1006 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 37, NO. 4, JULY 1991

Optimally Near-Far Resistant Multiuser Detection in Differentially Coherent

Synchronous Channels Mahesh K. Varanasi, Member, IEEE, and Behnaam Aazhang, Member, IEEE

Abstract -The noncoherent demodulation of differentially phase-shift keyed signals transmitted simultaneously via a syn- chronous CDMA channel is studied under the assumption of white Gaussian background noise. A class of noncoherent linear detectors is defined with the objective of obtaining the optimal one. The performance criterion considered is near-far resis- tance, which denotes worst-case multiuser asymptotic efficiency over near-far environments. It is shown that the optima1 linear detector is a noncoherent decorrelating detector. This detector is analogous to the coherent decorrelating detector that was ob- tained from similar considerations for the coherent CDMA channel by Lupas and Verdu. The commonality between the properties of the decorrelating detectors for coherent and nonco- herent channels is established. In particular, it is shown that no other DPSK multiuser detector achieves a higher near-far resis- tance than does the noncoherent decorrelator, i.e., the optimally near-far resistant linear detector is optimally near-far resis- tant.

Index Terms-Code-division multiaccess, differential PSK modulation, noncoherent detection, multiuser detection, mul- tiuser channels, multiaccess communications.

I. INTRODUCTION

I N a code-division multiple-access (CDMA) system, sev- eral information-bearing signals are simultaneously

transmitted over a common channel. Each signal is a result of digitally modulating a sequence of information symbols using distinct preassigned code or signature waveforms. The incoming signal at the receiver is there- fore a superposition of such signals. Upon observation of this composite signal and equipped with a knowledge of the signature waveforms, the receiver is required to de- modulate the information from each transmission. The

Manuscript received September 25, 1989; revised January 28, 1991. This work was supported in part by the Advanced Technology Program of the Texas Higher Education Coordinating Board under Grant 003604-018 and in part by the National Science Foundation under Grant NCR-8710844. This work was presented in part at the Conference on Information Sciences and Systems, Johns Hopkins University, Balti- more, MD, 1989, and at the IEEE International Symposium on Informa- tion Theory, San Diego, CA, Jan. 14-19, 1990.

M. K. Varanasi was with the Department of Electrical and Computer Engineering, Rice University, Houston, TX. He is now with the Depart- ment of Electrical and Computer Engineering, University of Colorado, Boulder, CO 80309.

B. Aazhang is with the Department of Electrical and Computer Engineering, Rice University, P.O. Box 1892, Houston, TX 77251.

IEEE Log Number 9144608.

study of such demodulation strategies is referred to as multiuser detection [21].

There has been considerable research on the coherent multiuser detection problem in recent years. The underly- ing assumption in this work is that the receiver is able to estimate and track the energy and phase of each compo- nent signal. Optimum coherent multiuser detection strategies were obtained and their performance thor- oughly studied in [19], [20], where it was established that a dramatic improvement over conventional single-user de- tection can be achieved. However, these optimum strate- gies are computationally intensive. Motivated by the need for high-performance, low-complexity detection schemes, a robust decorrelating strategy [81, [91 and a suboptimum multistage detection technique based on successive multi- ple-access rejection [15], [16] were obtained for syn- chronous and asynchronous channels, respectively. The reader is also referred to [21] for a lucid summary of recent activity on this problem, and the references therein, and also to [17] for additional results on multistage detec- tion.

Factors that dissuade or preclude the use of coherent single-user communication such as fading, oscillator phase instability at the transmitter, dynamically evolving posi- tions of transmitter and receiver in a mobile environment etc., are stronger deterrents in multiuser environments because the estimation and tracking of the energy and phase of each component signal has to be carried out not only in the presence of additive noise, but also in the presence of the interfering signals. It is for this reason that even in less turbulent channels, which may be viable for coherent single-user communication, the decision to employ complex estimation and tracking strategies could prove to be counterproductive in multiuser channels. The primary objective in this paper is therefore the study of a CDMA system where the modulation technique employed for each transmission allows noncoherent demodulation of the information, i.e., no knowledge of the energies and phases of any of the component signals is assumed at the receiver. In particular, attention is focused on CDMA channels where the modulation technique employed for each transmission is differential phase-shift keying (DPSK).

0018.9448/91/0700-1006$01.00 01991 IEEE

VARANASI AND AAZHANG: OPTIMALLY NEAR-FAR RESISTANT MULTIUSER DETECTION 1007

The conventional approach to differentially coherent DPSK detection in CDMA channels involves demodulat- ing each transmitted signal as if it were the only one present. The conventional noncoherent DPSK detector therefore consists of a bank of K decoupled single-user DPSK detectors, one for each transmission. Although each single-user demodulator is optimal in error probabil- ity in the corresponding single-user channel, this is no longer true in multiuser channels. Decision statistics are corrupted by multiple-access interference in addition to additive noise. Performance evaluation based on approxi- mate bit-error rates of the conventional noncoherent de- tector for a variety of spread-spectrum multiple-access (SSMA) systems has been undertaken in several works that include direct-sequence [3], frequency-hopped 121,151, and hybrid direct-sequence-frequency-hopped 141 signal- ing schemes. Acceptable performance for the conven- tional detector can be expected only in systems where the signal energies are not very dissimilar and the crosscorre- lations between the signature signals are kept low by employing spread-spectrum pseudonoise sequences or hopping patterns of long periodicity, i.e., in low band- width efficiency situations. When the received signal en- ergies are dissimilar (“near-far” environments), this de- tector is unable to demodulate the weak signals reliably even in the low bandwidth efficiency mode. In order to remedy the near-far problem, currently operational spread-spectrum systems rely on power control strategies in which transmitters are required to adaptively adjust power levels so that all signals arrive at the receiver with essentially similar energies. However, such a strategy is self-defeating [21] because it dictates a significant reduc- tion in most transmitter powers to accommodate the weakest transmitter, thereby diminishing the multiple- access capability of the overall system.

