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Code timing estimation in direct-sequence code-division multiple-access communication systems

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Code timing estimation in direct-sequence code-division multiple-access communication systems
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xi, 127 leaves : ; 29 cm.

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Matrices ( jstor )
Propagation delay ( jstor )
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Receivers ( jstor )
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Thesis (Ph.D.)--University of Florida, 1999.
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Includes bibliographical references (leaves 121-126).
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Typescript.
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Vita.
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by Ronald F. Smith.

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CODE TIMING ESTIMATION IN DIRECT-SEQUENCE
CODE-DIVISION MULTIPLE-ACCESS
COMMUNICATION SYSTEMS














By

RONALD F. SMITH


A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA


1999















I have been blessed with a wonderful family, and this work is dedicated to them.

To Amy, for all of her support, encouragement and love. To Sarah, for so readily

sharing her silly little giggles and big hugs and kisses.














ACKNOWLEDGMENTS

I would like to express my gratitude to my adviser, Dr. Scott L. Miller, for his

guidance and feedback throughout the course of my research. I also thank the mem-

bers of my Ph.D. committee, Drs. Leon Couch II, Jose Principe, Robert Fox, and

Richard Newman, for their interest and willingness to be a part of this effort.

While I have dedicated this work to my wife, Amy, she also deserves special

recognition. She has worked extremely hard supporting our family while I pursued

my graduate studies, and without her efforts this work wouldn't have been possible.

I thank my parents, Ernest and Joyce Smith, and my in-laws, Thomas and Peggy

Meneskie, for their continuous encouragement and support throughout this effort.

I also acknowledge my friends and colleagues, John Miller, Brad Rainbolt and Ali

Almutairi, who have made my time at the University of Florida an interesting expe-

rience.













TABLE OF CONTENTS

page
ACKNOW LEDGMENTS ............................. iii

LIST OF FIGURES ................................ vi

KEY TO ABBREVIATIONS ............................ viii

A BSTR A CT . . . . x

CHAPTERS

1 INTRODUCTION ............................ 1

1.1 Introduction to Spread-Spectrum Techniques . 1
1.2 Mathematical Notation . . 6
1.3 Outline of the Dissertation . . 7

2 SYSTEM MODEL FOR THE AWGN CHANNEL . 9

2.1 General Asynchronous DS-CDMA System Model for the
AW GN Channel . . . 9
2.2 Simplified Asynchronous DS-CDMA System Model for the
AW GN Channel . . ... .. 12

3 REVIEW OF THE LITERATURE . ... .. 14

4 THE MMSE RECEIVER AND SINGLE-USER CODE
ACQUISITION . . . ... .. 21

4.1 The MMSE Receiver . . ... .. 21
4.2 Advantages of Synchronizing the MMSE Receiver ..... 25
4.3 Adaptive Filters and the Estimation Algorithm ....... 29
4.4 Sum m ary . . . 35

5 CHARACTERIZING THE PERFORMANCE OF THE TIMING
ESTIM ATOR ............................. 37

5.1 Transient Statistics of the LMS Filter Weights ....... 37
5.2 Chip Selection Error Probability .................. 45
5.3 Approximation for the Conditional Variance of the Timing
Estim ate . . . 50
5.4 Coarse Acquisition Performance . ... 54








5.5 Sum m ary . . . 59

6 SIMULATING FADING CHANNELS ................... 62

7 CODE ACQUISITION IN NON-STATIONARY ENVIRONMENTS 71

7.1 Frequency Synchronization Errors ................ 75
7.2 Performance in Flat-Fading Channels .............. 78
7.3 Sum m ary . . . 84

8 INCREASED WINDOW OF OBSERVATION AND TIMING
ACQUISITION ............................ 86

8.1 Updated System Model ..................... 86
8.2 Updated Timing Estimation Algorithm ............. 88
8.3 Sum m ary . . . 93

9 SUMMARY AND FUTURE WORK ................... 94

9.1 Sum m ary . . . 94
9.2 Contributions ................. ... .. .... 96
9.3 Future W ork ........................... 96

APPENDICES

A PROBABILITY OF BIT ERROR USING THE GAUSSIAN
APPROXIMATION ......................... 99

B DERIVATION OF TRANSIENT EQUATIONS FOR THE LMS
FILT ER . . . . 102

B.1 Transient Weight Autocorrelation Matrix Using the Gaussian
Approximation ........................ 102
B.2 Transient Weight Autocorrelation Matrix Using the Known
Statistics of the Receiver's Input Vector .......... 106

C REVIEW OF QUADRATIC FORMS OF GAUSSIAN RANDOM
VARIABLES ............................. 112
D TYPOGRAPHICAL NOTES ........................ 119

REFERENCES ....................... ........ ... 121

BIOGRAPHICAL SKETCH ............................ 127












LIST OF FIGURES


Figure page
1.1 Power spectral densities of the unspread data sequence and the direct-
sequence spread-spectrum signals . . .. 3
4.1 Code acquisition with an adaptive receiver . .... .. 22
4.2 Minimum mean-squared error as a function of the propagation delay
of user # 1 . . . . .28
4.3 Plot of the cost function f(p, 6) for a single-user system ........ .. 33
4.4 Plot of the cost function f(p, 6) for a fifteen-user system ........ .. 34
5.1 Performance of equation (5.2) for a single-user system, a) The analyti-
cal mean filter weight vector norm, IIE[w(m)] II (*), and the simulated
mean filter weight vector norm IIw(m) I (solid line) plotted as a func-
tion of the training length; b) Comparing the error between the two
mean weight vectors, I|E[w(m)] wA(m)I . ... .. 43
5.2 Performance of equation (5.2) for a three-user system, a) The analyti-
cal mean filter weight vector norm, IIE[w(m)]II (*), and the simulated
mean filter weight vector norm II|w(m)II (solid line) plotted as a func-
tion of the training length; b) Comparing the error between the two
mean weight vectors, ||E[w(m)] wA(m)J . 44
5.3 Performance of equations (5.4) and (5.7) for a two-user system. The
Frobenius norm of the LMS filter's transient weight autocovariance
matrix; equation (5.4) (*), equation (5.7) (x) and simulation
results (+ ) . . . . .45
5.4 Comparing incorrect chip-selection probability to the Union-bound for
a three-user system. Simulation results (+), the Chernoff bound
using equations (5.2) and (5.7) (solid line), and the Chernoff bound
using simulated weight vector statistics (*) . .... .. 51
5.5 Comparing the simulated conditional variance of the 61 estimate to the
analytically derived approximation in absence of near-far MAI. 53
5.6 Comparing the simulated conditional variance of the 61 estimate to the
analytically derived approximation in presence of near-far MAI. 54








5.7 Average training bits required for correct acquisition as a function of
the number of system users in AWGN channel . ... .. 57

5.8 Average training bits for correct acquisition as a function of the level
of multi-access interference in AWGN channel . ... .. 58

6.1 Flat-fading channel simulator . . ... .. 68

6.2 Theoretical and simulated probability density function for the enve-
lope of the fading channel simulator output for two different Rician
parameter values . . . .... ..69

6.3 Magnitude and phase of a slow flat-fading Rayleigh process generated
with a carrier frequency of 1.8 GHz, a vehicle speed of 3 MPH, and
a data rate of 9600 BPS . . ... ..69

7.1 Modified adaptive receiver with channel compensation ... .. 74

7.2 Acquisition performance of the adaptive receiver in a five-user environ-
ment in presence of frequency offset errors . ... .. 77

7.3 Performance of the acquisition algorithm in a slowly flat-fading Rayleigh
channel using a third-order linear predictor for the channel estima-
tion algorithm . . . .. ..80

7.4 Performance of the acquisition algorithm in a slowly flat-fading Rayleigh
channel using a tenth-order linear predictor for the channel estima-
tion algorithm . . . .. ..81
7.5 Comparing the average training bits for correct acquisition as a func-
tion of the level of multi-access interference in a flat-fading channel. 82

7.6 Performance of the timing acquisition algorithm using the hybrid and
modified MMSE receivers . . ... .. 83

8.1 Plot of the cost function f(p, 6) for a fifteen-user system when the
receiver observes two bit-intervals . ... .. 91

8.2 Comparing the performance of the timing estimator when the receiver
observes 2-bit intervals (*) versus observing 1-bit interval (solid line)
as a function of the MAI . . ... .. 92

A.1 The MMSE receiver for the Gaussian approximation ... .. 99













KEY TO ABBREVIATIONS


AWGN: additive white Gaussian noise

BER: bit-error-rate

BPS: bits-per-second

BPSK: binary phase shift keying

CRB: Cramer-Rao bound

CTAN: Comprehensive TEJX Archive Network

DS-CDMA: direct-sequence code-division multiple-access

DS-SS: direct-sequence spread-spectrum

FH-SS: frequency-hopped spread-spectrum

IEEE: Institute of Electrical and Electronics Engineers, Inc.

ISI: inter-symbol interference

LFSR: linear-feedback shift-register

LMS: least mean squared

LOS: line-of-sight

LPD: low probability of detection

LPI: low probability of intercept

LSML: large-sample maximum likelihood

MAI: multi-access interference

MASE: multiple-antenna sensors-based estimator

MMSE: minimum mean-squared error

MOE: mean-output-energy

MPH: miles-per-hour

MSE: mean-squared error








MUSIC: multiple signal classification

PCS: personal communication system

PSD: power spectral density

RASE: rapid acquisition by sequential estimation

RLS: recursive least-squares

RMSEE: root mean-squared estimation error

SNR: signal-to-noise ratio













Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy



CODE TIMING ESTIMATION IN DIRECT-SEQUENCE
CODE-DIVISION MULTIPLE-ACCESS
COMMUNICATION SYSTEMS



By

Ronald F. Smith

May 1999
Chairman: Dr. Scott L. Miller
Major Department: Electrical and Computer Engineering

This dissertation considers code timing estimation for asynchronous direct-sequence

code-division multiple-access communication systems operating over additive white

Gaussian noise and flat-fading channels. Unfortunately, this type of parameter esti-

mation is difficult in the presence of multiple system users, and traditional methods

are known to be sensitive to the near-far problem. The near-far problem is a con-

dition which occurs when the received signal amplitudes of the the multiple system

users are very dissimilar.

A well-known receiver, referred to in the literature as the minimum mean-squared

error (MMSE) receiver, is used to form an estimate of the code timing for a single-

user. The MMSE receiver has several desirable properties. The MMSE receiver

is the optimal receiver for a single-user operating in an additive white Gaussian

noise channel, but is sub-optimal in a multi-user scenario. However, the IMMSE








receiver has been observed to be resistant to the near-far problem. Traditionally,

the MMSE receiver is implemented using an adaptive filter and can, therefore, learn

and adapt to the ambient channel conditions. It will be shown, that with very little

side information, it is feasible to form an estimate of a single-user's timing offset by

processing the weights of the adaptive filter. The performance of the proposed timing

estimation algorithm has been studied under several different scenarios, and has been

observed to be resistant to the near-far problem. The complexity of the proposed

timing algorithm is similar to conventional single-user estimation methods, and is

shown to be of lower complexity than other proposed timing estimation algorithms.

Others have studied the performance of the MMSE receiver in a flat-fading chan-

nel. It has been found that the MMSE receiver must be modified such that the

received signal is compensated to offset the dynamic phase changes induced by the

channel. In certain environments, this compensation algorithm is shown to degrade

the performance of the timing estimation algorithm. A modification to the operation

of the receiver is proposed to offset some of this performance degradation.














CHAPTER 1
INTRODUCTION

1.1 Introduction to Spread-Spectrum Techniques

In 1941, actress Hedy Lamarr and composer George Antheil filed for a patent

on a "Secret Communication System" [1]. Their device operated on the idea of fre-

quently changing the carrier frequency of a radio signal. If the carrier frequency was

updated fast enough, and in an apparently random fashion, an adversary would have

difficulty maintaining contact with the transmitting radio. This means that an ad-

versary would only be able to receive a small portion of a message, whether it was

sent coded or uncoded. And, therefore, it is much more difficult for an adversary

to discern the intent or contents of the transmitted message. Of course, this means

that the intended receiver has to dynamically synchronize its carrier frequency with

that of the transmitter. Lamarr and Antheil used paper rolls, similar to those found

in player pianos, as their synchronization method. Hole placement on the roll would

determine the carrier frequency used, and an exact duplicate of the roll used by the

transmitter was given to the intended receiver prior the "secret" communication.

Lamarr and Antheil gave their invention, and the rights to the patent, to the United

States government. Since this development occurred during World War II, this tech-

nique initially received a lot of attention. However, the U.S. government deemed the

mechanics of the synchronization method too complex for wide-scale military use.

The technique developed by Lamarr and Antheil, which is now known as frequency-

hopped spread-spectrum (FH-SS), is explained in the following few sentences. Let B

represent the bandwidth required to transmit the message signal, using a traditional

radio with a fixed carrier frequency. If we now let the carrier frequency hop through

N discrete carrier frequencies, each separated by B Hz, then the bandwidth of the








transmitted signal becomes NB. Therefore, the bandwidth of the transmitted sig-

nal using frequency-hopping has been spread to a wider bandwidth than what is

required to transmit the message signal. However, until the advent of the integrated

circuit, the problem of implementing a synchronization method limited the use of

such systems.

The dual of FH-SS is known as direct-sequence spread-spectrum (DS-SS). In DS-

SS, a digital signal is modulated by a digital spreading-sequence prior to transmission.

Consider a binary sequence of 1 data where each data bit has duration Tb. If this

data sequence is used to amplitude or phase modulate a carrier signal, the power

spectral density (PSD) of the transmitted signal will have a null-to-null bandwidth of

2/Tb Hz. A spreading sequence is comprised of a pseudo-random sequence of 1 chips

where each chip has duration Tc, and it is assumed that Tb = NTc. By multiplying

the data sequence by the spreading sequence prior to transmission, the PSD of the

transmitted signal now has a null-to-null bandwidth of 2/T, which is N times greater

than the corresponding bandwidth of the original unspread data sequence. The total

transmitted power in the unspread and the DS-SS signals is assumed to be equal,

since multiplication by 1 doesn't affect the transmitted power. This means that the

PSD for the DS-SS signal will have a much lower amplitude than the PSD for the

unspread signal as shown in Figure 1.1.

In the previous paragraph, we referred to the spreading sequence as being pseudo-

random. Ideally the chips of the spreading sequence would be randomly generated,

with +1 occurring with the same probability as -1. However, just as the intended re-

ceiver in a FH-SS system requires knowledge (and synchronization) of the frequency

hopping algorithm, the intended receiver in a DS-SS system must have knowledge

(and synchronization) of the spreading sequence. Therefore, some deterministic

method must be used to generate a spreading sequence. A pseudo-random sequence

can be generated with a linear-feedback shift-register (LFSR), which is just a bank of











1 PSD of Unspread
Signal







PSD of Spread



Frequency

Figure 1.1: Power spectral densities of the unspread data sequence and the direct-
sequence spread-spectrum signals.

flip-flops interconnected with the appropriate feedback. Using an LFSR to generate

the spreading sequence, the transmitter and receiver must have common knowledge

of the length of the LFSR, the feedback connections, and the initial state of the LFSR

prior to transmission. With the appropriate feedback, -1 will occur with a proba-

bility slightly higher than +1. In addition the sequence will be periodic. However, if

the period is sufficiently long, over a short observation interval the occurrence of +1

and -1 will appear random.

The goals of a covert communication system are to communicate privately with

an asset and to communicate reliably under adverse conditions, possibly caused by

an adversary. Over the last several decades, spread-spectrum systems have received

considerable attention for their use as covert communication systems. The attention

is due to three properties of spread-spectrum systems some of which are hinted at

in Figure 1.1. The first property of interest is the resistance to intentional jamming.

One way for an adversary to disrupt a communication system is to broadcast at a

high power level in the same frequency band. Due to the increased bandwidth utilized








in a spread-spectrum system, a jammer then only achieves partial band jamming or

has to spread the jamming signal over a wider bandwidth. In either case, the effect of

the jamming signal is reduced compared to the effect observed by the same jammer

in the unspread communication system. The second property of interest is the low

probability of detection (LPD). Even if an adversary knows you are transmitting,

without knowledge of the spreading sequence (or frequency hop sequence), it is very

difficult for that adversary to perform a good detection of the message signal. The

third property, which is especially important for clandestine communication, it called

low probability of intercept (LPI). Considering the reduced amplitude of the DS-SS

signal's PSD as shown in Figure 1.1, we see that with sufficient spreading the level of

the PSD can be made to approach the level of the background noise floor. This means

that an adversary with a spectrum analyzer that is actively looking for the spread-

spectrum transmitter will have difficulty detecting that the transmitter is active. In

other words, in order to locate the transmitter the adversary will have to be physically

located very near the transmitter when a signal is being transmitted. Under ideal

conditions the transmitter will be active for short time intervals, or the asset is able

to detect the presence of the adversary in which case he/she turns the transmitter off.

These conditions greatly decrease the probability of intercept and, therefore, increase

the "life expectancy" of the asset.

As the remainder of this work deals with what is essentially a multi-user direct-

sequence spread-spectrum system, we list several references which describe DS-SS

systems in more detail. Scholtz [2] and Pickholtz et al. [3] provide good tutorials

on the properties and use of spread-spectrum systems. Ziemer and Peterson [4] give

an extensive examination of these systems. Many references on the generation and

properties of spreading sequences are available, as this subject continues to be a

current research interest [5-7].








