Blind Equalization with Differential Detection for

Channels with ISI and Fading

Eloise Tse

A thesis subrnitted in conformity with the requirements

for the degree of Master of Applied Science

Graduate Department of Electrical and Cornputer Engineering

University of Toronto

@ Copyright By Eloise Tse (1997)

National Library 1*1 of Canada Bibliotheque nationale du Canada

Acquisitions and Acquisitions et Bibliographie Services services bibliographiques

395 Wellington Street 395, rue Wellington Ottawa ON K1A O N 4 ûttawa ON K1A ON4 Canada Canada

Yow Votre rdfemnce

Our fi& Notre refdrence

The author has granted a non- L'auteur a accordé une Licence non exclusive licence allowing the exclusive permettant à la National Library of Canada to Bibliothèque nationaie du Canada de reproduce, loan, distribute or sell reproduire, prêter, distribuer ou copies of this thesis in microform, vendre des copies de cette thèse sous paper or electronic formats. la forme de microfiche/^, de

reproduction sur papier ou sur format électronique.

The author retains ownership of the L'auteur conserve la propriété du copyright in this thesis. Neither the droit d'auteur qui protège cette thèse. thesis nor substantial extracts fkom it Ni la thèse ni des extraits substantiels may be printed or otheMrise de celle-ci ne doivent être imprimés reproduced without the author's ou autrement reproduits sans son permission. autorisation.

Blind Equalization wit h Different ial Detection for

Channels with ISI and Fading

Eloise Tse, M.A.Sc.

Graduate Depart ment of Electrical and Computer Engineering

University of Toronto, 1997

Supervisor: Professor Pas S. Pasupathy

Using coherent detection with carrier tracking and adaptive equalization with train-

ing sequence can achieve good performance in equalizing time-varying channels at the

expense of complexity and feasibility. Thus, differential detection and blind equaliza-

tion, which eliminate PLL and training sequence, are proposed. Decision feedback is

dso added to equalize null and fading channels. In this thesis, Godard and Modified

Constant Modulus Algorithms (MCMA) axe used. New systems are set up by com-

bining coherent and noncoherent detection with these two algorithms. It is found that

the use of noncoherent detection degrades the system performance. For MCMA, as it

can track the carrier, neither noncoherent detection nor PLL is required. Contrarily,

Godard needs either noncoherent detection or PLL to correct phase error. Thus,

the proposed system combining differentid detection, blind equalization and decision

feedback c m indeed equalize different channels, though the robustness of the system

is compromised. Further investigations must be done to deal with this problem.

Acknowledgment s

First and foremost, thanks to my supervisor, Professor Pas S. Pasupathy, for bis

unlimited patience and guidance.

Special thanks to Ali Masoomzadeh-Fard, for helping me to start my work,

and for his patience when answering my never-ending questions.

Last but not least, my family and friends, whose understanding and support

are forever appreciated.

University of Toron t O

April1997

. . . Ill

Acronyms

AWGN

BER

BPSK

CMA

DBPSK

DFE

DPSK

DQPSK

FSE

FT

GVA

IID

ISI

LE

LMS

MAP

MCMA

Additive White Gaussian Noise

Bit Error Rate

Binary Phase Shift Keying

Constant Modulus Algorit hm

Differential Binary Phase Shift Keying

Decision Feedback Equalizer

Different ial Phase Shift Keying

Differential Quadriphase Shift Keying

Fractionally S paced Equalizer

Fourier Transform

Generalized Viterbi Algonehm

Identical and Independently distributed

Intersymbol Interference

Linear Equ&zer

Leas t Mean Square

Maximum A-posteriori Probability

Modified Constant Madulus Algorithm

ML

MSE

PLL

PN

PSK

QPSK

RLS

SER

SNR

SVD

TEA

VA

ZF

Maximum Likeli hood

Mean Squared Error

Phase Locked Loop

Pseudo Noise

Phase Shift Keying

Quadriphase (Quadrature phase) Shift Keying

Recursive Least Squares

Symbol Error Rate

Signal to Noise Ratio

S ingular Value Decomposi tion

Triceptrurn Equalization Algorithm

Vit erbi Algori thm

Zero Forcing

Contents

1 Introduction 1

2 Coherent and Noncoherent Systems 4

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Coherent source 4

. . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Noncoherent source 7

3 Equalization with Differential Detection

4 Blind Equalization 17

. . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Probabilistic Approach 18

. . . . . . . . . . . . . . . . . . . . . . . . 4.2 Steepest Descent Approach 20

. . . . . . . . . . . . . . . . . . . . . 4.3 Higher Order Statistic Approach 21

. . . . . . . . . . . . . . . . . . . . . 4.4 Sequence Estimation Approach 24

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Application 27

5 Linear Godard and MCMA Systems 29

. . . . . . . . . . . . . . . . . . . 5.1 Linear Coherent Godard Algorithm 29

. . . . . . . . . . . . . . . . . . 5.2 Linear Differential Godard Algorithm 32

5.3 Linear Coherent MCMA . . . . . . . . . . . . . . . . . . . . . . . . . 35

5.4 Linear Differential MCMA . . . . . . . . . . . . . . . . . . . . . . . . 37

5.5 Simulation Results For Lincar Systems . . . . . . . . . . . . . . . . . 38

5.5.1 Channel Models and Parameters . . . . . . . . . . . . . . . . . 38

5.5.2 ltesults for Channel A . . . . . . . . . . . . . . . . . . . . . . 39

6 Godard and MCMA with Decision Feedback 45

6.1 Coherent Godard with Decision Feedback . . . . . . . . . . . . . . . . 45

6.2 Differential Godard with Decision Feedback . . . . . . . . . . . . . . 45

6.3 Coherent MCMA with Decision Feedback . . . . . . . . . . . . . . . . 52

6.4 Results for Decision Feedback Systems . . . . . . . . . . . . . . . . . 55

7 Conclusion 63

A Multipath fading channel 66

A. l Radio channel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

A.2 Mobile Channel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

B Adaptive Decision Based Equaiization 71

B.1 Derimtion of LMS for conventional systems . . . . . . . . . . . . . . 71

. . . . . . . . . . . . . . . . B.2 Derivation of LMS for Godard algorithm 74

C Definition of Performance criteria 77

vii

List of Tables

4.1 Table of nonlinear functions for Steepest Descent Method . . . . . . . 22

5.1 Parameters setting for systems simulated with SPWTM . . . . . . . . 38

5.2 System configurations for chamel A (for = forward equalizer, back =

backward equalizer) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

6.1 System codigurations for channel B . . . . . . . . . . . . . . . . . . . 56

6.2 Parameters for the fading filter . . . . . . . . . . . . . . . . . . . . . 58

6.3 System configurations for channel C . . . . . . . . . . . . . . . . . . . 59

6.4 List of systems simulated in three different channels . . . . . . . . . . 62

rr" C.1 Number of iterations required for systems to converge for Channel B . ( i

List of Figures

2.1 Coherent Decision Feedback equalizer (DFE) . . . . . . . . . . . . . . 6

2.2 Differential detection scheme for DQPSK . . . . . . . . . . . . . . . . 8

2.3 Linear Differentid equalizer (DIFF) . . . . . . . . . . . . . . . . . . . 9

3.1 Linear equalizer (LE) placed before differential detector . . . . . . . . 13

3.2 Nonlineat equalizer model for DPSK . . . . . . . . . . . . . . . . . . 15

4.1 General adaptive structure for Steepest Descent Approach . . . . . . 2 1

4.2 Block diagram for blind sequence estimation . . . . . . . . . . . . . . 26

5.1 Coherent System with Godard for LMS equalizer (CG) . . . . . . . . 31

5.2 Godard with Differential detection for LMS equalizer (DG-1 ) . . . . . 33

5.3 Differential detection with Godud for LMS equalizer (DG-2) . . . . . 34

5.4 SER vs SNR c u v e for coherent systems for Channel A . . . . . . . . 41

5.5 SER vs SNR curve for Godaxd systems for Channel A . . . . . . . . . 42

5.6 SER vs SNR c u v e for MCMA systems for Channel A . . . . . . . . . 43

. . . 6.1 Coherent Godard with feedback path for LMS equalizer (CGFB) 46

6.2 Differential Godard with feedback path for LMS equalizer (DGFB) . 48

6.3 Coherent MCMA with feedback path for LMS equalizer (CMFB) . . 52

. . . . . . . . . . . . . 6.4 SER vs SNR c w e for systems for Channel B 57

6.5 SER vs SNR cuve for systems for Channel C . . . . . . . . . . . . . 60

. . . . . . . A.1 Multiple signal paths due to reflections in fading channels 67

Chapter 1

Introduction

When detecting the data sequence from the received signal, two types of detection

can be used. One is coherent detection, the other is noncoherent detection. If coher-

ent detection is used, schemes such as phase-locked loop (PLL) are used to recover

the absolute phase of the received signal. However, this complicates the hardware

especidy when the data transmission rate is high and the phase is varying rapidly.

Thus, noncoherent detection, a simple structure where no phase tracking is required,

is desirahle. In mobile and data communications, differential detectiou is a commonly

used signaiLing scheme. Bandwidth efficient scheme like dgerent ia l quadriphase shift

keyzng (DQPSK), which utilizes both in-phase and quadrature axes, is used (11. How-

ever, with differential detection, nonlinear intersymbol interference (ISI) is generated.

Simple equalization methods are no longer feasible. Different schemes have been em-

ployed to deal with this non-linearity (21. However, this nonlinear component together

wi t h the charnel dis tortions, degrade the performances of the noncoherent receivers,

and make them inferior to the coherent decision feedback scheme. Furthermore, the

complexity for some of the schemes is very high. AU of these structures require some

knowledge about the channel characteristics which is not known for most practical

channels. Therefore, an efficient equalization algorithm is required for noncoherent

detection. In this thesis, new structures which combine blind equalization with dif-

ferential detection are presented. Also, to equalize null and fading channels, decision

feedback is added to these new structures.

Among the already known noncoherent receiver structures, sever al adapt ive solu-

tions are proposed for DQPSK. First, a linear equalizer is placed before the nonlinear

detector without decision feedback. This method fails to equalize fading channel

which requires decision feedback. Without decision feedback, nulls in the channel

response give rise to noise enhancement at the input of the slicer. Hence, poor perfor-

mance results. Then, the situation is remedied by introducing two modified adaptive

equalizers. In [2], these proposed algorithms are simulated. Acceptable performances

result . However, a testing sequence is still needed in the above systems where a known

sequence is transmitted and received before the actual data is sent. The equalizer

adapts to the channel by minimizing the enor between the known sequence and re-

ceived sequence. This known sequence is, thus, c d e d the training sequence. Usually,

a pseudo-noise (PN) sequence is used for this application. However, training sequence

imposes a certain amount of delay which must be taken into account if the actual

data transmission is short. Also, by using the training sequence, it is assumed that

the channel characteristics do not deviate a lot fiom its initial state after the sequence

is sent. This is certainly not true for time wying channels such as mobile channels.

In order to track the channel response, the training sequence has to be transmitted

periodically for the adaptation of the equalizer to the time variations. Sometimes

the transmission of such sequence may even be impossible in some communications

chasnels. This is why blind equalization is proposed. With blind equalization, no

such sequence is needed. Instead, the equalzer estimates and adapts itself imme-

diately when the data is received. This rnethod equalizes the channel based on the

information ob tained from the received signal, contrary to previous algorithms. Since

existing blind algorithms are applied to coherent detection, the challenge is to apply

differential detection to blind equalization. These two features together with decision

feedback result in new systems which c m be applied to mobile communications.

