Minimum mean-squared error adaptive antenna arrays for direct-sequence code-division multiple-access systems


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Minimum mean-squared error adaptive antenna arrays for direct-sequence code-division multiple-access systems
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viii, 129 leaves : ill. ; 29 cm.
Miller, John Earle
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Code division multiple access   ( lcsh )
Adaptive antennas   ( lcsh )
Antenna arrays   ( lcsh )
Electrical and Computer Engineering thesis, Ph. D
Dissertations, Academic -- Electrical and Computer Engineering -- UF
bibliography   ( marcgt )
non-fiction   ( marcgt )


Thesis (Ph. D.)--University of Florida, 1996.
Includes bibliographical references (leaves 117-128).
Statement of Responsibility:
by John Earle Miller.
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Copyright 1996


John Earle Miller

This work is dedicated to my wife Kim, my children Marissa and Christopher and my

parents Frank and Garnet.


I would like to thank all of the members of my committee for their involvement,

guidance and influence in this research. I would like to extend a special thanks to my

chairman, Dr. Scott L. Miller, for his support, many insights and thoughtful suggestions

which have influenced this work. The financial support provided by the Department of

Electrical and Computer Engineering is also gratefully acknowledged.


ACKNOWLEGMENTS ......................................................................................... iv

ABSTRA CT......................................................................................................... vii


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

Direct-Sequence Spread-Spectrum with Multiple-Access ............................
Adaptive Antenna Arrays......................................................................7
Previous W ork .................................................... ................................ 13
Dissertation Outline .................................................................................... 21

STEADY-STATE PERFORMANCE.................................. ............. 24

The Multiuser MMSE Processor........................................ ...... ........ 25
Spatially Orthogonal Users.............................. ........................................26
Nonorthogonal Users..................................................................................30
Single-User System....................................................................... 34
One Weak User, One Strong User...................................................34
Two Equal-Power Users ....................................................................36
Two Strong Users............................................ ......................... 36
Numerical Results................................................... ............................ 37

ADAPTIVE PERFORMANCE................................................................42

Mean-Based Performance Measure ........................................................43
Variance-Based Performance Measure ..................................... ............ 47
Sim ulation Results .................................................................................49

WITH IMPERFECT POWER CONTROL ...............................................52

System Description ..................................................................................... 53
A analysis ..................................................................... ............ ................ 54
Simulations and Results............................................................................... 61
Conclusions ........................................................................................ 65

MULTIPATH FADING AND SHADOW FADING................................... 68

A Single Cell with Signals Subjected to Rayleigh Fading .......................... 69
Multiple Cells with Signals Subjected to Rayleigh Fading
and Shadow Fading ............................................................................. 71
A nalysis...................................................................................................... 76
R esults.................................................................................................. 80
Conclusions/Discussion................. .................................................... 88
Summary ............................................................................................ 89

ADAPTIVE PERFORMANCE................................................................91

Signal and Channel Model............................. ...................................92
Adaptive Receivers .................................................................................. 94
Simulations .........................................................................................97
Conclusions/Summary ............................................................................... 102

7 SUMMARY ............................................................................................. 106

Areas for Future Work ............................................................................. 109


A THE OUTPUT SNR PERFORMANCE SURFACE ................................. 113

LIST OF REFERENCES ..................................................................................... 117

BIOGRAPHICAL SKETCH ............................................................................... 129

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



John Earle Miller

August, 1996

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

This dissertation examines the performance of direct-sequence code-division multiple-

access receivers which use a minimum mean-squared error adaptive antenna array as a

predetection spatial filter. The array attenuates multi-access interference prior to

conventional, direct-sequence matched-filter detection. Conventional detectors are

vulnerable to heightened levels of multi-access interference. Minimum mean-squared error

processing seems like a natural choice for optimization in a code-division multiple-access

environment since several efficient search algorithms exist which are compatible with

decision-directed equalizer structures.

Two array structures are examined. The first structure uses a single set of array

weights to equalize more than one desired signal. For two strong incident signals of

unequal power levels, the steady-state response gives output signal-to-noise ratios that are

leveled to a value near the maximum response of the weaker of the two users. The steady-

state adaptive performance of the multi-user array based on a least-mean-squares

algorithm is also examined. It is found that the maximum allowable step size based on

stability arguments will also give adequate output signal-to-noise ratio performance of the

multi-user processor.

The second structure uses a single set of weights per multiple-access signal and the

array output feeds a conventional detector. The array/detector performance measure is

outage probability and outage-based capacity as a function of the number of array

elements and the degree of power control error. A robust, incremental measure of

performance--the per-element capacity--is defined as the capacity per array element for a

given outage probability. Steady-state performance is evaluated for the case of directional

signals as well as for the case of signals subjected to multipath fading and shadowing. The

adaptive performance of the recursive least-squares algorithm is also investigated.


Wireless communication based on direct-sequence spread spectrum (DS-SS) has

received considerable attention as an efficient signaling format for code-division multiple-

access systems (CDMA). While DS-CDMA has a long history of use in defense

applications where jamming resistance and security are primary concerns,' some

proponents maintain that it will also serve as a high-capacity format for the next-

generation mobile-cellular and wireless systems for commercial use.23'4" System

specifications for domestic use have been endorsed by industrial agencies,6 the results of

field trials have been published7'8 and it has been studied as a possible format for third-

generation mobile radio systems in Europe.9'10

Despite a frenzy of activity and interest, DS-SS has a serious drawback: the near-far

effect This occurs when strong signals overwhelm a weaker desired signal during the

detection process. In a commercial mobile-cellular system the near-far effect can occur at

the cell-site base-station receiver. Incident signals originating from multiple-access, fixed-

power transmitters geographically distributed throughout a cell can have incident power

levels that change drastically as the transmitter positions vary. Signals originating from

transmitters near the base station may overwhelm received signals from transmitters on the

fringe of the cell. Vigorous proponents of commercial DS-CDMA systems have developed

transmit power-control formats to adjust each user's RF transmit power in real time so

that power levels of all multiple-access signals incident upon the base station are

approximately equal. Initially, it was stated that power control would need to be able to

adjust the transmit power over a 80 dB dynamic range in a mobile cellular scenario." Field

tests indicate that 50 dB is more likely.7'12

Another use for DS-SS is in the Global Positioning System (GPS). Originally

implemented as a positioning system for the Department of Defense, it has enjoyed

commercial applications in mapping, navigation, and surveying." The world-wide GPS

uses a DS-SS signaling format to obtain relatively accurate estimates of geographical

position. An earth-bound GPS receiver determines its position by measuring the path

delays of several DS-SS signals which originate from earth-orbiting satellites. The near-far

effect is not a critical issue for many commercial GPS applications, such as aviation or

shipping, since the signals are subjected only to free-space path losses. Some commercial

applications, such as surveying, suffer other forms of signal losses, such as multipath

fading or shadow fading, and these can indirectly lead to near-far limited performance.

This work examines receiver performance when an adaptive antenna array is used at

the DS-SS receiver. In such a system the array would act as a front-end spatial filter which

preserves the integrity of the desired signal or signals, attenuates interferers and

supplements the existing power control algorithm. Such an approach might be compatible

with existing cellular CDMA standards. The remainder of this chapter is divided into

several sections. The next section will give a qualitative review of DS-SS systems. The

second section will review quantities and expressions used in the analysis of the minimum

mean-squared-error (MMSE) beamforming antenna arrays. The third section will review

previous work germane to the research presented here and the last section will provide an

outline of the remainder of this dissertation.

Direct-Sequence Spread-Spectrum with Multiple-Access

Generating a DS-SS waveform involves multiplying a modulated baseband waveform

by a psuedorandom sequence sometimes known as a pseudonoise (PN) code. In an

asynchronous multi-access channel the incident DS-SS waveforms may be subjected to

random time delays and carrier phase delays and corrupted by noise. For a single-channel

receiver the multiple-access incident signals may be modeled by the expression:

Y(t)=Re Ac,(t -T,)b(t- T.)exp(j(ct+0,))+n(t) (1.1)

where the index m refers to the mth of K signals. The quantity Am is the incident signal

level of the mth signal, Cm(t-,m) represents the mth PN code waveform which consists of a

sequence (of length Nc) of psuedorandom square pulses, (c, (+1,-1 }) each with interval

Tc. The quantity b.(t-m) represents independent binary phase-shift keyed (BPSK)

modulation with equally-likely symbols (bi e (+1,-1 }) obtained via ideal, square pulses.

The time delay Tm is uniformly distributed over a symbol interval [0,Tb). The time delay Tm

arises because the channel allows asynchronous access; users may begin transmission at

any time. The carrier frequency is designated by ao, and 0, is the random carrier phase

uniformly distributed over [0,2x). The quantity n(t) is complex additive white Gaussian

noise (AWGN) with a one-sided power spectral density of No.

Note that in contrast to systems which use frequency-division multiple access

(FDMA), all users in a CDMA system share a common frequency o(. Frequency-division

channels are achieved by separating the carrier frequencies of multiple, bandlimited signals

to the point that they do not interfere with one another in the receiver. Code-division

channels in CDMA systems, on the other hand, are achieved through the low

crosscorrelation properties of the individual PN sequences. The PN sequences have many

interesting properties, including low crosscorrelation14 defined as

r (7) = J c.(t)c, (t + )dt<< Nc V m n. The limits of the integral are from zero to

NcTc. In a CDMA system the individual users share a common carrier frequency but are

assigned distinct PN sequences that permit code-division channeled links to a base station

or other receiver.

This work assumes that detection of DS-SS signals is accomplished by a correlating

detector also referred to as a conventional detector. The ith user's DS signal is multiplied

by a synchronized replica of its PN sequence, passed to an integrate-and-dump filter and

t = nTb + bT

x fJ( )dt -

Figure 1.1 Conventional detector, baseband model.

hard-limited to provide an estimate of the modulation symbol, as shown in Figure 1.1.

The conventional detector provides optimum performance for a single user in AWGN

but gives sub-optimum performance in a multi-access environment The performance of a

conventional detector in a multi-access environment depends roughly on the processing

gain which is a measure of the interference rejection capability of the detector and is

sometimes defined as the number of code symbols (or "chips") per modulation symbol (Nc

= Tb/Tc). For a fixed, finite processing gain it is always possible for strong interferers to

overwhelm the detection of a weaker signal and cause poor performance. The complex

baseband output of the integrate-and-dump portion of the conventional detector devoted

to user 1 consists of a desired signal component as well as multiaccess interference and

noise components:

K t, +T T +r
so ,,()= Ab, (t)+ A. c (t)c (t- )b(t- )dt+ fc,(t)n(t)dt
=2 T, (1.2)
=Ab, i(t)+ I A. [bi,,R,,, (,m) + b,,o.,i (Tm )]+ Jci (t)n(t)dt

Rm., (.m ) = Cm(t r. T lc(t)dt
where: 0

where bit 1 of user 1 is the detected signal of interest and Tr = 0. The quantities RP,, (r,)

and ki,, (r,) are the partial crosscorrelation functions. The quantities b,.l and bo

represent the contributions of the mth user's two bits which overlap the current bit interval

of the desired user. The values of crosscorrelations depend upon the sequences and their

relative time delays. The multi-access bit-error performance of the conventional detector

has been studied extensively; the dependence of the detector performance on PN-code

parameters has been well characterized.15'6 If one or more crosscorrelations are non-zero

and the corresponding input levels are large, those components may overwhelm the

desired signal in the detector. This is the near-far effect.

In mobile cellular or personal communication systems, signals incident on the base

station (also referred to as the uplink signal path) can take wide-ranging incident power

levels because of the random placement of users about the base station. The current

solution to this dilemma is to vary the transmitter power levels in real time via transmit

power control. In a system using closed-loop power control the base station receiver

monitors the detected power level of each multi-access signal. As the incident power

levels of the multi-access signals vary because of changing channel conditions, instructions

are sent from the base station to the mobile units, via an uncoded narrowband side

channel, to individually increase or decrease transmit power. The aim of the power control

is to force the effective bit-energy-to-noise spectral density ratio ((E/No),f) of all the

incident signals to the same threshold. In practice, however, the power control is unable to

perfectly track changing channel conditions because of doppler-induced multipath fading

and finite power control step size. Error in the power-controlled signal results in an

effective bit energy to noise spectral density ratio ((EbNo),r) at the detector output which

is approximately lognormal in distribution.17'1 Field trials indicate that high-mobility

mobile users (such as in fast-moving automobiles) have (Ed/No), standard deviations of oa

= 2.5 dB while lower values (or, = 1.5 dB) have been recorded for low-mobility users


Existing cellular CDMA specifications also have a contingency for open-loop power

control.6 In an open-loop power-controlled system the mobile or personal units use the

detected base-station carrier power as a reference to adjust their transmit power. No side

channel is required. Some researchers have proposed estimation algorithms which would

allow a mobile to make estimates of the base station carrier power. The researchers

present evidence that the carrier-measurement error in terrestrial mobile cellular systems is

well-approximated as lognormally distributed with a qc, 3 dB for vehicle speeds of up

to 60 mph and 8 dB of shadow fading.19 Other researchers working in the field of mobile

cellular systems based on low-earth-orbiting-satellites (LEOS) have also found open-loop

power control error to be well-approximated by a lognormally distributed random variable


If we generalize the previous work and assume that power control error may be

approximated by a lognormal r.v., then equation (1.1) may be rewritten slightly:

) =Re( )b)ep + (1.3)

where em is normally distributed with zero mean and variance equal to a,' (N(O, ce,2 )).

Note that in previous works the lognormal approximation of power control error applied

to the incident power levels for open-loop power control, and to the (EVNo)ef for the case

of closed-loop power control. To simplify analysis, this research will assume that power

control error results in lognormally distributed incident signal levels for closed-loop or

open-loop power-controlled systems.

Adaptive Antenna Arrays

This research assumes an adaptive antenna array is used as a front-end spatial filter to

attenuate direction-dependent cochannel interference prior to conventional detection of

multi-access DS-CDMA signals. The anticipated role of the adaptive array will be to

attenuate the strongest interferers and thereby lessen performance degradations due to the

near-far effect. The antenna arrays explored here will be limited to beamforming arrays -

linear combiners which use a minimum mean-squared error (MMSE) optimization

criterion. This technique is also referred to as Wiener filtering22 or optimum combining.23

A MMSE optimization is well suited to a mobile DS-SS-CDMA scenario since each user's

PN sequence, required for DS-SS detection and decoding, can provide a convenient

reference waveform. There are efficient MMSE adaptive algorithms which would require

no more side information than the code sequence and its timing.22

The array is composed of ideal isotropic sensors with no mutual coupling between the

elements. Incident signals are spatially sampled by the sensors as they propagate across the

array. The incident signals are assumed to be narrowband. This approximation is valid for

spread spectrum signals as long as signal bandwidths are a small percentage of the carrier

frequency. Current specifications6 call for bandwidths of 1.2288 MHz at a carrier

frequency of 850 MHz. The resulting double-sided signal bandwidth is much less than one

percent, so the narrowband approximation should be acceptable.

