Iterative multiuser receivers for CDMA channels: an EM-based approach

1700 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 44, NO. 12, DECEMBER 1996

Iterative Multiuser Receivers for CDMA s: An EM-Based Approach

Laurie B. Nelson, Member, IEEE, and H. Vincent Poor, Fellow, IEEE

Abstruct- Maximum-likelihood detection for the multiuser code-division multiple-access (CDMA) channel is prohibitively complex. This paper considers new iterative multiuser receivers based on the expectation-maximization (EM) algorithm and related, more powerful “space-alternating” algorithms. The latter algorithms include the SAGE algorithm and a new “missing parameter” space-alternating algorithm that alternately updates individual parameter components or treats them as probabilistic missing data. Application of these EM-based algorithms to the problem of discrete parameter estimation (Le., data detection) in the Gaussian multiple-access channel leads to a variety of convergent receiver structures that incorporate soft-decision feedback for interference cancellation and/or sequential updating of iterative bit estimates. Convergence and performance analyzes are based on well-known properties of the EM algorithm and on numerical simulation.

I. INTRODUCTION

HE expectation-maximization (EM) algorithm 111 pro- T vides an iterative approach to likelihood-based parameter estimation when direct maximization of the likelihood function may not be feasible. The generality of the algorithm, combined with its amenability for incomplete-data problems, has led to its widespread use for such diverse applications as speech recognition [2], identification of impulsive noise channels [3], quantum-limited imaging [4], and numerous others [5] in economics, medicine, psychology, etc. In the context of incomplete-data or missing-data problems, the basic approach of EM is to alternate between the following: 1) computing complete-data sufficient statistics based on the incomplete- data and current parameter estimates and 2) re-estimating the parameters based on the computed complete-data sufficient statistics. Although convergence of the algorithm to the maximum-likelihood estimator is not always guaranteed, EM does possess the intuitively pleasing property of producing estimates that monotonically increase in likelihood.

This report illustrates the use of EM algorithms for discrete parameter estimation-specifically, for joint data detection in

Paper approved by A Duel-Hallen, the Editor for Communications of the IEEE Communications Society Manusciipt received March 25, 1994, revised July 10, 1995, and April 18, 1996 This research was supported by the U S Army Research Office under Grant DAAH04-93-G-0219 and by an AT&T Bell Laboratories Ph D Scholarship This paper was presented in part at the IEEE International Symposium on Information Theory Trondheim, Norway, June 1994, and the IEEE and IMS Workshop on Information Theory and Statistics, Alexandria, VA, October 1994

L B Nelson is with the Department of Electrical Engineering, University of Minnesota, Minneapolis, MN 55455 USA

H V Poor is with the Department of Electrical Engineering, Princeton University, Princeton, NJ 08544 USA

Publisher Item Identifier S 0090-6778(96)09028-9

multiple-access channels. In applying the EM and related algorithms, our general approach is to treat the bits of interfering users as probabilistic missing data when updating the estimate for a given user’s bit. The resulting derivations lead to iterative multiuser receivers that use soft-decisions for interference cancellation andor sequential (rather than parallel) updates of estimates for the users’ data.

Several recent applications of EM to digital communications consider related problems of parameter estimation for superimposed signals in noise. In the context of direct-sequence code- division multiple-access (CDMA) over multipath channels, Fawer and Aazhang [6] propose multiuser receivers that iterate between EM-based amplitude estimation and multi-stage data detection. Poor also applies EM to amplitude estimation for direct-sequence CDMA 171; the approach differs from that of [6] in that the users’ data are not assumed known, but are instead treated probabilistically as missing data. In [8], Georghiades and Han consider maximum-likelihood sequence estimation (MLSE) with unknown random phase or fading parameters treated as missing data. The E-step of the resulting algorithms produces estimates of the unknown parameters; the M-step is then equivalent to MLSE for deterministic signals in additive noise.

The applications of EM to multiuser detection in the current paper demonstrate yet another approach. Similar to [8], EM is used for detection, rather than for estimation of continuous parameters as in [6] and 171. However, rather than use EM to facilitate detection of randomly faded signals, we assume the signals are known and apply EM to derive likelihood-based receivers of lower complexity than is required for optimum demodulation [9].

This paper is organized as follows: Section I1 establishes notation for the additive-noise, CDMA channel model. Pre- liminary material about the EM and SAGE algorithms is presented in the first parts of Section 111. A new EM-based algorithm, the “missing parameter EM’ (MPEM) algorithm, which altemately updates subsets of parameters or treats them as missing data, is also described in Section 111. Section IV contains derivations and analysis of receivers based on applications of the EM and MPEM algorithms with interfering users’ bits treated as missing data. Multiuser receivers based on the SAGE algorithm are considered in Section V. Some convergence and performance analysis of the EM- based receivers is included in the preceding sections; further performance comparisons based on numerical simulations are made in Section VI. Finally, concluding remarks are presented in Section VII.

0090-6778/96$05.00 0 1996 IEEE

NELSON ANI) POOR: ITERATIVE MULTIUSER RECEIVERS FOR CDMA CHANNELS: AN EM-BASED APPROACH 1701

11. SYSTEM DESCRIPTION The received signal in a K-user CDMA system is described

by K

.(t) = aksrc(t - 1T - ~ ) b k [ l ] + n(t) (1)

where { b k [ l ] } l , a k > 0 , and Tk are the respective bipolar bit stream, received amplitude and transmission delay for the kth user, and n(t) is additive Gaussian noise with spectral density cr2. The signature waveforms {SI,. . . , S K } have support on [O,T) and are normalized to have unit energy. For this Gaussian multiple-access channel, the sampled matched-filter outputs yk[Z] = ! ~ ( t ) ~ k ( t - l T - ~ k ) d t are sufficient statistics; hence, a discrete-time model equivalent to (1) can be defined in terms of { b k [ l ] } k , ~ and { y k [ l ] } k , ~ . When the users’ are synchronized (71 = 1 . 9 = 7K = 0), the discrete-time model has the form

k = l 1

a

y = R A b $ n (2)

where R is the K x K matrix of signature waveform cross- correlations, i.e., R J k = ( s J , s k ) = ! s k ( t ) s J ( t ) d t , A is the diagonal matrix of amplitudes a l , . . , U K , and b = [bl . . . b ~ ] ’ and y = [yl . . . YK]’ denote’ the input bits and output matched-filter samples, respectively. The filtered noise veclor n is Gaussian with zero mean and covariance matrix a2R. A discrete-time model similar to (2) , but with higher dimension, can also be derived for the asynchronous channel by considering a finite frame of bit intervals [lo].