The performance measure of primary interest in com- munication systems is probability of error. In multiuser systems, probability of error in the high signal-to-noise ratio region emphasizes degradation due to interfering users rather than that due to the additive noise. Multiuser asymptotic efficiency is a performance measure that not only captures this information but is analytically tractable as well. It was first introduced in the context of coherent multiuser communication by Verdu in [19]. For differen- tially coherent multiuser detection, asymptotic efficiency for the m th user whose bit-error rate is P,(v) can be formally defined as

rl,=sup OIr<l; (

J@ooP,,,(g)/exp[-rEm/2a2] < +a), (1.1)

where E, is the m th signal energy and cr2 denotes power spectral density level of the background white noise. Since the error probability in single-user channels em- ploying binary DPSK is O[exp( - E, /2a2>], vm is equal to the lim it, as (T -+ 0, of the ratio between the energy required by a single user to achieve an error rate P,(a)

and E,, the actual energy of the k th user. Asymptotic efficiency therefore characterizes performance loss when the primary degradation is due to the presence of inter- fering users.

A primary motivation for studying noncoherent detec- tion is to address the need for reliable communication in channels, such as in near-far environments in mobile communication, where signal energies and phases vary too rapidly for the receiver to estimate and track them. In order to find such detection strategies that are robust to near-far effects, we invoke the notion of near-fur resis- tance, a performance measure introduced by Lupas and Verdu in [8]. For noncoherent multiuser detection, we introduce a modification of this measure to reflect the lack of both energy as well as phase information at the receiver. In particular, the near-far resistance of a DPSK multiuser detector, the m th user asymptotic efficiency of which is vrn, is defined as

r, = inf vrn, E, P 0

(1.2)

where Ei and e1 denote the energy and phase of the ith transmission. A nonzero value of near-far resistance will therefore guarantee an exponential decay of error proba- bility with increasing signal-to-noise ratio, irrespective of the specific values of energies and phases of the interfer- ing signals.

The rest of the paper is organized as follows. In Section II, we model a general CDMA-DPSK system that encom- passes as a special case the hybrid direct-sequence- frequency-hopped spread-spectrum model (and hence di- rect-sequence and frequency-hopped SSMA). This allows a unified treatment of multiuser detection problems which traditionally have been dealt with separately. In this pa- per, we focus attention on the synchronous additive Gaussian CDMA channel because such a study promotes a fundamental understanding of key issues involved in a simple setting.

The goal in Section III is to motivate the need for detection strategies which are robust to near-far effects. This is done by characterizing the performance lim itation of the conventional noncoherent single-user detection strategy in a multiuser environment. In particular, it is shown that the conventional noncoherent detector is not near-far resistant.

The objective in Section IV is to alleviate the perfor- mance lim itation of the conventional detection scheme. To this end, a class of noncoherent linear detectors is introduced, and exact expressions for the bit-error rate and asymptotic efficiency of an arbitrary member of this class are obtained. In Section IV-A, we characterize the linear detector that is optimally near-far resistant, i.e., the linear detectors that achieve the highest worst-case asymptotic efficiency over near-far environments. Under m ild restrictions on the signature signal constellation, it is shown that noncoherent decorrelating detectors solve this m inimax optimization problem. The noncoherent decorre-

1008 IEEETRANSACTIONSONINFORMATIONTHEORY,VOL.37,NO.4,JULY1991

lating detector is an analog of the coherent decorrelating detector obtained from similar considerations in [8]. The commonality between the properties of the coherent and the noncoherent decorrelating detectors is established. Notable among these properties are that the bit-error rate of a noncoherent decorrelating detector (and hence its asymptotic efficiency) for each user is independent of the interfering signals energies and phases, thereby alleviating the near-far problem. Further, in Section IV-B, it is shown that no other DPSK multiuser detector, linear or nonlinear, has a higher near-far resistance than does the decorrelator, i.e., the optimally near-far resistant linear decorrelator is optimally near-far resistant. The paper is concluded in Section V.

II. SYSTEM MODEL

Noncoherent detection of multiple differentially phase-shift keyed transmissions of digital information made over a code-division multiple-access (CDMA) chan- nel is considered. It is assumed that a superposition of K transmissions arrive at the receiver in symbol synchronism perturbed by additive white Gaussian noise with noise spectral density u2. In the time interval [ - T, T], the complex envelope of the received signal is

k=l

t~[-T,T]. (2.3)

The finite-energy modulating signal {U,(t); t E [O, T]) can be expressed as (E,)“*f,(t)exp(jB,), where {fk(t); t E [O, T]} is in general a complex-valued, unit-energy signa- ture waveform assigned for the k th transmission. In DPSK modulation, information is encoded into phase differ- ences between successive symbol intervals. In binary DPSK, a “1” or a “0” is transmitted by shifting the phase of the carrier by r or zero radians relative to the carrier in the previous signaling interval. Therefore, b, = + 1 (or - 1) in the previous representation denotes transmission of a 0 (or 1) in the kth packet in the time interval [O,T]. Note that implicit in the signal model, is the assumption that the energies and phases of the signals remain con- stant over two successive symbol intervals. The problem can now be simply stated as the demodulation of the symbols b,, b,; * a, b,, given the received signal in the interval under consideration.

In general, the kth signature signal fk(t) is assumed to be of the form a,(t>exp(jo,(t)t). This general model encompasses as special cases, direct-sequence spread- SpWtrUm Sign& (with ok(t) = 0 and u,(t) as the kth user’s spreading waveform), frequency-hopped spread- spectrum signals (with a,(t) = 1 and w,(t) representing the k th user’s hopping pattern in the zeroth time interval), and hybrid direct-sequence-frequency-hopped signals be- ing represented by a combination of nontrivial a,(t) and w,(t). Results obtained from such signal models promote a unified understanding of each of these SSMA signaling techniques.