Among numerous blind equalization dgorithrns [3, 4, 51, two in particular are

of interest. They are Godard Algorithm (Constant Modulus Algorithm CMA) and

Modified Constant Modulw Algorithm (MCMA) from the Stochastic Gradient class.

The transmitter and channel mode1 are presented in Chapter 2. In Chapter 3, several

already proposed differential structures are discussed. Next, an overview of blind

equalization is given. Then in Chapter 5, the two blind equalizers are combined with

coherent and differential detections. Simulation results for channei without nuil are

presented. In Chapter 6, decision feedback is added to some of the structures in

Chap ter 5. The resulting systems are simulated for a charnel wit hout null, with nulis

and with fading. Lastly, conclusions about ail the systems and results are drawn.

Chapter 2

Coherent and Noncoherent

Systems

In order to compare the performances of the proposed equalizers, complex equivdent

baseband systems wit h coherent and diRerentia.1 detections are discussed.

2.1 Coherent source

For coherent systems, QPSK is used as the source, providing four possibilities ((1

+ j), (1 - j), (-1 + j), ( 1 - j)} for a = x + y . The symbols ai are shaped by

a raised cosine filter g( t ) , go through the channel p ( t ) (assuming it is symmetric)

and comipted by zero mean additive white Gaussian noise (-4WGN) n(t) with power

spectral density N 4 2 . Then they are received by a lowpass filter h(t) . At output of

this filter, the received signal is,

where û is an unknown angle introduced by the bandlimited channel, J is the memory

of the channel, f ( t ) is the convolution of g(t), p(t) and h( t ) , R(t) is the noise at the

output of the lowpass filter, and 1/T is the data transmission rate. Sampling r(t)

every T seconds yields,

where ri = r(iT), fj = f(jT) and hi = fi(iT). To equalize different channels, Least

Mean Square LMS equalizer with decision feedback is used in Figure 2.1. After ri has

passed through the forward equaiizer,

where 8 represents convolution, c,- and N are the tap coefficients a ~ d length of the

fornard equalizer respectively, and f i i is the noise after the equalizer. As for the

feedback equalizer, its output is,

where âi is the estimated symbol from the slicer, di is the tap coefficients of the

backward equalizer, and M is the length of backward equalizer. The s u r n of the

forward and feedback equdzers' output is fed to the slicer,

QPSK a i raised Channel Source cosine g(t) p(t) .

Figure 2.1: Coherent Decision Feedback equalizer ( D F E )

The slicer used is a complex quantizer which maps its input to the closest QPSK

constellation point. For conventional DFE, the error which is fedback to the equalizer

is defmed as the difference between the output and input of the slicer,

where âi and ai are the output and input of the slicer. In training mode, âi = ai; in

other words, known and correct data are fedback to adjust the equdizer's coefficients.

With LMS algorithm, the equalizer adjusts its tap coefficients y with the following

cri terion,

where p is the step size, V, is the gradient with respect to y, and Ji = ei in ( 2.6).

Thus, the coefficients of the fonvard and backward equalizers are adjusted using (see

Appendix B. 1 ),

If a fractionally spaced equalizer (FSE) is used instead of the symbol rate spaced one,

the performance of the equalizer improves significantly for the fading channel [6]. To

rnodie a symbol rate equalizer into a $-rate fractiondy spaced one, the sampling

rate of the forward equalizer is doubled (and/or doubling the number of taps).

2.2 Noncoherent source

For our noncoherent systems, differential phase shift keying (DPSK) is used. After

transmission through the communication channel, an unknown phase is often intro-

duced to the received signal. To compensate for this unknown phase, differential

encoding is combined with phase shift keying (PSK). We assume that the unknown

phase m i e s slowly, so that it is constant within a two-bits intenal. Instead of encod-

ing the information into absolute phase, the information is encoded by the change in

phase. Thus, for transrnitted symbol with the form Ak = e j O k , #& is determined by,

with the information embedded in Ah. Some common implementations are clifferen-

tial binary phase shift keying (DBPSK) and DQPSK where differential encoding is

combined with BPSK and QPSK.

To detect this differential scheme, coherent detection and noncoherent detection

can be used 161. To detect coherently, the actual phase of the received symbol is

Figure 2.2: Differentid detection scheme for DQPSK

determined. Then, the change in phase is caiculated by subtracting the phase of the

previous syrnbol from it, giving A*. Howeves, this scheme can be very difficult to

implement; especially when the phase characteristics of the channel vary so rapidly

that tracking of the carrier phase is an impossible task. Even if tracking can be done,

PLL has to be used, which complicates the hardware.

However with noncoherent detection, the phase is detected differentially. In other

words, only the change in phase between the previous and present symbols is of

interest. From Figure 2.2, ài contains Ak rather than the absolute phase. Thus,

the unknown phase is eliminated through multiplication without employing complex

hardware. However, this detection induces a 3 dB penalty compared to coherent

detection due to an increase of noise at the slicer input [6] (except for DBPSK which

is essentially the same as BPSK), shown by the bit error rate (BER) for DPSK,

1 Eb BER = Zexp(--)

N o

where Es, No are bit energy and noise power spectral density respectively. However,

this penalty can be easily compensated.

The differential scheme is particularly useful in fading channels where PSK is pre-

ferred. In t hese channels, the phase characteristics change rapidly. Ins tead of taking

Diff. & hardlirniter encoder

Channel cosine g(t)

A a i

4 sticer a a i ~ i . f f decoder

T

Figure 2.3: Linear Differential equalizer (DIFF)

on the difficult task of tracking this rapidly changing phase, DPSK with differential

detection is used. Therefore, with DQPSK, Figure 2.3 is setup. After the differential

encoder,

where ai , bi aze the QPSK and DQPSK symbols'.

The symbols bi are then transmitted through the same mode1 as the QPSK scheme.

Therefore, at output of the lowpass filter h(t) , the received signal is,

where f ( t ) is the convolution of the raised cosine filter g ( t ) , channel p ( t ) and lowpass

filter h( t ) , B is an unknown angle, J is the memory of the syrnmetric channel, and

n(t) is the noise at the output of the lowpass filter. Again r ( t ) is sampled every T

ltf ai, bi are cornplex, a hardlimiter is used to keep their magnitude to 1.0.

seconds to give,

where ri = r ( iT) , fj = f(jT) and f i i = fi(iT). After passing through the forward

equalizer,

where ci, N are the tap coefficients and length of the fonvard equalizer. The data is

then recovered by decoding the output of the equalizer hi using a diflerentid decoder.

This operation involves a nonlinear process,

This is where, if there is residual ISI, nonlinear ISI is generated. Since the data is

differentidly encoded, the function which is used to correct the equalizer's coefficients

must be derived by taking the gradient in ( 2.7). Same as before, Ji = ei where ei is

( 2.6) . It is found that ei has to be multiplied by the delayed output of the equalizer A

bi-l (see Appendix B.1). Therefore the equalizer's coefficients are adjusted by the

following equation, A

q + l = ci + bi-lei. (2.16)

This equation is different from ( 2.8) since ai is the output of the differential de-

coder. Then Z i is passed through the slicer which maps the data into one of the four

possibilities within the QPSK constellation.

In order to implement a decision feedback path, the estimated data âi must be

re-encoded into 6i before feeding into the backward equalizer,

Summing up the forward and feedbadc equalizer's output, bit input to the decoder is

rnodified into,

where the first convolution is ( 2.14), M, di are the length and tap coefficients of

backward equalizer. Same gradients are found for the LMS adaptations,

The above details the transmit ter, channel mode1 and two conventional sys tems,

while the blind receiver structures will be discussed in details in later sections. Next.

several previously proposed differential systems are discussed.

Chapter 3

Equalizat ion wit h Different ial

Detection

In most communication applications, the channel effects can be modeled by a discrete

linear time-invariant filter. Thus, the received signal is the convolution of the channel

impulse response and the input signal (after the transmit ter filter) as in ( 2.2) and

( 2.13). In order to compensate for this channel distortion, minirnize the noise, and

estimate the transmitted signal, a deconvolution is needed. There are many different

approaches as to how this deconvolution is done, such as linear equalization with zero

forcing (ZF) and mean squared error methods (MSE), decision feedback equalization,

adaptive equalization with LMS and recursive least square (RLS) algorithms, and

nonlinear equalization.

Previously [2], some work has been done on dxerential detection with adaptive

decision based equalizer on selective fading channel models. In this model, DQPSK,

as discussed in the last chapter, is used. The same transrnitter and channe1 mode1

Figure 3.1: Linear equalizer (LE) placed before differential detec tor

in Section 2.2 are used. The raised cosine filter has a roll-off factor of 1. Also, it is

assumed that the chamel characteristic is time-limited.

When differentid detection is used, nonlinear processing is done as shonm in Fig-

ure 2.2. Thus, any equalizer placed after this nonlinear processor has to deal with

nonlinear ISI. In order to incorporate a linear equalizer with differential detection, it

has to be placed before the nonlinear processor as in Figure 3.1. This way, the equal-

izer deais with linear (assuming that the channel is linear time-invariant ) rather than

nonlinear ISI. For channels with spectral n d s , such as fading channels, this equalizer

fails. This is due to the noise enhancement produced when the equalizer tries to invert

the channel effects. To combat this noise enhancement, decision feedback is required.

Therefore, another scheme with decision feedback is proposed. After the slicer, the

recovered data is re-encoded and fedback to a feedback equalizer 171. h this case

though, the fedback data does not have the unknown phase which the received data

has when entering the forward equalizer. Therefore, this decision feedback can only

be added to those equdzers which cari track phase variations and compensate for the

unknown phase. However, if this is so, there is no need to use differential detection;

since the whole purpose of noncoherent detection is to avoid phase tracking.

Thus, another proposal is to put the equalizer after the nonlinear detector [2].

Unlike the method above, here the equalizer deals with nonlinear ISI rather than

linear ones:

where 3, ta are memory of the channel and system

term and taking the last two terms as noise, the

be shown that the first term contains the desired

Using this result, an equalizer is setup to calculate

time delay. Ignoring the second

first term is expanded. It c m

response and nonlinear ISI [2].

the nonlinear ISI, resulting into

the structure in Figure 3.2. Same as linear equalizer, a coefficient adaptation, such

as LMS or RLS can be added to improve the performance. One advantage of this

nonlinear equalizer is that i t can equalize channels wi th spectral nulls, though this is

traded off with more complexity.

Then, the above linear and nonlinear methods are combined [2]. First, a linear

equalizer is placed before the differential detector, which is then followed by a non-

linear equalizer. Here, the predetection equalizer deals with the precursor ISI, while

the postdetection one deals with the postcursor ISI. This separation reduces the corn-

plexity of the nonlinear equalizer, which deds with both precürsor and postnirsor

ISI at the same time. Only half of the amount of nonlinear ISI is dealt with in this

system.

In a fading channel, its response is usually UiSU10wn. Thus, matched filtering can-

not be done with T-sampling. Also, an excess bandwidth pulse is used in practice for

transmission. To avoid aliasing, T-sampling again is insuscient. IR order to improve

ri- * 1 I

Cornpute ISI s Figure 3.2: Nonlinear equalizer mode1 for DPSK

-

the performance, FSE is proposed. Instead of sampling every T (a symbol duration)

seconds, the samples are taken every seconds where n is an integer. Simulations

are done in (21 using T-spaced equalizer on lineu phase and nonlinear phase channel.

It is found that for nonlinear phase channels, the input folded frequency spectnun

is varying too rapidly for the T-spaced equalizer to tradc. Rather, with FSE, better

performance results. FSE can be implemented into the stnictures discussed earlier

by replacing the original sampler at the receiver by a $ sampler.

