CODE TIMING ESTIMATION IN DIRECTSEQUENCE
CODEDIVISION MULTIPLEACCESS
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 inlaws, 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 SpreadSpectrum 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 DSCDMA System Model for the
AW GN Channel . . . 9
2.2 Simplified Asynchronous DSCDMA System Model for the
AW GN Channel . . ... .. 12
3 REVIEW OF THE LITERATURE . ... .. 14
4 THE MMSE RECEIVER AND SINGLEUSER 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 NONSTATIONARY ENVIRONMENTS 71
7.1 Frequency Synchronization Errors ................ 75
7.2 Performance in FlatFading 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 spreadspectrum signals . . .. 3
4.1 Code acquisition with an adaptive receiver . .... .. 22
4.2 Minimum meansquared error as a function of the propagation delay
of user # 1 . . . . .28
4.3 Plot of the cost function f(p, 6) for a singleuser system ........ .. 33
4.4 Plot of the cost function f(p, 6) for a fifteenuser system ........ .. 34
5.1 Performance of equation (5.2) for a singleuser 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, IE[w(m)] wA(m)I . ... .. 43
5.2 Performance of equation (5.2) for a threeuser system, a) The analyti
cal mean filter weight vector norm, IIE[w(m)]II (*), and the simulated
mean filter weight vector norm IIw(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 twouser 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 chipselection probability to the Unionbound for
a threeuser 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 nearfar MAI. 53
5.6 Comparing the simulated conditional variance of the 61 estimate to the
analytically derived approximation in presence of nearfar 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 multiaccess interference in AWGN channel . ... .. 58
6.1 Flatfading 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 flatfading 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 fiveuser environ
ment in presence of frequency offset errors . ... .. 77
7.3 Performance of the acquisition algorithm in a slowly flatfading Rayleigh
channel using a thirdorder linear predictor for the channel estima
tion algorithm . . . .. ..80
7.4 Performance of the acquisition algorithm in a slowly flatfading Rayleigh
channel using a tenthorder 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 multiaccess interference in a flatfading 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 fifteenuser system when the
receiver observes two bitintervals . ... .. 91
8.2 Comparing the performance of the timing estimator when the receiver
observes 2bit intervals (*) versus observing 1bit 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: biterrorrate
BPS: bitspersecond
BPSK: binary phase shift keying
CRB: CramerRao bound
CTAN: Comprehensive TEJX Archive Network
DSCDMA: directsequence codedivision multipleaccess
DSSS: directsequence spreadspectrum
FHSS: frequencyhopped spreadspectrum
IEEE: Institute of Electrical and Electronics Engineers, Inc.
ISI: intersymbol interference
LFSR: linearfeedback shiftregister
LMS: least mean squared
LOS: lineofsight
LPD: low probability of detection
LPI: low probability of intercept
LSML: largesample maximum likelihood
MAI: multiaccess interference
MASE: multipleantenna sensorsbased estimator
MMSE: minimum meansquared error
MOE: meanoutputenergy
MPH: milesperhour
MSE: meansquared error
MUSIC: multiple signal classification
PCS: personal communication system
PSD: power spectral density
RASE: rapid acquisition by sequential estimation
RLS: recursive leastsquares
RMSEE: root meansquared estimation error
SNR: signaltonoise 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 DIRECTSEQUENCE
CODEDIVISION MULTIPLEACCESS
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 directsequence
codedivision multipleaccess communication systems operating over additive white
Gaussian noise and flatfading 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 nearfar problem. The nearfar problem is a con
dition which occurs when the received signal amplitudes of the the multiple system
users are very dissimilar.
A wellknown receiver, referred to in the literature as the minimum meansquared
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 singleuser operating in an additive white Gaussian
noise channel, but is suboptimal in a multiuser scenario. However, the IMMSE
receiver has been observed to be resistant to the nearfar 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 singleuser'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 nearfar problem. The complexity of the proposed
timing algorithm is similar to conventional singleuser 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 flatfading 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 SpreadSpectrum 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 widescale military use.
