication systems use differential encoding that can accommodate for the
phase ambiguity [6].
38


Under the conditions discussed in section 4.8 for perfect equalization (PE) of
FSEs (i.e. no noise, sufficient equalizer length, subchannel disparity, and an iid
source), the CMA algorithm can achieve PE with the additional constraint of the
source having a constant modulus (e.g. BPSK or 4PSK). Surprisingly, CMA can
also successfully equalize non constant source alphabets such as QAM 16.
The CMA algorithm used in this thesis is the p=2 type proposed by Godard
[2] which minimizes the following cost function
JCM = E{(l d)(n) I2 -y)2} (6.1)
where y is defined as the dispersion constant and is the ratio of the fourth to the
second moments of the source sequence, thus
y = E{l co(n) l4}/E{l co(n) I2} = E{l co(n) 14}/c2. (6.2)
To minimize the CM cost function, a stochastic gradient descent (SGD)
search algorithm is used which is similar to the one used with cyclic LMS. It can be
shown that the gradient of JCM w.r.t. the equalizer coefficient vector is
VgJcM = E{l ffl(n) I2 -y}d)(n)yLg/2(n). (6.3)
The algorithm to adaptively update the equalizer coefficients g (at each
timestep N) is obtained in a manner similar to the cyclic LMS algorithm in (5.43).
The instantaneous approximation to the true gradient in (6.3) is found by dropping
the expectation operator. This results in
39


g(N) = g(N-l) + fi{y-1 (b(n) l2}&(n)y*Lg/2(N) (6.4)
which is the CMA SGD algorithm. The algorithm minimizes the CM cost by
starting somewhere on the CM cost surface (determined by the initialization of g)
and following the trajectory of steepest descent.
The similarity between the CM criterion and the MSE criterion is strong.
Minimizing the CM cost function is equivalent to minimizing the MSE cost function
in the vicinity of convergence [15].
CMA is also globally convergent to the MSE minimizing equalizer using a
finite length FSE or infinite length BSE under the PE conditions given in section
4.8. With channel noise present, JCM is a close proxy to JMSE [5].
6.2 Development of CMA from Godard [2]
The search for a carrier-phase independent algorithm led Godard to
consider blind equalization techniques based only on the signal modulus |yn|. [16]
In the following section the development of the CM criterion and the
algorithm used for its minimization (CMA) is presented. It is based primarily on
Godard [2],
Consider first the conventional zero delay MSE criterion derived from (5.14)
Using the widely-used LMS algorithm for adaptively minimizing, this leads to
(6.5)
40


g(n +1) = g(n) \ y(n)((n)e"J

(6.6)
J (6.7)
which has the disadvantage of typically requiring a training signal ro(n) and is
therefore non-blind. Alternatively it can be formulated to rely on a degree of
(cyclo)stationarity of the received signal y(n), as discussed in section 5.1.
Godard noticed that the minimization technique above required g(n) and
9 to be simultaneously close to their optimum value. This led him to
find a cost function that characterizes the amount of ISI at the equalizer output
independently of the data symbol constellation and of carrier phase [2],
The result was an algorithm that did not require carrier recovery, but instead
allowed the recovery, expressed in equation (6.7), to be carried out after the
equalizer in a decision-directed (DD) mode.
The criterion was minimization of dispersion functions D(p) of order p,
where p is an integer greater than zero,
D(p)=E(l&(n)lp-Rp)2. (6.8)
Rp is a positive real constant that is related to the modulus (magnitude)
I equalized by constraining the equalizer output signal to have constant magnitude.
41


The CM algorithm (CMA) is derived by applying the dispersion D(p) in the
classical steepest descent algorithm, similar to LMS
>HP>0 (6.9)
S~Sn
with the partial derivative term being expanded as
= 2pE|y*cb(n) I &(n) T2 (I a*n) I2 -Rp)], (6.10)
S=in
then dropping the expectation, which is usually done when minimizing the MSE.
The result is the stochastic approximation algorithm
g(n +1) = g(n) Vn(n) I fi In Godard [2] the constant Rp is determined by finding where the gradient of
the dispersion of order p with respect to g (see equation in (6.10)) is zero. This is
found to be where
9D(P)
3g
g(n + l) = g(n)-ji
8D(P)
dg
_ EI p ~ E I to(n) lp
(6.12)
From the definition of dispersion in (6.8) and equation (6.11) it can be shown that
the equalizer does not require carrier recovery.
For the selection of p, Godard reasons that p > 2 is not practical in
applications due to convergence, finite length arithmetic, and overflow
42