Through simulations [2], it is found that the linear equalizer performs better than

the nonlinear equalizer when little ISI is present. However, if spectral n d s are present,

the linear equalizer fails. Indeed, decision feedback is necessary for the equalization

of such channels. Thus, the nonlinear structure is able to equalize the null channel in

the simulation. As for the combination of linear-nonlinear stnictures, its performance

is almost the same as the nonlinear one, but with less complexity. However, these

Equalizer Y i

Slicer

A a i

'

equalizers' performances are still worse than that of the coherent DFE.

Another fact that needs to be considered is that ail of the techniques mentioned

above require either the input signal or the channel impulse response be known.

However, this is generdy riot available. Consequently, blind equalization is proposed.

Chapter 4

Blind Equalizat ion

Other names for this technique are blind deconuoktion and self-recouering equaliza-

tion. It is cailed blind or self-recovering as no training sequence is sent to assist the

adaptation of the equalizer to the channel. Even with little or no knowledge about

the input sequence and the channel, blind equalization can estimate the transmit ted

signal from the received signal. Thus, in situations where the channel characteristics

are time varying, with blind equalization, no extra delay is induced by the training

sequence for periodic update of equalizer's coefficients. One of the simplest blind

equalization algorithm is the decision directed algorithm. As implied by its name,

this algorithm adjusts the equalizer's coefficients by minimizing the error between the

output and input of the slicer. Due to its simplicity, it is unable to equalize channels

which sufTer severe ISI.

Assume that the equivalent baseband channel has sampled impulse response, fi,

identical and independently distributed (iid) input data, ai, sampled received data, ri,

and comipted by white Gaussian noise ni. The basic procedure for blind equalization

is to estimate fi from ri. Then deconvolve fi with ri to obtain âi. The following are

some of the more sophisticated methods which give better performances than the

decision directed method.

4.1 Probabilistic Approach

The f i s t class uses probabilistic methods based on maximum-likelihood criterion (ML)

or maximum a-posteriori estimation principles (MAP) . By M L criterion, the trans-

mitted data and the channel characteristics are estimated based on the maximization

of p(r If, a), the joint probability density function of received vector (r) conditioned on

channel impulse response (f) and input vector (a). Since this function is Gaussian in a

Gaussian noise channel, the rnaximization of the function is equal to minimization of

the exponent. Therefore, the metric for minimization is simplified to the following [3],

where N is the length of the data block, L is the channel response vector length, and

A is the data matrix,

al O 0 ... O

a* ai O O

O

a ~ a ~ - 1 a ~ - 2 --• a N - L 4

Since both f and a are unknown, it is diffidt to find the solution using this metric.

There are two approaches in finding the vectors f and a through the minimization

of the metric. One is by averaging the metric over d possible data sequences, and

thus, finding p(r1f). The f which maximizes this function is the solution. It is found

to be (31,

Once this optimal f is known, the most likely transmitted sequence â can be found

through Viterbi algonthm (VA) using the metric defined above.

The second method is to estimate the data and the channel impulse response

simultaneously. This can be done by calculating the estimation of the channel impulse

response for every data sequence. Then the sequence which gives the minimum metric

is selected. Generalized Viterbi algorithm (GVA) devised by Seshadri (1991) cm be

used in h d i n g the most probable sequence of data (31. If conventional VA is used,

computational complexity grows exponentially with the length of the data sequence.

In GVA, through the fkst L (where L is the length of data sequence) stages of the

trellis, the search is the same as the original VA. All the data sequences and their

corresponding channel impulse response estimates are stored. After that , K, instead

of one, surviving sequences and their channel estimates per state are retained. To

reduce the complexity, at each state, the channel estimates are updated recursively

by LMS algorithm. This method has pretty good performance at moderate SNR with

K = 4 [3], though its complexity is even higher than conventional VA.

Another approach is called the Quantized-channel algorithm [3,4]. In this method,

the channel impulse response assumes a

for this response is found by VA. And

certain value. Then the optimal data sequence

the initial channel estimate is updated using

this detected sequence. The algorithm repeats until the most likely data sequence is

found, or in other words, until the algorithm converges.

This class of blind equalization has the disadvantage of high computational com-

plexity due to the use of VA. However, they can be usefid for constellations that are

approximately Gaussian distributed. Also, this algorithm is optimal as it uses VA.

4.2 Steepest Descent Approach

The second class, based on the steepest descent, is called the Stochastic Gradient al-

gorithm. In this method, the equalizer's coefficients are initially set to certain values.

Thus, the output of the equalizer is the convolution of the received signal, channel

impulse response and the equalizer impulse response. This output includes three

elements-the desired response, the noise component and residual ISI (or sometimes

called convolutional noise). Then least MSE criterion is employed to estimate the de-

sired signal. The desired signal is a nonlinear function of the equalizer's output. This

noniinear function can be with memory or memoryless. The error is then caiculated

and fedback to the adaptive LMS equalizer as shown in Figure 4.1. There are many

different algorithms in this class. The difference between them lies in the nonlinear

function used. Some of the common algorithms are shown in Table 4.1 [3, 41. Note

that contrary to non-blind methods where the error is defined as the difference be-

tween the detected data and actual data, this equalizer calculates the error between

received data and output of the nonlinear function.

The main concern for this class is its convergence. Convergence is reached when

the average gradient of the cost function equals to zero. Typically, slow convergence

Figure 4.1: General adaptive structure for S teepes t Descent Approach

is expected when LMS is used, though it is easier to implement. To be able to

converge, the algorithms must satisfy the Bussgang property [3,4]. By this property,

the autoconelation of the equalizer's output ai equals to the cross-correlation between

the equalizer's output and the output of the nonlinear function g( i i i ) :

Thus, this class is sometimes known as the Bwsgang algorithm. As the nonlinear

functions in Table 4.1 are generdy multimodal, the LMS method may converge

to local equilibrium points rather than the desired point where the MSE is truly

minimized.

4.3 Higher Order Statistic Approach

The third class uses second and higher-order statistics of the received signal to equalize

the channel. It is known as the polyspectra approach. Recall that for Gaussian

distribution, only its &st and second order statistics are meanin@. As a result,

Godard (CMA p = 2)

Sato

GSSA

Benveniste-Goursat

S top-and-go

Nonlinear fvnction g(ai)

Table 4.1: Table of nonlinear functions for Steepest Descent Method

for signals that are comipted by Gaussian noise (provided that signals themseives

ore not Gaussian distributed), only their second order statistics are afFected. The

noise free higher order statistics can then be used to recover the transmitted signals.

Furthemore, this approach can be applied to channels with non-minimum phase

response whose true phase characteristics are not available in second order statistics.

The higher order statistics involved are nth order cumulants. Given a zero mean

sequence y@), its second and third order cumulants are dehed as follows [4],

And the polyspectra of the sequence is then the m-dimensional Fourier transform of

the (rn + l ) th order cumulant. One of the very cornmon polyspectra is the power

spectrum. It is the one-dimensional Fourier transform (1-d FT) of the second order

cumulaat, commonly known as autocorrelation. Some of the others which are used

in this class are bispectrum and trispectrurn, the two-dimensional FT of the third

order cumulant and three-dimensional FT of the fourth order cumulaut respectively.

Anotker function which is of interest is the inverse m-dimensional FT of the logarithm

of its mth-spectmm. Thus, for m = 2, 3, this function is known as biceptrum and

tricepstrum respectively. Proposed so far are three techniques, the pûrametric ap-

proach, the nonlinear least-squares optimization approach and the polycepstra-based

approach. The second approach is an adaptive one which rninimizes the cost function

derived from higher order statistics. And the third one calculates directly the poly-

cepstra of the received sequence, and uses its result to approximate the coefficients

of the equalizer.

An algorithm using the third approach mentioned above called Tn'cepstmm Equal-

ization Algorithm (TEA) has been proposed by Hatzinakos and Nikias (1991). In this

method, the tricepstrum and trispectrum is calculated. In (41, it is found that tricep-

strum is related to the minimum and maximum phase characteristics of the channel.

Therefore, by calculating the tricepstrum at different values of q, 1, r3, a set of equa-

tions is set up. Solving these equations, the characteristics of the channel can be

found. From the derived characteristics, the coefficients of the equalizer are then

calculated. Thus, any type of equalizer can be useci since channel characteristics and

equalizer's coefficients are found separately. This algorithm can also be implemented

adap tively by adj us ting the maximum and minimum phase charac teris t ics found from

the above procedure using LMS.

The problem with this approach is that large amount of data and high complexity

are involved due to the computation of higher order statistics, especially for the TEA

algorithm. However, TEA does have an advantage over the nonlinear least squared

approach, since the TEA adaptive algorithm guarantees convergence to the absolute

minimum of the cost function.

4.4 Sequence Estimation Approach

Most of the approaches discussed in the previous sections, except for the steepest

descent approach, estimate the channel response first. Then, using this estimate,

the channel effects are inverted to find the transmitted sequence. These methods

are thus, more applicable to situations such as imaging, where channel identification

is necessary [8]. However, for equalization in communication channels, recovering

the transmit ted sequence is more important t han channel identification. Also, some

channels are not identifiable due to the presence of spectral nulls. Therefore, a class

of algorithm which directly estimates the transmitted sequence arises [9, 101.

in the first algorithm [9], the transmitted sequence is estimated through the ex-

amination of the received signal. From this, the second order statistics of the source

are estirnated, and VA is employed to find the data sequence. Before the algorithm is

discussed, several assumptions are made. First, the channel response is assumed to

Se finite with length d symbol i n t e d s . And for N receivers, the channel response

forrns a N x d matrix with f d column r a d ; in other words, it has a finite impulse

response. The input is zero mean and iid. The correlations between symbols from

different receivers and different time instances are zero. The noise is assumed to have

zero mean, and the noise between different receivers are independent. Lastly, the

input and the noise are also independent.

In order to estimate the correlation of input symbols, the diamel ha . to be or-

thogonalized. This is done by Mahalanobis Orthogonalization. For received signal

where T is the transform matrix, which when multiplied by the channel response

matrix gives an orthogonal matrix. Consequently, when no noise is present, the

correlation of the orthogonalized preserves the correlation of the source. Thus, the

correlation of the source can be recovered using ri. Taking noise into account, the To

which gives the optimal input correlation estirnate is,

where d2 is the estimated variance of the noise; and ô2, A:, II,' are found from the

singular value decomposition (SVD) of the correlation of the received signals riri-1.

The whole algorithm is the following [9]. The SVD of &(O)' is computed to fmd

the impulse response dimension estimates d and noise variance estimates B2. With

dl â2, and ri, the optimal transform matrix T', is found. Then the correlation of the

transformed data yi is computed. Using the metric,

where &(i) = ~ L ~ a ~ - k - ~ ; VA is applied to find the transmitted sequence (Fig-

ure 4.2). By simulating this algorithm [9], good performance is obtained. Since the

estimation of the source correlation is simpler than channel identification, this al-

gorithm is less complex. Also, this algorithm applies to both single and multiple

receivers structures. For fading channels, spatial diversity can improve performances

of the receivers. Thus, this multiple receivers structure is very convenient .

Figure 4.2: Block diagram for blind sequence estimation

The second method uses MAP to estimate the transmitted sequence [IO]. Assum-

ing the input alphabet is finite, a channel response is calculated for every possible

sequence. Using MAP of the input sequence as the cost function, the most likely

sequences are selected. To reduce complexity, at each stage, only the most likely

sequences are retained; though, this only gives an approximation.

The mode1 used in this method is a time va,rying channel with additive noise. In

matrix form,

Ri = AiVdF + Ni (4.10)

where R, N, d are the received vector (d x 1), noise vector (d x l), and length of the

channel response; while,

%y MAP, p(a:lri) is computed for a.ll possible data sequences. Then only K sequences

with the largest ~ ( a : l r ~ ) (in other words, most probable) are retained. The computa-

tion repeats. Note that k counts the number of sequences, and takes on a value from

1 to M i where M is the alphabet size.