The physical displacements between the array sensors induce relative phase shifts on

our spatially sampled, narrowband signals. The phase-shifts are constant over frequency

and depend upon the array geometry and the signal's direction-of-arrival (DOA). The

complex signal component outputs of the N discrete antenna elements for the mth incident

signal can be described by a vector:

exp(j .o)

S= g.(t)exp(j(ky. t))= g (t)exp(- jot) exp(ij.,)

exp(j./ _i)
= g8(t)exp(- jco)t)u,
= [exp(jo.,) exp(ji,.) ... exp(jo.,_ )]l

where the vector y, (see Figure (1.2)) represents the physical displacements between

some reference point and the N array elements. An element of y. (yn n = O...N-1)

represents the physical distance between a reference point and the corresponding array

element. The parameter k is the free-space wave number of the incident plane wave ( k = 2

Tx/ ). Carrier phase-shifts have been absorbed into the complex, baseband function

g,(t).The vector u. contains electrical phase shifts resulting from the physical

displacements ym between the elements and the point of reference. The individual elements

of u, ( i.e. ~, ) represent the mth signal's phase shift between the reference point and the

nth array element. The vector u. will be referred to as the interelementphase-shift vector

or simply the phase-shift vector and will not be considered a function of time unless stated

otherwise. For the remainder of the report constants will appear as normal block or script

characters, vectors will be represented by lower-case boldface and matrices by upper-case


A diagram illustrating some of the physical quantities is shown in Figure 1.2. An

incident signal propagates as a plane wave s(t).The signals and noise are spatially sampled

by each element of the array; ignoring an explicit time dependence allows the samples to

be represented by x., n = O...N-1, and form the input vector x. The inputs are weighted

(via the weight vector w = [wo wi ...WN-I]T) and summed to form the beamformer output.

The quantity ky is the phase-shift of the propagating wave due to physical displacement y.

As the wave propagates across the array the phase-shifts due to displacement give rise to

interelement phase-shifts which are contained within the vector u. The lower part of the

figure illustrates how the direction-dependent phase-shift arises between two adjacent

elements. The quantity < is the phase shift between any two adjacent elements.

A change in array geometry would affect only the functional form of the relative

phase-shifts between the array elements. The quantity 0 = 2zd(sin0)/L in Figure 1.2 above

is the direction-dependent phase-shift between any two adjacent array elements for a linear

array. Arrays composed of individual elements arranged in a circle will be examined in

later chapters because, unlike linear arrays, they can resolve incident signals over the

direction-of-arrival interval [0, 2z). The phase-shift between any two adjacent elements is

given by 0 = (Rdcos[6-2,/Nl)/(1sin[/N]) for a circular array.

y = dsin0
foralinear s=ky='2yl/I
array: d = element spacing

the phase shift 9p i 2dsin
between the two 2 d sinQ
elements is then: A

Figure 1.2 A directional plane wave incident on a linear array of sensors.

The baseband array output vector is the sum of the noise and signal vectors:

x = n(t)+ st) = n(t)+ g.(t)u.
m=l m=l


where n(t) is the AWGN vector. The noise is spatially and temporally white. The

narrowband portion of the array autocorrelation matrix may be expressed as a sum of

outer products or as a product of matrices:

R = E x'xr= I+ A, u u, = r I+U'A2UT (1.6)

where U is the matrix of phase-shift vectors and A is a diagonal matrix of input

amplitudes, (*)* represents a complex conjugate and (O)T a vector transpose, respectively.

The expectation E[*] averages the AWGN, n(t) which is assumed to be N(O, 2). The

norm-squared of Um is:

iuJ2 =u u. = N (1.7)

independent of signal DOA where (O*) represents the hermitian transpose.

The MMSE weight vector minimizes the mean-squared error between the processor

output and the reference signal. The weight vector is given by:

wo = R-'p (1.8)

where p is the steering vector and is given by:

p = E[r(t)s;] = AdAu (1.9)

and Sd and Ud are the signal vector and phase-shift vector of the desired signal respectively.

The quantity r(t) is the time varying reference waveform with a peak amplitude of AR. The

MMSE weight vector maximizes the output signal-to-interference-and-noise-ratio27

(SINR) which is defined as the quotient of the desired signal power and the sum of the

interference and noise power SINR = Ps/(PJ+PN). The output signal-to-noise ratio (SNR)

for the mth signal is the quotient of the output power of the mth signal and the output

noise power. It is given by:27

SNRo =- -P = SNRi w u N SNRi, (1.10)
N, ollII2

The quantity SNRim = Am2/a2 is the incident SNR for the mth signal, Ps,m is the desired

signal output power, PI is the interference output power and PN is the output noise power.

For a single signal in AWGN, Wo = u*, and the upper bound becomes an equality. The

choice of w. = u,* represents the spatial equivalent of a matched filter (to the mth signal)

and is referred to as a conventional beamformer24, a classical beamformer2, or a

maximal ratio combiner.26

The sum total of the number of beams and nulls an array is capable of directing

simultaneously is N-1, one less than the number of array elements. This is also referred to

as the degrees-of-freedom27 of an array.

At this point, it may be important to explain the difference between the terms

adaptive array and diversity combiner. The term adaptive array implies that the processor

uses an array of sensors and exploits the phase and amplitude shifts between the array

elements, induced by directional signals, to make processing decisions. A diversity

combiner, on the other hand, exploits some degree of statistical independence between the

input samples to ensure signal integrity at the summed output. For example, in the case of

communication channels which contain multipath fading, an array of sensors might provide

statistically independent spatial samples. If a signal is faded at one sensor, it might not be

faded at another sensor; the statistical independence of the inputs is exploited by the

processor to improve the integrity of the overall output response. There are many diversity

combining strategies.28 A popular strategy for analytical purposes, mentioned previously

as maximal-ratio combining, maximizes the output SNR but does not allow adaptive

interference suppression. The diversity-combining counterpart to MMSE filtering is

referred to as optimum combining. It is sometimes possible to achieve statistically

independent samples through time sampling. Although not an issue in this research, a few

examples of time diversity will be cited in the following section.

Previous Work

This section is a survey of previous work in adaptive beamforming antenna arrays and

in CDMA that is pertinent to this research. The first part of the survey will give a

historical perspective to research in adaptive antenna arrays. Details of the earlier work

oriented towards radar and military communications is admittedly sparse. As the timeline

and focus become more current the coverage will become more detailed. The second part

of the survey will review the most recent work in which adaptive arrays function as an

integral component of DS-CDMA systems. The last part of the survey will examine

previous work in cellular CDMA ( without antenna arrays) as it applies to this research.

A beam-steered array was investigated by Applebaum29 but the results were not

published in open literature until a decade later.30 Widrow et al.31 reported on an antenna

array using a least-mean-squares (LMS) processor in 1967. A special issue of the IEEE

Transactions on Antennas and Propagation devoted to active and adaptive antennas was

published in 1964,32 197633 and 1986.34 Other processors were investigated by Frost35,

Griffiths36 and Schorr.37 Much of the initial adaptive array work focused on the

performance of particular processors or radar-oriented applications. One exception to this

was the maximal ratio diversity combiner investigated by Brennan.26

Subsequent work explored the role of adaptive arrays in communication systems.3"39

In particular, Compton40 presented a qualitative evaluation of an experimental adaptive

array in a DS communication system. The evaluation focused on a single desired signal

and a limited number of jammers. One of the conclusions reached by Compton is that the

array makes appreciable contributions to interference suppression. Winters41 studied the

acquisition performance of an LMS adaptive array in a DS system using four-phase

modulation with two PN codes, a short code for rapid acquisition and a long code for

protection against jammers. Ganz42 evaluated the bit-error-rate (BER) performance of a

receiver employing an adaptive array and one of several detectors for binary-phase-shift-

keying (BPSK), quadrature phase-shift keying (QPSK), or differential phase-shift keying

(DPSK) modulation. The receiver was subjected to continuous-wave (CW) jamming.

Several authors have investigated the use of antenna arrays for mobile or personal

communications. Bogachev and Kiselev43 evaluated optimum-combining diversity arrays

for the case of a single interferer. Winters23 conducted a comparative study of optimum

combining and maximal ratio combining base station arrays in a multiuser mobile

environment with multipath fading but no shadow fading. The results quantify the possible

improvement in SINR if optimum combining is selected over maximal ratio combining.

Like optimum combining, maximal ratio combining preserves the integrity of the desired

signal, but unlike optimum combining maximal ratio combining has no ability to adaptively

suppress interference. Winters did propose the use of psuedonoise codes to generate the

LMS reference, but transmit power control was mentioned only for its effect on the

convergence properties of the LMS array. He did not investigate in any detail the

condition when the number of users greatly exceeds the number of array elements. Winters

also explored the use of adaptive arrays on base stations for in-building systems using

dynamic channel assignment4 He again considered a PN-coded PSK modulation and

circumvented power control considerations by assuming interferers were of equal power

and much stronger than the desired signal. A more recent study45 investigated the

acquisition and tracking performance of LMS and sample-matrix-inverse (SMI)

beamformers in a time-division multiple-access system.

Yeh and Reudink4 examined the contributions made by spatial diversity combiners to

spectral efficiency in FDMA cellular systems when the arrays are located on the base

station and on the mobiles. Glance and Greenstein47 also examined the contributions of

diversity order (the number of array elements) on average BER in a mobile FDMA system.

Vaughan4 discussed the benefit of MMSE combining at the mobile in an FDMA system

and concluded with the comment that for MMSE combining to be successful wide

bandwidth signals, such as those found in spread-spectrum systems, are necessary.

More recently, adaptive array research has examined commercially-oriented

applications as CDMA and non-CDMA wireless systems gain popularity. At this time,

antenna arrays are under investigation as a means of providing space-division channels in

multiple-access systems. The technique, called space-division multiple access (SDMA) by

some authors, utilizes the spatial filtering properties of the array to selectively receive

signals which share the same time slot and the same frequency band. The SDMA

technique might apply to either time-division multiple-access systems (TDMA) or CDMA

systems. Swales et al.49 established that a steerable, multi-beam antenna array can increase

the capacity and spectral efficiency of a cellular system. Suard and Kailath50 studied the

upper bound of the information-based capacity of a wireless system which used a base-

station antenna array in the uplink path. Experimental studies have been conducted. Xu et

al.51 and Lin et al.52 have examined algorithms based on direction-finding techniques

MUSIC" and ESPRIT.5 They have concluded that SDMA techniques based on DOA

estimation will not be effective in multipath-rich environments. Xu and Li55 developed an

SDMA/TDMA protocol. Ward and Compton56'57 examined the contributions that a

receiving array can make to the performance of an ALOHA system.

Arrays which include spatial and temporal adaptive processing nested within LMS

feedback loops were proposed by Kohno et al.58 and Ko et al.59 Kohno used an LMS array

in conjunction with adaptive temporal filtering which successively canceled DS signals due

to multi-access interference. Ko described a null-steering beamformer nested within an

LMS loop, not necessarily restricted to CDMA applications. Both considered limited

multi-access scenarios with deterministic interference parameters.

Diversity combining is not necessarily restricted to space diversity. Balaban and

Salz60'61 examined in detail the performance of a general multi-channel MMSE combiner

working in conjunction with a decision-feedback equalizer. They established an upper

bound on BER which is a function of the MMSE. A great deal of work has also been

devoted to temporal diversity combiners which consist of a single input to a bank of

matched filters, the outputs of which are coherently combined via maximal ratio

combining. This is the basis for the RAKE62 receiver as well as variations studied by other

authors. Several structures were examined by Turin63 while Lehnert and Pursley64

examined diversity combining in multiuser CDMA system in which successive bits are

spread with different PN code subsequences. Wang et al. consider diversity for an indoor

DS-CDMA system with Rician fading.6

Some authors have studied the possible contributions made by arrays to the uplink

path performance in cellular systems. Liberti and Rappaport have studied the effects of a

directive, steered-beam base-station receiving array on uplink performance in a CDMA

cellular system with perfect power control. The study focused on the effect of beam shape

and beam width on the average BER. It was found that beam width has the greatest

impact on performance and that adding a three-element array to the base station can

improve BER performance by three orders of magnitude. Tsoulos, Beach and Swales have

recently examined the role of adaptive antenna technology in large "umbrella" cells

overlaying smaller microcells in cellular CDMA systems ; they also examined the outage-

based capacity enhancement due to an adaptive antenna array using a recursive least-

squares (RLS) processor in a multi-cell CDMA environment.6 The latter study was

confined to simulations of the total interference to calculate outage. The authors

concluded that an antenna with 6 dB of directivity gain can increase capacity by a factor of


Winters, Salz and Gitlin6 studied the effects of optimum combining spatial diversity

arrays on the capacity of a TDMA system in which the total number of incident signals

was less than or equal to the array DOF. They applied previous workw'' which resulted in

an upper bound on BER. The assumption of high input SNR allowed a zero-forcing

approximation to the optimum combining solution. Using these assumptions, and

examining analytical expressions for BER, they found that optimum combining with N

antennas and K interferers gives the same results as a maximal ratio combiner with N-K+1

elements and no interferers. Their theoretical results, as they point out, no longer apply

when the number of interferers exceeds the number of antenna elements.

A structure which combines the spatial filtering of an antenna array and the temporal,

diversity-combining properties of a RAKE receiver have been proposed and investigated

by Khalaj in concert with several other authors.6970'71 The structure is intended for use in

channels with frequency-selective multipath fading. The structure allows the resolution of

identifiable multipath rays by the time-filtering properties of the RAKE combiner and by

the spatial filtering properties of the antenna array.

A number of authors have conducted brief simulation-based studies of adaptive arrays

for DS-CDMA systems. Yoshino et al.72 examine the simulation performance of two

RLS-based spatial diversity combiners operating in concert with (Viterbi) sequence

estimators. One processor subtracts estimates of the cochannel interference from the

output prior to estimation of the desired user's data sequence while the other processor

does not. Wang and Cruz73 examine the pattern behavior and BER of a six-element arrays

based on the RLS and ESPRIT algorithms with six active users with well-separated

DOAs. Liu74 examined the performance of an LMS array with a scenario-dependent

matrix preprocessor which aids in interference cancellation. Hanna et al.75 investigated the

BER performance of a two-element LMS array which operates in conjunction with an

adaptive equalizer.