considerations. In practice, p=2 is a popular choice which results in the Godard p=2
CM algorithm (CMA)
g(n +1) = g(n) X.2y*d)(n)(l cb(n) I2 -y) (6.13)
with
EI E I (6.14)
A good summary of Godards paper is given as Johnson [16] summarises:
Godard showed that for sub-gaussian sources, D(p) is minimized by system
responses generating zero IS I, and that choosing y (as in equation (6.14))
ensures that local minima of JCM exist at the perfectly-equalizing system
responses. [16]
6.3 More General Criteria
Shalvi [17] develops a criterion for equalization and blind deconvolution that
is more general than the CMA. It uses the fact that a sufficient condition for
equalization is the requirement that the probability distribution function (pdf) of the
input symbols co(n) be equal to the pdf of d)(n). This criterion is more flexible than
the CM criterion because co(n) may be real or complex random variables with
arbitrary continuous or discrete distribution.
The equalization criterion in [17] is
Maximize I K(cb(n)) I subject to E{l cb(n) 1} = E{ I co(n) 1}. (6.15)
where K(6(n)) is the kurtosis of cb(n).
43


The constrained maximization in (6.15) is solved by using a stochastic
gradient algorithm. However, the first development of this algorithm requires
spectral pre-whitening of the system (channel) output. Later in [17 equations (60)
and (61)] a constraint free criterion is developed that removes the need for
whitening. A special case of this maximization is shown to be equivalent to the
minimization of the dispertion function D(p)in Godard [2]. Also, a special case of
the stochastic gradient algorithm solution of the unconstrained criterion is shown to
be equivalent to the Godard p=2 CMA in equation (6.13).
6.4 CMA Cost Surface
One useful way to visualize the CM criterion cost function of (6.1) is by
simulating a simplified communication system with only two fractionally spaced
equalizer coefficients g(n). By doing this this the magnitude of JCM can be plotted
versus the values of the two equalizer coefficients. The result is a three dimensional
cost surface plot that often resembles and egg carton (for an example see Figure
6.1). Note that the number of equalizer coefficients is limited so the surface can be
visualized. Typically the surface is multidimensional because there are more than
two equalizer coefficients.
44


Figure 6.1 JCM|BPSK for a well behaved noiseless real-valued channel
In mathematical format, to derive a 3-D visualizable cost surface, start with a
4-tap fractionally spaced channel h = [h0 h, h2 h3]T The downsampled
convolution matrix (as defined in (4.20)) for the 2-tap FSE becomes
H,=
h, h0
h3 h2
(6.16)
with the fractionally spaced equalizer given by g = [g0 g1 ]T. This g is of sufficient
length for PE.
45


Johnson et. al. [5] gives many examples of the CM cost surface under
various violations to the PE conditions in section 4.8 and shows the CM criterions
cost surface similarity to the JMSE cost surface. One example, repeated here, is for
the case of a binary valued (+/- 1) BPSK source on a well behaved (i.e. absense of
common or nearly common subchannel roots) noiseless channel. This is shown in
Figure 6.1.
With a BPSK source, the CM cost function becomes
JCM,BPSK=E{(l-ld)(n)l2)2} (6.17)
and this can be expanded into a function of the system and equalizer coefficients (f
and h) and the source and noise statistics (in a real valued channel).
9 CMIBPSK
2
2llll2
-2ifi ++E+k, -3)£s*+3 i=0 i=0
where kv = E{l v(n) I4}/a* is the normalized noise kurtosis of v(n) and
(6.18)
Lf-l
||f||2 = I f (n) I2 is the squared l2 -norm of f(n).
n=0
In the example shown in Figure 6.1, the channel is well behaved with
h = [-0.0901 0.6853 0.7170 -0.0901]1
(6.19)
and noiseless (a2 = 0). Therefore many terms in equation (6.18) will drop off.
46


A generalized JCM cost function in terms of the system/equalizer coefficients
(f and h) and source/noise statistics is available in [5, equation (61)]. This
generalized form is a rather long equation, but shorter versions are also available in
[5, Appendix A] for PAM, PSK, and QAM sources.
6.5 Robustness of CMA and Effects on the Cost Surface
CMA appears to have some positive robustness properties when dealing with
violations of the PE conditions of equalizer length, channel disparity, noise, and
source distribution. In [6] these robustness properties of CMA are compared to the
relative lack of robustness of SOS methods in practical situations. [3] also contains
a robustness comparison of CMA to the indirect SOS methods of Tong, et. al. [12]
and Moulines et. al. [14].
Though the intent of this thesis is not to delve into a detailed comparison
study of the robustness properties of CMA, some general comments will be
presented regarding the CMAs behavior under violation of the four PE conditions.
The reader is referred to the paper by Johnson et. al. [5] which contains a detailed
categorization of the plethora of recent papers dealing with CMA robustness,
convergence, and SGD behavior. Another useful resource for experimentation with
the CMA is a MATLAB-5 based software environment called The BERGULATOR,
written by P. Schniter of Cornell Universitys Blind Equalization Research Group
[11]. It is available on the web site http://backhoe.ee.comell.edu/BERG.
47