In direct-sequence code-division multiple-access (DS-CDMA) systems, each user

has a unique spreading sequence and all users occupy the same frequency band trans-

mitting independently of all other users. Consider a K user DS-CDMA system in

which each user transmits 1 times its spreading code. We will communicate with

one of the users, called the desired user, using a receiver that is optimized for a

single-user operating in the presence of additive white Gaussian noise (AWGN). For

this example, the optimum receiver is a filter matched to the desired user's spreading

code. Therefore, the output of the matched filter will depend on the correlation of

the desired user's code with all of the other K 1 users' codes. If all of the users'

spreading codes are orthogonal for any amount of time shift between the codes, the

output of the matched filter will not depend on the presence of the other system

users. A more realistic condition is that the set of spreading codes is chosen so that

the cross-correlation between any two codes is well-behaved. This means the output

of the matched filter will be dominated by the desired user, but some component of

the output will be due to contributions of interfering users.

It is possible to receive each of the transmitted signals at a different power level.

For example, this condition could occur if any of the K 1 interfering users is located

physically closer to the receiver than the desired user. In this case the output of

the matched filter may become dominated by the interference. This type of multi-

access interference (MAI) is referred to as the near-far problem. The challenge for

communication systems engineers is to develop receiver structures that are insensitive

to the near-far problem.

In this dissertation, we are concerned with the problem of achieving synchroniza-

tion with the spreading sequence of the desired user. Moon et al. [8] studied the

effects of MAI on traditional techniques (energy detector) that are used in single-

user spread-spectrum communication systems. It was found that the average time

to achieve acquisition quickly increases when a near-far scenario exists with only two








system users. Moon et al. [9] presented a multi-user synchronization method, called

the rapid acquisition by sequential estimation (RASE) algorithm, which essentially

tries to estimate the contents of the LFSR used to generate the spreading sequence for

each user, based on observations of the received signal. They assumed that all users

transmit only their spreading code and that the receiver knows the received ampli-
tude of each user. For a total of K users, this algorithm has a complexity of O(K2K)

which is prohibitive for K > 10. A similar single-user approach was presented by

Barghouthi and Stiiber [10, 11], under the assumption that all users are received at
the same power level. Madhow and Pursley [12], considered the effects of achieving

code synchronization in the presence of MAI on the system capacity. They found

that the system capacity for achieving code synchronization with a matched filter

was much less than the system capacity measured with respect to the bit-error-rate
(BER). These results form the motivation for researching code acquisition techniques

for DS-CDMA systems.

1.2 Mathematical Notation

Matrices and vectors are typeset in a bold face, for example v or V. In general,
a lowercase letter will be used to denote a vector, while an uppercase letter denotes

a matrix. Unless otherwise noted, all vectors are assumed to be column vectors. In

order to denote the nth element of a vector, a subscript will be used such as vn. The
(i, j)th element of a matrix V is denoted by Vij. A subscript containing the variable

k denotes an element belonging to the kth user in the DS-CDMA system.

The transpose, the conjugate (or Hermitian) transpose, and the 2-norm of a
column-vector v are denoted by vT, vH and lv|| = /vHv, respectively. The conju-

gate of a scalar quantity will be denoted as a*. For a matrix V we will use iiViIF to








denote the Frobenius-norm of the matrix defined as


iiv i12 -=-i2 (Vl)2.
IlVll; ~ ZI ,jl2 (1.1)
i 3

The Kronecker delta function is defined as

1 if m = 0,
6K(m) = (1.2)
0 otherwise.

The expectation of a random variable, /3, either a scalar or non-scalar quantity, will

be denoted as E[/3] or /3. We use the notation O(N) to denote that an algorithm has
numerical complexity on the order of N. When a is a parameter to be estimated, we

denote the estimate as &.

1.3 Outline of the Dissertation
In the next chapter, a system model that is widely used to simulate a DS-CDMA

communication system will be presented. The system model is expressed in a conve-
nient vector-matrix notation which facilitates simulating a DS-CDMA communication
system on a computer. The third chapter provides a brief review of the literature
concerning DS-CDMA receivers and timing acquisition techniques. While it is impos-

sible to provide a complete review of these subjects, an effort is made to provide the
reader with enough of a review to understand several timing acquisition techniques

and their associated complexities. In the fourth chapter, a low-complexity single-user
timing estimation algorithm is presented. The algorithm is based on processing the

weights of an adaptive filter, which is commonly referred to as the MMSE receiver.
The MMSE receiver is a single-bit single-user detector, which is a sub-optimal receiver

for a multi-user DS-CDMA communication system, and is receiving considerable at-
tention in the literature. As several papers in the literature ignore the effects of
operating the MMSE receiver asynchronous to the intended user, several examples








of why synchronization is important are presented. In the fifth chapter, the per-

formance of the timing estimator is characterized. Several analytical techniques are

used to characterize the performance of the timing estimator, based on a developed

statistical model of the transient statistics of the MMSE receiver weights. Simulation

results are used to study the performance of the timing acquisition algorithm for

a general asynchronous DS-CDMA system when two commonly used adaptive algo-

rithms are used in the MMSE receiver. It will be shown that for one of these adaptive

algorithms, the performance of the timing acquisition algorithm is near-far resistant.

In the sixth chapter, the effects of fading on a communication system are discussed

and a commonly used implementation of a flat-fading channel simulator is presented.

The performance of the timing acquisition algorithm when the receiver is operating

in two non-stationary environments (frequency-offset errors and flat-fading channels)

is studied in the seventh chapter. Barbosa and Miller [13] have presented a modified

version of the MMSE receiver that can be used in flat-fading channels. The perfor-

mance of the timing acquisition algorithm for this receiver will be studied. Based on

simulation results, it will be shown that a slight improvement in the performance of

the timing estimator is feasible, through a slight modification in the training cycle of

the modified MMSE receiver. In earlier chapters, the system model and the timing

estimation algorithm were developed such that only one of the DS-CDMA system

users can be in the mode of acquiring timing acquisition. In the eighth chapter, a

modification to the system model and the timing estimation algorithm is presented

such that more than one system user can be in the timing acquisition mode. In the

final chapter, a summary of the work presented in this dissertation, and the contri-

butions to the area of DS-CDMA research, as well as several areas of future research

are provided.













CHAPTER 2
SYSTEM MODEL FOR THE AWGN CHANNEL

2.1 General Asynchronous DS-CDMA System Model for the AWGN Channel

In this section, a general system model for the case when the receiver operates

in an AWGN channel and observes only one symbol interval of information at a

time will be presented. After this general model is developed, a simplified version,

which will eventually be shown to lead to a reduction in the numerical complexity

of evaluating the timing estimator, will be presented. The only difference between

these two models, is how we interpret or apply the effect of carrier phase offset to the

input of the receiver for the desired user.

In this work, a binary phase shift keying (BPSK) communication system is used.

There are a total of K system users, each operating at the same carrier frequency fc,

but asynchronous to and independent of the other users. The kth user is assumed to

transmit a signal whose complex envelope [14], Sk(t), is a polar data sequence, dk(l) E

{+1, -1}, which has a unique signature or spreading waveform, Ck(t), superimposed

upon it such that
00
Sk(t) = E dk(l)ck(t lTb) k e {1,2,..., K}. (2.1)
1=-00
Therefore, the signal actually transmitted by the kth user is given by


Sk(t) = Re[Sk(t)exp{j27frt}] = Re[Sk(t)exp{jwct}] k E {1,2,...,K} (2.2)

where Re[] returns the real part of its argument. Without loss of generality, it is

assumed that the first user is the desired user and all other users act as interference.

The data bits have a duration Tb while the chips of the spreading sequence have

duration Tc. Each spreading sequence has a period of N = Tb/Tc chips. That is, one








period of a spreading sequence is equal in duration to one bit interval. While this is

not a requirement for DS-CDMA systems in general, it is required for the receiver

structure studied in this work. Let the vector Ck = (Ck,O, Ck,1,... Ck,N-1)T represent

one period of the kth user's spreading sequence so that the spreading waveform can

be written as
N-1
Ck (t) = Ck,n17 (t- nT,) (2.3)
n-=O

where pT (t) is the chip pulse shape which is taken to be 1 over the interval [0, Tc)

and zero otherwise.

We assume that the receiver is asynchronous in time and phase to the transmitted

signal of the desired user, as well as being asynchronous to all of the interfering users.

Due to the K system users, the received signal is of the form

K
r(t) = v"Pk Re [Sk(t Tk) exp {jwct + jOk}] + n(t). (2.4)
k=1
In the above expression, Tk and Pk are the propagation delay and the average received

signal power, respectively, for the kth user, 0k is the phase-offset for the kth user,

while n(t) is the additive white Gaussian noise which is assumed to have a one-sided

power spectral density of A.M.

In order to process the received signal, the receiver converts r(t) from a bandpass

signal to a baseband signal, R(t), which is then passed through a filter matched to

the chip pulse shape and the output of this matched filter is sampled at the chip rate.

It is assumed that any double-frequency terms created in the conversion to baseband

are not passed through the chip matched filter. During the receiver's mth bit interval

the nth chip sampled output can be expressed as

mTb+(n+1l)Tc
rm,n =- R(t)dt e C. (2.5)
2P-P1 llc j
mTb+nTc








During each bit interval, the N chip samples are stored in a received vector,
r(m) = (rm,0, rm,1, ... rm,N-1)T and it is this vector quantity that will be processed
by the receiver. Using the above definitions for how each user contributes to r(t) we
can express how each user contributes to r(m) using a notation similar to that used
by Miller [15]. It can be shown that
K Pk
r(m) = P Jk(m)exp{jOk} + Nm E CNx (2.6)
k=l P
where Jk(m) is the contribution to the received vector from the kth user during the
mth receiver bit interval given by

Jk(m) = [z2k-1(m)a2k- (pk, bk) + Z2k((m)a2k(pk,6k)] (2.7)

where

Z2k-i(m)=[dk(m)+dk(m -1)]/2 {-1,0,1}

Z2k(m) = [dk(m)- dk(m 1)]/2 e {-1,0,1} (2.8)

a2k-l(pk,(k)= (1 k)Ck) +kCk+)

a2k(pk,bk) =(1- .k) + 6kCk (2.9)

Cn (Ck,N-n,Ck,N-n+l, ... Ck,N-l, Ck,O, Ck,l, ... Ck,N-n-1)T (2.10)

k = (-Ck,N-n, -Ck,N-n+l, ...- -Ck,N-l, Ck,O, Ck,li ... Ck,N-n-l)T (2.11)

E {+1,-1} if m E {0,,...,N- 1},
Ck,m = (2.12)
0 otherwise.

and Tk =PkTc + kTc with pk E {0,1,..., N 1} and 0 _< k < 1.
Since the receiver is asynchronous to any of the K system users and only observes
one bit interval of information, it is possible to have a data bit transition occur during
the receiver's bit interval. This fact is accounted for in the previous equations. The
quantities Z2k-i(m) and z2k(m) are used to indicate if adjacent data bits for the kth








user are similar. If adjacent data bits are the same, then Z2k-l(m) will be non-zero,

otherwise z2k(m) is non-zero. The vector quantities ck7 and 'n) denote the nth right

cyclic and the nth modified right cyclic shifts of the kth user's spreading sequence,

respectively. As shown in the previous equations, Jk(m) is just a function of two

slightly offset cyclic shifts of the spreading sequence for the kth user.
In the remainder of this work, we will refer to the propagation delay in terms

of the integer part, pk, and the fractional part, 6k, only. The noise vector, Nm, in

equation (2.6) consists of independent complex Gaussian random variables whose real

and imaginary parts are also independent having zero-mean and equal variances of
a2 = N/(2EIb/MAo), where Eb is the average received energy per bit of the desired

user. In the remainder of this work, we define the signal-to-noise ratio (SNR) as

SNR = Eb/lAo.

2.2 Simplified Asynchronous DS-CDMA System Model for the
AWGN Channel
In the previous section, the receiver was assumed to be phase asynchronous to any

of the K systems users. This is a reasonable assumption, since it is highly unlikely

that all of the oscillators for the users will be phase synchronous as viewed by the

receiver. Certainly, it could be possible for all of the transmitted signals to be phase

synchronous, a situation that would arise at a transmitting base station in a cellular
type communication system where one oscillator can be used to transmit all of the

K user's signals. At any of the mobile receivers, then one could say that all users

experience the same propagation delay and carrier phase shift in the channel from

the base station to the mobile receiver. However, in the communication link from the

mobile user to the base station, different propagation delays and carrier phase shifts

will exist due to the different physical locations of the mobile users.








However, we can model the system as being equivalently phase-synchronous as

follows. We can express equation (2.6) as

K Pk
rm=E (cos(0k)+Ijsin(0k))Jk(m) + Nm eCNx. (2.13)
k=1
We then can create a modified receiver input by only processing the real part of r(m).

That is,

K I
i(m) = Re [r(m)] = PkJ cos()Jk(m) + Nm e RNxl (2.14)
k=l 1
where the noise vector, Nm, in equation (2.14) consists of real-valued Gaussian

random variables that are independent having zero-mean and equal variances of

a2 = N/(2Eb/Aro).

Now the cos(Ok) term can be absorbed into the ratio of Pk/P1. Therefore, by

adjusting the level of the SNR appropriately one can equivalently model the effects of

the carrier phase offset for the desired user. To account for the effects of the carrier

phase offset for the other users, the value of Pk/PL for each interfering user is adjusted

as appropriate.

As a result of using this procedure to form an equivalently phase-synchronous

system, we will eventually show that the numerical evaluation of the timing estimator

for the propagation delay for the desired user is simplified. In fact, we will see that

for this model the estimate of 1, the fractional part of the desired user's propagation

delay can be expressed in closed form, which is not the case when the model of

Section 2.1 is used.














CHAPTER 3
REVIEW OF THE LITERATURE

Since the area of DS-CDMA has received so much attention in the literature over

the last several years, it is impossible to review all contributions made in the recent

past. However, in this chapter, we will try to summarize any major contributions

as well as those that are directly related to code acquisition in DS-CDMA systems.

We will discuss the complexity of the timing estimation algorithms, so that we may

later compare them to the complexity of the timing estimator presented in the next

chapter.

Verdi [16,17] provided a major breakthrough in DS-CDMA research by proving

the existence of an optimum multi-user receiver that is near-far resistant. The receiver

is comprised of a bank of matched filters, one matched filter for each user, followed

by a Viterbi decoder. The receiver requires a lot of side-information, in order to

create the matched filters and form a decision metric for the Viterbi algorithm. The

receiver requires knowledge of each users' spreading code, and must be synchronized

with each user. In addition the receiver must know the received signal energy for each

user. The complexity of the receiver is 0(2K) for each binary decision made. The

receiver is considered impractical due to this complexity, which provides motivation

for investigating sub-optimal receiver structures.

Lupas and Verdi [18,19] have proposed a sub-optimum multiuser receiver which

is known as the linear decorrelating receiver. The receiver is similar to the optimum

receiver [16,17] in that it is comprised of a bank of matched filters, but does not

require knowledge of the received signal energy for each user. The outputs of the bank

of matched filters are processed by a linear transformation (called the decorrelating

filter) that removes the multiuser interference from the output of each filter. The








decorrelating filter cancels the MAI by accounting for the known correlation between
the users' spreading codes. The complexity of the receiver is dependent on taking the

inverse of a matrix, which has complexity 0(N3), to form the decorrelating filter. If

this operation can be done once, the complexity of the receiver is then linear in the

number of users, which is a great improvement over the exponential complexity of

the Viterbi based decoder.

Str6m et al. [20] studied the impact of timing errors on the decorrelating receiver.

It was found that the decorrelating receiver loses its near-far resistance with the pres-

ence of timing errors. That is, the BER of the receiver reaches a non-diminishing

floor even as the SNR increases, when errors are present in the estimates of the

timing offsets. In addition, they observed that the required variance of a timing

estimate is inversely proportional to the level of the MAI. This means as the MAI in-

creases, making timing estimation more difficult, a better timing estimate is required

in order for the receiver to maintain its near-far resistance. Zheng and Barton [21]

perform a similar analysis on the decorrelating receiver. In addition to considering

time synchronization errors, they studied the effects of phase synchronization errors

and frequency synchronization errors. They found that if these quantities are limited

to small, and possibly unrealistic, values the decorrelating receiver still offers a sig-

nificant advantage over the conventional matched filter. These results motivate the

need for the development of timing estimators that produce unbiased estimates with

low variances, even in the presence of a severe near-far environment.

Varanasi and Aazhang [22] have proposed a suboptimal multiuser detector whose

complexity is also linear in the number of users. The receiver requires knowledge of

the received signal strengths of each user, the code sequence of each user, and time and

phase synchronization with each user. The receiver is comprised of a bank of matched

filters (one per user) followed by a multistage interference rejection algorithm. The

interference rejection algorithm is summarized in the following sentences. Consider








the output of one of the matched filters. If a decision is made on the filter output, it

may be corrupted by the interference caused by the other users. If the receiver has

perfect knowledge of the other users' received signal energies and transmitted data

sequences, the receiver could directly calculate the interference at the output of the

matched filter and subtract this quantity from the filter's output. Then the receiver

could make a decision on an interference-free quantity, and the decision made for

each user would be optimal. However, the receiver does not have perfect knowledge

of such quantities and, therefore, can only form imperfect estimates of the MAI term

at each filter output. The receiver reconstructs the estimate of the MAI at each

filter output during the mth interval, based on the decisions formed in the (m 1)th

interval. Since the reconstruction of the MAI is based on these decisions, perfect

cancellation of the MAI term from the filter output will not occur (and hence the

receiver is suboptimal).

