5716 ieee transactions on signal processing, vol. 55,...

5716 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 55, NO. 12, DECEMBER 2007

Successive Interference CancellationUsing Constellation Structure

Ananya Sen Gupta and Andrew Singer

Abstract—An approach to successive interference cancellationis presented that exploits the structure of the combined signalconstellation in a multiuser system. The asymptotic conditional ef-ficiency of a successive detector is defined, based on the conditionalprobability of error at high signal-to-noise ratio (SNR), as a quan-titative measure for evaluating detector performance at each stageof successive detection. The joint successive interference canceller(JSIC) that jointly detects consecutive users in an ordered setis proposed as an improvement over the conventional successiveinterference canceller (SIC). The maximal asymptotic conditionalefficiency successive interference canceller (MACE-SIC) and itsJSIC equivalent (MACE-JSIC) are also derived as the multiuserdetectors that achieve the highest asymptotic conditional mul-tiuser efficiency at each stage of successive detection among allpossible SIC and JSIC detectors, respectively, given any partic-ular ordering of user signals. The ordering of users achievingthe highest asymptotic conditional efficiency at each stage ofsuccessive detection is derived. Performance bounds based on thesignal constellation structure are derived to quantify the gain ofthe MACE-JSIC detector compared to the MACE-SIC detector.

Index Terms—Asymptotic efficiency, convex optimization, detec-tion, interference cancellation, multiuser detection.

I. INTRODUCTION

THE goal of multiuser detection is to correctly demodu-late the information bits of mutually interfering users in a

multiple-access communication system. A performance boundfor joint detection is given by the maximum likelihood (ML)detector, which determines the most likely bits sent over thechannel. While the ML detector achieves the lowest probabilityof error for joint detection, it has a complexity that is exponen-tial in the number of users. The problem of low complexity highperformance detection has been extensively studied in the pasttwo decades, with numerous approaches proposed in the liter-ature [1]. These include linear detectors, such as the matchedfilter or conventional detector [1], decorrelating detector [2], andlinear minimum mean square error detector [3], [4], which si-multaneously demodulate all user bits by applying a linear trans-formation to the received signal. Another popular approach is

Manuscript received April 29, 2006; revised December 17, 2006. The as-sociate editor coordinating the review of this manuscript and approving it forpublication was Prof. Karim Drouiche. This work was supported in part by theNational Science Foundation under Grant CCR-0092598 (CAREER).

A. Sen Gupta was with the University of Illinois at Urbana-Champaign, Ur-bana, IL 61801 USA. She is now with Microsoft, Redmond, VA 98052 USA(e-mail: [email protected]).

A. Singer is with the University of Illinois at Urbana-Champaign, Urbana, IL61801 USA (e-mail: [email protected]).

Digital Object Identifier 10.1109/TSP.2007.898779

successive interference cancellation (SIC) in which the inter-fering signal of a particular user is cancelled out after makinga decision on that user’s bit. Many variants of interference can-cellation exist in the literature, such as multistage interferencecancellation [5]–[10], parallel interference cancellation (PIC)[11]–[13], and parallel arbitrated successive interference can-cellation (PASIC) [14], among others. Most interference can-cellation techniques use a linear detector as a kernel to decideon or update a user’s bit before canceling its contribution to thereceived signal. Interference cancellation is attractive because itis a low-complexity solution that works well in many practicalmultiaccess systems, particularly when the number of users isless than the number of dimensions of the signal space, i.e., forunder-loaded communication systems.

Tutorial articles on multiuser detection are available with ex-tensive reference lists [8], [15], [16]. Among the many subop-timal lower complexity alternatives to maximum-likelihood de-tection, the sequential decoding algorithm [17], cyclic decisionfeedback sequence detection [18], group detection [19]–[23]and detection based on interference cancellation from tentativedecisions [12], [24]–[27] deserve mention. An extensive litera-ture on multistage detection can be found in [11], [28]–[32] andthe references therein.

In this work, we introduce a joint successive interferencecancellation technique (JSIC) that gives improved performanceover that of conventional successive interference cancellation(SIC). The key idea behind JSIC is to exploit the structuralproperties of the sub-constellation formed by the signals oftwo consecutive users in an ordered set to gain improvementin detector performance. The asymptotic multiuser efficiency[1], which measures the rate of decay of the bit error rate(BER) in the high SNR regime, has been used as a benchmarkwith which to compare the performance of detection methods.We introduce asymptotic conditional multiuser efficiency as aquantitative measure for comparing the performance of mul-tiuser detectors that specifically employ successive interferencecancellation. The optimal linear detector, in terms of achievingthe highest asymptotic efficiency among all linear detectors,was proposed by Lupas and Verdu [2]. We provide an inter-pretation to maximal asymptotic efficiency linear detectionin terms of the constellation structure and use this insight toderive maximal asymptotic conditional efficiency successivedetection. For a given ordering of user signals, we derive themaximal asymptotic conditional efficiency SIC detector thatachieves the maximum asymptotic conditional efficiency ateach stage of successive detection among all possible SIC de-tectors. We also derive the optimal ordering of user signals thatmaximizes the asymptotic conditional efficiency at each stage

1053-587X/$25.00 © 2007 IEEE

Authorized licensed use limited to: University of Illinois. Downloaded on July 27,2010 at 06:24:36 UTC from IEEE Xplore. Restrictions apply.

SEN GUPTA AND SINGER: SIC USING CONSTELLATION STRUCTURE 5717

of successive detection. We extend the concept of maximalasymptotic conditional efficiency (MACE) detection to the caseof joint successive interference cancellation (JSIC), where ateach stage of successive detection, the corresponding symbolis detected, taking into account the interference of the nextuser’s signal in an ordered set of users. Both MACE detectionalgorithms proposed are robust against strong correlation ofuser signals, e.g., in a multiuser system where the user signalsare linearly dependent. Simulation results demonstrate thatthe maximum asymptotic conditional efficiency approachsignificantly improves detector performance, particularly athigh SNR. The focus in this paper is on understanding andmitigating the driving effects of multiuser interference (MUI)in MUI-dominated environments. As such, we consider theeffects on bit-error-rate in the limit of high SNR to capture thedriving factors from MUI on performance. The contributionsin this paper are six-fold:

i) introduce joint successive interference cancellation asa method of combining the joint information betweenconsecutive users in an ordered set to attain higherperformance;

ii) formulate the maximal asymptotic linear detectionproblem in terms of the constellation structure;

iii) introduce asymptotic conditional efficiency as a mea-sure of successive interference cancellation (SIC)performance;

iv) quantify what can be regarded as a “good ordering” ofusers, i.e., specifically derive the optimal ordering of usersthat maximizes the asymptotic conditional efficiency ateach stage of successive detection;

v) derive the successive interference canceller that optimizesthe asymptotic conditional efficiency at any given stage ofdetection for a random ordering, and;

vi) derive performance bounds to quantify the tradeoff be-tween design complexity as well as detection complexityand performance of maximal asymptotic conditional effi-ciency successive detection.

