IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 41, NO. 4 ...ecee.colorado.edu/~varanasi/pvarit1995-07.pdf · 1084 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 41, NO. 4, JULY 1995

IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 41, NO. 4, JULY 1995 1083

Group Detection for Synchronous Gaussian Code-Division Multiple-Access Channels

Mahesh K. Varariasi, Member, IEEE .

Abstract- The concept of group detection is introduced to address the design of suboptimum multiuser detectors for Code- Division Multiple-Access (CDMA) channels. A group detection scheme consists of a bank of P group detectors, one each for detecting the information symbols of users in each group of a P group partition of the li simultaneously transmitting users. In a parallel group detection scheme, these group detectors operate independently, whereas in a sequential scheme, each group detector uses the decisions of the previous group detectors to successively cancel the interference from those users. Group detectors based on the generalized likelihood ratio test (GLRT) are obtained for the synchronous Gaussian CDMA channel. The complexity of these detectors is exponential in the group size, whereas that of the optimum detector is exponential in I<:. Since the partition of users is a design parameter, group sizes can be chosen to satisfy a wide range of complexity constraints. A key performance result is that the GLRT group detectors are optimally group near-far resistant. Furthermore, upper and lower bounds on the asymptotic efficiency of the sequential group detectors are derived. These bounds reveal that the sequential group detectors can, under certain conditions, perform as well as GLRT group detectors of much larger group sizes. Group detection provides a unifying approach to multiuser detection. When the users are partitioned into I< single-user groups, the GLRT, a modified form of GLRT, and the sequential group detectors reduce to previously proposed suboptimal detectors; namely, the decorrelator, the two-stage detector, and the decorrelating decision-feedback detector, respectively. For the other nontrivial partitions, the group detectors are new and have a performance that is commensurate with their complexity.

Index Terms-Signal detection, code-division multiaccess, min- imax methods, digital communication, direct-sequence spread spectrum, Gaussian noise.

I. INTRODUCTION

I N SYNCHRONOUS Code-Division Multiple-Access (CDMA) communication, several digitally modulated

waveforms that are encoded by distinct signature signals are simultaneously transmitted in a symbol-synchronous fashion over a shared channel. This paper deals with the additive white Gaussian CDMA channel over which pulse-amplitude modulation (PAM) or quadrature-amplitude modulation (QAM) is used [ 11. The received signal is modeled as a superposition of the K simultaneously transmitted

Manuscript received February 9, 1993; revised January 5, 1995. This research was supported by the National Science Foundation under Grant NCR-9206327. The material in this paper was presented in part at the 26th Annual Conference on Information Sciences and Systems, Princeton University, Princeton, NJ, March 1992.

The author is with the Department of Electrical and Computer Engineering, University of Colorado, Boulder, CO 80309 USA.

IEEE Log Number 9412066.

waveforms and the additive noise, so that its complex baseband representation is given by

K

r(t) = CbkUklLk(Q + n(t) (1) k=l

where bk is the kth user’s information symbol, Uk = fiej+, is the received amplitude of the kth user’s transmission, and Uk(t) iS the kth user’s complex-valued, unit-energy signature signal which is time-limited to [0, T]. Direct sequence (DS), frequency-hopped (FH), and hybrid DS-FH spread-spectrum signals can be modeled by real, imaginary, and complex- valued signature signals, respectively. The complex additive white Gaussian noise process n(t) has a power spectral density of height g2. The symbol bk belongs to Fk, which is the M- ary signal constellation of the kth user. For convenience, when the signal constellation of the users are all two-dimensional (QAM) or all one-dimensional (PAM), we refer to the resulting multiple-access channel (MAC) as a QMAC or PMAC, respectively.

A succinct description of the problem is facilitated by using matrix notation. The K-length column vectors of the signature signals, the signal amplitudes and energies, and the information symbols are denoted as u(t), a and w, and b, respectively. The Cartesian product FI x. . . x FK of admissible values of b is denoted as F and the signature signal correlation matrix as

R= .I

T u*(t)uT(t) dt

0

while the superscripts * and T denote complex conjugation and matrix transposition, respectively. The signature signals are linearly independent, and hence R is positive-definite.

For the PMAC, where the symbols transmitted by all the users are real-valued, it is convenient to introduce the passband representation of the received signal so that

+(t) = -&i$&(t) + ii(t) k=l

(2)

where bk E Fk is now the kth user’s real-valued information symbol, and tik (t) is the kth user’s unit-energy signature signal which is time-limited to [0, T] and expressed in terms of uk (t) as

Further

&(t) = Re [&k&(t)e’(wct+‘k)].

A(t) = Re [fin(t)ejuc”]

001%9448/95$04.00 0 1995 IEEE

1084 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 41, NO. 4, JULY 1995

is an additive white Gaussian noise process with a power spectral density of height g2. Let O(t) denote the vector [‘CLl(t),‘.‘,~K(t)] T of signature signals. The correlation matrix of these signals is given as

kZ .I’

T ii(t)&‘(t) dt = $[O*RO + OR*@*] (3)

0

where 0 is the diagonal matrix formed by the complex expo- nentials of the phases of the users. Since (3) is valid even when the signals {t&(t)} are complex-valued, the aforementioned PMAC includes the bandwidth-efficient PAM-SSB or PAM- VSB modulation schemes as well.

Centralized coherent multiuser detection refers to the si- multaneous detection of all the users when their signatures and amplitudes are known at the receiver. Since the additive noise is white and Gaussian, and all the realizations of b are equiprobable, the optimum multiuser detection rule that achieves the minimum probability of error for the joint decision on b, selects 6 according to the minimum energy realization of the noise process. Consequently, with A = diag {a}, we have a baseband representation for the optimum detector given as

6 = arg F>; 2 Re {Q*~As} - s*~A*RAs (4)

where

cl= s

T r(t)u* (t) dt

0

is the sufficient static vector obtained as the sampled outputs of a bank of matched filters, matched to each of the signature signals. Unfortunately, the combinatorial maximization problem in (4) is NP-hard in K [2]. As described in [3], the MK values of the likelihood function in (4) can. be computed in a tree structure. However, the time complexity per symbol (TCS) of 0 (MK/K) is too high for all but small values of K. The design and analysis of low-complexity and suboptimum multiuser detectors has therefore emerged as a key problem in multiuser detection. For instance, in [2], the linear decorrelating detector was proposed and shown to be optimally near-far resistant. In [4], a multistage detector based on a nonlinear multiple-access interference cancellation technique was proposed, bit error probability formulas for the two-stage detector with the conventional and decorrelating first-stages were derived, and it was shown that substantial improvements over the first stage were possible. Neural- network-based multiuser detection was investigated in [5], and a decorrelating decision feedback detector was proposed in [6].

The rest of the paper is organized as follows. In Section II, we derive group detectors for the QMAC and the PMAC based on the generalized likelihood ratio test (GLRT). Using these GLRT group detectors as building blocks, parallel and sequential group detection schemes are proposed. In Section III, a performance analysis of the two schemes is undertaken. This includes the characterization of the asymptotic efficiencies of GLRT group detectors, a proof of their optimality in group near-far resistance, a comparison between the PAM and QAM group detectors, the tradeoff between their performances and

implementation, the derivation of upper and lower bounds on the asymptotic efficiency of the sequential group detectors, and results pertaining to the choice of good ordered partitions for the sequential group detection scheme. Section IV concludes the paper, and the Appendix introduces a modified form of the GLRT group detector.