As essential parameter of the system is the K X K matrix of normalized complex cross-correlations R of the complex-valued signature signals whose elements are de- fined’ by

so that the diagonal elements are all equal to unity.

III. CONVENTIONALNONCOHERENT SINGLE-USER DETECTOR

In this section we consider binary DPSK modulation and obtain the exact bit-error probability of the conven- tional noncoherent single-user detector in multiuser channels. The performance lim itation of this detector is characterized in terms of near-far resistance. The con- ventional detector consists of K decoupled noncoherent single-user detectors where the m th single-user detector is optimum in the absence of all but the m th signal. In a single-user channel, it is well-known that the decision variable evaluates the phase difference between the re- ceived signal in the current and previous symbol intervals and selects the symbol whose phase is closest to this difference. In the binary DPSK multiuser environment, therefore, the conventional noncoherent detector decision on b, for each m are made as follows (cf. [12]):

6, = en [ Re {zm(-l)zm(~)}], (3.4) where

z,Ji) = ;/(i+l’Tl(l)f,,(t - iT)dt. IT

The bank of decoupled single-user detectors based on matched filter realizations is shown in Fig. 1 where, for simplicity, we have chosen to represent all operations over complex low-pass representations of real bandpass signals. Consider the following lemma, which is a stan- dard result in communication theory and can be found in several references (e.g., see [13]).

Lemma 1: Let X and Y be uncorrelated, complex-val- ued Gaussian random variables with means pLx and pLy, respectively, and a common variance u2. Then, the proba- bility that the decision variable D = Re(XY) is less than zero is given as

where Q(a, b) denotes Marcum’s Q-function2 and the parameters a and b are defined as

up @qlIpx - pyl and

‘The symbol denotes complex conjugation. ‘The Marcum Q-function [lo] has the integral representation

lFxexp(-(x2 + b2)/2)10(ax)dx, where lo(x) is the zeroth order modi- fied Bessel function of the first kind.

VARANASI AND AAZHANG: OPTIMALLY NEAR-FAR RESISTANT MULTIUSER DETECTION 1009

Matched Filter 1

$1

Matched Filter 2

6

Fig. 1. Conventional detection: A bank of K single-user DPSK detectors.

Next, we will use this lemma to obtain the error probabil- ity of the conventional single-user DPSK detector in mul- tiuser Gaussian channels, the decision of which is ex- pressed in (3.4). Since the m th user error probability conditioned on the m th information bit b, is equal for b, = 1 or - 1, we have3

Pz)( a) = Pr [ Re (Z,J - l)Z,(O)) < Olb,,, = + I]

= 21eK c Pr [Re(Z,( -l)Z,(O)) <Olb], (3.5) bsB,

where B, denotes the set {b E { - 1, l}K; b, = 1) and the second equality follows by averaging over equiprobable interfering bit combinations. From (2.3) and the defini- tion of Z,(i), the statistics Z,(- 1) and Z,(O) can be written as

Z,( - 1) = ; E;‘* eXP(.Sj)Rjm + Ym( -I>, j=l

Z,(O) = t E;12 exp (@j)bjRj, + Y,(O) 7 (3.6) j=l

where y,(i) are complex-valued zero-mean Gaussian ran- dom variables defined as

t)f,( t - iT) dt.

Conditioning on the interfering bit-combinations and the energies and phases of the different transmissions, the statistics Z,( - 1) and Z,(O) are complex-valued Gaussian

3The superscript (c) in Pz) identifies the analysis of conventional noncoherent system.

random variables with means

pL,( -1) = E E;/2exp(j0j)Rj, j=l

and

p,(O) = 5 Ej/2 exp ( jej)b,Rjm, j=l

and variances

(3.7)

u:(i) = E[ Iym( i)y,( i)] = 2a2, for i= -l,O.

Further, these statistics are uncorrelated because they are obtained from the received signal over nonoverlapping time intervals. Therefore, we can apply Lemma 1 to the probability inside the summation in (3.5) to obtain

P,$‘(,) =21-K c beBm

Q(a,,b,) - ;I,,(a,b,)

.exp( -q)), (3.8)

where

and a, = (2a) -‘lpc( - 1) - PC(0)I

b, = (2~) -‘I/-4 - 1) + ~$91, which in turn can be evaluated from the expressions for the means in (3.7). In the following proposition, we show that the conventional single-user detector in a multiuser environment is not near-far resistant.

Proposition I: The near-far resistance of the conven- tional noncoherent detector for the m th user, denoted as ?j$‘, is identically equal to zero unless the m th user’s signature signal is orthogonal to the subspace spanned by the other K - 1 signature signals.

Proof: From the definition of asymptotic efficiency in (1.1) and the expression for the bit-error probability for

1010 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 37, NO. 4, JULY 1991

the conventional noncoherent detector in (3.8), we have

Pr [Re (.?,A - VXO)) < ($1

qE)=sup

i

OIr11; lim 21pKb FB

m <+m CT-0 exp(-rE,/2a2)

Pr [ Re (-%-A - Wm(0)) < ($1 bsB, exp( - rE,/2a2) ’ ’ -\

The second equality follows from the fact that, in a low background noise region, the summation of error proba- bilities in P$‘(a> is dominated by the term corresponding to the transmission of least-favorable bits of the interfer- ing users. The last equality defines conditional asymptotic efficiency that we denote as 4’,“‘(b), as the asymptotic efficiency conditioned on the vector of transmitted bits. Therefore, the near-far resistance of the conventional detector can be bounded above as

yp) = m inf E, t 0

min +2’(b) = min bEB,

&f. bs4n 1

d?( b > @,E[--,Tl eit[-a,al

i=#m i#m

I ,‘“,fo 4??(b) > VbEB,.