The previous paragraphs give a brief overview of what has been done in terms of

multiuser detectors for DS-CDMA. As mentioned the complexity of the these tech-

niques, and the amount of required side-information, makes them infeasible. However,

all of the receiver structures mentioned require some form of synchronization with the

user spreading sequences. In the remainder of this chapter, we will focus our attention

on techniques that have been proposed to address the issue of code synchronization.

One area of parameter estimation that is receiving considerable attention in the

literature are estimation techniques based on subspace methods [23-26]. While Strdm

et al. [23] consider the more general case of sampling the received signal more than

once a chip-interval, we restrict our attention to the system model given in Section 2.1.

We modify the system model slightly, by assuming that all users can transmit random

data sequences. Using this system model, the receiver is operating in a stationary

environment. The autocorrelation matrix of the received signal, which is defined as








R = E[r(m)rH(m)], can be written as

Kp
kR = P [a2k- (pk, 6k)a2ki(Pk,6k) + a2k(pk, k)ak(pk, 6k)] + 2u2I (3.1)
k=1

where I is an N x N Identity matrix. Note that R is symmetric, and is also positive
definite as long as a2 > 0. When 2K < N, the eigen-decomposition of R can be
expressed as


R= E8 En E, E, H. (3.2)
0 An

As is a diagonal matrix of the 2K largest eigenvalues of R and E, is an N x 2K
matrix of the corresponding eigenvectors. Likewise, An is a diagonal matrix of the
(N-2K) smallest eigenvalues (all equal to ao2) of R and En is an N x (N-2K) matrix
of the corresponding eigenvectors. The signal subspace is defined to be the subspace
spanned by the set of a2k-l(Pk, 6k) and a2k(pk,6k) vectors Vk G {1, 2,..., K}. The
noise subspace is defined to be the orthogonal complement to the signal subspace. The
columns of A, form an orthonormal basis for the signal subspace, and the columns
of An form an orthonormal basis for the noise subspace.
The basis of the multiple signal classification (MUSIC) algorithm, is that since
the set of {al, a2,..., a2g} vectors are in the signal subspace, they are orthogonal
to the noise subspace. Therefore, given perfect knowledge of the noise subspace, one
could find Trk as the solution to En a2k-1 (pk, 6k) = 0. Typically, the receiver does not
have perfect knowledge of the noise subspace or R. However, it can form an estimate
of R using the sample correlation matrix defined as

1 M
M r(m)r'(m)- (3-3)
m=1l
Given RM, the receiver can form an estimate of the noise subspace, En, an operation
that has complexity 0(N3). Strdm et al. [23] use this idea to form a timing based on








the MUSIC algorithm. They find an estimate of the kth user's propagation delay as

arg minm a2k ll2 IHI1H 2 (3.4)
Tre[O,Tb) la2k-l(T)jI + jla2k(rT)12

The independent work presented by Bensley and Aazhang [24] is similar, but also

addresses the problem of estimating the channel gain and phase in a time-invariant
multi-path channel. As a natural extension of their earlier work, Str6m et al. [25]
consider using the MUSIC based estimator in a time-varying channel.

The MUSIC-based timing estimator requires very little side information in order
to form its estimate. The number of system users, as well as the spreading code (to be

used in equation (3.4)) is required. While the complexity of the algorithm is 0(N3),
it requires no training period, and can be used to estimate each user's propagation

delay. However, the algorithm does not work for 2K > N, as the noise subspace
of R has zero rank. It has been noted that for low signal-to-noise ratios and little

near-far effect, that the traditional correlator performs better than the MUSIC-based

algorithm. However, as the MAI interference increases the MUSIC-based algorithm
outperforms the correlator. It has also been noted that the MUSIC-based algorithm
is resistant to the near-far problem [23,24].

Zheng et al. [27] have proposed a single-user propagation delay estimator, using
the system model of Section 2.1, that is known as the large-sample maximum likeli-
hood (LSML) method. This method models the received signal as a known training
sequence (the desired signal) and all other signals including the interfering signal
and thermal noise as unknown colored Gaussian noise that is uncorrelated with the
desired signal. The resulting timing estimate is found by rooting a second order poly-

nomial. The coefficients of the polynomial are dependent on a matrix inverse, Q-',

where Q is related to the RM matrix given in equation (3.3). In order to evaluate
the matrix inverse, Q must be full rank. This means that the LSML timing estimate
is unavailable until the receiver has observed at least M = N samples of r(m).








As an extension to the LSML timing estimator, Liu et al. [28] have proposed a

similar timing estimation scheme known as the multiple-antenna sensors-based esti-

mator (MASE). That is, the estimator is derived using the known training signal and

modeling the MAI and the additive noise as an unknown colored Gaussian random

process. In the MASE algorithm, the receiver observes the outputs of an arbitrary

antenna array of L sensors. In fact, when L = 1 the MASE algorithm reduces to the

LSML algorithm. Like the LSML, the MASE algorithm requires a training sequence,

and the timing estimate is formed by rooting a second order polynomial. Once again,

the coefficients of the polynomial are dependent on the inverse of a matrix. In the

LSML algorithm, the receiver has to acquire at least M > N samples of r(m) in

order for the Q matrix to have full rank. By using L antenna sensors, the MASE

algorithm is able to average over the outputs of the sensors and hence only requires

LM > N samples of r(m) in order for the Q matrix to have full rank. Since the

MASE algorithm requires fewer training symbols than the LSML algorithm, under

conditions where the channel is time-varying, the MASE has the advantage over the

LSML algorithm [28].

In the next chapter, a timing estimation algorithm is presented that is based on

processing the weights of the MMSE receiver. In this dissertation, it is assumed

that the MMSE receiver uses an initial training sequence to adapt the weights to

minimize the MSE between the filter output and the data sequence for a single user.

This means that a known data sequence is used to train the MMSE receiver, prior

to performing data detection. Honig et al. [29] have developed a single-user detector

based on minimizing the mean-output-energy (MOE) of an adaptive filter. Note

that the term MOE in actuality is used to denote the variance of the filter output.

The proposed receiver [29] requires knowledge of the timing and spreading sequence

of the desired user. The weights of the filter are expressed in a canonical form as

w(m) = C1 + x(m), where ci is the spreading code of the desired user and x(m) is








orthogonal to cl. The vector x(m) is updated to minimize the variance of the filter

output. This adaptive algorithm does not require knowledge of the desired user's

data sequence, and, therefore, is known as a blind adaptive algorithm.

Madhow [30] uses the blind adaptive receiver in an ad-hoc fashion to form an

adaptive receiver which only requires knowledge of the desired user's spreading se-

quence. Madhow uses a system model similar to that presented in Section 2.2, but

observes two bit-intervals (2Tb) of the received signal, and, therefore, has 2N sam-

pled outputs of the chip-matched filter comprise the received vector. The prop-

agation delay for the desired user is assumed to be one of 2N hypotheses, T1 E

{0, 0.5Tc, 1Tc, 1.5Tc,..., (N 0.5)To}. The received signal is processed by a bank of

2N parallel blind adaptive filters (one for each 7-1 hypothesis) during a blind acquisi-

tion cycle. Let the ith filter be the adaptive filter that has the lowest MOE at the end

of the acquisition cycle. At the end of the acquisition cycle the receiver chooses two

filters, the ith filter and either the (i 1)th or (i + 1)th filter, to form the adaptive

receiver. The output of the ith filter is combined with the other filter in an ad-hoc

fashion in an attempt to combine the timing hypotheses that are closest to the true

propagation delay for the desired user. Using this method, Madhow is attempting

to create a receiver structure that performs joint acquisition and detection of the de-

sired user's signal. However, this method is very complex and doesn't really achieve

synchronization with the desired user. In the next chapter, we demonstrate why syn-

chronization is important by considering the effects of being a half-chip asynchronous

to the desired user.













CHAPTER 4
THE MMSE RECEIVER AND SINGLE-USER CODE ACQUISITION

As an alternative to using the complex estimation techniques presented in the

previous chapter, we present a single-user estimation algorithm which is based on

processing the weights of an adaptive receiver. One benefit of this receiver structure

is that the only side information the receiver requires in order to form its estimate is

the spreading code of the desired user. In addition, the same structure may be used
for data detection. This receiver structure, which has commonly been referred to as

the MMSE receiver, has received significant attention in recent literature.


4.1 The MMSE Receiver
In this section, using certain system assumptions, we will discuss how the weights
of the MMSE receiver, shown in Figure 4.1, may be used to estimate the propagation
delay of the desired user. The weights of the receiver are chosen to minimize the mean-

squared error E [|e(m) 12], where e(m) is the difference between a desired response

and the filter's output. The vector which minimizes the mean-squared error (MSE)

is well-known to be given by the Wiener-Hopf [31] equation


w(m) =R-l(m)p(m) (4.1)

where R(m) = E [r(m)rH(m)] and p(m) = E[d(m)r(m)] are the autocorrelation

matrix and the steering vector, respectively.

The received vector r(m) corresponds to chip-samples of one period of the spread-
ing sequence. Since the receiver is initially asynchronous to the desired user, it is

possible for data bit transitions to occur anywhere within the length of the received

vector. In order to avoid these bit transitions for the desired user, we assume that the








Cl I
10Estimation i
w(m)* algorithm

Convert to H
r(Ft) baseband r(m) Adaptive r (a)w(n) ptdi(m)
tr i a+chip a enr p of code sgn[Re(.)] A -ch
l tch.ied filter, w(m) S 2.e1
matched
filter + -( +

--_ A daptive 1 e(m )
algorithm I I (M)

Figure 4.1: Code acquisition with an adaptive receiver.

desired user's data bit is a constant di(m) = 1. In other words, the desired user will
transmit an all ones data sequence for purposes of code acquisition. A side-channel
is used to control the addition of new users into the system, so this restriction is not
unrealistic. Based on the system model presented in Section 2.1, we see that the filter
is operating in a stationary environment. It is assumed that the users' data sequences
and carrier phases are independent of each other and also independent of the additive
white Gaussian noise. The autocorrelation matrix and the steering vector are given
by the following equations.

R= E [r(m)rH(m)] = a,(pi, 61)aT(pi, 61) + 2a21
K (4.2)
+ P [a2k- 1(pk, 6k)a2k _i(pk, 4k) + a2k(pk, 6k)ak(pk, 6k)]
k=2P
p = a (pi, 61) exp(j01) (4.3)

where I is an N x N Identity matrix. The MMSE occurs when the filter weights are
optimal, and is given by


Jmin = 1 pHR-lp.


(4.4)








In order to solve the Wiener-Hopf equation for the optimal weight vector, Wopt,

we have to invert an N x N matrix, an operation that has complexity of O(N3).

Since R is a symmetric matrix, it can be expressed as


R =VAVT, (4.5)

where V is a matrix whose columns contain the normalized eigenvectors of R and

A is a diagonal matrix that contains the corresponding eigenvalues. This notation is

referred to as the eigen-decomposition of the R matrix. Using the above notation,

we can then express the inverse of the autocorrelation matrix as

R-1 = VA-1VT. (4.6)


Instead of calculating R1 directly, we note that it can be shown that an eigenvector

of P = R 2oa21 is also and eigenvector of R. For each eigenvector of P, adding

2ca2 to the corresponding eigenvalue produces the corresponding eigenvalue for the

matrix R. Therefore, if we find the eigen-decomposition of P we immediately have

the eigen-decomposition of R.

In equation (4.2), we note that each asynchronous user contributes at most two

outer product terms in the expression for P. If 2K < N, then the P matrix will

not be full-rank and will have at most 2K non-zero eigenvalues. Miller [32,33] used

this fact to simplify the calculation of the eigen-decomposition of P, by working

on a 2K x 2K matrix instead of on a N x N matrix. For large ratios of N/(2K)

this technique produces a significant reduction in the complexity of calculating R-1.

However, in that work the desired user was assumed to be synchronous (ri = 0) with

the receiver, so one must slightly modify Miller's notation in order to process the

desired user in the more general asynchronous case. Of course when 2K > N, this

technique is not valid and one must find the eigen-decomposition of an N x N matrix

(either R or P).








In order to illustrate the dependence of the filter weight vector on the amount
of multi-access interference, we will observe the optimal filter weight vector for two
cases. For the first case let there be only one chip-synchronous system user, which
implies that r(m) = c) exp (jOi) +Nm. Using the Wiener-Hopf equation, it is found
that

c. exp(ji) (47
Wopt = ICi2 +22

Assuming cl has good auto-correlation properties, this result shows that we could
conceivably form an estimate of Pi, the propagation delay of the desired user, by
finding the peak of the magnitude of the cross-correlation between cl and Wopt.
For the second case, a second synchronous user is added to the system such that
r(m) = c1 exp (j01) + d2(m/) /P2/P1c2 exp (j02) + Nm. For this second example,
the optimal filter weight vector is

(c1) --^YC2)exp (j01)
Wopt = C- 112 2- T (4.8a)
S+20 ^2 2 C1
cT (Pi)
2 1221 (4.8b)
Ic2H +2 P2

Note that 7y is bounded in magnitude by the value of the correlation coefficient of
the two spreading codes, and is therefore expected to be much less than one. In
fact, in the limit P2/P1 --+ oo, we see that Wopt is the projection of c(P' onto a
(pl)~~~ ~ ont orhgnat 2,tercie
vector orthogonal to c2. In the ideal case that c(P1' is orthogonal to C2, the receiver
completely rejects the second user even though it is received at a much higher power
level. In the general case, for codes with good correlation properties (c2 c1 is small),

the filter should be able to suppress the second user. In the general case, the addition
of the second user causes a small perturbation of the filter weight vector about the
solution given by equation (4.7). We should still be able to form an estimate of pi as
described above.








Equation (4.8a) is used as a basic example of why a timing estimator based on

processing the filter weight vector should even be considered. As more users are in-

cluded in the DS-CDMA system, additional terms due to the spreading codes are

added to the expression for Wopt. In order to illustrate the dependence of wopt on the

desired user's spreading code in the above example, and to simplify the resulting ex-

pressions, we assumed that the interfering user was chip-synchronous to the receiver.

In the general case, each asynchronous interfering user contributes two similar terms

to the expression for Wopt. However, if the set of spreading codes has sufficiently good

cross-correlation properties one would expect that Wopt is closely related to cl such

that a reasonable estimate of P, is still feasible.

4.2 Advantages of Synchronizing the MMSE Receiver

Consider using the MMSE receiver in a single-user system. One might ask why
synchronization of the MMSE receiver is even an issue that must be considered. If

the MMSE receiver is used for detection, are the weights optimized to give us the best

performance? The answer is maybe. The weights are optimized for the given value of

T-1, the propagation delay of the desired user relative to the receiver. It is conceivable

that Jmin at T1 = 0 is less than the resulting Jmin when Tr1 i 0. Therefore, we should

prefer to operate the receiver synchronous to the desired user's received signal. In

order to expand on this idea, we present three brief cases that demonstrate some of

the costs associated with using the MMSE receiver asynchronous to the desired user.

For the first case, consider using the MMSE receiver to demodulate data in a

single user system where

r(m) ai(pi, 61) + Nm (4.9)
a,(pi, 61) (4.10)
Wopt a,(p, 6)112 + 2210







Consider the signal-to-noise ratio of the output statistic z = rH(m)Wopt, which is
defined as SNRo, = E[z]2/Var(z). Note that E[z] = af(pi,6i)wopt, and the expres-
sion for Var[z] is evaluated using Var[z] = E[wOtr(m)rH(m)wopt] E[z]2. Direct
substitution of the appropriate terms into this expression produces

Vr2 22 Ial(pi, 61)112
Var~z] 2
(Ilal(pi, 61)112 + 2a2) (4.11)
2o2E[z]
Il ai(pi, 61)112 + 2U2
The result of these expressions is that SNRo = Ilal(pi, 61)112/(2o2). If we assume
that the spreading code has good autocorrelation properties such that the terms
involving c"c1 are negligible when compared to other terms in |Iai(pi, 1)1|2, we can
approximate the output signal-to-noise ratio as

SNRo [(1- 61)2 ]1 (4.12)
2o2
where o72 = N/(2Eb/.fAo) as previously defined. When 61 is non-zero, there is an
effective loss in the output SNR of the output statistic z. If 61 is uniformly distributed
over the interval [0,1), then the average loss in the output SNR is 2/3 (1.8 dB). This
loss will adversely affect the capacity of the communication system which we typically
want to maximize. Note that this effect can be offset by sampling the received signal
more often than once a chip interval. However, increasing the sampling rate directly
increases the length of the received vector, which leads to an increase in the complexity
of the receiver.
Madhow [34] presents a method where an adaptive receiver is used to perform
joint detection and acquisition by increasing the length of the received vector and the
filter to 2N chip samples. By increasing the length of these vectors to 2N elements,
the desired user does not have to transmit an all ones training sequence. In addition
it may be possible for more than one user to be in the training mode at the same
time. The receiver structure performs joint detection and acquisition without actually








achieving chip synchronism with the desired signal [34]. It is important to ask how

the performance of the MMSE receiver is affected by the lack of chip synchronism.

To address this issue, we calculated the Jmin of the MMSE filter as a function of the

propagation delay of the desired user when the SNR was 10 dB and the length of

the received vector and filter was set to 2N chip samples. This result is shown in

Figure 4.2 for propagation delays in the range of [0, 5) chip intervals. Similar results

are obtained for propagation delays in the range of [5,31) chip intervals but were

omitted for clarity of the plot. There are two things to notice in this plot. If one

were to consider a system that is nearly chip synchronous then it appears that the

performance of the filter, based observing the MMSE, is unaffected by the propagation

delay of the desired user. This result could lead one to the incorrect conclusion that

achieving code synchronization is not important for the MMSE receiver. However,

when the filter is not chip synchronous with the desired signal, the value of 61 falls

in the region (0,1), the MMSE of the filter is sensitive to the propagation delay of

the desired user. Therefore, some method is required for code acquisition and code

tracking even when 2N chip samples are used in the MMSE receiver structure.