II. GROUP DETECTION

A natural strategy for multiuser detection involves partitioning the K users into P disjoint groups and detecting the information symbols using a bank of P group detectors, one for each group in the partition. Each group detector performs the joint detection of symbols in the corresponding group. A parallel group detection scheme consists of group detectors that operate independently and in parallel, and a sequential group detection scheme consists of group detectors that are connected sequentially to enable successive interference cancellation.

A. The PAM and QAM Group Detectors

In this section, we derive group detectors for the QMAC and the PMAC based on the generalized likelihood ratio test, and we describe the implementation of parallel group detection schemes that naturally result from them.

Notation: Let R = { 1, . . . , K}, G, H C R, and the complement of G with respect to R be denoted as G. For any x E CK, XG E (cIGl is obtained from x by striking out Xk !f’lc E G. For any X E (CKxK, XGH E a!GlxlH1 is obtained from X by striking out the ith rows and the jth columns V’i E G and Vj E I?.

Consider the detection of bG, the symbols of users indexed by the set G. Assume that the signal amplitudes of all interfering users in G are unknown. This means that if only bG has to be detected as in decentralized detection, ac need not be acquired. However, for H C G, if by has to be detected as well, then aH has to be acquired to detect bH, but its knowledge is not used in detecting bG. With this assumption, we propose to detect the symbols of the users in group G using the generalized likelihood ratio test [7]. For a given s E F, let Q = As so that the generalized likelihood function is the maximum value of the likelihood function over a~. The QAM group detector therefore selects iG that maximizes the generalized likelihood function over

rI Fk = FG LEG

so that

6G = arg sm2;G”,“-p 2 Re [q*Ta] - CX*~RCX G G

= argsm;lii$ v*~Rv G

RGG RGG VG

RcG RGG I[ 1 UC (5)

where the second equality is obtained from the change of variable v = a-&q, with Q = R-l, and the third equality follows from rearranging terms in the quadratic form v*~Rv. Using

VARANASI: GROUP DETECTION FOR SYNCHRONOUS GAUSSIAN CDMA CHANNELS 1085

the block triangular decomposition of partitioned matrices [S], we obtain

by = arg sm~gc'v"Gf[V~T v~*RGGR-~GG + vk*] G 7

! RGG - RGGR-~G~RGG 0

0 RGG 1 I "'G

R-l~gRgG~~ + VG 1 = arg,m~;G{V&T(&G - RG~R-~~GRGG)vG} (6)

c

where, because RGG is positive-definite, the second equality results from the likelihood function being uniquely minimized at vG = -R-IGGRGGvG. Further, (RGG - RG~R-~cGRGG), the Schur complement of RGG, is equal to Q&. Given that VG = AGGSG - [Q~]G, and letting 2~ = Q&[QQ]G, the QAM group detector in (6) reduces to

in = arg~m~~~{2Re(2~TA~~sG)-s~TA~GQC~A~a~~}. G

(7)

Consider next the detection of bG in a PMAC. The QAM group detector derived above is of course applicable. However, when the signal phases of all the users can be acquired, it is possible to improve upon the performance of the QAM group detector by considering an alternative strategy. Let W be the diagonal matrix formed by the vector w, and let 4 be the vector of sufficient statistics for the PMAC defined as

J T

4X i(t)&(t) dt = ;[O*q + Oq*]. 0

The PAM group detector is derived from the generalized likelihood ratio test in a manner similar to that of the QAM group detector but with the difference that only the signal energies of the interfering users in the set G are assumed to be unknown. The PAM group detector for the PMAC therefore takes the form

iG = argsm3$cG{2k~~~~sG - s~~jj~i$,~~($jsG} (8) G

where 0 = A-’ and ?G = Q&[&]G. The implementation and complexity properties of the group

detectors are listed below. l A parallel group detection scheme for a two-group parti-

tion of a five-user QMAC can be implemented as shown in Fig. l(a). The vector of sufficient statistics q, obtained at the outputs of the bank of K matched filters, forms the input to each of the bank of group detectors, where the group G detector transforms it by a linear transformation to produce the vector ZG = Q($[Qq]G, which in turn forms the input to the decision algorithm (7).

l A more efficient implementation can be obtained as in Fig. l(b) by noticing that the vector %G can be obtained at the output of a bank of IGI matched filters that are matched to the vector of signals defined as the complex conjugate of Q,& [Qu* (t)]G. These signals can be thought of as group decorrelating signature signals because the space spanned by them is orthogonal to Span

MU&G. BY using these matched filters, the linear operations required to transform q to ZG, which must be performed once every symbol duration for each group in the implementation of Fig. l(a), are now eliminated. They are replaced by a one-time precomputation of the group decorrelating matched filter impulse responses. Furthermore, only as many matched filters as the number of users to be detected are required in this case. The structure, implementation, and complexity properties of the PAM group detector are similar to those of the QAM group detector. However, the PAM group detector is specified for the passband model; & is obtained from a single bank of JGI matched filters, matched to the group decorrelating signals o,&[@* (t)]G, and hence the realization of these filters requires the precise knowledge of the signal phases of all the users. The implementation of either group detector requires the combinatorial maximizations in (7) or (8) that have the same functional form as that required for the maximum- likelihood detector in (4) (the special case of the two group detectors where G = R reduce to the maximum- likelihood detector). The group detector is therefore NP- hard in ICI, the cardinality of G. Fortunately, however, IGI is a design parameter. In practice, if design consider- ations allow the implementation of an optimum detector for up to N users, the set of all active users has to be partitioned into groups of size no greater than N each because the time complexity per symbol for the group detection scheme is dominated by the largest group size. Consider the detection of user k alone, i.e., G = {k}. The QAM group detector reduces to the QAM decorrelating detector,

(9)

The significance of this detector is that it is computation- ally the most efficient, consisting of only a decorrelating matched filter followed by a QAM slicer and not requiring the signal amplitudes of any of the interfering users. For MPSK modulation, it does not require the energy of the desired signal either and, for Fk = { $1, -l}, it becomes the antipodal QAM decorrelator

h+ = w [Re (G(Qq)lc)l. (10)

The special case of the PAM group detector with G = {I;} yields the PAM decorrelating detector which is the real- arithmetic version of (9), given as

h = args:i;k([Li4]k - SkdGJ2. (11)

Furthermore, if Fk = { $1, - l}, it coincides with the decorrelator, ir, = sgn [&]k, that was obtained in [2] by optimizing the near-far resistance among linear detectors.

B. Sequential Group Detection

In this section, we propose a sequential group detection strategy where the users to be detected are partitioned into an


,-----------------~

1 G - group detector; .------t [ -. . i$‘=‘(1,2,3) ,

r(t)

t

U vt=iT

+I- MF3 t = iT I

u vt=iT’ . . . . . .-em-( & - group detector!

_____-_______----I

----------i )_ - - - - - - - ~ - -~-roupdetector j+$L,-+

(4

----------i , )_ - - - - - - - ~ - --groupdetector

I

I

,

9 . , ,

-y% , I I ,

E i 5-c .z.z ;

ky 0 -$ ;

I b

4 -if ; I I I \ 1

P j-4 ,

c___-_-___-_-______-__________) ;---‘-‘----“-‘-‘-------------.