6.5.1 Robustness to Equalizer Length
Having good robustness to non ideal (i.e. shorter) equalizer length is a very
practical property of any blind equalization method.
As hardware advances permit increased baud rates, physical channel delay
spreads remain unchanged and the relative length of the channel impulse
response grows proportionately. To combat ISI there is a corresponding
need to increase equalizer length. Therefore the desire for higher
communication rates will always stress the equalization task. [5]
CMA contributions from unmodeled portions of the channel, as long as they
are not significant, typically result in only mild cost surface deformation and the
CM minima stay in close proximity to the Weiner MMSE solutions. For example,
with the AST database empirically measured microwave channels used in the
simulations to follow, only the channel coefficients that are greater than about 15%
of the magnitude of the largest tap need to be covered by the CMA FSE [6].
6.5.2 Robustness to Channel Disparity
As discussed in [3], while most indirect methods based on SOS have
problems when the T/2 spaced channel loses disparity (i.e. has symmetric T/2
spaced root w.r.t. reflection through the origin or nearly common subchannel roots),
the CMA effectively solves for the baud spaced inverse of these common
subchannel roots.
In the presence of common subchannel roots the Diophantine equation of
(4.17) can be rewritten as
48


f = hcom + (h0 go + hi Si) (6.20)
where hcom contains the zero locations common to both subchannels. Since the
desired system response is a pure delay, recently proposed SOS algorithms require
an infinte length equalizer when inverting hcom. However, as discussed in [6], the
CMA has no problem providing an approximate baud spaced inverse of
hCOm provided the equalizer is long enough, which allows the extra FSE taps to solve
for the remaining part of the Diophantine equation.
Loss of channel disparity becomes more likely as channel length increases,
as can be seen in plots of T/2 spaced root of a realistic length-300 SPEB channel in
[3]. Also, a long FIR approximation to a pole in a physical channel would also
generate nearly reflected roots. For these reasons, robustness of a blind algorithm to
loss of channel disparity is a practical benefit.
6.5.3 Robustness to Noise
Noise deforms the CM error cost surface and causes the minima to move
toward the origin. Johnson et. al. [3] shows a good example of this for the well
behaved channel similar to the one presented in equation (6.19). Figure 6.2 shows
a top-down contour plot of the CM cost surface for a well-behaved noiseless
channe. The *s represent the points of global MSE (i.e. JMSE) minima. In the noise
free case shown, these points exactly match the CM cost surface minima.
49


4
Figure 6.2 JCM contours (solid) and JMSE overlay (dashed) for a well behaved noiseless channel.
Global MSE minima given by *s.
When an additive white channel noise v(n) is introduced to the system, the
Jmse and JCM both move to the origin by different amounts. The minimums still
remain very close, which allows the JCM to be a close match to JMSE when channel
noise is present.
50


6.5.4 Robustness to a Non Constant Modulus Source
For the CMA to achieve PE, one of the necessary conditions is for the source
to be iid, zero mean, and constant modulus (CM). An example of this would be
BPSK in which the source values always have an absolute magnitude of one.
However, many popular modulation techniques such as QAM 16 are not CM. Non
CM sources increase the source kurtosis Km to a value greater than one. The effect
of this increased kurtosis is to raise the CM cost surface as shown below in Figure
6.3, which was presented in [5]. This example uses the same well-behaved channel
presented in equation (6.19) and Figure 6.1, but the source of real-valued 32PAM is
far from being CM.
Figure 6.3 Effect of Non-CM 32PAM source ( Km = 1.8) on JCM for a well behaved noiseless
channel
51


The flattening makes the CMAs convergence to the minimum slower and
the DC offset (i.e. raised floor) introduces an additional asymptotic error level.
Shaping or correlating the symbols of the source (even if the constellation
itself is CM) for the benefit of increasing the coding gain also increases the source
kurtosis which also raises and flattens the cost surface.
52


