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IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 48, NO. 9, SEPTEMBER 2002 2507

Large-System Performance Analysis of Blind andGroup-Blind Multiuser Receivers

Junshan Zhang, Member, IEEE,and Xiaodong Wang, Member, IEEE

Abstract—We present a large-system performance analysisof blind and group-blind multiuser detection methods. In thesemethods, the receivers are estimated based on the received signalsamples. In particular, we assume binary random spreading,and let the spreading gain , the number of users , and thenumber of received signal samples all go to infinity, whilekeeping the ratios and fixed. We characterize theasymptotic performance of the direct-matrix inversion (DMI)blind linear minimum mean-square error (MMSE) receiver, thesubspace blind linear MMSE receiver, and the group-blind linearhybrid receiver. We first derive the asymptotic average outputsignal-to-interference-plus-noise ratio (SINR) for each of thesereceivers. Our results reveal an interesting “saturation” phe-nomenon: The output SINR of each of these receivers convergesto a finite limit as the signal-to-noise ratio (SNR) of the desireduser increases, which is in stark contrast to the fact that theoutput SINR achieved by the exact linear MMSE receiver canget arbitrarily large. This indicates that the capacity of a wirelesssystem with blind or group-blind multiuser receivers is not onlyinterference-limited, but also estimation-error limited. We thenshow that for both the blind and group-blind receivers, the outputresidual interference has an asymptotic Gaussian distribution,independent of the realizations of the spreading sequences. TheGaussianity indicates that in a large system, the bit-error rate(BER) is related to the SINR simply through the function.

Index Terms—Blind multiuser detection, group-blind multiuserdetection, large-system analysis.

I. INTRODUCTION

I N this paper, we present a large-system performance anal-ysis of blind and group-blind linear multiuser detection tech-

niques for the following canonical discrete-time-user syn-chronous code-division multiple-access (CDMA) channel

(1)

(2)

Manuscript received March 12, 2001; revised March 15, 2002. This work wassupported in part by the U.S. National Science Foundation under Grants CCR-9875314, CCR-9980599, and DMS-0073651. The material in this paper waspresented in part at the 35th CISS’01, The Johns Hopkins University, Baltimore,MD, March 21–23, 2001.

J. Zhang is with the Department of Electrical Engineering, Arizona State Uni-versity, Tempe, AZ 85287 USA.

X. Wang is with the Department of Electrical Engineering, Columbia Univer-sity, New York, NY 10027 USA.

Communicated by V. V. Veeravalli, Associate Editor for Detection andEstimation.

Publisher Item Identifier 10.1109/TIT.2002.801468.

where , , and are the received am-plitude, data bit, and unit-energy spreading signature sequenceof the th user, respectively, and

is the additive white Gaussian noise, withdenoting anidentity matrix. These are collected in vector forms as

and

In a direct-sequence spread-spectrum system with spreadinggain , the signature sequence of theth user is of the form

(3)

A number of recent works [2]–[4], [6], [5], [8], [13], [14],[16], [20], [24], [25], [27], [28], [32], [36], [35], [37] haveanalyzed the asymptotic performance of various CDMAreceivers for systems with random spreading sequences, whenthe number of users grows without bound and the ratio ofthe number of users to the spreading gain is kept fixed. Inparticular, a capacity analysis of large CDMA networks withlinear multiuser receivers is provided in [24], and a fundamentalresult—the Tse–Hanly equation—is obtained, the solutionto which is the output signal-to-interference-plus-noise ratio(SINR) of the linear minimum mean-square error (MMSE)multiuser receiver. Generalization of the results in [24] tomore complex systems with asynchrony, multipath fading, andantenna diversity can be found in [4], [13], [25]. The optimalnear–far resistance for synchronous dual-rate CDMA systemsis studied in [3]. The performance of a reduced-rank linearfiltering receiver using the linear MMSE criterion is evaluatedin [8], and an important conclusion is that the rank needed toachieve a desired SINR does not scale with the system size.In [16], a random matrix model is introduced to describe thespatial and temporal multipath propagation between transmit-ting and receiving antenna arrays, and this model enables thecalculation of SINR for large antenna systems. In [27], a largesystem analysis for a single-class system is carried out to derivethe Shannon-theoretic performance, where all users have thesame received powers and the same rate requirements. Theimpact of fading on the admissibility and network capacity isstudied in [34], [35]. The Gaussianity of the output interferenceof linear MMSE receivers in a large network is established in

0018-9448/02$17.00 © 2002 IEEE

2508 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 48, NO. 9, SEPTEMBER 2002

[37], via exploiting random matrix theory and martingale limittheory.

On the other hand, motivated by the recent development ofblind and group-blind multiuser detection techniques [7], [29],[30], [19], [31], the analytical performance assessment for thesemultiuser detection methods has been developed in [10]–[12].where the effect of estimation error (due to the finite number ofreceived signals, based on which these receivers are estimated)on the receiver performance. These studies on blind receiverperformance analysis (referred to later as BRPA) differs fromthe works on large-system performance analysis (LSPA) men-tioned earlier in the following perspectives.