I

-2 , 1

- $4 E :

g-c ! , ,

‘Z.E , ’ I ‘G 0

am I -Om 1

l +B ! , G, - group detector b , I .__________-_-______-----------

(b) Fig. 1. (a) A parallel bank of two group detectors for a five-user CDMA channel with G1 = (1, 2, 3) and Gz = (4, 5). (b) Group decorrelating matched filter (GDMF)-based implementation of a parallel group detection scheme for detecting five users of a CDMA channel.

ordered set of P groups, Go, . . . , GP-1. The users in group Go are detected by the GLRT group Go detector specified by (7) (or (8) in the case of the PMAC). These decisions are then used to subtract the multiple-access interference due to users in Go from the remaining matched filter outputs q@,. The resulting decision statistics are used to detect the users in the next group Gi using the group Gi detector designed under the assumption that the multiple-access interference cancellation is perfect. The process of interference cancellation and group detection is thus carried out sequentially for users in group G2,... , G+i, with the group detector for group G, taking advantage of the decisions made by the group detectors for Go,. . . , Gp-l.

Let us introduce the initializing definitions before we em- bark on a formal description of the sequential group detector for the QMAC.

q(O) = q R(O) I R Q(O) ZT Q. (12)

Fig. 2. The sequential group detector for a two-group partition R = Go UG1 with the sequential G1 group detector performing the operations of (14) and (15) with p = 1.

Sequential Group Detection:

1)

2)

3)

4)

5)

The decisions b& on the symbols of users in Go are made according to (7). Set p = 1. Define

P-1

HP = R - u Gi q(p) = QH P

i=l

Rep) = RH,H, and Q(p) = [R(p)]-l.

Obtain the decision statistics z(P) by subtracting the multiple-access interference contribution due to users in l?, to q(P) so that

&‘) = q(P) - RHPHpAH H bH P P P’ (13)

Form the vector of decision statistics z$: that is required by the group G, detector as

z$$‘: = [QgiG,lml [Q(P)~(P)]~, (14)

where Q$‘jGp is obtained by retaining the G,-rows and G,-columns of Q(p) with the rows and columns of Q(p) indexed according to the elements of HP rather than from 1 to 1 HP I. The same convention applies to [Q(P).&)]G,. The group Gp detector is designed under the idealized assumption that the multiple-access interference cancellation in step 3) is perfect, and as a result

&, = arg sGm2;G 2 Re (LC$~**AG,G,SG,) P P

-SG *T&&G~ [Q~~G,l-lA~,~,~~, . (15)’

6) Augment &, to b~~+~ by appending to it the decisions && obtained in step 5). Increment p by 1. If p < P - 1, go to step 1); else, stop.

An implementation of the sequential group detector for a five-user CDMA channel with Go = { 1, 2, 3) and Gi = (4, 5) is depicted in Fig. 2.

The sequential group detection scheme for the PMAC can be derived in an analogous manner. The special case of the PMAC sequential group detection strategy when all the groups


in the partition are single-user groups can be shown to coincide with the decorrelating decision feedback detector proposed in [6] where the users are arranged in the decreasing order of their energies. In describing the sequential group detection strategy, we have not specified a particular choice of ordered group partition. The problem of choosing good ordered partitions will be dealt with after the performance of the detection scheme is characterized.

III. PERFORMANCE ANALYSIS

The performance of the parallel and sequential group detection schemes is analyzed, and the performance-complexity tradeoff issue is studied. The asymptotic efficiency of group detectors is derived, the group near-far resistance measure is introduced, and the optimality of group detectors in this measure is established. Upper and lower bounds on the performance of sequential group detectors are derived. Several numerical examples are provided to illustrate these results.

A. Bit Error Probability and Asymptotic Eficiency

Dejinition: The asymptotic eficiency of any user Ic employing the detector 4 whose error probability for that user is Pk(a, 4) is defined as

vk(4) = Sup 1

0 5 T 5 1; lim h(a, 4) ,+o,p> < O” (16)

1

This performance criterion was introduced in [9], and there are several reasons for considering it over the bit error rate. The asymptotic efficiency of the maximum-likelihood detector is known to tightly hug the bit error probability performance in medium to high signal-to-noise ratio regions (cf. [4], [9]). At the same time, unlike the bit error rate, it is analytically tractable and results in a succinct characterization of the loss in performance due to the presence of interfering signals.

Denote the, QAM group detector as 4~ and the PAM group detector as 4~ so that their asymptotic efficiencies for user k E G are denoted as vr~ (4~) and nk (&), respectively. Define the set of error vectors of user Ic as EI, = (6 E (-1, 0, l}lGl, Sj, = l}, with jk being the index of the kth user within the group G. Consider the following proposition.

Proposition 1: The asymptotic efficiency of any user k E G employing the QAM group G detector is

and the asymptotic efficiency of any user Ic E G employing the PAM group G detector is

n Proof The statistical characterization of the vector 2~ =

A&XG is obtained by describing its dependence on the received signal

2~ = A&Q&[Q[RAb + 711~

= A&GQ&AGG~G + A (19)

where X = AkGQ&[Qq]G is a zero-mean Gaussian random vector, with a covariance matrix that is given as E[XX*T] = LT’A&Q&AGC. Now consider a fictitious IG]-user QMAC with a normalized signature signal vector v(t) and signal amplitudes c given by

v*(t) = D-1/2Q&[Q~*(t)]G and c = D112ac (20)

where D is a diagonal matrix whose diagonal elements are the same as those of Q&. It is evident that 2~ in (19) can be seen as having been generated as the scaled outputs of the bank of IGI matched filters matched to C*v(t) over this fictitious channel. Furthermore, the group detector in (7) is the maximum-likelihood detector based on 2~. These facts taken together establish a statistical equivalence between the group G detector in the K user QMAC and the maximum-likelihood detector in the fictitious IGI-user QMAC specified above. The equivalence between the two detectors is depicted in Fig. 3.

The asymptotic efficiency in (17) can be obtained by invok- ing the aforementioned equivalence. The bit error rate of the group G detector is equal to that of the maximum-likelihood detector in the fictitious ]G]-user QMAC, which in turn can be asymptotically tightly upper- and lower-bounded as in [3]. Hence

pQ(d lc,min/a) 5 pk((7, b) < c 2~~(6)Q(JZllvT(t)CSll/a)

6EIk < c 2-1”(6’Q(~ll~T(t)Csll/a) (21)

6EEk

where p E (0, 11, w(S) is the weight of 6, and the minimum distance parameter d, min is defined as

dZ, min = mEyl(vT(t)Cs2 I

= minSTA&Q&A~GS. (22) 6EEk

In the tighter of the two upper bounds, the set 1, C El, denotes the set of indecomposable error vectors, obtained from E,+ by discarding the decomposable error vectors that can be decomposed into two nonzero vectors S’ and S” such that a) S = S’ + S”, b) Sj = 0 + Si = Sy = 0, and C> Re(S’TAT=GQ,$ A GG S”) > 0. The expression for the asymptotic efficiency in (17) follows by noting that, in the high signal-to-noise ratio region, the sum of terms in the upper bounds is dominated by the error vector corresponding to least energy di, min.