Perhaps the most in-depth study of the possible contributions of MMSE adaptive

beamforming arrays to the performance of mobile cellular CDMA systems using closed-

loop power control has been made by Naguib in concert with other authors. Initial work76

focused on steady-state performance of an array of sensors, each followed by a DS

conventional detector, which functions as a post-detection combiner (termed code-

filtering by the authors). Analysis resulted in a simple expression for capacity. The signal

model was for unfaded signals originating in an isolated single cell. Power control error

was modeled by assigning a single interferer an incident signal level 10 dB higher than the

other signals.

Naguib, Paulraj and Kailath77 extended the steady-state model in order to determine

the outage probability in a cellular system with BPSK modulation, perfect power control,

shadow fading, multipath fading, and cochannel interference equivalent to two tiers of

surrounding cells. Assuming that the array pattern response consisted of a main lobe and

no sidelobes resulted in a simple, closed-form expression for an upper bound on outage

probability which simplified to the single-channel results of Gilhousen et al." when the

array is reduced to a single element. Modeling the MMSE processor as a maximal ratio

processor and assuming the interference was Gaussian resulted in a simpler expression for

the outage probability upper bound.78 Naguib and Paulraj then modified the simulation

model to include DPSK-modulated signals and determined the Erlang capacity.79 The

uplink performance with M-ary orthogonal modulation was examined as well. 0

A recursive beamforming algorithm was proposed by Naguib and Paulrajs8 and

simulated results were presented. The algorithm performed recursive updates on the

matrix square root of the inverse of the covariance matrix. The authors claimed that the

accumulation of numerical and quantization errors may cause the covariance matrix

inverse to cease being hermitian definite and that updating the matrix square root will

allow the covariance matrix inverse to remain hermitian definite even when the matrix

square root is not. Thus, say the authors, numerical instabilities are avoided.

Naguib and Paulraj82 continued their investigation of cellular base station antenna

arrays by examining the tradeoffs in coverage area, mobile transmit power, and capacity

that are available when an array is used and the users are subjected to perfect power

control. Using simplifying assumptions the authors derived expressions which generalized

the effect of the antenna array on performance. This research will attempt to extend some

of the results to the case of imperfect power control.

Later investigations have resulted in detailed, simulation-based studies of an IS-95

system which uses orthogonal signaling, forward error-correction coding, closed-loop

power control and a base-station antenna array. Unlike their previous studies Naguib and

Paulraj applied a model for imperfect closed-loop power control. They studied the

standard deviation of power control error, although where the error is defined is not clear

in the paper. Using simulation results they show that power control error dependence on

the power control step size, the number of array elements, and the maximum doppler

frequency and the spread in DOA of the multipath rays.83 The dependence of BER on the

same parameters was the topic of a subsequent paper."

A multiuser LMS array for use in GPS receivers was examined by Beach et al.85 in a

simulation study. The results were confined to plots which show the evolution of the

adaptive array pattern over time in the presence of CW jammers; no steady-state results

were presented. The near/far effect was not an issue in the study.

A variety of authors have investigated the performance of cellular CDMA systems

with imperfect power control and no antenna array. Simpsom and Holtzman6 used

simplified analytical models in order to provide insight into the interactions between power

control, coding and interleaving. Stuber and Kchao87 examined a multiple-cell CDMA

system and evaluated the dependence of BER as a function of the distance from the base

station. Jalali and Mermelstein" conducted a simulation study of a microcellular CDMA

system which included imperfect closed-loop power control and antenna diversity with

square-law combining. Milstein et al.89 studied the average BER performance of a

multiple-cell system in which users where subjected to power control error which was

described by a uniform random variable. Newson and Heath90 examined a CDMA system

which suffered from imperfect sectoring and lognormally-distributed power control error

and made capacity comparisons to TDMA and FDMA systems.

Dissertation Outline

The remainder of the dissertation will be divided into six chapters. Chapter 2 will

examine some aspects of the steady-state performance of a multiuser MMSE array; some

limited aspects of an adaptive LMS version were investigated by Beach et al.8 The

near/far performance will first be investigated for strong, spatially orthogonal users and

will then be extended to the case of two users with any DOA spacing. The analysis will

focus on the ability of the array to confine the signal outputs to the same output SNR. The

results will show that the array may be suitable when signals are well-separated in DOA

and do not outnumber the array DOFs. The analysis will borrow some of the analytical

techniques used by Gupta9" and apply them to evaluate the steady-state performance of

the multi-user MMSE array. Unlike either previous study, however, the analysis will focus

on the performance of the array in a near/far environment.

The third chapter will examine the adaptive performance of the multiuser MMSE

array when it is implemented as an LMS processor. Analysis will show that the rule-of-

thumb for picking step size to ensure stability also serves as a limit to reduce the spread in

output signal levels due to LMS misadjustment

The fourth chapter will examine the steady-state performance of an MMSE array

feeding a conventional detector. Unlike the array of the first two chapters, this array will

be dedicated to a single user among many multiple-access users. The single-cell users will

not be subjected to many of the influences normally found in a mobile environment, such

as multipath fading, shadow fading or interference which originates from surrounding

cells. This will simplify the analysis: the intent is not to provide an analysis fraught with

mathematical rigor, but to use simplifying assumptions to aid in the derivation of closed-

form analytical expressions which accurately predict the uplink performance and which

clearly show the dependence of uplink capacity on the system parameters, especially the

array and the degree of power control error. The performance measures will be outage

probability and outage-based capacity.

The analysis of chapter four will result in a simple closed-form expression for the

capacity which is linear in the product of processing gain and the number of array elements

and which decays exponentially with increasing power control error. The slope of the

capacity line (the per-element capacity) will serve as a robust measure of the array's

incremental contribution to capacity. Although some of the assumptions are highly

idealized (i.e. no multipath fading), they will allow some well-understood quantities, such

as array gain, to be exploited in the analysis. Some of the work in this chapter is related to

that of Padovani7 and also to Naguib and Paulraj.82 The details will be discussed during the

chapter's derivations and discussions.

Chapter 5 will extend the steady-state multi-access model of Chapter 4 to include the

effects of multipath fading, shadow fading and outer-cell interference. The performance

measures will again be the steady-state outage probability and outage-based capacity.

The sixth chapter will examine the recursive-least-squares (RLS) adaptive

performance of the MMSE adaptive array in a mobile cellular scenario in a nonstationary

channel. The discrete-event simulation model includes time-varying multipath fading,

stationary shadow fading and time-varying outer-cell interference.

The last chapter will give a brief summary and some conclusions of this research. It

will also identify some areas of future work.

This research makes contributions in two areas. First, analysis and numerical solutions

provide more insight into the steady-state and the adaptive behavior of the multi-user

MMSE array (chapters 2,3) than was provided by previous authors. Analysis shows that

the array can remove the near-far effect and level the output SNRs to nearly the same

level, under certain circumstances. Second, the analysis and simulation of a single-user

array operating in conjunction with a conventional detector results in simple expressions

for capacity when the signal sources are subjected to imperfect power control (chapters

4,5,6). The results characterize the incremental contributions an adaptive array can make

to uplink performance. This is in contrast to previous works which examine the effects of

closed-loop power control error only in simulation for limited scenarios.


This chapter examines the ideal, steady-state performance of the multiuser MMSE

array proposed by Beach et al.5 The authors conducted a simulation study of the pattern

behavior of a multiuser LMS adaptive array with and without directional interference.

They did not provide any analytical results which give general insight into the array's

steady-state performance. In related work, Gupta91 examined a multi-user steered-beam

array for non-CDMA applications and simplified his analysis by assuming the special case

of spatially orthogonal users. Gupta was interested in using the array's effective aperature

to improve the output SNR of the desired signals and in using the adaptive properties to

reject interference. Other authors have shown that steered-beam arrays and LMS arrays

give the same output SNR performance as long as the steering vectors differ by a

constant.92 Using this rationale it might seem that Gupta's work could provide some

insight to this problem. However, he was trying to configure the array processor to avoid

the power leveling effect that this work is attempting to exploit.

This chapter will focus on the array output SNR performance for strong incident

signals. The analysis will deemphasize the role of the spread-spectrum signalling other

than to note that it provides a relatively easy way to provide separation and detection of

the multiple, desired signals present at the single array output. The remainder of this

chapter will be divided into several sections. The first section will provide some qualitative

information about the multiuser MMSE array processor and give some refinements to the

general array equations given in the introduction. The second section will develop

analytical expressions of the output SNR for K _N-1 spatially orthogonal users. The third

section will present analytical expressions for two users which have arbitrary DOAs and

are therefore not necessarily spatially orthogonal. The last section will compare the

analytical and the numerical results and discuss their implications.

The Multiuser MMSE Processor

The multiuser array treats each of the K signals as a component of a single composite

desired signal. This is accomplished via a reference signal which is a sum of the modulated

PN sequences of the desired incident signals. A block diagram is shown below in Figure

2.1. In the figure all functions of time are ignored. For analytical simplicity we will assume

that perfect estimates of the modulation and code waveforms are available for the

generation of the reference waveform. This will not usually be the case. Generation of the

reference signal can be a challenging issue and it has been investigated by other


The array will force the outputs of the individual signals to the levels of that signal's

component of the reference waveform. To avoid the near/far effect the individual

components comprising the reference waveform have the same peak amplitudes. Since all

of the incident signals share the same beamformer weights, the output SNRs will be

leveled to approximately the same value. The analysis will show that the array will force

the output SNRs to values very close to the maximum output SNR of the weakest user

under certain conditions.

The steering vector defined by equation (1.9) represents the crosscorrelation between

the reference signal and the multiple, independent input signals. It becomes a sum of

single-user steering vectors:

p= p. = R Au'. (2.1)

where A, and Um are the incident amplitude and phase-shift vector of the mth signal,


Figure 2.1 Multiuser MMSE processor

Spatially Orthogonal Users

Consider the limiting case of multiple users which are spatially orthogonal to one

another (uifuk = 0 V ik ). The minimum source separation (in DOA) which allows

spatial orthogonality corresponds to the Rayleigh limit.22 The Rayleigh limit is generally

To a bank of
Output DS detectors
(t) no(t)+ on()
+ ,=1
', bI (t TI )c, (t T)
+ 4'Lb 2 (t 2r )c (t T K )

b( V- T )C K (t T K


considered to be the minimum amount of source separation which still allows resolution of

two sources by a beamformer.24 If the users are orthogonal and the array elements are

isotropic the inverse of the autocorrelation matrix is:91
1 ( K N-SNRi, '
R-'1 I- u '-uT (2.2)
aO =1 +N-SNRi -

and the resulting MMSE weight vector, w, = R'p, is given by:

wO R i= .u' (2.3)
o ,=, 1+N.SNRi,

Substituting these quantities into the general expression for the n'th users output SNR

(equation (1.10)) gives:

f SNRi, K N-SNRi
SNRo, = 1+NSNRi,,) ^ (1 +N SNRi ,)2 (2.4)

Some coarse judgements on performance for multiple users can be made using

equation (2.4). If {SNRim >> 1 V m =1 ... K) the output SNRs for all of the users are

approximately equal to:

SNRo = N- S (2.5)
..= SNRi.

If there are K users with the same input SNR = SNRi then equation (2.5) simplifies to:

SNRo = N i (2.6)

which shows that the users will all have output SNRs equal to the maximum possible SNR

divided by the number of active users. If the array parameters remain constant, the

performance will degrade as the number of users increases.

If there are S users with the same input SNRi = ki.SNR and K-S users with SNRi =

SNR then the output SNR for the K users simplifies to:

SNRo = N- SNR k (2.7)
S+k, .(K-S)

If k-(K S) >> S (a possible near/far scenario) the expression for output SNR

simplifies further:

SNRo .- KS (2.8)
k,-N-SNR K

S = no. of strong users with input SNRi = ki SNR, k >> S
K = total number of users
K S = no. of weaker users with input SNRi = SNR

From the equation immediately above it is apparent that one weak user (K-S = 1) will

dominate the overall performance of the array. This might make intuitive sense, even for

the general case. A minimum MSE processor will try to force each output signal to have

the same value as that signal's component of the reference waveform. This strategy will

favor weaker incident signals which need large weight values to cause their array output to

match their component of the reference waveform ( i.e. minimum error power). Large

incident signals will be attenuated via phase shifts and the noise power the denominator

of equation (2.4) will be large thus causing a lower than maximum output SNR even for

strong input signals.

Equation (2.8) above establishes that the weakest incident signal drastically affects the

overall performance of the array. The worst array response towards an arbitrary signal

(SNRon) for a different input signal (SNRi,) can be found by finding the value of SNRim

which forces dSNRon/dSNRim, the derivative of the nth signal's output SNR with respect

to the mth signal's input SNR, to zero:

dSNRo, N SNRi, (- N SNRi,. N SNRik)
dSNRi, I++N-SNRi +N -SNR k1 (1+ N SNRi)

From the equation above its is evident that a minimum occurs for SNRim = 1/N.

Substituting SNRim = 1/N into equation (2.4) gives (for SNRin >> 1, n i m):

SNRo, 4. N SNRi,, 4
1 + N. SNRi, (2.10)
SNRo, = 1

For SNRim = 1/N the value of the mth eigenvalue of R is twice that of the noise-only

eigenvalues. At this input level the array is barely able to resolve the m'th input signal

from the input noise.

It might be useful to define an output SNR "spread" which bounds the output SNRs

for all of the users. From equation (2.4) it can be seen that the largest and smallest input

signals result in the largest and smallest output SNRs respectively. We can predict the

output spread by the difference between the largest and smallest output SNRs:


= f NSNRi, f NSNRi, V, NSNRij
[ +N SNRi) +N- SNRi,, (1+N SNRi,2 )


If we let SNRimax pass to infinity the output spread becomes:

lim I N SNRi Y D A N SNRiA
Ri oo:ASNRo = 1 (2.12)
SNRi Rii + S--N-i. + (I + N SNRij)2

The equation above simplifies to:

ASNRo = 2 + NSNRin (2.13)
(1 +N-SNRi )2 D N-SNRij
1+ 2
N SNRi, ,,m*a. (1+ N-SNRi )

An upper bound on the output SNR spread results:

ASNRo < 2 + (2.14)

In summary it would appear that for N-SNRim >> 1, m=l...D, the user outputs will

be near the value of the maximum output SNR for the weakest user and the output SNR

spread will be equivalent to 2.

Nonorthogonal Users

Analyzing the general case of multiple narrowband users which are not spatially

orthogonal is difficult because of the matrix inverse in equation (1.8). When the users are

not orthogonal the analytical matrix inverse (via the matrix inversion lemma) quickly

becomes intractable as the number of users increases beyond unity. The general case of

only two users was examined in detail. The approach taken here is to derive expressions

for the two output SNRs. The expressions are then interpreted as transfer functions with

poles and zeros which are dependent upon the input levels and the DOAs. Critical points

of the functions are then evaluated.