o,E[1-T,?71 i#m

In order to show that the near-far resistance of the conventional detector is zero, it is sufficient to show that the expression on the right-hand side of this inequality-henceforth referred to as conditional near-far resistance-is equal to zero for some appropriately cho- sen value of b E B, because near-far resistance is non- negative by definition. Let us consider the probability of error given that the transmitted bits of the interfering users are all equal to + 1, i.e., when b = u = [l, 1; . *, llT. Substituting b = u in (3.8), we have

Pr[Re(Z,(-l)Z,(O))<Olb=u] =Q(L?~,~,)

where a’,, 6, are equal to a,, b, evaluated at b = u, re- spectively. It is easily seen that 6, is identically equal to zero. Therefore, the error probability conditioned on b = u is given as

Pr (0:) < Olb = u) = Q(O,&,) - :1,(O) exp (- @/2)

(3.9)

where equality follows from the property of the Q-func- tion that Q<O, X) = exp(- x2/2> and also from the fact that I,(O) = 1. From the conditional error probability ex- pression in (3.9) and the definition of conditional asymp-

totic efficiency, we have

l/2 exp(j(% - %))Ri,

=min{l,laTr,12}, where the second equality is obtained by evaluating bz and the last equality by denoting the mth column of R as rrn and defining the K-dimensional normalized amplitude vector

a= (E,/E,)“2exp(j(8,-t),),...,(E,/E,)1’2 [

‘exp(j(eK-em)].

Hence, corresponding to the set of admissible values of interfering energies and phases is the set of all admissible values of a denoted by A, and equal to A,,, = {a E CK; am = 1). Now, the conditional near-far resistance, condi- tioned on b = u can be written as

where the last equality is obtained by observing that the objective function in the minimization problem is nonneg- ative and choosing a vector a, normalized so that a, = 1, that lies in the null space of r,-the existence of which is guaranteed, provided that r,,, is not equal to the mth unit vector, i.e., when the mth user’s signature signal is not orthogonal to the other signature signals-thereby yield- ing a minimum that is equal to zero. Since 7:) is nonneg- ative and is no greater than inf,, A, +$‘(u>, the result follows. 0

The near-far resistance of the conventional noncoher- ent detector is bounded away from zero only when the corresponding signature signal is orthogonal to the sub- space spanned by the other signature signals. In practice, however, it is more by chance than by design that the signature signals are orthogonal, because of bandwidth restrictions, lack of synchronism and other design con- straints. A similar result for single-user detection in the coherent multiuser channel was obtained by Lupas and Verdu in [8].

VARANASIANDAAZHANG:OPTIMALLYNEAR-FARRESISTANTMULTIUSERDETECTlON 1011

Matched Filter 1 h

1) fl(T-t)

<z(i), > t=iT Rec.)

r(t)

Matched Filter 2 f,jT-t) t=iT Rec.)

Fig. 2. Linear multiuser DPSK detector.

In the next section, we would like to remedy the inabil- ity of the conventional noncoherent detector to cope with the uncertainties associated with the transmissions of the interfering users.

IV. NONCOHERENTLINEAR MULTIUSER DETECTORS

We consider a class of noncoherent linear multiuser detectors. The computational complexity of these detec- tors is independent of the number of users. We obtain the exact bit-error probability for an arbitrary member of this class. The objective is to characterize the linear detector which maximizes near-far resistance, or equivalently, maximizes the worst-case asymptotic efficiency over the energies and phases of the interfering signals. Such a detector would then be near-far resistant and at the same time retain the computational simplicity of the conven- tional detector.

A noncoherent linear multiuser detector for the mth user, denoted by a nonzero transformation h’“’ E CK, is defined by the decision

6, = sgn Re I i

5 h’,“)z,( - 1) 5 h$“)~I(0) 11 . (4.10) k=l I=1

The class of linear detectors will be denoted as 0 and membership to this class will be denoted by h(‘“) E 0. Note that the conventional noncoherent detector for the mth user is a degenerate member of the class fi with a representation h’“’ = urn, the mth unit vector. Further- more, this class is the noncoherent counterpart of the class of coherent linear detectors introduced by Lupas and Verdu in [8].

A centralized matched-filter based structure of a linear detector is shown in Fig. 2. For simplicity, we show operations over equivalent low-pass representations of actual signals. Note that all matched-filter outputs now enter into the decision variable in the demodulation of each user via the corresponding transformation. The vec- tor of complex inner product evaluations represent a

computational complexity per demodulated symbol of O(K) in complex operations.

An alternative structure of the linear detector, sug- gested in the coherent context in [21], can be obtained by observing that the decision on the mth user’s symbol can be expressed as

where 6, = w [ Re {xm(-xm(0)}],

~,~(i) = ilCi+‘)‘r(t)gm(t -iT)dt IT

and g,(t) is a function that lies in the subspace generated by the signature signals with a representation

g,(t) = 5 hi%(t). (4.11) k=l

The resulting structure is identical to that of the conven- tional detector in Fig. 1 with the impulse response of each user’s matched filter, say f,,( T - t), replaced by g,( T - t). Since the use of more complex matched filters replaces the O(K) software complexity, the choice of structure depends solely on the tradeoff between compu- tational complexity and ease of realization. In decentral- ized reception, the second realization is minimal in the sense that it requires only as many matched filters as the number of signals to be demodulated.

Proposition 2: The asymptotic efficiency of user m for an arbitrary linear detector h is given as4

(h) = 1 Tm 4E,hTR%

.max2 O,min l,Br$; leT(Z+B)RhI ( i

-,eT(Z- z3)$}, (4.12)

4The superscript (h) in 7:) and Pch) identifies the analysis of a m noncoherent linear multiuser detector.

1012 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 37, NO. 4, JULY 1991

where e denotes the vector of complex amplitudes with kth entry as Ei/” exp( j0,) and where B, is the set of admissible values of the diagonal information symbol ma- trix B restricted by b, = + 1.