7. Performance Measures of Equalization Algorithms
Equalization is used to lower the probability of an incorrect bit (or symbol)
decision at the receiver. This measure is commonly called bit error rate (BER) or
symbol error rate (SER). Equalization based on lowering BER or SER as the
criteria usually involves highly nonlinear functions. For example, for a system with
additive noise the analysis of SER involves the erfc complimentary error function
[6]. In practice, simpler performance measures, such as mean squared error (MSE)
and residual ISI, are used.
In general, the lower the MSE or ISI, the better the algorithms are
performing. Zero is the ideal. Most real systems, however, cant achieve this.
Instead, the blind algorithms are designed to reach a certain target level. In [6] and
[3] it is stated that one of the goals of the blind algorithm is to perform the initial
cold startup of the receiver system to sufficiently open the eye so the equalized data
stream can then be transferred to a Decision Directed (DD) mode to provide the
necessary tracking and further error rate reduction. For QAM 16 (the modulation
scheme used in the simulations in this thesis) a SER of approximately 0.04 (which
corresponds to and MSE of approx. 0.076 for QAM16) opens the eye sufficiently
for transfer to a DD mode.
53


In this thesis, the various methods of blind equalization are compared via
their MSE and residual ISI performance. Typically the graphs show the MSE
and/or ISI plotted versus the symbol number index. The plots usually achieve an
asymptotic steady-state level after a sufficient number of symbols (e.g. 2500 is
typical). Also it is useful to look at the input and output (unequalized and
equalized) symbol constellations.
7.1.1 MSE
The mean squared error, used as a performance measure, has a similar form
to the MSE-criterion cost function in (5.14)
MSE = E{ I d)(n d) to(n) I2}. (7.1)
In practice this equation (7.1) can be replaced by a consistent sample
estimate based on N observations
MSE(N) = 0 (n d) co(n) I2) (7.2)
N n=0
and this equation (7.2) is used for most of the simulation plots in this thesis.
However, it is possible to derive an expression for MSE equivalent to (7.1) that uses
the system parameters f, equalizer parameters g, noise variance a2 and input
symbol variance o2.
For this derivation, start by assuming the system has no noise. The MSE can
then be expressed as
54


(7.3)
MSE, =E{l^f(^)oXn-^)-co(n)l2}
e=o
and expanded using the relation I a 1= a a*
f,=o
E{^f(£1)co(n-£1)^f,(£2)(ot(n-£2)-(o(n)^f(£)(o(n-£)
i7=o
e=o
(7.4)
- (0* (n)^ f (f)co* (n £) + ci^n) (n)}.
?=0
After this, the separate £x and l2 summation indices drop out due to the expectation
operating on an iid source. This, with more simplification, yields
= ]£f{£)i*(l)E{co(n-l)tP (n -0} -E{ co* (n)^ f (^)(o(n -0}
- E{to(n)^f *(^)co*(n £)} + E{co(n)o)*(n)}.
<=o ' ^ '
(7.5)
= a:
£lf(f) I2 -f(0)-f*(0) + l
^=0
(7.6)
Now noise will be introduced into the system by using baud-spaced
subchannel noise v0(n) and v,(n). The overall MSE now becomes
MSE = E{ IX f (^)co(n -£) + £ g0(/)v0 (n -1) + £g, (*)v, (n -1) co(n) I2}. (7.7)
(=0 (=0 (=0
When this is expanded all the expectations that involve the noise and signal drop out
because they are uncorrelated. The terms that involve the noise will temporarily be
55


called MSE2 in this derivation. The terms involving the signal were already done in
(7.6) and called MSE,. The overall MSE = MSE, + MSE2.
MSE2=E{Jg0(^)v0(n-£1)£g;^2)v0(n-£2)
<1=0 f2=0
L. is,
+Ssi(^i>vi(n^i)2Lsr(^2)vi(n^2)>
f,=0 ^2=0
= G
2
v0
+ 0:
(7.9)
The final form for the MSE combines (7.6) and (7.9)
MSE = <
£ifC012 -f(0)-f*(0)+i
(=0
(7.10)
This equation is very similar to that found in Giannakis and Halford [1], and has
been shown to give equivalent results to (7.2) in simulations.
7.2 ISI
The residual inter-symbol interference, used as a performance measure, is
given by
ISI =
£lfCOI-maxlf(f)l
e=o______________
(7.11)
max I f (£) I
This equation is the ratio of the sum of residual terms in the system to the largest
term. It is the same as the equation given in [1] with the squares removed. The
56


author is not sure why the form in [1] has squared terms because using (7.11) in
simulations achieves results that match the plots in [1].
The idea behind this equation can be seen if one considers the equation for
the output symbol estimate
<5Xn) = co(n)*f(n) (7.12)
and assume that the ideal system function f(n) is an impulse response with f(0)=l
and zero elsewhere. However, a more realistic system function f(n) will have some
residual terms at f(l), f(2), etc, though in general theyll be smaller than f(0). With
this in mind, (7.12) can be rewritten as
tb(n) = co(0)f (0) + ^ coff )f (n-l) (7.13)
i
where the second term is the intersymbol interference. The contribution to from this ISI term, taken as a value between zero and one, is probably equivalent to
the equation in (7.11). This is left to the reader for further investigation. The author
was only able to verify a connection between (7.13) and (7.11) for a positive and
real f(n) and co(n).
In simulations of blind equalization algorithms to follow, the residual ISI is
compared using equation (7.11).
7.3 Constellation Plots
An interesting and sometimes useful plot is the constellation diagram
(sometimes referred to as an eye diagram) at the input and output of the equalizer.
57