• LSPA treats the spreading sequences as independent andidentically distributed (i.i.d.) random variables with zeromean and unit variance; whereas in BRPA, the spreadingsequences are considered as deterministic vectors.

• LSPA considers the performance of exact linear multiuserreceivers; whereas BRPA treats the performance of blindand group-blind multiuser receivers estimated from thereceived signals.

• LSPA considers the asymptotic performance when boththe number of users and the spreading gain tend toinfinity; BRPA focuses on the case where bothandare fixed, while considering the performance as a functionof the sample size .

In this paper, we focus on large-system asymptotic perfor-mance analysis for blind and group-blind multiuser detectionmethods. In particular, we assume binary random spreading,and we let , , and go to infinity while keeping the ratios

and fixed. (We note that similar approacheshave been adopted in recent work [32], [33].) We also assumethat , and that is reasonably large. In such a sce-nario, we characterize the asymptotic performance of variousblind and group-blind multiuser receivers. We summarize ourmain findings as follows.

1) As the processing gain increases, the output SINR of eachreceiver considered here converges with probability oneto some deterministic limit, independent of the spreadingsignatures.

2) Based on the limiting SINR of the blind linear MMSEreceivers, we find that as the signal-to-noise ratio (SNR)of the desired user increases, the output SINRachieved by the direct-matrix-inversion (DMI) blindreceiver converges to , and that achieved by the sub-space blind receiver approaches ; this is in sharpcontrast to the fact that the asymptotic SINR achieved bythe exact linear MMSE receiver can get arbitrarily large.This observation reveals a “saturation” phenomenon ofthe SINR achieved by the blind receivers. That is, beyondcertain input SNR, the estimation error dominates andhence determines the output SINR. This indicates thatthe capacity of a wireless system with blind multiuserreceivers is not onlyinterference-limited, but alsoestima-tion-error limited. Furthermore, the effect of estimationerror can be quantified, as shown in Section III.

3) The group-blind linear receiver outperforms the subspaceblind linear receiver because the asymptotic SINR of theformer is strictly higher than that of the latter. Moreover,the capacity of a wireless system with group-blind linearreceivers is also estimation-error limited.

4) We also establish that the output residual interference ofthese receivers is asymptotically Gaussian, for almostevery realization of the spreading signatures. Therefore,the asymptotic bit-error rate (BER) is related to theasymptotic SINR simply through the function. As aside result, we obtain that the estimation error of thesereceivers is asymptotically orthogonal to the spreadingsignature sequences.

The rest of this paper is organized as follows. In Section II,we provide some background material on blind and group-blind linear multiuser receivers, and the analytical performanceresults of these receivers obtained in [10]–[12]. In Section III,we summarize the main results of this paper, namely, thelarge-system performance of the blind and group-blind linearmultiuser receivers, in terms of asymptotic SINR, distributionof residual interference, and BER. In Section IV, we providesome numerical examples to illustrate the theoretical results.The detailed proofs of the main results in this paper can be foundin Section V. Finally, some technical theorems and lemmasthat are useful in proving the main results are collected inthe Appendix.

II. BACKGROUND

Consider the signal model (1). A linear receiver for user 1 is a(deterministic) vector such that is demodulatedaccording to

(4)

Suppose that an estimate of the linear receiver is obtainedfrom the received signals . Denote

(5)

Then, the average SINR at the output of the estimated linearreceiver is given by

SINR

(6)

with

(7)

(8)

ZHANG AND WANG: PERFORMANCE ANALYSIS OF BLIND AND GROUP-BLIND MULTIUSER RECEIVERS 2509

and . For the following estimated linearreceivers, namely, a) DMI blind linear MMSE receiver [7], [26],b) subspace blind linear MMSE receiver [31], and c) subspacegroup-blind linear hybrid receiver [29], it is shown in [10]–[12]that for fixed number of users , fixed spreading gain , andfixed set of spreading sequences , we have

in distribution, as

The limiting error covariance has been found in closed-formfor each of the above receivers, based on which the averageoutput SINR can then be computed according to (6) [11], [12].

A. Blind Linear MMSE Receivers

Consider again the signal model (1). The linear MMSE re-ceiver for user 1 is defined as

(9)

where is some positive constant. Because the linear detectionrule (4) is invariant to a positive scaling, the linear receiver in(9) is invariant to the positive constant . For simplicity, wechoose so that .

In blind multiuser detection, the linear MMSE receiver is esti-mated from the received signals , with the prior knowl-edge of the desired user’s signature sequence, say. Let theeigendecomposition of in (8) be

(10)

where contains the largest eigen-values of ; contains the eigenvec-tors corresponding to the largest eigenvalues in ;

contains the eigenvectorscorresponding to the smallest eigenvalueof . The linearMMSE receiver for user 1 is then given by [7], [31]

(11)

(12)

Corresponding to the above two forms of the linear MMSE re-ceiver, there are two approaches to its blind implementation. Inthe DMI method, the autocorrelation matrix is replaced bythe corresponding sample estimate. That is,

(13)

(14)

In the subspace method, the eigencomponentsand arereplaced by the corresponding eigenvalues and eigenvectors ofthe sample autocorrelation matrix . That is,

(15)

(16)

For large , the average output SINRs of the above two formsof the blind linear MMSE receiver are characterized by the fol-lowing result [11], [12].