Proof: Let us obtain the m th user bit-error rate, denoted as P$‘(v), of the linear multiuser detector h E R (we drop the superscript for convenience). Since this error probability, conditioned on the m th user’s bit, is equal for b, = + 1 or b, = - 1, we have

P(h)(c) m

5 hkZk(-1) :x$,(O) k=l I=1

= 21pK C Pr [Re (X,( - 1)X,(O)) < oJ~] , B E 4,

(4.13)

from straightforward calculations as

‘a;(i) = E[ (x,(i) - Ph(i))(Xm(i) - ph(i))]

= E f hkyk(i) ; [ - k=l I=1

= 2cr2hTRit i= -l,O. Therefore the random iariables X,( - 1) and X,(O) have a common variance. Further, they are uncorrelated be- cause they are linear combinations of statistics obtained by processing the received signal over nonoverlapping ‘time intervals [ - T,O] and [O, T], respectively. We can apply Lemma 1 to obtain the probability inside the sum- mation in (4.13) so that

P,$‘)(a) = 2l-k c je(uh,bh)

1 - y’“( a/$,) exP (-fy )}. (4.14)

where the last equation results from averaging over the where ah and b, are given as restricted set B, of admissible values of B that are the equiprobable interfering bit combinations. Conditioning

ah = (2&?%) 7erZ#Z - eTBR%I,

on the bit combinations and the energies and phases of different transmissions, X,<- 1) and X,(O) are linear

b, = (2~di%-~1e~Rh + e’BI&l. (4.15)

combinations of Gaussian random variables and hence We obtain asymptotic efficiency by substituting (4.14) into (1.2),

nE)=sup Osrll; lim u-0 exp( - rE,/2a2)

= m in sup B E 4 exp( - rE,/2a2)

are Gaussian. Further, the means and variances of the Gaussian random variables X,(i) are denoted as ph(i) and a;(i) for i = - l,O. Each of these quantities is easily obtained as

K

ph(-l)=E [ -

c h,Z,(-1) =eTRx, k=l 1

where equality follows from the first equation. of (3.6). Similarly,

~~(0) = E ; h,Z,(O) = eTBR%, [ 1 I=1

where equality follows from the second equation of (3.6). The variances of X,(- 1) and X,(O) can be obtained

where the second equality follows from the fact that in the lim it as (T + 0, the summation of conditional error probabilities in (4.14) is dominated by the term corre- sponding to the least favorable interfering bit combina- tion. The penultimate equation is obtained by using the fact that Q<a, b) - 1/2Za(ab) exp (-[a* + b2]/2> - O(U exp[ -(b - aj2/2]> for b > a [l]. The result in (4.12) is obtained by substituting the expressions for ah and b, from (4.15). The proof of Proposition 2 is therefore com- plete. 0

We note here that in the hypothetical scenario where the received signal phases are all equal, and where the signature signals of all the users are real-valued, as in direct-sequence spread-spectrum signaling, and with a further restriction on the linear transformation h to be real-valued, the expression for asymptotic efficiency in

VARANASI AND AAZHANG: OPTIMALLY NEAR-FAR RESISTANT MULTIUSER DETECTION 1013

(4.12) is identical to that of the linear detector for coher- ent CDMA channels employing binary phase-shift keyed modulation obtained in [S]. It is in this sense that the problem considered here is a complex arithmetic general- ization of the real-valued problem considered in that paper. However, in contrast to the situation for the coher- ent problem, where the discrete m inimization over the interfering bit-combinations was immediate, no closed form expression is available for the m inimization in (4.12) in the general K-user case.

It is instructive to consider the two-user case. A closed form expression for asymptotic efficiency can be obtained in this case by using the fact that the magnitude of the sum of two complex numbers is not less than the differ- ence between their magnitudes. For instance, the asymp- totic efficiency of user 1 in a two-user system is given as

(h) = 71 &maxz(O,min(l,min{~e,[R%]l

=max2 O,min i i

1, I[J5wl~2ml2I

Ii m ) where a2 = \IE2/EI exp(j(0, - 0,)) and [.li denotes the ith element of the vector in square brackets. Consider the special case of the two-user conventional detector. It is easily verified that

where R,, is an element of the normalized complex crosscorrelation matrix R defined in Section II. There- fore asymptotic efficiency of the conventional detector is identically equal to zero when the interfering user is sufficiently strong relative to the desired user, i.e., when JE2/EI 2 1/ IR,,l.

Having obtained the asymptotic efficiency of a nonco- herent, linear multiuser DPSK detector, we now proceed to the next section with the objective of finding the optimal detector from the class of linear detectors, the optimality criterion being near-far resistance.

A. Optimal@ of Decorrelating Detectors

A noncoherent decorrelating detector for user m is defined by the decision in (4.10) with the linear transfor- mation h = d where d denotes the complex conjugate of the m th column of a generalized inverse R’ of R. If the m th user is linearly independent, it can be shown (cf. [8, Lemma 11) that Ra= u,, the m th unit vector. If all the signature signals are linearly independent, then R-’ ex- ists and the decorrelating transformation d is uniquely characterized as the complex conjugate of the m th col- umn of the inverse of R. In the sequel, we do not need the latter assumption.

The definition of the class of decorrelating transforma- tions is similar to that introduced in 181 for coherent

CDMA channels. This point is further clarified. From (4.10) and the expressions in (3.61, it is seen that the desired signal plus multiuser interference component of the decision statistic of any linear detector is identically equal to Re(eTRd;tTRBZ). For the decorrelating linear detector d, and assuming that the m th user is linearly independent, we use the fact that Rd= u,, whence the desired signal plus multiuser interference component is simply E, b,, i.e., it corresponds to a detector in 1R that eliminates the multiuser interference from its decision statistic. Equivalently, note from (4.11) that the m th user matched filter of the decorrelator is matched to g:(t) = Cfz,dim)fk(t). Since g:(t) is orthogonal to each of the interfering signature signals f,(t), it is able to effectively tune-out the multiple-access interference. Therefore, ow- ing to its striking similarity with its counterpart in coher- ent multiuser detection [8] and in keeping with the nomenclature suggested therein, the linear detector d will be referred to as the noncoherent decorrelating detector.