As a final example, we will consider the effect of asynchronous operation (with

respect to the desired user) on the resulting probability of bit error. This example

will show that code synchronization is required for the MMSE receiver if one is to

maximize the system capacity. The following briefly explains the outline of the analy-

sis used to derive the probability of bit error. Since we are interested in bit errors, we

assume that the filter is at steady-state, and that the desired user is transmitting a

random data sequence. We then model the output of the MMSE receiver, wotr(m),

as a complex Gaussian random variable, and derive an expression for the probability

of bit error. The details of this derivation are given in Appendix A. Modeling the

output of the MMSE receiver as a Gaussian random variable has been recently jus-

tified by Poor and Verdi [35], who have compared the bit error rate of the MMSE








0.1 --

0.09

0.08

I0.07 \

0.06

0.05

0.04 -- I-
0 1 2 3 4 5
Propagation Delay of User #1 in Chips

Figure 4.2: Minimum mean-squared error as a function of the propagation delay of
user #1.

receiver versus the analytical bit error rate of the MMSE receiver when the receiver
output is modeled as a Gaussian random variable. It was found that the Gaussian
approximation was quite good for various number of users and various levels of multi-

access interference. In addition, several others have found that this approximation

provides reasonable results [13,36].

Once the DS-CDMA environment is configured, # users, delays, phases and power
levels, we can numerically solve for Jmin. We then use equations (A.9) and (A.10)
to find the probability of bit error. By varying the DS-CDMA environment, we may

observe the effects of asynchronous operation on the system capacity. For purposes

of this example, we set the SNR to 7 dB and let all interfering users be received at
a power level 10 dB above the desired user. A set of spreading codes for 33 users
were selected at random from a set of Gold codes [5] using N = 31 chips/bit. A set

of 33 propagation delays were chosen from a uniform distribution over the interval

[0, 31). A set of 33 carrier phase-shifts were selected from a uniform distribution over








the interval [0, 27r). If we wanted to analyze a K-user system, the first K elements

of each of the spreading code, delay, and phase-shift sets would be used to create the

DS-CDMA environment.

For the case when the receiver was one-half chip out of synch with the desired

user, using the Gaussian approximation we found that 4 system users would produce

a 2% bit-error rate at the receiver. A total of 9 system users would produce a 5%

error-rate. If the receiver was synchronous with the desired user, a total of 14 users

would produce a 2% bit-error rate. Likewise, a total of 24 users would produce a 5%

bit-error rate. This shows that a significant improvement in the system capacity can

be achieved if the receiver is synchronous with the desired user. While these results

are only valid for case we examined, they are sufficient to demonstrate the costs of

operating the receiver asynchronous to the desired user.

4.3 Adaptive Filters and the Estimation Algorithm

The complexity of finding the optimal weights, given by the Wiener-Hopf equation,

is 0(N3). However the receiver requires knowledge of the autocorrelation matrix R

and the steering vector p. Typically, these parameters are estimated by the receiver

based on observations of the received signal. In practice, the weights of the filter

are chosen adaptively according to some implementation which usually takes the

form of either the least mean square (LMS) or the recursive least-squares (RLS)

algorithm. The LMS algorithm has low complexity, 0(2N), does not require any

matrix inversion operations, and is easy to implement. The RLS algorithm is more

complex, O(N2), and performs an iterative matrix inversion by using the Matrix

Inversion Lemma [31]. Since the RLS algorithm is more complex, it is not surprising

that it typically converges faster than the LMS algorithm. While these algorithms

will not produce the same weight vector given by the Wiener-Hopf equation, as the

filter approaches steady-state, they can provide reasonable approximations to the

optimal weight vector.








Our basic idea is to derive a code timing estimation algorithm for a single-user

system, and then study the performance of the estimator in a multi-user environ-
ment. Since the receiver is initially asynchronous to the desired user, r(m) =
a (pi, 61) exp (j1) + Nm. In order to derive a code estimation algorithm, we would
like to have an exact representation of the statistics of the filter weight vector as

the filter adapts. This appears to be intractable. Instead, we model the filter coef-
ficients as independent jointly complex Gaussian random variables with a mean of

/3a (pi,5 61) exp (ji) and equal variances, where /3 is some unknown constant. We
choose this expression for the mean since for this scenario the optimal weight vector
is proportional to a (pi, 61) exp (j01). Clearly, this is not an exact description of the

statistics of w(m), and our resulting timing estimator will not produce an optimal es-
timate of the desired user's propagation delay. The statistics of the weight vector have
been observed (through simulations) as a function of time when an LMS algorithm

was used, and the LMS step-size was chosen such that the filter was convergent in the
mean-square. We observed that the components of the filter weight vector become

highly uncorrelated as the filter approaches steady-state. Therefore this assumption
on the statistics of the filter weights seems to be a reasonable approximation and

hence can be used to derive a meaningful timing estimator.
Based on our assumed statistical model for the filter weight vector, we find that
we should choose as our estimates the set {8, /3, p, 6} which minimize


g(, 0, p, S) = I|w(m) -/ ai (p, 6) exp(jo)112 (4.13)

with p E {0,1,...,N 1} and 0 < S < 1. Note that even though our assumed
mean is based on optimal values, we will use this function to make estimates after
every receiver bit interval. This approach is taken because it can be shown that

when the LMS algorithm is used, the multi-user transient mean filter weight vector

E[w(m)] is highly correlated with the single user optimal weight vector when a small







adaptation step-size is used. Due to the constraint on the value of p, g(/13, 0, p, 6) is not
differentiable with respect to p. To minimize g(3(0, 0,p, 6), we must find the solutions
to 9g(3, 0,p, 6)/9x = 0 (for x =/3, 0,6), for all possible values of p. Therefore, each
value of p produces a corresponding set of candidates for the subset {0, /3, 6}. Our
estimation algorithm iterates through all possible values of p and produces a list of
all of the corresponding candidates. By choosing the set {0, /3, p, 6} out of this list
which minimizes g(/3, 0,p, 6), we find the global minimum of g(/3, 0,p, 6).
The solutions to Og(/3, 0,p, 6)/9x = 0 for x = 0,/3 are given by

0i (p, 6) = arg (aT(p, 6)w(m)) (4.14a)
/(p~, 6)- ',a(p, 6)w(m)I
0lai(p, 6)112 (4.14b)

Substitution of these expressions into the previous equation produces the following
two expressions.

g(o(p, 6),0(p, 6),p,6) = IIw(m)112- f(p,6) (4.15a)
f(p,6) = (p,6)w(m)12 (4.15b)
Ilal(p, 6)112
These equations show that minimizing g(3(p, 6), 9(p, 6),p, 6) corresponds to finding
the set {p, 6} which maximize the cost function f(p, 6).
Since the estimate of pi must be an integer, the cost function f(p, 6) is a piece-
wise continuous function. Maximizing f(p, 6) with respect to 6 produces a quadratic
equation whose coefficients depend on the value of p. Therefore, for each possible
value of p we must find the roots of the quadratic equation D2 (p) t2+ D1 (p) + D0o(p),
where the coefficients are given by the following set of equations.

Do(p) = NA1(p) + 2CAo(p) (4.16a)

D1(p) = 2NA2(p) 4CAo(p) (4.16b)
D2(p) = 2C(A2(p) + Ai(p)) (4.16c)








C = N c c1) (4.16d)

Ao(p) = Iw (m)c P)2 (4.16e)

Ai(p) = 2 Re [(cP))Tw(m)w(m)c]P+1) 2Ao(p) (4.16f)
(P+1) 12
A2(p) = IwH(m)ci 2 Ai(p) Ao(p) (4.16g)

By definition, the value of 6 must satisfy the condition 0 < 6 < 1, so we must ignore
any solutions from the previous equation which fall outside of this region. In order
to clarify how we form our estimate of the propagation delay for the desired user, we
present the following algorithm.
Let 7T represent the set of costs corresponding to the set of candidate estimates
of (p, 6), denoted by U. We can find the maximum of f(p, 6) as follows:
Step 1. Let T = {f(0,0),f(1,0),f(2,0),..., f(N 1,0)}.
Step 2. Let U = {0,1,2,..., N 1}.
Step 3. For p = 0,1,... N 1, do the following:
a) Compute the coefficients D2(p), Dj(p), Do(p).
b) Solve for the two roots of the quadratic equation, pi and A2.
c) If 0 < pi < 1 for i {1,2}, add the cost f(p, p) to the set T, and add

(p + pi) to the set U.
Step 4. Let Uk denote the kth element of the set U. Then f = Uk, where k = maxTj.
3
Step 5. The estimates of p, and 61 are given by Pil(m) = [fj and Si(m) = L-fj,

where [xj rounds x to the nearest integer towards zero.
We plot the cost function, f(p, 6), for two different scenarios in Figures 4.3 and 4.4,
under the assumption that the receiver is operating using the optimal filter weights.
Figure 4.3 shows the cost function for a single-user system when r1 = 15.25T, and
Eb/.Afo was 7 dB. This plot shows that the timing estimate will be f, = 15.25Tc, which
is a perfect estimate. Since we have developed the timing estimator for a single-user
system this result is not surprising. Of course, when the filter weights are from the








0 1 I I I I I I I I I I I I I I I I I I


0.08


0.06
too

^ 0.04


0.02 -


0 ./. ,,y
0 5 10 15 20 25 30
Tested Propagation Delay

Figure 4.3: Plot of the cost function f(p, 6) for a single-user system.

LMS or RLS algorithms we expect some noise in this estimate due to the noise in the

filter weights. Also, the rate of convergence of the adaptive filter to its steady-state

value will determine how long the filter has to adapt before a reliable estimate of T1

can be made. This issue will be addressed in the next chapter.

Figure 4.4 shows the cost function, f(p, 6), for a more severe environment. In this

figure 14 interfering users have been added to the system, each received at a power

level 10 dB above the desired user. Since the maximum value of f(p, 6) does not

occur near 15.25Tc, the timing estimate will be wrong. In this example f- = 7.43Tc.

We have intentionally heavily-loaded the DS-CDMA system, to demonstrate what

will happen to the cost function. In general, we can say that at some level of MAI,

the estimation algorithm completely breaks down. We will study the effects of the

level of the MAI on the performance of the timing estimator in the next chapter.

When the simplified system model presented in Section 2.2 is used, the timing

estimator takes on a slightly different form. Recall that the simplified model assumes

that all users are received phase-synchronous, and the weights of the adaptive filter








0 .0 3 1 , i

0.025

0.02 -

0.015

0.01

0.005

0
0 5 10 15 20 25 30
Tested Propagation Delay

Figure 4.4: Plot of the cost function f(p, 6) for a fifteen-user system.

are purely real. Using this assumption the autocorrelation matrix and the steering

vector for the adaptive filter are written as

R = E [r(m)rH(m)] = a(P, (l)a (P, i) +
K^ (4.^17)
+ K Pk [a2k -1(pk, 6k)ak -(Pk, 6k) + a2k(pk, 6k)ak(pk, 6k)] (4.17)
k=2
p = a(pi, 651) (4.18)

where I is an N x N Identity matrix.

Having the weights of the filter real leads to a small reduction in the complexity

of forming a timing estimate for the desired user. The derivation of the algorithm is

similar to that previously presented. Maximizing the cost function, f(p, 6), produces

a quadratic equation whose roots may be written in closed form. However, one of the
roots produces a trivial result and therefore can be ignored. The other root is given








by the following equation.

NwT(m)c(P+l) T (1) T M)C(P)
S(m-c1 1 (4.19) m)
S(N -"I"TC(1))(wT(m)c(P+1)'+ wT(m)c~p))
S(N 1 1rcKw 1? + W I)^

Based on the above equation, the timing estimation algorithm is then modified as
follows:

Step 1. Perform steps 1 and 2 of the previous algorithm.
Step 2. For p = 0,1,..., N 1, do the following:

a) Use equation (4.19) to compute the coefficient p1.
b) If 0 set U.
Step 3. Perform steps 4 and 5 of the previous algorithm.

4.4 Summary
The main contribution of this chapter is the presentation of a timing estimator for
a single-user, based on processing the weights of an adaptive filter. We motivate why
these weights may be used to form a timing estimate by studying how the optimal
weights depend on the DS-CDMA environment for two simple cases. The MMSE

receiver has received considerable attention in the literature. One fact that seems to
be ignored by several authors is that there are performance penalties to be paid for

operating the receiver asynchronous to the desired user [34]. Therefore, one may come
to the wrong conclusion that no advantage is offered by synchronizing the receiver
with the desired user. We present three examples as to why operating the MMSE

receiver synchronous to the desired user is important. Certainly, when the system

capacity is considered we must synchronize the receiver with the desired user, if we
want to support as many users as possible in the DS-CDMA system.

One may argue that the form of the timing estimator is complex since it depends
on, N, the number of chips/bit. However for large N, even using the traditional

correlator for timing estimation becomes complex. In fact, the correlator uses the






36

exact same estimation algorithm that we have presented in this chapter. The differ-

ence between our technique and the correlator, is that we form an estimate based on

observing the filter weights while the correlator forms an estimate by observing sam-

ples of the received signal. The additional complexity added to the receiver by our

technique is that of running the adaptive filter (LMS O(2N), RLS 0(N2)). Since the

same adaptive-filter receiver structure can be used for detection, and we have demon-

strated that synchronization is desired, including the timing estimator in the receiver

should be an acceptable increase in the over-head of implementing the receiver.













CHAPTER 5
CHARACTERIZING THE PERFORMANCE OF THE
TIMING ESTIMATOR
In this chapter, several methods will be presented in an attempt to characterize the

performance of the adaptive filter-based timing estimator in an AWGN channel. We

are interested in developing analytical tools that can be used to obtain performance

measures on the quality of the timing estimate in hopes of avoiding long Monte-

Carlo simulations to get such performance metrics. We first consider a bound on the

probability of correct acquisition when we are only interested in correctly estimating

the chip level timing. Next using traditional techniques, an approximation to the

lower bound on the conditional variance on the timing estimator will be derived. This

expression will be developed by deriving the Cram&r-Rao bound (CRB) for a fictitious

timing estimator, once several assumptions are made with respect to the statistical

properties of the LMS filter weights. In order to test the quality of these analytical

expressions, they will be compared to simulation results under several different multi-

user scenarios. Finally, the coarse acquisition performance of the timing estimator

will be observed through Monte-Carlo simulations. The acquisition performance of

the timing estimator will be observed for both the LMS and RLS based adaptive filter

architectures. The effects of multi-access interference on the timing estimator will be

observed for several different scenarios.


5.1 Transient Statistics of the LMS Filter Weights

In order to derive analytical performance metrics on the timing estimator, we

first require knowledge of the statistics of the filter weights as a function of time.

When we developed the timing estimator, we assumed that the filter weights were








independent and jointly Gaussian random variables. Using these assumptions, one

could then completely define the assumed statistics of the filter weights by defining

the filter weight mean vector and the covariance matrix for the filter weights. Since

the filter is adaptive, these quantities will be dependent on the initial conditions of the

filter as well as the length of time that the filter has adapted (and of course somehow

depend on the statistics of the input signal). The next few paragraphs describe how

the statistics of the the LMS filter, using the system model presented in Section 2.2,

may be derived using several assumptions about the filter and its input signal.

One area that has received a fair amount of attention in the literature is the

tracking performance of the LMS filter [31,37-42]. The main subject that has been

addressed in these works, is what bounds must be placed on the LMS step-size, A, such

that the filter has desirable convergence properties. One of the properties of interest,

is the convergence of the filter weights about the Wiener-Hopf solution. The filter

is said to be convergent in the mean if lim _. E[w(n)] = wopt, the solution to the

Wiener-Hopf equation. As an extension of this idea, even if the filter is convergent in

the mean, it would be very desirable to have a finite variance of the filter weights about

Wopt. The variance of the filter weights about the Wiener-Hopf solution determines

the level of the mean-squared error at the filter output. As one would expect, the

bound on p required to minimize the mean-squared error of the filter output is less

than the value of p required for convergence in the mean. An intrinsic part of the

analysis on the limits of p for such convergence properties, and what we are really

interested in, is a set of equations that describe the characteristics of the filter weights

as a function of adaptation time.

Senne [37] presented a set of iterative equations to describe the time-dependent

filter weight mean vector and the filter weight correlation matrix for an LMS filter

when the filter input was a vector of zero-mean Gaussian random variables. However,

in the system model used in Section 2.2, due to the desired user's contribution to the








receiver input, the filter input signal is not zero-mean. We can apply the model used

by Senne to derive a similar set of equations to describe the transient statistics of

the filter weight vector. In order to begin the derivation, we make several assump-

tion regarding the relationships between the filter weight vector and the input signal.

These assumptions are referred to as the fundamental assumption [37] or indepen-

dence assumption on the filter statistics and are included here for convenience. The

fundamental assumption consists of three parts which are:

1) Each receiver input vector r(m) is statistically independent of all previous input

vectors and all previous desired filter outputs di (m).

2) Each desired output dj(m) is dependent on the corresponding input vector r(m)

but is statistically independent of all previous desired outputs and received vectors.
3) All desired quantities and received vectors are mutually Gaussian-distributed ran-

dom variables.