Proposition 2.1: For large-signal sample size , the av-erage output SINR of the blind linear MMSE receiver, denotedSINR , is well approximated by

(17)

with

(18)

(19)

(20)

(21)

where

(22)

is the exact MMSE receiver for user, and

DMI blind receiver

subspace blind receiver.(23)

B. Group-Blind Linear Hybrid Receiver

In group-blind multiuser detection, it is assumed that the re-ceiver has the knowledge of the spreading sequences of somebut not all the users. Without loss of generality, assume that thefirst users’ spreading sequences are known to the receiver,whereas those of the rest users’ are unknown. Denote

It is assumed that has full column rank. Denote as theth unit vector in . The group-blind linear hybrid receiver

zero-forces the interference caused by theknown users andsuppress that from the remaining unknown users ac-cording to the MMSE criterion. In particular, such a receiver foruser 1 is given by the solution to the following constrained op-timization problem:

s.t.

(24)

Note that the cost function in (24) parallels that in (9) with. (We have chosen for convenience because the

detection rule (4) is invariant to positive scaling.) The solutionto (24) is given in terms of the signal subspace components as[29]

(25)

Such a linear group-blind multiuser detector can also serve asthe first stage in an iterative nonlinear group-blind multiuser


detector [22]. Replacing and in (25) by and , re-spectively, we obtain the estimated group-blind linear hybrid re-ceiver as

(26)

For convenience, define the partition of the followingmatrix:

(27)

where the dimension of is . Denote

and

(28)

The following result gives the average output SINR of thegroup-blind linear hybrid receiver for given signatures in a fi-nite-size system (i.e., fixed and ) with large sample size[11], [12].

Proposition 2.2: For large-signal sample size, the averageoutput SINR of the estimated group-blind linear hybrid receiver,denoted by SINR , is well approximated by

(29)

with

(30)

(31)

(32)

(33)

where is given by (25).

III. M AIN RESULTS

Throughout this paper, we impose the following regularityconditions on the signal amplitudes.

Condition 1: The empirical distribution function of the se-quence converges weakly to a distribu-tion function as .

Condition 2: The sequence isbounded above and below by positive numbers.

Next, a few words on the above conditions. Condition 1 is astandard assumption in the literature on the large system anal-ysis. Condition 2 is applicable to many practical systems of in-terest, and will be used in proving the almost sure (a.s.) conver-gence of the SINR.

Denote the eigenvalues of the random matrix by(they are random, depending on the realization of

), and the empirical distribution of the eigenvalues by .We know from Theorem A.1 in Appendix A that convergesweakly to some deterministic distribution function , withprobability one [21]. Define

(34)

It has been shown that is the unique positive solution to thefollowing fixed-point equation (cf. [24]):

(35)

We note that the quantity can be interpreted as the SINRachieved by unit received power [24], [37].

A. Asymptotic SINR

Our results on the SINR indicate that in a large system withmany users and large processing gain, the performance depen-dence of the blind and group-blind linear receivers on the sig-nature sequences vanishes. More precisely, as noted before, weassume that and are fixed when .We also assume that is reasonably large so that the SINR ap-proximation given by (17) and (29) hold. Note that this assump-tion is indeed reasonable because in many practical systems thedata packet size is much larger than the processing gain. Ourresults reveal that the limiting output SINR achieved by variousblind and group-blind linear receivers is deterministic, and is in-timately related to defined in (34).

The following theorem gives the average output SINR of theblind linear MMSE receivers in the asymptotic regime (as

).

Theorem 3.1:Suppose Conditions 1 and 2 hold. Assumethat is large enough so that Proposition 2.1 holds. Then as

, for almost every realization of signature sequences,the output SINR achieved by the DMI blind linear MMSEreceiver is well approximated by

(36)

and that achieved by the subspace blind MMSE receiver is wellapproximated by

(37)

where

(38)

with

(39)

(40)

As noted before, we focus on the case where in thispaper. It is worth noting that as , the subspace blindMMSE receiver is identical to the DMI receiver, and (36) still


holds. (It should be pointed out that the result in (36) can bederived from [32, eqs. (27), (53), and (54)], with herebeing replaced by . Clearly, the difference between

and is negligible since is assumed to belarge.) Indeed, in many practical systems, the packet sizeison the order of hundreds of bits and therefore is reasonablylarge, indicating that the results in (36) and (37) provide a goodapproximation of the output SINR.

Several important observations based on these new findings(36) and (37) are in order.