It is important to note the distinction between the coherent and the noncoherent decorrelators. The normal- ized signal correlation matrix in [8] is independent of the signal energies but depends on the signal phases, and so does the coherent decorrelating detector. In contrast, the noncoherent decorrelating detectors introduced here are independent of both the signal energies and phases.

Proposition 3: If the signature signal of user m is lin- early independent, the bit-error rate of the m th user noncoherent decorrelator is independent of the complex amplitudes of the interfering users and is equal to5

p.I:“(V)=;exP(- 2V:;,.,)? (4.16)

where R,:, is the m th diagonal element of the unique Moore-Penrose generalized inverse R+ of R. This prop- erty is of special significance since it implies that no matter how strong the interfering signals are or what their carrier phases may be, the decorrelating detector has an error rate equal to that of a single user with energy E,[R:,l- ‘.

Proof: We evaluate the bit-error probability of d from (4.14) and (4.15) that characterizes the bit-error probability of any linear detector. Since the m th user is linearly independent, the decorrelating transformation satisfies Ra= u,, and we have

Pid’( a) = 2’-K c Q( ad, bd) B E 4,

1 2 - zZo( a,b,) exp (-a’+b ) y

=2-K C B E &

exp(-P)=iexp( -&),

where the penultimate equation follows from the fact that ad = 0 and the last equality from the fact that b, is

5The superscript (d) in q!$) and PA’) identifies the analysis of noncoherent decorrelating multiuser detectors.

1014

y q r-----jh f,(t)

y,’ q h fp (a)

IO cj I

IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 37, NO. 4, JULY 1991

0 2 4 6 8 10 12 14

SNR(1) in dB (b)

Fig. 3. (a) Direct-sequence signature signals derived from Gold se- quences of length 7 assigned to the four users of a four-user DS-SSMA system. (b) Bit-error rate of first user as a function of the first user’s signal-to-noise ratio. These error rates are independent of interfering signal energies and phases.

independent of B and is equal to a-Id-, where d, is the m th element of d and hence the m th diagonal element of any generalized inverse of R, which in turn can be shown to be equal to RL, [a]. The expression for error probability in (4.16) follows. 0

Fig. 3(a) shows four direct-sequence signature signals derived from Gold sequences of length 7 assigned to a four-user direct-sequence SSMA system. Fig. 3(b) depicts the bit-error rate of the decorrelator for the first user as a function of the first user’s signal-to-noise ratio, as the second, third and the fourth user’s become active. These error rates are independent of the interferers’ signal energies and phases.

Corollary 1: The asymptotic efficiency of a’ny decorre- lating detector for a linearly independent user is given as

(4 - 1 Tm -R’>o.

m m (4.17)

The same expression can also be obtained by using Rd= urn in the expression for asymptotic efficiency for an arbitrary linear detector in (4.12). In order to obtain (4.17), we have to show that RL,k 1, a fact that we establish in the Appendix. This completes the proof of Corollary 1. 0

The asymptotic efficiency of the decorrelator in com- Since asymptotic efficiency of a decorrelating detector is parison with the conventional detector for the two-user independent of interfering signal energies and phases, it case is depicted in Fig. 4 and for the four-user direct- is also equal to its near-far resistance. This implies that sequence SSMA system of Fig. 3(a), is depicted in Fig. 5.

z 0.9

& 0.8 m

Decorrelating detector

-- L --- 2 0.7-

2 0.6-

.: 2 0.5

g 0.4 \\\

.o 0.3- \ Single-user detector

5 'L 0.2- E \t $ O.l- a 0.0 . I ’ I ’ I I ’ I I ’ I ’ I ’ I ,-

IO -8 -6 -4 -2 0 2 4 6 8 10 E2/El(in dB)

Fig. 4. Asymptotic efficiency comparison between the single-user and decorrelating detectors in the 2-user case when the correlation is 0.5.

Decorrelator(Z-user) Decorrelator(3-user)

4; --

0.6

Single-user detector(2

0.3

Single-user detector(3-user) :I:: -10 -8 -6 -4 -2 0 2 4 b 8 10

Ei/El(in dB)

Fig. 5. Asymptotic efficiency comparison of the conventional single- user and the decorrelating detectors for the four-user DS-SSMA system as the first two, three, and finally four users become active.

the decorrelator has a nonzero near-far resistance and is therefore robust to interfering signal uncertainties.

Proof: Substituting the bit-error rate of a decorrelat- ing detector from (4.16) of Proposition 3 into the defini- tion of asymptotic efficiency in (1.2), we have

VARANASI AND AAZHANG: OPTIMALLY NEAR-FAR RESISTANT MULTIUSER DETECTION 1015

The performance measure of interest is bit-error proba- bility. The set of design strategies is the set of linear detectors 0. The uncertainty for the detector designer arises from two quantities; the bit-combination of the interfering users and their energies and phases that are both determined by the transmitter. These two quantities determine the operating point for the receiver. The objec- tive is to find the linear detector in R that is robust to operating point uncertainties. Therefore we will consider a m inimax approach, where the design goal would be to find an optimal detector from R that optimizes the worst-case performance over all admissible operating points. Consider the next proposition.

Proposition 4: For a linearly independent user, a decor- relating detector achieves the highest near-far resistance from among the class of linear detectors, i.e.,

d = arg ~z; e 54, r$‘), (4.18)

where E,,, denotes the uncertainty set for demodulation of the m th user’s signal which is the set of admissible values of interfering signal energies and phases and can therefore be written as 5m = {e E CK; [el, constant}.