In many applications of the LMS filter, the filter input vector is sampled at the

same rate as the output of the filter. Therefore, the first part of the fundamental

assumption is clearly not valid in this situation since the elements of r(m 1) are

shifted by one position in creating r(m). This weakness in the fundamental assump-

tion under these conditions is what has driven the study of the convergence properties

of the LMS filter. Of noticeable interest is the work by Douglas [42], in which a nu-

merical algorithm is presented to observe the exact transient statistics of the MSE

convergence of the filter. However, this algorithm is very complex in that a set of

linear equations must be dynamically created and solved based on the number of taps

in the LMS filter. As an example of the complexity of this algorithm, for a five-tap

filter, a set of three-thousand equations were required to solve for the transient MSE

of the filter. Based on this example, applying to this method to a filter with more

taps does not seem reasonable.








For the adaptive receiver, we have assumed that the input to the filter is sampled

at a rate N times faster than the filter output. This means that r(m) could truly be
statistically independent from all previous receiver inputs. Certainly the contribution
to r(m) from the AWGN is independent from all previous receiver inputs. However,

two symbol intervals of each asynchronous user contribute to r(m). So the first part of

the fundamental assumption, while not being true, will still be asserted and we use the

fundamental assumption to derive the desired equations. We begin the derivation of
the filter transient weight mean vector by manipulating the general LMS filter update

equation given below

w(m + 1) = w(m) PU(m)uT(m)w(m) + pIu(m)d(m) (5.1)

where u(m) is the filter input signal. Using the system model defined in Section 2.2,
u(m) = i(m) defined in equation (2.14), with the caret dropped for notational sim-
plicity in future equations.

We note that the desired filter output is not time-dependent, and will therefore

drop the dependency on m for this quantity. Taking the expectation of both sides of
equation (5.1) will produce the desired filter transient weight mean vector

E[w(m + 1)] = w(m + 1)
(5.2)
= {I- LRrr} w(m) + pfrd

where I is an N x N identity matrix, and Rr = E[r(m)rT(m)] is used for notational

convenience. Once we define the initial conditions on the filter wV(0), realize that
r = a (pi, 61) and let d = 1, we can use the previous equation recursively to describe
the filter transient weight mean vector.
In order to develop an iterative equation for the filter transient weight covariance

matrix, equation (5.1) will be used to find an iterative equation for the filter transient

weight autocorrelation matrix defined as Rw,(m) = E[w(m)wT(m)]. This derivation







is described in Appendix B, but the final result is listed below.

Rww(m + 1) = Rww(m) + Pw(m)iTd- lR.(m)R,,

+ trwT (m)d Rrr..ww(m) + 2rrd2
2p2o2 {w1 (m)rI + T (m) + w(m)T I} d
2A2 -r*T(m)rT d + 22a02Rww(m)rrT (5.3)

+ 22o2a4R1,(m) + P2a4 trace(Rl,,(m))I

+ l a22TpRu(m)I + 4A2a2 trace(Rl,,(m))rrT

2p2a2rrTR.w(m) + i2rrTRw,(m)rrT

Given the above equation, we can easily define the transient filter weight covariance
matrix using the following equation once we define w(0), R1,(O0), r and Rrr.

Cww(m) = R (m) *(m)wT(m) (5.4)

As an alternative to equation (5.3), we also consider using the transient weight
error covariance matrix which describes the deviation of the filter weights about the
Wiener-Hopf values as a function of time [31]. This is the same equation used by
Miller [32,33] to study the transient MSE response of the LMS filter. The transient
weight error covariance matrix is defined as

K(m) = E [(w(m) Wop0t)(w(m) woPt)T] (5.5)

and can be evaluated at the mth interval using the following iterative equation.

K(m + 1) = K(m) + p[RrrK(m) + K(m)R.,] + '2Rrr trace(RrrK(m))
(5.6)
+ p2 RrK(m)Rrr + A2JminRrr

In order to use equation (5.6) we note that K(0) = WptWoptw, when w(0) = 0, and
Jmin = 1 Woptali(Pi, 61). The transient weight covariance matrix can be found as
shown in equation (5.7).








C.(m) = E [(w(m) *(m))(w(m) w(m))T]
L J (5.7)
= K(m) w(m)*T(m) WoptWoTpt + (m)W Woptw T(m)


Several different DS-CDMA environments were simulated, and the transient mean
weight vector and transient autocorrelation matrix of the LMS filter weights were
observed at the mth interval of adaption using the following equations.
1 M
w(m) = ^w(m) (5.8)
i=i
1 M
Rw(M) = M w(m)wT(m) (5.9)
i=1
These quantities were then used to form an estimate of the transient autocovariance
matrix of the LMS filter weights as


Cww(m) = ,(n) W(m)W T(m). (5.10)

Of course, for each trial of a given test, the number of users, the spreading sequences
and propagation delays are fixed. The data sequences for the interfering users and
the AWGN were random during each trial, and independent from trial to trial. The

LMS step-size, [, was chosen such that the filter was convergent in the mean-square.
By comparing the results produced by the above equations to the corresponding
analytically derived quantities, we can in some sense measure the quality of the
analytical expressions.
For the first comparison, a single-user environment was simulated using 31 chips/bit
and a SNR of 10 dB, and 71 = 15.5Tc. A set of 3,000 independent trials was used
in computing equations (5.8) through (5.10). Figure 5.1 compares the IIE[w(m)]||
produced by equation (5.2) to the observed simulation results. In order to compare
these two quantities directly, the error is plotted as IIE[w(m)] w(m)|I in the second
part of the figure.









0.3

0.25

0.2

0.15

0.1


a) Magnitude of Mean vectors

r- -- -I -- -




IAf' 2 Of i


IU ZU JU V iU ou
b) Magnitude of Difference


Ar


IU 1ou U iU


0 10 20 30 40 50 60 70 80 90 1
# of Training Bits


Figure 5.1:


Performance of equation (5.2) for a single-user system, a) The analytical
mean filter weight vector norm, ||E[w(m)]|| (*), and the simulated mean
filter weight vector norm I|w(m)|I (solid line) plotted as a function of the
training length; b) Comparing the error between the two mean weight
vectors, IIE[w(m)] wA(m)11.


As a second test, a three-user system was simulated. As before the desired user

was received at a SNR of 10 dB and the two interfering users were received at a

power level 10 dB above the desired user. The interfering users had propagation

delays of Tr2 = 6.257Tc and T3 = 28.0125Tc. This environment was simulated using

3,000 independent trials. Figure 5.2 compares equation (5.2) to the simulation results.

Figures 5.1 and 5.2 show that the analytical expression for the transient mean weight

vector of the LMS filter given by equation (5.2) closely tracks the corresponding

simulation results. Several other multi-user scenarios with various levels of SNR and

MAI were simulated, and the performance of equation (5.2) was similar to those

shown in Figures 5.1 and 5.2. Therefore, we use equation (5.2) in the sequel when we

require use of an expression for the LMS filter's transient weight mean vector.


1
" 1(4


4


3.5

























0 20 40 60 80 100 120 140 160 180 200
# of Training Bits

Figure 5.2: Performance of equation (5.2) for a three-user system, a) The analytical
mean filter weight vector norm, IIE[w(m)]|| (*), and the simulated mean
filter weight vector norm IIw (m)|| (solid line) plotted as a function of the
training length; b) Comparing the error between the two mean weight
vectors, IIE[w(m)] wA(m)11.

A similar test was used to characterize the performance of equations (5.4) and (5.7),
which are the analytical expressions for the LMS filter's transient weight autocovari-
ance matrix. We observed that for a single-user environment, both equations gave
similar results. However when multiple-users were present, equation (5.4) initially
tracks the simulation results for C1w(m), but at some point diverges from the simu-
lation results. However, equation (5.7) was observed to produce similar results, when

compared to Cw,(mn), regardless of the number of users and the level of the MAI. A
two-user system with the desired user received at a SNR of 10 dB and the interfering
user received at a power level 10 dB above the desired user was simulated for 3,000
trials. The results of comparing the Frobenius norm of equations (5.4) and (5.7) to
the Frobenius norm of the simulated Cww,(m) matrix are shown in Figure 5.3. Note

that similar results were observed for other multi-user scenarios. We have observed








x 10-4
4

3K 3K *
3.5 yt I--------- '
3 W )K Method #1
Sxx x x X Method #2
3 '. "x x + Simulation Results

2.5 x

2

1.5

I

0.5 ''
0 20 40 60 80 100
# of Training Bits

Figure 5.3: Performance of equations (5.4) and (5.7) for a two-user system. The
Frobenius norm of the LMS filter's transient weight autocovariance ma-
trix; equation (5.4) (*), equation (5.7) (x) and simulation results (+).

that the Frobenius norm of the difference between equation (5.7) and the simulated

Cw(m) is on the order of 10-20% over several different multi-user scenarios.

We have derived a set of analytical expressions, equations (5.2) and (5.7), that

we will now use to describe the transient model of the LMS filter weights under the

assumption that the weights are jointly Gaussian. Since we are using the weights of

the filter to form a timing estimate of the desired user's propagation delay, we are

interested in how the transient response of the filter weights determines the quality of

the resulting timing estimate. In the next two sections, we use our analytical model

for the transient weight statistics, in an attempt to characterize the relationship

between the filter weights and the quality of the timing estimate.


5.2 Chip Selection Error Probability

We are interested in developing an analytical expression for the chip selection

error probability when the fractional part of the propagation delay is ignored. That








is, we want to find an expression for Pr(P1i pi) when the value of 61 is ignored. In

this case, the propagation delay estimate at the ith iteration of the adaptive filter is

formed as shown below.


Pi,=argmaxF(p) p E (0,1,2,...,N-1) (5.11)
p
P
F(p) = IwT(i)c1P)I2 (5.12)

Therefore, the chip selection error probability can be expressed using a union bound

[43] as

N-1
Pr(fi 5 Pi) = E Pr(F(pi) F(k) < 0). (5.13)
k=O
kt-pi
Note that each term in the previous equation can be expressed as

F(pi) F(k) = w(i) (cic (ck) w(i) (5.14)
(5.14)
= wT(i)Akw(i).

Using our assumption that the filter weights are jointly Gaussian, the expression in

equation (5.14) is just a quadratic form of Gaussian random variables. This type

of expression is common in radar applications, weapon control systems and is also

intrinsic to any discussion of variances of random variables [44]. So we now must

concern ourselves with how to evaluate Pr (wT(i)Akw(i) < 0). This type of evalua-

tion has received considerable attention by many researchers [45-50]. The next few

paragraphs will summarize various methods that may be used to handle this problem.

As shown in Appendix C, the distribution of wT(i)AkW(i) can be shown to have

the same distribution as

n
Y = Z i(Wi bi)2 (5.15)
i=
where the Wi are independent Gaussian random variables of zero mean and unit

variance and the terms Ai and bi depend on the transformation matrix Ak as well the






47

covariance matrix of the filter weights. Deriving the distribution of equation (5.15)

directly does not appear to be mathematically tractable. However, deriving the

corresponding characteristic function is relatively simple. For a random variable that

has a probability density function of fx(x), the characteristic function is given by

+00
x(w) = E[exp (jiwx)] =f fx(x) exp(jwx)dx. (5.16)
-00
Likewise, if one is given the characteristic function Dx(w) then the corresponding

probability density function is found using the inversion formula stated below.
+00
fX (x) x- (w) exp (-jwx) dw (5.17)
-00
The characteristic function for equation (5.15) is found to be

( 1 n 2) 1i n b2 n 1 (5 18
Y(w) = exp -2 b exp 2 1- 1(5.18)
2 i=1 \2 i=1 -2jwA/=1 v1 2jwAi

Johnson and Kotz [45] provide an extensive summary of techniques that can be
used to find fy(y) or the Pr(y < t), where t is some given threshold, when the transfor-

mation matrix Ak is positive definite. For specific restrictions on the bi and A, in the

characteristic function, several expansions of Dyu(w) into power series representations

are presented. Then, the inversion formula in equation (5.17) is applied to each term

in the series to produce fy(y). A matrix A is positive definite when xTAx > 0 for

any non-zero vector x. Therefore, if the transformation matrix Ak is positive definite

then a chip-selection error can never occur and, therefore, Pr(Y < 0) = 0. We must
determine if the transformation matrix is positive definite. A sufficient condition for

positive definiteness, is that all of the eigenvalues of the matrix Ak are positive [51].

Without actually computing these eigenvalues, we can test for the possible existence
of negative eigenvalues. It is known that the sum of the eigenvalues is equal to the

trace of the matrix. Since all of the diagonal elements of Ak are zero, there must








exist negative as well as positive eigenvalues and therefore the transformation matrix

is not positive definite.

Rice [46] provides several numerical techniques that can be used to evaluate

Pr(Y < t) through applications of equations (5.18) and (5.17) for a general trans-

formation matrix. However, it is noted that these techniques can exhibit slow con-

vergence due to the oscillatory nature of the exponent in equation (5.18). In order

to improve the convergence rate of the numerical integration techniques, a change of

variables that produces a tilting of the integration path is presented. However, the

change of variables is inversely dependent on the desired threshold t, and is therefore

not valid when t = 0, which is the case at hand.

A quadratic form of independent Gaussian random variables, such as shown in

equation (5.15), can also be shown to have the same distribution as a linear sum of

independent non-central X2 random variables [44]. That is, equation (5.15) can also

be represented by

r
Y = jXj+ aX0, (5.19)
j=1
where Xj are independent random variables, having non-central X2 distribution with

nj degrees of freedom and non-centrality parameter 6? for j = 1,..., r and X0 having

a zero-mean unit-variance Gaussian distribution. Davies [50], based on inverting

the corresponding characteristic function [49], presents a numerical technique that

can be used to find the distribution of a linear sum of independent non-central X2

random variables, when the original quadratic form has a general (not necessarily

positive definite) transformation matrix. The technique involves evaluating a series

of exponential terms, where the truncation error is dependent on the number of

terms included in the series. The number of terms required for the integration is

determined approximately by the total number of degrees of freedom and the sum

of the non-centrality parameters as well at the value of t, at which the distribution








function is to be evaluated. Davies presents a table that list the number of terms

required for integration as a function of these parameters. This table shows that as t

approaches zero, the number of terms required for integration quickly increases from

several hundred to tens of thousands.

Therefore, the techniques of Rice or Davies appear to be applicable to a situation
where the Pr(Y < 0) has to be evaluated very infrequently. Recall that at the
receiver's ith interval, we have to evaluate Pr(Y < 0) a total of (N-l) times to form a

bound on the chip-selection error probability. If we want to evaluate the chip-selection

probability as a function of the LMS filter's adaptation time, these techniques become

numerically prohibitive. That is, we found that Monte-Carlo simulations of a specific

DS-CDMA environment produce results for the chip-selection error probability in less

time than is required to evaluate the same quantity using one of Rice's techniques for
the same environment.

As a means of evaluating Pr(F(pi) F(k) < 0), we instead turn to the moment

generating function. Instead of trying to evaluate the desired expression explicitly,
we use a Chernoff bound to find an upper bound on Pr(F(pi) F(k) < 0) for each

term in equation (5.13) (see Appendix C for details). We then sum these (N 1)

Chernoff bounds to find an upper bound on Pr(pi 54 pl). An example of the resulting
bound using this method is shown in Figure 5.4. A three-user DS-CDMA environment

was simulated with a SNR of 10 dB, and both interfering users were received at a

power level 10 dB above the desired user. The propagation delays for the users were

71 = 15T7, T2 = 6T, and T3 = 28Tc. The LMS step-size, M, was set to 0.1/(the total
input power) such that the filter was convergent in the mean-square.

There are three curves present in Figure 5.4. The first curve is the simulated

results for the incorrect chip-selection probability, which is plotted using a plus sign

(+). The other two curves are the resulting upper bound for this probability, for
two different applications of the Chernoff/Union bound technique. The solid line,






50

represents the upper bound when equations (5.2) and (5.7) are used to describe

the transient mean vector and autocovariance matrix of the LMS filter weights as a

function of the training interval. Note that the resulting upper bound is not very

tight. It seems appropriate to wonder if the looseness of the bound is due to the

Chernoff/Union bound technique, or due to errors present in the model given by

equations (5.2) and (5.7). As a control for this test, the third curve (*) represents the

upper bound found by using the transient mean vector and autocovariance matrix

of the LMS filter weights obtained directly from the simulation. This is slightly

closer to the simulation results, but the upper bound is still very loose. Therefore,

is seems that the application of the Chernoff/Union bounds are the main reason
that the derived upper bound is very loose when compared to the actual simulation

results. Note that this behavior was observed for several other multi-user DS-CDMA

scenarios. Unfortunately, based on this result this technique is not very useful in

terms of characterizing the performance of the timing estimator.

5.3 Approximation for the Conditional Variance of the Timing Estimate

A traditional technique that is used to characterize the performance of an esti-

mator is to compare the variance of the estimate to the Cramer-Rao bound (CRB).

The CRB is a lower bound on the variance of any unbiased estimator for a given log-

likelihood function [52,53]. In this section, we will use the traditional CRB technique

along with several assumptions, in hopes of forming an approximation to the perfor-
mance of our timing estimator. Since we are interested in characterizing the transient

performance of our timing estimator, we base our technique on the time-dependent

statistical model for the LMS filter weights given in equations (5.2) and (5.7).