1) It is known that the asymptotic SINR of the exact linearMMSE receiver given by (11) or (12) satisfies [24]

SINR (41)

Comparing (41) with (36) and (37), we see that the per-formance loss in the asymptotic regime due to the estima-tion error inherent to the blind receiver is captured by thefactor

for the DMI receiver; and by the factor

for the subspace receiver.

2) When the SNR of the desired user increases (as), by (36), the asymptotic SINR achieved by the

DMI blind receiver converges to ; and by (37), thatachieved by the subspace blind receiver approaches(since by (38), as ), i.e.,

SINRDMI

subspace

as (42)

This is in stark contrast to the fact that the asymptoticSINR achieved by the exact linear MMSE receiver givenby (41) can get arbitrarily large, as increases.This observation reveals a “saturation” phenomenonof the SINR achieved by the blind receivers. That is,beyond certain input SNR, the estimation error dom-inates and hence determines the output SINR, andtherefore, the performance gain obtained by increasingthe user transmission power diminishes. This saturationphenomenon of SINR indicates that the capacity of awireless system with blind multiuser receivers is notonly interference-limited, but alsoestimation-error lim-ited. Furthermore, the effect of estimation error can bequantified as shown earlier.

3) From (36) and (37), it is clear that in a large system, therelative performance of the DMI blind receiver and thesubspace blind receiver hinges on the parameter. Thatis, if , then the subspace receiver outperforms theDMI receiver; otherwise, the DMI receiver outperformsthe subspace receiver. Moreover, it is seen from (42) thatin the high-SNR region, the asymptotic output SINR of

the subspace blind receiver is higher than that of the DMIblind receiver by a factor (recall that ).Therefore, in the high-SNR region, the subspace blindreceiver outperforms the DMI blind receiver. On the otherhand, in the low-SNR region, e.g., as , then by(38), if , then , hence the subspace receiverperforms better; and if , then , hence theDMI receiver performs better.

4) Heuristically, we can approximate the output SINRachieved by the blind linear MMSE receivers in a largeCDMA system (with many users and large processinggain ) as follows. For the DMI blind receiver

SINR (43)

and for the subspace blind receiver

SINR (44)

Our next result gives the expression for the average outputSINR of the group-blind linear hybrid receiver in the asymptoticregime (as ). Define .

Theorem 3.2:Suppose Conditions 1 and 2 hold. Assumethat is large enough so that Proposition 2.2 holds. Then as

, for almost every realization of signature sequences,the output SINR achieved by the group-blind linear hybridreceiver is well approximated by

(45)

We have the following observations.

1) Comparing (41) with (45), it is seen that the performanceloss in the asymptotic regime due to the estimation errorinherent to the group-blind linear hybrid receiver is cap-tured by the factor

2) Comparing (37) with (45), it is clear that the asymptoticSINR of the group-blind linear hybrid receiver is strictlyhigher than that of the subspace blind linear MMSE re-ceiver, since . On the other hand, com-paring (36) with (45), it is seen that the group-blind linearhybrid receiver outperforms the DMI blind receiver in theasymptotic regime if and only if .

3) The asymptotic SINR achieved by the group-blind linearhybrid receiver saturates when the SNR of the desireduser, , increases, i.e.,

SINR as (46)

Accordingly, the capacity of a wireless system withgroup-blind receivers is also estimation-error limited.Comparing (46) with (42) we see that in the high-SNRregion, the group-blind receiver achieves the best perfor-mance, followed by the subspace blind receiver, which


in turn outperforms the DMI blind receiver. On the otherhand, in the low-SNR region, e.g., , thenby (38), if , then , hence thegroup-blind receiver outperforms the DMI blind receiver;otherwise, the DMI receiver outperforms the group-blindreceiver.

4) Heuristically, we can approximate the average outputSINR achieved by the group-blind linear hybrid receiverin a large CDMA system as

SINR

(47)

B. Receiver Output Distribution and BER

Using (5), the output of an estimated linear receiver is givenby the following generic form:

(48)

where is the overall residual interference, with

(49)

(50)

Note that the term in the preceding expressions correspondsto the sum of the multiple-access interference (MAI) and noisecomponents at the output of theexactlinear receiver , and

represents the MAI enhancement and the noise enhance-ment due to the estimation error .

The BER is an important performance measure in commu-nications, and is determined by the distribution of the overallresidual interference consisting of the MAI, the estimation errornoise, and the ambient noise. For CDMA systems with mul-tiuser receivers, it is of interest to consider a model—the samerandom signature—is repeated from symbol to symbol (see,e.g., [7], [24], [31], [37]). Accordingly, the more interestingquantity in this case is the conditional distribution of the re-ceiver output given the signatures [36], [37]. Based on (48), onewould expect that the output interference distribution dependson the choices of the signatures. As revealed in the followingtheorem, however, the dependence on the signatures vanishesas the processing gain becomes large. In [37], it is shown thatfor almost every realization of the signature sequences, the con-ditional distribution of the residual interference at the output ofan exact linear MMSE receiver (i.e., the term in (48) when

denotes the blind linear MMSE receiver) converges weaklyto the same Gaussian distribution for largeand , indepen-dent of signature sequences. In Section V, we provide a parallelstudy to establish the Gaussianity of the residual interferenceat the output of an exact group-blind linear hybrid receiver. Tostudy the distribution of the estimated receiver output (48), itthen suffices to characterize the distribution of the interferenceand noise enhancement term in (48), and the correlation of

and . As shown in Section V, is also asymptoticallyGaussian, independent of the signature sequences. We note that

the techniques required to establish the Gaussianity ofare dif-ferent from those used in [37].