With a glance at equalizer output plot, it is usually easy to see if the algorithm is
working.
Taking the input modulation formation QAM 16 as an example, Figure 7.1
shows in (a) the original input symbols, and in (b) the same signal after passing
through an example system h(n), before the equalizer and with noise added. Finally
(c) shows the equalizer signal after passing through g(n).
1 o O O 0 * o o o ** m m
o o o o o o o o 2 os # ap <# S* m .
-1 o o o o - m m>
(a) Input QAM 16 symbols 5 -3 -2 0 J 3 4 \ys 05 (b) Unequalized with SNR=30dB (c) Equalized 750 symbols Figure 7.1 Example constellation plots
58


8. Channel Models Used in Simulation
The following channel models are presented for simulation purposes.
8.1 Channel Class RC: Raised Cosine
Raised cosine filters are commonly used in digital communication systems
because they eliminate ISI at baud spaced intervals. Some systems rely on
equalizing the pulse-shaping dynamics, so channel models with only the pulse
shaping response are of practical use.
In the simulations of this thesis, a casual approximation to a two-ray multi-
path radio environment is used that includes a raised-cosine response and is the
same channel used in [1, experiment 1]. This continuous time channel is described
by
hc (t) = e~i2n(0- 15)rc (t 0.25T, (3) + 0.8e"j2n<0 6,rc (t T, (3) (8.1)
which spans four symbols. rc(t,P)is the raised-cosine given in [7, p. 546] as
(t,(3) = sinc(7tt/T)
cos(7tpt/T)
l-4p2t2/T2
(8.2)
with P = 0.35 used in the simulations. The discrete time equivalent channel hc(t)is
found by sampling (8.1) at a rate of T/2 or h(n) = hc(nT/2)for n=0,l,...7. Figure
8.1 shows the magnitude of the h(n) impulse response, the zeros of h(n), and the
zeros of subchannels h0(n) and h,(n).
59


Two Ray Multipath: Raised Cosine Pulse Shape
Figure 8.1 Two ray multi-path channel with raised cosine pulse shape. h(n) impulse response, h(n)
zeros, and subchannel zeros. Plots first presented in [1],
8.2 Channel Class AST: AppSigTec
This channel class, used in simulations to follow, is taken from empirically
measured T/2 spaced digital microwave channel data (see [1], [3], and [6] for
references to the source of this data). The channel data, in raw form, is a vector of
complex-valued impulse response coefficients with greater than 200 taps. However,
shortened versions are derived using linear decimation of the FFT of the full-length
T/2 spaced data. The resultant length-16 frequency domain data is inverse FFTed
to obtain length 16 impulse responses. This process reduces time domain
decimation aliasing but maintains most of the frequency characteristics.
!
i
i
|
60
i


These shortened responses have near common subchannel roots as well as
roots near the unit circle (in the SISO T/2 space model). Thus, this channel class
shows how real-world channels can have the characteristics that aggravate the
performance of blind equalization methods.
8.3 Channel Class NRR: Nearly Reflected Roots (i.e.
Loss of Disparity)
This channel class is constructed to have nearly common subchannel roots or
symmetric T/2 spaced root w.r.t. reflection through the origin. Channels of this
nature are used to analyze how the blind equalizers respond to a loss of disparity.
In the simulations to follow, the NRR channel used in [1, experiment 3] has
zeros as shown in Table 8.1. The resulting lh(n)l impulse response, h(n) zeros, and
subchannels zeros are shown is Figure 8.2.
Subchannel Roots
h0(n) h, (n)
0.0074-0.49991 0.0070-0.5094]
-0.2736-0.3719j -0.2777-0.3799]
-0.4560+0.2706] -0.4383+0.2598]
-0.2472+0.38701 -0.2328+0.3628]
0.5213-0.05391 0.4902-0.0545]
0.1961+0.7765] 0.1008+0.2539]
0.4149+0.1422] 0.3789+0.1586]
Table 8.1 Nearly common subchannel roots
61