Since we have assumed that the filter weights are jointly Gaussian, equations (5.2)

and (5.7) are sufficient to completely define the time-dependent statistical model for
the LMS filter weight vector. We do not intend to use this model to derive a new

timing estimation algorithm. However, we use this model to derive a CRB, hoping













+++++
J ulf + + + + + +


10-3-



20 25 30 35 40 45 50 55 60
# of Training Bits
Figure 5.4: Comparing incorrect chip-selection probability to the Union-bound for a
three-user system. Simulation results (+), the Chernoff bound using equa-
tions (5.2) and (5.7) (solid line), and the Chernoff bound using simulated
weight vector statistics (*).

that the result can be used to characterize the conditional variance of our estimate
for 61. Using equations (5.2) and (5.7), the time-dependent log-likelihood function at
the mth iteration (ignoring constants) is

9(w161, m) = -1 [(w(m) *(m))TC (m)(w(m) w(m))]. (5.20)

The CRB for the above log-likelihood function is then given by the following equation
[52].
V{r "02 (wl( lm) i-1 (.1
Var( & _1) > t E --g 3-- ^ (5.21)

In order to evaluate the above expression, we require knowledge of how Cww(m)
depends on 6i, in order to evaluate the required partial derivatives. This does not
appear to be mathematically tractable since, as seen in equation (5.7), we need to
evaluate E[Ow(m)/c61wT(m)]. In order to continue with this approach, we assume








that OC,(,(m)/O6i = 0. Using this assumption, after straightforward algebraic ma-
nipulations, our approximation to the variance of the 61 estimate is given below.


E[( 61)2I, = P] > (-1) (1 Cw(m) 0 1 (5.22)

Several auxiliary equations and initial conditions are required in order to evaluate
equation (5.22). The required auxiliary equations are
w(m + 1) rrr o (m) 9
w6--) (m) + (I- ARrr) 1 + A (5.23)

Oar -cp ) + c+1) (5.24)
0+c1
9Rrr ar _T a QT
__r + r (5.25)
961 -9651 r 9S 1)

We hasten to stress that we have used the mechanics of the CRB technique to derive
equation (5.22). However, due to the various approximations used along the way our
result is not a lower bound, and is only intended to provide a rough approximation
to the performance of our estimator.
Several different DS-CDMA environments were simulated in order to verify the
usefulness of equation (5.22) as an approximation to the conditional variance of the
61 estimate. For the following results, the users' spreading codes were selected from
a set of Gold codes [5] with 31 chips/bit and the desired user was received at a SNR
of 7 dB. The propagation delay for each user was chosen from a uniform distribution
over the interval [0, Tb). Once the spreading codes and delays were selected, they
were considered fixed and the DS-CDMA environment was simulated for 2,000 trials.
Therefore, the variables that were considered random between the individual simu-
lation trials were the AWGN and the data sequences for the interfering users. Note
that the LMS step-size, p, was always chosen to be 0.1/(the total input power) such
that the filter was convergent in the mean-square.








10o0--

a: 1 User
ib: 5 Users
\ c: 2 Users
S i\,, d: 10 Users
0


-1


C
r a
b

10-21-----
0 20 40 60 80 100 120
# of Training Bits

Figure 5.5: Comparing the simulated conditional variance of the 61 estimate to the
analytically derived approximation in absence of near-far MAI.

Figure 5.5 shows the comparison between equation (5.22) (dashed line) and the

standard deviation of the 61 about 61, conditioned on P, = Pi (solid line) for several

different scenarios. In this figure, all interfering users are received at the same power

level as the desired user. The results shown in this figure look very promising. The

largest error between the simulation results and equation (5.22), which occurs for

10 users, is 6%. Unfortunately, the usefulness of equation (5.22) quickly disappears

as more users are added to the system, or the level of Pk/P1 is increased (or some

combination of these two conditions). As an example of this statement, Figure 5.6

shows how the approximation fails for several multi-user scenarios where all interfering

users are received at some level above the desired user. Upon review of the derivation

of the approximation, the weakness of the approximation appears to stem from the

assumptions that were made to facilitate the derivation, and not from equations (5.2)

and (5.7) as these have been observed to agree very well with simulation results.








10
a: 3 Users,N/F= 15dB
b: 5 Users, N/F = 20 dB
c: 7 Users, N/F = 15dB
U

10"-


Cl) -C



a b -. -
10-2 I I I I
0 100 200 300 400 500
# of Training Bits

Figure 5.6: Comparing the simulated conditional variance of the 61 estimate to the
analytically derived approximation in presence of near-far MAI.

5.4 Coarse Acquisition Performance
In this section, we present numerical results for the acquisition performance of the

timing estimator. We compare the performance of our timing estimator with a well

known conventional technique. One approach to timing estimation in a single-user

spread spectrum system is the correlator. The received signal is correlated with time

delayed versions of the known spreading code, and the timing estimate is given by the

amount of time delay that maximizes the correlation. The receiver forms its estimate

as shown below:

la(I 6) 11I2 1 M
(,,)=argmax I' ra(p- )r12M (5.26)
(p,b) I|ai(p,6)112 J M =-

where M is the number of r(m) samples observed before computing the timing es-

timate. As shown by the above equation, this technique also requires an all ones

training sequence. It is also known to be optimal for a single-user in the presence of

AWGN only, but can be sub-optimal in the presence of multi-access interference.








Our timing estimator will be computed by processing the weights of either an
LMS filter or RLS filter. For the LMS adaptation results, the LMS step-size P was

chosen to be 0.1/trace(R), where trace(R) is the total filter input power. With
this step-size, the LMS filter was convergent in the mean and also convergent in the
mean-square. As the metric for measuring the performance of the timing estimator,
we observed the average acquisition time of the timing estimator. The acquisition

time is defined as the number of training bits required so that the probability that the
timing estimate is within one half-chip of the true propagation delay of the desired
user is greater the 90 percent. That is, we recorded the smallest value of m such that


Pr (1i- Ti| < T,/2) > 90%. (5.27)

In addition, based on the discussion about the advantages of synchronization, we

were interested in observing the root mean-squared estimation error (RMSEE) given

correct acquisition.

RMSEE = VE [(f, r\)21 (1f rlTj
The value of the RMSEE gives a measure of how well the receiver could be synchro-

nized if the timing estimate was used to update the receiver's timing relative to the

desired user.
Each DS-CDMA environment was simulated for 500 independent trials. For each
trial, the user spreading codes were selected at random from a set of Gold codes using

31 chips/bit. For each trial, the propagation delay for each user was chosen from a

uniform distribution over the interval [0, 31), independent from the other users. The
users' phase-shifts were chosen from a uniform distribution over the interval [0, 27r),
independent from the other users. The desired user was received at a SNR of 7 dB.
Also at the start of each trial, the received power level Pk for each interfering user

was selected from a log-normal distribution that had a mean and standard deviation








of 10 dB. That is, Pk/Pi = 100/1 where is a Gaussian random variable with a mean

of 10 and a variance of 100. The log-normal power distribution was used to simulate

environments where different power levels exist due to shadowing or system power

control error. By choosing the simulation parameters at the start of each trial, and

running many independent trials for each environment, the output of this test is an

estimate of the number of training bits required to achieve coarse acquisition.

Figure 5.7 shows the coarse acquisition performance using the correlator, LMS

filter, and the RLS filter in forming a timing estimate as a function of the number

of users. As shown in this figure, for a few number of users the LMS based timing

estimator offers no advantages over the conventional correlator estimator. However

as the number of users increases, the LMS based timing estimator does perform much

better than the correlator. In addition, the maximum RMSEE for all of the points

on the LMS curve was 0.11To, which shows that the timing estimator does produce

a reasonable estimate of the desired users propagation delay. But with 15 system

users, on average 240 training bits are required to achieve coarse acquisition. It is well

known that the convergence rate of an LMS filter, which will affect the performance

of the timing estimator, is dependent on the ratio of the largest eigenvalue of R

to the smallest eigenvalue of R. As the level of near-far interference increases this

eigenvalue ratio quickly increases. This plot shows a weakness in using the LMS

adaptation algorithm, not a weakness in the timing estimator.

The results shown in Figure 5.7 for the RLS filter support the previous statement.

For the same DS-CDMA environment, the RLS filter based timing estimate performs

much better than the correlator or the LMS filter. The RLS algorithm requires an

initial positive definite estimate of the data autocorrelation matrix. For this work, the

soft-constrained initialization RLS algorithm was used, in which the initial estimate

of the data autocorrelation matrix is just a positive constant times an identity matrix.

The maximum observed RMSEE was 0.17Tc, which is only slightly worse than the








10 3


+
+
So102 +



10
1 0 -- ----""""""""""""'
|?. 10 Correlator
I + MMSELMS
-MMSE RLS


1 0 0 I
2 4 6 8 10 12 14 16
# Users

Figure 5.7: Average training bits required for correct acquisition as a function of the
number of system users in AWGN channel.

results for the LMS based timing estimator. With 15 users (50% system capacity)

the timing estimator can achieve coarse acquisition with 45 training bits, which is

not an unreasonably long training period. For 10 system users, the RLS-based tim-

ing estimator achieves acquisition about 6 times faster than the LMS based timing

estimator. Clearly, the timing estimator based on the RLS filter performs better, in

terms of acquisition time, then the timing estimator based on the LMS filter.

While it is interesting to observe the performance of the timing estimator as a

function of the number of system users, it is equally important to consider the near-

far resistance of the timing estimator. In other words, for a fixed number of system

users, we would like to see how the number of training bits required to achieve correct

acquisition depends on the level of the multi-access interference. For this test, the user

spreading sequences, phases and propagation delays were chosen using the methods

previously described. The desired user was received at a SNR of 7 dB. However,

the received power level Pk for each interfering user was selected from a log-normal












S 3 MM I-t iL +
a 10 ~'--'S


2. +
15 Users + +


101 2 2 Users


10 '
0 2 4 6 8 10 12 14

XindB

Figure 5.8: Average training bits for correct acquisition as a function of the level of
multi-access interference in AWGN channel.

distribution that had a mean and standard deviation of X, where X was allowed

to vary over the range of (0,13) dB. For a given value of K, the number of system

users, and a specific value of X, 500 independent trials were used to find the coarse

acquisition performance of the timing estimator. This scenario was used to simulate

an environment in which there is very loose power control, and the desired user is

received at a power level which on average is one standard deviation below the mean

of the power levels of all the interfering users.
Figure 5.8 shows the coarse acquisition performance of the 3 timing estimation

algorithms for 2 system users and 15 system users as a function of the the value of

X. From this figure it is seen that none of the timing estimators are truly near-far

resistant. Clearly, the correlator is very sensitive to the amount of near-far interfer-

ence. As was observed in the previous plot, the LMS-based timing estimator performs

better than the correlator, and the RLS-based timing estimator performs better than








the LMS-based timing estimator. For 2 users, the RLS based timing estimation algo-

rithm could be considered as near-far resistant. When the value of X was set to 0 dB,

only 5 training bits were required to achieve correct acquisition. When the value of

X was set to 13 dB, a much more severe near-far environment, only 7 training bits

were required. With 15 system users, the corresponding number of required training

bits changed from 10 to 75, which we consider as being not enough of an increase in

the training period to label the estimation algorithm as not being near-far resistant.

However, it is clear from this plot that the length of the training period required to

achieve correct acquisition in a system with a severe near-far problem when either

the correlator or the LMS based timing estimator is used is unacceptable.


5.5 Summary
In this chapter, we have attempted to characterize the performance of the timing

estimator. An equation for the transient weight mean vector of the LMS filter weights

was derived and tested under several DS-CDMA environments. Simulation results

have shown that this equation is in close agreement with the true transient mean of

the LMS filter weights. A second equation was derived and tested for the transient

weight autocovariance matrix of the LMS filter weights. This equation works well

for a single-user environment, but does not follow the observed response of the filter

weights for a multi-user situation. Therefore, the weight error covariance matrix was

used to derive a second equation for the transient weight autocovariance matrix. This

equation has been observed to agree within about a 10-20% error of what is observed

through direct simulation. We then used these two iterative equations to derive two

analytical techniques to characterize the performance of the timing estimator.

The goal of the first technique was to provide a means of observing how quickly

the timing estimator can correctly estimate the propagation delay of the desired user,

in a chip-synchronous environment. The idea was to use the analytical expression as








a means to observe the timing estimator, without running long Monte-Carlo simula-

tions, or at least to get a sense for how to set up such simulations. Unfortunately,

evaluating the required probability was a difficult task at best, so we used several

methods to form an upper bound on the desired probability. We observed that the

upper bounds were very loose, and so this technique while interesting is of very limited

use.

The second technique was more along the lines of the traditional Cramer-Rao

bound. We desired an analytical expression that could be used as an approximation

to the variance of the 61 estimate (about the true value of 61) conditioned on getting

the integer part of the timing estimate correct. We observed that the resulting ap-

proximation was useful for few users, with perfect power control, but was not useful

for more general cases.

Lastly, we characterized the performance of the timing estimator using Monte-

Carlo simulation methods. We compared the performance of the adaptive filter-based

timing estimator to a conventional correlator-based timing estimator. The correlator-

based timing estimator is well known to be optimal for a single-user operating in an

AWGN channel, but can be sub-optimal for a multi-user DS-CDMA environment.

The metric used to compare the timing estimator, was the average number of training

bits required such that fi was within one-half a chip of the true value of T1 > 90%

of the time. For few users, with low levels of MAI, we observed that the adaptive

filter-based timing estimator offered a slight improvement over the correlator-based

timing estimator. However, as the number of system users increases, both the LMS-

based and RLS-based implementations of the timing estimate perform better than

the correlator, with the advantage going to the RLS-based algorithm. The near-far

resistance of the algorithm was tested, by simulating an environment that could occur

in an open loop system, or a system with very loose power control. Neither the LMS-

based or the RLS-based estimation algorithms are truly near-far resistant. However,








the change in the length of the required training interval for the RLS-based algorithm

was so minor that we consider it to be near-far resistant.

While it is difficult to directly compare our timing estimator with other techniques,

as some of the other techniques estimate more than a single-user's parameters, one

observation can be made. The complexity of the estimation algorithm is dependent

on N, the number of chips in one period of a spreading sequence. The complex-

ity of the estimation algorithm is 0(N2) above the complexity of the traditional

correlator-based system when the RLS algorithm is used. However, note that for

some environments reliable estimates of rT1 are available with training sequences that

are shorter than N symbols long. In a previous chapter, other estimation algorithms

were presented. Most, if not all, had higher complexities and some required training

sequences that were at least N symbols long before a timing estimate could be made.

Therefore, our timing estimator has the advantage over these other estimators when

short training sequences are required, under the condition that our timing estimator

provides reliable estimates.













CHAPTER 6
SIMULATING FADING CHANNELS
Previously, the communication channel was modeled as an AWGN channel. A

more realistic channel model must be used to account for the dispersive effects ob-

served in urban environments. In urban environments, it is possible for multiple

versions of a transmitted signal to arrive at a receiver due to reflections off of ad-

jacent buildings, cars or other obstacles. If a total of L versions of the transmitted

signal are received, then the channel is said to have L paths. The signal received from

each path can experience its own time delay, phase shift and attenuation. Therefore,

at the receiver these L signals may add constructively or destructively, a condition

which is known as fading. If relative motion exists between the transmitter and the

receiver (or the reflecting obstacle is in motion), then the received signal is also ob-

served to have a shift in its carrier frequency. Also, due to relative motion between

the receiver and the transmitter, it is easy to conceive that the characteristics of the

channel change as a function of time and, therefore, a more realistic channel model

will account for the time varying effects observed on the received signal.

Rappaport [54] provides a summary of the various types of fading, as well as pre-
senting several techniques that can be used to simulate fading channels. Proakis [55]

also presents a characterization of fading channels and discusses the effect of fad-

ing on the performance of communication systems. The type of fading induced by

a communication channel is dependent on the time/frequency characteristics of the

channel with respect to to the time/frequency characteristics of the transmitted sig-

nal. Sklar [56] presents several block diagrams that summarize how these relationships

between the channel and the transmitted signal determine the type of fading induced

by the channel. In addition, Sklar presents a summary of traditional techniques that








have been used to mitigate the effects of fading [57]. In the remainder of this chapter,

we will briefly review the various types of fading and discuss how a fading channel

can be simulated.

Consider a multi-path communication channel where there are L distinct paths

with unique propagation delays T1 > T2 > .. TL-1 > TL. The quantity TL T1 is

defined as the excess delay spread. As expected, this multi-path delay spread causes

time dispersion in the received signal. In fact, if the excess delay spread is significant

when compared to the symbol interval, the channel can be viewed as inducing severe

inter-symbol interference (ISI). That is, one transmitted symbol will contribute to

several symbol intervals at the receiver. Since the number of paths, and hence their

corresponding propagation delays, are assumed to change as a function of time, the

excess delay spread is a random variable. The statistical characteristics of the excess

delay spread relative to the symbol interval along with the frequency characteristics

of the channel and transmitted signal determine if the fading is flat or frequency-

selective.

If a communication channel has a constant gain and linear phase response over

a bandwidth that is greater than the bandwidth of the transmitted signal, then all

frequency components of the transmitted signal's spectrum will be equally attenuated

by the channel. In this condition, the channel is said to be flat. However, due to the

time varying nature of the multi-path channel, the channel's impulse response and

hence its frequency response are time variant. In order to characterize the frequency

response of the channel, the coherence bandwidth is defined. The coherence band-

width of a channel denotes a bandwidth in which different frequency components

experience the same amplitude attenuation. The coherence bandwidth of a multi-

path channel is related to the excess delay spread in some manner. While no exact

relationship between these two quantities exists, there are several well known rules

of thumb [54]. Let the transmitted signal have a bandwidth, Bs, a symbol duration








Ts, while the channel has a coherence bandwidth Bc and root-mean-squared excess
delay spread of at. The following statements regarding classification of the fading

channel with respect to the transmitted signal will be described by Ts, in order to

describe a general communication system. With Ts used to denote the duration of the

transmitted symbol, we must remember that for a DS-CDMA system Ts is equal to

To, the duration of one-chip interval, when applying the following statements. If the

bandwidth of the transmitted signal is much less than the coherence bandwidth of the

channel, Bs < Be, and the symbol duration is much greater than the RMS excess

delay spread, Ts at, then the channel induces flat fading. If the bandwidth of the

transmitted signal is greater than the coherence bandwidth of the channel, Bs > Be,

and the symbol duration is less than the RMS excess delay spread, Ts < at, then the

multi-path channel induces frequency-selective fading. Note that flat fading channel

models are more prevalent since a frequency-selective fading channel can be viewed as

the composite sum of a multi-path channel where each path induces flat fading, but

the propagation delays between the individual paths are set such that the composite

channel induces frequency-selective fading.