We have the following result on the distribution of the residualinterference at the output of the blind linear MMSE receivers(cf. [37]).

Theorem 3.3:Suppose Conditions 1 and 2 hold. Then theconditional distribution of the overall interference at the outputof the blind linear MMSE receiver , given the signa-tures sequences, converges with probability one (as )to a Gaussian distribution with zero mean.

It is worth noting that the variances of the limiting Gaussiandistributions corresponding to the DMI and subspace blindreceivers are different (the variances will be presented inSection V).

Similar results hold for the group-blind linear hybrid receiver,as given by the following theorem.

Theorem 3.4:Suppose Conditions 1 and 2 hold. Then theconditional distribution of the overall interference at the outputof the group-blind linear hybrid receiver , given thesignatures sequences, converges with probability one (as

) to a Gaussian distribution with zero mean.

The proofs of Theorems 3.3 and 3.4 are given in Section V.In essence, we show that the estimation error of these linear re-ceivers is asymptotically Gaussian and orthogonal to the signalsubspace. It is interesting to note that in a large CDMA net-work with many users and large processing gain, the estimationerror approximately lies in the subspace orthogonal to the signalspace. (However, one must be cautioned that the components ofthe estimation error are correlated, which is evident from the ex-pressions for [cf. (116) and (139)].) We state this importantfact in the following proposition.

Proposition 3.1: The estimation error of the blind linearMMSE receiver (DMI or subspace) is asymptoticallyorthogonal to the spreading signature sequences, that is,

(51)

This property also holds for the estimation error of the group-blind linear hybrid receiver.

The proof of Proposition 3.1 is relegated to Appendix B. Wehave the following remarks in order.

1) In light of Theorems 3.3 and 3.4, assuming antipodalmodulation, we conclude that in a large CDMA system,where blind linear MMSE receivers or group-blind linearhybrid receivers are employed, the average BER can bewell approximated by

SINR (52)

whereSINR is the asymptotic average output SINR of thecorresponding to a specific receiver, given by (43) or (44)or (47).

2) We also note that it is not difficult to generalize the pre-ceding result, via the techniqueconditional weak con-vergence[23], [36], to establish the Gaussianity of the


residual interference at the output of blind linear mul-tiuser receivers in systems where the signatures are com-plex-valued.

IV. NUMERICAL EXAMPLES

In this section, we provide some numerical examples to illus-trate the theoretical results outlined in the preceding section.

A. Asymptotic AverageOutput SINR

We consider the special case when the received powers of allusers are the same, i.e., . In thiscase, in (34) can we written in closed form as (cf. [24])

(53)

It is shown in Appendix C that in this case the parametersand defined in (39) and (40) are given, respectively, by

(54)

(55)

To illustrate the convergence of the random SINR to the asymp-totic limit in the equal power case for different receivers, Fig. 1compares the actually realized SINRs from randomly generatedspreading sequences to the corresponding asymptotic limits forthe three blind receivers. (For the purpose of comparison, wealso plot the SINR corresponding to the exact MMSE receiver.)For different spreading lengths, and for each value of, 200samples of realized SINR for user 1 are obtained from randomlygenerated normalized spreading sequences. We can see thatfor each receiver, as the processing gain increases, the spreadaround the asymptotic limit becomes narrower, a phenomenonalso observed in [24] for the exact linear MMSE receiver. More-over, it is seen that the group-blind linear hybrid receiver out-performs the subspace blind linear MMSE receiver, which out-performs the DMI blind linear MMSE receiver at the SNR valueconsidered here.

B. The Output SINR Saturation Phenomenon

We next illustrate the SINR saturation phenomenon due to theestimation error for the DMI/subspace blind and group-blind re-ceivers discussed in this paper. We consider a CDMA systemwith , , , and (whenthe group-blind linear hybrid receiver is employed). In Fig. 2,we plot the average output SINR for each of the three receivers(as well as the exact MMSE receiver) as a function of the inputSNR (i.e., ), with randomly generated signature sequences,compared with the asymptotic limit. For each SNR value, 200random realizations of are generated. In each figure, we alsoplot the limit SINR value corresponding to (as), as the dashed line. The SINR saturation effect is clearly ob-

served from Fig. 2. In contrast, the SINR corresponding to theexact MMSE receiver grows without bound, as is evident in thefigure.