Near Common Subchannel Roots
Channel Zeros
Subchannel Zeros
Real Axis
Figure 8.2 NRR channel
8.4 Channel Class NUC: Near Unit Circle
This channel class has roots that form a ring of zeros near the unit circle, but
none form reflected pairs. Channels like this tend to have nulls in the frequency
response. According to [6] and [3], the open literature at the time (1997) contained
little work addressing the significance of such dynamics for the fractionally sampled
case. This class is not analyzed in simulations in this thesis.
8.5 Channel Class B: Benign
This channel class does not have any channel class NUC or NRR
characteristics or symmetric impulse responses the RC class. Therefore, it doesnt
62


expose any of the problems with current popular blind adaptive equalization
algorithms. This class is not analyzed in simulations in this thesis.
63


9. Simulations
The simulations presented here are based upon experiments 1-4 in [1]. The
equalization techniques include direct blind MMSE, optimal Weiner MMSE, direct
blind ZE-MMSE, and blind p=2 CMA. The calculation methods include batch,
recursive non-adaptive, cyclic LMS, cyclic RLS, and CMA. The channel models
used (see chapter 8) include two-ray multi-path RC, empirically derived AST, NRR,
and time varying channels. The noise added ranges in power to achive SNRs from
12.5 to 30dB.
The following four sections include details, such as graphs and results, from
the experiments. For each experiment, the calculation methods are listed in a sub
section. Under each subsection the following information is given:
Matlab file: The name of the Matlab file. The listing of the Matlab code
can be found in chapter 10.
Equalizer Type: The type of equalizer used, such as Blind MMSE.
Input: The input data used
Channel Type: such as two-ray multi-path RC, AST, or others.
Noise Added: The type of noise and SNR.
Equalizer Calculation Methods: such as batch, Cyclic LMS, or others.
64


Monte Carlo Runs: The number of averages across multiple simulations,
if any.
Results: Comments that are primarily intended to compare the results to
the experiments in [1].
9.1 Experiment 1: Performance in Noise with RC
Channel
9.1.1 Batch Method
Matlab File: el l.m
Equalizer Type: Blind MMSE FSE, Zero delay (d=0)
Input: QAM-16, 2500 symbols
Channel Type: Channel Class RC two-ray multi-path mobile radio. Lg=Lh=3.
Channel equation can be found in the Giannakis and Halford paper [1 pg. 2285].
Noise Added: Zero mean, Gaussian to achieve SNRs of 30, 20, 15, and 12.5.
Equalizer Calculation Method: Batch. Equalizer calculated every 100 symbols
from accumulated sum
Monte Carlo Runs: Averaged across 10 simulations
Results: See Figure 9.1. MSE and ISI match Giannakis and Halford [1 Fig.5
and Fig.6], The figure parts (a) and (c) are from the simulations for this thesis.
Parts (b) and (d) are the corresponding figures from [1], which were smoothed
across 100 Monte Carlo runs.
65


Mean-Square Error {Symbols)
WVO-01 'A(ea-0JS2: :36WW pu!|0
symbols
Mean-Square Error (Symbols)


Residual ISI
Blind MMSE:Zero Delay, 16-QAM
symbols
(c)
(d)
Figure 9.1 Experiment 1 blind MMSE batch method. Plots (b) and (d) are results from [1],
67


9.1.2 Recursive Non Adaptive Using Equations (5.32)
and (5.29)
Matlab File: el ll.m
Equalizer Type: same as above
Input: same as above
Channel Type: same as above
Noise Added: same as above
Equalizer Calculation Method: Recursive Non Adaptive using Equations (5.32)
and (5.29) that uses prior C2u (A) estimate. This equation is
ClCN) = ^C'2u(N-\)+y(N~ 1}Ny(N~1}; g(N) = Pinv{Cl{N))alH{:X) (9.1)
Monte Carlo Runs: same as above
Results: same as above
9.1.3 Optimal MMSE Weiner Equalizer from Equation
(4.32)
Matlab File: el2.m
Equalizer Type: same as above
Input: same as above
Channel Type: same as above
Noise Added: same as above
68


Equalizer Calculation Method: Direct to find optimum MMSE FSE after 2000
symbols using Equation (4.32). The MMSE Wiener filter which has exact
knowledge of the channel.
Monte Carlo runs: same as above
Results: The resultant minimum MMSE closely matches the results of [ 1, Table
I]. See a side by side comparison in Table 9.1 below.
SNRfdBl Optimum MMSE (MSE1
30 0.0035
20 0.0258
15 0.0656
12.5 0.1025
SNR i MSE
in dB Blind FSE Optimum MMSE
30 0.0060 0.0033
20 0.0274 0.0237
15 0.0652 0.0618
12.5 0.1026 0.0993
(b) Simulation results
(a) Optimum Weiner MMSE Result from
Giannakis and Halford [1, Table I].
Table 9.1 Optimal Weiner MMSE equalizer comparison
9.2 Experiment 2: Channel Class AST Empirical
Channel
9.2.1 Batch Method
Matlab File: e21 l.m
Equalizer Type: Blind ZF-MMSE FSE, delay d=4
Input: same as experiment 1
Channel Type: Channel class AST empirically measured P=2 digital microwave
channel with Lh=Lg=7, duration spanning eight symbols. Channel data was
downloaded from BERG website at Cornell U., chan 1.mat.
69