The autocorrelation matrix R for two independent, narrowband users in AWGN

(equation (1.6)), is given by:

R = a2 -I+ A':u;u + A'u2u
= 2 [I + SNRi2u u + SNRi2u;u (

The inverse of R (R-1) may be calculated via the matrix inversion lemma:

R-1 1 1 SNRi T
R = IU.U
a. a2, 1+ SNRi,
u u;SNRi uI' uSNRi U
SNRi 1+ NSNRi, 1 +NSNRi,
o uu;1 SNRi, SNRi2
1 + SNRi, -
2 1+SNRi,

The steering vector p which results from equation (2.1) is:

p = RAu; + RA2u; = aR S uiu + aRR SR-u (2.17)

Substituting equations (2.16) and (2.17) into equation (1.8), wo = R'1p, gives the MMSE

weight vector:

w = R- =R [pi(1+ SNRi2)- uTu;SNRi, SNRi2]u; +

R [ (NRi2 ( + SNRi) )- uluSNRi, JSNRi ]u;

The quantity Do is defined by:

D, =1+ N(SNRil + SNRi) + (N -uI u;2 SNRiiSNRi2 (2.19)

Two vector inner products are required to find the output SNR defined in equation

(1.10). If we assume the point of reference for the array is the same as the physical center

of the array, the inner products will be real.24 An inner product between two phase-shift

vectors may then be expressed in dot-product form: uTu* = Ncosa 2 where al2 is the

angle between the two vectors in signal space. Substituting this relationship into equation


11w.112 = N(SNRiI +SNRi2) +2N SNRizSNRi2 cosa,2 +

R2 N 2SNRi SNRi, sin2 a2 (4 + NSNRi, + NSNRi, 2N SNRi SNRi2 cosa 2)
a ,20,D

where: Do = 1+ NSNRi + NSNRi2 + N2SNRiNSNRi, sin2 a,2 (2.20)

Expressions for the numerator of equation (1.10), the output SNR, are possible for the

two users:

loUw. = R2 [NSNRi2l + NSNRi22 sin2 a12)+ SNRi cosa1]2 (2.21)

Substituting equations (2.20) and (2.21) into equation (1.10) will result in the steady-state

output SNR for the two users at the point of minimum mean-squared error.

An additional substitution might simplify the expressions further. If we substitute the

expression SNRij= a2 SNRi2 into equations (2.20) and (2.21) the output SNR can be

expressed as a quotient of polynomials in a with coefficients that are functions of NSNRi2

and a12. The numerators of the quotients are:

nums = N2SNRi2 ( + NSNRi, sin2 a2,,) a+ cosa12
S+ 12NSNRi2 sin2 a12
num2 = N 4SNRi sin a2 a, a+a +cos2_+
num2 = NNSNRi, sin2 a21 NSNRi2 sin2 a12

for user 1 and user 2 respectively. The denominator of both biquadratic terms is:

Dens2 = a2N2SNRi2 sin2 a [4 + NSNRi2(a2 2acosa,2 +1)] +
NSNRi4(a2 + 2acosa,2 +1)

and the expressions for the two output SNRs become:

SNRo = a2SNRi2 numn SNRo2 = SNRi um2 (2.24)
Dens, Densm

The expressions for output SNRs may be interpreted as transfer functions in the

variable a. The numerators may be treated as products of second order polynomials with

easily determined roots which may be interpreted as real zeros of the output SNR "transfer

function." The roots of the numerators are:

Z cosa12
1 + NSNRi2 sin2 a2

= cosa12 [1 +Jl-4NSNRi tan2a I
Znu 2NSNRi2 sin' a12

The denominator is a fourth-order polynomial in a and analytical expressions for its roots

are not particularly useful in the general case. To circumvent this difficulty the analytical

expressions for output SNR will be examined and approximated for specific cases of a2

(the near-far ratio) and a12 (the angle between the phase-shift vectors in weight-space)

and results will be compared to numerical solutions. Note that a12 is determined

exclusively by the users' DOAs for a given antenna array configuration. The output SNR

expressions will be examined for four scenarios of the variable a:

1. a2 = 0 (single user system)
2.0 < a2 << 1 (1 weak, 1 strong user)
3. a2 = 1 (2 strong, equal power users)
4. 1 < a2 < oo (2 strong, unequal powers)

Two cases of DOA separation will be examined for each of the four scenarios listed

immediately above. One case (referred to as case a) will be that of spatially orthogonal

users (cos2k12 = 0, sin2a12 = 1). This will allow comparisons with the the results in the

previous section. The second case (case b) will be for DOAs with spacings greater than

the Rayleigh limit (0 < cos2a12 << sin2c12 < 1) and will also be referred to as well-


The following analysis will consider approximations to equations (2.22)-(2.25) for

some combinations of the intervals of a listed above. Rather than being exhaustive the

analysis will focus on critical points which will reveal trends in performance.

Single-User System

When a = 0 we have a single user system and :

SNRo = a2SNRi2 nun = 0
SNRo2 = SNRi2 Dens = NSNRi2

One Weak User. One Strong User

For the case of one weak user and a moderately strong user which are well separated

(case b) the numerators and denominator of equation (2.24) can be approximated by:

num, = N 2SNRi 2(+ NSNRi2 sin ao)2 a + cosa12 2
1 + NSNRi, sin2 a,2

num2 N4SNRi3 sin4' a 2 + NSNR sin2 1
2 1 NSNRi, sin2 a21

Dens, = N 2 SNRi2 (4+ NSNRi,2)sin' a a2 a2 +N+ N2
2 1 NSNRi 2(4+ NSNRis) Sin 2 a12

If we can further assume that NSNRi2 >> 4 then the output SNRs may be approximated

[a + (cosa12 )z2b ]2
by: SNRo, = a2N SNRi2 sin 2 [a2 + P2b

2 [a + Z2b j

SNRo2 = N SNRi, sin2 a,2 [a +z b (2.28)
[a +P2b]

where: 1 1 Z2b
where: NSNRi sin2 a P2b N2 SNRi22 sin a,2 N SNRi,

Where the notation z2b represents the numerator zero for scenario 2 and DOA case b.

Note that P2b < Z2b. Also note that for scenario 2, SNRoI is small since it is multiplied by a

second order zero at a = 0. As a increases, SNRoj increases with some possible wrinkles

in the magnitude curve added by the numerator zeros (a z2bcosal2) and denominator

poles ( a = 4p,). Note that since the cosa12 term in the numerator of equation (2.28)

can be less than zero the possibility exists that SNRoj could equal zero for nonzero a. The

most interesting behavior is displayed by SNRo2. For very small a, SNRo2 is near the

maximum value SNRo2= NSNRi2. As a increases away from zero, the SNRo2 function rolls

off at the -3 dB point of a2 = P2b and continues to decrease until a2 = Z2b ( SNRil =

[Nsin2a,12-1 ). At a2 = z2b the SNRo2 roll-off attenuation is halted and SNRo2 begins

increasing since the pole at p2b is canceled by the zero at a2 = z2b. Over the range of a

specified by this scenario, the point a2 = z2b represents a minimum of SNRo2. Substituting

a2 = z2b into equation (2.28) gives SNRo1 = 1 and SNRo2 = 4NSNRi2/(NSNRi2 + 5)= 4.

Note that similar expressions for the case of orthogonal users ( case a above) can be

readily found by simply setting sin2a12 = 1 and cos2a12 = 0 into equation (2.28).

In summary, for most of scenario 2 the array is unable to resolve the low-power user 1

input signal from the input noise. The array can just begin to resolve the user 1 input signal

from the input noise when a2 = z2b ( SNRi1 = [Nsin2al2-1 ). At this point, SNRoI = 1 and

SNRo2 4 which represents a minimum value of SNRo2 over this scenario.

Two Equal-Power Users

If the two users have equal input power levels the output SNRs are equal:

NSNRi (1 + cosa2 + NSNRi sin2 a,,)2
SNRo (2.30)
2 1+ cosao2 + NSNRi sin2a 2 (2+ NSNRi(1 cosa12))

and for the case of well-separated users (case b: 0 < cos2a 2 << sin2a12 < 1) the output

SNRs may be approximated as:

NSNRi (NSNRi sin'2 a2)2
SNRo -------
2 NSNRi sin2 a12(NSNRi(l- cosa2)) (2.31)
2S (l+cosa12)

If the users are spatially orthogonal (case a) then sin2a12 = 1 and cos2al2 = 0 and the

output SNRs are exactly equal to SNRo = NSNRi/2, which is consistent with equation


Two Strong Users

For this scenario (#4) we have two strong users with unequal input power levels.

For case b ( 0 < cos2aI2 << sin2a12 < 1 ) the numerator terms of equation (2.26)

may be modified by multiplying all SNRi2 terms by an additional sin2a12 term. The


denominator (equation (2.23))changes slightly:

2[ 1 4 1
D4,b = a +a N2NR2 sin2 + 4 + 1 + (2.32)
NaSNRgi sin' a,, NSNRi2 N2SNRi sin a12

Terms containing higher powers of a dominate the numerators (equation (2.22)) and

denominator (equation (2.32)) of the output SNR expression (2.24). This condition

represents the presence of two strong users of unequal power: SNRo, >> SNRo2. The

expressions for the (scenario 4, case b) output SNRs may be approximated as:

(1+ NSNRi2 sin2 a2)2
NSNRi, sin a12 (2.33)

SNRo2 = NSNRi2 sin2 a12

Note that for large input SNR the expressions in equation (2.33) should be nearly equal.

The difference between the output SNRs for this scenario is:

1 1
ASNRo = SNRo, SNRo, 2 + NS 2 = 2 + 1 (2.34)
NSNRi, sin2 aa2 SNRo,

Which is very similar to the expression for output spread for orthogonal users given by

equation (2.14).

Numerical Results

This section will first present numerical solutions to corroborate the two-user analysis

of the previous section. The data will be presented in plots of the output SNRs (equation

(2.26)) versus the input near-far ratio (a2). Array performance will be evaluated for

spatially orthogonal users and well separated users evaluated over a wide range of a2.

Specific comparisons between analytical and numerical solutions will be made for critical

points identified in the analysis.

Performance for a linear MMSE array was investigated by plotting the output SNRs

of two signals while varying one signal input level and holding the other at a fixed value of

input SNR. For each point the MMSE weights and corresponding output SNRs were

calculated using equations (1.6)-(1.10) and equation (2.1). The DOAs were fixed to

specified values. The plot below shows the output SNRs for the case of a three element

array with two users. User 1 is positioned broadside to the array (DOA = 0 degrees) with

varying input SNR, and user 2 is positioned at 41.8 degrees from broadside with an input

SNR of 10. This DOA condition gives the least amount of source separation which allows

the sources to be spatially orthogonal.

Figure 2.2 shows the (N = 3) array output SNRs versus a2 for a two-signal scenario

over the range 10-5 < a2 <105. The users are spatially orthogonal, al2 = 90 degrees and

the user 2 input SNR is 10. For a2 = 10-5 the array is in essence a single user system,

SNRo2 NSNRi2 = 30 which is the maximum possible output SNR for user 2 and SNRo1

0. As a (SNRi1) increases, SNRo2 begins its roll-off attenuation: SNRo2 rolls off 3 dB by

the time a2 = .0011, as predicted by equation (2.28). As a increases the SNRo2 roll-off

continues until it reaches the minimum value of 3.33 at a2 = 0.033. At this point the array

is just able to resolve signal 1 from the input noise and SNRol = 0.89. These numbers

agree well with equations (2.10) and (2.28).

o *. ..............

10 F 10d Near-Far Ratio: a2 10s

Figure 2.2 Output SNRs versus near-far ratio for spatially orthgonal incident

For the scenario a2 = 1 the output SNRs are equal to 15 (see eqn. (2.6)). As a2

becomes very large the output SNRs lose their dependence on a2 and approach nearly the

same value. This is shown in equations (2.33) and (2.34). At large near-far ratios (a2 =

105) SNRol = 32.03 and SNRo2 = 30. The difference between their values is ASNRo = 2.03

which agrees precisely with equations (2.14) and (2.34).

Figure 2.3 shows the performance for DOA spacings greater than the Rayleigh limit.

For this scenario user 1 remains at array broadside and the DOA for user 2 moves to 90

degrees (in line with the array). This case gives the vector quantities UJTU2*= Ncosa12= -

1, cos2ai2 = 1/9 and sin2 al2 = 8/9. When a2 is small (a2 = 10"5) SNRo2 = NSNRi2 30

as in the previous case. As a2 increases to 1.18x10-3 SNRo2 rolls off approximately -3 dB


to a value of 15 and SNRo, increases from nearly zero to 6.5x1073. Note that equation

(2.28) approximates SNRol = 5x103 and a -3 dB roll-off point of a2 = 1.25x103.

At a2 = 0.042, SNRoi = 0.798 and SNRo2 takes on a minimum value of 2.807.

Equation (2.28) predicts a minimum value of SNRo2= 3.87 will occur for a2 = zzb =

0.0375 and SNRo, = 1.01. From the plot SNRo, = SNRo2 = 9.94 when a2 = 1 which

agrees with the approximations given by equation (2.31). When a2 increases beyond unity

the outputs quickly level to values near the maximum of SNRo2: SNRol = 28.64 and

SNRo2 = 26.61 at a2 = Ix105. This gives an output spread (eqn. (2.34)) of ASNRo = 2.03.

Figure 2.4 shows what happens when the DOA spacings are less than the Rayleigh

limit In this case, signal source 1 remains at broadside while source 2 moves to DOA2 =

Figure 2.3 Output SNRs versus near-far ratio for nonorthogonal incident signals
with DOA spacings greater than the Rayleigh limit.


100 Near-Far Ratio: a2 10'

20 deg., about half the Rayleigh limit. The outputs are degraded for a2 >> 1, compared to

the two previous plots.


10D Near-Far Ratio: a2 1

Figure 2.4 Output SNR versus near-far ratio for incident nonorthogonal signals
with DOA spacings less than the Rayleigh limit


Consider for a moment that filter weights in an adaptive process are random variables

with salient statistical properties. Adaptive processors are unable to perfectly track the

corresponding steady-state solution; this induces a penalty known as misadjustment93 The

goal of this chapter is determine the effect of the step size of the complex LMS

algorithm" on the steady-state SNR-leveling performance of the multi-user processor.

