asymptotic multiuser efficiencies for decision-directed multiuser detectors

502 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 44, NO. 2, MARCH 1998

Asymptotic Multiuser Efficiencies forDecision-Directed Multiuser Detectors

Xuming Zhang,Member, IEEE, and David Brady,Member, IEEE

Abstract—The asymptotic multiuser efficiencies (AME’s) arederived for various classes of decision-directed multiuser de-tectors, including multistage detectors, and decision-feedbackdetectors. Novel classes of soft-decision multistage detectors areproposed and analyzed in this paper. Each class is specifiedin part by a soft-decision nonlinearity, such as a symmetricquantizer or a linear clipper. Closed-form expressions for two-user AME’s are derived for soft-decision two-stage detectorsand can be used as a design criterion to optimize the soft-decision nonlinearities. For a special case of two synchronoususers, the soft-decision two-stage detector using an optimizedlinear clipper with either conventional or decorrelated tentativedecisions is shown to achieve optimum AME. Upper and lowerbounds on the AME are obtained for decision-feedback detectorsusing either conventional or decorrelated tentative decisions. Itis demonstrated that decision-directed multiuser detectors withconventional tentative decisions have low near–far resistancecompared to those with decorrelated tentative decisions.

Index Terms—Asymptotic multiuser efficiency, code-divisionmultiple access, decision-feedback detection.

I. INTRODUCTION

I N the modest-to-high signal-to-thermal-noise power ratio(SNR) regime, the symbol error rate of most multiuser

receivers exhibits a constant exponential rate of decay withthe inverse noise level. This behavior mimics the symbolerror rate of optimal demodulation for equally-likely, binaryantipodal signaling in the single-user white Gaussian noisechannel which is bounded by

SNR SNR

where SNR is the signal-to-noise ratio of the isolated user.1

Since this upper bound is exponentially tight, the performanceof a given multiuser detector may be compared to an optimal,single-user scenario via an error rate exponent

SNR(1)

Manuscript received July 21, 1993; revised June 4, 1997. The material inthis paper was presented in part at the 1993 International Symposium onInformation Theory, San Antonio, TX, and in part at the 1993 31st AllertonConference on Control, Communication and Computing, Monticello, IL.

X. Zhang was with the Department of Electrical and Computer Engineering,Northeastern University, Boston, MA 02115 USA. He is now with RockwellInternational, Newport Beach, CA 92660 USA.

D. Brady is with the Department of Electrical and Computer Engineering,Northeastern University, Boston, MA 0211 USA..

Publisher Item Identifier S 0018-9448(98)01631-9.1

Q(x) =1

x

(1=p2�) exp [�(t2=2)]dt:

where is the symbol error rate of the particulardemodulation scheme for theth user and is the noisepower level.2 From this expression, , the AME of user ,may also be interpreted as an attenuation of the received energyper symbol due to interference suppression for the particularmultiuser receiver. If , for example, thenthe asymptotic multiuser efficiency is zero. Conversely, apositive asymptotic multiuser efficiency implies that the errorprobability of the multiuser detector decays exponentially fastwith the decreasing noise level. implies that the symbolerror rate has the same logarithmic slope as that of optimaldemodulation for isolated transmission.

Several important features of AME have been established.First, the AME has the relative advantage of analyticaltractability and exhibits simple, closed forms for severalchannels and several multiuser receivers of interest [1],[2]. Further, the AME determines the performance of anydetection algorithm in the presence of strong interference,i.e., the near–far resistance[1], [2]. Specifically, a detector isnear–far resistant if remains positive for all combinationsof interfering energies, implying that a near–far resistantdetector exhibits an exponential rate of decay (versus SNR)of the symbol error probability regardless of the strengths ofinterfering signals. Lastly, the AME may be used as a criterionfor designing multiuser detectors, since the ordering of theAME’s of two detection schemes determines the ordering ofthe corresponding symbol error probabilities for sufficientlylow thermal noise levels via (1).

Many multiuser detectors have been proposed and analyzed[3], including decision-directed receivers which reconstruct theinterference for each symbol via decisions fed back throughtime [4]–[6] or through stages of processing [7]–[9]. Thedecision statistics involved in these demodulation schemesare nonlinear functions of the sufficient statistics. Since thecomputational complexity is linear in the number of users,these detectors retain the computational advantage of lineardetectors.