111. EXPECTATION-MAXIMIZATION AND RELATED ALGORITHMS

Toward describing the EM algorithm, consider the general problem of estimating a parameter 0 E A, given a realization of Y with probability density f (y; 0). The maximum-likelihood (ML) estimator of 0 is defined as (e.g., [ 111)

(3)

In many situations for which (3) is intractable or has high complexity, the EM algorithm [ 11 provides an attractive alternative to computing ML estimates. The algorithm postulates the existence of some “missing” data 2 that would aid in estimation of 0, but that may not be observable. Then, the key element of EM is to replace (3) with an iterative maximization of the new objective function

(4)

Referring to X = { Y , Z } as the complete data, (4) may be viewed as a smoothed version of the complete-data log likelihood. Given an initial estimate O0, the ith iteration of the EM algorithm is described by

Q ( e ; e ) 2 E{logf(Y,Z;O)IY = ?/;e}.

E(xpectation)-step : Compute Q(B; 0’) ( 5 ) M(aximization)-step : = arg max Q(0; 0’). (6)

‘For the synchronous model at hand, one-shot detection IS optimum and

e a

the time index I may be disregarded.

A judicious choice for the missing data 2 is such that the computations in (5) and (6) are simpler than the maximization in (3). However, such advantages are often accompanied by notoriously slow convergence of EM. In general, the trade-off between ease of implementation and asymptotic convergence rates hinges on how much information about 0 is embodied by the complete data (see, e.g., [ll, [121, [14]).

On letting L(0) e log f (y; e ) , (6) and a simple application of Jensen’s inequality lead to the relations

Le., the EM estimates mono1.onically increase in likelihood. In addition, ML estimates (3) are fixed points of the algorithm and, under mild conditions, B2 converges to a local maximum [l], [15]. As for any deterministic iterative maximization of functions with local maxima, the ability of EM to find the global maximum (3) depends, in general, on the initialization 00.

For the multiuser detection problem, [SI considers two optimum detectors, based on the two criteria of maximizing either the joint likelihood of b or the marginal likelihood of b k . (The latter maximization is equivalent to minimization of the kth-user error probability when b k is equally likely to be $1 or -1.) For either optimality criterion, however, the O ( a K ) complexity of optimum detection precludes its implementation in practical CDMA systems (for which K may be large). Consequently, a variety of lower-complexity receivers, including several that use decision feedback techniques to combat multiple-access interference (MAI), have been proposed. In a sense, decision feedback receivers have an inherent estimate- maximize structure: currently available bit decisions are used to estimate MAI; then, following cancellation of the estimated MAI, subsequent decisions for other (or possibly the same) bits are made-this last step is tantamount to a likelihood maximization given side information about the M,4I. More formally, the EM algorithm is applied in Section IV to multiuser detection by letting 0 = b k and 2 = {bm: m # k } , where 2 is distributed uniformly on {kl}K-l; note in this case that the parameter 0 is single-dimensional, implying demodulation of only one user’s data. Algorithms for K - dimensional 0 and joint demodulation are based upon the SAGE algorithm, considered next.

A. SAGE Algorithm Recently, Fessler and Hero introduced a space alternating

generalized EM (SAGE) algorithm [ 131 that can improve upon EM convergence rates for multidimensional ,parameter estimation. At each iteration, the SAGE algorithm updates only a subset of the parameter components. This adlows the use of multiple, less informative “complete” data sets, thereby improving convergence rates without incurring the otherwise attendant increases in computational complexity for tlhe M-step maximization.

On letting 0s denote those parameter components indexed by the set S , the ith iteration of the SAGE algorithm is


described by2

(8) Choose a parameter index set S. Choose the missing-data 2’. Defn-step:

E-step: Evaluate QS(Qs; Qi) (9)

where As denotes the restriction of A to those dimensions indexed by S and S is the complement of S. The objective function in (9) is defined by

Qs(Qs;8) e E{logf (Y ,ZS;Bs ,B~)JY = y;8} (11)

and is similar to that for the EM algorithm. One difference in the algorithms is that, at each iteration, the SAGE M- step maximizes QS over only those parameter components currently indexed by S.3 As such, the maximization may be tractable for “complete” data {Y , Z s } (complete for 0s) that is less informative than would be required for tractability in the standard EM algorithm. The space-alternating structure described above provides significant flexibility in formulating the SAGE algorithm for specific applications.

Similar to EM, the SAGE estimates monotonically increase in likelihood, and maxima of the likelihood function L(0) are fixed points of the algorithm [13]. To illustrate the SAGE algorithm, [13] compares EM and SAGE algorithms for parameter estimation for superimposed signals in noise and demonstrates improved convergence for SAGE. This superimposed signal example is equivalent to that considered in [6] for CDMA multipath channels, and it suggests that the EM-based amplitude estimation in [6] could be improved by implementing the SAGE algorithm instead.

In Section V, the SAGE algorithm is applied to multiuser detection with 0 =: b. This particular application is almost trivial in that it requires no missing data model. The resulting receiver algorithm is similar to the multistage receiver [16], except that joint decisions for the users’ bits are updated sequentially rather than in parallel. Unlike the previously mentioned application of EM to multiuser detection, the SAGE multiuser receiver does not incorporate probabilistic models for the users’ bits (the parameter). The next section considers new EM/SAGE-related algorithms that incorporate such models.

B. Space-Alternating “Missing-Parameter ” EM Algorithms In many applications, an a priori distribution for the param-

eter 0 may be available. (For applications to digital communications, the parameter may represent discrete data, for which the uniform distribution is often appropriate.) In such cases, EM and SAGE can be extended to address maximization of the posterior mode rather than the likelihood. Beyond such

2Notation is taken from [I31 with some simplifications allowed by restrict- ing attention to complete data of the form X” = { Y , Zs}. The more general presentation of the SAGE algorithm in [I31 is based on “hidden” dum spaces X s that are stochastic functionals of the incomplete data Y .

’Prudent choices for the index sets are such that S covers all parameter components infinitely often. When S = 0 for every iteration, @-( lo) and the EM algorithm are equivalent.

simple extension, however, the space altemating approach of the SAGE algorithm provides a unique framework for a novel incorporation of the parameter statistics-namely, the parameter components may be alternately updated and treated as probabilistic missing data. This section presents two such “missing parameter” algorithms. The first considered is an obvious, albeit limited, modification of the SAGE algorithm. Next, a less straightforward, but more useful (for multiuser detection), related algorithm is presented. The second algorithm is especially relevant when the complete-data log likelihood is quadratic in 8.