This concern is motivated by two factors. First, from an intuitive standpoint, it seems that

the multi-user processor might cease to level the output SNRs if the misadjustment

becomes too large. Second, the multi-user array processor might share some similarities to

an LMS automatic line enhancer (ALE) for multiple time signals. Fisher and Bershad97

studied the misadjustment performance of an LMS-ALE for the case of multiple sinusoids

in AWGN and found the equalizer misadjustment to be especially sensitive to the step size.

This work will rely heavily of the work of Senne95 who developed expressions for the

time-dependent and steady-state weight covariance matrix of a real LMS adaptive

processor. Other authors have investigated the transient and steady-state behavior of the

complex LMS weight covariance matrix.96'97 and have found that the steady-state

eigenvalues are very similar to those derived by Senne for the real LMS algorithm. This

detail will be applied in a later section of this chapter.

Other authors have examined the performance of adaptive processors subjected to

hard and soft constraints on SNR as well as optimizations of SNR itself subject to

nonlinear constraints.98'9

This chapter will be divided into three sections. The first section will give the

development of a performance measure (cost function) which is based on the mean output

SNR of the adaptive array. The second section develops a performance measure based on

the variance of the array output SNR. The last section presents numerical results which

allow comparisons between the analysis and simulations. The cost functions will be used

to find acceptable bounds on the LMS step size. It is found that the upper bound on step-

size arising from stability arguments also gives acceptable output SNR performance for

the multi-user processor.

Mean-Based Performance Measure

If the adaptive weights of an LMS processor are considered to be time-varying

random variables, w(n) = wo + v(n), the mth user's time-dependent output SNR results

from substituting the expression for w(n) into equation (1.10):

(w + v(n)) u.,
SNRo(n), = SNRi, (w0 +(3.1)
IIwo + v(n)11

The random, time-varying weight component is represented by v(n), the mean of the

weight vector is the steady-state MMSE weight vector, wo = R7'p, and the discrete time-

dependence is introduced by the variable n. Note that the phase-shift vector Um and SNRi,

are constants.

In order to derive an expression for the performance measure we need to find the

mean of the output SNR. Some assumptions facilitate this effort:

1. Ilv(n)+ w.II 1 I1wol ; instantaneous deviations of norm(w) from
norm(w.) are negligible because of small step size gt.
2. v(n) are independent from sample-to-sample.
3. Input signals and noise are i.i.d., stationary and ergodic.
4. The elements of w(n) form a jointly Gaussian process.

The resulting mean of the output SNR is:

E[SNRo(n), ] = SNRi, I + SNRi uC (3.2)
IIW,1I2 .wl3
where the first term on the right-hand side is SNRom, the steady-state output SNR defined

earlier in equation (1.10). The matrix Cw = cov(v(n)) is the steady-state weight covariance

matrix given by Senne.9 The matrix is not a function of time, n, since v(n) components are


The last term on the right-hand side will cause the average output SNR for user m to

increase as the adaptive weight covariance increases. This might have little effect for weak

users; the term is not negligible for strong users. The average output SNRs may no longer

be leveled if the term becomes too large. This point is clarified in Appendix A which

presents some general characteristics of the SNR performance surface.

The first measure of performance is the difference between the average and steady-

state output SNRs divided by the steady-state output SNR:

E[SNRo(n)]-SNRo. SNRi, uT Cu:
SC (3.3)
SNRo. SNRo, |w0 112

Where SNRo, is the steady-state output SNR for user m and is defined by the first term on

the right side of equation (3.2) above. The quantity CI o- 1/(NSNRi2) << 1 is a constant

which serves as an upper bound which will limit the excursions of the time-dependent

SNR (resulting from the adaptive process) from the steady-state solution. The constant

will be defined in more detail later in the analysis. If the adaptive process is allowed to

stray too far from the steady-state solution the output SNRs may no longer be held to

nearly the same level and near/far limited performance may result. Using this relationship

with equation (3.2) above gives an expression in which step size dependence is expressed

indirectly through C,:
T "'2 SNRo(3
u:C,,u' cllw SNR (3.4)

The quantity on the left-hand side may also be lower-bounded by using the maximin


1min mirN(qC q"
in (a,) = q e (3.5)
q#0 ( q

where CN denotes the complex N-dimensional space, q is an N-length vector and Xw =

min{ all eigenvalues of C,}. Let q of equation (3.5) equal Um of equation (3.4) where

UmTUm* = N. Equation (3.5) may then be used to form a lower bound for equation (3.4).

The left-hand term of equation (3.4) would then be bounded by:

min u 2C .112 SNRo.
SNRi, (3.6)

The next task is to find a useful expression for the lower bound of equation (3.6).

Senne95 has shown that the weight covariance matrix is diagonalized by the eigenvector

matrix of R, the input autocorrelation matrix. For an N = 3 array, the diagonal of A, the

eigenvalue matrix of Cw, is:

diagIA, 1 (1pA ^-1 I- (3.7)
2 (1P r 21 Ip 2 1-A 2 1-pA2, 2 1-pA3J

where A1, A2, and A3 are the eigenvalues of R, and p. is the LMS step size. Compton'00 has

shown that when SNRil >> SNRi2 > 1, the signal and noise eigenvalues of an N-by-N

autocorrelation matrix R are approximated by:

N' 2 V 1 (3.8)
2 1+( SNRi( +uNu;21)

1, = a m=3...N

where the largest eigenvalue A, is established by the strongest incident signal and A, are

the noise-only eigenvalues.

If the step size is small (.u < 1/1A)the smallest A, from equation (3.7) may be

approximated by:

S) I-w (-- 1
2 _2 I-pA t 2 1-pA, 2 1-p J, (3.9)


The simplification in equation (3.8) results directly from our assumption of small step size.

If p. is small the term in brackets and the term in parentheses outside of the brackets are

equivalent to unity.

This establishes the upper and lower bounds on uC,,u :

1NpE2 < u C,,u < CJwo2 SN (3.10)
2 SNRi,

If all the incident signals are well-separated, strong and perfectly correlated to the

reference waveforms the minimum MSE should be well-approximated by the output noise

power: e'. a 1w 11,2. If user m = 1 is the strongest user then SNRil can be determined

in terms of the eigenvalues via equation (3.8) (N SNRi = ( x/~ ) -1 = I /a2 ). The

constant C, from equation (3.3) establishes a bound on the average excursions from the

steady-state output SNR due to misadjustment, so it must be made suitably small. The

strongest user user 1- should have a steady-state output SNR leveled to approximately

the same value as user 2, therefore SNRol = SNRo2 = N-SNRi2. Since even SNRi2 >> 1

then C, = p /(2N*SNRi2) should give acceptable results where go is a constant (0 < pX < 1)

and represents the unormalized step size This results in an upper bound on ut:

U- (3.11)

which is similar to the upper bound on the LMS step size derived from stability

arguments,22,93 and is consistent with our earlier assumption of a small step size.

Variance-Based Performance Measure

The second cost function is the variance of the adaptive output SNR divided by the

squared mean of the adaptive output SNR. Choosing gt to keep this measure small will

limit excursions from the mean of the adaptive process. The second moment of output

SNR is:

s2] +1(n))T U. 42
E[SNRo(n),] = E(w + v())U m (3.12)

Senne assumed that the filter weights, w(n) = w. + v(n), form a jointly Gaussian process.

In the following analysis this assumption is used to expand fourth-order joint moments in

terms of the second-order moments via the Gaussian moment factoring theorem.101

If we assume the vector v(n) is multivariate and complex Gaussian we can express the

fourth order moments as functions of the second-order moments.'0' Expanding the

bracketed vector term and discarding terms that would give odd-ordered moments of v(n)

results in several terms:

EI(w + vf)u = w u,. 4 + 4. lw UI u.C,,u,

+ E[(w'Um )2(v~u +)2 (Wu )2(TU)2] (3.13)
+ Y Ju, u;u,,uq;I E[vv,v;,v,* i]
11,=1, =1 1=114 =1
where we have dropped the time-dependent notation from the weights. The third term on

the right is zero. The fourth-order moment may be expanded in terms of the second-order


E[v, vvv, v ] = E[v v, ]Ev v +E[v vv ]E[v v, ] (3.14)

The summation term becomes:

XX auI,,u,u,; E[v v V, V ]= 2 (uTC,,u,)2 (3.15)
I=i 4,=4l = -1

The second moment of the output SNR is:

2 (IWOTU.2 C+uC,, 2u -lw.U.14
E[SNRo(n)2 ]= SNRi' 2(w (3.16)

and the variance of the output SNR is (from equation (3.2)):

2 i12 I +u(CMr*, V"2-w 4u
aSNR() = SNRi I 14 (3.17)

= E[SNRo,(n)]2 SNRo.

where the second line of equation (3.17) follows from (3.2) and SNRo. is the steady-state

output SNR of the m'th signal. The performance measure becomes:

2T SNRo02
SNR=o1 < < C2 (3.18)
E[SNRo(n), ] E SNRo(n),

where C2oc 1/(NSNRi2) << 1 is a constant to be defined later.

Substituting quantities from equation (3.2) into equation (3.18) results in the


S .SRo[ 1I
uCu: Vu 5 Iw|1 2SNRo -C 1 (3.19)
SNRi. C2

Since C2 < 1 the radical term may be approximated by a two-term series
(1- C2)2 = 1+ C2/2 and the inequality takes the same form as equation (3.4). The same

procedure used to determine the step size upper bound for the first performance measure

may be followed for this second performance measure with the same results: pt /,/At1.

Simulation Results

This section will use simulations to corroborate the steady-state analytical results of

previous sections.

The next two plots show curves of the quantities obtained from equations (3.2) and

(3.18). Figure 3.1 shows the average output SNRs versus the step size coefficient for

means resulting from simulation and from expression (3.2). The input scenario is SNRi,=

10, SNRi2 = 106, DOAi = 0 deg., DOA2 = 90 deg., N = 3. The initial weight for the LMS

simulation was MMSE weight vector from equation (1.8), w, = R1p.

Plots of the variance-based performance measure are shown in Figure 3.2. The input

scenario is identical to the one described in Figure 3.1.The curves show the ratio of

variance divided by the squared mean of the output SNR obtained through analysis

Average output SNR vs. step-size coefficient

SNRi2 = 60 dB
DOA1 = 0 deg
DOA2 = 90 deg

CL User2: sim


User 2: calc

User 1: sim. calc
25 -

20 ..... ..--. --.. ...
10" 10" 10- 100
Step-size coefficient

Figure 3.1 Output SNR versus step-size coefficient go. The step size is
normalized by the input power

(equation (3.18)) and simulation. The curves show, as we might expect, that the

performance will be worst for the strongest user. Selecting step size for acceptable strong

user performance will lower-bound the performance for the remaining weaker users.

2nd moment performance measure vs. step size coefficient

Figure 3.2 Variance-based estimator versus step-size coefficient Po.
The step size is normalized by the input power. The input scenario is
identical to the one described in Figure 3.1

Selecting a small performance measure will limit the point-by-point excursions of the

output SNR from the steady-state value and will give results which are acceptable in the

mean. If 1% is selected as an acceptable value of the performance measure then from the

curves for user 2 above it would appear that p 0. 1/Tr(R) is the maximum allowable step

size. This is in agreement with existing rules-of-thumb for selecting step size for stability

and convergence. For these conditions, it would appear that upper bounds on LMS step

size derived from convergence arguments will give acceptable performance of the

multiuser processor.







10- 10- 10- 10"
Step size coefficient


This chapter examines the steady-state outage probability and outage-based capacity

of a single cell containing multiple directional signal sources transmitting to a central base

station. The only fluctuation in the signal levels is due to lognormally-distributed power

control error in the multiple transmitters. The central receiver consists of an array of N

isotropic sensors, K minimum mean-squared error single-user beamforming processors

and a bank of conventional detectors. The performance measures are outage probability

and outage-based capacity. The goal is to find simple expressions which relate the outage-

based capacity to the antenna array parameters, the number of active signal sources and

the degree of power control error. The analytical results are expressed in terms of the

number of users per array element which may be supported for a given outage probability.

Analytical results are found to agree closely with those obtained from Monte Carlo


This work is unique in several respects. First, most previous work examined uplink

performance for a traditional single-channel receiver. Second, the authors which

considered a base station antenna and imperfect power control presented simulation

results for the case of closed-loop power control.4'85 As was mentioned in Chapter 1,

power control error will be modeled by incident power levels which are lognormally

distributed. This model has gained some acceptance for open-loop power-controlled

systems. Its suitability as a model for closed-loop power-controlled systems remains an

open issue.

The remainder of the chapter is divided into several sections. A qualitative description

of the array and the receiver is given in the next section. The third section outlines the

development of the analytical model and the derivation of the expressions for uplink

capacity. The fourth section describes the Monte Carlo simulations and compares the

analytical and the simulated results. The last section presents the conclusions.

System Description

The incident DS-CDMA signals are independent and originate from independent

transmitters which are arbitrarily placed about the base station receiving antenna. The K

active transmitters (located in a single cell) result in incident signals with independent

directions-of-arrival (DOA) uniformly distributed over the interval [0, 27). The

transmitters' output levels are continuously adjustable and have an infinite dynamic range.

The output power adjusted by an unspecified power control algorithm results in output

power which is independent between sources and which is also lognormally distributed.

The modulation format is BPSK.

The base station receiving array consists of N ideal isotropic sensors arranged in a

circle. Adjacent sensors are separated by a distance with the electrical equivalent of one-

half wavelength. The sensor array feeds K separate banks of N complex weights

controlled by K MMSE beamforming processors. The optimum steady-state weight

vector for each beamformer is given by equation (1.8), wo = R'1p. Each of the K

beamformer outputs feeds a distinct conventional DS-SS detector which in turn provides

an estimate of the demodulated output The receiver is assumed to be capable of perfect

carrier tracking of the desired signal. Unlike the multiple-user array of the previous

chapters, the array/detector combination is devoted to detection of a single desired signal

amidst multi-access interference. A diagram is shown in Figure 4.1.

Figure 4.1 A bank of baseband single-user beamformers and conventional detectors
sharing an array of sensors


The stationary, complex base-band output of the sensor array is found by combining

equations (1.3) and (1.4):

Y(t) = AI1O" 20 c.(t T.)b(t T,)u, + n(t) (4.1)

where A is the peak amplitude and em is N(O, pc2). The sensor outputs are weighted and

summed by the beamformer and then processed by the conventional detector which forms

an estimate of the current output bit. The signal from source 1 is considered the desired

signal. The remaining K-l signals are considered multi-access interference.