It is well known that error propagation usually plaguesdecision-directed multiuser detectors. An important problemfor decision-directed multiuser detectors is performance evalu-ation that accurately reflects this error propagation. In previouswork, the error propagation problem has been addressed byMonte Carlo simulations or ignored via the assumption ofperfect cancellation [4]–[6].

2This expression defines the asymptotic multiuser efficiency (AME), asimple yet accurate performance measure of multiuser detectors, originallyintroduced in [1].

0018–9448/98$10.00 1998 IEEE

ZHANG AND BRADY: ASYMPTOTIC MULTIUSER EFFICIENCIES FOR DECISION-DIRECTED MULTIUSER DETECTORS 503

In this work, a unified approach for the derivation of theAME’s for a variety of decision-directed multiuser detectors ispresented. One of the important conclusions from this analysisis that decision-directed detectors with decorrelated tentativedecisions yield higher AME than those with conventionaltentative decisions and that hard decision-directed detectors[4]–[8] exhibit poor near–far resistance. The degradation isprimarily due to incorrect tentative decisions, which inducea doubling of the corresponding interference for antipodalsignaling. Motivated by this observation, we have proposed in[9] classes of soft-decision-directed multiuser detectors whichreduce the effects of error propagation while retaining thesame demodulation complexity as their hard-decision-directedcounterparts. It will be shown here that for a special case oftwo synchronous users, soft-decision two-stage detection withan optimized nonlinearity achieves the optimum AME usingeither conventional tentative decisions or decorrelated tentativedecisions. For the case of two asynchronous users, it is shownthat in strong contrast to hard-decision detectors, the soft-decision two-stage detectors with optimized nonlinearities arenear–far resistant. It is also shown that the AME improvementof multistage detection by increasing the number of stages islimited. Finally, a set of decision-feedback detectors whichuse final decisions to cancel the causal part of interferenceare also analyzed and bounds on AME for these detectors arepresented.

The rest of the paper is organized as follows. In Section IIthe code-division multiple-access (CDMA) system model anddecision-directed multiuser detectors are introduced. Classesof soft-decision nonlinearities and tentative decision strategiesare specified. In Section III, we derive and illustrate the AME’sachieved by soft-decision multistage detectors. The optimalityof these nonlinearities is also examined. In Section IV an upperbound and a lower bound on the AME for decision-feedbackdetectors are derived. Conclusions are given in Section V. Allproofs are relegated to an appendix for ease of exposition.

II. PRELIMINARIES

A. System Model and Notation

Consider a set of unit-energy waveforms

which are assigned to asynchronous and co-channel binaryphase-shift keying (BPSK) transmitters in an additive whiteGaussian multiple-access channel. Let the data symbols bearranged in the vector sequence

the normalized waveforms in

and let the received baseband equivalent symbol waveforms

be arranged similarly in . With this notation, the receivedsignal may be written as

(2)

where

and the additive noise is modeled as white Gaussiannoise with spectral level . In this equation , , andrepresent the relative delay, carrier phase, and received symbolenergy for user , respectively. We shall assume that accurateestimates of the constant amplitude, phase, and relative delayare possible for all users. The collision of theasynchronoussignals spans symbols, and without loss ofgenerality we order the users according to increasing delays,

. Further, let denote all thefuture symbols which interfere with directly, i.e.,

shall denote tentative decisions for these quantities, andshall denote final decisions.

The sufficient statistics for the detection ofare the sampled output sequence of filters

matched to . The real part of the matched-filter bankoutput sequence3 has the form

(3)

where is the real signal cross-correlation matrix4

with the th element denoted by . Due to the length-support of for all . The real noise

vector in (3) is Gaussian with zero mean vector and co-variance matrix .

B. Decision-Directed Multiuser Detectors

To motivate the decision-directed multiuser detectors,we shall decompose with respect to the influenceof the desired signal and the interfering symbols

(4)

where and denote the multiple-access interference(MAI) vectors due to past symbols and future symbols, re-spectively, the th components of which are represented by

3The imaginary part of the matched-filter output will be removed for furtherdemodulation. This may be done without a loss of performance for decision-directed detection with conventional tentative decisions. A small performanceloss occurs in decorrelated tentative decisions at low SNR levels.