The technique developed by Lamarr and Antheil, which is now known as frequency
hopped spreadspectrum (FHSS), 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 frequencyhopping 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 FHSS is known as directsequence spreadspectrum (DSSS). In DS
SS, a digital signal is modulated by a digital spreadingsequence 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 nulltonull bandwidth of
2/Tb Hz. A spreading sequence is comprised of a pseudorandom 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 nulltonull 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 DSSS signals is assumed to be equal,
since multiplication by 1 doesn't affect the transmitted power. This means that the
PSD for the DSSS 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 FHSS system requires knowledge (and synchronization) of the frequency
hopping algorithm, the intended receiver in a DSSS system must have knowledge
(and synchronization) of the spreading sequence. Therefore, some deterministic
method must be used to generate a spreading sequence. A pseudorandom sequence
can be generated with a linearfeedback shiftregister (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 spreadspectrum signals.
flipflops 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, spreadspectrum systems have received
considerable attention for their use as covert communication systems. The attention
is due to three properties of spreadspectrum 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 spreadspectrum 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 DSSS
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 multiuser direct
sequence spreadspectrum system, we list several references which describe DSSS
systems in more detail. Scholtz [2] and Pickholtz et al. [3] provide good tutorials
on the properties and use of spreadspectrum 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 [57].
In directsequence codedivision multipleaccess (DSCDMA) 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 DSCDMA 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
singleuser 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 crosscorrelation between any two codes is wellbehaved. 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 nearfar problem. The challenge for
communication systems engineers is to develop receiver structures that are insensitive
to the nearfar 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 spreadspectrum communication systems. It was found that the average time
to achieve acquisition quickly increases when a nearfar scenario exists with only two
system users. Moon et al. [9] presented a multiuser 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 singleuser 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 biterrorrate
(BER). These results form the motivation for researching code acquisition techniques
for DSCDMA 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 DSCDMA system.
The transpose, the conjugate (or Hermitian) transpose, and the 2norm of a
columnvector 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 Frobeniusnorm 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 nonscalar 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 DSCDMA
communication system will be presented. The system model is expressed in a conve
nient vectormatrix notation which facilitates simulating a DSCDMA communication
system on a computer. The third chapter provides a brief review of the literature
concerning DSCDMA 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 lowcomplexity singleuser
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 singlebit singleuser detector, which is a suboptimal receiver
for a multiuser DSCDMA 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 DSCDMA 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 nearfar resistant.
In the sixth chapter, the effects of fading on a communication system are discussed
and a commonly used implementation of a flatfading channel simulator is presented.
The performance of the timing acquisition algorithm when the receiver is operating
in two nonstationary environments (frequencyoffset errors and flatfading 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 flatfading 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 DSCDMA 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 DSCDMA research, as well as several areas of future research
are provided.
CHAPTER 2
SYSTEM MODEL FOR THE AWGN CHANNEL
2.1 General Asynchronous DSCDMA 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 DSCDMA systems in general, it is required for the receiver
structure studied in this work. Let the vector Ck = (Ck,O, Ck,1,... Ck,N1)T represent
one period of the kth user's spreading sequence so that the spreading waveform can
be written as
N1
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 phaseoffset for the kth user,
while n(t) is the additive white Gaussian noise which is assumed to have a onesided
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 doublefrequency 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)
2PP1 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,N1)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) = [z2k1(m)a2k (pk, bk) + Z2k((m)a2k(pk,6k)] (2.7)
where
Z2ki(m)=[dk(m)+dk(m 1)]/2 {1,0,1}
Z2k(m) = [dk(m) dk(m 1)]/2 e {1,0,1} (2.8)
a2kl(pk,(k)= (1 k)Ck) +kCk+)
a2k(pk,bk) =(1 .k) + 6kCk (2.9)
Cn (Ck,Nn,Ck,Nn+l, ... Ck,Nl, Ck,O, Ck,l, ... Ck,Nn1)T (2.10)
k = (Ck,Nn, Ck,Nn+l, ... Ck,Nl, Ck,O, Ck,li ... Ck,Nnl)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 Z2ki(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 Z2kl(m) will be nonzero,
otherwise z2k(m) is nonzero. 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 zeromean 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 signaltonoise ratio (SNR) as
SNR = Eb/lAo.
2.2 Simplified Asynchronous DSCDMA 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 phasesynchronous 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 realvalued Gaussian
random variables that are independent having zeromean 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 phasesynchronous
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 DSCDMA 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 DSCDMA 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 DSCDMA research by proving
the existence of an optimum multiuser receiver that is nearfar 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 sideinformation, 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 suboptimal receiver structures.