For this algorithm, convergence and complexity depend on the identifiabili ty of

the channel and the properties of the input. If input is from a finite alphabet, and is

persistently exciting of order 2d - 1, the channel and the data sequence is identifiable.

And the complexity is bounded by the first time instant (that is, Mi0 where to is the

system time delay) if the input is persistently exciting of order d where d is the length

of the channel impulse response [IO]. The performûnce of this algorithm shows fast

convergence, low BER and good tracking proper ties.

4.5 Application

BLind equalization can be applied to transmission monitoring, deblurring of astro-

nomical images, multipoint network communications, echo canceling in wireless tele-

phony, digital radio links over fading channels, and identification of the channel re-

sponse [4,11,12,13]. Among al1 types of blind equalization mentioned above, the ML

method is optimal though its computational complexity is very high due to the use

of VA. Therefore, this algorithm is suitable for channel where the span of ISI is short;

while other approaches, though suboptimal, can deal with Channel with a long span

of ISI. If tracking of carrier phase is required, then the steepest descent method can

provide this tracking along with equalization. Moreover, blind equalization is most

applicable to channels where the transmission of a training sequence is impossible.

By modification, new algorithms can be derived for a p a r t i d a r application. But

so far, blind equalization has only been applied to coherent detection. Thus, in the

following chapters, two blind equalizers, Godard and MCMA, are used with the LMS

algori thm and differential detection to equalize different channels.

Chapter 5

Linear Godard and MCMA

Systems

5.1 Linear Coherent Godard Algorithm

The Godard cost function selected for LMS is independent of carrier phase, and is

the dispersion of order p [14],

where R, is a positive real constant,

Then this cost function is used in the LMS algorithm to adjust the equalizer's coeffi-

cients ci by the following equation,

where p is the stepsize, and VCi is the gradient with respect to ci. One of the most

common application is with p = 2 and is used here.

When Godard is used with the LMS equalizer, the cost function D(2) replaces the

conventional error to be evaluated for the adjustment of tap coefficients [14]. In order

to expand ( 5.3), the gradient V, of the cost function magnitude squared with respect

to tap coefficients has to be evaluated. This gives the necessary error expression ei.

Thus, assuming coherent detection is used, the Godard cost function is,

where âi is the output of the equalizer, and

where ai is the source symbol [14]. Assume that QPSK source is used and it is

followed by a hardlimiter. Therefore, lail is always 1. As a result, the constant R2 in

the cost function is found to be 1. After calcdating the gradient, the error fed into

the equalizer is (see Appendix B.2),

With this expression, the coherent system CG using Godard with LMS is setup in

Figure 5.1. The coefficients c,- are adapted with the following equation,

where p is the step size, ri is the complex conjugate of the received signal in ( 2.2)

and the remaining expression is e; in ( 5.6). This Godard algorithm belongs to

the Stochastic Gradient class (see Section 4.2). Godard is blind in the sense that

QPSK Source a i raiseci Channet & hardimiter cosine g(t) ~ ( t )

1 Slicer

Figure 5.1: Coherent System with Godard for LMS equalizer (CG)

it does not require prior knowledge of the Channel or transmission of any training

sequence. The cost function used is a nonlinear, multimodal function. This implies

that it contains local and global minima. When the equalizer converges, there is a

possibility that it converges to a local rather than global minimum. There are many

papers written on the convergence issue of Godard [15, 5, 161. Simulation by Godard

(1980) shows that this algorithm converges with only an order of magnitude more

iterations than the equalization scheme with a training sequence. And the smailer the

step size, the longer the convergence period. In [14], it is found that the convergence

of the cost function depends on the initial tap values of the equalizer. It must be

setup such that the energy at the output of the equalizer must be sufticient for it to

converge to a global minimum. Therefore, the center tap c, m u t be initialized to a

value greater than the threshold below [14],

where a; is the source symbol and po is a sample of the channel response with the

largest magnitude. By exaaining the cost function, D(*) is dso found to be phase

blind as it takes the absolute squared of the information Ci. In other words, the phase

information is not used when the cost function is evaluated for ( 5.3). And carrier

recovery is independent of the convergence of the system. This blindness leaves the

equalized data with a phase error if phase rotation or frequency offset is introduced

during transmission through the channel. To eliminate these distortions, the carrier

must be tracked with a PLL. Thus, Godard has proposed a joint equalization and

carrier recovery structure in [14]. The carrier phase is tra&d by,

where p+ is the step size, ej = r ; - âi, zi = Ziexp-j" is the equalized output with

phase error correction, and âi is output of the slicer. Next section, an altemate

structure is proposed to compensate this phase blind problem.

5.2 Linear Differential Godard Algorithm

Ins tead of using PLL to correct the residue phase error, two new structures are pro-

posed here using differential detection. To combine Godard with different ial encoding,

two stmctures DG-1 and DG-2 are possible. In DG-1, the output of the equalizer ii is used for the evaluation of Godard cost function before it is fed into the differential

decoder as shown in Figure 5.2. The only clifference between this and Figure 5.1 is

QPSK Source ia i Diff. b i a . Chanel & hdlirnittr encoder cosint g(t) ~ ( t )

A a i

4- Diff g i LMS

S licer r- I

decoder n-1

[cilo

e

Figure 5.2: Godard with Differential detection for LMS equalizer ( D G 1 )

the addition of differential encoder and decoder. At the decoder,

where hi, ai are the input and output of the decoder. Since the differential decoder (a

nonlinear processor) is placed after the calculation of the Godard cost function, the

residue ISI is linear and the error ei is found to be the same as CG'S (see Appendix

B.2), with replacing âi in ( 5.6). That is, for LMS equalizer, the tap coefficients ci

are adjusted as,

where p is the step size and rt is the complex conjugate of the received signal in

( 2.13).

In DG-2, the signals are differentidy decoded before they are used to evaluate

the enor as shown in Figure 5.3. Hence, the decoded symbol ai is used to calculate

QPSK Source & hrudlimitcr

Figure 5.3: Differentid detection with Godard for LMS equdizer (DG-2)

A a i -

the error, and it contains nonlinear residue ISI. The gradient in ( 5.3) has to be re-

evaluated. It is found that the error in ( 5.6 ) , multiplied by the delayed version of A

the equalizer's output, bi - l , forms the new error (see Appendix B.2)'

This is consistent with the conventional linear differential equalizer which multiplies

the conventional error by the delayed equalizer output [l?]. Therefore, equation for

the adjustment of tap coefficients is modified into,

" i

SLiccr

The difference between ( 5.14) and ( 5.7) is the multiplication of bi-L . With these two structures, the phase blind problem of Godard is solved as dif-

ferential detection, which doesn't detect any information fiom the absolute phase, is

used. Therefore, even if 6 # O in ( 2.13), the systems c m still converge. However,

several dBs are traded off for the use of differential detection which introduces error

propagation and noise enhancement. In the simulation, the trade off is determined.

Diff. c n d a

b i

9

;yT-p

r

raised Chruincl cosine g(t) PO)

a i *

Godard

4 .

Diff LMS

decoda n-1

[cib

lowpass "

And the performances of DG-1 and DG-2 are compared to investigate whether the

positioning of the differential decoder is of importance. Since Godard is blind, no

training sequence is required to bring the equalizer into convergence. However at low

SNR, when most of the symbols are in enor, the error propagation and the blindness

of Godard slow down the rate of convergence, and decrease the performance of DG-1

and DG-2. As mentioned in the last section, convergence of the equalizer is related

to the initial tap values. This sensitivity of the cost function to the change in initial

tap values continues to be an issue when differential detection is used. There exists

a threshold for the initial value of the center tap above which the equalizer is able

to converge, same as the CG, as shown in the simulation. Next, the second blind

equalizer, MCMA, is discussed.

5.3 Linear Coherent MCMA

In this algorithm, the Godard cost function is split into real and imaginary parts [Ml.

With the equalizer's output splits into real R;,R and imaginary âcr, the cost function

becomes,

where

To implement into the LMS dgorithm, assuming coherent detection, the gradients

( 3 ) 2 Vci of IDiTRl and 1 DI: l2 are evaluated. Thus, the following error expressions are

found [18] ,

where ai is the output of the equalizer. With this error ei, the LMS equalizer adjusts

its coefficients with ( 5.11). MCMA exhibits similar properties as Godard with less

complexity. Therefore, to setup the coherent MCMA system (CM), the Godard block

in Figure 5.1 is replaced by MCMA block. Both Godard and MCMA are sensitive

to the variation of the initial tap values. Thus, convergence is an important issue,

and depends upon the initial set ting of the tap values. However, contrary to Godard,

MCMA cost function is not phase blind. This is achieved by splitting the Godard cost

function into real and imaginary parts to form the MCMA cost function in ( 5.15).

As a result, it is sensitive to both modulus and phase of the equalizer's output.

Even without the carrier tracking loop, phase recovery is done sirnultaneously with

equalization; though there will be a limitation as to how fast it can track [18]. This

certainly improves over Godard as no PLL is required for carrier recovery. In fact,

MCMA performs just as good as Godard with PLL as shown in [18]. To eliminate the

limitation in carrier tracking, differential MCMA systems are proposed and discussed

next.

5.4 Linear Differential MCMA

When setting MCMA up with differential detection, simila structures as Godard's

are derived. To observe the behavior of differential MCMA, the detector is placed

after and before the calculation of the error. Similar to previous sections, the gradient

of magnitude squared of MCMA cost function is evaluated to form the error ei for

LMS equalizer. The ei are similar to that of the differential Godard systems. Thus,

to setup D M 4 and DM-2, simply replace the Godard block in Figure 5.2 and 5.3

with MCMA. For DM-1, where differential decoder is placed after the calculation of

ei , the data used in calculating ei carries linear residue ISI. The tap coefficients are

adjusted using ( 5.11) with ei in ( 5.17) replacing ai with Bi (output of the equalizer).

For DM-& the symbols bi are decoded into âi before the calculation of ei. As a result,

the residue ISI is nonlinear, and the error ei in ( 5.17) muse be multiplied by a delayed

version of the equalizer's output &i-i.

Just as CM, the two differential systems are sensitive to the wiat ion of the

initial tap values. The center tap must be initialized above a threshold in order for

the equalizer to converge which is confirmed by simulation. Also, the convergence

rate of the equaiizers is slow due to error propagation at the differential decoder.

However, this propagation is only of one symbol duration, and thus, is not severe.

Since MCMA is able to track the carrier by itself, difFerentia1 detection is a redundant

operation. And by switching from coherent to differential detection, several dBs of

SNR are traded off as shown next. In the next section, the simulation results for all

of the above systems are discussed.

1 Sampling frequency 1 19200 bps 1 1 Nurnber of samples per symbol 1 8 ( Baud rate 1 2400 1 1 Raised cosine roll off factor 1 0.5 1 1 Number of taps of raised cosine filter 1 33 1

- - 1 Windowing of raised cosine filter 1 hamrniq- 1 1 Noise bandwidth 1 1800 Hz 1

1 Number of tops of lowpass fiter 1 33 1 Table 5.1: Parameters set ting for systems simulated wi th SP wTM

5.5 Simulation Results For Linear Systems

Before the simulation results are discussed, the parameters and models used in the

simulation are presented.