4The superscriptH denotes complex conjugate of a vector or a matrix.


where for with , andfor .

In this work we focus on multistage detection [7], [8]and decision-feedback detection [4]–[6], which use tentativedata decisions to construct MAI estimates. These estimatesare subtracted from the final predecision variable. To ex-plain these decision-directed detectors concisely, we define

as the tentative decision statistics for and in whatfollows we consider the demodulation of without lossof generality. We are particularly interested in two strate-gies: conventional (matched-filter outputs, ) anddecorrelated (decorrelator outputs [2], ) ten-tative decisions. In either case, the tentative decision variablesequence may be described as

, where is the transmitted data vector,is a known matrix, and is azero-mean Gaussian vector with known covariance matrix.We will define . In order tocompletely specify the symbol error rate for , we shall

need the distribution of ,that is, the joint distribution of noise components involvedeither in the final decision or in the tentative decisions

[9].We shall focus on two classes of nonlinearities which

admit tractable and explicit AME’s. The first class consistsof the mid-step, symmetric quantizers. We shall describe thesenonlinearities as

where for each

For this class of nonlinearities, it is convenient to denotethe range of as and the range of the tentative

decision vector as . Forsmall , practical detectors may be implemented, whilelarger permits an accurate estimate of most memorylessnonlinearities, including theth-law limiter ( odd)

(5)

where is the saturation threshold for user, as wellas the arctangent function .To approximate these nonlinearities, we can calculate all

’s according to where we set. Note that for each nonlinearity and a fixed

there is only one parameter which needs to beoptimized. Also, we shall analyze the performance of softdecision-directed multiuser detectors utilizing linear clippernonlinearities ( in (5)) [9].

To conclude this section, we summarize the decision-directed multiuserdetectors as

conventional preprocessingdecorrelated preprocessing [2]

two-stage detection [7], [8]

decision-feedback detection [4], [5].

The structures of these two classes of decision-directed de-tectors are easily illustrated for a special case of two asyn-chronous users as given in Fig. 1(a) and (b), respectively.

III. AME FOR MULTISTAGE DETECTION

A. Derivation of AME

We begin with a detailed derivation of AME for soft-decision two-stage detection using a symmetric quantizer. Inwhat follows we focus on error events for the demodulation of

, and to permit a unified presentation we will introducea scalar if conventional (matched-filter) preprocessingis used and if decorrelated preprocessing is used. Webegin by transforming the noise vector to a Gaussianvector with independent and identically distributed (i.i.d.) com-ponents. Consider a square-root factorization of the covariancematrix of , and let the th row of be denotedby . Let be a Gaussian vector with zero mean vectorand covariance matrix such that has the samedistribution as . Consequently, the probability of the errorevent given may be expressed as

(6)

where is the indicator function of the event ,denotes the unconditional expectation with respect to thedistribution of , and

is the group of transmitted symbols5 that coincide temporallywith those in but do not belong to

. In (6) the event is specified by

(7)

5Provided that the cross correlationsf�ijg are known, one can easily locateTTT 1(0) with respect to the worst case error events givenBBB1(l) and ~BBB1(l) forthe case of conventional tentative decisions.


(a)

(b)

Fig. 1. Block diagrams of decision-directed multiuser detectors for a two-user asynchronous system. (a) Soft-decision two-stage detector and (b)decision-feedback detector.

where denotes the standard inner product of the vectorsand , and . In (7),

denotes an estimate of using tentative decisions.Since the probability of error can be written as a weighted sumof the -dimensional Gaussian cumulative distributionfunctions, we may find the exponential rate of each term, andtherefore the AME, using the following result [10].

Lemma 1: Let denote a zero-mean Gaussian vector withcovariance matrix , and let denote an event ofinterest, where does not depend on . Then

In the -user case, the constraint region (a polytope) isthe intersection of a positive half-space (hard final decision)and space segments determined by two combinedinequalities (soft tentative decisions). Each segment is de-termined by a vector , and the constantsand . The vectors form alinearly independent set if the random components of arestatistically independent (the dependent case is even easier tohandle), and is completely specified by the method of tentativedecisions. Closed-form solutions for the AME will follow froma well-known characterization of this constrained minimumdistance problem given below [11].