Lupas and Verdi [18,19] have proposed a suboptimum 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 nearfar resistance with the pres
ence of timing errors. That is, the BER of the receiver reaches a nondiminishing
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 nearfar 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 nearfar 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 interferencefree 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 DSCDMA. As mentioned the complexity of the these tech
niques, and the amount of required sideinformation, 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 [2326]. While Strdm
et al. [23] consider the more general case of sampling the received signal more than
once a chipinterval, 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 eigendecomposition 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
(N2K) smallest eigenvalues (all equal to ao2) of R and En is an N x (N2K) matrix
of the corresponding eigenvectors. The signal subspace is defined to be the subspace
spanned by the set of a2kl(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 a2k1 (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) (33)
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) la2kl(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 timeinvariant
multipath channel. As a natural extension of their earlier work, Str6m et al. [25]
consider using the MUSIC based estimator in a timevarying channel.
The MUSICbased 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 signaltonoise ratios and little
nearfar effect, that the traditional correlator performs better than the MUSICbased
algorithm. However, as the MAI interference increases the MUSICbased algorithm
outperforms the correlator. It has also been noted that the MUSICbased algorithm
is resistant to the nearfar problem [23,24].
Zheng et al. [27] have proposed a singleuser propagation delay estimator, using
the system model of Section 2.1, that is known as the largesample 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 multipleantenna sensorsbased 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 timevarying, 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 singleuser detector
based on minimizing the meanoutputenergy (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 adhoc 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 bitintervals (2Tb) of the received signal, and, therefore, has 2N sam
pled outputs of the chipmatched 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 71 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 adhoc
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 halfchip asynchronous
to the desired user.
CHAPTER 4
THE MMSE RECEIVER AND SINGLEUSER CODE ACQUISITION
As an alternative to using the complex estimation techniques presented in the
previous chapter, we present a singleuser 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 meansquared error (MSE)
is wellknown to be given by the WienerHopf [31] equation
w(m) =Rl(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 chipsamples 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 sidechannel
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 pHRlp.
(4.4)
In order to solve the WienerHopf 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 eigendecomposition of the R matrix. Using the above notation,
we can then express the inverse of the autocorrelation matrix as
R1 = VA1VT. (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 eigendecomposition of P we immediately have
the eigendecomposition 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 fullrank and will have at most 2K nonzero eigenvalues. Miller [32,33] used
this fact to simplify the calculation of the eigendecomposition 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 R1.
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 eigendecomposition 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 multiaccess interference, we will observe the optimal filter weight vector for two
cases. For the first case let there be only one chipsynchronous system user, which
implies that r(m) = c) exp (jOi) +Nm. Using the WienerHopf equation, it is found
that
c. exp(ji) (47
Wopt = ICi2 +22
Assuming cl has good autocorrelation 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 crosscorrelation 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 DSCDMA 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 chipsynchronous 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
crosscorrelation 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 singleuser 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
T1, 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 signaltonoise 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)12, we can
approximate the output signaltonoise ratio as
SNRo [(1 61)2 ]1 (4.12)
2o2
where o72 = N/(2Eb/.fAo) as previously defined. When 61 is nonzero, 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 steadystate, 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 meansquared 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 DSCDMA 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 DSCDMA 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 phaseshifts were selected from a uniform distribution over
the interval [0, 27r). If we wanted to analyze a Kuser system, the first K elements
of each of the spreading code, delay, and phaseshift sets would be used to create the
DSCDMA environment.
For the case when the receiver was onehalf chip out of synch with the desired
user, using the Gaussian approximation we found that 4 system users would produce
a 2% biterror rate at the receiver. A total of 9 system users would produce a 5%
errorrate. If the receiver was synchronous with the desired user, a total of 14 users
would produce a 2% biterror rate. Likewise, a total of 24 users would produce a 5%
biterror 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 WienerHopf 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 leastsquares (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 WienerHopf equation, as the
filter approaches steadystate, they can provide reasonable approximations to the
optimal weight vector.