The array tends to steer a pattern lobe towards the desired signal and also tends to

steer nulls of finite depth towards the N-2 strongest interferers when K 2 N-1. Because

the strongest signals are attenuated, no single signal dominates the output statistics and the

output may be well-approximated as Gaussian. An additional assumption of long codes

(codes which span more than one data symbol) allows the PN sequences to be modeled as

random codes. The random code model of Pursley'02 will be used here. More refined

models exist, but the additional complexity they introduce tends to improve accuracy

when only a few users are present. 103,104.05

From Pickholtz et al.' 6 we know that the effective bit energy to noise spectral density

ratio may be approximated by (Eb/No)eff= Nc-SINR. The random code approximation may

be modified to account for varying input amplitudes among the incident signals due to

their power control error. Using one of the intermediate steps (equation (14) from

Pursley'M) allows a convenient expression for (Eb/No)f while retaining the cross-

correlative properties of the PN codes:

E = NcSINR = 1 NcSNRo, (4.2)
1 ./,^ .___ ^ -- (4.2)
3N2_ SNiRo. r.i +

where SNRo. is the array output SNR of the mth incident signal and r,j is a

crosscorrelation parameter between the first and mth PN codes. If SNRo, is replaced by its

sample mean and the remaining sum of ri terms is replaced by the random code

approximation (equation (16) from Pursley102) the expression becomes:

(E = NSNRo, (43)
SNo X SNRo, +1l
3 m=2

The quantity SNRok may be rewritten as:

SNRo, = 10e1"0 Gp, SNRi (4.4)

where Gp. = Iwu, 2AwoE is the normalized power gain of the array towards the mth

signal and SNRi = A 2/c2.

The power gain is a complicated function of the input scenario and the array

geometry. In order to simplify the analysis we will use a simplifying assumption that the

array power gain with respect to the first user, Gpl = Gpd, will be approximately equal to

its upper bound N, the number of array elements. The MMSE array gain towards the

interferers (Gpm,m = 2 ... K) is difficult to characterize. Intuitively, we might expect that

the avg{Gpm } = Gpi might be well-approximated by the average (over DOA) normalized

power gain (Gpg) of a classical beamformer (wo = ul) with no adaptive null-steering

capability. The value of Gpg ranges from unity for a single element to 1.6 for a 30-

element circular array. Simulated results presented shortly will show that these

assumptions with regard to the individual quantities (Gpd, Gpi) are not always valid over

the conditions of interest. The average of the ratio GpdGpi ,however, does provide a

reasonably good fit to the simulated results when the above assumptions (Gpd N, Gpi

N) are used.

If there are many interfering signals the sum in the denominator of equation (4.3) may

be approximated by averages:

(E, = NNSNRilOx(o
=1,o, (4.5)
N\ 2 ~ 1.
v -(K-1)SNRiGpIO10-2W +1

Further explanation is required. The sum in the denominator of equation (4.3) is almost a

sample mean and may be approximated by an ensemble average. Assume the power gain

and the exponential of equation (4.4) are independent: the expectations apply separately.

This results in the base-10 exponential term, the (K-1) term and Gpi in the denominator of

(4.5). This approximation causes a slight overestimation of interference: by breaking a sum

of products into a product of sums we have invoked Schwartz's inequality twice in

succession to get from (4.2) to (4.3) and to get from (4.3) to (4.5). Our simplifications

have equated sample means and ensemble averages and also ignored the complicated

nature of the array response by treating the power gain as an averaged quantity.

Simulations of the power gain show our assumptions regarding Gp may give

acceptable results. Figure 4.2 shows the results of Monte Carlo simulations of averaged

values of (MMSE) Gp for a desired signal (Gpd), an interferer (Gpi) and their quotient for

a varying number of users, 3 dB of power control error and a 30-element array. The

curves show that as the number of users exceeds the degrees-of-freedom of the array our

simulated values for the individual gains are not very close to approximated values of Gpd

- N = 30 and Gpi, = Gpav, = 1.6 but their quotient Gpd/Gp, = N = 30. So, for this

circumstance it might be useful to approximate the gains in equation (4.5) as Gpd = N and

Gpi, = 1.

Why will this work? As the number of users increases and the system becomes

interference-limited the "1" term in the denominator of equation (4.5) becomes negligible

leaving a good approximation of the quotient of the gains as Gpd/Gpi N. When there are

few users the multi-access interference is negligible, Gpd = N and the approximation will

still hold. These gain approximations do not hold separately to predict the desired signal

output power or the interference output power, but their combination might prove useful

in calculating outage.

What about more extreme scenarios? Figure 4.3 shows the gains and the gain ratio for

a power control error of 10 dB. The individual approximations Gpd N and Gpi, Gp,

= 1.6 are even worse than before. However, the simulated gain ratio Gpd/Gpi N/1.3 for

K > N. This is a little closer to the ratio of the individual gains Gpd IGpag. = N/1.6. This,

GpGp=N 30




power control error = 3 dB N =30
101 --- ---------,- -----,-
0 20 40 60 80 100 120
Number of Incident Signals
Figure 4.2 Gain versus the number of incident signals. Power control error is

.................... ................... .....................
Gpd/ Gi
101 Gpd



power control error = 10 dB N=30
10"1 ,,,-
0 20 40 60 80 100 120
Number of Active Users

Figure 4.3 Gain versus the number of incident signals. Power control error is
10 dB.

and other simulations, indicate that choosing Gpd = N and Gpi = Gpa, = 1.3 for this

model is a good approximation for outage calculations over the ranges of power control

error examined here (0 5 O < 10). Note that Gpi = Gpav = 1.3 also arises from a

sample mean (over N) of the average power gain (over DOA) for N = 1,2,4,8,15 and 30-

element arrays for a classical beamformer.

Approximating the interference as an average quantity in equation (4.5) eliminates

complex scenario-by-scenario interactions between the desired signal and the interference

in the analytical model. It also leaves the lognormally distributed desired signal as the only

random variable in the model. The resulting distribution of the (Eb/No),y in equation (4.5)

is therefore log-normal and results in simple expressions for the outage probability and the

capacity. Outage occurs when the (Eb/No)e, is less than some threshold. Converting

equation (4.5) to dB and noting that ej is N(O,&p,.) results in a simple expression for the

outage probability:

Pr., = Pr((< Eb ZdB -d (4.6)

where Q(x) is the complementary Gaussian CDF, a is the desired threshold in dB and:

N- "
ZdB = 10 logo (4.7)

The quantity D is the denominator of equation (4.5). The quantity E/No (Ti/c)SNRi

=NcSNRi represents the equivalent bit energy to noise density ratio for a single incident

signal and a single array element. Note that Naguib et al.78 has also developed a Q-

function upper bound for outage probability for the case of perfect power control. For the

case of imperfect closed-loop power control Naguib et al.81 presented the simulated means

and variances of (EI/No),.f

Some simple algebraic manipulations result in the average capacity as a function of

desired outage probability:

K = N -1 (E +1

where: (4.8)
3,1 :- (I10a 10
C= 23M 10(.)

where for high EANo the capacity is approximately linear in processing gain, Nc, or the

number of array elements N but decays exponentially as the power control error increases.

Note also that the threshold is no longer in dB. Note that if we interpret equation (4.8)

as being linear in N, we can define a per-element capacity by noting the slope of the line.

Note also that the capacity asymptotically approaches a finite maximum as Eb/No

increases. A similar effect was noted by Naguib and Paulraj82 for the case of a 2-D RAKE

combiner at the base station receiver and perfect power control in the mobiles. They

examined the capacity as the equal incident signal levels went to infinity and named the

parameter asymptotic capacity. A simple expression for uplink capacity was also

formulated by Suard et. al.76 for a post-detection combiner. The model for power control

error was restricted to a single user with an incident power level 10 dB higher than the

other users.

Simulations and Results

Monte-Carlo simulations were used to corroborate the analytical results given by

equation (4.6). Autocorrelation matrices for the desired signal, interferers and noise were

generated by ensemble averages as in equation (1.6). Signal DOAs were uniformly

distributed over [0, 27). The processing gain was 127 and Eb/No = 7 for a single antenna

element and no power control error. For a single trial the MMSE weights were calculated

via equation (1.8) and the desired signal, interference and noise power out of the array

were then used to calculate (Eb/No),f via equation (4.3). An outage condition was judged

to exist for that trial if (Eb/No)ef < 7 dB (i.e. from equation (4.6), '5 = 7 dB). The

quantities were averaged over 20,000 trials for each combination of power control error,

number of users and number of array elements.

Curves of the outage probability Pr((Eb/No)4ff < 7 dB) versus the number of users

are shown in Figure 4.4. The figure contains curves from equation (4.6) as well as the

Monte Carlo simulations. Power control error is 4 dB. Note that for 20,000 trials, curves

in Fig. 4.4 might be inaccurate for outage less than 102.

Figure 4.5 shows the total array/receiver capacity versus the number of array elements

with outage probability as a parameter. The curves are formed by plotting constant

contours of a three-dimensional surface formed by Pr((Eb/No)ff < 7 dB) as it varies over

K and N. The solid curves show simulated results; dashed lines show the constant-value

contours of equation (4.6) for power control error equal to 4 dB.

The curves of Figure 4.5 show the user capacity of the array/detector is roughly linear

in N. We may therefore use as a performance measure per-element capacity, the number

of users per array element which may be supported for the given values of outage

probability and power control error. As noted earlier the analytical expression for the per-

element capacity may be obtained by noting the slope of the total capacity line in equation

(4.8) with N as the independent variable. A point of note: the average power gain (Gpg)

in equation (4.8) has a slight dependence on N: it is equal to unity for a single element and

is equal to 1.6 for a 30-element array. For the sake of simplicity, this slight dependence is

ignored and a mean value of Gpa, = 1.3 is assumed, which was noted earlier when

comparing simulated Gpd Gpi curves in figures 4.2 and 4.3.

The per-element capacity versus the power control error for Pr((Eb/No)ff < 7 dB) =

0.02 is shown in the upper plot of Figure 4.6. For simulated data, the normalized capacity

was determined by extracting the approximate slopes of the capacity curves (as shown in

Figure 4.4) via a linear least-squares curve fit The analytical results were calculated from

Figure 4.4 Outage probability versus the number of incident signals. Power
control error is 4 dB. The number of array elements N is a parameter

110 ....... ..................... .. --.. .. ......
100 ........... .......... ..... ...... ... ........
90 .. ... ... .. ""---------- -- -. .... ..........
Sr(Z< dB)= 0.07 / .......
80 -------- ------*.. .-. z.- .--o.o
Capacity .... .'. .'""'-|Pr(Z < 7dB)= 0.03

0 ... ........... Equati.......on ..........
140 '. ".. -- i..---*.------- --.---- ---- ---- .. ...........
50 ........ /7 _.,. .......... .......... ...........

.'//" --Simulation
20 ------ .. V...... .......... .... ....... ......................
20 ..---.----. Equation (8)
10 -.. / ......... ....... .............. ......... ...........

5 10 15 20 25 30
Number of Array Elements

Figure 4.5 Capacity versus the number of array elements with outage probability
as a parameter. The power control error is 3 dB.



120 140

0 20 40 60 80 100
Number of Incident Signals


Per 30
Element Simulations
Capacity --------- Analysis, eq. (4.8)
20 -----..................--..* -..........-- -

0 -- -----------

0 2 4 6 8 10


-20 ......... ..... .......... ........ ......

-10 --------- --------------------------------
10 ...... a

0 --------- Aniysis, eq. (4.8)

0 2 4 6 8 10
(a) Power Control Error (dB)

lower plot of Igure 4.6 shows the intercept point of the line in equation (4.8). Obviously
-10 ---- ----- -- -------- '-----.--- --------- S

the analytical and simulated results are not in as close agreement as the upper plot(4.8) Note
0 2 4 6 8 10
(1) Power Control Error (dB)

Figure 4.6 Normalized array capacity versus power control error in dB.

the slope of the system capacity line given in equation (4.8). As the curves show, the

analytical and simulated results for the per-element capacity are in close agreement. The

lower plot of Figure 4.6 shows the intercept point of the line in equation (4.8). Obviously

the analytical and simulated results are not in as close agreement as the upper plot. Note

that to determine the overall system capacity given an array of N elements it would be

necessary to know the per-element capacity as well as the intercept.


In this chapter we attempted to develop accurate, simple analytical expressions for the

outage-based uplink capacity in an idealized single-cell DS-CDMA system with multiple,

possibly near/far, signals incident on a base-station receiving array. Power control error

was assumed to be lognormally distributed. Some simplifying assumptions regarding the

interference and the array response allowed an approximate expression for the uplink

capacity that was compared with results from Monte Carlo simulations.

The approximate expression given in equation (4.8) shows that a roughly linear

relationship exists between the capacity and the number of array elements. The overall

capacity K consists of two components. The first component is the slope of the line in N

and has been defined as the per-element capacity, the number of users per array element

which may be supported for a given level of outage probability and power control error.

Equation (4.8) indicates that the per-element capacity is not dependent on the nominal

input level (i.e. the input level without power control error: SNRi) but decreases

exponentially with increasing power control error. For the levels of power control error

examined here the term which dominates the exponential roll-off is opeQ'l(Prouage)/10 of

equation (4.8). The decrease in the per-element capacity is therefore dependent on the

outage requirements and the standard deviation in dB of the power control error. Since

the per-element capacity is insensitive to the nominal input levels and the array size, but is

keenly dependent on the degree of power control error and the outage, it might serve as a

useful asymptotic measure of the performance for this array/receiver.

The second component of equation (4.8) is the intercept term equal to 1-C(Eb/No)'

where C is defined in (4.8). The intercept is inversely proportional to the nominal input

levels. As the input levels decrease this term becomes larger, diminishing the overall

capacity. In a plot of capacity (K) versus array elements (N), the capacity line moves away

from the origin along the horizontal N-axis as the nominal input levels decrease. The

intercept is weakly dependent on the power control error (via C) and also on the number

of array elements (via Gpa,,) and is not dependent on the outage probability. Analytical

and simulated results do not agree as closely as those for per-element capacity.

The agreement or lack of it between the analytical and simulated results for per-

element capacity and the intercept can be interpreted in terms of outage probability curves

shown in Figure (4.4). The analytical model can predict well the horizontal spacing

between the continuous outage probability curves. The per element capacity predicts the

incremental increase in capacity with increasing array elements and is a measure of the

horizontal displacements of the outage curves relative to one-another. The less reliable

prediction made by the model is the horizontal placement of the outage curves relative to a

point on the horizontal axis. This enters into the model via the intercept parameter

described above.

The simple model presented here was based on some simplifying assumptions

regarding the array response towards the incident signals. In spite of this, the analytical

model accurately predicts some aspects of the array/receiver performance when directional

signals are employed with lognormally distributed power control error. The directional

signals originated in a single cell and were not subjected to any kind of environmentally-


induced fading. The system performance in the presence of fading and interference

resulting from outer cells is the topic of the next chapter.