Lemma 2. Kuhn–Tucker Theorem:Let

where

and

are constant. The unique solution to

is given by

(8)

if or if

(9)

if (10)

Note that if the region contains the origin, then theerror rate exponent (AME) is zero. Otherwise, the solutionvector lies on one of the boundaries of and the prob-lem reduces to the selection from a finite set of possiblesolutions. A particular finite set of candidate solutions given

is the set of closest points(to the origin) of faces6 of the convex polytope and we have

sets to search for the AME. Note that the computationalcomplexity of AME for soft-decision two-stage detection is

6There are three kinds of faces, i.e., facet (plane), vertex (point), and edge(line).


exponential in the number of users and in the (common)number of quantizer levels.

We now turn to the derivation of the AME for soft-decisiontwo-stage detection using a linear clipper. Perhaps the simplestexplanation of the performance of this detector is to begin witha partition of the space of tentative decision variables

for

Since each component of or may lie in eitherthe “linear” or “clipped” domain of the respective nonlinearity,the sample space of , , may bepartitioned into regions. Let denote a subsequenceof which index the users whose tentativedecisions are clipped in , and define such that

. For a given ,we define as the rectangle of supportfor which admits the clipped components

and .For fixed subsequences and , the final decision

variable for may be decomposed as shown in (11)at the bottom of this page, where , for

and if conventionalpreprocessing is used.

When for fixed then the finaldecision statistic has a similar form to that using quantiz-ers. Important modifications to the final decision statistic whichare reflected in (11) include: a reduction of the desired signalamplitude (first term, first line), a known change in the noisecovariance matrix (second term, first line), the introductionof other interfering symbols (lines 4–8), and a change in the

amplitude of each interfering signal (line 2). If the probabilityof error for the soft-decision detector using a linear-clipper isexpressed as

then each term may be computed using the identical techniqueas in (6).

It should be mentioned that the proposed technique canbe generalized to -stage detection with . Thegeneralization does not involve conceptual difficulties andtherefore will not be addressed here. However, a set ofnumerical examples for will be given in Section III-C.

B. Two-User AME’s

In order to visualize the basic principles of soft-decisiontwo-stage detection, we shall concentrate our attention ona special case of two users. Although general closed-formexpressions do not exist for , we shall show that aset of explicit formulas for two-user AME are possible. Webegin with the fact that for and for binary antipodalmodulation the error probabilities and AME’s are independentof the signs of cross correlations, and we may assume that

and

if (11)


Fig. 2. AME’s of two asynchronous users versus relative energyw2=w1 (�12 = 0:2; �21 = 0:6): solid line—hard-decision two-stage detector withconventional tentative decisions; dashed–dotted line—soft-decision two-stage detector with an optimized dead-zone limiter and conventional tentative decisions;dashed line—conventional detector; dotted line—soft-decision two-stage detector with an optimized dead-zone limiter and decorrelated tentative decisions.

without loss of generality. Given and the previousassumption, the dominating error event is indexed by

. For clarity, we define

and

We now present a set of closed-form expressions of thetwo-user AME’s. To contrast the weakness of hard-decisiontwo-stage detection with conventional preprocessing, we firstpresent the following results.

Proposition 1: Define

(12)

(13)

(14)

where , and (15) and (16) at the bot-tom of this page. Then, the AME for hard-decision two-stage detection with conventional tentative decisions is

.An illustration of this result is given in Fig. 2 (solid line).

At this point, two remarks are in order. First, it should be

if

if (15)

if

if

(16)


emphasized that if , the hard-decisiontwo-stage detector employing conventional preprocessing isnear–far limited because for

. Second, note that the conventional detectoroutperforms two-stage detection for small.

These two drawbacks can be eliminated by using softtentative decisions. To see this, let us look at the simplest,nontrivial symmetric quantizer: a dead-zone limiter

, and . For this device, there isa single parameter which determines the AME for user 1.We are interested in maximizing the AME for a given relativeamplitude by selecting such that

The near–far resistance of the optimized strategy is found byminimizing over the relative symbol amplitudes

Proposition 2: For , the soft-decision two-stagedetector using conventional preprocessing and an optimizeddead-zone limiter: a) is near–far resistant and b) it exhibitsuniformly larger AME versus relative energy than the conven-tional detector.