The proposed techniques are most useful in stable multiusersystems with well-known parameters, such as in satellite wire-less communications. Statistical models for fading channels areextensively treated in [33]–[37], among others. A rich litera-ture exploring the problem of mitigating intersymbol interfer-ence introduced by fading channels include channel estimationtechniques such as [38]–[41] and references therein, as wellas techniques for mitigation of frequency-flat, frequency-selec-tive fading [42], [43], adaptive equalization of multipath [44],among others. We have assumed that a reliable channel esti-mation or equalization algorithm, such as the decision-feed-back equalizer (DFE) ([33], [45], and references therein), or thesage algorithm [46], is used to nullify the effect of any inter-symbol interference and the CDMA model is assumed to besynchronous. It has been shown [1, p. 25], that an asynchronousmultiuser system can be treated as a synchronous system with ahigher number of users.

The organization of the paper is as follows. The system modeland list of commonly used notation are given in Section II.Section III introduces the basic idea of joint successive in-terference cancellation. Section IV reformulates the maximal

asymptotic efficiency linear detection problem in terms of theconstellation structure. Section V introduces asymptotic condi-tional efficiency as a measure of SIC detector performance, andthe maximal asymptotic conditional efficiency SIC and JSICdetectors are introduced in Sections V and VI respectively.Section V-A derives the optimal ordering that maximizes theasymptotic conditional efficiency at each stage of SIC detec-tion. The tradeoff between complexity and performance gainis specifically treated in Section VII. Finally simulation resultsare presented in Section VIII.

II. SYSTEM MODEL AND NOTATION

We assume a synchronous BPSK-signaling CDMA model,where represents the set of user signals,

are the user symbols. The equivalentbaseband received signal given as

(1)

where is a stationary white Gaussian noise process ofnoise power . It is assumed that the number of users aswell as the user signals are precisely known, as is the casein stable multiuser systems such as satellite communicationsystems. The user signals in can be equivalently repre-sented as vectors in a vector space, generated by anorthonormal basis that spans . Whereverapplicable in the sequel, represents the standard Eu-clidean inner product and represents the standardnorm. Thus and

. The received signal can be written as avector in signal space,i.e.,

(2)

where is the matrix of signal vectors,is the vector of user symbols and

is a white Gaussian random noise vector. The energy of eachuser’s signal is given as , where is often referredto as the amplitude of the th user’s signal. Another CDMAmodel similar to (2) and used in the literature assumes the signalspace generated by the basis set , which need not be linearlyindependent. An equivalent CDMA model using instead ofas the basis is given by

(3)

where is the autocorrelation matrix of the user signalsin , and .

Wherever applicable, we shall use the following notation. Ad-ditional notation will be introduced as needed in the sequel.

• denotes the constellation point corresponding to thebit vector ;

• denotes the point corresponding to the vector insignal space. Unless stated otherwise, any vector is as-sumed to be centered at the origin with the tip at ;

• denotes an estimate of the bit ;• denotes the norm;



• denotes the inner product in an appropriate Hilbertspace;

• denotes the sub-constellation formed by, ;

• denotes the sign ( or ) of , i.e.,if , and if ;

• For a set of user signals, the notation

denotes the set , i.e., theset without the user signal ;

• denotes the sub-constellation formed by, e.g., denotes the sub-constellation

;

• denotes the subconstellation formed

by , e.g., denotes

the subconstellation ;• For a given ordering of the user signals,

denotes the sub-constellation cor-responding to all bit vectors with the th bit set to , the

th bit set to , and bits set to theirestimates at the th stage of detection. For exampleif , denotes the subcon-stellation , , being theestimate of at the 1st stage of successive detection;

• For any ordering of user signals and someindex set , let be somesubset of user signals. Then , denotes thesub-constellation formed by ,

e.g., denotes the point sub-constella-tion ;

• denotes the convex hull of the constellation ;• denotes the minimum Euclidean distance

from the vector to ;• denotes the minimum Euclidean distance

from the hyperplane to .

III. JOINT SUCCESSIVE INTERFERENCE CANCELLATION

In this section we present joint successive interference can-cellation in which the structural information of the sub-constel-lation formed by two consecutive users in an ordered set is ex-ploited to improve the performance of conventional successivedetection. The motivation behind this approach is to account forthe effect of the “closest” interferer while detecting the bit of aparticular user in each stage of successive detection. We firstintroduce the two-user joint successive interference canceller(JSIC) and then extend it to the -user case.

A. The Two-User JSIC Detector

In this section, we describe a decision-driven detector thatyields maximum-likelihood (ML) decisions for the two-usercase. Fig. 1 shows a two-user signal constellation formed by theuser signals along with the maximum-likelihood deci-sion regions. The two-user JSIC detector exploits the geometryof the constellation to give the maximum-likelihood solution.If is the inner product between the user signalsand , represents the signal . Note that

Fig. 1. ML Decision regions for two users with signal vectors s and s andsgn(�) > 0.

the four-point constellation generated by the two-user systemforms a parallelogram as shown in Fig. 1. The four

points , , and correspond to theconstellation points , , , ,

respectively. The ML decision regions are formedby the perpendicular bisectors of the four sides of the paral-lelogram and , the perpendicular bisector of ,the shorter diagonal. The proof that is indeed the shorterdiagonal is given in [47]. The vector aligned with the shorterdiagonal, i.e., the vector can be written ,in terms of the user signals. The two-user JSIC detector esti-mates from the received signal vector using three innerproduct operations as shown in Table I. Fig. 2 demonstrateshow the two-user JSIC detector reaches a decision given aparticular received vector . The point representing insignal space is shown as an unfilled circle at the tip of the vector

. The darkened hyperplanes , and , labeled as stepsin the algorithm represent the inner products taken in order ofprecedence to reach the final bit vector estimate, , forthe given received vector .

B. The -User JSIC Detector

Consider a -user system in which is an or-dered set of user signals, arranged according to some appro-priate criterion. For example, one popular approach is to orderthe users in a decreasing sequence of received powers. In thesequel we will also discuss an appropriate ordering of usersin terms of multiuser efficiency. The extension of the two-userJSIC algorithm to the -user case is given in Table II. The

-user JSIC detector estimates from the received signalin steps.