Noise Added: Zero mean, Gaussian to achieve this 30dB SNR.
Equalizer Calculation Method: Batch. Equalizer calculated every 100 symbols
from accumulated sum.
Monte Carlo Runs: Averaged across 10 simulations
Results: See Figure 9.2. This algorithm works well for chan 1.mat with the MSE
converging to about 10~2 with performance exceeding that of Giannakis and
Halford [1]. For chan4.mat, it works poorly.
70


Mean-Square Error (Symbols)
symbols
(a)
Blind MMSB: 16-QAM Empirical Channsl
(b)
Figure 9.2 Blind ZF-MMSE FSE batch method. The straight line in (a) can be ignored. Plot (b) is
result from [1],
71


9.2.2 Cyclic RLS Using Equation (5.38)
Matlab File: e221.m
Equalizer Type: Blind ZF-MMSE FSE, delay d=4
Input: same as above
Channel Type: same as above
Noise Added: same as above
Equalizer Calculation Method: Cyclic RLS using equation (5.38). Initial
equalizer was found from 100 samples in a similar way to el l.m. A, = 0.98.
Monte Carlo Runs: None
Results: Erratic performance, it can hover above MSE= 10-1. Combinations of
chan 1.mat and chan4.mat were tried because its not known which AST channel
is the correct model. Also combinations of calculating gd(T) based on H*(:,l)
or H*(:,d+1) were tried. The equations for these are different, and the author
derived the former.
9.2.3 Cyclic LMS Using Equation (5.39)
Matlab File: e213.m
Equalizer Type: Blind ZF-MMSE FSE, delay d=4
Input: same as above
Channel Type: same as above.
Noise Added: same as above.
72


Equalizer Calculation Method: Cyclic LMS using equation (5.39). Initial
equalizer was found from 100 samples in a similar way to el l.m. Stepsize
H = 0.0025.
Monte Carlo Runs: None
Results: See Figure 9.3. Performance is better than [1] when using chan 1.mat if
the gd(T) equation is based on H*(:,d+1). If gd(T) is based on the authors
derivation using H*(:,l) performance is poor. Chanl.mat and chan4.mat were
tried because its not known which AST channel is correct model. Calculating
gd(T) based on H*(:,l) or H*(:,d+1) results in equations that are are different.
The author derived the former.
73


(a)
Blind WM3S: 19-QAM empirical Channal
(b)
Figure 9.3 Blind ZF-MMSE FSE cyclic LMS method. The title on the left graph should read LMS.
Plot (b) is result from [1],
74


9.2.4 CMA Using Equation (6.4)
Matlab File: cmal2.m
Equalizer Type: Blind CMA FSE, delay, p=2 type from Goddard [2]. See
equation (6.4).
Input: same as above
Channel Type: same as above.
Noise Added: same as above.
Equalizer Calculation Method: CMA
Monte Carlo Runs: Averaged across 10 simulations
Results: See Figure 9.4. The performance was equal or lower than the MSE line
of the CMA method in Giannakis and Halford [1, Figure 8], This may be due to
some differences in the formula used to derive the MSE. Three methods were
tried. The first MSE calculation method was from a CMA Matlab program from
the BERG website. This method is not fully understood, but achieves an MSE
convergence that is more similar to the CMA line in the Giannakis and Halford
paper [1 Fig.8]. The second method is per equations (7.1) and (7.2), repeated
here:
Finally the third method per equation (7.10) is very similar to [1, equation 53],
repeated here:
(9.2)
75


(9.3)
MSE = a* XM2 1=0 -f(d)-f*(d)+i +<> X|go0)|2 .1=0 + vl £mi! .1=0
The second and third methods have exactly equal results and achieved a lower
MSE (by about 0.02) than the first method.
76


Mean-Square Error (Symbols)
(a)
Blind MMSS: 16-QAM Empirical Channsl
(b)
Figure 9.4 Blind CMA FSE with three MSE calculation methods. Plot (a) is result from [1],
77