The importance of this result follows from a previous ob-servation that neither conventional detection nor hard-decisiontwo-stage detection is near–far resistant as .The AME improvement of this detector over the conventionaldetector (dashed line) and two-stage hard-decision detectoris also shown in Fig. 2 (dashed–dotted line). It should bementioned that the conclusion here holds for any soft-decisiontwo-stage detector with an optimized symmetric quantizerusing either conventional or decorrelated tentative decisions.

Next, we demonstrate that the AME for soft-decision two-stage detection with symmetric quantizer using decorrelated

preprocessing is superior to its conventional counterpart. Thefollowing formulas are useful for computing the AME for thistype of soft-decision detector.

Proposition 3: The AME for soft-decision two-stage de-tection with decorrelated tentative decisions and symmetricquantizer is given by

where, as shown in (17)–(20) at the bottom of this page, and

is the AME of the decorrelator, and

In Fig. 2, we also plot the AME (versus ) for soft-decision two-stage detector using an optimized dead-zonelimiter and decorrelated preprocessing. It is clearly shown thattwo-stage detectors with decorrelated preprocessing uniformlyoutperform their counterparts with conventional preprocessing.

We now illustrate the use of Proposition 3 to compare ath-law limiter and the arctangent function. Fig. 3 shows the

AME’s for soft-decision two-stage detection with decorrelatedpreprocessing and four different nonlinearities correspondingto the best choice of the saturation thresholds for

. The intermediate ’s can be calculated accordingto the method specified in Section II-B. It may be noted fromFig. 3 that an optimized linear clipper outperformsall others in this example. Interestingly, for two synchronoususers the optimum AME of the maximum-likelihood sequence(MLS) detector can be achieved by a soft-decision two-stagedetector with an optimized linear clipper as we formally statein the following proposition.

(17)

(18)

(19)

if

if

(20)


Fig. 3. AME’s of two asynchronous users and soft-decision two-stage detection using a�2; l-optimized, symmetric quantizer withN2 = 20(�12 = 0:6; �21 = 0:2).

Proposition 4: If the optimized saturation threshold fora linear clipper satisfies

for decorrelated preprocessing

(21)

for conventional preprocessing

(22)

then the AME for the soft-decision two-stage detector witha linear clipper in a two-user, synchronous system coincideswith that of the optimum MLS detector.

The expressions (21) and (22) for the optimal saturationthreshold of are quite intuitive. For small , should belarge and thus the soft-decision two-stage detector is equivalentto the conventional detector. For large’s, must be smalland the performance of the soft-decision two-stage detector isclose to that of the hard-decision detector. More interestingly,the optimum in (21) sometimes requires a small negative

. Fig. 4 depicts the typical linear clippers with nonnegativeand negative ’s. However, a simple justification showsthat such a modification of is necessary unless

and generally it does not offer much performanceimprovement. Therefore, we shall restrict ourselves to the caseof nonnegative scalar .

Proposition 5: In a two-user, asynchronous system, thenear–far resistance for soft-decision two-stage detection usingan optimized linear clipper satisfies

for decorrelated preprocessing

(23)

for conventional preprocessing.

(24)

(a)

(b)

Fig. 4. Typical linear clippers with (a)�k � 0 and (b)�k < 0.

Actually, (23) and (24) can also be achieved by detectorswhich employ adaptive amplifiers (without clipping the sig-nals) with gain for both conventional and decorrelatedpreprocessing.

C. Two-User AME for -Stage Detection with

We now examine the possible performance improvement of(hard-decision) multistage detection by increasing the numberof stages [7], [8]. Geometrically, as the worst case constraintregion increases in size, the AME decreases. Or, conversely,as the number of constraints increases, so would the minimumEuclidean distance, and the AME may increase. Based on thisobservation, it is straightforward to verify that the followingstatement is true for all -stage detection with and

.


(a)

(b)

Fig. 5. AME’s of two asynchronous users and hard-decisionM -stage de-tection with M = 1; 2; 3. (a) Conventional tentative decisions and (b)decorrelated tentative decisions(�12 = 0:2; �21 = 0:5).

Proposition 6: The AME for -stage detectionand is uniformly larger than or equal to that of

-stage detection using the same tentative decisiontechnique.