It is easy to verify that the JSIC detector performsinner product operations to estimate the bit vector ,

and therefore has computational complexity , i.e., linearin the number of users. The key idea behind the JSIC detectoris to account for a significant interferer in an ordered set using atwo-user locally ML kernel. As with all SIC detectors, the per-formance of the JSIC detector will be a function of the orderingof the users in the algorithm. Improved estimates might be ob-tained for each user at each stage of successive detection througha different (more favorable) ordering. Fig. 3 demonstrates how



TABLE ITWO-USER JSIC DETECTOR

TABLE IIK-USER JSIC DETECTOR

Fig. 2. Illustration of the two-user JSIC algorithm for a given received vector.

the -user JSIC algorithm works, step by step, for a given a re-ceived vector. Similar to 2, the darkened hyperplanes representthe inner products labeled in order of precedence to reach thefinal bit vector estimate, in this case, . The receivedvector is given by the vector pointing to the unfilled circle,which represents the received point in signal space.

IV. MAXIMAL ASYMPTOTIC EFFICIENCY LINEAR DETECTOR:A STRUCTURAL PERSPECTIVE

In this section, we formulate maximal asymptotic efficiencylinear detection in terms of the structural geometry of the mul-tiuser signal constellation, and use this perspective in subse-quent sections to develop SIC and JSIC detectors optimized interms of the slope of the conditional probability of error. To eval-uate the performance of the detectors proposed in this work, wefocus on the asymptotic multiuser efficiency of the multiuserdetector. Asymptotic multiuser efficiency was first proposed by

Fig. 3. Illustration of theK-user JSIC algorithm for a given received vector.

Lupas and Verdu [2] and is a quantitative measure of detectorperformance in terms of its bit error rate (BER) at high SNR. Thebit error rate of the th user is denoted and is a function ofall the user signals interfering in the channel, the multiuser de-tector used and the noise variance, . The multiuser efficiencyof a detector for the th user in a multiuser detector, denoted

, is given as the ratio of the effective energy, , thatthe th user would require in order to achieve the same bit errorrate in a single-user channel, to the actual energy of the

th user, , in the multiuser channel, i.e.,

where and is the amplitude of theth user.In the absence of interfering users, the effective energy will

be the same as the actual energy. Because of multiuser inter-ference, the bit error rate is increased, and hence the effective



energy is less than the actual energy of the th user. The mul-tiuser efficiency is therefore always less than unity. Weare interested in the asymptotic behavior of and hencethe effect of multiuser interference as the noise variance van-ishes. Mathematically, the asymptotic multiuser efficiency of adetector for the th user can be calculated to be [1]

(4)

We state without proof the following well-known results [1,p. 352]

Theorem 4.1: Let denote the probability thatthe multiuser detector outputs given was transmitted. Letus denote the minimum distance from the constellation point

to (decision region corresponding to ) by. If is a convex poly-

tope that includes its boundary and does not depend on , then

(5)

For completeness we include this extension here, though theresult parallels other well known results from large deviationstheory [48], [49, Theorem 5.2, p. 77].

Theorem 4.2: Let the decision region of bethe union of convex polytopes . Let

. Then

(6)

It follows from Theorem 4.2, that for detectors with convexdecision regions, or decision regions that comprise a union ofconvex sets, the asymptotic multiuser efficiency of the detectorfor the th user is given by the shortest distance squared from aconstellation point to the decision boundary separating it fromanother point that differs in the th bit, normalized by . Thiscan be written

(7)

This has a number of implications for optimizing the perfor-mance of the JSIC detector that we will explore in the sequel.

The “best” (or optimal) linear detector introduced by Lupasand Verdu in [2] achieves the highest asymptotic multiuser effi-ciency among all linear detectors. By a linear detector, we meanany detector that outputs bit estimates as ,

, for some vector . The asymptotic efficiencyof the optimal linear detector for the th user is given in [2] as

(8)

where the matrix is as defined in the CDMA model from (3),and .

Note that, in general, the best linear detector must be found bysolving the convex optimization (8) over all linear detectors .Using (2), when the signal space is generated by an orthonormal

basis , if is a vector repre-sented with , and is the corresponding representation withrespect to , then . Therefore, . There-fore, in the model (2), the maximum asymptotic efficiency of alinear detector for the th user is given by

(9)

In this paper, we interpret the above convex optimizationproblem in terms of the structure of the constellation. Weanalyze the detector given in [2] in terms of the constellationstructure and apply this perspective to extend the idea of max-imal asymptotic efficiency multiuser detection to successiveinterference cancellation and derive an appropriate ordering ofusers for successive detection.

Theorem 4.3: For a linear system represented as in (2), themaximum asymptotic efficiency of a detector for the th user,denoted , achievable among all possible linear detectorsoutputting the decision of the th bit as ,

, is equal to the minimum distance squared from theorigin to the convex hull of the sub-constellation with the thuser’s bit set to 1, i.e., , normalized by the energy of the

th user, i.e.,

(10)

where is the minimum distance squaredfrom the origin to the convex hull of the sub-constellation

, and is the amplitude of the th user. The maximumasymptotic efficiency linear detector will output bit estimates

(11)

where is proportional to the minimum distancevector from the origin to , such that

.Proof: For notational simplicity, we will denote and

as and respectively. Also, for clarity, we denoteas the point corresponding to the vector . The asymptotic

efficiency of a detector with probability of error for theth user, , is given by [1]

(12)

With BPSK signaling, any pair of constellation points and, and being bitwise complements, will be radially sym-

metric about the origin . Thus the sub-constellations andwill also be radially symmetric about . Therefore, from sym-metry considerations, we must have

We must also have that the conditional probability of error given, denoted , is equal to the conditional

probability of error given , . We denote, as the probability that was sent

and assume that , . The bit-error-



rate (BER), , of a given linear detector that estimates theth bit as is given by

(13)

(14)

whereis the conditional probability that givenand given that , were the other users’ bits.There are possible combinations for the other users’bits and hence the summation has been taken over all pos-sible configurations of the bit vector such that

. For each bit vector combination , ,will

be given by , since thismeasures the probability that , is greater than

in each of the cases that an erroris made on . We index the possible combinationsof as and denote the corre-sponding bit vector as and the normalized inner product

as . We define

(15)

i.e., is the minimum normalized inner product betweenand any constellation point belonging to the sub-constellation

. Note that will depend on . From large deviationtheory [48], it can be shown that [49, Section 3.2] as ,

is dominated by the term that has a -function with theminimum argument. The following derivation in the context ofasymptotic efficiency for equalizers [49] is reproduced here forconvenience.