5 .S. 1 Channel Models and Parameters

AU simulations are done with spwTM. Table 5.1 shows the values of the parameters

used in the simulation1. Six different noise seeds are used during six consecutive

simulations which average out as one point on the SER versus SNR curve. Thus, when

averaged out, the cuve will have sufficient accuracy (191. Convergence is defined as

the instance when the MSE falls into its steady state value. The variable parameters

are initial center tap value, number of taps and step size. To test the blind system,

'As raised cosine filter is noncausal and has infinite number of coefficients, it needs to be trun-

cated. Thus, Hammingwindow is used to reduce the frequency distortions induced by the truncation.

it is first fed with known data (in training mode) to assure the feasibility of the

structure. Theu recorded simulations start over in blind mode and are fed with

estimated data. Due to the sensitivity of the systems to the above parameters, several

sets of parameters are simulated before the one with the best performance is chosen.

The performance of the systems is elaluated based on four criteria-symbol error

rate (SER) versus S NR, convergence rate, robustness and stability. Robustness is

defined as the ease witk which a system can adapt to miations of noise and channel

conditions. Stability is defined as the ability to stay in convergence regardless of the

wiations of channel conditions (see Appendix C).

For cornparison, the coherent (DFE) in Figure 2.1, f-rate fractionally spaced

coherent DFE (FS-DFE) and linear differential equalizer (DIFF) in Figure 2.3 are

built. Since these structures are not sensitive to the change in initial tap values, their

tap coefficients are set up with a fixed center tap value and al1 other taps as 0.0.

Aiso, they are left in training mode for the entire run, as their results are used for

cornparison only. However, for all the blind systems (CG, DG-1, DG-2, CM, DM-1,

DM-2), initial setting is not as easily defined. For the initialization of tap coefficients,

al l taps except the center one are initialized to 0.0, while the center one is set above

a threshold in order for the cost function to converge.

5.5.2 Results for Channel A

Channel A is a fixed linear channel without nulls and little ISI. Its impulse response is

ak6(t - kT) where T is the symbol duration, a0 = 0.304, c r ~ = 0.903, cr? = 0.304.

The values of the parameters used in simulation for ail the systems are detailed in

Table 5.2. To compare the performances of a l l the systems, Figure 5.4 shows all

Systems

DFE

no. of taps

(for, back)

DIFF

DGFB 1 ( 5 4

(5,3)

CGFB

0.011 varies

step

size

7

0.01

(7,l)

Table 5.2: System configurations for chônnel A ( for = forward equalizer, back =

backward equalizer)

initial center tap

(for, back)

0.005

CM

DM-1

D M-2

CMFB

the coherent systems, Figure 5.5 shows ail Godard systems, and Figure 5.6 shows

all MCMA systems. .U of them are able to equalize Channel A. As expected, the

coherent systems perform better than the differential ones. At SER = le-2, there is

about 3 dB difference between the two groups of systems due to error propagation

and noise enhancement in the differential systems. As the channel has little ISI, the

gain

control

(1.0,l.O)

0.005

error propagation is not severe, and thus, the convergence rate is not affected. While

exarnining Figure 5.4, it is found that indeed Godard and MCMA exhibit very similar

performances. CG and CM are only 2 dB worse than DFE (at SER = le-2). At

no

2.0

7

7

7

(771)

no

varies no

0.005

0.005

0.01

0.005

1.5

2.5

1 .O

varies

no

no

Yes

no

1 04[ I I I I I I I I 1 2 4 6 8 10 12 14 16 18 20

SNR in dB

Figure 5.4: SER vs SNR curve for coherent systems for Channel A

low SNR, the performances of these two equalizers approach that of the DFE. As

discussed in Section 5.2, the only difference between DG-l and DG-2, DM-1 and

DM-2 is the positioning of the differential decoder. From Figure 5.5 and 5.6, it is

concluded that the positioning has no effect on the performance. When the decoder

is placed before the cost function (for DG-2 and DM-2), gain control is needed to

prevent the error from shooting to very large d u e . This is due to the fact that the

noise in the data used in calculating the error is enhanced by the decoder (which

carries out an multiplication). Other than this, the systems are very similar in terms

of robustness. As shown in Table 5.2, the non-bhd systems (DFE, DIFF) are very

...............+................ . r . . . . . . . . . - I . . . . . , . . . . ? . . . ...... I . . . . . . .... 1 ...................... S.......... I ...S...... ,......-....

) .........., ........... . . . . . ........... ,

.............................. ......................................................................

1 o - ~ I I 1 1 I 1 1 1

2 4 6 8 10 12 14 16 18 20 SNR in dB

Figure 5.5: SER vs SNR c w e for Godard systems for Channel A

robust. None of their parameters need to be varied for them to converge. Contratily,

the blind systems require careful setting of the parameter (initial tap values) in order

to obtain the optimal point. However, since there is little ISI in the channel, this

optimal point is easily located, as it is reached when the initiai center tap value is

above a threshold. AU the other parameters in Table 5.2 are fwed2. Thus, the blind

systems are robust for Channel A. Moreover, with little ISI presents, they are able

*For system with decision feedback, the forward and backward equalizers' settings are shown. In

others, only the forward equalizer's setting is shown. When the parameter is said to be 'varies', it

means that it has to be .manipulated before the opt'mal point is found.

1 1 1 1 1 1 1 1 I 2 4 6 8 10 12 14 16 18 20

SNR in dB

Figure 5.6: SER vs SNR curve for MCMA systems for Channel A

to stay in convergence once they have converged, which imphes that the systems are

stable for channel A.

As one of the reason for using differentid detection is to elirninate any angle

rotation introduced during transmission, an unknown angle 6 is added to all of the

systems after the channel, and simulated using Channel A. The results are shown

in Table 6.4. For CG, it is not able to converge as it is phase blind and needs PLL

to correct the angle rotation. When differential detection is used, the rotation is

eliminated. Both DG-1 and DG-2 are able to converge when 19 # O, even though the

phase blind Godard is used. For MCMA systems, as they can track carrier phase,

they are able to converge without PLL for coherent or differential detection. However,

when CG, DG-1, DG-2, CM, CM-1 and CM-2 are simulated with a channel which

has nulls, none of the systems is able to equalize the channel. This is expected as

the nulls are enhanced when the linear systems try to invert the channel distortions.

In order to deal with spectral nulls, decision feedback must be added to the above

systems. Thus, three new systems are proposed next.

Chapter 6

Godard and MCMA with Decision

Feedback

For all systems discussed in the previous chapter, none of them is able to converge

when the channel has nulls. To use blind equalization for such channels, modification

must be made to the systems [23, 24, 25, 261.

6.1 Coherent Godard with Decision Feedback

In order to equalize channels with nulls and fading, a decision feedback path has to be

added to CG. Therefore using QPSK, Figure 6.1 is setup. For this proposed coherent

system CCFB, the estimated signals âi are fed into a backward LMS equalizer. This

structure is different from a conventional DFE in two ways. Beside using Godard cost

function D(*) in ( 5.4) instead of conventional error function, the feedback path is also

in a blind mode. In other words, no training sequence (no known correct data) is fed

into the backward equalizer to help the equalizers to converge. With the addition of

QPSK Source ai -. raised . Channel & hardlimi ter cosine g(t) ~ ( t )

Figure 6.1: Coherent Godard with feedback path for LMS equalizer ( CGFB)

the feedback path, the output of the forward equalizer is added to the output of the

backward one. The sum ai is,

where the upper cases represent vectors, Ci = [ ~ ( i ) , c l ( i ) , . . . , ~ ~ - ~ ( i ) ] ' is the fomaxd

equalizer tap coefficients vector, Ri = [ r ( i ) , r(i - i ) , . . . , r ( i - N + 1)It is the received

vector in ( 2.2), Di = [do@), dl (i), . . . , dnf-, ( i) jC is the backward equalizer tap coeffi-

cients vector, and Âi = [ â ( i ) , â ( i - l), . . . , â ( i - M + l)jt is the output vector of the

slicer. N and M are the number of taps for forward and backward equalizers respec-

tively. The s u m ai is then used to calculate error ei. To use LMS for this modified

system, not only the gradient with respect to tap coefficients of the forward equalizer

s has to be found, but also the gradient with respect to the backward ones di must

be found for,

where p is the step size and D(?) is the Godard cost function ( 5.4). After evaluating

the gradients (see Appendix B.2), ei are found to be the sarne for both fornard and

backward equalizers, in the form of ( 5.6). Here, ai is ( 6.1).

For this system, since no training sequence is transmitted due to the blind nature

of the algoritlim, the fedback data may contain error. These errors propagate to affect

future symbols. The number of symbols afFected by the error propagation depends

on the memory of the backward equalizer. Thus, to prevent severe propagation, the

number of taps of the backward equalizer is limited. Another parameter of interest is

the initial tap values. Since the first portion of the fedback data has a high probability

of error, the equalizers will not be able to converge if both equalizers start with the

same energy. To solve this problem, the fonvard equalizer is initialized with a larger

tap d u e (more energy) than the backward equalizer (less energy). This reduces

the effect of the enoneous feedback data, and d o w s both equalizers to adapt to the

charnel. Also, since two sets of tap values are involved, the variation of the tap values

becomes a two dimensional problem. With the use of Godard cost function, this

variation affects the convergence behavior of the equalizers. As mentioned earlier ,

the cost function contains both local and global minima. Together with the blind

feedback path, the two center tap values must be carefdy initialized in order for the

cost function to converge to the global minimum. Through simulation, it is found

QPSK Source a i Diff. b i - raised Channel & hardlimitcr encoder cosinc g(t) p(t)

Figure 6.2: Differential Godard with feedback path for LMS equalizer (DGFB)

that indeed the equalizers are very sensitive to the initialization. In the next section,

a differential decision feedback systern is proposed.

6.2 Differential Godard with Decision Feedback

Finally, the three features-differentid encoding, blind equalization and decision feed-

back are combined into one system (DGFB) in Figure 6.2. Bere, the earlier structure

DG-1 in Figure 5.2 is modified by adding the output of the feedback equalizer to the

output of the forward one,

where ri is the received signal in ( 2.13), zi is the fedbadc data, and the upper

cases are vectors. Ci = [ ~ ( i ) , cl (i), . . . , cNml (i)]' is the forward equalizer tap coef-

ficients vector, Ri = [r(i),r(i - 1), . . . , r ( i - N + 1)It is the received vector, Di =

[do(i), dl (i), . . . , d M - l (i)Jt is the backward equalizer tap coefficients vector, and Zi =

[ ~ ( i ) , z(i - l), . . . , r( i - A4 + 1)It is the feedback vector. N and M are the number of

taps of the fomard and backward equalizers. The sum bi is fed into the Godard cost

function same as CGFB for the evaluation of the error e;. The ei for both forward

and backward equalizers are found by taking the gradients of the magnitude squareci

of the cost function with respects to forward and backward taps. It is derived as the

previous sections, and the results are found to be the same for both equalizers. ei is

( 5.6) with bi in ( 6.3) replacing üi (see Appendix B.2). Hence, the taps are adjusted

a,

where ci, di are the forward and backward tap coefficients, p is the step size, ri is the

received signal, and 1 is the fedback data.

Since the data are differentially encoded at the transmitter end, and decoded

before the slicer, the recovered symbols âi are in QPSK form. For consistency, âi

must be re-encoded before feeding into the badcward equalizer whose output is added

to the forward one which is in DQPSK form. Therefore, the feedback data zi is

These recovered symbol âi does not contain any unkiown phase which has been

eliminated by the differential decoder. Therefore, ri and Y i in ( 6.3) do not have any

phase rotation. However, the output of the forward equalizer x i , which hasn't been

decoded yet, is rotated. When xi is added to Yi, the phase blind feature of the Godard

algorithm is destroyed. Indeed, through the following analysis, it is shown that the

output of the equalizers and thus, the error function, depend on the unknown angle

B. Expanding ( 6.3),

where bi is the symbol from DQPSK source, f ( t ) is the convolution of transmitter

filter, channel and receiver filter, rii is the noise at the output of the forward equalizer,

N, M are the number of taps of the fonvard and bacha rd equalizers, and J is the

memory of the channel. After differential decoder,

N-L J M-1

(e-je C C b;j-n-lf,-n~i +fi;-, + C -r-m-2c)

Clearly, the second and third terms in the above equation contain the unknown angle.