Similar to the time dispersion induced by the multi-path delays, relative motion

between a transmitter and its receiver induces dispersion in the frequency domain.

Consider a single-path channel over which a constant amplitude and frequency sinu-

soidal signal is transmitted. If there is relative motion between the transmitter and

receiver then there is a frequency shift observed in the received signal. The maximum

observable frequency shift, fD, is known as the Doppler frequency or Doppler spread

and is given by

f V = (6.1)
fD = _f (6.1)








where v is the relative speed between the two devices in m s-1, C is the speed of light

in free space and fc is the carrier frequency in Hz. When a time-varying multi-path

channel is considered, the observed frequency shift becomes random and time-varying.

The relative speed between a transmitter and receiver determine the rate at which

the impulse response of the channel changes. In order to characterize this rate of

change the coherence time of the channel, Tc, is defined. The coherence time of the

channel is used to characterize the correlation of samples of the channel output. That
is, if two samples of the channel are separated in time by less than the coherence time

of the channel, they will be highly correlated (similar). As before, there are several

known rules of thumb that are used to relate the coherence time of a channel to

the Doppler spread. If the symbol duration is greater than the coherence time of

the channel, Ts > Tc, and the bandwidth of the transmitted signal is less than the

Doppler spread, Bs < fD, then the channel impulse response changes many times

during a symbol interval and the channel induces fast fading. If the symbol interval
is much less than the coherence time of the channel, Ts < Tc, and the bandwidth

of the transmitted signal is much greater than the Doppler spread, Bs > fD, then

the channel is essentially time-invariant when viewed over several symbol intervals.

In this case, the channel is said to induce slow fading.

The Rayleigh distribution is commonly used to characterize the time-varying
statistics of the envelope of the received signal in a flat-fading environment. It is

well known that the random variable z = Vx2 + y2 when x and y are independent

zero-mean Gaussian random variables with equal variances a2 follows a Rayleigh

distribution. The probability density function for a Rayleigh distribution is given by

r ex 2 0 < <
fR(r) = 0e h a (otrwi
0 otherwise.








If however there is a dominant path in the communication channel, such as the case
when a line-of-sight (LOS) path exists, the envelope of the received signal is modeled

by a Rician distribution and has a probability density function of

r exp r 2+A2) (AO 0,
f(r) ) 2a2 (6.3)
0 otherwise.

where Io() is the zero-order modified Bessel function of the first kind. Since the

Rayleigh distribution is just a special case of the Rician distribution, it is easy to
create a fading channel simulator that generates both distributions for the envelope
of the received signal.
The fading channel simulator used in this work is based on Clarke's [58] channel
model. In that work, the fading was due to the scattering of electro-magnetic waves in
such a way that all paths had the same propagation delay. This means that all paths

form one composite path, and since there is no delay spread the channel induces flat
fading. The received envelope was shown to be a random process whose amplitude
follows a Rayleigh distribution and has an autocorrelation function given by

R(r) = Jo(27rfDT) (6.4)

where Jo() is the zero-order Bessel function of the first kind. Therefore to simulate
a channel based on this model, we must create a random process whose amplitude is
Rayleigh distributed, in such a way that the autocorrelation of the random process
is similar to equation (6.4).
A third-order filter has an transfer function of

3
H(s) = (65Woo
(s2 + 2CwoS + w2) (s + Wo) (6.5)








and a corresponding impulse response of

h(t) = (exp(-at) [A sin(3t) B cos(/3t)] + C exp(-wot)) u(t) (6.6)

where

A Wo B= C WO
2v/1 -(2 2(1-

a =wo = o w/1 C2.

By setting wo and C based on the value of the Doppler frequency as shown below,


Wo= 1-rfD (6.7)
1.2
C = 0.175 (6.8)

the convolution of h(r) with h(-rT) closely follows equation (6.4) over the first major

lobe. This means that if a complex white Gaussian noise process is used as the input

to the third-order filter, the autocorrelation of the output process closely follows

equation (6.4) for values of r where samples of the process are highly correlated.

And hence, we use this third-order filter to simulate a flat-fading channel based on

Clarke's model.

Figure 6.1 shows a block diagram of a fading channel simulator based on the

third-order filter that can be used on a digital computer. The continuous-time filter

is converted to a digital filter where the sampling rate is specified, typically equal

to one bit interval. The input to the filter are samples of complex Gaussian noise,

whose real and imaginary parts are independent, have zero-mean and equal vari-

ances of 1/2. The block diagram accounts for the presence of an LOS, or specu-

lar component, such that the envelope of the simulator output can follow both the

Rayleigh and Rician distributions. Therefore the output of the channel simulator is








Samples of 3rd Order Sampled
Complex er (p)+ Flat Fading
Gaussian Noise Process


A(p)exp(j14) LOSor
LOS or
Specular Component

Figure 6.1: Flat-fading channel simulator.

a sampled complex Gaussian process having a mean of A exp(jq) and a variance of
a2 = E [|Ak exp(jOk) Aexp(j)12].

In Figure 6.1, two parameters are shown to be a function of a variable called p.

The ratio p = A/a is referred to as the Rician parameter and the pair (A, a) are
normalized so that the fading process has unit power.

1
a(p) p> 0 (6.9)
1+ p2
A(p) P p p>0 (6.10)
V/l+p2
A plot comparing the theoretical probability density functions of the envelope of

the fading process given in equations (6.2) and (6.3), to the those observed through

simulations for two values of p is displayed in Figure 6.2. As shown in this figure,

when p = 0 the envelope of the fading process is Rayleigh distributed, while for

other values the envelope of the fading process follows a Rician distribution. As a
final demonstration of the fading channel simulator, Figure 6.3 displays a typical

realization of a flat Rayleigh fading process. The magnitude of the envelope and

the phase of the fading process are shown when the carrier frequency was 1.8 MHz,

the data rate was 9600 bits-per-second (BPS), and the relative speed between the

transmitter and the receiver was 3 miles-per-hour (MPH). The third order digital
filter, as well as the output of the fading channel simulator where sampled at the

data rate.

























0 11 1,,_ I I I
0 0.5 1 1.5 2 2.5 3 3.5
r

Figure 6.2: Theoretical and simulated probability density function for the envelope
of the fading channel simulator output for two different Rician parameter
values.


2000 4000 6000 8000


2000 4000 6000 8000
Sample Number


10000


10000


Magnitude and phase of a slow flat-fading Rayleigh process generated
with a carrier frequency of 1.8 GHz, a vehicle speed of 3 MPH, and a
data rate of 9600 BPS.


-20 L
0


200


-200'
0


Figure 6.3:








Consider simulating a single-user DS-CDMA system operating in a fading channel

environment. For a given Doppler frequency, the fading channel simulator is created

and assume that both the third-order filter as well as the simulator output are sam-

pled at the symbol rate. In a non-fading channel the contribution to the receiver's

input during the mth symbol interval due to the user is given by J1(m) as shown
in equation (2.7). Multiplication of Ji(m) by the mth sampled output of the fading
channel simulator, A(m) exp(j0(m)), at each symbol interval will produce an input

to the receiver whose envelope experiences fading. In order to simulate a multi-user
fading channel DS-CDMA system, each user should have its own fading process that
is independent from all other users. We could implement K independent fading chan-
nel simulators, and then multiply Jk(m) by the output of the corresponding fading
channel simulator. However, a simpler approach was used in this work. A single
fading channel simulator was created for a specified Doppler frequency. Then a very

large sequence of samples of the fading process was created. Consider two samples
of this sequence, A(n) exp(jO(n)) and A(n + m) exp(jO(n + m)). We know that if m
is chosen appropriately then the correlation between the samples becomes negligible,
and therefore we can treat the samples as being independent. Using this idea, the

original sequence was partitioned into K sequences such that each sequence is con-

sidered to be independent of each other. That is, the sequence for the first user was
created using samples (1, 2,..., m), while the sequence of the second user was created

by using samples (m + 1, m + 2,..., 2m), and so on for each user.













CHAPTER 7
CODE ACQUISITION IN NON-STATIONARY ENVIRONMENTS

It is well-known that training an adaptive filter in a non-stationary environment

can be difficult [31]. Since our timing estimator is based on processing the weights

of the adaptive filter, a study of how the estimator is affected by such environments

is required. In this chapter, we will consider timing acquisition in two such environ-

ments. The first environment occurs due to non-ideal down-conversion of the received

signal due to frequency synchronization errors. When this condition occurs, the input

to the adaptive filter has a time-varying phase induced on it. The second condition

to be studied is when the received signal has experienced fading, in which case both

the amplitude and phase of the received signal are time-varying.

Barbosa and Miller [13] studied the adaptive detection of DS-CDMA signals in

fading channels. In particular, they studied the performance of the adaptive receiver

shown in Figure 4.1 operating in frequency non-selective fading channels. They found

that this receiver structure does not work in a Rayleigh-fading channel. When the

desired user's signal experiences a deep fade, the receiver frequently loses lock on

the desired signal. When the desired user's signal emerges from the deep fade, the

receiver may emerge into any of 3 possible states:

1. correctly locked on phase to the desired user's signal;

2. locked 180 out of phase to the desired user's signal; or

3. locked either in phase or 180 out of phase to any of the interfering user signals.

In a single-user environment, only the first two of these conditions are possible. The

first condition is desired. The second condition is not due to the filter's inability to

track the rapid phase changes during a deep fade. It is due to the fact that since the

decisions are unreliable in a deep fade (many errors occur in the feedback loop) the








filter is essentially running "blind" during the deep fade. Once the filter locks onto the

desired user's signal at 180 out of phase, errors will continue to occur. This problem

can be handled by differential encoding and decoding of the data sequence. The third

condition is catastrophic, and is what renders this receiver structure useless in this

type of environment. Barbosa and Miller proposed a modified receiver structure to

alleviate these problems. The next few paragraphs describe this receiver structure

and its derivation.

The main problem with the adaptive receiver structure shown in Figure 4.1 seems

to be loss of phase lock during deep signal fades. Therefore, it stands to reason

that if reliable estimates of the fading process can be made, these phase variations

could be removed from the input to the adaptive filter, Then, maybe the adaptive

receiver could perform adequately, even in a Rayleigh-fading channel. In a near-far

environment, the signal-to-interference-plus-noise ratio for the desired user's signal

would probably be too small to allow for phase estimation without a complicated

estimation procedure. However, the filter suppresses the multi-access interference

such that the signal-to-interference-plus-noise ratio at the filter output is higher than

the same quantity at the filter input. Therefore, the filter weights are used in the

channel estimation procedure.

By taking the real part of the received signal prior to entering the adaptive filter,

the weights of the filter are real and, therefore, the filter makes no attempt to track

the phase of the received signal. Noisy estimates of the amplitude and phase of the

fading process during the mth bit interval for the desired user may be formed as

7y(m) = di(m)wT(m)r(m). The weights of the filter are updated at the bit rate,

which means that -y(m) can be updated at the bit rate. Therefore, we must predict

the channel conditions for the current bit interval based on previous values of "y(m).

An Lth-order linear predictor is used to form the current estimate of the channel








conditions, based on past estimates as shown in equation (7.1),

L
=(m) = (m i) (7.1)
i-i

and the estimate of the phase during the mth bit interval is found by 0i,m = Z/(m).

The coefficients of the Lth-order linear predictor are chosen to minimize the mean-
squared error E[laIm exp(j'i,m) (m)12], where a,1,m and 01,m are the amplitude and

phase induced on the desired user by the channel during the mth bit interval. The

solution of these coefficients depends on the weights of the MMSE filter, which depend

on the multi-access interference. In order to remove this dependence, the coefficients

were calculated for a single-user system, and then are used for all other environments.
Not surprisingly, the coefficients of the Lth-order linear predictor, a, are found by

a = C-1v (7.2)


where


{v}i = 1 (i,7rfDTb)2 i {1,2,..., L} (7.3)

{B},,j = 1 ((i j)rfDTb)2 ij E {1,2,..., L} (7.4)

C = B + (Eb/A/o)-I1 (7.5)

and I is an L x L Identity matrix and fD is the maximum Doppler frequency induced

by the channel. It is important to remember that these equations were derived for

a single-user and, therefore, in the presence of multi-access interference the channel
estimator may be very far from optimal.

The modified form of the adaptive receiver, using the Lth-order linear predictor
designed by Barbosa and Miller (equations (7.2) through (7.5)) is shown in Figure 7.1.

Now that we have presented the background of this receiver, we summarize several








C1
w(m) Estimation Ti
wm) algorithm

Convert to
r(t) baseband Adaptive nnn ci
i r+ chip x Rebos filter, w(m) tepo-
matched (eco
filter j -?+ 6

--__Z Adaptive e(m)
algorithm

Channel phase 44 di (M)
estimator 4 w(m)

Figure 7.1: Modified adaptive receiver with channel compensation.

important results presented by Barbosa and Miller [13]. They found that the perfor-

mance of the receiver (detection) is insensitive to L, the order of the linear predictor.

Also, unlike the unmodified form, the performance of the detector is independent of

the rate of the fading process as the detector performed equally well in slow-fading as

well as fast-fading. However, the most important result is the effect of the modified

receiver structure on the system capacity. For a given level of probability of bit error

rate, it was found that the structure in Figure 7.1 could support more users than the

conventional receiver (which is a correlator). In fact, it was shown that the system

capacity when using the modified receiver structure can be made to approach 100%

(N users), while the conventional receiver typically restricts the system capacity to

on the order of 10-20%.

Several comments about the modified receiver structure as it relates to timing

acquisition are required at this point. Barbosa and Miller were interested in detection,

and the weights of the adaptive filter were initialized close to their steady-state values.

The result is that the Lth-order linear predictor can make reliable estimates of the

channel conditions at the start of their simulations. In this work we are interested in

timing acquisition, and therefore the weights of the adaptive filter are initialized to







a vector of all zeros. This means that most likely the initial estimates of the channel
phase will be incorrect. As seen in Figure 7.1, the phase compensation operation,
Re [r(m) exp(-j1,m)j, causes a delay in the build of energy in the filter since when
k1,m 01(m) is significant, the phase compensation operation rejects some of the
desired user's input signal from the input to the adaptive filter. The time to achieve
correct acquisition will be dependent in some manner on the transient response of
the channel estimation algorithm.

7.1 Frequency Synchronization Errors
Consider the situation that occurs when the frequency of the kth user's transmit-
ting oscillator is not equal to the frequency of the receiver's oscillator. In this case
after down-conversion, the input to the adaptive filter for the kth user will have a
linearly time-varying phase. In order to model this effect, we assume that the fre-
quency offset for the kth user, Awk, can be expressed as Awk = 27r/(NTcMk) where
Mk is a positive integer and NTc is the bit interval.
Using this notation, we can model the input to the adaptive filter using equa-
tion (2.6) once we re-define Jk(m) as shown below.

Jk(m) .NMk exp 27rm\ ]
Mkr (7.6a)
[Z2k-1 (m)v2k-1 (Pk,tk) + Z2k(m)v2k(pk, 6k)

where the nth components of v2k-1(pk, 6k) and V2k(pk, 60k) are given by:

/ c \ j27rn N
V2k -l,n(Pk, 6k) = exp N-- X
\NMk /
{ exp 0Z2r d 1c(Pk+1)
{ NMk ) k,n (7.6b)

+ [exp ( 27_ -exp (J2"']c(Pk)
NMk exP {NMk ] c k,,







t j21rn
V2k,n(Pk, 60k) = exp N X

ex 1] NM&) k (7.6c)
-NMJk 1 k )
[ (j27r (J275k] e(Pk
+[exp (KNMk)- exp ~NMkJ k,n

As a sanity check of these equations, note that in the limit Mk -+ 00 the above
equations reduce to the previous definition of Jk(m).
The results in Figure 7.2, show the acquisition performance of the estimation
algorithm in the presence of frequency synchronization errors. The desired user was
received at a SNR of 10 dB, and 4 interfering users were each received at a power level
10 dB above the desired user. The desired user was assumed to have a frequency offset
of 1% of the data rate (Mi = 100), while all other users had a frequency offset of 2%
of the data rate (M2-5 = 50). The timing offsets and initial phase-offsets for each user
were chosen at random, for each of the 250 trials. For each trial, 3 independent RLS
filters were run in parallel. The first filter was uncompensated, while the second filter
used the modified receiver structure with a 10th-order linear predictor to estimate
the channel phase. As a control, the third filter was compensated with the known
phase induced on the desired user at each bit interval. By doing this, the effects of
the transient response of the channel estimation algorithm on the timing estimator
can be avoided.
As shown in Figure 7.2, the uncompensated filter was initially able to form a good
estimate of propagation delay for the desired user. However, the filter was unable
to track the phase of the input signal and therefore the timing estimation algorithm
eventually breaks down. When the modified receiver structure is used, and the input
signal is compensated using perfect knowledge of the phase induced on the desired
user, the timing estimation algorithm works very well. Once again, this is due to the
fact that the transient response of the channel estimator and the phase compensation
operation are not causing a delay in the build-up of the desired user's signal in the










1-

S0.8

VI
0.6

0.4

0.2

0-
0


Figure 7.2: Acquisition performance of the adaptive receiver
ment in presence of frequency offset errors.


in a five-user environ-


weights of the adaptive filter. The 10th-order linear prediction compensated filter,
unlike the uncompensated filter, is able to achieve correct acquisition at the cost of

an increase in the length of the training interval.