This chapter examines base station receiver performance when incident signals are

subjected to frequency-nonselective Rayleigh multipath fading and lognormally distributed

shadow fading. The effects of multi-access interference from outer cells subjected to

shadow fading will also be included in the incident signal model. The power control

error model will continue to be described by a lognormally-distributed random variable

and the receiver still consists of an array of N ideal isotropic sensors, K minimum mean-

squared error single-user array processors and a bank of conventional detectors (see

Figure 4.1).

The goal of this chapter is to determine the performance dependence on the number

of active signal sources, the number of array elements and the degree of power control

error. As before, the performance will be expressed by outage probability and per-element

capacity. Unfortunately the introduction of fading and outer cell interference further

complicates the development of simple analytical models. In spite of these complications

the simulated results closely follow some general trends established by the model in

Chapter 4. In particular, curves of outage-based capacity continue to be linear in N with

slopes that decrease exponentially with increasing power control error.

Previous authors have investigated the performance of optimum combining from the

standpoint of interference rejection. Winters23'44 and other authors60o61 have examined

contributions to TDMA system performance while Naguib et al.7684 have studied CDMA

systems. Unlike previous work, this research examines the outage-based capacity for the

case of imperfect power control and attempts to provide some analytical models which

would allow easy assessment of performance.

The remainder of this chapter is divided into six sections. The first section will present

the receiver model for single-cell signals subjected to Rayleigh fading. The second section

will extend the model to the multi-cell case. The third section will quickly revisit the model

of chapter four and introduce a second analytical technique. The fourth section will give a

brief description of the simulation parameters and compare the simulated results with the

analytical results.. The fifth section entitled Conclusions and Discussion will review the

results in some detail. The last section gives a summary.

A Single Cell with Signals Subjected to Rayleigh Fading

Multipath fading arises when propagating electromagnetic waves originating from a

single signal source arrive at a receiver via different propagation paths. The individual

paths may include line-of-sight propagation as well as paths resulting from reflection off of

one or more surfaces. The individual waves as well as their sum are highly dependent

upon the frequency of the propagating waves, their path lengths as well as the reflective

properties and geometric arrangement of the encountered surfaces.

This research will exploit the assumption that multipath fading of a signal from a

single source results from the sum of many reflected waves. The individual waves have

roughly equal power as well as independent amplitudes and independent phases. This

allows the fading component of the incident signal to be modeled as a complex Gaussian

random variable with uniformly distributed phase and an envelope which is Rayleigh

distributed.107 If the channel is well-approximated by a constant frequency response

characteristic then relative time delays between arriving wave fronts are negligible. This is

known asfrequency-nonselective Rayleigh fading or flat Rayleigh fading. The flat-fading

condition is probably an accurate approximation for indoor communication systems with

large path losses and mobile systems with scatterers located in close proximity to the

mobile. It represents a worst-case condition from the standpoint of detection since it will

not allow the use of a RAKE receiver62 to provide resolution of individual time-delayed


The complex base-band output of a sensor array outwardly resembles the expression

given in equation (4.1):

y(t) = AClOe'20. c(t zm)bm(t r)u, + n() (5.1)

where the elements of the phase shift vector u. are complex Gaussian random variables

which result from Rayleigh fading. This is in contrast to equations (1.4) and (4.1) in which

the components of um are complex exponentials resulting from directional, unfaded


The components of Um may have any permissible degree of correlation. The study of

spatial diversity combiners is dedicated to antenna array structures which force the

correlation of fading components between array elements to be low (ideally zero).'08 A

base station diversity array must have larger element spacings than the customary half-

wavelength spacings used in a mobile radio receiver.28 Lee conducted an empirical study

of fading correlations in a two-element base station array. He concluded that, for low

correlation, the interelement spacings must be 15X 20X if the signal arrives from

broadside and 70k 80X if the signal arrives along the axial direction. Salz and Winters'9

examined a linear array and developed closed-form expressions for the direction-

dependent fading correlations between array elements when multipath rays are "dense"

throughout a range of DOAs. Raleigh et al."0 proposed an analytical model which

describes the spatially-dependent correlations of the fading process. Verification of the

latter two models through experiment remains an open issue, as does more refined

spatially-dependent channel models. Naguib and Paulraj3 examined the effects of the Salz

and Winters fading model on a base-station diversity array in an IS-95 system using

closed-loop power control.

Because spatial channel models remain an open issue this research will exploit the

assumption that the fading process is independent between antenna elements and the

elements of u= in equation (5.1) are complex i.i.d. N(0,1). Spatial dependence of the

incident signals via the interelement phase-shifts no longer exists and the array

geometry is critical only in that it results in independent fading between elements.

Winters23 has shown that even when directional information is not used by the processor

(i.e. the array functions as a diversity combiner) an optimum combiner will outperform, in

steady state, a maximal ratio combiner because it is able to adaptively attenuate cochannel


Multiple Cells with Signals Subjected to Rayleigh Fading and Shadow Fading

The models and results from the last subsection will be extended here to include the

effects of outer-cell interference. Like the previous section of this chapter the incident

signals will be subjected to flat Rayleigh fading. Unlike the previous section, however, the

cell which contains the desired user will be surrounded by several layers of cells containing

sources of multi-access interference (i.e. other DS CDMA users). The interference will be

subjected to multipath fading and shadow fading. Shadow fading occurs when structures

(such as buildings, hills or mountains) attenuate propagating signals. Shadow fading varies

more slowly than the multipath fading component of the signal and is interpreted as the

time-varying mean of the rapid, multipath-induced signal fluctuations.2

The incident signal model may be modified slightly:

y(t) = A XOe-'20 c(t Z)b,(t rz)u, +
K-NO 10'.20 (5.2)
Y A,,10''20 o c,(t-.)b,(t- )u. +n(t)
n1 rY,0 )

where the first summation is for center-cell and the second summation (with index n)

results from the outer-cell interference. All users have lognormally distributed power

control error where em is N(0, pce2). The quantity No is the number of outer cells while K

remains the number of users/cell. The variable s, is N(0,64) which in turn specifies

lognormally distributed shadow fading with 8 dB standard deviation. The quantity rn, is

the distance-dependent, fourth-order propagation loss between the n'th interferer and the

center cell base station. The phase-shift vectors u. and u, are, as in the previous

subsection of this chapter, composed of i.i.d. random variables which are N(0,1). Note that

the incident amplitude An is an indexed quantity unlike the first summation representing

the center-cell users. This is because the outer-cell amplitudes are determined by a hand-

off to the outer-cell base station with the least path loss. This will be discussed in more

detail shortly. The quantity n(t) is a vector of complex AWGN with power o2.

Figure 5.1 Spatial region for simulation of outer-cell

Figure 5.6 below shows a diagram of the arrangement of the hexagonal cells, each

with a unit radius ( cell area = 3 13/2). The region of interest is circular and contains the

equivalent area of N, = 21.67 cells, excluding the center cell. The small circle at the

center of each cell designates the position of each cell's base station. The wedge-shaped

region within the larger circle contains the equivalent area of 3.61 cells and it is over this

region that the spatial distribution of outer cell interferers is uniform, excluding the portion

containing the center cell. The interference from this wedge-shaped region will be

determined for each scenario and the results replicated to generate the interference

components for the remainder of the circular region.

The path loss which occurs between an outer-cell user and an outer-cell base station

contains a deterministic propagation loss (o r4) as well as lognormally distributed

shadowing. The random components complicate the hand-off or membership of a user

to a cell: a user is not necessarily serviced by the closest base station. Viterbi et al."'

investigated the properties of outer cell interference for the case of perfect power control

and lognormal shadowing while Lee et al.12 investigated the effects of imperfect power

control and lognormal shadowing. These previous works examined the case of up to 4

base stations involved in the hand-off. Increasing the number of base stations in the hand-

off beyond Ns = 4 will only slightly decrease the outer cell interference for a given degree

of shadowing and power control error.

The hand-off was computed by selecting the base station path with the minimum

base-station-to-mobile path loss. For the mth outer-cell user, this results minimizing a

convenient ratio of path losses between the outer cell mobile and the outer cell base

station and the outer cell mobile and the center-cell base station:

min r4 10 10
L n.= N[ ~ where: m = ...K N (5.3)
n = l... N, 2 -
r. 10 10 -10 10

where m represents the index over the outer-cell users in the wedge-shaped region and n is

the index over the NB = 11 base stations denoted by the solid black circles in Figure 5.6

above. The quantities r, and 10'-~.0 are the propagation loss and lognormal shadowing

respectively between the mth mobile and the nth base station. The quantities r4

10-'.po and are the propagation loss and shadowing between the mth mobile and the

center cell at the origin. The quantity 10-'-/I0 is the power control error of the mth user

with respect to the base station chosen for hand-off. As the figure shows, base stations just

outside of the wedge-shaped region are utilized in the hand-off calculations for the outer

cell interferers. The shadowing and power control error components are assumed to be

independent from path-to-path. Some authors"3'"4 maintain that spatial correlations exist

in shadowing components, but those effects are not incorporated into this model. The

incident levels (A.'s of equation (5.3)) resulting from hand-offs in the wedge-shaped

region of Figure 5.6 were replicated to generate interference levels for the remaining

portions of the large circular region. The model used in this research agrees precisely with

the results of Viterbi et al."1 which reported that for NB = 3 the ratio of the outer cell to

inner cell interference is 0.57.

It was assumed that power control error correlates perfectly between hand-off base

stations and does not enter into interference calculations until hand-off is chosen based on

minimum path loss. This is in contrast to the work of Lee et al."2 which assumed closed-

loop power control error was uncorrelated between hand-off base stations. The authors

minimized the quantity:

min r4 -0 10 .10 10
4 = I .= ...N .-- where: m = ...K N, (5.4)
r4,- 10 10

where 10 10 is the power control error between the mth user and the nth hand-off base

station. They assumed that NB = 3; hand-off occurred to one of the three closest base



This section will briefly revisit the analytical results of chapter four which gave an

asymptotic expression for capacity when signals were directional. Another analytical

model will then be introduced which exploits the assumption of lognormal interferers.

The expression given in equation (4.3) for the (Eb/No),g may be rewritten slightly

using equation (4.4) for a single cell:

(Eb N SNRiGpO 100(5.5)
N ef SNRi Gp. 0le+1(
m- m

where SNRi, = (AZ/o ).10'-/10 is the input SNR and Gp, = Iw u. 12llw.2 is the

normalized array gain towards the mth signal. In chapter four some assumptions (via

equation (4.4)) allowed the sum term in the denominator of the expression for (Eb/No),ff

to simplify into products of average terms, some of which are not explicit functions of the

input parameters (i.e. avg(Gpm ) = Gpi, m 1, the interference signal gain). In addition,

the gain towards the desired user Gpj = Gpd was modeled by its upper bound N, the

number of array elements. While these approximations were not accurate when considered

separately, their quotient resulted in a convenient form and gave acceptable results ( see

Figures 4.2 and 4.3). The per-element capacity was the slope of equation (4.8):

( 3NNc In(10)Q-'(Pro,) In210 )
KE = exp a (5.6)
2 Gp, 10 200

where the term containing the Q-function is dominant for the range of power control error

considered here. The term analagous to a single-pole roll-off- resulted from modeling

the multi-access interference as an averaged quantity. The averaging operation left the

desired signal as the only random variable and resulted in a lognormal distribution of

(Eb/No)rf in which the standard deviation (in dB) is determined solely by the power

control error standard deviation o, (see equations (4.5), (4.6)). Under this simplified

analytical model, the number of users K and the number of array elements N affect the

mean of (Eb/No)4 but not the standard deviation.

This series of assumptions gave acceptable results for the asymptotic case of strong

nominal input levels if the signals were directional and had a moderate degree of power

control error. With the addition of multipath fading the results of chapter four are less

precise. The roll-off of a diversity combiner's per-element capacity is slower for low

values of power control error because the standard deviation of (Eb/No)rf is not

determined solely by the power control error op.

A different analysis technique which might approximate the standard deviation of

(Eb/No)f more accuately has been examined by other authors. The model assumes that a

sum of lognormally distributed random variables is also lognormal. Along this line,

Schwartz and Yeh developed an iterative version of Wilkinson's method15 and then used

their model to evaluate the outage probability of a multi-cell AMPS system.16 Beaulieu et

al. examined several methods for approximating a sum of lognormal random variables and

concluded that the "best" choice of model depends upon the system parameters (i.e. the

degree of shadowing and the magnitude of the outage probability).

This research will supplement the analysis of previous sections by approximating the

multi-access interference as a sum of lognormal r.v.'s. This method will result in a closed-

form expression for outage probability. An explicit equation for the per-element capacity

will not be possible.

If the array gain towards the interference may be modeled by a constant, the resulting

noise and multi-access interference is approximated by:

K x. K Y.
IuA =.110- + F 10-, + 3 (5.7)
m-2 Gp, ,.I 2GpSNRi

where the left-most summation represents the inner-cell interference and the second

summation represents the outer-cell interference. The right-most term results from the

presence of noise. The outer-cell interference has been modelled as a quantity normalized

by the inner-cell interference: the outer-cell summation is over K rather than KNoc. This

approach has been used for incident signals by previous authors"'11" and will be extended

to the diversity array output via the gain constant F.

Wilkinson's method begins with the assumption that the multi-access interference is a

lognormal r.v. The first two moments of the In are then matched to the first two

moments of the sums of lognormal random variables. The mean and second moment of In

are given in equation (5.8) below:

El,] = E exp ln(10) = exp(3mz + 22 o/2)

=(K-1)exp(p82 C/2)+ F Kexp(2 /2)+ 3 =
Gp; 2Gpi SNRi

E[ E exp 21n(10)[j = exp(2fi +2p j)

SE[(. exp(xe.)+ F exp(Iy)+ 3 )2]
m-2 Gpi X-1 2Gp, SNRi

(K- 1)exp(2ga'2u) + (K lXK 2)exp(p/a2)

+ (-- J (Kexp(2P2) + K(K 1)exp(I 2F))

E[I = + 2( )K(K -1)exp(2 ( + 2)/2)+ (2G NRi2 =b2 (5.8

+ 3 (K l)exp(/f2 of/2)

+ 3F Kexp 2 /2)

where the variable z is N(m,, o2) and f = ln(10)/10. Note that the exponential terms of
the inner and outer cell interference are specified as N(O, 2) and N(0, 02) respectively.
The logarithms of the moments allow linear solutions of m, and %o2 in terms of the
moments of the sums (bi,b2):

m = 21n(b,)- ln(b,)
a = ln(b,)-21n(b))
Once the interference moments are determined then (Eb/No)ff may be expressed as:

E- N 3 P 10 t (5.10)
N, 2 Gp

and the outage probability is given by equation (5.11) below where log(e) denotes the

base-10 logarithm. This method does not allow an explicit closed-form expression for the

per-element capacity. It will, however, allow that quantity to be extracted from outage

probability contours as was done for the simulated results. The expression for the outage

probability from the approximated moments:

E Pr f-
(3 Gp 3 N
7 1= 10*log3 N, d 10*log 3 NN (5.11)
(2 Gpi 2 Gpi

= + 100log2(exp(1))a.