Our basic idea is to derive a code timing estimation algorithm for a singleuser
system, and then study the performance of the estimator in a multiuser 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 stepsize was chosen such that the filter was convergent in the
meansquare. We observed that the components of the filter weight vector become
highly uncorrelated as the filter approaches steadystate. 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) = Iw(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 multiuser transient mean filter weight vector
E[w(m)] is highly correlated with the single user optimal weight vector when a small
adaptation stepsize 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) = Lfj,
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 singleuser 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 singleuser
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 singleuser 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 steadystate
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 heavilyloaded the DSCDMA 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 phasesynchronous, 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 fifteenuser 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(mc1 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 singleuser, 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 DSCDMA 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 DSCDMA 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 adaptivefilter 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 overhead 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 filterbased 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&rRao 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 MonteCarlo simulations. The acquisition performance of
the timing estimator will be observed for both the LMS and RLS based adaptive filter
architectures. The effects of multiaccess 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 [31,3742]. The main subject that has been
addressed in these works, is what bounds must be placed on the LMS stepsize, A, such
that the filter has desirable convergence properties. One of the properties of interest,
is the convergence of the filter weights about the WienerHopf solution. The filter
is said to be convergent in the mean if lim _. E[w(n)] = wopt, the solution to the
WienerHopf 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 WienerHopf solution determines
the level of the meansquared error at the filter output. As one would expect, the
bound on p required to minimize the meansquared 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 timedependent
filter weight mean vector and the filter weight correlation matrix for an LMS filter
when the filter input was a vector of zeromean 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 zeromean. 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 Gaussiandistributed 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 fivetap
filter, a set of threethousand 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 timedependent, 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
WienerHopf 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 DSCDMA 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 stepsize, [, was chosen such that the filter was convergent in the meansquare.
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 singleuser 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 singleuser system, a) The analytical
mean filter weight vector norm, E[w(m)] (*), and the simulated mean
filter weight vector norm Iw(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 threeuser 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 multiuser 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 threeuser 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 singleuser environment, both equations gave
similar results. However when multipleusers 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
twouser 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 multiuser scenarios. We have observed
x 104
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 twouser 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 1020% over several different multiuser 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,...,N1) (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
N1
Pr(fi 5 Pi) = E Pr(F(pi) F(k) < 0). (5.13)
k=O
ktpi
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 [4550]. 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 nonzero vector x. Therefore, if the transformation matrix Ak is positive definite
then a chipselection 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 noncentral 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 noncentral X2 distribution with
nj degrees of freedom and noncentrality parameter 6? for j = 1,..., r and X0 having
a zeromean unitvariance 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 noncentral 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 noncentrality 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 (Nl) times to form a
bound on the chipselection error probability. If we want to evaluate the chipselection
probability as a function of the LMS filter's adaptation time, these techniques become
numerically prohibitive. That is, we found that MonteCarlo simulations of a specific
DSCDMA environment produce results for the chipselection 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 threeuser DSCDMA 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 stepsize, M, was set to 0.1/(the total
input power) such that the filter was convergent in the meansquare.
There are three curves present in Figure 5.4. The first curve is the simulated
results for the incorrect chipselection 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 multiuser DSCDMA
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 CramerRao 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 timedependent
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 timedependent 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 + + + + + +
103
20 25 30 35 40 45 50 55 60
# of Training Bits
Figure 5.4: Comparing incorrect chipselection probability to the Unionbound for a
threeuser 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 timedependent loglikelihood 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 loglikelihood function is then given by the following equation
[52].
V{r "02 (wl( lm) i1 (.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 DSCDMA 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 DSCDMA 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 stepsize, p, was always chosen to be 0.1/(the total input power) such
that the filter was convergent in the meansquare.
10o0
a: 1 User
ib: 5 Users
\ c: 2 Users
S i\,, d: 10 Users
0
1
C
r a
b
1021
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 nearfar 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 multiuser 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 . 