The product of the asymptotic efficiency of the detector forthe th user and the energy of the th user, sometimes referredto as the asymptotic multiuser energy, can be expressed as

(16)

The last logarithm can be expanded in a Taylor series andvanishes in the limit . Therefore, we obtain

(17)

where .Using the well-known inequality [1, p. 98]

and taking the limit as on both sides, it is easy to verifythat

(18)

Therefore

(19)

Thus to maximize the asymptotic efficiency, , for the thuser, we maximize the minimum normalized inner product be-



Fig. 4. Illustration of property 4.1.

tween and a constellation point in . Denoting the vectorthat maximizes as , we have

(20)

Let us denoteand . Therefore,

. Let be the minimumdistance vector from the origin to the convex hull of thesub-constellation , i.e., . Wenow show that is the minimum distance from the originto the convex hull of , i.e., . The convex hullof any set of points is the minimum convex set containing thosepoints. Any point is given by ,

, . Before we proceed, we state the followingwell-known property of convex sets [50, Theorem 1, p. 69],which will be used throughout the proof.

Property 4.1: If be the minimum distance vector fromthe origin to a convex set and is a point in ,then . Equality is achieved only when thepoint lies on the hyperplane orthogonal to . This is graphicallyillustrated in Fig. 4, where, for the point , we have

and for the point , we have

Case 1: Let us first consider the case when ,i.e., itself is a constellation point in the sub-constellation

. Since, , if lies along , i.e., ,, we must have

(21)

It follows from Property 4.1, that for any , ,we must have . By construction, andare in the same direction. Therefore, we must have

(22)

Fig. 5. Illustration of the case where the minimum distance vector is not a con-stellation point.

since . Therefore, if , for any

(23)

If , , then the vectors and will subtendsome angle with each other, i.e.,

(24)

Therefore if , , we must have

(25)

From (23) and (25), we infer that is proportional to theminimum distance vector from the origin to the convex hullof and that .

Case 2: Now let us consider the case where is not aconstellation point, as shown in Fig. 5. Let be the projection ofthe origin onto the convex set , i.e., is the unique pointin that is geometrically closest to the origin . Letbe the -dimensional hyperplane orthogonal to the vector

and passing through . Therefore, by construction,, . For any constellation point and any

vector , , we can express the normalized innerproduct between and as

(26)

Therefore, for any constellation point , i.e.,, but does not lie along , we must have, from



Property 4.1, that . Hence, for , ,we have

(27)

As such, for any , we must have that

(28)

Suppose does not lie along , i.e., ,. Let where and are parallel and

perpendicular to respectively. Consider .Since the intersection of a hyperplane with a convex hull is aconvex set, is convex. Let be a constellation pointlying on . We can break into components perpendicular andparallel to as . Since , .

Now, by construction, the projection of the origin is thepoint , which is an interior point of . If was exterior to

, then will not be the closest point on to . Ifwere not strictly interior to , but lay on the boundary of ,then would be a constellation point in , which was treatedin Case 1. Note that evaluating is geometricallyequivalent to passing an -dimensional hyperplanethrough the point orthogonal to and determining to whichside of the point lies. Since is a convex set, anyhyperplane passing through an internal point will bi-partitionit into two convex subsets. Therefore, there will always exist atleast one point such that . Hence, wemust have

(29)

and that

(30)

We then obtain that for any ,

(31)

From (28) and (31), it follows that . Combiningcases 1 and 2, the result follows.

From Theorem 4.3, we observe that the maximum asymp-totic efficiency of the detector for the th user, will bezero if the minimum distance from the origin to the convexhulls of the sub-constellations is zero, i.e., if

Fig. 6. Three-user system with ordering O = fs ; s ; s g.

overlap. We will show that this will happen if and only ifbelongs to the convex hull of the sub-constellation

.Lemma 4.1: The convex hulls of sub-constellations

and overlap if and only if .Proof: Since the signaling scheme used is BPSK, i.e., it is

symmetric about the origin , the constellation pointscorresponding to any bit vector and its bitwise inverse will be ra-dially symmetric about . Thus the origin will be the midpointof the line segment joining them. Therefore, the convex hullsof the signal constellations will also exhibit radial symmetryabout the origin. Consider the -user system with ordering

, e.g., as shown in Fig. 6 for .Without loss of generality we will prove the stated result for theuser signal . Though the proof uses Fig. 6 as an illustrative ex-ample, no assumption has been made that the number of usersis three, or that the signal space is two-dimensional. The convexhull of sub-constellation is shownby solid lines and the convex hulls of the sub-constellations

are represented by dashedlines. Wherever applicable in the sequel denotes trans-lation of the point along the direction . Note that, by con-struction, and hence,

, where denotes translation along the di-rection of the vector .

Proof of lemma 4.1 (i): If , thenoverlap.

Let and we extend the line along the di-rections and to intersect at and re-spectively. Because of radial symmetry around the origin ,

. By construction , which implies. Thus ,

which implies,. Similarly, . Since

and both contain , they must overlap. Therefore,if then and overlap.

Proof of lemma 4.1 (ii): overlap only if.

Suppose and overlap. Then theremust exist at least one constellation point ,



, where such that . By con-struction, the constellation point corresponding tothe vector also belongs to . Fromthe definition of convex sets [50, p. 17], any point lying onthe line segment between and belongs to .Therefore, from radial symmetry of linear constellations, ,the midpoint of and lies inside . Similarly,

, i.e.,.

Combining parts (i) and (ii), the result follows.It is noteworthy that must be a linear combination of

, from the definition of convex sets [50, p. 17],for to lie in the convex hull of the sub-constella-tion generated by the other signals. Therefore, the condition

, and hence , is triviallysatisfied when the set of user signals are linearly independent.

V. MAXIMAL ASYMPTOTIC CONDITIONAL EFFICIENCY

SUCCESSIVE INTERFERENCE CANCELLATION

In this section, we will introduce the maximal asymptoticconditional efficiency (MACE) approach to successive detec-tion. We derive the maximal asymptotic conditional efficiencysuccessive interference canceller, henceforth referred to as theMACE-SIC detector that achieves the highest asymptotic con-ditional efficiency for a particular user among all possible SICdetectors for a given signal constellation and a given signalordering. We define the asymptotic conditional multiuser effi-ciency for a given ordering of user signals as a quantitative mea-sure of SIC detector performance based on the conditional prob-ability of error at each stage of successive detection. Withoutloss of generality, let be a given orderingof user signals for successive interference cancellation. Denoteby the conditional probability that the th user’s bit iscorrectly detected at the th stage of successive detection, giventhat users with signals have been correctly de-tected in the first stages, i.e.,

Definition 5.1: Given an ordering , and a detector with con-ditional probability of error for the th user, the asymp-totic conditional multiuser efficiency for the user, denoted as

, is

(32)

where is the energy of the th user. Accordingly, theasymptotic conditional effective energy is defined as the quan-tity .