9.3 Experiment 3: Channel Class NRR
9.3.1 Batch Method
Matlab File: e311 .m
Equalizer Type: Blind ZF-MMSE FSE, delay d=4
Input: same as above
Channel Type: NRR Channel with near common subchannel roots described in
section 8.3 taken from channel in [1, experiment 3].
Noise Added: Zero mean, Gaussian to achieve this SNR=30dB.
Equalizer Calculation Method: Batch. Equalizer calculated every 100 symbols
from accumulated sum.
Monte Carlo Runs: Averaged across 10 simulations
Results: See Figure 9.5. Works well with performance exceeding [1,
experiment 3],
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Mearv-Squaje Error (Symbcte) Mean-Square Error (Symbols)
(a)
Blind MMSE: 16-QAM Near Common Roots
(b)
Figure 9.5 Blind ZF-MMSE FSE batch method. The straight line in (a) can be ignored. Plot (b) is
result from [1].
79


9.3.2 Cyclic RLS Using Equation (5.38)
Matlab File: e321.ni
Equalizer Type: Blind ZF-MMSE FSE, delay d=4
Input: same as above
Channel Type: same as above
Noise Added: same as above
Equalizer Calculation Method: Cyclic RLS using equation (5.38). Initial
equalizer was found from 100 samples in a similar way to el 1 .m. X = 0.98.
Monte Carlo Runs: None
Results: Does not work and will not converge. More work is needed here.
9.3.3 Cyclic LMS Using Equation (5.39)
Matlab File: e312.m
Equalizer Type: Blind ZF-MMSE FSE, delay d=4
Input: same as above
Channel Type: same as above
Noise Added: same as above
Equalizer Calculation Method: Cyclic LMS using equation (5.39). Initial
equalizer was found from 100 samples in a similar way to el l.m. Stepsize
p = 0.0025.
Monte Carlo Runs: None
80


Results: See Figure 9.6. Performance is better than [1], but gd(T) equation had

to use H*(:,d+1). Also the calculation of y conj(y) seemed backwards but is
needed otherwise it wont work.
81


r~Sqoare Error (Symbols} Mean-Square Error (Symbols)
(a)
Blind MMSE: 16-QAM Near Common Root*
lb)
Figure 9.6 Blind ZF-MMSE FSE cyclic LMS method. Plot (b) is result from [1],
82


9.3.4 CMA Using Equation (6.4)
Matlab File: cma31.m
Equalizer Type: Blind CMA FSE, d=4 (or 3?) from Godard paper [2] (see ref.
[14] in the Giannakis and Halford paper [1])
Input: same as above
Channel Type: same as above
Noise Added: Zero mean, Gaussian to achieve SNRs of 30dB only.
Equalizer Calculation Method: CMA
Monte Carlo Runs: None
Results: See Figure 9.7. Regardless of MSE calculation method, the MSE
performance is a little worse then the CMA line in [1]. This is probably due to
the rotation in the output constellation due to CMAs inability to operate on
phase. This effects the calculation MSE. See the next experiment for a non
rotated version which significantly reduces the MSE.
83


Mean-Square Error (Symbols)
CMA p=2 Experiment 2, 16-QAM, SNR=30dB, AST Channel 4
$.v ¥^505






BERG Method
-- Mod. Giannakis (53) Method, Gordon derrive
f > Radenkovic Direct Method
______i_____i____i i i i i r ~i
0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000
symbols
(a)
Blind MMSE: 16-QAM Near Common Roots
(b)
Figure 9.7 Blind CMA FSE with three MSE calculation methods. Plot (b) is result from [1],
84


Figure 9.8 CMA constellation rotation. The small dots show the original QAM16 input. The fuzz-
balls are the equalizer output symbols. The dots at the origin are artifacts due to algorithm
initialization.
9.3.5 CMA with Output Rotation Tweak
Matlab File: cma31r.m
Equalizer Type: same as above
Input: same as above
Channel Type: same as above
Noise Added: same as above
Equalizer Calculation Method: CMA with output manually rotated to match
input constellation phase.
85


Monte Carlo Runs: None

Results: See Figure 9.9. The MSE is much lower than the CMA line in [1] when
using the direct MSE calculation method per equations (7.1) and (7.2).
86


Mean-Square Error (Symbols)
symbols
(a)
Blind MMSE; KW3AM Near Common Roots
(b)
Figure 9.9 Blind CMA FSE with three MSE calculation methods. Plot (b) is result from [1],
87


Figure 9.10 CMA constellation with manual rotation to match input QAM16 phase. The dots at the
origin are artifacts due to algorithm initialization.
9.4 Experiment 4: Time Varying Channels
9.4.1 Cyclic LMS Using Equation (5.39)
Matlab File: e41.m
Equalizer Type: same as above
Input: same as above
Channel Type: Time varying channel is same as described in [1, experiment 1]
and section 9.1.1, but at t>=2250 the channel in equation (8.1) changes its
parameters to
88