The asymptotic efficiency of the PAM group detector given in (18) is obtained in a similar manner. The equivalence result for the PMAC is that the PMAC group G detector is equivalent to a maximum-likelihood detector in a fictitious IGI-user PMAC with normalized signature signal vector c(t) and signal energy vector i: with

ti(t) = B-1/2Q&[$&(t)]G and i: = &IJ (23)

where the diagonal matrix B is defined to have diagonal elements which are equal to those of the matrix s,k. n


An easily computable, amplitude-independent upper bound on the asymptotic efficiency is given in the next proposition. Sufficient conditions under which the bound is achieved are given in terms of the matrix V = ALGQ&A~~.

Proposition 2: The asymptotic efficiency of the QMAC group G detector is upper-bounded as

where the upper bound is achieved if, VI # jk

Proof: Since the matrix V is Hermitian, the asymptotic efficiency of the group detector can be written as

’ vk($G) = - minST(V + V*T)S. 2)akj2 ~EEI,

(26)

The bound in (24) is easily obtained by evaluating ST(V + V*T)S at S = ujb, the jkth unit vector. Consequently, the bound is achieved if and only if t/S E { - 1, 0, l}lGl, we have

2~S~Re(Kj,)+ ~~SdRe(k) 20. (27) l#jk I#jk;#jk

A sufficient condition for (27) can be obtained by lower- bounding the expression on the left-hand side of the inequality by taking the worst case approach [l l] of replacing the correlations by the negative of their magnitudes and excluding -1 from the possible values of SI. Therefore, the upper bound is achieved if VSG E (0, l}IGl

2~S@(K~,)I - ~6% + c c Sd;JReKiI IO. l#jk lflik l#jk i#jk, 1

(28) The above inequality is in turn satisfied if the upper bound on the expression on the left-hand side

-XL + IRe (Kj,)l + ClRe (%>I if1 is nonpositive, for which the conditions in (25) are sufficient.

n The analog of Proposition 2 for the PMAC is that the

asymptotic efficiency of the PMAC group detector is upper- bounded as qk(dG) 5 [o,k]j,j,, with equality holding if, with the matrix U defined as Wkgo& W$?, the following inequalities hold Vl # jk:

From these sufficient conditions, it can be seen that the lath user’s asymptotic efficiency is equal to the upper bound (unity when IGI = K) provided that the equivalent energy of that user is sufficiently low, a conclusion that was obtained for the maximum-likelihood detector achieving unit asymptotic efficiency in [12].

maximum likelihood ---) ‘0

decision algorithm :

.

Fig. 3. The equivalence of the QAM group G detector in the K-user QMAC and the maximum-likelihood detector in the fictitious IGI-user QMAC with signature signals and amplitudes defined in (20).

J-/J-~ u, (t)

Fig. 4. Direct-sequence signature signals of the four-user CDMA system used in Examples 1-3.

Example I: A four-user CDMA channel is considered where the signature signals are real-valued direct-sequence spread-spectrum signals derived from Gold sequences of length 7 and are shown in Fig. 4. These signals were previously considered in the computation of numerical examples in the context of multistage detection in [4], decorrelating decision feedback detection in [6], and differentially coherent decorrelating detection in [lo]. Consider the asymptotic efficiency of user 1 and the possible improvement in performance over the decorrelating detector under the constraint that no more than two users can belong to a single group. The first user can be paired with one of the other three users, and in each case, the asymptotic efficiency is depicted in Fig. 5. While no single strategy uniformly outperforms the other, the pairing of user 1 with user 4 yields the maximum performance gain for the additional complexity in the near-far region. The result of Proposition 2 is also illustrated in this figure. The upper bound on the asymptotic efficiency given by (24) is achieved when the interfering user within the group is sufficiently strong compared to the desired user.

B. Near-Far Resistance

In this section, we consider the performance of the PAM and QAM group detectors in terms of the neat-far resistance


0.6

Amplitude ratio (ak/al)

Fig. 5. The asymptotic efficiency of user 1 for the three two-user group detectors and the decorrelator (see Example 1 for details).

measure. In a key result of this section, we establish the optimality of the GLRT group detection schemes in this measure.

Definition: The Q(P) near-far resistance of user k employing a detector 4 is defined as the worst case nk(4) over the signal amplitudes (energies) of all other interfering users and is denoted as ?jk(4)(ek($)).

The concept of near-far resistance was introduced in [2] through the P neat-far resistance measure in the context of coherent multiuser detection, and the Q near-far resistance was adopted in the context of differentially coherent multiuser detection in [lo].

Proposition 3: For any group G C R, and for each Ic E G, the QAM (PAM) group detector has a Q (P) near-far resistance that is equal to the lath user Q(P) near-far resistance of the maximum-likelihood detector. Therefore, for the QMAC, we have

7]k(4G) = ~k(b,t) = & (29)

and for the PMAC, we have

c$kk

Proof The Q near-far resistance of the QAM group detector admits the following equalities:

The second equality is obtained by combining the two mini- mizations over the complex amplitudes and the error vectors into one minimization over the set of admissible values of

t = ailA~~S. The third equality results from an application of the Cauchy-Schwarz inequality, and the last equation is the result of [ 10, Proposition 51. The result for the PMAC can be obtained in an analogous manner. n

The following corollary orders the P and Q near-far resistances of the PAM and QAM group detectors. The proof is left to the reader.

Corollary 1: The P near-far resistance of the PAM group detector is uniformly higher than the P near-far resistance of the QAM group detector, which in turn is higher than the Q near-far resistance of the QAM and the PAM group detectors, the last two being equal, and hence implying the optimality of the PAM group detector in Q-near-far resistance, so that

rjk($G) > +k($‘G) > qk($G) = qk($G). (32)

Further, equalities in both inequalities hold if the phases of all users coincide, and equality holds in the second inequality if the phases of the users in group G are equal. n

Proposition 3 states that, independently of how the users are partitioned, the Q (P) near-far resistance of any user is as high as that of the optimum detector for that user. Therefore, according to the near-far resistance criterion, there is no dis- tinction between any of’the QAM (PAM) group detectors; they are all optimally Q(P) neat-far resistant. However, near-far resistance is a conservative performance measure and gets more conservative as the number of users in the overall system of fixed bandwidth increases. Consequently, the difference in performance between the decorrelating detector and the optimum detector, while being zero at one operating point, becomes more pronounced with an increase in the total number of users for operating points outside its small neighborhood. This behavior can be attributed to the fact that the difference between the exponentially complex optimal decision regions and the linear hypercone approximations of those regions grows rapidly with increasing K. A finer performance measure is considered in the next section.

C. Group Near-Far Resistance

Dejinition: The Q(P) group near-far resistance with respect to group G of user Ic E G employing a detector 4, denoted as rk($, G) (+k($, G)), is defined as its Worst case lath user asymptotic efficiency over the signal amplitudes (energies) of users in G.

A nonzero value of group near-far resistance testifies to the robustness of a detector to variations in the amplitudes (energies) of the signals in G, in that it assures the exponential decay of the bit error probability, independently of what the amplitudes (energies) of users in group G may be, at a rate that is at least equal to that achieved by the user in question when transmitting in isolation with energy reduced by a factor equal to the group near-far resistance.