First, consider the space-alternating algorithm (8)-( 10) with the altemative definition4

QS(6’s;e) !! E{logf(Y, Zs, 0s jQs) lY = y, Os = es}. (12)

Analogous to analysis for the EM and SAGE algorithms (e.g., [13]), it is simple to show that

Qs(Qs;e) - QS(es;G) =LS(es) - Cs(es) - D(O,y((es) (13)

where C s ( Q s ) is the marginal likelihood

LS(Qs) 5 1% f(YlQS) = 1% f (Y,Q3lQs) dQ3 s and the divergence, or Kullback-Leibler discrimination, is defined by

Jensen’s inequality implies that

~(Qs l les ) 2 D(esll8s) = 0. (14)

If Si denotes the index set at the ith iteration, it follows from (13), (14), and (10) that

CSZ ( @ ) - csz (e,) 2 0 (15)

i.e., each new estimate increases the marginal likelihood of the currently indexed parameter components. However, it is not in general true that, for some fixed set S,, the sequence { Cs* (6);- )} is monotonic, since there may be iterations during which the index set S is chosen to be different from S,, but with S n S, # Q.

If the index sets are chosen from a partition { SI, S2, . . . , SK }, this scenario is avoided and (15) does imply the monotonicity of { C S F ( Q ~ J } ~ , for k = 1, . . . , K. The monotonicity may also be understood by noting that, for this special case, the algorithm reduces to a bank of K EM algorithms that individually estimate Qs, with 03, treated as missing data for IC = 1, . . . , K. This decoupling highlights the fact that (12) does not take advantage of current estimates Q> for the missing data parameters when computing 8;”. One solution to this shortcoming is motivated by the form of the Gaussian likelihood function and is described next.

41n the sequel, B denotes a realization of the random parameter 0

NELSON AND POOR ITERATIVE MULTIUSER RECEIVERS FOR CDMA CHANNELS: AN EM-BASED APPROACH 1703

Suppose momentarily that the complete-data log likelihood, logf(y, 2, @,Ids), is quadratic in [Z’ 0’s 8g]. To evaluate (12), any additive terms in the complete-data log likelihood that are independent of 19s may be ignored (since the M- step maximization is over 8s only). The preceding quadratic assumption implies that each of the remaining terms is either 1) quadratic in B s , or 2) linear in [Z’ Oil (for fixed 8s). For the former case, the E-step expectation is trivial. For the latter case, the E-step requires evaluation of

E{ [Z’ Ol,]IY = y, Os = e,}. (16)

To better utilize the current estimates e:, one might replace (16) with the estimates

E { Z ( Y = y, os = os, os = eg j (17) E{@,IY = y, os = Os, = O2- j (18)

S-1,)

for all m E S. Note that (17)-(18) probably provide better estimates of the missing datdparameters than does (16), due to the additional conditioning in (17)-( 18).

The modification motivated above is achieved by redefining the objective function Qs as

-

Qs(Qs; 8s) 1 log f(y, z, ej.les)h(z, 0s; Y, 0) dz

(19)

where h is given by the following product of marginal conditional densities?

h(z,Q;y,$) f(.ly,8) rI f(QmlY,Z,$&). (20) m€S

Although such formulation does not lead to a simple expression for Qs as a conditional expectation, one may expect that the missing parameter algorithm with Qs defined as in (19) performs better than the algorithm with Qs defined by (12) whenever the complete-data log likelihood has the quadratic form described above.

Despite this intuitive advantage, it is not possible to show, in general, that the MPEM algorithm defined by (8)-( 10) and (1 9) produces estimates that monotonically increase in likelihood. Even for the relatively simple case when 1) 2 = 0 (Le., Os comprises the entire missing data), 2) the index sets are chosen from a partition {S~,...,SK}, 3 ) O s 1 , 0 s 2 , are mutually independent, and 4) the complete-data log likelihood function has the specified quadratic form, to obtain (15) would require the nonnegativity OF

f(QmlY, &-{??%))

mES E- slogfi(s,.,lY, G+l, k { , } l f(QmIY, Qk, e;-,,}) dQm (21)

5 M o ~ generally, one could replace h with f(zlv,@)Ih s,gsf(@q,l~.z,8i,) where {SI. ..SA} 1s a partition from which the index sets are chosen

6Two cases for which (21) is nonnegative for all S are 1) when_the parameter subsets are conditionally independent given Y and 2) when S 15

a singleton for all S in the partition (The latter case holds when 0 is two- dimensional (2-D) ) In both cases, the algorithm reduces to that considered earlier with Q’ defined by (12)

for all S E {SI,..*,SK}, where - A e3 = E{O,IY = ?/,Os 8 : , 0 ~ - ( ~ ) = 8%- j , b‘j E S S - { J }

The integrals in (21) have a form similar to that of a (nonnegative) divergence, except that 8s- (in the numerator density) differs from Qz- (in the integrating density). Arguing that Q s - { ~ ) = 8’- - since both quantities are

S - { m ) estimates for Os-{,i-,,one may expect that (21) is “approximately nonnegative and, consequently, that the MPEM estimate sequence increases the marginal likelihoods. This approximate argument helps to explain subsequent simulation results that show convergence of multiuser receivers derived from this algorithm.

For similar reasons, it is also not possible to show that maxima of the marginal likelihoods Cs (Os) are fixed points (of iterations that update 0s). However, this fact is not necessarily disadvantageous for applications to discrete parameter estimation, since maxima of the likelihood function over a continuous set are not necessarily achieved near to the most-likely discrete estimates.

S- im)

Iv. MULTIUSER RECEIVERS FOR “MISSING DATA” INTERFERENCE

This section derives EM-based receivers that treat the interfering users’ bits as missing data when estimating the bit of a particular user. First, an iterative receiver for user k is derived from the standard EM algorithm with 8 b k . Next, a multiuser receiver that jointly detects b is described. This second receiver is motivated by a simple and intuitive modification of the preceding EM receiver, and it is equivalent to an MPEM algorithm with 0 b.

A. kth-User EM Receiver

Let bg denote the bits of all but the kth user. Application of the EM algorithm to estimation (detection) of bj, with b, treated as missing data provides the iterative receiver derived below.