C. Orthogonality Between Estimation Error and Signal Space

Proposition 3.1 reveals that for the three linear receivers con-sidered here, the estimation error is asymptotically orthog-onal to the spreading signatures. Note that (51) is equivalent tothe following:

as (56)

In Fig. 3, we plot the value of as a function of for theblind linear MMSE receiver (DMI or subspace) and the group-blind linear hybrid receiver, with randomly generated spreadingsequences. For each value of, 200 random realizations ofare generated. It is clear from these figures that thedecreases monotonically asgrows, corroborating (56).

V. PROOFS OFMAIN RESULTS

A. Technical Preliminaries

Define

(57)

It is easy to see that the linear MMSE receiver given by (11) or(12) can also be written as

(58)

Thus,

(59)

It has been shown in [24] that

(60)

where is given in (34). It follows that

(61)

We need the following lemmas in proving our main results.The proofs of these lemmas are relegated to Appendix B.

Lemma 5.1:Suppose as . Assumethat exists for any and exists aswell. Then

Our next lemma gives the asymptotic limit expression ofas , where is given in (27).

Lemma 5.2:As

B. Proof of Theorem 3.1

The proof of Theorem 3.1 boils down to evaluating the in-terference power. In what follows, we calculate the interference


Fig. 1. The average output SINR of four linear receivers (DMI blind MMSE, subspace blind MMSE, group-blind hybrid receiver, and exact MMSE) with randomlygenerated signature sequences, compared with the asymptotic limits (shown as the dashed lines) in the equal-power regime, for� = 10, = 15 dB, withN = 16 andN = 64.

power (i.e., the denominator in (17)) term by term under the as-sumption of random signatures.

First, we have that

[by (18)]

[by (58)]

[by (59)] (62)

Also, it is clear from (19) and (61) that

(63)

Next we evaluate in (20). Using Condition 2, we assume thatthe sequence is bounded above and belowby and , respectively. Analogously to (61), we have

(64)

Observe that


Fig. 2. The output SINR saturation effect of the blind linear receiver.N = 32, � = 10, and� = 0:5. The solid lines are the given by (36), (37), and (45),and the dashed lines are given by (42) and (46).

(a) (b)

Fig. 3. Illustration of the orthogonality between the estimation error�www and the signal spaceSSS for the blind and group-blind linear receivers. (a) (DMI orsubspace) blind MMSE receiver,� = 10,� = 0:5, and = 15 dB. (b) Group-blind hybrid receiver,� = 10,� = 0:5,� = 0:25, and = 15 dB.

[by (58)]

(65)

By Condition 2, we use [9, p. 470] to obtain that1

1By matrix inequalityAAA � BBB we mean thatAAA�BBB is positive semidefinite.

(66)

which implies that

[by (60)] (67)

It then follows by combining (64), (65), and (67) that

(68)

which indicates that

(69)


In the following, we calculate the limits of in (23). For theDMI blind receiver, we have by (61)

(70)

For the subspace receiver, we know that

(71)

It is straightforward to see that

(72)

For convenience, define

(73)

(74)

After some algebra, we have that

(75)

(76)

Combining (71), (72), and (76) yields

(77)

Note that both and are independent of . Next weshow that

(78)

To this end, observe that

For any , replace in (78) by . Letting, we can show, via a continuity argument, that

(79)

It is clear that the eigenvalues of arebounded above by , which indicates that for a given, the

spectral radius of is bounded above.Using Theorem A.2 in Appendix A and the Borel–Cantellilemma, it follows that

(80)

Appealing to [17, Theorem 7.11], we obtain the result in(78). Alternatively, we can use [1, Theorem 2] to show thatthe spectral radius of is bounded above with prob-ability one as gets large. Without loss of generality, let

denote the largest eigenvalues of. Observe that

(81)

and that the smallest nonzero eigenvalue of is equal to thesmallest eigenvalue of . Furthermore, by [1, Theorem2], the smallest eigenvalues of converge to . Itthen follows that the minimum among isbounded below with probability one as . (This obser-vation will be used to obtain and defined later.)

Note that by [21], the eigenvalues of and are asymp-totically the same as . Then it follows that

(82)

Also note that

(83)

and for . Thus, it follows that

(84)

(85)

where (84) follows from [21] and the bounded convergencetheorem.

Recall that

(86)

(87)

(We note that as , exists while .) Asshown above, the minimum among isbounded below with probability one as , (hence themaximum among is bounded abovewith probability one), therefore, converges withprobability one. We use Lemma 5.1 to conclude that

(88)

Observe that

(89)


Using [17, Theorem 7.11] and Lemma 5.1, we conclude that

(90)

Combining (77), (78), (82), (83), (85), (88), and (90), we obtainthat

(91)

It then follows immediately that for the subspace blind MMSEreceiver

(92)

which implies that

(93)

Combining (62), (63), (69), (70), and (93), we have that for theDMI MMSE receiver

(94)

and for the subspace blind MMSE receiver

(95)

where

This concludes the proof of Theorem 3.1.