Proposition 4: The Q(P) group near-far resistance of any user k E G employing the QAM (PAM) group G detector is equal to the Q (P) group near-far resistance of the maximum- likelihood detector. Therefore, for the QMAC, we have

‘-Yk(4G, G) = ‘?k(+optr G) ‘dk E G (33)


and for the PMAC, we have

‘?k($G, G) = +k(dopt, G) \Jk E G’. (34)

Hence, the QAM (PAM) group G detector is optimally Q(P) group neat-far resistant in the sense that each user in G achieves the highest achievable Q(P) group near-far resistance. n

Proof: The result is proved for the QMAC. The corresponding result for the PMAC can be proved analogously. The Q group near-far resistance of the maximum-likelihood detector can be obtained by minimizing its asymptotic efficiency over ac. We have the following self-evident equalities:

‘-fk(bpt, G) = 1

-inf hi2

min eTA* RAE UE es{-l,O, l}K Ck=+l

0.2 0.4 0.6 0.8 1.0 1.2 1.4

Signal ampli tude (user 1)

Since the last minimization problem is identical to the one Fig. 6. Worst case asymptotic efficiency of group detectors and optimal

solved in the derivation of the GLRT group detector, we have near-far resistance (see Example 2 for details).

= qk($G) = ‘-Yk($G, c)

where the last two equations follow from Proposition 1 and the fact that the asymptotic efficiency of &! is independent of ac. Therefore, the group near-far resistance of the group G detector is equal to the group near-far resistance of the maximum-likelihood detector. Since the latter is optimal in asymptotic efficiency, the group G detector is optimally group neat-far resistant in the sense that, for each Ic E G, no other detector achieves a higher group near-far resistance. n

Example 2: Consider the four-user system from Example 1. The optimality of group detectors in terms of near-far resistance is demonstrated in Fig. 6. The asymptotic efficiency of all users employing group detectors for any nontrivial choice of partition must, according to Proposition 4, lie between the corresponding optimum and decorrelating detector performances. Consider the proof of Proposition 3 for the case where G = 0. The vector t that solves the minimization problem in (31) yields the set of amplitudes for which the asymptotic efficiency of the optimum detector achieves its worst case value, which in turn is equal to the asymptotic efficiency of the decorrelating detector. Since the latter two measures meet at this point, so will the asymptotic efficiency of any group detector. In Fig. 6, this phenomenon is displayed for the first user by fixing the amplitudes of the interfering users at the worst case levels normalized so that the amplitude of user 1 is unity. The abscissa then denotes the variation in the amplitude of the first user, and the ordinate depicts the first user’s asymptotic efficiency for the four different groups considered in the previous example. For this choice of interfering user amplitudes, it is seen that all the curves meet at the worst case asymptotic efficiency point.

The result of Proposition 4 states that the Q(P) group neat-far resistance of the QAM (PAM) group detector, which has an exponential complexity in ICI, is as high as that of the optimal detector whose complexity is exponential in K.

Furthermore, this measure becomes less conservative with an increase in the size of the group G, which in turn, however, increases the complexity of the corresponding QAM (PAM) group detector.

The next corollary draws a comparison between the PAM and QAM group G detectors and makes the case for the PAM group detector if all signal phases can be acquired.

Corollary 2: The worst case asymptotic efficiency of the PAM group detector over the interfering signal phases is identically equal to the asymptotic efficiency of the QAM group detector, i.e.

(35)

Consequently, the Q group near-far resistances of the PAM and the QAM group G detectors are equal for all users k E G, and hence the PAM group detector is optimally Q group near-far resistant. n

As a consequence of Proposition 4, we also have the result that, for GLRT group detection, increasing the group size results in a uniform improvement in asymptotic efficiency.

Corollary 3: Suppose that Gr, Gs s R are such that k E Ga 5 Gr. Then the asymptotic efficiency of the QAM (PAM) group Gi detector is uniformly higher than that of the QAM (PAM) group Ga detector so that

and

qk($G~) > qk($G,) &fJ E WT. (37)

In particular, the asymptotic efficiency of any user employing a group detection scheme for any partition of the set of K users is uniformly lower-bounded by the asymptotic efficiency of the decorrelating detector for that user. n


Proojl The second part of the corollary follows from the first part by specializing it to the case where Ga = { Ic}. The first part can be proved as follows:

where the first equality is the result of Proposition 4 applied to the group G2 detector, the penultimate inequality follows from the fact that G2 G Gi, and the last equality is again a result of Proposition 4 applied to the group Gr detector. The corresponding result for the PAM group detector is proved similarly. n

Example 3: Considering again the four-user system of Ex- ample 1, Fig. 7 depicts the asymptotic efficiency of user 1 for the GLRT group detection schemes corresponding to the groups G = {l}, G = (1, 4}, G = (1, 2, 4}, and G = (1, 2, 3, 4}, the first of these being the decorrelating detector and the last being the maximum-likelihood detector. The asymptotic efficiency is shown as a function of the relative amplitudes of the interfering users with respect to the first user, and they are chosen for convenience to be real-valued (since the signature signals are also real-valued, the PAM and QAM detectors are identical) and equal. The result of Corollary 3 is illustrated in this figure; the larger the group size, the higher the performance.

D. Pegormance Bounds for Sequential Group Detection

While an exact performance analysis of the sequential group detector is intractable, one can obtain upper and lower bounds on its bit error probability and hence on its asymptotic efficiency. Criteria for the selection of good ordered partitions are evolved based on these bounds.

Consider the asymptotic efficiency nk(&, ), of the sequential group detector for the ordered group partition R = U,‘<’ Gi, for user k E G,. A simple upper bound on nk ($G, ) is given by the asymptotic efficiency under the assumption that the interference cancellation from previous groups Go,... , Gp-i is perfect. This perfect interference cancellation (PIC) bound for the QAM sequential group detector, denoted as &Gp3 is therefore given as

’ ?‘:($G,). (38)

For group Go, the PIC bound is equal to the asymptotic efficiency for each user in Go, but as p increases, it becomes increasingly weak. It is easy to show that, for every value of p > 0, the PIC bound on the asymptotic efficiency is uniformly higher than the asymptotic efficiency of the GLRT group G, detector. Due to the optimality of the GLRT detector in group near-far resistance, however, this implies that, for each p > 1, there exist operating regions where this bound ex- ceeds even the asymptotic efficiency of the optimum detector. Furthermore, these operating regions increase uniformly with increasing values of p. In order to avoid this deficiency, the

h 0.9

t

2 0.8

6 El ‘U .- 0.7 z W

0 ‘% z

0.6

E 2 a 0.5

0.4

1

L- 0 i i 3 4

i 5

Amplitude ratio ak/al

Fig. 7. Asymptotic efficiency of user 1 for four group detectors of increasing complexity and performance (see Example 3 for details).

PIC bound in (38) is easily modified as

vk($G,) 5 min {df(JG,), vk(dbt)>. (39)

Consider next the fact that, even though the bound in (38) is independent of the signal amplitudes of users in Gr, the actual performance is independent only of signal amplitudes of users in the future groups H,+r = UF=;:, G;. Consequently, nk(4G,) admits another upper bound which is the asymptotic efficiency of the GLRT group detector for the union of the past and current groups i?,+i. In fact, this bound can be shown to be uniformly tighter than the modified PIC bound of (39). Consequently, we have the following proposition.