Most of the details of the simulation procedure are outlined in the section Simulations

and Results of Chapter 4 with one difference: the faded, incident signals were no longer

directional and a DOA was not specified. Each signal was specified by its power control

error (and shadowing for the outer cell interferers), and its phase-shift vector which

consisted of complex, i.i.d. unit-variance Gaussian random variables. For each trial the

(Eb/No),f was determined via equations (1.8), (1.10),(5.3) and (5.5) and was then

compared to an outage threshold of 7 dB. A sample mean was tabulated over the trials

for each combination of users and array elements. Even with fading the resulting outage

contours were roughly linear in N with slopes which decrease with increasing power

control error.

Figure 5.2 shows Pr((Eb/No),f < 7 dB) versus the number of users with the power

control error equal to 4 dB and with no outer cell interference. As in previous plots, the

curves are plotted parametrically. The different curves represent outage probability for

different numbers array elements N. For this value of power control error the performance

is not drastically different than for the nonfaded case shown in Figure 4.4. The nominal

input signal levels (without power control error) are such that Eb/No = 7 ( or SNRi = 7/Nc

= 7/127 = -12.6 dB) for a single signal incident on a single array element feeding a




0 20 40 60 80 100
Number of Incident Signals

120 140

Figure 5.2 Outage probability versus the number of incident signals with power control
equal to 4 dB. The number of array elements N is a parameter.

conventional detector. When comparing Figures 4.4 and 5.2 it is easy to see the faded and

unfaded systems have roughly the same outage performance. This similarity ceases for

other values of power control error.

Figure 5.3 shows simulated outage contours as the power control error varied. The

top two plots show contours of Pr((Eb/No)ff < 7 dB)= 0.02 as the power control error

varied from 0 to 7 dB with no outer cell interference. The upper plot (a) is for a nominal

input condition (without power control error) of SNRi = 7/Nc = -12.6 dB while plot (b)

shows contours for a nominal input condition of SNRi = 108/Nc = 59 dB. Plots (c) and (d)

show the same conditions but include outer cell interference. The plots show that, even

with fading, outage-based capacity continues to be linear in N. Note also that as the

nominal input levels increase, the intercepts of the approximately linear contours move

towards the origin. This general behavior was predicted for the nonfaded case by equation


The slopes of the lines in Figure 5.3 were extracted using a least-squares curve fit and

plotted as the per-element capacity in Figure 5.4. Note that the per-element capacity is less

than that for the case of no Rayleigh fading as long as the power control error is less than

approximately 1.5 dB. As long as the power control error is greater than 1.5 dB the

receiver employing a diversity array combiner in the presence of fading outperforms a

receiver using a beamformer to equalize signals which are not subjected to multipath



10 15


1 2 3

20 25 30
Number of Array Elements

5 6 7
Number of Array Elements


i 2dB .3dB :4rB SNRi=:-12.6 dB

. .....*.........5 ..

....... ....... .. ............ .pce.=i. dB

5 10 15 20 25 30
(c) Number of Array Elements

Capacity SNRi =59 dB dB / 1 dB 2 dB
3 dB
:pce = 0 d .
10 ----------....... ... ..... .. ... ......- .

5 ---------- .. ...-:pce -

1 2 3 4 5 6 7
(d) Number of Array Elements
Figure 5.3 Capacity versus the number of array elements with power control error
as a parameter. Outage probability = 0.02. Plots (a), (b) are without outer cell
interference. Plots (c), (d) include outer cell interference.

SNRi = -12.6 dB
pee=0 :1 2dB: 3dB 4dB

.. .. .. . .

SNRi = 59dB

......... .

-- -- -- -' --

Note that in Figure 5.4 there is degradation in the per-element capacity as the nominal

input levels increase and outer cell interference is present (plot (b)). When the nominal

signal level is small (SNRi = -12.6 dB) the individual outer cell signals incident on the

inner cell are very weak (compared to ambient noise and the inner cell interferers) due to

the path loss. In this circumstance the outer cell interference probably just adds to the

ambient AWGN. When the nominal input signals are large (SNRi = 59 dB), outer cell

interferers can overcome the outer-cell to center-cell path loss and can have signal levels

much higher than the ambient noise, even at the center cell. In this case some percentage

of individual outer cell signals compete with the inner cell interference for attention from

the array processor, and performance suffers.

Element '
Cap.......---.. SNRi = -12.6 dB
--------- SNRi = 59.6 dB

0 --------------------------------------

0 1 2 3 4 5 6 7
Per- 1
Element SNRi =-12.6 dB
10 ::* *........ ... ... -- -___ C!. An

0 1 2 3 4 5 6 7
S(b) Power Control Error (dB)

Figure 5.4 Per-element capacity (users/array element) versus power control error
(dB) for an outage probability = 0.02. The upper plot is for the case of no outer
cell interference. The lower plot includes outer cell interference.

Figure 5.5 below shows the per-element capacity from the simulated results of Figure

5.4b (with outer cell interference) and the approximation given in equation (5.2) with < =

7 dB and Gp, = 2. Agreement between (5.2) and the simulated results for the case no

outer cell interference was poor, and the results are not presented here. The value of Gpi

accounts for the fact that the output power of the outer cell interference is about equal to

the output power of the inner cell interference when averaged over most of the possible

combinations of user population, array elements and power control error. The rapid roll-

off of the curve from equation (5.2) is due to the use of averaged multi-access interference

when computing (Eb/No)', as discussed in a previous section.

Figure 5.6 shows the per-element capacity resulting from the application of

Wilkinson's method to model the multi-access interference as a sum of lognormal

variables (equations (5.7) through 5.(11)). The lognormal statistics of the inner cell

interferers are assumed to be due to power control error only while the outer cell

interference combines power control error and 8 dB of shadowing (x, of equation (5.7) is

N(0, pc2)) while ym is N(0, oc,2 + 64)). Simulations show that the average interference

gain Gpi for the inner cell interferers can vary from zero to unity depending on the

number of elements, the number of users and the nominal input level. Interestingly enough,

Gpi does not seem sensitive to the power control error. Averaging over these conditions

results in Gp, = 0.7. The gain constant for the outer cell interferers F of equation (5.7) -

is taken to be unity. Simulations have shown that the array attenuates the interference so

that the average output power due to the inner and outer cell interferers is about equal

0 1 2 3 4 5 6 7
Power Control Error (dB)

Figure 5.5 Per-element capacity versus power control error (dB) with outer cell
interference. The capacity is from equation (5.2).

over many scenarios of interest. Interestingly enough, simulations show the ratio of the

incident signal power from the outer cell and inner cell sources is 0.57.

The upper two plots of Figure 5.6 show that Wilkinson's method does not provide a

particularly useful approximation when there is no outer cell interference. The lower two

plots include the effects of outer cell interference and tend to agree more closely with

simulated results for a range of power control error from 1 to 6 dB. For power control

error = 7 dB the simulated per-element capacity is almost an order of magnitude greater

than the analytical results.


Per 30
Element : SNRi= -12.6 dB
Capacity ------- Analysis
20 %; .......... :................ ................ ................ A
20 4.....
,--8 e Simulation

10.- ---l....- ----

1 2 3 4 5 6 7
Element SNRi = 59 dB
-Capacity ... Analysis
20 -5 ......... .-.................... ..
8 8- Simulation
10 ... .. ..... .. ....... ....... -- ........... -.........

o0 ----.-
1 2 3 4 5 6 7

Per 15
Element SNRi = -12.6 dB
Capacity ------ Analysis
10 .. .. .......... ............ ............
.-e--e Simulation
5 ........... ........ ........ .......... ........... ...........

1 2 3 4 5 6 7
Per 15 ,
Element : SNRi = 59 dB
Capacity ------ Analysis
S10 .. .. .. ..... "... ...........i ........ ... y ?..........
~e----e Simulation

1 2 3 4 5 6 7
(d) Power Control Error (dB)

Figure 5.6 Per-element capacity versus power control error (dB) for outage
probability = 0.02. Plots (a) and (b) are without outer cell interference. Plots (c)
and (d) include outer cell interference. Curves from analysis result from equations
(5.8) (5.11).


Simulations showed that even in the presence of multipath fading, the per-element

outage-based capacity might serve as a useful performance measure for a wide range of

conditions when considering a single-cell with multi-access interferers. As the nominal

input levels increase for the multiple-cell case the per-element capacity deteriorates for pee

< 3 dB compared to the single-cell case. The reason: for low nominal input levels the array

treats the outer cell signals like AWGN. When the nominal input levels are high all

interferers incident on the base station are well above the noise and the array must

dedicate some processing to attenuate the outer cell interferers, at a loss in performance.

The simulated results presented in this chapter show that for low nominal input levels

(SNRi = 7/127 = -12.90 dB) and power control error 2 1.5 dB a single-cell system with

multipath faded signals will have a higher per-element capacity than a system with

directional, unfaded signals. The addition of outer-cell cochannel interference

degrades the per-element capacity by about 80-90% for p.c.e. 2 dB and 60-70% for

p.c.e. > 2 dB. For strong nominal input levels (SNRi = 59.6 dB) the addition of outer cell

interference degrades the per-element capacity by a factor of 2/3 to 3/4 for power control

error 5 2.5 dB.

The two analytical models for per-element capacity gave mixed results. The model

developed in Chapter 4 which averaged multi-access interference degrades too rapidly

with increasing power control error to be of much use in a single-cell scenario. In a multi-

cell scenario with low input levels the model gives a good fit to simulated results for

power control error 2 1 dB. When input levels are high, the array output power due to the

outer cell interferers can be as much as twice the power of the inner cell interferers (there

are 21.67 times more signals originating in the outer cells than the inner cells). This

accounts for the decrease in the capacity in Figure 5.5 as the nominal input levels increase.

This model is limited in several ways. First, we are modeling the interference as an

averaged quantity so the complex trial-to-trial interactions between the interference, the

desired signal and the noise are lost. Second, we are attempting to absorb the relatively

complicated behavior of the processor into two parameters, Gpi.and N, which do not vary

with the number of users or the power control scenario.

The model based on Wilkinson's method of evaluating sums of lognormal random

variables was introduced in an attempt to resolve the first of these two issues. It was

hoped that the accuracy of the analytical model might be improved by approximating the

interference as a sum of variables rather than an average. The plots in Figure 5.6 show that

this model might offer some improvement in accuracy compared to simulations for the

select case of multiple cells. If more accuracy is required for the case of a single-cell

system or power control > 6 dB, then refinements of the models would be necessary.


This chapter presented simulation results of the outage-based capacity for incident

signals subjected to frequency-nonselective Rayleigh multipath fading and lognormally

distributed power control error. The models included single-cell and multi-cell scenarios

where outer cell interferers were subjected to lognormal shadow fading. The multipath

fading model assumed that the spatial interactions between the faded incident signals and

the antenna array allowed the fading process to be fully decorrelated between array

elements. Under these conditions the array functioned as a diversity combiner rather than a

beamforming antenna array.

The per-element capacity the number of users/array element which may be

supported for a required outage probability was evaluated via simulations. The

motivation behind the use of per-element capacity was to find a performance measure

which was robust to the variations in user population and array size and which would

reflect the possible contributions a receiving array would make to the capacity of a power-

controlled CDMA system.. An additional goal was to formulate approximations which

would allow easy assessments of the per-element capacity.

Physical arguments in conjunction with semi-empirical curve fits resulted in two

analytical models. The first model was adapted from the model in Chapter 4 in which the

signal, the interference and the noise were averaged. The complex interactions between

even these quantities was overlooked for the sake of simplicity. Array outputs were

modeled as averages of input quantities and gain constants. Values of these constants were

derived empirically from simulations.

The second model exploited Wilkinson's method of approximating lognormal

variables so that averaging of the interference could be avoided. This model resulted in

slightly improved accuracy compared to the first model for power control error 5 2 dB

but was somewhat worse for power control error 2 6 dB.


This chapter examines the steady-state performance of the Recursive Least Squares

(RLS) adaptive algorithms for a base station diversity array receiving faded incident

signals with power control error. A simulation approach is used since the scenarios are too

complicated to allow useful analytical solutions. The simulations were discrete-event

simulations of transmitted bits through a nonstationary AWGN channel with time-varying

multipath fading as well as stationary power control error and shadow fading. The

simulation results will be given in terms of histograms of the effective Eb/No which can

facilitate outage calculations. The performance of an RLS array in the presence of

cochannel interference has been examined by several authors,67'72'73 but of these only

Tsoulos et al.67 considered near/far scenarios with fading and outer cell interference. They

used simulations to determine the outage probability when the (N = 8) array functioned as

a beamformer, not a diversity combiner.

Simulation results show that the RLS algorithm can track the time-varying solution

effectively and that the adaptive solution is close to the steady-state solution. The

structure of this chapter is similar to the two previous chapters. The first section revisits

the now-familiar signal model from previous chapters and also introduces the time-

dependent fading process. Approximations of the fading process used in this research are

also presented. The second section outlines the RLS adaptive algorithms and the receiver

structure used in the simulations. The third section gives the results of the discrete-event

simulations and provides some discussion. The last section gives the conclusions.

Signal and Channel Model

The signal model is the same as in the previous chapter:

Y(t) = A 10 lO' c.(t -T.)b.(t -T.)u +
KNoc f(6.1)
A,. 10'-'20- 10 c,(t -,)b.(t-,)u, +n(t)

where the first summation is for center-cell and the second summation (with index n)

results from the outer-cell interference when it is taken into account. Power control error

and shadow fading are lognormally distributed (e, is N(O,pc2), sx is N(0,64)). The

quantity Noc is the number of outer cells while K remains the number of users/cell. The

quantity r, is the distance-dependent, fourth-order propagation loss between the nth

interferer and the center cell base station. As in Chapter five the amplitude A, accounts

for the hand-off with the least path loss when outer cell interference is taken into account.

The quantities c( ) and b( ) represent the spreading code and the BPSK modulation with

ideal, square pulses. We assume the beginning and end of a PN sequence corresponds to

the beginning and end of an information bit. In this chapter the PN sequences, no longer

random, are Gold codes of length 127. The quantity n(t) is a vector of complex AWGN

with power On2 which is temporally and spatially white. The power control error and

shadow fading are assumed to be constant over the observed interval.

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