Many SIC detectors output bit estimates as, i.e., at the th stage, a linear detector is used on

the received vector after subtracting off the interference of theuser signals detected in previous stages. Thus, the maximumasymptotic conditional efficiency at the th stage of detection issimply the maximum asymptotic efficiency achieved by a lineardetector for the th user in the reduced system .

Given an ordering , the maximum asymptotic conditional effi-ciency achievable by a SIC detector at the th stage of detection,denoted as , will be given as

(33)

where wherever applicablein the sequel.

From Theorem 4.3 it follows that (33) represents a convexoptimization problem that seeks the minimum distance fromthe shifted origin to the convex hull of the

-point sub-constellation .The computational complexity of solving this convex opti-mization problem will be exponential in [51], yielding

and

(34)

Thus, the maximum asymptotic conditional efficiency succes-sive interference canceller, given an ordering , henceforth re-ferred to as the MACE-SIC( ) detector, will output its bit esti-mates as

(35)where , is proportional to the min-imum distance vector, , from the shifted origin tothe convex hull .

The asymptotic conditional multiuser efficiency of theMACE-SIC( ) detector at the th stage, , can thereforebe expressed as

(36)

Fig. 7 shows the complete decision regions for theMACE-SIC detector with the ordering . We nowintroduce an algorithm to generate the optimal ordering of usersignals that maximizes the asymptotic conditional efficiency ateach stage of successive detection.

A. Maximum Asymptotic Conditional Efficiency Ordering ofUser Signals

The maximum asymptotic conditional efficiency ordering, is the ordering that maximizes , defined in (33)

among all possible orderings for each stage insuccessive detection. Let denote the maximizedasymptotic efficiency for a linear detector for the user withsignal in a multiuser system with as the set of user sig-nals. We can derive according to the following greedyalgorithm:

1) Among possible choices for the user to be decoded first,choose the user signal from the set such that



Fig. 7. Decision region of optimized SIC detector for the ordering fs ; s ; s g.

2) Choose the second user signal among thechoices for the reduced system such that

3) Continue in this fashion such that at the th stage we choosethe user signal among the choices left in

such that

4) At the th stage, the algorithm terminates yielding

Note that maximizes the asymptotic conditional efficiencyat each stage of successive detection, given the ordering this far,and does not necessarily maximize the overall asymptotic effi-ciency, or the asymptotic conditional efficiency for each (or anyparticular) user. However, it follows from (36) and (7) that theasymptotic efficiency for the th user in the ordered setwill be lower bounded by the minimum asymptotic conditionaleffective energy, , among all users pre-ceding it, normalized by . Therefore

(37)

The ordering maximizes this lower bound by construction.Complexity: It is noteworthy that the computational com-

plexity of deriving is exponential in , the number ofusers. A motivation behind this paper is to tradeoff the com-plexity of setting up the SIC detector with the performance

gain achieved by optimizing the asymptotic conditional effi-ciency at each stage of detection. The run-time complexity ofthe MACE-SIC( ) detector is always linear in the numberof users, i.e., . As such, the MACE-SIC( ) detector issuitable for stable communication systems with power controland a stable system of users, e.g., in satellite communications.In more dynamic multiuser systems, e.g., mobile cellular net-works, the high complexity of setting up the detector when thesystem changes may render it impractical without modification.

VI. MAXIMAL ASYMPTOTIC CONDITIONAL EFFICIENCY JOINT

SUCCESSIVE INTERFERENCE CANCELLATION (JSIC) DETECTION

In this section, we extend the MACE approach to JSICdetection and derive the JSIC detector that achieves themaximal asymptotic conditional efficiency at each stageof successive interference cancellation. Consider an or-dering of user signals. We set up themaximum asymptotic conditional efficiency JSIC detector,henceforth referred to as the MACE-JSIC( ) detector, for agiven ordering as follows. We use to denotethe maximized asymptotic efficiency for a linear detectorfor the user with signal in the reduced system . Con-sider the four sub-constellations , where, e.g.,

, atthe th stage of successive detection. Letand . Let , andbe the optimized vectors for linear detection that achieve

, and respec-tively, in the reduced systems ,

andrespectively. The MACE-JSIC( ) detector uses the vectors

, and instead of , andrespectively at the th stage of interference cancellation. Sincewe are using sub-constellations, and not constellation pointsin the two-user MACE-JSIC kernel, we need to consider both

and , rather than in the JSIC kernel and therefore,have three, instead of two inner product operations in theMACE-JSIC kernel as shown in Table III.

From Theorem 4.2 it can be shown that the asymptotic condi-tional efficiency of the MACE-JSIC( ) detector, willbe given by

(38)

The ordering that achieves the highest asymptotic condi-tional efficiency at each stage among all possible orderings canthen be achieved through a similar algorithm to that given inSection V-A. At each stage , we choose the two user signals

and over the possible choices such that thequantity

(39)

is maximized , , where, , and



TABLE IIITHE MACE-JSIC KERNEL AT THE ith STAGE

Fig. 8. Decision regions of MACE-JSIC(O) detector with orderingfs ; s ; s ; s g for four-user system.

. Fig. 8 shows the decision regions for a MACE-JSIC de-tector for a given ordering in a four-user system in two dimen-sions. Though the run-time complexity of MACE-JSIC( ) andMACE-SIC( ) detector are linear in the number of users, theMACE-JSIC( ) detector performs three inner product opera-tions at each stage compared to the MACE-SIC( ) detector,which performs only one inner product operation. The designcomplexity of the MACE-JSIC( ) detector is exponential in

, the number of users, and its run-time complexity, thoughlinear in the number of users, is greater than the run-time com-plexity of the MACE-SIC( ) detector. We now show that thisincrease in run-time complexity is compensated by an increasein the asymptotic conditional multiuser efficiency at each stageof successive interference cancellation.

Lemma 6.1: For a given convex hull ordering , theMACE-JSIC( ) detector always achieves higher asymptoticconditional efficiency for every user than the correspondingMACE-SIC( ) detector.