Proposition 5: The asymptotic efficiency of the QMAC sequential group detector for the ordered user partition R = UF=i’ Gi for user k E G, admits the following bounds:

n The corresponding result for the PMAC sequential group

detector can be similarly deduced, and it can be verified that, for the particular case where all groups consist of single users, i.e., where the PMAC sequential group detector reduces to the decorrelating decision feedback detector, the PMAC counterpart of (40) yields bounds that are both uniformly tighter than the bound given in [6] because the latter is a special case of the PMAC counterpart of the PIC bound.

Example 4: The perfect interference cancellation (PIC) bound can be weak. Consider, for instance, a K-user CDMA system with equal energy, time-limited signature waveforms of duration T designed to maximize the determinant of the correlation matrix under a constraint on the maximum rms bandwidth as in [13]. Nontrivial solutions to this problem exist for the time-bandwidth product 2BT E (1, z/(K + 1)(2K + 1)/6] where B denotes the maximum rms bandwidth allowable for each signal. The upper end of this interval yields orthogonal signals for an optimum solution. Consider an eight-user example with a value of

1092 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 41, NO, 4, JULY 1995

2BT = fl that is much smaller than that required to achieve orthogonality. The optimum signature signals are not unique, and we pick the Hadamard solution described in [13]. These signals have the convenient property of yielding equal optimum detector asymptotic efficiencies for all users (see [ 131 for a proof). The resulting signal correlation matrix is given as

R=

1 .oooo 0.2365 0.4453 0.1238 0.2365 1.0000 0.1238 0.4453 0.4453 0.1238 1 .oooo 0.2365 0.1238 0.4453 0.2365 1 .oooo 0.6865 0.1895 0.3564 0.1050 0.1895 0.6865 0.1050 0.3564 0.3564 0.1050 0.6865 0.1895 0.1050 0.3464 0.1895 0.6865

0.6865 0.1895 0.3564 0.1050 - 0.1895 0.6865 0.1050 0.3564 0.3564 0.1050 0.6865 0.1895 0.1050 0.3564 0.1895 0.6865 1 .oooo 0.2365 0.4453 0.1238 0.2365 1.0000 0.1238 0.4453 0.4453 0.1238 1.0000 0.2365 0.1238 0.4453 0.2365 1.0000 _

In the bar graph of Fig. 8, the lower bound (to be derived in Proposition 6), the tight upper bound of Proposition 5, and the perfect interference cancellation bound of (38) on the asymptotic efficiency for each user of the sequential group detector for the ordered signal user group partition {8>{4}{2>{6}{7>{3>{5>(1) are depicted as the first three bars (an algorithm for obtaining good ordered partitions is described towards the end of this section). Notice that the PIC upper bound grossly overbounds the performances of users 4, 6, 3, and 1, whereas the tight upper bound of Proposition 5 remains much closer to the lower bound. The next two bars depict the lower and upper bounds for the sequential group detector with group size of 2 for the same example with the ordered group partition Go = (4, 8}, G1 = (2, 6}, GZ = (3, 7}, and Gs = { 1, 5) with the last bar denoting the optimum asymptotic efficiency. This detector can be seen to perform much like the optimum detector for all the users.

In the next proposition, we derive a lower bound on the asymptotic efficiency of the sequential group detector by obtaining an upper bound on its bit error probability.

Proposition 6: The lower bound on the asymptotic efficiency for user k E Gr of a sequential group detector for the ordered group partition R = Ur=i’ Gi is given as

where $($G,) is the PIC upper bound defined in (38). n

0 Low bnd SGD(1) q High bnd SGD(l) w PIC bnd SGD(1) q Low bnd SGD(P) 0 High bnd SGD(P) n ML Detector I

2 3 4 5 6 7 6

User Index

Fig. 8. Upper and lower bounds from Propositions 5 and 6 on the asymptotic efficiency of the sequential group detector with single-user groups (SGD(l) in legend) are compared with the PIC bound for that detector for each user of an eight-user CDMA system described in Example 4. The near-optimality of the sequential group detector with groups of two users (SGD(2) in legend) is also shown.

Proof Let the bit error probability of user Ic E G, be denoted as P~(JG,). Conditioning on the binary event of the decisions 6~~ of the sequential group detector on the symbols of the users in the past groups i?, = UyIt Gi being correct or incorrect, and bounding from above with unity a) the conditional probability that the lath user’s symbol is in error conditioned on &n, being incorrect and b) the marginal probability that &,p is not erroneous, we have

The first term in this bound is the probability of the union of events UT:: {by, # bG, }, which in turn is equal to the probability of the complement of the intersection nzzi { bGz = bG,}. Using the chain rule to express the probability of this intersection, we have

p-1

Pr (&BP # bB,) = 1 - nPr (bc, = bGi I&Hi = bBi). (43) i=o

Noting that the ith term in the product can be written as 1 - Pr (&G, # k, I&H, = b,%), and that, for a set of nonnegative numbers {pi}r=r satisfying the condition C&pi < 1, we have the inequality

- fi(1 -Pi) 5 -&i. i=l i=l

We deduce the result that, for sufficiently small g,

P-1

Pr(h, # b,) < )$+-(hGt # bG, I&, = b,). (44) i=o

The ith term in the above sum can be expressed as

Pr U {bl # bl}(&~, = bgi 1~Gi >


Applying the indecomposable error vector upper bound to each term in the union of error events, we have

pr (%G, # bG, lb, = bg,) 5

c 2-w(6) Q

2STA&;,; [Q~)G;]-~AG,G,~

where jr(i) (with I E Gi) is a subset of the set El of error vectors S of length ]Gi] with elements in { -1, 0, l} and with Sj, = 1 where jr is the index of user 1 within the set Gi. This subset is formed by removing from El the decomposable error vectors S that can be decomposed into two nonzero vectors S’ and S” such that a) S = S’ + S”, b) Sj = 0 + 65 = 8; = 0, and c) Re (S’.I.AT;,,G,[Q~iG,]-1AG,G;6”) > 0. Finally, substituting the inequality of (44) into (42), and making use of the result in (43, we have

P--l

i’O6E u &(i) LEG,

.Q d 2STA&Gs [Q$iG, I-~AG,G, 6 0

+ c 2-746)

6E& (P)

.Q \/ 2STAT, G [Q$J”G ]-lAG G 6 ppapp ‘~7

The result of the proposition follows by noting that the sum of terms in the above upper bound is dominated in the high signal-to-noise ratio region by the Q(.) function with the smallest argument. n

Proposition 6 has the following interpretation. The lower bound on the asymptotic efficiency is determined by the minimum of the effective energy (product of actual energy and the asymptotic efficiency) of the user under consideration and those of all the users in the past groups, with each such effective energy being computed under the assumption that the corresponding past users are perfectly detected. If for some user 1 in group G, it was found that the effective energies of all the users in the past groups Go, . . . , G,-1, each evaluated under the corresponding assumptions of perfect interference cancellation (PIC), were greater than the effective energy of the user under consideration under its PIC assumption, then the lower bound on its asymptotic efficiency in (41) coincides with its PIC upper bound and is hence equal to its exact asymptotic efficiency. In such cases, the tighter upper bound of Proposition 5 will a forteriori coincide with the PIC bound. This implies that the performance of the sequential group detector for the user in question rivals the performance of the GLRT “super group” detector 4~~+~. Consequently, in a system with sufficiently disparate energies, there exist ordered group partitions for which the lower bound on the asymptotic efficiency will coincide with the upper bound for every user,

0 Decorrelator q SGW) H ML detector

3 4 5 6 User Index

7 a

Fig. 9. Asymptotic efficiency comparison of the decorrelator and the sequential group detectors of group size 1 (SGD( I) in legend) for an eight-user equal-energy system described in Example 5.

thereby yielding the performance of a super group detector for every user with the super group HP+1 becoming successively larger for users in later groups.