The key to this proposition is to show that for each worstcase constraint region in the error rate expression for the

-stage detector, there exists a superset in the error-rateexpression for -stage detector. However, increasingthe number of constraints does not guarantee an improvementin the performance. It can be shown that for the case oftwo synchronous users the AME of hard-decision-stagedetection saturates at (for even ) and (forodd ) for conventional tentative decisions and at (foreven ) and (for odd ) for decorrelated tentativedecisions. It is observed that the terminating AME is far fromoptimum—it does not converge to the AME of MLS detection.Fig. 5 depicts the typical AME’s for hard-decision -stage

detection (with ) using (a) conventional tentativedecisions and (b) decorrelated tentative decisions. It is alsoobserved that -stage detectors with even do better forweak users while those with odd can do better for strongusers.

IV. AME FOR DECISION-FEEDBACK DETECTORS

In this section, we derive upper and lower bounds onAME for decision-feedback multiuser receivers where the finaldecisions are used to cancel the causal part of the MAI. Weshall focus attention on the demodulation of withoutloss. For this class of detectors, the entire past decisionhistory (final decisions and associated tentative decisions) isrequired to calculate the true AME. Additionally, the coupled(correlated) noise components (seen at matched-filter outputsand preprocessing filter outputs) make the derivation of thetrue AME difficult. However, as we shall show, tight boundson the AME are tractable. In the following we assume thatboth tentative and final decisions are carried out via a hardlimiter.

A. Bounds on the AME

Analogous to the derivation of the AME for two-stagedetection, conditioning on all available transmitted symbolsup to time epoch , , , and the

noise vectorwhere , the final and tentativedecisions are deterministic. Therefore, the AME can becomputed in a similar way to that of two-stage detectionalthough for decision-feedback detection the feasible regionis defined by all past final and tentative decisions.

To derive these bounds, we define and such thatthe tentative decision variables and the final decision variablescan be written as, respectively, and

. Note that each of the other noise termswhich do not appear in but are involved in the tentativedecisions, i.e.,

can be expressed as a linear combination of becausethere are only independent noise terms (recall that

is the transmission horizon.). Further, we define thehalf-horizon final and tentative decision error sequences as,respectively,

and

where and .Similarly, we may transform the colored noise vector

to a Gaussian vector with i.i.d. componentsand expressthe set of the feasible region of the vector given alltentative decision and final decision error sequences as shownat the bottom of the following page, where are thesubsequent representations of , for instance,if conventional tentative decisions are used. Obviously, the


exact calculations of AME might not be feasible because thedimension of may be infinite. However, arbitrarily tightlower and upper bounds can be obtained as follows.

Proposition 7: a) A lower bound on the AME can beobtained by directly deleting part of the inequality constraintsfrom the entire set of constraints. b) An upper bound on theAME can be derived by using any subset of final decision ortentative decision error sequences.

The tightness of the lower bound on the AME is propor-tional to the number of constraints involved in the calculation.For the upper bound, one may select subsets of final decisionerror sequences which have a fixed length nonzero “simple”sequences with all trailing zeros. By “simple” sequences wemean those sequences containing no more thanconsecutive zeros between nonzero elements (see [1], [12] fordetails). The larger the set of “simple” sequences, the tighterthe upper bound.

B. Examples for

It is a common belief that a decision-feedback detectoroutperforms its tentative decision-aided counterpart becauseof the “accuracy” of the final decisions. However, we shallshow that in general it is not true in a near–far situation. Weshall illustrate this result for the case of two asynchronoususers. The first example considers the conventional decision-feedback detector, in which the precursor MAI is canceledusing conventional tentative decisions [4]. Fig. 6(a) showsupper and lower bounds on the AME for different settings ofcross correlations . In this example, we consider theerror propagation up to , i.e., the final decision errorsequences having the form of

In particular, letting gives the upper bound on theAME while setting leads the lower boundon the AME. It is clearly demonstrated that this detector yieldspoor near–far resistance (NFR), because the error propagationintroduces severe performance degradation. Also shown is theAME for the soft-decision two-stage detector with an opti-mized linear clipper and conventional tentative decisions. Incontrast to the two-stage detection, whose AME is independentof the order of cross correlations, the decision-feedback detec-tion favors heavy trailing (postcursor) interference when theinterfering user is weak while favors heavy leading (precursor)interference when the interfering user gets strong.

(a)

(b)

Fig. 6. Bounds on AME of two asynchronous users and decision-feedbackdetection with a different order of cross-correlations. (a) Conventional tenta-tive decisions and (b) decorrelated tentative decisions.