For angles in different quadrants, the equalizers exhibit different convergence prop-

erties. This is further supported by simulation. Though no longer phase blind, the

feedback path nonetheless operates in a blind mode. The recovered data, in error or

not, is fed back to the backward equalizer. Therefore, same as the case for CGFB, the

backward center taps need to be initialized with a s m d e r d u e than the forwaxd one.

Then the forward equalizer can dominate during the initial phase of the equalization,

and this reduces the effect of the erroneous feedback data.

For DGFB, error propagation is an important issue. First there is enor propa-

gation of one symbol duration in the differentiai decoder as shown in ( 2.15). Then,

when the recovered symbol is re-encoded in ( 6.5), error again propagates to the next

symbol. Lastly, there is propagation in the feedback path where the number of future

symbols affected is determined by the memory of the backward equalizer. Thus, due

to these three folds of error propagation, extra effort must be placed to keep the prop-

agation within control such that the initial burst of errors will not be too large for

the equalizer to converge. This, together with the properties of Godard cost function,

contribute to the sensitivity of the system to change in parameters such as number

of taps, step size, and initial tap values. First of all, the number of backwards taps

becomes very limited in DGFB. C o n t r q to conventional DFE with training mode

where large number of taps (for more accuracy) is allowed, only small number of taps

is feasible in this system. This is to avoid severe error propagation. The larger the

number of backward taps, the longer the error propagation, and thus, more wrong

decisions are used in the feedback equalizer. To pinpoint the location of the global

minimum, the step size also plays an important role. Through simulation, it is found

that the system is not only sensitive to the initial tap values, but dso to the step

size. O d y limited combinations of step size and initial tap values give the fastest

convergence with the least number of errors. Next, a modified MCMA system which

QPS K a i rn raised Channel cosine g(t) ~ ( t ) 1 Source (

1 Slicer

Figure 6.3: Coherent MCMA with feedback path for LMS equalizer (CMFB)

can equalize channel with nul1 is proposed.

6.3 Coherent MCMA with Decision Feedback

Same as Godard, in order to equalize channels with n d s , linear MCMA is not suf-

ficient. In fact, a feedback path must be added as shown in Figure 6.3. Assuming

coherent detection, the symbol ai used in cdculating the error ei is,

where ri is the received signal in ( 2.13), âi is the recovered data, ci, di are the forward

and backward tap coefficients, and the upper cases are vectors same as ( 6.1). After

calculating the gradient of the magnitude squared of the MCMA cost function with

ai, similar result as CGFB is found (except for replacing the Godard error function

in ( 5.6) with MCMA error in ( 5.17)). The fonvard and backward taps are adjusted

with,

where ei is in ( 5.17).

With the addition of a feedback path, the equalizers become very sensitive to

both initial tap values and step size. Same as CGFB, the feedback equalizer must

be initialized with a smailer energy than the forward one. Or else, the equalizers

will never be able to stabilize and converge. Thus, the center tap values are set

such that the forward one is greater than the backward one. This sensitivity to

the change in initial tap values increases when channel is fading or with nulls, both

involve severe channel distortions as encountered in simulations. While simulating

CMFB with fading and null channels, carefd manipulation of the step size, number

of taps and initial center tap values is required. All of these parameters contribute

to the convergence behavior of the equalizers. In fact, the step size becomes as

significant a factor as the initial tap values. This presents a four dimensional problem

for convergence. Nonetheless, since there is less error propagation in CMFB than

DGFB, it is able to converge faster than DGFB. Another limitation is the memory of

the backward equalizer (number of taps of feedback equalizer). Again this is due to

the error propagation. During the initial phase of equalîzation, the blind equalizers

attempt to adapt itself to the channel without the benefit of a training sequence.

Consequently, most of the fedback data (which is the output of the slicer) are in

error. If the memory of the feedback equalizer is large, the erroneous data will affect

a large block of data. This keeps the equalizers from reaching the stable convergent

point. Therefore, the number of backward taps is limited to a small number.

The use of MCMA eliminates the phase blind problem with Godard. In theory, this

implies that no PLL is required following the equalizer to recover from angle rotation

and frequency shift, again contrary to the Godard algorithm [18]. The only limitation

for CMFB is how large a normalized Doppler shift (or how fast the speed of movement )

of the fading channel it can equaüze. If the variations in the channel response are

too fast, its self tracking ability may fail. Regardless of how fast the self tracking

mechanism can track phase variations, this rnechanism nonetheless eliminates the

need for differential detection which is superfiuous if used. With coherent detection

and decision feedback, CMFB is already very sensitive to four parameters-step size,

forward and backward initial center tap values and number of taps, as it has to

ded with the error propagation in the feedback equalizer. If differential detection

is used, error propagation will be too severe due to the feedback of erroneous data,

the decoding and re-encoding of symbols. This together with the blind nature of the

algorithm degrades the system so much that differential detection is not advisable.

With all the proposed systems, three channel models are simulated. The results are

presented in the next section.

6.4 Results for Decision Feedback Systems

With decision feedback, there are infinite combinations of values for the forward and

backward center taps. For convenience sake, either the initial center tap of forward

equalizer is set to 1.0, while the backward one is set to less than 1.0; or the forward

equalizer's center tap is set to values greater than 1.0, while the backward one is

set to 1.0. .&O, the sensitivity of the decision feedback systems to the change in

parameters is enhanced, since the step size, number of taps and two initial center tap

values are all contributing to the convergence of the system. Hence, convergence of

the systems become a four dimensional problem. With a wrong combination of the

four parameters, the output of the slicer can be a whole block of one's or zero's, a

regular pattern of one's and zero's, or a random mix of one's and zero's with most

of the symbols in error. This happens when the erroceous fedback data propagate

and dominate in the sum of fonvard and backward equalizers' outputs throughout

the simulation nrn.

First the threc decision feedback systems are simulated with Channel A (Ta-

ble 5.2). Both CGFB's and CMFB's performances approach that of the DFE (Fig-

ure 5.4). In fact, above 10.5 dB, these two systems perform better than DFE whose

performance is our lower bound. This is due t o the fact that their settings axe care-

W y adjusted before the optimal point is found, while DFE is simulated with one

setting only. However, at low SNR (6 dB or lower), the blindness of the equalizers

dominates. The erroneous fedback symbols negate the improvement brought by deci-

sion feedback. And thus, their performances are about the same as the coherent blind

systems without decision feedback, which is slightly worse than DFE's (less than 1

dB). F indy , the addition of feedback path in DGFB improves the performance over

-- -

Table 6.1: System configurations for channel B

Systems

DFE

CGFB

DGFB

CMFB

other differential systems for SNR above 13 dB (Figure 5.5). For SNR lower than

this threshold, the noise enhancement and error propagation in DGFB override the

improvement of the feedback path. As Channel A has little ISI, the error propaga-

tion in CGFB, CMFB and DGFB is not severe, and thus, the convergence rate is not

affected. Similax to the linear systems, CGFB, DGFB and CMFB, though sensitive

to the change in system parameters, the converging point cas still be easily found by

varying the two initial center tap values. Therefore, they are quite robust for Channel

A. They are also stable as Channel A is a fixed channe1.

Then, an unlcnown angle is added to the system. Except for CGFB which is phase

blind, the other two systems are able to equalize the chamel. One important note

about DGFB is that although it is able to converge, its converging behavior depends

on the value of the angle. By varying the unknown angle, different tap initialkations

are required for the system to converge. This is due to the fact that DGFB is no

longer phase blind as shown in Section 6.2.

Channel B is a fixed channel with nulls and severe ISI (with channel coefficients

cro = 0.407, ai = 0.815, a2 = 0.407). For those systems without decision feedbadc,

no. of taps

( ' T , back)

(5~3)

( 7 J )

(511)

(773)

initial center tap

(for, back)

(1.0,l.O)

varies

varies

varies

step size

0.01

varies

varies

varies

1

gain control

no

no

no

no

............... .,. .............. .,. .m............ .F. .........S... .................................................................. .................................................. C . . . . . . . . . . . . . . .................................................................. .................................................................. .................................................................

................ ............ ............. ...... ............. L > . ..: .-.:.- * >................+................. ................................. . .................................,......................................................*....... . . . .

.....................................................................................-.................

IO+ I 1 I I I

O 5 1 O 15 20 25 30 SNR in dB

Figure 6.4: SER vs SNR cuve for systems for Channel B

they are not able to converge with any parameter settings. Therefore, only four

SER curves are obtained in Figure 6.4. The system with the best performance is

the conventional DFE. Then CMFB and CGFB axe worse than DFE by 1 and 2

dB respectively at SER = le-2. Though CMFB is slightly bet ter than CGFB at

high SNR, below 6 dB, both behave with the same SER. All three coherent systerns

perform similarly with the blind equalizers slightly worse in performance and slower

in convergence due to the feedback of enoneous data. Also, CGFB and CMFB are

quite robust. Though they are sensitive to the change in step size and the two initial

tap d u e s (Table 6.1), the optimal settings are easily found. By changing the step

1 Normalized Doppler frequency 1 1.0e-4 1 -- - - 1 Maximum path delay 1 16 samples (2 syrnbo;) 1

Table 6.2: Parameters for the fading filter

Delayed pat h relative power

Fading fdtsr lengt h

size at one set of initial tap values, the optimal region can be found. Then the initial

center tap values are varied to locate the optimal point. Once they converge, they are

able to stay in convergence as the channel is fixed. Thus, the systems are stable for

chamel B. However, this is not the case for DGFB. First, its performance is about 10

dB worse than the coherent systems as shown in Figure 6.4. This is due to the severe

noise enhancement and error propagation discussed in Section 6.2. Second, it takes a

long period of time to converge (Table C.l). Under the three folds of error propagation

and severe ISI distortions, DGFB becornes so sensitive to its parameters in Table 6.1

that for SNR lower than 25 dB, it is extremcly dificult to h d the specific optimal

setting. Therefore, DGFB is not robust for channel B. However, once it converges, it

is able to stay in convergence, and thus, is stable.

Channel C is a Rayleigh selective two-paths slowly fading channel with Doppler

frequency at 0.24 Hz. In frequency domain, this channel has deep nuils in i ts ampli-

tude response (see Appendix A) [6]. The selective Rayleigh fading channel block in

S P W * ~ is used with parameters set to values in Table 6.2. Thus, simulation shows

that decision feedback is required in order for the equalizer to counter the dis tortions.

However, with the time variations in channelys magnitude and phase responses due

to fading, CGFB is not able to converge even with a feedback path. Whereas CMFB

0.5

256

Table 6.3: System configurations for channel C

Systems

DFE

FS-DFE

DGFB

CMFB

is able to equalize the channel as it can track the carrier phase simultaneously with

equalization. Its performance is only about 2.5 dB worse tban DFE (the lower bound)

at the SER of le-2. Even though CMFB is sensitive to the wiations of step size and

initial center tap values, the seriousness of this sensitivity is much less than DGFB.

Its convergence rate is faster than DGFB as well. Moreover, it is very stable since it

can either deal with a channel fade when converging or recover quickly from it.

To use Godard algorithm for fading channel, differentid detection is required.