The complete-data log-likelihood function is given by

logf(bg,ylbk) = logf(bi,Ibk) + logf(ylb) (22)

where the last density is multivariate normal with mean RAb and covariance a2R. Any additive terms in the complete-data log likelihood that are independent of bk do not affect the M-step maximization. Dropping these terms from (22) leaves

The only random quantities in the above expression are the missing data, b, for m # k . Since these terms appear linearly, it is easy to show

1704 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL 44, NO 12, DECEMBER 1996

where analytically. However, simulation results in Section VI support this conjecture, and the corollary below verifies it in the

In the asymptotic regime, performance is aptly characterized by the rate at which a receiver’s error probability decays exponentially to zero. A normalized measure of this rate is given by the asymptotic multiuser efficiency

A

Then, the M-step update is defined by

bk++’ = arg maxQ(bk:;bi)

bm(i) = E{b,lY = ?/; bl: = b“,. (24) asymptotic (infinite SNR) regime.

(25) b k E A

\ \

A where g A ( r ) = arg maxbcA[-b2 + 2baI. (EM receivers for specific models A are analyzed later in this section.)

Using Bayes’ formula, it is straightforward to show that (24) is given by

bm(i) = tanh ( %(ym - R,,,kakbi))). (27)

Thus, in view of the shape of the function tanh, the E-step involves computing soft-decision estimates of the interfering users’ bits (the missing data). Note that the component of ym corresponding to user m’s signal is ambm; then it be- comes clear that user m’s signal-to-noise ratio (SNR) u:/o2 determines the “softness” of the E-step estimator for 6,.

Note from (27) that there is no attempt in the E-step to cancel interference from users j # IC prior to soft-limiting. When there are more than two users in the system, this component may adversely affect the E-step estimate bm(z ) . Joint detection based on the MPEM algorithm avoids this shortcoming. Before considering the MPEM receiver, however, some interesting analytical results for specific EM receivers are presented.

B. EM Receiver with Hard-Decision M-Step An obvious model to consider for the parameter space is

A = {fl}. In this case, the M-step update (26) is given by the hard-decision

and we obtain the following simple results. Theorem 1: Let L ( b k ) = log E { f ( y , b i l b k ) } and 6ipt =

arg maxbhE{-l,l)C(b~).7 Let b“, E {&I} denote the EM receiver estimates for bk given some initial estimate b; E {kl}. Then

1) the EM receiver converges in one iteration (with prob-

A A related measure, the near-far resistance 7, = inf a3 > 0 rjk , provides a worst-case asymptotic performance measure. Opti- mum near-far resistance is achieved by the linear decorrelating receiver [ 101. This receiver does not require information about the users’ amplitudes, and, as long as s k ( t ) is not spanned by the other users’ signals, its near-far resistance is strictly positive. The following results are immediate from Theorem 1.

Corollary 1: Let r jgm denote the AME of the kth-user EM receiver with hard-decision M-step (28). Let 7; denote the AME of the receiver chosen to provide the initial estimate b: E {&l}. Then,

J f k

Corollary 2: The EM receiver with hard-decision M-step (28) and decorrelating detector initial stage (37) is optimally neat-far resistant.

C. EM Receiver with Linear-Clipper M-step In deriving EM-based multiuser receivers, one may also

consider EM algorithms for nondiscrete parameter spaces A. (A final hard-limiter would be used to provide binary-valued bit decisions.) In a sense, this extends the “search” of the M-step maximization beyond the space { f l}. Consequently, the EM receiver no longer converges immediately and its performance may benefit from implementation of additional stages (iterations).

For the model A = [-c, c], c > 0, the M-step update (26) uses the unit-slope linear-clipper nonlinearity’

e, r > c

-c, 5 < -c. (30)

ability 1); 2) PrjbL # b k ] 5 Pr[@ # b k ] + p r [ p # b k ] .

The proof (shown in the Appendix) is a simple consequence of the binary dimension of the parameter space and the monotonic likelihood of EM estimates. In particular, the monotonicity property implies that the EM receiver chooses

Convergence analysis of the resulting EM receiver follows the standard analyses of [15] and [I], since A is continuous in this case. The following lemma is the key to such analysis, the results of which are listed in Theorem 2. (Proofs are in the Appendix.)

the Optimum decision at least as Often as the stage receiver does. One may expect this to imply that the EM receiver bit-error rate (BER) is upper bounded by that Of

the initial stage receiver. This intuition 1s difficult to JUStlfY

7Assuming bl, is equiprobable on {*l},b;f’t is the minimum-error-

*In (29), which is equivalent to the definition in 1171, the normalizing term

9For reference, linear clippers with arbitrary slopes and maximum absolute output equal to unity were considered for soft-decision multistage detection in [ 181 In contrast, the M-step linear clippers are constrained in slope (unity) but not in maximum output As c i 00, (30) tends to an “infinltely-soft” decision

a i / 2 represents the optimum decay rate for the single-user channel

probability (viz optimum) decision for bh

NELSON AND POOR: ITERATIVE MULTIUSER RECEIVERS FOR CDMA CHANNELS: AN EM-BASED APPROACH I705

Lemma 1: Let {bi}gO denote the sequence of estimates generated by the kth-user EM receiver with linear-clipper M-step (30). Then

Q(b",+'; b k ) - Q(&; b k ) 2 X(bk+' - b t ) 2

show that each iteration of the MPEM receiver has complexity comparable to that of one stage of the multi-stage receiver. Thus, the MPEM receiver complexity per binary decision is K times that of the multi-stage receiver-in other words, is O ( s K 2 ) , where s is the number of implemented stages.

A where X = aE/2g2 > 0. Theorem 2: Let {b",p=, denote the sequence of estimates

generated by the kth-user EM receiver with linear-clipper M-step (30). Then

1) C(bk) converges monotonically to C* - lim C(b",). 2) { b i } converges to a stationary point b; of C ( b k ) with

Consequently, the sequence of hard-decisions { sgn( b",} also converges, but this sequence is not necessarily monotonic in likelihood (unlike the case for A = {hl}).