C. Proof of Theorem 3.2

The proof of Theorem 3.2 again boils down to evaluating theinterference terms. Define . Note thatby (27)

(96)

By (24), the group-blind linear hybrid receiver can also bewritten as

(97)

It follows that

[by (30)]

[by (97)]

[by (96)]. (98)

A direct application of Lemma 5.2 allows us to conclude that

(99)

and by (31)

(100)

Using the same techniques as in proving Lemma 5.2, we canshow that

(101)

Furthermore, along the same line as in showing (65), (67), and(68), we have that

(102)

Substituting (101) and (102) into (32) yields

(103)

In what follows, we proceed to calculate the quantity.Denote

(104)

After some algebra, we can show that [11]

(105)

where

(106)

By (96)

(107)

and similar to (75), we have

(108)

It then follows that

(109)

Using the same techniques as in showing (78), (82), (92), andproving Lemma 5.2, we conclude that for

(110)


and ,

(111)

Then we have that

(112)

By using (112) and Lemma 5.2 in (109), we have

(113)

Combining (99), (100), (103), and (113), we conclude that theasymptotic output SINR achieved by the group-blind linear hy-brid receiver is given by

(114)

where is given by (38). This concludes the proof of The-orem 3.2.

D. Proof of Theorem 3.3

We prove Theorem 3.3 by the following three-step procedure.

1) For almost every realization of, converges to aGaussian distribution as .

2) For almost every realization of, converges to aGaussian distribution as .

3) For almost every realization of, the covariance ofand vanishes as .

Step 1: A direct application of [37, Theorem 3.2] shows thatin (49) converges to a Gaussian distribution for almost

every realization of .

Step 2: The main portion in the interference enhancementdue to the estimation error in (50) is . In the fol-lowing, we show that for almost realization of,has a limiting Gaussian distribution. Observe that for fixedand , is Gaussian. Then, it suffices to show thatfor almost every and , the variance of the corresponding

is the same as . To this end, note that forfixed and

(115)

We next show that the spectral radius of the matrixis bounded with probability one. Recall

that the covariance of the estimation error for the blindlinear MMSE receiver is given by [11], [12]

(116)

where

(117)

DMI

subspace.(118)

It is straightforward to see that

(119)

Then it follows that

(120)

By Condition 2, we use [9, p. 470] again and have that

(121)

Note that

It follows that the spectral radius of the matrix

is bounded in , with probability one. Because ,for , it is clear that

(122)

where the last step follows from Condition 1. Then we have that

(123)

As shown in (121), the spectral radius of the matrix

is bounded above, it follows that the spectral radius of

is also bounded above. Note that

is a rank-one nonnegative matrix, and


which indicates that the spectral radius of isbounded above. Using Weyl’s theorem [9, p. 181], we have thatthe spectral radius of the matrix is bounded withprobability one. Therefore, by combining Theorem A.2 in Ap-pendix A and the Borel–Cantelli lemma, we conclude that

(124)

Observe that

(125)

It is straightforward to see that converges to the samelimit for almost every . Using the results on , , and , weconclude that converges to the same limit foralmost every (the limits in the DMI and subspace cases aredifferent). Thus,

converges with probability one, where the expectation is takenwith respect to .

Observe that is the inner product of two Gaussianvectors. As the processing gain gets large, by the centrallimit theorem, this term also approaches Gaussian distribution.Therefore, we conclude that for almost every, convergesto a Gaussian distribution as ; for the DMI MMSE re-ceiver, the variance of

(126)

and for the subspace blind MMSE receiver, the variance of

(127)

Step 3: We proceed to show that the covariance of andvanishes as gets large for almost every. The covariance

of and is

(128)

Then it suffices to show that , which followsdirectly from Proposition 3.1.

Built on Steps 1–3, a simple application of Lemma B.2 inAppendix B completes the proof of Theorem 3.3.

E. Proof of Theorem 3.4

Recall that the exact group-blind linear hybrid receiver is

(129)

By definition, for . Then, the outputresidual interference of the estimated group-blind receiver isgiven by , with

(130)

(131)

We now prove Theorem 3.4, using the same three-step approachas in the proof of Theorem 3.3.

Step 1: The proof of Step 1 hinges on the use of dependentcentral limit theorem for martingale difference arrays [15], andfollows essentially the same line as that of [37, Theorem 3.2].We outline the necessary details as follows. Define

(132)It is clear that the array is amartingale difference array with respect to the-algebra

[15]. Using the central limit theorem in[15], it suffices to show that for almost every realization of,the following three conditions hold:

a) is bounded in norm;

b) converges to in probability as;

c) converges to the same limit in probabilityas .

To this end, it is straightforward to see that condition a) holdsbecause is bounded. We proceed to establishcondition b). Using [37, Lemma 4.3], we have that

(133)

Along the lines of proving [37, Lemma 5.1], we combineMarkov’s inequality and the Borel–Cantelli lemma to concludethat

(134)

It follows that

(135)

Therefore, we have that

(136)


and condition b) follows. Now, observe that

(137)

By Lemma 5.2

(138)

Combining the above result with (99) yields condition c),thereby completing the proof of Step 1.