Example 5: The sequential group detector has the potential of outperforming the parallel group detection scheme when the group sizes are the same in both schemes. We illustrate this for the difficult example of a system with equal energies. For instance, it can be shown that, for a correlation matrix of the form Rkl = olk-‘l, the asymptotic efficiency of the decorrelator is equal to 1 - a2 for the first and last users and (1 - cy2)/( 1 + cr2) for the rest, these results being independent of, the number of users. The proof is left to the reader. The asymptotic efficiency of the sequential group detector when the users are arranged in the order of their indices is equal to 1 - a2 for all but the last user, and the last user’s asymptotic efficiency is at least equal to 1 - 02. Fig. 9 illustrates these results for the example of a CDMA system with eight users of equal energy and a correlation matrix given as RM = a! P-11 with QI = 0.75.

The problem of ordered group partitioning for sequential group detection is considered next. Since the minimum effective energy among all users in previous groups tends to limit the performance of users in the current group, it is in general advantageous to let ]Go] > ]Gr] >,...,> ]GP-11. Once the group sizes are chosen, there are two factors that must be considered in evolving algorithms for determining good ordered partitions. First, the algorithm must rely on the bounds on the asymptotic efficiency given in (40) and (41). Second, since any measure of total performance based on these bounds will depend on the ordering of the groups, the total number of admissible ordered partitions is too large to permit an exhaustive search. A method that simultaneously addresses both factors involves an efficient sequential search. Among

P-l

K = zIGi1

i=o

users, select the first I Go I users which maximize the worst case effective energy (from (17)) among those users, and assign them to group Go. After groups Go, . . ’ , G,-l have been chosen, the group G, is chosen from the remaining users


by selecting IG,) users that have the maximum worst case upper bound on effective energy (from (40)). This procedure is continued until only IGp-11 users remain which are finally assigned to GP-1. This strategy is a natural one since the performance measure at each step in the sequential procedure is based on the performance of the corresponding sequential group detector which depends only on the current group and the previous groups which are already determined. By choosing the best group Go, the assumption of perfect cancellation is best ensured for any choice of Gi. Similarly, by selecting the best group Gi from HI to maximize the worst case effective energy, the perfect cancellation assumption is best ensured for any choice of Gz, and so on.

Example 6: Consider an eight-user CDMA system where the correlation matrix is of the form Rkl = al!+‘1 with a! = 0.8. The signal energies are unequal now and given as w = [3 9 7 6 8 5 4 21. The decorrelating and the optimum detector effective energies are shown in Fig. 10 along with the lower and upper bounds on the effective energies of the sequential group detector for the optimum sequential ordered partition of single-user groups given as {2}{3>{1}{4}{5}{6>{8}{7~. Note that this ordered partition is not the same as the decreasing order of energies. Further, the four weakest users 1, 6, 7, and 8 achieve a near-optimal performance. This is not the case for the four strongest users 2, 3, 4, and 5. However, since the strong users’ energies are high, their effective energies for the sequential group detector are all high as well. In general, it may be less important for the strong users to achieve optimal performance than it is for weak users. The sequential group detection scheme can therefore be seen as aiding the users that need it the most while compromising the performance of the strong users who are better able to withstand this loss. Furthermore, the ordering of users according to decreasing energies is an easy partition, but it can perform poorly relative to the optimum sequential partition suggested above. This is also illustrated in Fig. 10, which includes the lower and upper bounds on the effective energy of the sequential detector for this partition where users 1, 7, and 8 are severely degraded because of the suboptimal ordering.

Example 7: Consider again the eight-user example where the correlation matrix is of the form Rkl = ~l~-‘l with Q = 0.6, and consider signal energies

w = [l.Ol 1.9 1.75 1.6 1.8 1.4 1.2 1.651

that are less disparate than in the previous example. Suppose that it is of interest to find a detection scheme which gives a minimum effective energy for every user that is equal to 1. Fig. 11 shows the effective energies of all the users for the decorrelator and for the optimum detector and the upper and lower bounds of Propositions 5 and 6 on the effective energies of the two sequential group detectors with all group sizes equal to 1 and 2, respectively. The sequential group detector for single-user groups has users arranged in the order {8}{2}{3}{4}{5}{1}{6}{7} and the sequential group detector for group sizes equal to 2 has users arranged in the order (2, 3}{4, 5}{1, 8}{6, 7}, both obtained by sequentially optimizing the effective energy upper bound of Proposition 5. Notice that neither the decorrelating detector nor the first

3.U q High bnd SGD(1) n LB SGD(1) E ord

6 2.5 HB SGD(1) E ord

i5

I5 2.0

al 2 1.5

Tz z 1 .o

-.- 12 3 4 5 6 7 8

User Index

Fig. 10. Asymptotic efficiency comparison of the decorrelator, the maximum-likelihood (ML) detector, and two sequential group detectors of group size 1 corresponding to an optimal sequential ordered partition (SGD(l) in legend) and a suboptimal ordering according to decreasing energies (SGD(l) E ord in legend) for an eight-user disparate energy system described in Example 6.

I .-f

2 iii 1.2

15 1.0

(I) 2 0.8

g 0.6 z= W 0.4

0.2

0.0 12 3 4 5 6 7 8

User Index

Fig. 11. Asymptotic efficiency comparison of the decorrelator, the maximum-likelihood (ML) detector, and the sequential group detectors corresponding to two optimal sequential ordered partitions of equal group sizes of 1 and 2 for an eight-user unequal energy system described in Example 7.

sequential group detector meet the specification that the minimum acceptable effective energy for any user is unity, even though the latter markedly improves the performance of the decorrelator. The second sequential group detector, however, does meet the required specification.

IV. CONCLUSIONS

Group detection is a unifying approach to multiuser detection. It allows a rich tradeoff between performance and complexity and includes as special cases most of the previously proposed suboptimum multiuser detectors such as the optimum detector, the decorrelating detector, the two-stage detectors, and the decorrelating decision feedback detector. The optimality of a group detector in group near-far resistance testifies to its optimal robustness to the energy levels of the users not lying in the group under consideration. The modified group detector introduced in the Appendix can further improve upon this performance. When group detectors are


qi q2 q3

04 Gp

+ -

- gG (.I - R%G2 2

4 - II

(4

&Ilk

(b)

I

P [ [?&k]

gk[6%k]

Fig. 12. (a) The modified group G1 detector for a five-user CDMA channel with G1 = (1, 2, 3) and Gz = {4. 5). The decision algorithm performs the operation of (51). (b) Hard decision nonlinearity. (c) Soft decision linear clipper.

connected sequentially so that a particular group detector in the chain has access to the decisions made by the previous group detectors, these decisions can be used to successively cancel the multiple-access interference from the users in those previous groups before the joint detection of users in the current group is performed. Upper and lower bounds on the asymptotic efficiency of this sequential group detector indicate that, under certain conditions, it can have a performance that rivals the performance of a super group detector, with the super group being the union of the previous and current groups, thereby optimizing the worst case asymptotic efficiency over the signal amplitudes of only the users in future groups. The efficient sequential algorithm for obtaining good ordered group partitions for sequential group detection attempts to best satisfy those conditions.