In the second example, we use decorrelated tentative de-cisions for the precursor cancellation [5]. In addition, weconsider the same length of the error propagation as that in thefirst example. It can be seen that the noise vector involved infinal decision and tentative decisions contains four independent


Fig. 7. Bounds on AME of two asynchronous users and the hybrid decision feedback detector with a different order of cross correlations(�12 = 0:6; �21 = 0:2) and (�12 = 0:2; �21 = 0:6).

terms, i.e.,

where is the noise term seen at the second branchof the decorrelator at time and the other noise terms,

(from tentative decision statistics),can be expressed as a function of . They can be derivedby equating the corresponding covariance terms. Fig. 6(b)illustrates upper and lower bounds on the AME for twodifferent settings of cross correlations. For comparison, wealso plot the AME for the soft-decision two-stage detectorusing an optimized linear clipper and decorrelated tentativedecisions. It is observed from Fig. 6 that the AME of thedecision feedback detection with decorrelated tentative deci-sions (Fig. 6(b)) dominates that using conventional tentativedecisions (Fig. 6(a)). Also of interest is to note that a largedifference between upper and lower bound appears when therelative energy of interfering user gets large. If tighter boundsare needed for this region, more constraints have to be involvedin the calculation.

To finish this section, we finally examine a hybrid decisionfeedback detector proposed in [6] in which the precursorMAI intends to be canceled using a partial decision-feedbackdetection scheme. By partial decision feedback, we meanthat only postcursors are compensated for but precursor MAIstands the same. Fig. 7 concludes that this type of detector isnear–far limited for this narrowband application.

V. CONCLUSIONS

The asymptotic multiuser efficiencies for a variety ofdecision-directed multiuser detectors were derived, providinginsight into their performance. The symbol error probabilitiesfor the decision-directed multiuser detectors were derived andshown to be difficult to compute. On the other hand, the

AME’s for several soft-decision two-stage detectors admita simple explicit form and have been used as a designcriterion to optimize the receivers. It has been establishedthat soft-decision multistage detectors, proposed and analyzedin this paper, exhibit substantially higher AME than theirhard-decision counterparts by using a set of appropriatelyoptimized memoryless nonlinearities as tentative decisionfunctions. Moreover, the proposed soft-decision two-stagedetector with an optimized linear clipper and decorrelatedtentative decision statistics yield the highest AME of any othersuboptimum detectors with the same complexity constraints.For the case of two synchronous users, it achieves the AMEof the MLS detector. Another important conclusion fromthe AME comparison is that the decision-directed detectorswith conventional tentative decisions produce low near–farresistance compared to those with decorrelated tentativedecision. Finally, the AME’s for -stage detectors werealso presented for larger asynchronous user populations tocompare some of the previous results.

APPENDIX

Proof of Lemma 1:The probability can beexpressed as

where and is the -dimensional Gaussian density function given by

Let


and . The following inequalities are immediate:

(A.1)The left-hand side term in (A.1) can be written as

LHS

(A.2)

We have used the fact that where isthe -dimensional Dirac–delta function. The right-handside term in (A.1) is given as

RHS

where is a polynomial function of and

Taking the natural logarithm of all sides in (A.1), and thelimit as

Proof of Lemma 2:See [11].Proof of Proposition 1:

a) Consider exactly one tentative decision error, say,and assume . Because

the correct tentative decision resultsin an unrestrictive constraint, the problem reduces tofinding the minimum-norm vector in a two-dimensionalpolytope defined by andwhere

and

(see Fig. 8). As increases from zero, the location ofthe minimum-norm vector progresses as follows: i) onthe facet of ; ii) on the vertex (or the

Fig. 8. Illustration of the constraint region for Proposition 1.

origin if ; and iii) on the facet of. From Fig. 8 the squared distance to the

vertex can be decomposed into

The following rules guarantee the solution in the feasibleregion. disappears when is below and the point

becomes admissible if it is in the feasible region.In addition, if the feasible region includes the origin,then the AME is zero (both and disappear).Therefore, we arrive at (15) where

and

where is defined in (12).b) Consider both and . As

increases, the location of the minimum-norm vectorprogresses as follows: i) on the facet of ;ii) on the edge of (assume

); iii) on the vertex (or on the origin if); and iv) on the edge .