Proof: At the th stage of interference cancella-tion, the asymptotic conditional efficiency of usergiven is the minimum of ,

and , normalized by , i.e.,

(40)

Let denote the minimum distance vector betweenand . Then

(41)

Since , and the minimum dis-tance between subsets of convex sets is always greater than orequal to the distance between the convex sets, we must have

(42)

Combining (40), (41), and (42) the result follows.

VII. PERFORMANCE BOUNDS ON MULTILEVEL JOINT

SUCCESSIVE INTERFERENCE CANCELLATION

We have so far considered joint successive interference can-cellation considering two users at a time from an ordered setof user signals . We can extend this con-cept and derive a family of maximal asymptotic conditional ef-ficiency successive interference cancellers where, at each stage,we consider more than two users at a time. In the previous sec-tion, we have shown that the increase in run-time complexityby choosing MACE-JSIC over MACE-SIC is compensated bya corresponding increase in asymptotic conditional efficiency.In this section, we derive a design measure that quantifies thistradeoff between complexity and performance by providing abound over the performance gain achieved by considering moreuser signals in the MACE-JSIC kernel.

At the th stage of successive detection, we consider thesub-constellations, being a design parameter, denoted asand given as, . Let us denote the sub-constellations belonging to that have the th user’s bit set to

1 as . The th user’s bit is decoded based on the jointinformation between users with signals .The asymptotic conditional efficiency for the th user for theMACE-JSIC( ) detector, denoted as will be given by theminimum of the minimum distance between the convex hulls ofany two sub-constellations belonging to andrespectively, normalized by the energy, of the thuser. Since sub-constellations belonging to are a subsetof , i.e., as increases, we partition each sub-con-stellation into smaller and smaller sub-constellations. Since theminimum distance between the subsets of convex sets is always



greater than the minimum distance between the convex sets, wemust have

(43)

However, the run-time complexity of the MACE-JSIC( ) de-tector will have inner products, i.e., it increases expo-nentially with . Also, the design complexity of the MACE-JSIC( ) detector will be exponential in and . Therefore,the gain in as increases is offset by an exponentialincrease in complexity. This motivates us to derive bounds on

and determine a quantitative measure to compare theperformance of the MACE-JSIC( ) detectors as increases.Let us start with the basic MACE-JSIC(2) detector and com-pare its performance with respect to the MACE-SIC, i.e., theMACE-JSIC(1) detector.

Let us denote , whereis the energy of the th user. To compare between and

it is sufficient to compare and . Letbe the minimum distance from the origin to the convex hull ofthe sub-constellation . Therefore

(44)

Now, is the solution to the convex optimization

(45)

Let denote the minimum distance betweenthe convex hulls of the sub-constellations and

. Similarly, we define ,and as the minimum distance between the corre-sponding sub-constellations. For linear systems, we must have

from symmetry considerations.Therefore, we must have

(46)Let . For linearsystems, is the minimum distance from the origin to theconvex hull of the sub-constellation

that has , i.e.,

(47)

Also, by construction, we must have . Now,let be the set that provides the

solution to (45). Let us denote ,

, . Note that without loss of gener-ality, we have assumed . Therefore, we havethat

(48)

where , , and. Therefore, we obtain

(49)

Combining the above inequality with (44) and (43) we have thefollowing inequality:

(50)

Similarly, we can set up the sequenceand arrive at

the relation:

(51)

which leads to the relationship

(52)

Thus the ratio can be used as aquantitative measure of the performance gain and as a metric todetermine whether or not it is worth the increase in design orruntime complexity by further increasing .

VIII. SIMULATION RESULTS

In this section, we present numerical evidence supportingour theoretical assertions in previous sections. All user signalsrandom Gaussian sequences of length , and all bit streamswere random BPSK sequences. The performance of the JSIC,MACE-SIC and MACE-JSIC detectors have been tested againstthe conventional SIC detector, and the performance of the MLdetector has been provided as a lower bound. Numerical simu-lations were run on several multiuser systems and the bit-error-rate (BER) of each user was averaged over the user signals andbit streams. Figs. 9 and 10 show the average BER (i.e., the biterror rate averaged over all users) for a random ten-user systemwith and respectively. Fig. 11 shows thejoint error rate (JER) respectively in a random ten-user systemwith . We observe that the MACE-SIC and MACE-JSIC consistently outperform the conventional SIC and JSICdetectors and the performance of MACE-JSIC is generally su-perior to that of MACE-SIC, though at high SNR their per-formance is quite close. We also observe that on average the



Fig. 9. Single-user BER averaged over all users in a representative ten-user system with N = 64.

Fig. 10. Single-user BER averaged over all users in a representative ten-usersystem with N = 128.

conventional SIC and JSIC detectors are not near-far resistant,but the MACE-SIC and MACE-JSIC detectors show robustnessagainst multiuser interference since the conditional multiuserefficiency has been optimized at each stage. Fig. 12 shows thesingle user performance for a particular over-loaded four-usersystem, i.e., systems where the number of users, , is greaterthan the number of dimensions , and therefore, the user sig-nals form a linearly dependent set. Fig. 13 also shows simu-lations in which the SNR is held fixed at 15 dB and the BER ofthe proposed detectors are observed over an increasing numberof users.

IX. CONCLUSION

We have introduced joint successive interference cancella-tion as an improvement over conventional successive interfer-ence cancellation. We also presented an approach to successiveinterference cancellation that exploits the structure of the mul-tiuser signal constellation to annul the effect of the closest in-terferer in an ordered set of users. Defining the conditional ef-ficiency of a successive interference canceller, we determinedthe SIC and JSIC detectors with the highest asymptotic con-ditional efficiency for a given ordered set of users. We have

Fig. 11. Joint error rate (JER) in a representative ten-user system with N =

128.

Fig. 12. Single-user BER in a representative four-user system with N = 2.

also derived the maximum asymptotic conditional efficiencyordering of user signals that achieves the highest asymptoticconditional efficiency at each stage of successive interference



Fig. 13. BER comparison for SIC, JSIC, MACE-SIC, MACE-JSIC and MLdetectors for SNR = 15 dB and N = 64 plotted against number of users.

cancellation. The performance of the MACE-JSIC detector isshown to be strictly higher than that of the MACE-SIC detector.Based on this principle, we also proposed a family of joint suc-cessive interference cancellers that achieve higher performancewith an increase in complexity. By quantifying the tradeoff be-tween detector complexity and performance gain, in terms ofthe signal constellation structure, higher performance gains inconditional probability of error can be achieved at high SNR.The successive interference cancellation techniques introducedin this paper assume perfect knowledge of the signal constel-lation, and are practical for implementation in stable multiusersystems with good power control, e.g., in satellite communica-tions. However, in dynamic multiuser systems the signal con-stellation may be continuously changing due to mobility withabrupt changes as users enter or leave the system. The user sig-nals also may not be perfectly synchronized or may change overtime due to fading or imperfect power control. The high de-sign complexity of MACE detection can also be significantlyreduced for multiuser systems with reasonable restrictions onthe signal constellation, e.g., when the user signals are linearlyindependent.