Proof: We show that the highest near-far resistance achievable by any linear detector is upper bounded by the near-far resistance of a decorrelating detector. Using the expression for asymptotic efficiency in (4.12) and the definition of near-far resistance in (l.l), an upper bound on the highest achievable near-far resistance is obtained as follows:

i-p) = 111 sup inf max2 hEi Ed%?

pend only on the information bit of the desired user. The specific choice of this dependency is dictated by the dual-objective of obtaining a tight bound and at the same time retaining a mathematically tractable problem. Fortu- nately, these two objectives are served simultaneously by the choice b = u, i.e., when all the information bits are equal to that of the desired user. In this case, the m ini- max problem has a natural setting in the finite-dimen- sional complex semi-inner*product space CK with semi-in- ner product defined as (x, Y)~ = xTRj. Note that it denotes a valid semi-inner product since R is a nonnega- tive definite Hermitian matrix. Therefore, choosing &$i)(u> as an upper bound for q$r), we have

where the second equality follows from the nonnegativity of the objective function +(h, a) e ](a, hjR12/(h, h>R and the last inequality follows from the m inimax inequality [6]. Interestingly, the penalty function q?(h,a> is identical to the signal-to-noise ratio functional encountered in the robust matched filtering problem [ll], [18]. The inner

1 1, m in le’(Z+ B)ti

24s BE% jlr’(Z-B)zy

e&y(b), VbeB,, (4.19)

where the normalized amplitude vector a and the set of its admissible values A, were defined in Section II and 0 is chosen as the set {h E CK; hTZ8z # 0). The first inequal- ity is obtained from the m inimax inequality [6] and the second inequality holds for each interfering bit-combina- tion b E B, and the expression on the right side of this inequality, which we have defined as &z)(b), can be interpreted as the highest near-far resistance achievable by a linear detector in a hypothetical communication environment. This environment consists of co-operative interfering transmitters sending information bits that de-

maximization in the last equation of (4.20) is easily ob- tained by using the Cauchy-Schwarz inequality (which can be shown by a simple extension of the Cauchy-Schwarz inequality for inner products [6]), since R can be chosen to contain A, so that

l(a,h)R12 sup ch,hjR =(a,a)R,

hECK h’Rh#O

(4.21)

with the result that if the m th user is linearly indepen-

1016

dent,

m (a,a)R}=min

1 i 1 ‘l,-

R+ ’ mm(4.22)

The last equality is obtained by noting that the m inimum-norm optimization problem in the first of the above equations is the complex;arithmetic version of that solved in Proposition 1 of [8] and that the proof of that proposition extends to the complex case under considera- tion here. However, a simpler proof involves restating [9] the m inimum-norm optimization problem as

inf(a,a)R subject to (a,djR = 1,

where d is a decorrelating transformation. The existence of d is guaranteed by the linear independence of the m th user. We invoke the Cauchy-Schwarz inequality for semi-inner product spaces again, so that

IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 37, NO. 4, JULY 1991

m in (a,a)R= &=bw (4.23) LZEA,

Summarizing, from (4.191, (4.201, (4.20, and (4.23), and using the fact that R L, 2 1 which was obtained in Propo- sition 3, we have the following:

T$fk &f”‘(u) < [RLm] -I. (4.24) From Proposition 3, a decorrelating detector achieves a near-far resistance equal to the upper bound above, whence the desired result follows. 0

We remark here that equality in the inequality of (4.20) holds because the functional $(h, a) has a saddle point, a fact that we now establish. The seminorm in (4.23) is m inimized uniquely for the least-favorable operating point aL = [ RL,]-‘d. The optimal detector for the least-favor- able operating point can be obtained from the condition for equality in the maximization problem in (4.211, which is h, = aaL = a’d, and we assume LY’ to be unity for convenience. The pair (d,[RL,]-‘d) can be shown to be a saddle point if it can be verified that [R~,]-‘d is the worst operating point for d, i.e., if

[RLm]-‘d=argarnl; $(d,a), m

or equivalently, that

I(d,[R~,]-‘d),l~I(d,a)RI, v’aEA,> which can be immediately confirmed.

Corollary 2: A linear detector that is not a decorrelator is not near-far resistant.

Proof: From (4.201, the conditional near-far resis- tance of a detector h that is not a decorrelator is given as

4,“) = m in 1, inf l(a,h)Ri2

a=A, (h,h)R

since the penalty function is nonnegative and it is always

possible to find a vector a with a, = 1 such that it is orthogonal to the vector & when the latter is not equal to the m th unit vector. Further, since it constitutes an upper bound for near-far resistance, which is nonnega- tive by definition, we have the desired result. 0

This corollary together with the characterization of the near-far resistance of the decorrelating detectors in Proposition 3 could have been used to establish Proposi- tion 4. However, the game-theoretic approach that we followed brings out the close parallels between the opti- mum near-far resistance problem formulations for coher- ent and noncoherent channels.

Corollary 3: For a linearly dependent user, there is no linear detector that is near-far resistant.

Proof If the m th user’s signature signal is linearly dependent, it can be shown following [81 that inf a E A (a, a)R = 0, since there is a linear combination of the colimns of R with a nonzero coefficient for the m th column that is identically equal to zero. Therefore, it is possible to find an a with a, = 1 such that Rii = 0. Together with the first equation in (4.22), we have $?)“‘<u) = 0. Since optimum near-far resistance, Yjr)---nonnega- tive by definition-is bounded from above by &F)(U), we have q(o) = 0, which completes the proof. m 0

B. Optimum Near-Far Resistance

The discussion thus far does not exclude the possibility of the existence of detection schemes, that are not of the linear type considered so far, which may achieve a higher near-far resistance than that of the noncoherent decorre- lators. Consider the following proposition.