Therefore, DGFB is used to equalize the fading channel. When the channel fades out

(amplitude response decreases) and noise dominates the received signal, errors occur

and the equalizer is bounced out of convergence. The equalizer then needs to re-adapt

and re-converge. Due to three types of error propagation, once error occurs, due to

propagation, a big burst of errors follows. Whsn it tries to re-converge from the errors,

the converging period is so long that it has to be taken into consideration for the SER

c w e . This greatly degrades the convergence behavior and convergence rate of the

equalizer. Also, with the severe distortions of the channel and the limitations of the

system, the parameter setting becomes even more specific than CGFB and CMFB.

no. of taps

(for, back)

(593)

(3~3)

(571)

( 5 4

step size

0.01

mies

varies

varies

initial center tap

(/Or, back)

(1.0,l.O)

(1.0,l.O)

varies

varies

gain control

no

no

yes

no

... P... .

.......

....... ....... ....... ....... ....... .*.-... .......

........

...\.. ..

1 041 i I I l I I I 1 4 6 8 10 12 14 16 18 20

SNR in dB

Figure 6.5: SER vs SNR cuve for systems for Channel C

Gain control must be added to keep the enor from shooting up to astronomical

values during the initial phase of equalization. As a result, in addition to step size,

initial tap values and number of taps, one more parameter (gain) contnbutes to

the converging behavior. Especidy at low SNR, only a few combinations are able

to bring the equalizer into convergence. For the backward equalizer, only 1 tap is

dowed. It is impossible to bring the equalizer into convergence if more than 1 tap

is used. Allowing time for DGFB to reconverge, the SER curves in Figure 6.5 are

obtained. Indeed, DGFB's performance is worse than the coherent systems'. And its

performance can be much worse if re-converging period is not dowed. Nonetheless,

DGFB ca. equalize the fading channel, even though the system is unstable and not

as robust as the coherent systems. Although DFE's pedormance is the best in this

group, it can be improved by using fractiondy spaced equalizer (FS-DFE), as it is

well known that FSE is better in dealing with phase distortions. In the simulation,

T-rate FSE is used, and it is about 1.5 dB better than DFE at le-2. 2

Table 6.4 is a list of dl the systems, the channel models which they have simulated

with, and a summary of the resultsl.

L~ check mark means the system is able to equaLze the ehannel, rvhile a cross means it's not.

The systems (DFE, FS-DFE, DIFF, CG, CM) are systems (*) found from the literature and are

used to compare with the remaining new systems.

Systems 1 Channel A 1 Channel A & Angle 1 Channel B Channel C

DIFF* 1 J 1 d I

1

d

d

DFE*

FS-DFE*

CG*

D G 1

D G-2

4 J

CGFB

DGFB

CM*

D M- 1

Table 6.4: List of systerns simulated in three different channels

d d

J J

V'

CMFB

d d

d

d d J

x

d d

J 1 J

x

d 4

J

x

x

x

d

x

x

x

d

4 x

x

4

x

d x

x

Chapter 7

Conclusion

For channels that are bandlimited, angle rotation and frequency offsets arc intro-

duced. In mobile communications, when the receiver is in motion, a time varying

component is introduced through the Doppler shift. This phase variation, for CO-

herent systems, can be tracked by using PLL. However, if noncoherent detection is

used, a much simpler structure is resulted. This detection, though relieves the bur-

den of tracking the phase variations, introduces nonlinear ISI to the system. Several

algorithms have been proposed to deal with this nonlinearity. However, the cornplex-

ity and performance of these structures induce the seeking of yet another algorithm.

Thus, in this thesis, new structures which combine blind equalization with differential

detection and decision feedback are presented for the application of mobile and data

commirncations,

Since most practical channels are UISU1own and time Msying, to track the varia-

tions, adaptive equalization with training sequence is used. In many situations, the

transmission of such sequence is not desirable. Thus, a new solution is proposed.

hstead of transmit ting a training sequence and using PLL, two blind equalization al-

gorithms, Godard and MCMA, are used with differential detection. These algorithms

estimate the transmit ted signal through the received signal with virtudly no informa-

tion on what is transmitted and what the channe1 characteristics are. As mentioned

before, fading channels introduce unknown time-varying phases to the received signal.

Therefore, blind equalization with differential detection and decision feedback is the

proposed new algorithm to solve both problems with the goal of achieving the same

or even better performance as the coherent DFE. The systems are then simulated

with threc channels-without null, with nulls, and with fading.

Al1 of the systems are able to equaiize channel without n d . If angle rotation

is added, coherent Godard is not able to converge without PLL whereas differential

Godard is able to equalize the channel. For MCMA, the equalizer is able to eliminate

the angle rotation with coherent or differential detection. As Channel A has little

ISI, al1 blind systems converge easily when the initial center tap value is set above a

threshold. Also, they are stable since the channel is fixed.

For channel with nulls and with fading, decision feedback is required due to the

presence of nulls in frequency domain. Thus, CGFB, CMFB, and DGFB are able

to equalize Channel B. Good performances are achieved wi th the coherent systems.

Contrarily, due to error propagation, DGFB performs much worse. However, even

with decision feedback, coherent Godard is not able to equalize fading channel since

it is phase blind. The combination of blind equalization, differential detection and

decision feedback can equalize Channels B and Cl though it exhibits severe error

propagation and noise enhancement. This greatly limits the number of taps, value of

step size, and initial values of center tap of forward and badcward equalizers. In fact,

only specific combinations of these parameters are able to bring it into convergence.

Also, the convergence rate is slow due to error propagation. As for the time varying

fading channel, DGFB becomes unstable such that a fade can cause the system to

corne out of its convergent state. It then takes a long time before it re-converges.

MCMA, on the other hand, is able to track the carrier phase by itself. Thus, only

coherent MCMA with feedbadc is used as differential detection greatly degrades the

system. Though careful settings of the parameters are required, the set tings are no t

as specific as DGFB and the convergence rate is not as slow. CMFB is stable for

both fixed and time varying channels.

On the whole, the coherent systems perform better than the differential ones.

Al1 of the blind equalizers equaiize the unknown channels without training sequence.

With differential Godard, no PLL is required with a trade off in performance. With

coherent MCMA, there is only a s m d trade off in performance for the benefit of self

carrier tracking. To improve the pedomance, stability, and reduce the sensitivity

of the system, an optimal detection method has to be derived. In this paper, a

simple slicer and least MSE criterion are used. However, an optimal criterion can be

derived by h d i n g the probability of error and minimizing it. Another alternative is to

use a different detection method [27,28]. The idea is to reduce the error propagation

which is the main limitation of the DGFB system. Also, possible improvement can be

achieved by using FSE instead of the curent symbol spaced one. Some preliminary

work was done in combining FSE with the above systems. However, due to the

limitation of the number of backward taps, the result was not satisfactory. More

involved analysis and simulation have to be done to figure out the best method for

combining these two features.

Appendix A

Multipat h fading channel

In channels such as microwave radio channels and mobile communication channels,

an important phenomenon must be acknowledged-multipath fading [6]. This is the

frequency ~ s y i n g attenuation of a signal along different signal paths during trans-

mission. Since this luge attenuation often occurs within a narrow bandwidth, it is

also known as selective fading. As indicated by its name, several signal paths are

induced for a single transmission, shown by Figure A. 1, due to reflection of the radio

waves from objects such as buildings and ground, and to variations of the index of

refiaction from transmitter to receiver. Thus, the resulting effect is similar to the

mode distortion in optical fibres.

A.1 Radio channel

For radio chasne1 with point to point transmission ( f ied ends), the twwpaths model

can adequately describe its characteristics, as shown in Figure A.l with the direct

path and any one of the other paths present. In this model, the eEect of multipath

Figure A.1: Multiple signal paths due to reflections in fading channels

fading is the superposition of two path responses. These two paths have attenuations

of Al and A 2 , and propagation distances of dl and d2 . Since the speed of propagation

for radio wave is the speed of light, c = 3.0 x 108m/s, the propagation delays are

TI = d l / c and rz = dz/c. Therefore, the complex baseband equivalent channel transfer

function is defined as,

where k = y is the propagation constant, Ar = rl - r 2 is the delay spread, and

Ad = d l - d2 is the path difference. The first term on the right side of the equation

is the response of the path 1, while the term in parentheses shows the fiequency

dependence introduced b y the constructive and destructive interferences of two signals

arriving at the receiver. Two cases result in this model depending on the fiequencies

of interest. For IwArl < R, the second term in the parentheses vanishes. Thus, the

remaining response is that of a single path. This is called the narrowband model.

The second case is the broadband model for IwArl 2 R, where there is a strong

frequency dependence component in the transfer function. Thus, the constructive

and destructive interferences of two signds will create either a gain or a notch in the

channel response.

Another effect of the fading is a time-varying component due to some atmospheric

phenomena. This variation, though slow for fixed end to end transmission, is signifi-

cant for transmission through large distances (30 km or

of a fade is as important as its depth. And the larger

greater). Thus, the duration

the distance, the deeper the

fade.

A.2 Mobile Channel

When one or both of the transmit ting and receiving devices are in motion in a radio

channel, the two-paths model used above is not sufficient in describing the charac-

t eristics of the channel. This is due to the fact that there are rapid time wiations

in channel characteristics even for short distance transmission. Also, there is the

shadowing effect which is the variation of received signal power due to the obstacles

that are present in the propagation path. In addition, these obstacles cause multi-

ple reflections. As a result, the M-paths model has to be employed, resulting in the

following complex baseband channel output,

where Ai, ri and di are attenuation, propagation delay, and propagation distance

for path i respectively; u( t ) is the transmitted pulse entering the channel. Due to

shadowing effect, the power distribution (in dB) c m be approximated by a Gaussian

distribution. As mentioned in the last section, multipath fading results in large

attenuation due to destructive interference of the paths and gain due to constructive

interference; though the fluctuation is much slower when both the transmitting and

receiving ends are fixed.

To examine the transfer function for the mobile channel, the time varying propa-

gation distance for path i is approximated as,

where u is the initial distance from between the transmitter and the receiver, v is the

velocity of the moving object, and Bi is the angle of incidence of the propagation patk

i relative to the velocity vector. The output of the channel thus becomes,

From this equation, it is obvious that the phase and delay at the output of this

channe1 are time varying. And the frequency offset of the carrier signal due to motion,

proportional to velocity of the moving object, is known as the Doppler shi' The shift

for the ith path is

wdi = kv COS Bi. (A.5)

When two or more such paths, each with a different Doppler shift, are superimposed

onto each other, rapidly changing phase and amplitude pulses are received. For the

narrowband model, the delay spread between different paths is s m d ; thus, the delays

for all paths are assumed to be Ti = r. The overall received s i p a l is thus,

where 4i is the initial phase. Due to the rapidly changing amplitude and phase, the

function r ( t ) can be modeled as a complex random process R(t). As the number of

paths increases, by Central Limit Theorem, the random process R(t) with R(t,) =

~ e j ' can be approximated by a zero mean Gaussian distribution. And the envelope

of this complex Gaussian random variable is Rayleigh distributed,

while the phase is uniformly distributed between [O, 274

Since any multiple reflections off a building is a superposition of several reflections

with a s m d delay spread. Thus, each of these multiple reflections is a narrowband

model. The broadband model can then be modeled as a superposition of these nar-

rowband models, whose envelope is Rayleigh distributed with delay ri. Throughout

the development of the Rayleigh model, it is assumed that there is no dominant path,

meaning the attenuation of all paths is about the same order of magnitude. If there is

a path with a strong attenuation relative to other paths (a spectular component), the

above model has to be modified. The strong component in the channel response r ( t )

with attenuation A. is modified to a non-zero mean Gaussian, while the remaining

components in r ( t ) are zero mean Gaussian distributed. The envelope of this new

model is thus, Rician distributed with the following normalized envelope distribution,

where p = p = $ and Io (pp) is the modified Bessel function of zero order1.