A - 2 1 0 0

C(bE) = C i a

D. Multiuser MPEM Receiver

v. SAGE MULTIUSER RECEIVERS

The SAGE algorithm can be applied to multiuser (detection with 0 = b E {+l}K and with the index sets cycling: through 1, .. . , K . In this way, the algorithm is easily implemented without any missing data Zs . Consequently, the ]E-step is trivial and may be ignored. Each iteration of the resulting algorithm comprises the following steps:

Defn-step: Let k = 1 + ( i mod K )

A multiuser receiver that jointly detects the users' data can be derived from the MPEM algorithm with 0 E b, 2 s = 0 for all S, and the index sets chosen cyclically as S, = { 1 + (z mod K ) } . Like the kth-user EM receiver, this MPEM receiver treats all but one of the users' bits as missing data during each iteration. Since both receivers depend on similar complete-data models, derivation of the MPEM receiver is

The SAGE multiuser receiver is simply a coordinate descent, or greedy, algorithm. It is similar to the multistage receiver of [19], except that updates of the bit estimates are made sequentially, rather than in parallel. Since the SAGE estimates monotonically increase in likelihood and the log likelihood function

A

similar to that for the EM receiver in Section IV-A. For iterations that update the estimate bk, i.e., k = 1 + ( i mod K ) , the MPEM objective function (19) is given by (23), except that the soft-decisions gm(i) are defined by

(31)

rather than (27). The ith iteration of the resulting MPEM receiver is described by

- A b,(i) = €{bmlY = y,b% = bk}

Defn-step: Let k = 1 + ( i mod K ) . E-step: Compute for all m # k ,

A hard-limiter would be used to produce final bit decisions. When there are only two users, (27) and (3 1) are identical, and the EM and MPEM receivers are equivalent. When K > 2, however, the MPEM receiver is expected to outperform the EM receiver since current parameter estimates bz are used to cancel a12 multiple access interference (MAI) in the E-step above. In the latter case, analysis and evaluation of the MPEM receiver is limited to the arguments in Section 111-B and to numerical simulation (Section VI).

In light of terminology for the multistage receiver of [16], we shall identify as "one stage" of the MPEM receiver the completion of K consecutive iterations such that the bit estimate b i for each user is updated once. It is simple to

L(b) = log f ( y l b ) = 2y'b - b'ARAb + C (34)

is bounded above, the SAGE multiuser receiver is convergent. The parallel-update multistage receiver [ 191, on the other hand, does not always converge.

Alternative SAGE receivers may be derived by modeling b as taking values in, e.g., A = [-c,cIK or A = R K . In either case, the (upper-) boundedness and monotoinicity of the sequence {L(b ' ) } guarantee convergence of the SAGE multiuser receiver. For the former case, the M-step updates use the unit-slope linear-clipper (30) in place of the signum function in (33). For the latter case, the M-step update. is given by

and the convexity of (34) ensures convergence of the SAGE algorithm to the unique point (RA)-ly that maximizes (34). Hard-limiting this estimate provides the final bit decisions

sgn((RA)- 'y) = sgn(K1y)

where the right-hand side expression is simply the decorrelating detector [20]; thus, the SAGE receiver with M-step defined by (35) provides an iterative implementation of the decorrelating detector. For this particular implementation of the SAGE algorithm, the general convergence rate analysis in [13] is simplified considerably and may be used to show that the algorithm converges at a rate inversely proportional to the weakest user's SNR (min, a f f /a2 ) .

1706 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL 44, NO 12, DECEMBER 1996

&M, conv. inits : 3-stage

D

0

0 -, Single-User Lower Bound

6 8 10 12 14 16 18 1 0-4i SNRl (dB)

Fig. 1. EM (equivalently MPEM) receiver performance for the two-user system with hard-decision M-step, SNR2 = 11 dB, and ( s i , SL) = 0.7.

VI. ERROR PROBABILITY SIMULATIONS The decision feedback inherent to the EM-based receivers

complicates performance comparisons based on exact analyses; for this reason, Monte Carlo simulations (of 2 million runs) were used to approximate the receivers' error proba- bilities. Two- and four-user CDMA systems with signature waveforms characterized by the respective cross-correlation matrices

r 7 -1 3 3 7

are considered. In particular, the four-user example is based on Gold sequences that were also used in analysis of decision- feedback receivers in [16] and [21]. Similar studies for the EM-based receivers are of interest here.

two-user systems, the MPEM receiver is equivalent to two parallel EM receivers, one for each user. The performances of this receiver with hard-decisions in the M-step (A = {hl}) are plotted in Fig. 1 for two different initializations, the conventional receiver decisions"

1) EM and MPEM Receiver Pe rformance: For

bo = sgn(y) (36)

bo = sgn(K1y) . (37)

and decorrelator decisions

As expected from analysis in Section IV, Fig. 1 shows that this EM receiver converges in one iteration and it outper- forms the initial-stage decisions (regardless of the interference power). Also seen in Fig. 1 is that performance is improved significantly by using decorrelator, as opposed to conventional, decisions in the initial stage. Similar comments apply to plots of the EM receiver performance for the four-user channel in Fig. 2.

For the four-user channel, however, the EM receiver performance is limited by the fact that the EM receiver does not

"Actually, the kth-user EM receiver requires only the single estimate bp

' 0 0

, ' O conv inits d

x

Single-User Lower Bound

I" 4 6 8 10 12 14 16 18 SNRl = SNR2 = SNR3 (dB)

Fig. 2. EM and MPEM receiver performances for the four-user system with hard-decision M-step and SNR4 = 11 dB.

"track" estimates for all users' bits (and consequently does not attempt to cancel all MA1 before computing the E-step soft-decisions). By using joint detection, the MPEM receiver overcomes this limitation, as illustrated in Fig. 2. Note that the improvement of MPEM over EM is especially pronounced for the case of conventional receiver initialization (36). Results of Fig. 2 indicate convergence after one stage ( K iterations) and performance improving upon that of the initial-stage decisions for both the MPEM and EM receivers (with hard-decision M-step).

Fig. 3 plots the performance of the MPEM receiver with the "unit-clipper'' M-step (corresponding to A = [ - 1,1]) versus the interference SNR. To aid comparison with Fig. 2, the initial stages (36) and (37) are considered. Fig. 3 shows that, for either initialization, the MPEM receiver with unit-clipper M- step achieves excellent performance by the third stage, and it converges by the fourth stage.

2) SAGE Receiver Performance: The performance of the SAGE (or sequential-update multistage) receiver with hard- decision M-step and either a decorrelator or conventional first stage is plotted in Fig. 4. For these examples, the SAGE and MPEM receivers achieve comparable performance results; see Fig. 2.