102 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 nearfar 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 singleuser
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) Iai(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 singleuser in the presence of
AWGN only, but can be suboptimal in the presence of multiaccess 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 stepsize P was
chosen to be 0.1/trace(R), where trace(R) is the total filter input power. With
this stepsize, the LMS filter was convergent in the mean and also convergent in the
meansquare. 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 halfchip 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 meansquared 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 DSCDMA 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' phaseshifts 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 lognormal 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 lognormal 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 nearfar 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 DSCDMA 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
softconstrained 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 RLSbased 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 multiaccess 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 lognormal
S 3 MM It 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
multiaccess 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 nearfar
resistant. Clearly, the correlator is very sensitive to the amount of nearfar interfer
ence. As was observed in the previous plot, the LMSbased timing estimator performs
better than the correlator, and the RLSbased timing estimator performs better than
the LMSbased timing estimator. For 2 users, the RLS based timing estimation algo
rithm could be considered as nearfar 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 nearfar 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 nearfar 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 nearfar 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 DSCDMA 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 singleuser environment, but does not follow the observed response of the filter
weights for a multiuser 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 1020% 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 chipsynchronous environment. The idea was to use the analytical expression as
a means to observe the timing estimator, without running long MonteCarlo 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 CramerRao
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 filterbased
timing estimator to a conventional correlatorbased timing estimator. The correlator
based timing estimator is well known to be optimal for a singleuser operating in an
AWGN channel, but can be suboptimal for a multiuser DSCDMA environment.
The metric used to compare the timing estimator, was the average number of training
bits required such that fi was within onehalf 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
filterbased timing estimator offered a slight improvement over the correlatorbased
timing estimator. However, as the number of system users increases, both the LMS
based and RLSbased implementations of the timing estimate perform better than
the correlator, with the advantage going to the RLSbased algorithm. The nearfar
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 RLSbased estimation algorithms are truly nearfar resistant. However,
the change in the length of the required training interval for the RLSbased algorithm
was so minor that we consider it to be nearfar resistant.
While it is difficult to directly compare our timing estimator with other techniques,
as some of the other techniques estimate more than a singleuser'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
correlatorbased 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 multipath communication channel where there are L distinct paths
with unique propagation delays T1 > T2 > .. TL1 > TL. The quantity TL T1 is
defined as the excess delay spread. As expected, this multipath 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
intersymbol 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 multipath 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 rootmeansquared 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 DSCDMA system Ts is equal to
To, the duration of onechip 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
multipath channel induces frequencyselective fading. Note that flat fading channel
models are more prevalent since a frequencyselective fading channel can be viewed as
the composite sum of a multipath channel where each path induces flat fading, but
the propagation delays between the individual paths are set such that the composite
channel induces frequencyselective fading.
Similar to the time dispersion induced by the multipath delays, relative motion
between a transmitter and its receiver induces dispersion in the frequency domain.
Consider a singlepath 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 s1, C is the speed of light
in free space and fc is the carrier frequency in Hz. When a timevarying multipath
channel is considered, the observed frequency shift becomes random and timevarying.
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 timeinvariant 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 timevarying
statistics of the envelope of the received signal in a flatfading environment. It is
well known that the random variable z = Vx2 + y2 when x and y are independent
zeromean 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 lineofsight (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 zeroorder 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 electromagnetic 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 zeroorder 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 thirdorder 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= 1rfD (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 thirdorder 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 thirdorder filter to simulate a flatfading channel based on
Clarke's model.
Figure 6.1 shows a block diagram of a fading channel simulator based on the
thirdorder filter that can be used on a digital computer. The continuoustime 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 zeromean 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: Flatfading 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 bitspersecond (BPS), and the relative speed between the
transmitter and the receiver was 3 milesperhour (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 flatfading 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 singleuser DSCDMA system operating in a fading channel
environment. For a given Doppler frequency, the fading channel simulator is created
and assume that both the thirdorder filter as well as the simulator output are sam
pled at the symbol rate. In a nonfading 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 multiuser
fading channel DSCDMA 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 NONSTATIONARY ENVIRONMENTS
It is wellknown that training an adaptive filter in a nonstationary 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 nonideal downconversion of the received
signal due to frequency synchronization errors. When this condition occurs, the input
to the adaptive filter has a timevarying 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 timevarying.
Barbosa and Miller [13] studied the adaptive detection of DSCDMA signals in
fading channels. In particular, they studied the performance of the adaptive receiver
shown in Figure 4.1 operating in frequency nonselective fading channels. They found
that this receiver structure does not work in a Rayleighfading 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 singleuser 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 Rayleighfading channel. In a nearfar
environment, the signaltointerferenceplusnoise 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 multiaccess interference
such that the signaltointerferenceplusnoise 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 Lthorder 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)
ii
and the estimate of the phase during the mth bit interval is found by 0i,m = Z/(m).