REFERENCES

[1] S. Verdú, Multiuser Detection. Cambridge, U.K.: Cambridge Univ.Press, 1998.

[2] R. Lupas and S. Verdú, “Linear multiuser detectors for synchronouscode-division multiple-access channels,” IEEE Trans. Inf. Theory, vol.35, no. 1, Jan. 1989.

[3] U. Madhow and M. Honig, “Mmse interference suppression for direct-sequence spread spectrum CDMA,” IEEE Trans. Commun., vol. 42, pp.3178–3188, Dec. 1994.

[4] H. Poor and S. Verdú, “Probability of error in MMSE multiuser detec-tion,” IEEE Trans. Inf. Theory, vol. 43, pp. 858–871, May 1997.

[5] E. Dahlman and K. Jamal, “Multi-stage serial interference cancella-tion for DS-CDMA,” presented at the IEEE 46th Vehicular TechnologyConf., Atlanta, GA, Apr. 28–May 1, 1996.

[6] A. Johansson and A. Svensson, “Successive interference cancellationschemes in multi-rate DS/CDMA systems,” in Proc. 5th WINLABWorkshop on 3rd Generation Wireless Information Networks, 1995,vol. 3, pp. 155–174.

[7] A. Johansson and A. Svensson, “Multi-stage interference cancellationin multi-rate DS/CDMA systems,” in Proc. 6th IEEE Int. Symp. Per-sonal, Indoor and Mobile Radio Communications, Sep. 1995, vol. 3,pp. 965–969.

[8] A. Duel-Hallen, “Decorrelating decision feedback multiuser detectorfor synchronous code-division multiple-access channel,” IEEE Trans.Commun., vol. 41, pp. 285–290, Feb. 1993.

[9] A. Duel-Hallen, J. Holtzman, and Z. Zvonar, “Multiuser detection forCDMA systems,” IEEE Trans. Commun., vol. 43, no. 2/3/4, p. 4658,Apr. 1995.

[10] A. Duel-Hallen, “A family of decision feedback multiuser detectors forasynchronous code-division multiple-access channels,” IEEE Trans.Commun., vol. 43, no. 2/3/4, pp. 421–434, 1995.

[11] D. Divsalar, M. Simon, and D. Raphaeli, “Improved parallel interfer-ence cancellation for CDMA,” IEEE Trans. Commun., vol. 46, no. 2,pp. 258–268, Feb. 1998.

[12] M. Varanasi and B. Aazhang, “Near-optimum detection in synchronouscode-division multiple access systems,” IEEE Trans. Commun., vol. 39,no. 5, May 1991.

[13] H. Sari, F. Vanhaverbeke, and M. Moeneclaey, “Combined TDMA/OCDMA with iterative multistage detection,” in Proc. IEEE Int. Conf.Communications, New Orleans, LA, Jun. 2000, vol. 2, pp. 899–903.

[14] G. Barriac and U. Madhow, “PASIC: A new paradigm for low-com-plexity multiuser detection,” presented at the CISS, Baltimore, MD,Mar. 21–23, 2001.

[15] S. Verdú, “Multiuser detection,” in Advances in Statistical Signal Pro-cessing: Signal Detection, H. Poor and J. Thomas, Eds. Greenwich,CT: JAI Press, 1993, pp. 369–410.

[16] S. Moshavi, “Multiuser detection for DS-CDMA communications,”IEEE Commun. Mag., pp. 124–136, May 1996.

[17] Z. Xie, R. Short, and C. Rushforth, “A family of suboptimum detectorsfor coherent multiuser communications,” IEEE J. Sel. Areas Commun.,vol. 8, pp. 683–690, May 1990.

[18] E. Fain and M. Varanasi, “Cyclic decision feedback multiuser sequencedetection,” in Proc. 32nd Annu. Allerton Conf. Communication, Con-trol, and Computing, Oct. 1994, vol. 3, pp. 372–381.

[19] M. Varanasi, “Group detection for synchronous Gaussian code-divi-sion multiple-access channels,” IEEE Trans. Inf. Theory, vol. 41, pp.1083–1096, Jul. 1995.

[20] Y. Sun, “Local maximum likelihood multiuser detection,” in Proc.CISS, Baltimore, MD, Mar. 21–23, 2001, pp. 7–12.

[21] S. Buzzi, M. Lops, and G. Ricci, “A new group detection strategy forDS-CDMA systems,” in Proc. IEEE Int. Symp. Information Theory,Sorrento, Italy, Jun. 2000, p. 357.

[22] S. Buzzi, M. Lops, A. Pauciullo, and G. Ricci, “Group detectors forDS-CDMA systems with multipath fading channels,” in Proc. AllertonConf. Communications, Control, and Computing, Monticello, IL, Oct.2000, pp. 816–825.

[23] J. Luo, K. Pattipati, P. Willet, and G. Levchuk, “Optimal grouping algo-rithm for a group detection feedback detector in synchronous CDMAcommunications,” IEEE Trans. Commun., vol. 51, pp. 341–346, Mar.2003.

[24] M. Varanasi and B. Aazhang, “Near-optimum demodulation for co-herent communications in asynchronous Gaussian CDMA channels,”in Proc. CISS, Mar. 1988, pp. 832–839.

[25] M. Varanasi and B. Aazhang, “Probability of error comparison of linearand iterative multiuser detectors,” in Proc. ComCon’88, LA, Oct. 1988.

[26] M. Varanasi and B. Aazhang, “Multistage detection for code-divisionmultiple access communications,” IEEE Trans. Inf. Theory, vol. 38, pp.509–519, Apr. 1990.

[27] R. Kohno, H. Imai, and M. Hatori, “Cancellation techniques ofco-channel interference in asynchronous spread spectrum multipleaccess systems,” Electron. Commun. Jpn. (English Version), vol. 66,no. 5, pp. 20–29, May 1983.

[28] C. Hegarty and B. Vojcic, “Two-stage multiuser detection for non-co-herent CDMA,” in Proc. 33rd Allerton Conf. Communication, Controland Computing, Oct. 1995, pp. 1063–1072.