Group detection has also been studied for the asynchronous Gaussian CDMA channel [14] and the frequency-selective Rayleigh fading CDMA channel [ 151. It is particularly relevant for bandwidth-efficient CDMA communications where the gap between the performances of the linear and optimum detectors is significant.

APPENDIX MODIFIED GROUP DETECTION

In this appendix, we propose modified group detectors which take advantage of the knowledge of the signal amplitudes of the users in the interfering groups to improve upon the performance of the corresponding GLRT group

detector. These modified group detectors can be used in a parallel or a sequential group detection scheme in place of the corresponding GLRT group detectors. The specification of the modified group detectors also serves to provide a unifying framework for both linear and nonlinear interference cancellation strategies.

The modified group detector is specified for the PMAC channel starting with the PMAC group detector since it already optimally incorporates the knowledge of the phases of the interfering signals. Note that the group detector in (8) can be equivalently written as

hi = args~~;G{@&&~(liG - ~~GE[&]G)]~

with the interpretation that it is the maximum-likelihood detector based on & = & - fi GG[&]E. The vector $G represents the decision statistics obtained from i& after the multiple- access interference contribution to it from the interfering users in G has been subtracted based on decorrelating soft decisions on the interfering symbols. The maximum-likelihood decision is then made, taking into account the joint uncertainty associated with the soft-decision interference cancellation.

If the signal energies of the interfering users are known, it is clear that this additional information can be incorporated by passing soft decisions [dj@]~ through a simple-to-implement vector nonlinear transformation (denoted as g) before the interference subtraction is performed. The modified statistic that results is given as

3/G = @ G - &,g([&]d~ (48)

Ideally, the specification of the modified group detector based on eG must take into account the resulting noise properties due to the nonlinear transformation. In general, however, this is analytically intractable. We take the conservative approach of designing the best possible detector for the decorrelating soft-decision case so that

The implementation of the modified group detector for the group G = { 1, 2, 3) of a five-user CDMA channel is shown in Fig. 12(a). The vector nonlinear transformation could be a simple bank of ICI scalar M-level quantizers for M-PAM modulation, which correspond to hard-decision limiters for binary phase-shift keying as shown in Fig. 12(b). Another example for this binary signaling is a bank of ICI scalar linear clippers as shown in Fig. 12(c). The latter transformation in the case when ck = 0 v’k E G effectively clips each component of the noise in [&]G before it is used to produce YG. This noise clipping therefore results in a better performance of the modified group detector as compared to the corresponding GLRT group detector. By appropriate positive-valued choices of the constants tj (in proportion, for instance, to the standard deviation of the jth noise variable in [&]G), it is possible to obtain further improvements.

IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 41, NO. 4, JULY 1995

Next, we seek connections betwCen the special case of the modified group detector for the singleton group G = {k} with previously proposed multiuser detectors. For the PMAC with binary antipodal signaling and with g denoting an identity transformation, (49) reduces to the decorrelating detector of [2]. W ith G = {Ic} and g denoting a bank of scalar hard-decision limiters, the modified group detector reduces to the two-stage detector of [4] with a decorrelating first stage. Moreover, when G = {k} and K = 2 and gG denotes a scalar linear clipper or a multilevel hard-decision quantizer with a dead-zone nonlinearity, the modified group detector corresponds to those studied in [ 161. The performance analysis of the modified group detectors when g is a nonlinear transformation is difficult and appears to admit analytical results for only a few special cases (cf. [4], [16]).

Cl1

PI

r31

[41

REFERENCES

S. Benedetto, E. Biglieri, and V. Castellani, “Digital Transmission Theory,” Englewood Cliffs, NJ: Prentice-Hall, 1987. R. Lupas and S. Verde, “Linear multiuser detectors for synchronous code-division multiple-access channels,” IEEE Trans. Inform. Theory, vol. 35, no. 1, pp. 123-136, Jan. 1989. S. Verdh, “Multiuser detection,” in Advances in Statistical Signal Pro- cessing, Vol. 2: Signal Detection, H. V. Poor and .I. B. Thomas, Eds. New York: JAI Press, 1992. M. K. Varanasi and B. Aazhang, “Near-optimum detection in synchronous code-division multiple access systems,” IEEE Trans. Con- mm., vol. 39, no. 5, pp. 725-736, May 1991.

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[I61

B. Aazhang, B. Paris, and G. Orsak, “Neural networks for multiuser detection in code division multiple-access communications,” IEEE Trans. Commun., vol. 40, no. 7, pp. 1212-1222, July 1992. A. Duel-Hallen, “Decorrelating decision feedback multiuser detector for synchronous code-division multiple-access channel,” IEEE Trans. Commun., vol. 41, no. 2, pp. 285-290, Feb. 1993. H. L. van Trees, Detection, Estimation, and Modulation Theory. New York: Wiley, 1968. L. L. Scharf, Statistical Signal Processing. Reading, MA: Addison- Wesley, 1991. S. Verdd, “Optimum multiuser asymptotic efficiency,” IEEE Trans. Inform. Theory, vol. IT-32, no. 5, pp. 890-897, Sept. 1986. M. K. Varanasi and B. Aazhang, “Optimally near-far resistant multiuser detection in differentially coherent synchronous channels,” IEEE Trans. Inform. Theory, vol. 37, no. 4, pp. 1006-1018, July 1991. R. R. Anderson and G. J. Foschini, “The minimum distance for MLSE digital data systems of limited complexity,” IEEE Trans. Inform. Theory, vol. IT-21, no. 4, pp. 544-551, July 1975. S. Verdli, “Minimum probability of error for asynchronous Gaussian multiple-access channels,” ZEEE Trans. Inform. Theory , vol. IT-32, no. 1, pp. 85-96, Jan. 1986. D. Parsavand and M. K. Varanasi, “RMS bandwidth constrained signature waveforms that maximize total capacity of PAM-synchronous CDMA channels,” to appear in IEEE Trans. Commun. M. K. Varanasi, “Group sequence detection for asynchronous Gaussian code-division multiple-access channels,” in Conf Rec. Communications Theory Mini-Cot& GLOBECOM, ‘94 (San Francisco, CA, Nov. 1994). -> “Group detection for synchronous CDMA communication over frequency-selective fading channels,” in Proc. 3 1st Annual Allerton Cof on Communication, Control, and Computing, Sept. 1993, pp. 849-858. X. Zhang and D. Brady, “Soft-decision multistage detectors for asynchronous AWGN channels,” in Proc. 27th Annual Con5 on Information Sciences and Systems (Baltimore, MD, Johns Hopkins University, 1993).

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