For the unit-clipper M-step, however, the SAGE receiver performance differs significantly from that of the MPEM receiver. As shown in Fig. 5, SAGE performance converges to an approximately constant level, that is very nearly optimum for interference energies yielding worst-case optimum performance. It is interesting to note that this apparent min- imax robustness to MA1 is similar to that achieved by the decorrelating receiver: in the limit of infinite SNR's (0 -+ 0), the decorrelator achieves worst-case optimum performance; the simulation results suggest that the unit-clipper SAGE receiver may have similar optimality, but for nonasymptotic SNR's. However, this robustness comes at the expense of limited performance in the presence of strong interferers-the unit-clipper M-step is too "soft" for cancellation of strong interferers (whereas the E-step tanh nonlinearity approaches a hard-decision in such scenarios).

NELSON AND POOR: ITERATIVE MULTIUSER RECEIVERS FOR CDMA CHANNELS: AN EM-BASED APPROACH 1 IO1

Conventional Detector Initialization

* * * * r e * *

Decorrelating Detector Initialization 10.‘

+ 4-stage

2 % I O P

/

0 0 0 0 0 0

Optimum

x o


6 8 10 12 14 16 18 6 8 10 12 14 16 18 I l04A

SNRI = SNR2 = SNR3 (d6) SNRl = SNR2 = SNRB (dB)

Fig. 3. or decorrelator hard-decision initial estimates.

MPEM receiver performance for the four-user system with unit-clipper M-step ( A = [-1. l]I’), SNR4 = 11 dB, and either conventional

In order to show convergence, the plots of Fig. 5 consider many more stages than are practically feasible. To improve SAGE performance with fewer stages requires careful selec- tion of the initial estimates.

3) Initial-Stage Decisions: For the preceding examples, and in general, the decorrelating detector initialization provides significant gains over the conventional detector initialization. The use of soft-decision initializations may also improve the performance of iterative receivers.

As a first example, consider the following almost-trivial initialization

bo = A-ly. (38)

Fig. 6 shows that the resulting MPEM receiver achieves near-optimum performance, even for such a seemingly poor initial estimates! In fact, the MPEM performance is comparable to that obtained in Fig. 2 for the same receiver with a hard-decision decorrelator initial stage. The reason for these somewhat surprising results is that interference cancellation based on the estimates (38) is effective in approximately decorrelating the users’ signals, as shown below: the E-step in the first MPEM iteration uses (38) to cancel MA1 prior to tanh soft-limiting. In matrix form, this cancellation may be described by

y - ( R - I ) A b o = y - ( R - I ) y = 2 y - R y (39)

(40) = (2R - R2)Ab + noise.

Given that R is symmetri9 with unit diagonal, one can show that

For practical systems, it is reasonable to expect that the signal correlations R k j = ( s k , s j ) are small and, consequently, that (2R - R2) is diagonally dominant. (For a two-user system, 2R - R2 is always diagonal.) Then, the E-step soft-decisions

conv. iniits

- -

Conventional, ’ 10.’ ~

0 0

d

, ‘ O 2 ,‘s

s- Optimum

a -

6 8 .io 12 14 16 18 SNRI = SNR2 = SNRB (dB)

Fig. 4. Sequential-update multistage (SAGE) receiver performance for the four-user system with hard-decision M-step updates (n = { -1. l}“ ), hard-decision initializations, and SNR4 = 11 dB.

{gm(0)} are obtained by scaling and soft-limiting the signal (41). Thus, the good performance in Fig. 6 is explained partially by the fact that the first E-step estimates are derived from approximately decorrelated signals.

The preceding observations motivate consideration of tunh soft-decision decorrelator initializations:

The performance of the SAGE receiver with this initial stage is plotted in Fig. 7 for three different M-steps: the unit-clipper, hard-decision, and tunh soft-decision nonlinearities. ((The last nonlinearity does not correspond to a particular A, but is instead chosen heuristically.) The soft-decorrelator initialization leads to excellent performance for any of these M-steps

“I t is interesting to note that (42) is the conditional-mean estimator of hr, given [ I i - ’ y ] r , ; hence, it is optimal in the sense of minimizing the mean-square-error E{ ( 6 ~ . - bk)”J[R-ly]r,}.

IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 44, NO. 12, DECEMBER 1996

? w 10Qr 3 : i g

1708

Fig. 5

pi 3-stage # . Decorrelator :

3

Conventional Detector Initialization

3 :

e - h i

c . 0

c - ii 3 1 0 . ~ ~ Q

lo-', J

. Decorrelator .

- -o- -0- -o- 'D- 0 . _ _ _ _ _ - _ - _ ---.- 0 0 0 0 O 0 0 0 0 5) 1 -

0 - - - - - - _ _ _ _ _ _ - .- - - - - - - - - ~ i r n " n l

4 Single-User Lower Bouud

+ + + + + + + + +

0

Q L 1

( Single-User Lower Boqnd

7 4 ,

3 x X X x X x x i r . + Y X Decorrelator : i .

--.. Single-User Lower Bound

I 6 8 10 12 14 16 18

SNRl = SNR2 = SNR3 (dB)

Decorrelating Detector Initialization 10.' .

__ Single-User Lower Bound

6 8 10 12 14 16 18 SNRl = SNR2 = SNR3 (dB)

SAGE receiver performance for the four-user system, with unit-clipper M-step updates ( A = [-1, l]"), SNR4 = 11 dB, and either conventional (left-hand graph) or decorrelator (right-hand graph) hard-decision initial estimates

'" 4 6 8 10 12 14 16 18 SNRl = SNRP = SNR3 (dB)

Fig. 6. MPEM receiver performance for the four-user system; unit-clipper M-step, matched-filter initial stage (Le., bo = A-ly), and SNR4 = 11 dB.

when the interference is weak. For stronger interference, good performance requires the use of "harder" M-step nonlinearities, such as the signum function or the EM-based tunh nonlinearity (the softness of which scales with individual SNR's of the interferers).

As a last example, consider the performance results shown in Fig. 8 for MPEM receivers with soft-decorrelator initial stage (42) and various M-step nonlinearities. For the two- stage MPEM receivers in this example, the choice of M-step nonlinearity has little effect on receiver performance (for the last user). For the MPEM receiver with unit-clipper M-step, however, addition of a third-stage moderately improves performance to a level very nearly optimum. In general, the MPEM receivers with soft-decorrelator initialization achieve excellent performance over a wide range of interference powers and a variety of M-step nonlinearities.