Considering the results of Figure 7.2, it seems reasonable that a hybrid of the

uncompensated and compensated filters may be useful in forming a timing esti-

mate. That is, it may be possible to start acquisition using an uncompensated filter

(with complex weights) and at some point switch to the compensated filter (with real

weights) such that we get better acquisition performance. We will address this idea

in the next section where we study the performance of the acquisition algorithm in a

flat-fading channel.


50 100 150
Time (Symbols)


SCompensated, using perfect knowledge
-of phase induced on desired user.

/J\ Compensated,
S10th order linear
Prediction




4- Uncompensated


200


250


300








7.2 Performance in Flat-Fading Channels

We model the input to the adaptive filter using equation (2.6), by defining the
contribution of the kth user during the mth bit interval as


Jk(m) = ak(m) exp(jOk(m)) [z2k-1(m)a2k-1(pk, 6k) + Z2k(m)a2k(pk,S k)] (7.7)

where ek (m) and Ok (m) are the amplitude and phase induced on the kth user's signal

by the fading process during the mth bit interval. The fading process for each user is
assumed to be independent of the fading process induced on each of the other users.

The fading parameters, ak(m) and 0k(m), are taken from samples of the flat-fading

channel simulator described in the previous chapter.
As a first test of the acquisition algorithm, a ten-user DS-CDMA environment
was created with a SNR of 10 dB and all interfering users were received at the same
power level as the desired user (in the absence of fading). A slowly flat-fading channel
was created for each user, using a carrier frequency of 1.8 GHz, a data rate of 9600
BPS, and a speed of 3 MPH. These parameters are commonly used to simulate a

personal communication system (PCS) environment in which the users are mobile at
walking speeds. The resulting Doppler frequency, fD, is 8 Hz which is a normalized
rate (normalized to the data rate) of 8.4 x 10-4.

To test the acquisition algorithm in this environment, a bank of four parallel RLS
filters was used. One filter, was uncompensated as shown in Figure 4.1, and processed

the received signal using complex weights. Two of the filters, used the architecture
shown in Figure 7.1 with a Lth-order linear predictor for the channel estimator. In
the following, we will refer to each of these filters as the modified MMSE receiver.
One of the modified MMSE receivers compensated the received signal using only

the phase estimate of 7(m) (as shown in Figure 7.1). The second modified MMSE
receiver compensated the received signal using both the amplitude and phase of 7(m).
The fourth filter, referred to as the hybrid receiver, was an ad-hoc mixture of the








two receiver structures. The idea of the hybrid receiver is to jump-start the channel

estimation algorithm by pre-loading the adaptive filter's weight vector, w(m), and the

linear predictor vector, 7, hopefully bypassing some of the learning curve required to

form reasonable channel estimates otherwise. The hybrid receiver uses the structure

of Figure 4.1 for 20 bit intervals without making timing estimates. After the 20th bit

interval, the receiver forms an estimate of r1 and then makes an estimate of 01 using

equation (4.14a). This phase estimate is loaded into the L-elements of 7, and the

imaginary parts of the filter's weight vector and the inverse of the correlation matrix

are removed. The receiver then switches to the modified receiver structure shown in

Figure 7.1, and starts the timing and channel estimation algorithms using the (now

real) weights of the adaptive filter.

The results of this test when a third-order linear predictor was used for the channel

estimation algorithm are shown in Figure 7.3. As seen in this figure, the acquisition

algorithm based on processing the weights of the uncompensated filter performs bet-

ter than any of the other techniques. In general, this situation will not occur as the

level of the MAI changes and/or the Doppler frequency changes, and results similar

to that shown in Figure 7.2 can be expected. Barbosa and Miller [13] observed the

detection performance of the modified MMSE receiver was insensitive to L, the order

of the linear predictor used in channel estimation. In addition, they compensated the

received signal by only using the phase of the channel estimate, '(m), as no advan-

tage was observed by also using the amplitude of 7(m) to compensate the received

signal. Since they were interested in detection, they did not observe the start-up

transient of the channel estimation algorithm as the weights of the adaptive filter

were initialized to their steady-state values. In Figure 7.3, we observe the effects of

the start-up transient of the channel estimation algorithm on the performance of the

timing estimation algorithm for the modified MMSE receiver. Unlike the observations

of Barbosa and Miller, it is observed that including the channel amplitude estimate











0.8 -X ^
-.0.8 / T Uncompensated

S0.6 Hybrid
V1
t Modified MMSE
k 0.4 (phase)


0.2- Modified MMSE
(phase and amplitude)


0 100 200 300 400 500
Time (Symbols)

Figure 7.3: Performance of the acquisition algorithm in a slowly flat-fading Rayleigh
channel using a third-order linear predictor for the channel estimation
algorithm.

in the compensation operation imposes a severe penalty in the performance of the

timing estimate when compared to the performance of the timing estimate when only

the channel phase estimate is used. To study the effects of the value of L on the

timing estimation algorithm, the same DS-CDMA environment was observed when

a tenth-order linear predictor was used in the channel estimation algorithm. These

results are shown in Figure 7.4. Once again, unlike the observations of Barbosa and

Miller, the performance of the timing estimation algorithm is sensitive to the value of

L. When a tenth-order linear predictor is used in the channel estimation algorithm,

the timing estimation algorithm performs better than when a third-order linear pre-

dictor is used as noted by the difference between the rates of convergence of the two

probability curves in Figures 7.3 and 7.4. For a variety of DS-CDMA environments

(Doppler frequency, MAI, and number of users), it has been observed that no signif-

icant improvement in the performance of the timing estimator is achieved by using

a higher order linear predictor in the channel estimation algorithm. Therefore, in








1 1


0.8


,0.6
vi J Modified MMSE
/ J (phase)
S0.4

kf ~Modified MMSE
0.2 (phase and amplitude)

I Hybrid
OI I II
0 100 200 300 400 500
Time (Symbols)

Figure 7.4: Performance of the acquisition algorithm in a slowly flat-fading Rayleigh
channel using a tenth-order linear predictor for the channel estimation
algorithm.

the sequel when we refer to the modified MMSE receiver we are using a tenth-order

linear predictor and only the phase of the channel estimate is used to compensate

the received signal. As seen in Figures 7.3 and 7.4, even-though the hybrid receiver

has been formulated in an ad-hoc fashion the resulting performance of the timing

estimator clearly shows that the idea has its merits.

In order to further test the hybrid receiver, a study of the effects of the level of

MAI on the performance of the timing estimation algorithm similar to that shown

in Figure 5.8 was performed. The carrier frequency was 900 MHz, the data rate was

9600 BPS, and the relative speed between the receiver and all of the system users was

assumed to be 65 MPH. This scenario is used to depict a DS-CDMA system in which

the users are traveling at highway speeds. The Doppler frequency induced by the

flat-fading channel was 87 Hz, which is a normalized Doppler rate of 9.1 x 10-3. Each

DS-CDMA environment was simulated for 500 independent trials. For each trial,

the user spreading codes were selected at random from a set of Gold codes using










.200- ".
200 10 Users
^ ^^ /^.Hybrid
1-
=,100 srMdfe MME



5 3 Users, Hybrid

0 I I I I
0 2 4 6 8 10 12 14
XindB
Figure 7.5: Comparing the average training bits for correct acquisition as a function
of the level of multi-access interference in a flat-fading channel.

31 chips/bit. For each trial, the propagation delay for each user was chosen from a
uniform distribution over the interval [0, 31), independent from the other users. The
desired user was received at a SNR of 10 dB. Also at the start of each trial, the received
power level Pk for each interfering user was selected from a log-normal distribution
that had a mean and standard deviation of X dB. That is, Pk/P1 = 100/1 where
is a Gaussian random variable with a mean and standard deviation of X dB. By
varying the number of system users and the level of the MAI, it is possible to study
the performance of the hybrid receiver in conditions where the uncompensated filter
shown in Figure 4.1 doesn't perform well, but the hybrid initialization technique may
improve the acquisition performance over that offered by just the modified MMSE
receiver alone.
The results of this test are shown in Figure 7.5 for a three-user and ten-user DS-
CDMA system where the level of the log-normally distributed interference (X) ranged
from 0 dB to 13 dB. For both the three-user and ten-user systems, the hybrid receiver
performs better than the modified MMSE receiver in terms of the average training











0.8


^0.6
Vi 0 % Uncompensated

U 0.4

0.2


0 IL
0 200 400 600 800 1000
Time (Symbols)

Figure 7.6: Performance of the timing acquisition algorithm using the hybrid and
modified MMSE receivers.

bits required in order to achieve acquisition to within a half-chip of Ti. When the

hybrid receiver was used in the three-user environment, the performance of the timing

acquisition algorithm appears to be independent of the level of the MAI. This result

is due to the length of the training cycle used in the hybrid receiver before timing

estimates are actually made (20 bits). For a three-user system, the uncompensated

filter is able to make reliable timing estimates prior to 20 training bits and, therefore,

it may be possible to use a shorter training interval for the hybrid receiver initial-

ization cycle without degrading the performance of the timing acquisition algorithm.

The length of the training cycle used to initialize the hybrid receiver is sensitive to

the number of system users and the level of the MAI. However, over the range of users

and log-normally distributed MAI depicted in Figure 7.5, 20 bits of training provide

the hybrid receiver with a definite advantage over the modified MMSE receiver in

terms of timing acquisition performance.








A situation in which the hybrid receiver offers no advantage over the modified

MMSE receiver is shown in Figure 7.6. This result is for the same DS-CDMA envi-

ronment tested in Figure 7.5, however there are fifteen system-users and X = 1 dB for

all of the log-normally distributed interfering users. This plot readily demonstrates

how the performance of the timing acquisition algorithm using the hybrid receiver

is "modulated" by the performance of the uncompensated filter and the length of

its training cycle before switching to the modified MMSE receiver structure. While

decreasing the length of the training cycle below 20 bits might not affect the per-

formance of the timing acquisition algorithm, certainly increasing the length of the

training cycle to 100 bits would be disastrous.


7.3 Summary
Barbosa and Miller [13] studied the performance of the MMSE receiver in a flat-

fading channel, and discovered that the MMSE receiver doesn't work in a fiat-fading

Rayleigh channel. They proposed a modified receiver structure which compensates

the received signal such that the adaptive filter doesn't attempt to track the phase

of the desired user's signal. In this chapter, the modified MMSE receiver structure

was used to study the performance of the timing acquisition algorithm in two en-

vironments in which the input to the adaptive filter was non-stationary. While the

timing acquisition algorithm is applicable to use with the modified MMSE receiver, it

has been observed that the start-up transient of the channel estimation algorithm de-

grades the performance of the timing estimation algorithm by increasing the length

of the training interval required to achieve correct acquisition. A hybrid receiver

structure has been presented to circumvent some of the start-up transient response of

the channel estimation algorithm. The hybrid receiver uses a short training interval

to form an estimate of the phase induced on the desired user, 01, and then uses the

modified MMSE receiver with the channel estimator pre-loaded with 91 to continue

the timing acquisition cycle. Through simulation results of a DS-CDMA system, it






85

has been observed that the timing acquisition algorithm based on the hybrid receiver

structure can offer an improvement over that of the modified MMSE receiver. How-

ever, the initialization cycle for the hybrid receiver is based on an uncompensated

MMSE receiver and therefore with an inappropriate choice of the length of the initial

training cycle, it is possible for the performance of the timing acquisition algorithm

for the hybrid receiver to be worse than that offered by the modified MMSE receiver

alone.














CHAPTER 8
INCREASED WINDOW OF OBSERVATION AND TIMING ACQUISITION

In previous chapters, the adaptive receiver was used to form a timing estimate of

the propagation delay for the desired user. Since the receiver only observed one bit

interval of information at a time, the desired user is forced to use an all ones training

pattern during the code acquisition cycle. However, it is possible to use a known

sequence of ones and zeros to train the adaptive receiver, if the receiver observes more

than one bit interval of information at a time. By letting the observation window of

the receiver be two bit intervals, a full data bit (and possibly two bit transitions) are

guaranteed to occur within the observation window. This increase in the observation

window also allows for more than one user to be in the code acquisition mode at the

same time.


8.1 Updated System Model

In this section, the system model for an observation window of two bit intervals will

be presented. This model is just a special extension of that presented in Section 2.1,

however a notation similar to that presented by Madhow [59] will be used. As with

the previous system model, the receiver converts r(t) to a baseband signal and passes

R(t) through a filter matched to the chip pulse shape and the output of the filter is

sampled at the chip rate. As before, we let the quantity rm,n represent the nth chip

sampled output during the receiver's mth bit interval.

mTb+(n+1l)Tc
rmn -:-- R R(t)dt E C. (8.1)
r2- mTb+nTc
mTb nT,







During each bit interval, a total of N chip samples are gathered and grouped with
the N chip samples from the previous bit interval such that

r(m) = (rm-i,o, rml,, I .. r-1,N-1, rm,o0,rm,1, ..., tim,N-i) (8.2)

and it is this vector quantity that will be used as the input to the adaptive receiver.
As with the previous system model,
K rfkC2,
r(m) = E Jk(m)exp(JOk) +Nm E C2NX1 (8.3)
k=1l P

where Jk(m) is the contribution to the received vector from the kth user during the
mth receiver bit interval. Let the dk,m represent the data bit for the kth user that falls
completely in the two bit observation interval during the receiver's mth bit interval.
Unlike before, we now define the propagation delay for the kth user to be Tk E [0, Tb]
such that Trk = pkTc + kTc with Pk E {0,1,..., N 1} and 0 < 6k < 1, or Tk = NTc.
Let ak represent a vector of length 2N consisting of the N elements of the spreading
sequence for the kth user followed by N zeros.

ak = (Ck,o, ck,1,. .., Ck,- 1, 0,..., O)T (8.4)

Let T(L) represent the nth left acyclic shift of a vector, while T() represents the nth
right acyclic shift of a vector. Both functions operate on and produce vectors with
2N elements. Using a general vector x = (X0o, X1,. .. X2N-1)T these functions are
defined below.

T()L (x) = T() (x) = x (8.5)

(X)= (xl, X2,..., x2 1,0) (8.6)


TL(-1)) (X2N-I,..., O) (8.7)

Tl(x) = (0,xo,x,... x2-2T (8.8)










T(R2N-1)(X) = (0,...,,) (8.9)
T(L(X) T(x) (0,...,0)T V n > 2N (8.10)

Using the above definitions we can now define the kth user's contribution to the
received vector as

Jk(m) = dk,m-lVk1 + dk,mV + dk,m+lV (8.11)

where

Vk1 (1 6k)TW -(ak) + kT. -- (ak]) (8.12)

V = (1 6k)TR (ak) + kR +(ak) (8.13)

vk = (1 4k)Tw Pk)(ak) + ikTR+Pkl(ak) (8.14)

Using this notation, the autocorrelation matrix of the adaptive filter's input and the
steering vector are evaluated as shown below.

R E [r(m)r H(m)]

1 ( + V0(V)T + V(V) ) (8.15)
k=l1
+ 2or2I

p = E[r(m)dl,m] = v exp(j0,) (8.16)

8.2 Updated Timing Estimation Algorithm
We are interested in developing a new timing estimator for the propagation delay
of the desired user, now that we have increased the length of the filter. We will use
the same procedure as before. That is, the weights of an MMSE filter for a single-
user will be used to form the timing estimator. Therefore, we need to calculate the
weights of the optimal filter for a single-user operating in an AWGN channel. In







order to study the weights of the Wiener-Hopf filter, we consider the case where we
apply the equivalent phase-synchronous model for the equations listed above. As in
the other system model, this means the filter weights become real and the complex
exponential term is removed from the steering vector expression. In this case, the
optimal weight vector is found using the Wiener-Hopf equation with Rwopt = v,
where we let Wopt = avj-1 + bv + cv1.
The coefficients of the optimal weight vector, a, b and c, are given by the solution
to the following equation.
[ + iv-1]2 (vIV-1)Tv (v-)Tv a 0
(vi-v0 + IIV112 (v?) 1 b = 1 (8.17)
(vi-a) v(0v ) v2
(v1)Tvi1 (vl)TV v 02 + 1v11J10

Using the above equation, we see that the a and c coefficients can only be zero when
both vi-1 and v1 are orthogonal to v. This condition occurs when 61 = 0. However,
in the general case when 6j # 0, the a and c coefficients will be much smaller in
magnitude than the b coefficient, since v1 and v1 are only non-zero in one chip-
position where v? is non-zero also.
Based on this result, we will ignore the contributions of the vi1 and v, vectors on
the Wiener-Hopf weights when we develop the timing estimator. Using the weights of
the adaptive filter, we estimate the propagation delay for the desired user by finding
the set (/3,O,p,6) that minimize ||w(m) /3v exp(j0) l2. As in the case where the
receiver only observes one bit of information, the estimate of -1 is given by the set
{p, 6} which maximize the cost function f(p, 6) given below.
f(p,) = v0)Tw(m) (8.1)2
= ilV 112 (8,18)

Maximizing f(p, 6) with respect to 6 produces a quadratic equation whose coeffi-
cients depend on the value of p. Therefore for each possible value ofp E {0, 1,..., N-
1} we must find the roots of the quadratic equation D2 (p)2 + D, (p)p + D0o(p), where




Full Text

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