[29] V. Ghazi-Moghadam, L. Nelson, and M. Kaveh, “Parallel interferencecancellation for CDMA systems,” in Proc. 33rd Allerton Conf. Com-munication, Control and Computing, Monticello, IL, Oct. 1995, pp.216–224.

[30] Z. Shi, W. Du, and P. Driessen, “A new multistage detector for syn-chronous CDMA communications,” IEEE Trans. Commun., vol. 44,pp. 538–541, May 1996.

[31] R. Buehrer and B. Woerner, “Analysis of adaptive multistage interfer-ence cancellation for CDMA using an improved Gaussian approxima-tion,” IEEE Trans. Commun., vol. 44, pp. 1308–1321, Oct. 1996.



[32] X. Zhang and D. Brady, “Asymptotic multiuser efficiency for deci-sion-directed multiuser detection,” IEEE Trans. Inf. Theory, vol. 44,pp. 502–515, Mar. 1998.

[33] J. Proakis, Digital Communications, 3rd ed. New York: McGrawHill, 1995.

[34] R. Clarke, “A statistical theory of mobile-radio reception,” Bell Syst.Tech. J., vol. 47, pp. 957–1000, Jul.–Aug. 1968.

[35] W. Jakes, Microwave Mobile Communications. New York: Wiley,1974.

[36] C. Loo, “A statistical model of land-mobile satellite link,” IEEE Trans.Veh. Technol., vol. 34, no. 3, pp. 122–127, Aug. 1985.

[37] B. Sklar, “Rayleigh fading channels in mobile digital communicationssystems–Part II: Mitigation,” IEEE Comm. Mag., vol. 35, pp. 102–112,Jul. 1997.

[38] G. Turin, “Introduction to spread-spectrum antimultipath techniquesand their application to urban digital radio,” Proc. IEEE, pp. 328–353,Mar. 1980.

[39] R. Mowbray, P. Grant, and R. Pringle, “New antimultipath techniquefor spread spectrum receivers,” Electron. Lett., vol. 29, no. 5, pp.456–457, Aug. 1993.

[40] T. Zemen, J. Wehinger, C. F. Mecklenbrauker, and R. R. Muller, “Iter-ative detection and channel estimation for MC-CDMA,” in Proc. IEEEInt. Conf. Communications (ICC), May 2003, vol. 5, pp. 3462–3466.

[41] A. Paulraj and C. Papadias, “Space-time processing for wireless com-munications,” IEEE Signal Process. Mag., vol. 14, pp. 49–83, Nov.1997.

[42] S. Stein, “Fading channel issues in system engineering,” IEEE J. Sel.Areas Commun., vol. 5, pp. 68–89, Feb. 1987.

[43] B. Sklar, “Rayleigh fading channels in mobile digital communicationssystems—Part I: Characterization,” IEEE Comm. Mag., vol. 35, pp.136–146, Sep. 1997.

[44] Y. Sanada, A. Kajiwara, and M. Nakagawa, “Adaptive rake receiverfor mobile communications,” IEICE Trans. Comm., vol. E76-B, pp.1002–1007, Aug. 1993.

[45] E. Lee and D. Messerschmitt, Digital Communication. Norwell, MA:Kluwer, 1994.

[46] A. Kocian and B. H. Fleury, “Em-based joint data detection andchannel estimation of DS-CDMA signals,” IEEE Trans. Commun.,vol. 51, pp. 1709–1720, Oct. 2003.

[47] A. Sen Gupta and A. Singer, “Linear complexity multiuser detectorusing joint successive interference cancellation,” in Asilomar Conf. Sig-nals, Systems and Computers, Pacific Grove, CA, Nov. 9–12, 2003.

[48] J. Bucklew, Large Deviation Techniques in Decision, Simulation andEstimation. New York: Wiley, 1990.

[49] J. Nelson, “Mitigating the effects of intersymbol interference:Algorithms and analysis,” Ph.D. dissertation, Univ. Illinois at Ur-bana-Champaign, Urbana-Champaign, Oct. 2005.

[50] D. G. Luenberger, Optimization by Vector Space Methods. NewYork: Wiley, 1988.

[51] A. Sen Gupta, “A Structural Approach to Multi-User Detection,” Ph.D.dissertation, Univ. Illinois at Urbana-Champaign, Urbana-Champaign,Dec. 2006.

Ananya Sen Gupta was born in Kolkata, India, in1976. She received the B.E. degree in electronics andtelecommunication engineering from Jadavpur Uni-versity, India, in 1994 and the M.S. and Ph.D. de-grees, both in electrical engineering, from the Uni-versity of Illinois at Urbana-Champaign, in 2001 and2006, respectively.

Currently she works at Microsoft, Redmond, WA,as a Program Manager for Office Shared Services.

Dr. Sen Gupta is a member of Tau Beta Pi. Shewas awarded the Harriett and Robert Perry Fellow-

ship and the Sundaram Seshu Fellowship from the Department of Electricaland Computer Engineering, University of Illinois at Urbana-Champaign, in aca-demic years 2004-2005 and 2005-2006, respectively. She was also a recipientof the Jagadis Bose National Science Talent Search (JBNSTS) National Awardof India, presented for academic excellence for the years 1994 to 1998.

Andrew Singer was born in Akron, OH, in 1967. Hereceived the S.B., S.M., and Ph.D. degrees from theMassachusetts Institute of Technology, Cambridge,in 1990, 1992, and 1996, respectively, all in electricalengineering and computer science.

Since 1998, he has been on the faculty of theDepartment of Electrical and Computer Engineering(ECE) at the University of Illinois at Urbana-Cham-paign, where he is currently an Associate Professorin the ECE Department and a Research AssociateProfessor in the Coordinated Science Laboratory.

During the academic year 1996, he was a Postdoctoral Research Affiliate in theResearch Laboratory of Electronics at MIT. From 1996 to 1998, he was a Re-search Scientist at Sanders, a Lockheed Martin Company, in Manchester, NH.His research interests include statistical signal processing and communication,universal prediction and data compression, and machine learning.

Dr. Singer is a member of Eta Kappa Nu and Tau Beta Pi and the MIT Ed-ucational Council. He was a Hughes Aircraft Masters Fellow and the recipientof the Harold L. Hazen Memorial Award for Excellence in Teaching in 1991. In2000, he received the National Science Foundation CAREER Award, in 2001 hereceived the Xerox Faculty Research Award, and in 2002 he was named a WillettFaculty Scholar. He serves as an Associate Editor for the IEEE TRANSACTIONS

ON SIGNAL PROCESSING.


5716 ieee transactions on signal processing, vol. 55,...

Documents