4) Comparisons to Multistage Receivers: Lack of space prevents explicit performance comparisons

between the EM-based receivers and the parallel-update multi-

1 O'I I 1

t j t I , j

6 0 10 12 14 16 18 1 0-41

SNRl = SNR2 = SNR3 (dB)

Fig. 7. SAGE receiver performance for the four-user syste with soft-decorrelator initial stage, various M-step nonlinearities, and SNR4 = 11 dB. The implementation of additional stages (beyond those represented in the graph) did not noticeably affect receiver performance; these results are not plotted.

stage receiver [19] (with hard- or soft-decisions). Some results of comparisons in [22] are summarized briefly here. In general, the hard-decision multistage and SAGE receivers can achieve comparable performance levels, depending on the number of stages andor the order of sequential updating; the key difference, however, is the quick convergence of the SAGE receiver and the lack of convergence for the multistage receiver. When soft-decisions (unit-clippers or the tunh nonlinearity) were used in either the initial or updating stages, both receivers converged to approximately the same performance levels; however, the SAGE receiver converged in a few stages, whereas the multi-stage receiver often exhibited slower convergence and oscillatory behavior (as additional stages were added). The best results for both receivers were obtained with a tunh soft-decision decorrelator initialization. When the tanh soft-decisions were also used in subsequent (M-

NELSON AND POOR: ITERATIVE MULTIUSER RECEIVERS FOR CDMA CHANNELS: AN EM-BASED APPROACH 1709

lo-' 1 I I

hard M-step, 2-st - - tanh M-step, 2-st

unit-clipper M-step, 2-st

- - unit-clipper M-step, 3-st

e b


6 a 10 12 14 16 18 1 o . ~

4 SNRI = SNR2 = SNR3 (dB)

Fig. 8. MPEM receiver performance for the four-user system with soft-decorrelator initial stage, various M-step nonlinearities, and SNR4 = 11 dB. The implementation of additional stages (beyond those represented in the graph) did not noticeably affect receiver performance: these results are not plotted.

step) updates, the receivers' performances were only slightly worse than the MPEM performance in Fig. 8.

VII. CONCLUSION

This paper presents an approach to multiuser detection based on application of expectation-maximization algorithms to detection of one or several users' data. The resulting receivers have multi-stage-like structures that use sequential bit-estimate updates and/or intermediate soft-decisions for interference cancellation. Analytical properties of the EM algorithm and the related SAGE algorithm guarantee convergence for some of the receivers and provide a bound on the asymptotic multiuser efficiency of the EM receiver with hard-decision M-step. An algorithm similar to EM and SAGE that incorporates probabilistic models for the parameter was introduced; application of this MPEM algorithm to multiuser detection led to a receiver that incorporates both the E-step soft-decisions of the EM receiver and the sequential bit- estimate updates of the SAGE receiver. The MPEM receiver has complexity K times that of the SAGE or multi-stage receivers, and it achieves good performance for a variety of initializations. The SAGE (and multi-stage) receivers are more dependent upon reliable initial estimates; when soft-decision decorrelator initialization and soft-decision updates are used, these receivers can nearly achieve the best performance for the MPEM receiver.

In principle, the EM approach to multiuser detection herein is straightforward to extend to asynchronous channels, since such channels can be represented similarly to (2), but with higher dimension. In this case, the choice of the missing data (corresponding to bits of the interfering users) may be influenced not only by the M-step complexity, but also by causality or delay considerations. The performance and convergence advantages of soft-decision feedback (based on, e.g., tunh conditional-mean estimates) and of sequential updating of bit estimates are expected to be equally relevant for the

asynchronous case. Lastly, we note that the asynchronous multiuser channel can be described as a hidden Markov model (HMM). (This Markovian structure is the basis for dynamic programming implementations of optimum multiuser detection [9].) Statistical inference for HMM has been analyzed exten- sively in the work of Baum et ul. [23] (see also the: tutorial [2] and references therein), wherein the the ideas underlying the EM algorithm for HMM originate. Further investigation of these analyzes in the context of asynchronous multiuser channels (with interfering users' data modeled as missiing data) may yield new algorithms and insight into the use of decision- feedback techniques for detection and related prohlems in signal acquisition and tracking.

APPENDIX

Theorem 1: For the first result, we wish to show that12

bk = arg max Q(bk; bk) . b k G { f l )

(43)

If bk = b i , the result is trivial. Suppose b: # b i , and note that

(44) arg max Q(bk; bk) = arg max [L(bk ) - D(bkllbk)]

where

The monotonicity property of the EM estimates and the event b! # b i imply that b; = maxbkE{fl) L ( b k ) (i.e., bk = &ipp">. In addition, bk minimizes D(bkllbk). Hence, b; solves (44) and (43) follows.

To show the second result, the error probability may be written as

Pr[bk # bk] = Pr[bk # bk, bt # bk] + Pr[bk # bk, b:i = bk].

Since bk # b i implies bk = p, we have

Pr[bk # bk] = Pr[bk f bk, b; # bk] + Pr[bk # bk, b i = b k , p # b ~ ]

5 Pr[bi f b k ] + P r [ p # bk]

where the inequality follows from Pr[A n B] 5 Pr[EI]. Lemma 1: Let

Then, from (23)

Q(b",+'; b i ) - Q ( b i ; b i ) = -[(b",' 4 - L [ ( b i + ' U 2

- (b , i+l ) 2 - 2 ( b i - G+')z] - 6:)' + 2(bi+' - b:)(x - b;'-')].

2 0 2

2a2 -

Given the definition of b"c1 [see (26) and (30)], it is easy to show that (bi+' - b",)(z - bki-') 2 0 by considering se:parately the cases x > e, z < -e, and x E [-e, e] . The lemma follows..

'*The function arg max Q(br;; b i ) is defined uniquely except when yh - YnL+h Rkn,ambnZ(i) = 0. This event occurs with probability zero.


Theorem 2: Since b i E [-C, C] and C ( b k ) iS COntinUOUS,

of {L(b“,}. For the second result, Lemma 1, (7) (with L(0) = C ( b k ) in this case) and the first result imply (hi+’ - b”,’ 4 0; hence, b i b;. That b; is a stationary point follows from noting that 508-519, Apr. 1990.

0 = -Q(b; db b z ) l b = b ; = --C(b)lb=b; db - -D(bllb;)lb=b; db 496-508, Apr. 1990.

[I71 S. Verdli, “Optimum multiuser efficiency,” IEEE Trans. Commun., vol. 34, no. 9, pp. 890-897, Sept. 1986.

chronous AWGN channels,” in 31st Annu. Allerton Con$ Commun., Control Computing, Monticello, IL, Sept. 1993.