The coefficients of the Lthorder 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 multiaccess interference. In order to remove this dependence, the coefficients
were calculated for a singleuser system, and then are used for all other environments.
Not surprisingly, the coefficients of the Lthorder linear predictor, a, are found by
a = C1v (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 singleuser and, therefore, in the presence of multiaccess interference the channel
estimator may be very far from optimal.
The modified form of the adaptive receiver, using the Lthorder 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 slowfading as
well as fastfading. 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 1020%.
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 steadystate values.
The result is that the Lthorder 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 downconversion, the input to the adaptive filter for the kth user will have a
linearly timevarying 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 redefine Jk(m) as shown below.
Jk(m) .NMk exp 27rm\ ]
Mkr (7.6a)
[Z2k1 (m)v2k1 (Pk,tk) + Z2k(m)v2k(pk, 6k)
where the nth components of v2k1(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 (M25 = 50). The timing offsets and initial phaseoffsets 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 10thorder 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 buildup 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 fiveuser environ
weights of the adaptive filter. The 10thorder 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
flatfading 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 FlatFading 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)) [z2k1(m)a2k1(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 flatfading
channel simulator described in the previous chapter.
As a first test of the acquisition algorithm, a tenuser DSCDMA 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 flatfading 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 104.
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 Lthorder 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 adhoc mixture of the
two receiver structures. The idea of the hybrid receiver is to jumpstart the channel
estimation algorithm by preloading 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 Lelements 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 thirdorder 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 startup
transient of the channel estimation algorithm as the weights of the adaptive filter
were initialized to their steadystate values. In Figure 7.3, we observe the effects of
the startup 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 flatfading Rayleigh
channel using a thirdorder 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 DSCDMA environment was observed when
a tenthorder 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 tenthorder linear predictor is used in the channel estimation algorithm,
the timing estimation algorithm performs better than when a thirdorder 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 DSCDMA 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 flatfading Rayleigh
channel using a tenthorder linear predictor for the channel estimation
algorithm.
the sequel when we refer to the modified MMSE receiver we are using a tenthorder
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, eventhough the hybrid receiver
has been formulated in an adhoc 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 DSCDMA system in which
the users are traveling at highway speeds. The Doppler frequency induced by the
flatfading channel was 87 Hz, which is a normalized Doppler rate of 9.1 x 103. Each
DSCDMA 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 multiaccess interference in a flatfading 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 lognormal 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 threeuser and tenuser DS
CDMA system where the level of the lognormally distributed interference (X) ranged
from 0 dB to 13 dB. For both the threeuser and tenuser 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 halfchip of Ti. When the
hybrid receiver was used in the threeuser 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 threeuser 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 lognormally 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 DSCDMA envi
ronment tested in Figure 7.5, however there are fifteen systemusers and X = 1 dB for
all of the lognormally 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 fiatfading
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 nonstationary. While the
timing acquisition algorithm is applicable to use with the modified MMSE receiver, it
has been observed that the startup 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 startup 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 preloaded with 91 to continue
the timing acquisition cycle. Through simulation results of a DSCDMA 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) = (rmi,o, rml,, I .. r1,N1, rm,o0,rm,1, ..., tim,Ni) (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,. .. X2N1)T these functions are
defined below.
T()L (x) = T() (x) = x (8.5)
(X)= (xl, X2,..., x2 1,0) (8.6)
TL(1)) (X2NI,..., O) (8.7)
Tl(x) = (0,xo,x,... x22T (8.8)
T(R2N1)(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,mlVk1 + 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 singleuser operating in an AWGN channel. In
order to study the weights of the WienerHopf filter, we consider the case where we
apply the equivalent phasesynchronous 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 WienerHopf equation with Rwopt = v,
where we let Wopt = avj1 + bv + cv1.
The coefficients of the optimal weight vector, a, b and c, are given by the solution
to the following equation.
[ + iv1]2 (vIV1)Tv (v)Tv a 0
(viv0 + IIV112 (v?) 1 b = 1 (8.17)
(via) 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 vi1 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 nonzero in one chip
position where v? is nonzero also.
Based on this result, we will ignore the contributions of the vi1 and v, vectors on
the WienerHopf 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

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