Proposition 5: No detector, linear or otherwise, achieves a higher near-far resistance than that of a noncoherent linear decorrelator.

Proof: The central idea here is to find an upper bound on the highest near-far resistance achievable by analyzing the performance of an optimum receiver that, in addition to observing {r(t); t E [ - T, T]}, has additional side information. In particular, suppose that the receiver has a perfect knowledge of the signal energies and phases. In this case, the observation of the received signal in the time interval [ - T,O] provides no information about the information symbols in [O, T]. Therefore, the m inimum error probability in the demodulation of the m th user’s bit is equal to the error probability of the coherent multiuser m inimum-error probability. detector, exponen- tially tight bounds for which have been obtained in [19]. In particular, it was shown that for any 6 > 0, there exists a0 > 0 such that for all (+ < uo, the m inimum-error proba- bility of the m th user satisfies

C,“Q(A,i,(m)/u> ~f’$)~ C,“(l+ ~>Q(A,i,(m)/u), (4.25)

where C,” and CE are positive constants and A,Jm> is the Euclidean distance between signals corresponding to

VARANASI AND AAZHANG: OPTIMALLY NEAR-FAR RESISTANT MULTIUSER DETECTION 1017

the two closest hypotheses that differ in the m th user’s bit, i.e.,

= m in ETHE) EE{-l,o,+l)K

Em=+1

where Hkl = (E,E,)‘/‘Rk, exp(j(0, - 6,)). Using the up- per and lower bounds for the complementary error func- tion [14], we have from (4.25) that for any 6 > 0, there exists a ut, > 0 such that for all u < (TV, the error probabil- ity of the m th user satisfies

Therefore, the asymptotic efficiency of the m inimum er- ror probability coherent multiuser T& is given as

,t=min(l, A’::) )

detector, denoted as

m in E, ‘e*He cE(-1,0,+1)K

Em=+1

= m in EE{-l,O,+l)K

{1,2Az&}, c,=+l

where A = diag(a). The near-far resistance of the m ini- mum-error probability receiver with side information is obtained from the following equations that are reminis- cent of those that yield the near-far resistance for coher- ent channels (note that the definitions of asymptotic efficiency and near-far resistance for coherent and non- coherent channels are not identical) of the m inimum- error probability detector in [8]

YjL=aF{ m in 1, m i

m in ETARAE l E{-1,0,+1JK

Em=+1 1

=min 1, i

m in inf E*& EE{-l,O,+ljKaE&

Em=+1 1

=rnin(l,~~~+v*R~)

l/R;,,,,’ if m th user is linearly independent, = 0, if m th user is linearly dependent,

where the last equation was obtained from the solution to the m inimum-norm‘ optimization problem of (4.22) and the fact that R+ m m 2 1, both of which were obtained in Proposition 4. Finally, since the near-far resistance of the m inimum error probability receiver with side information overbounds the near-far resistance of any DPSK receiver (without this side information), and since from Proposi- tion 3 the near-far resistance of the decorrelator achieves this upper bound, we have the sought result. •1

V. CONCLUSION

The main contribution of this paper is finding the decorrelating linear multiuser detector for noncoherent demodulation of differentially phase-shift keyed transmis- sions in a synchronous CDMA channel. This detector was obtained as a solution to the m inimax optimization prob- lem of finding the linear detector that optimizes the worst-case asymptotic efficiency over near-far environ- ments. It was also shown that not only is the decorrelator optimal in terms of near-far resistance among the class of linear detectors, but it also achieves the highest near-far resistance achievable by any noncoherent DPSK detector. The only restriction for the linear decorrelator to be near-far resistant for a particular user is that the corre- sponding signature signal be linearly independent of the rest of the interfering signature signals-a m ild restric- tion compared to the requirement of the conventional detector that it be orthogonal to the other signature signals. The performance invariance of the decorrelator to interfering signal uncertainties together with its ease of realization-neither requiring elaborate energy and phase estimation and tracking schemes nor any computation in excess of the conventional detector-make it eminently suitable for practical applications.

While it is conceivable that transmissions emanating from a central transmitter in multicast networks may arrive at a receiver synchronously, it is difficult to achieve this synchronization in multipoint-to-point communica- tion. Of considerable interest, both from the viewpoint of theory and practice, is the generalization of this work for the noncoherent asynchronous CDMA channel.

ACKNOWLEDGMENT

The authors would like to thank the reviewers for their helpful comments and one reviewer in particular for providing detailed and insightful suggestions for improv- ing a previous version of this paper in both style and content.

APPENDIX

In this appendix, we show that R,&,, 2 1, a result that was required to prove Corollary 1 of the paper. The proof given here is an extension of the corresponding result in [7] for real and invertible matrices. Since R is a nonnega- tive definite Hermitian matrix, it is unitarily similar to the diagonal matrix A = diaglh,, . . *, AK} of its nonnegative

1018 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 37, NO. 4, JULY 1991

eigenvalues so that R = UAGT, where U has the eigenvec- tors as its columns. Further, the Moore-Penrose inverse can be written as R+ = UA+ UT, where A = diag{h:; . ., hi}with A:=hL:’ when iEZ with Z={~E u,2,*. ., K}; Ai > 0) and A,? = 0 when j E Z. Therefore, R+ = l&G, = Ci E IAil~,i12 (real and positive) with uz bz:g the m th row of U. Since the diagonal elements of R are all equal to unity, we have Cit IAil~,i12 = 1, so that

where we use the fact that n + x-l 2 2 for x > 0. Since UaT= I, we have C~rl~,~l~ = 1. For some k E I, i.e., when A, = 0, we have Ru, = 0. A linear combination of the columns of R that is equal to zero cannot include the linearly independent columns, i.e., u,~ = 0 for every m E I. Therefore Ci t II~,i12 = 1 whence it follows that Rz, 2 1.

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