Appendix B

Adaptive Decision Based

Equalizat ion

In practical channels, channel response is often unknown or partially known, and time

~ s y i n g . This is why adaptive eqiialization is used. This is a class of algorithms which

adjusts the coefficients of the equalizer as time progresses so that the noise and ISI

at the equalizer's output are minimized.

B. l Derivation of LMS for conventional systems

Assuming coherent detection with decision feedback, the adaptive LMS algorithm is

derived. The s u m of the output of forward and b a c h a r d equalizers at ith iterations

(in other words, at iT where T is the symbol period) is ai. In vector form, Ai =

C,%, + Df Ai-, where Ci = [ ~ ( i ) , q( i ) , . . . , c ~ - ~ (i)It is the forward equalizer tap

coeEcients vector, Ri = [r(i), r(i-1), . . . , r(i-N+1)It is the receivedvector in ( 2.2),

Di = [Q(i ) , d+), . . . , dM-i(i)]t is the backward equalizer tap coefficients vector, and

Ai = [a ( i ) , n(i - 1), . . . , a(i - M + 1)It is the fedback vector which is equal to the

source symbols in training mode. The error function e; is d e h e d to be the difference

between the slicer's output âi (in training, âi = a;) and input ai. This error function

is used for coeficients adaptation. The coefficients are updated according to the

following equation,

where is the coefficient at i + lth iterations, p is the step size, Vci is the gradient

with respect to forward equalizer's coefficients d e h e d as,

Since VciÛi = 0, thus, the remaining expression in vector form is (61,

Note that VciDf Ai-l = O. Expanding the first term,

Combining the second term,

where Ri is the received vector. Similady, the above operations are repeated by taking

the gradient with respect to di to derive the LMS adaptation for backward equalizer.

Same result as forward equalizer's is found. The equations for coefficients adaptation

are,

where p is the step size, ri is the received signal, and ai is the fedback data which is

the source symbol in training mode.

When differential detection is used before the calculation of error, the signal used

in the calculation is the decoded symbol,

where hi is the output of the equalizer assuming no decision feedback. Therefore,

( B.2) becomes,

*

with Vci Rf-l Ci-l = O , Bi = C;LR,, and Ci in the same vector form as the coherent

system. Expanding the first tenn [U],

Cornbining the second te=,

vcileiI2 = 2 ~ ; & ~ - ~ i ï ~ - 2 â ~ ~ : b , - - ~

Substituting ( B.lO) into ( B.l) gives,

where p is the step size and ri is the received signal.

(B. 10)

(B.11)

( B.10) is different from ( B.5)

with the multiplication of hi-i.

B.2 Derivation of LMS for Godard algorithm

For Godard, the cost function is ( 5.4). Taking the gradient for ( 5.3), assuming

coherent detection, and R2 = 1 with no decision feedback,

where â; is the output of the equalizer. After some manipulation, it is found that,

Thus, only VCi(âi12 needs to be found,

Putting ( B.14) back into ( B.12),

(B. 14)

Substituting ( B.15) into ( 5.3) gives ( 5.7).

If decision feedback is added, the input of the cost function becomes,

where Ci = [ ~ ( i ) , cl (i), . . . , cNN1 ( i ) l t is the fonvard equalizer tap coefficients vec-

tor, Ri = [r(i), r ( i - 1), . . . , r(i - N + l)It is the received vector in ( 2.2), Di =

[do (i), di (i), . . . , dnfei (i)lt is the backward equalizer tap coefficients vector, and Ai =

[â(i), â(i - l), . . . , â(i - M + 1)It is the output vector of the slicer. Substituting this

into ( B.12), again only gradient of Iüilz needs to be calculated. The result is same

as ( B.15) except that ai is the sum of fonvard and backward equalizer's output.

( B.15) dso applies to the noncoherent system (with or without decision feedback)

with differential decoder placed after the Godard function.

For differential Godard with decoder placed before the Godard function without

decision feedback, ( B.7) is fed into the cost function. Therefore, the gradient becomes,

Thus, ~ ~ ~ l i ~ k - ~ 1 ~ as derived in ( B.9) is substituted into ( B.17). The result is,

This equation is substituted back into ( 5.3) to give ( 5.14). Again the difference be-

tween the above equation and ( B.15 ) is the multiplication of Similu derivation

can be done for MCMA.

Appendix C

Definition of Performance criteria

The performance of the systems is evaluated based on fou. criteria-symbol error rate

(SER) versus SNR, convergence rate, robustness and stability. The SER versus SNR

are presented quantitatively in the gaphs. However, the remaining three criteria are

not as clearly defined.

To compare the convergence rate, the SNR is fked at 30 dB, and the initial center

tap is fixed at a value where a l l the systems are able to converge. Then the step size

(the main parameter which affects the convergence rate) is varied, and the number of

iterations the systcrn needs to converge (the point where no more error occurs from

then on) is noted. The fastest convergence point is noted for each system. Then this

fastest convergence point of ail systems are compared as shown in Table C.1.

Table C.l: Number of iterations required for systems to converge for Channel B

CGFB CMFB DGFB

Robustness is defmed as the ease with which a system can adapt to variations

of noise and channel conditions. As more parameters are varied to reach system's

convergence, the less robust the system is found to be. To clarify this criterion, the

following definition is used to judge the robustness ifrom most robust to least),

1. fix set of parameters

2. one parameter is varied (one dimensional variation)

3. two puameters are varied (two dimensional variation)

4. three parameters are varied (three dimensional variation)

5. more than t hree parameters (higher dimensional variation).

S tability is defined as the ability to stay in convergence regardless of the variations

of channel conditions. In this t hesis, if the system is bounced out of convergence and

need to re-converge as the channel varies, then it is unstable. If it is able to stay in

convergence regardless of channel variation, then it is stable.

Bibliography

[l] S. Haykin, Communication S y s t e m . New York: Wiley, 1994, 3rd edition.

[2] A. Masoomzadeh-Fard and S. Pasupathy, "Nonlinear Equalization of Multipath

Fading Channels with Noncoherent Demodulation", IEEE Journal o n Selected

Areas in Communications, vol. 14, no. 3, pp. 512-520, April 1996.

[3] J. G. Proakis, Digital communications. New York: McGraw-Hill, 1995, 3rd edi-

tion.

[4] D. Hatzinakos, "Blind Deconvolution: Channel Identification and Equalization" ,

Control and Dynamic System, vol. 68, pp. 279-331, 1995.

[5] S. Haykin, Adaptive Filter Theory. Englewood Cliffs, N J : Prentice Hall, 1991.

[6] E. A. Lee and D. G. Messerschmitt, Digital Communication. Bosten: Kluwer

Acadernic Publishers, 1988, 2nd edition.

[7] W. Huang, T. S. Rappaport and M. J. Feuerstein, 'Performance of Decision

Feedback Equalizers in Urban and Indoor Mobile Channelsn, IEEE Veh. Technol.

Conference, vol. 3, pp. 368-3'71, May 1992.

[8] H. H. Zeng and L. Tong, "Blind Channel estimation: cornparison studies and a

new algorithm", IEEE ICC, vol. 1, pp. 12-16, 1995.

[9] L. Tong, "Blind Scquence Es timationn, IEEE Tbansactions on communicatiorrp,

vol. 43, no. 12, pp. 2986-2994, Dec. 1995.

[IO] F. Gustafsson and B. Wahlberg, 'Blind Equalization by Direct Examination of

the input Sequences", IEEE Tkansactions on commsnications, vol. 43, no. 7, pp.

2213-2222, July 1995.

[ I l ] R. F. H. Fischer, W. H. Gerstacker and J. B. Huber, "Dynarnics Limited Precod-

ing, S haping, and Blind Equalization for Fast Digit al Transmission over Twisted

Pair Lines", IEEE JSAC, vol. 13, no. 9, pp. 1622-1633, Dec. 1995.

(121 L. B. White, S. Perreau and P. Duhamel, 'Reduced Computation Blind Equal-

ization for FIR channel input Markov Models" , IEEE ICC, vol. 1, pp. 993-997,

1995.

[13] K. H. Chang and C. N. Georghiades, "Iterative Joint Sequence and Channel

Estimation for Fast Time-Varying Int ersymbol Int erference C hannels" , IEEE

ICC, vol. 1, pp. 357-361, 1995.

[14] D. N. Godard, "Self-Recovering Equalization and Carrier Tracking in Two-

Dimensional Data Communication Systemsn , IEEE Tkarrpactions on Commu-

nications, vol. 28, no. 11, pp.1867-1875, Nov. 1980.

[15] S. Haykin (Editor), Blind Deconvolution. Englewood Cliffs, NJ: Prentice Hall,

1994.

[16] S. Bellini and F. Rocca, "Blind Deconvolution: Polyspectra or Bussgang Tech-

niques?", Digital Communications, North-Holland: Elsevier Science Publishers

B. V., pp. 251-262, 1986.

[17] P. Sehier and G. Kawas Kaleh, "Adaptive equalizer for differentially coherent

receivern, IEE Proceedings, vol. 137, pp. 9-12, Feb. 1990.

[18] Kil Nam Oh and Yong Ohk Chin, "Modified constant modulus algorithm: blind

equalization and carrier phase recovery algorithm", IEEE ICC, vol. 1, pp.498-

501, 1995.

[19] M. C. Jeruchim, P. Balaban and K. S. Shanmugan, Simulation of Communication

systems. New York: Plenum Press, 1992.

[20] U. W. Pooch and J. A. CVd, Dbcrete Event Simulation: A practical approach.

Florida: CRC Press, 1993, pp. 259-282.

[21] A. M. Law and W. D. Kelton, Simulation Modelling and Analysis. New York:

McGraw-Hill, 1982, pp. 287-307

[22] P. Bratley, B. Lw Fox and L. E. Schrage, A guide to Simulation. New York:

Springer-Verlag, 1987, 2nd edi tion, pp. 48-49.

[23] Y. Sato, "A Method of Self-Recovering Equalization for Multilevel Amplitude-

Modulation Systemsn , IEEE Transactions on Communicatiotls, vol. 23, pp. 679-

682, June 1975.

[24] J. R. Treichler and B. G. Agee, &A New Approach to Multipath Correction of

Constant Modulus Signalsn, IEEE Transactions on Acoustics, Speech, and Signal

Processing, vol. 31, no. 2, pp. 459-471, April 1983.

[25] G. Picchi and G. Prati, "Blind Equalization and Carrier Recovery Using a Stop-

and-Go Decision-Directed Algorithm", IEEE Tkansactions on Communications,

vol. 35, no. 9, pp. 877-887, Sept. 1987.

[26] K. N. Oh and Y. O. Chin, "Blind Decision Feedback Equalizer using Dual-

Variance Gaussian Clust ering Algori thm For Wireless Communications", IEEE

ICC, vol. 1, pp. 696-700, 1996.

(271 N. Hamamoto, "Differential Detection with IIR Filter for Improving DPSK De-

tection Performancen, IEEE fiansactions on Communicatiow, vol. 44, no. 8,

pp. 959-966, August 1996.

[28] F. Adachi and M. Sawahashi, "Decision Feedback Differential Detection of Dif-

ferential Encoded 16 APSK signais", IEEE Tbarrclactions on Communications,

vol. 44, no. 4, pp. 416-418, April 1996.

TEST TARGET (QA-3)

A P P L E D IMAGE, lnc - 1653 East Main Street - -. - - Rochester. NY 14609 USA -- -- - - Phone: 71 6/482-O3OO -- -- - - FW 71 6/28&5989