The result (16) follows from these considerations.

Proof of Proposition 2: Part a) is true because the simplechoice can keep theconstraint region away from the origin for all possible.Part b) follows because a dead-zone limiter includes the zerofunction as .

Proof of Proposition 3: Since the noise componentsand are uncorrelated, the contribution to the AME fromthe final decision constraint can be separated and given as

We only need to consider a two-dimensional minimum Eu-clidean distance search problem.


Fig. 9. Illustration of the constraint region for Proposition 3.

a) Consider . The con-straint region is specified by

where

(see Fig. 9). The admissible solution must be on the linesegment , i.e., either on , or on , or on thefacet of (see Fig. 9). Therefore, we have (17).

b) Consider . The result(18) follows because the solution must be on the segment

.c) Consider . Similar to

part b), the result (19) follows because the solution mustbe on the segment .

d) Consider and as-sume . If , then the solution issimilar to those of parts b) and c) and is given in the firsttwo lines of (20). If , it is not difficultto verify that the solution lies on the segment ofand the next two lines in (20) follow. Finally, forthe solution is either on , or on , or the origin. Thiscompletes the proof.

Proof of Proposition 4:

a) Consider decorrelated preprocessing. For (inclipped domain), the corresponding admissible solutioncan be obtained from Proposition 3 (with ) as

(A.3)

For (linear domain), we have the followinginequality constraints:

(A.4)

(A.5)

where are derived from the following covariancematrix:

It can be shown that the minimum-norm solution is onthe facet of . The admissible candidateis given by

(A.6)

The extremal , in (21)7 can be derived from (A.6)by setting the derivative of .Next, we need to check that the value of in (A.6)is admissible, or equivalently

The validity of the above inequalities requires, which is true. Moreover, with

for all . Also, it is not difficult (but somewhat te-dious) to exclude any other admissible solutions. Finally,substituting (21) into (A.6) we have

and

Therefore, we have proved (21).b) The proof of (22) for conventional preprocessing is

exactly same as that of (21). The validity of the optimumAME requires

or, equivalently, whichis true for any . Therefore, is extremal.

Proof of Proposition 5. Sketch:A complete proof is too in-volved to be presented here. For illustration, we shall highlightthe important procedures.

a) Let us focus on decorrelated preprocessing. First, weneed to show that (23) is attainable with . Itcan be accomplished by solving for the AME explicitlywith . For the admissiblesolution satisfies where the inequalityholds for small and the equality is true if is large.Therefore, one simple choice of leads to

. Next, we need to show that . Towardthat end, we again consider the both tentative decisionsin the linear domain, i.e., . It can

7For notational simplicity, the explicit dependency of the optimum�2 on�2 is omitted.


be shown that one constraint (final decision) is activeand its solution is given by

(A.7)

which has a saddle point at

and the corresponding minimax solution is . It isstraightforward to check that the saddle point of (A.7) isadmissible. Therefore, and (23) follows.

b) For the conventional tentative decisions with optimizedlinear clipper, the proof follows in a similar manner.The saddle point reaching (24) is shown to be

Proof of Proposition 6: Consider two asynchronous usersand hard-decision multistage detection. Let be the th-stage decision for user 2 at time. To obtain -stage detectionof , we need two -stage decisions from user 2,viz., and, in turn, we need three

-stage decisions from user 1, viz.,

and so on. If any one of -stage decisions is inerror, then the entire set of constraints for in-cludes at least one subset of constraints forfor some . Due to time invariance of the error probabilities,the AME for is independent of . Therefore, wehave the desired result because for the case where all three

-stage tentative decisions for user 1 are correct (perfectcancellation), the admissible AME for -stage detectionexceeds one (don’t care).

Proof of Proposition 7: Part a) follows from the fact thatthe deletion of constraints (the number of inequalities) fromthe entire set is equivalent to enlarging the constraint region.The larger the constrain region the smaller the minimumEuclidean distance andvice versa. Part b) is obvious becausewe search the minimum among a subset of final decision errorsequences.

ACKNOWLEDGMENT

The authors wish to thank Dr. Z. Zvonar and M. Kocic fortheir helpful suggestions.

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asymptotic multiuser efficiencies for decision-directed multiuser detectors

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