[ 191 M. K. Varanasi and B. Aazhang, “Multistage detection in asynchronous CDMA communications.” IEEE Trans. Commun., vol. 38, no. 4, pp.

[20] R. Lupas and S. Verdh, “Near-far resistance of multiuser detectors in d a d asynchronous channels,” ZEEE Trans. Commun., vol. 38, no. 4, pp.

[21] A. Duel-Hallen, “Decorrelating decision-feedback multiuser detector for synchronous code-division multiple-access channel,” IEEE Trans.

{ C ( b k ) } is bounded. Then ’) follOws from the monotonicity [I81 X, Zhang and D, Brady, “Soft-decision multistage detection for asyn.

where the divergence is defined in (45). Properties of the - divergence imply that the last derivative is zero; hence, (d/db)L(b)/b=bl; = 0.

Commun., vol. 41, no. 2, pp. 285-290, Feb. 1993. [22] L. B. Nelson, “Multiuser detection for radio-frequency and optical

CDMA communications,” Ph.D. dissertation, Princeton University, NJ,

REFERENCES

A. P. Dempster, N. M. Laird, and D. B. Rubin, “Maximum likelihood from incomplete data via the EM algorithm,” J. Roy. Stat. Soc., Ser. B, vol. 39, no. 1, pp. 1-38, 1977. L. R. Rabiner, “A tutorial on hidden Markov models and selected apulications in speech recognition,” Proc. IEEE, vol. 77, no. 2, pp. _ _ 2’53-286, Feb. 1989. S. M. Zahin and H. V. Poor, “Efficient estimation of class A noise parameters via the EM algorithm,” IEEE Truns. Inform. Theory, vol. 37, no. 1, pp. 60-72, Jan. 1991. T. J. Schulz and D. L. Snyder, “Imaging a randomly moving object from quantum-limited data: applications to image recovery from second- and third-order autocorrelations,” .I. Opt. Soc. Am. A, vol. 8 , no. 5, pp. 801-807, May 1991. X. Li Meng and S. Pedlow, “EM: a bibliographic review with missing articles,” in ASA Proc. Stat. Computing Sect., 1992, pp. 24-27. U. Fawer and B. Aazhang, “A multiuser receiver for code division multiple access communications over multipath channels,” IEEE Trans. Commun., vol. 43, no. 2/3/4, pp. 1556-1565, Feb./Mar./Apr. 1995. H. V. Poor, “On parameter estimation in DS/SSMA formats,” in Advances in Communications and Signal Processing, W. Porter and S. Kak, Eds. New York: Springer-Verlag, 1988, pp. 59-70. C. N. Georghiades and J. C. Han, “Sequence estimation in the presence of random parameters via the EM algorithm,” pre-print, 1994. S. Verdh, “Minimum probability of error for asynchronous Gaussian multiple-access channels,” IEEE Trans. Inform. Theory, vol. IT-32, no. 1, pp. 85-96, Jan. 1986. R. Lupas and S. Verd6, “Linear multiuser detectors for synchronous CDMA channels,” IEEE Trans. Inform. Theory, vol. 35, no. 1, pp. _ _ 123-136, Jan. 1989. H. V. Poor, An Introduction to Sijinal Detection and Estimation, 2nd ed. New York: Springer-Verlag, 1994. A. 0. Hero and J. A. Fessler, “Convergence in norm for alternating expectation-maximization (EM) type algorithms,” Siatistica Sinica, vol. 5, no. 1, Jan. 1995. J. A. Fessler and A. 0. Hero, “Space-alternating generalized EM algorithm,” IEEE Trans. Signal Processing, vol. 42, no. 10, pp. 2664-2677, Oct. 1994. X. Li Meng and D. B. Rubin, “On the global and componentwise rates of convergence of the EM algorithm,” Linear Algebra & Its Applicat., vol. 199, pp. 413425, 1994. C. F. J. Wu, “On the convergence properties of the EM algorithm,” Ann. Statistics, vol. 11, no. 1, pp. 95-103, 1983. M. K. Varanasi and B. Aazhang, “Near-optimum detection in synchronous code-division multiple-access systems,” IEEE Trans. Com- mun., vol. 39, no. 5, pp. 725-736, May 1991.

Jan. 1995. [23] L. E. Baum, T. Petrie, G. Soules, and N. Weiss, “A maximization

technique occurring in the statistical analysis of probabilistic functions of Markov chains,” The Ann. Math. Statistics, vol. 41, no. 1, pp. 164-171, 1970.

Laurie B. Nelson (S’89-M’95) received the B.S.E.E. degree (with highest distinction) from Purdue University, West Lafayette, IN, in 1990 and the M.A. and Ph.D. degrees in electrical engineering from Princeton University, NJ in 1992 and 1995, respectively.

Since January 1995, she has been an Assistant Professor of Electrical Engineering at the University of Minnesota, Minneapolis. Currently, she holds a Visiting Research position with the Institute for Defense Analyses, Princeton, NJ. Her research interests include statistical signal processing and communications theory with applications to synchronization and demodulation of spread-spectrum multiple-access signals, CDMA cellular networks, and optical communications.

Dr. Nelson was a recipient of the NSF Creativity Award Fellowship and the AT&T Ph.D. Scholarship while at Princeton University, NJ.

H. Vincent Poor (S’72-M’77-SM’82-F’87) received the Ph D degree in electrical engineermg and computer science from Princeton University, NJ in 1977

He is Professor of Electrical Engineering, Princeton University. From 1977 until he joined the Princeton faculty in 1990, he was a Faculty Member at the University of Illinois, Urbana. He has also held visiting and summer appointments at several universities and research organizations in the United States, Britian, and Australia. His

research interests are primanly in the area of statistical signal processing and its applications His publications in this area include the graduate textbook, An Introduction to Signal Detection and Estimation (New York. Springer-Verlag, 1988 and 1994)

Dr Poor received the Terman Award from the American Society for Engineering Education in 1992 and the Distinguished Member Award from the IEEE Control Systems Society in 1994 He is a Fellow of the Acoustical Society of America, and of the American Association for the Advancement of Science. He has been involved in a number of IEEE activities, including serving as President of the IEEE Information Theory Society in 1990, and as a member of the IEEE Board of Directors in 1991 and 1992.

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Iterative multiuser receivers for CDMA channels: an EM-based approach