Step 2: We establish in what follows that for almost everyrealization of , has a limiting Gaussian distribution.To this end, we show that for almost everyand , the varianceof the corresponding is the same as . Again,note that for fixed and

and recall [11], [12] that the covariance of the estimation errorof the group-blind hybrid receiver is

(139)

where

(140)

(141)

(142)

(143)

Plugging (139) into yields that

(144)

where

We next show that

for . Recall that


(145)

(146)

Also, note that

(147)

where and

(148)

Then, based on Theorem A.2 in Appendix A and the Borel-Cantelli lemma, we have that it suffices to show that the spectralradii of the following three matrices are bounded above:, ,and .

To this end, we need the following results on the spectral ra-dius of square matrices [9].

Lemma 5.3: If is any matrix norm, then the spectralradius of any square matrix is bounded above by .

The next result shows that the spectral radius of any squarematrix is the greatest lower bound for the values of all matrixnorms of .

Lemma 5.4:Denote the spectral radius ofas . Givenany , there is a matrix norm such that

We have shown in the proof of Theorem 3.3 that the spectralradius of is bounded above. Define

(149)

Then

(150)

Observe that is idempotent, and can be writtenas [18, p. 48]

(151)

where is some unitary matrix. It follows that

(152)


Because has bounded spectral radius, and bothandare nonnegative definite, we have that the spectral radius ofis bounded above. Built on the boundedness of the spectral radiiof and , we use Lemmas 5.3 and 5.4 to conclude that thespectral radius of is bounded above. It then follows that

(153)

Along the same line as in proving Theorems 3.1 and 3.2, wecan obtain that converges to the same limit foralmost every . It then follows that con-verges with probability one, where the expectation is taken withrespect to . We conclude that for almost every, con-verges to a Gaussian distribution with zero mean and variance

(154)

Step 3: It is straightforward to use Proposition 3.1 to proveStep 3, thereby completing the proof of Theorem 3.4.

APPENDIX ATWO RANDOM MATRIX RESULTS

The first result is on the empirical distribution functionof eigenvalues of large random matrices [21]. Denote theeigenvalues of the random matrix by(they are random, depending on the realization of), and theempirical distribution of the eigenvalues by . We restate[21, Theorem 1.1] in the following.

Theorem A.1:Under Condition 1, converges weakly toa distribution function with probability one, and the Stieltjestransform of is the solution of the following functionalequation:

(155)

where the Stieltjes transform of any distribution is de-fined as

for .

We also use the following result repeatedly, which followsdirectly from [21, Lemma 3.1]:

Theorem A.2:Let , , be an matrixwith spectral radius bounded in, and ,where the ’s are i.i.d. with .Then

where the constantdoes not depend on and .

APPENDIX BPROOFS OFTECHNICAL LEMMAS

Lemma B.1:Suppose and as ,where both and are finite constants. Then

It is straightforward to establish the above lemma via standardreal analysis techniques.

Lemma B.2:Suppose both and have limitingGaussian distributions , and and are asymp-totically uncorrelated. Then, has a limiting Gaussiandistribution.

The proof of Lemma B.2 follows by using an argument viacharacteristic functions.

A. Proof of Lemma 5.1

Outline of the proof: Let

Let . It is clear that . Observe that forany , as

(156)

It then follows immediately that for any

(157)

thereby completing the proof.

B. Proof of Lemma 5.2

We know that for

(158)

Next we show that for

(159)

To this end, it suffices to show that


Note that is independent from , and is independentfrom both and . Using the matrix inverse lemma, it can beshown that

(160)

It follows that

(161)


Since is a rank one matrix, and

the spectral radius of is bounded above. ByLemma A.2

(162)

Combining (161) and (162) yields that

Let denote the set on which converges. It isstraightforward to see that

That is,

(163)

Then, it can be shown that

(164)

Along the same lines, we can obtain the other elements in.The proof is completed.

C. Proof of Proposition 3.1

In the following, we show that for the blind linear MMSEreceiver case, , which is equivalent to showingthat for almost every realization of signatures, .To this end, it suffices to show that

(165)

Observe that for the blind linear MMSE receiver (DMI or sub-space), by (116), we have

(166)

Because , , , and converge totheir finite limits with probability one, we conclude that

(167)

Next we also show that , where is theestimation error of the group-blind hybrid receiver. Since by(139)

(168)

using the findings on the boundedness of the spectral radii ofthe matrices in , , and (cf. Section V-E, Step 2),we apply Theorem A.2 in Appendix A to conclude that for thegroup-blind hybrid receiver

(169)

It then follows that , completing the proof.

APPENDIX CDERIVATIONS OF (54) AND (55)

We first derive in (54). Substituting (53) into (39), we have

(170)

We next derive in (55). Using (53), we have


(171)

After some manipulations, we have

(172)

(173)

where

(174)

Using (40) we obtain

(175)

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