Widely Linear Versus Linear Blind Multiuser Detection With Subspace-Based Channel Estimation: Finite...

1426 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 57, NO. 4, APRIL 2009

Widely Linear Versus Linear Blind MultiuserDetection With Subspace-Based Channel Estimation:

Finite Sample-Size EffectsAngela Sara Cacciapuoti, Giacinto Gelli, Luigi Paura, and Francesco Verde

Abstract—In a recent paper [A. S. Cacciapuoti et al., “Fi-nite-Sample Performance Analysis of Widely Linear MultiuserReceivers in DS-CDMA Systems, IEEE TRANSACTIONS ON SIGNALPROCESSING, vol. 56, no. 4, pp. 1572–1588, Apr. 2008], we presentedthe finite-sample theoretical performance comparison betweenlinear (L) and widely linear (WL) minimum output-energy(MOE) receivers for direct-sequence code-division multiple-access(DS-CDMA) systems, worked out under the assumption that thechannel impulse response of the desired user is exactly known. Themain scope of this paper is to extend such an analysis, taking intoaccount not only autocorrelation matrix (ACM) estimation effects,but also the accuracy of subspace-based blind channel estimation(CE). We aim to answer the two following questions: Which ofthe two estimation processes (ACM or CE) is the main source ofdegradation when implementing the receivers on the basis of a finitesample-size? Compared with the L-MOE one, is the finite-sampleWL-MOE receiver with blind CE capable of achieving the perfor-mance gains predicted by the theory? To this goal, simple and easilyinterpretable formulas are developed for the signal-to-interfer-ence-plus-noise ratio (SINR) at the output of the L- and WL-MOEreceivers with blind CE, when they are implemented using eitherthe sample ACM or its eigendecomposition. In addition, thederived formulas, which are validated by simulations, allow oneto recognize and discuss interesting tradeoffs between the mainparameters of the DS-CDMA system.

Index Terms—Blind multiuser detection, channel estimation, di-rect-sequence code-division multiple-access (DS-CDMA) systems,linear and widely linear filtering, minimum output-energy (MOE)criterion, perturbation analysis, proper and improper randomprocesses, subspace methods.

I. INTRODUCTION

I N multiuser communication systems, such as non-or-thogonal direct-sequence code-division multiple-access

(DS-CDMA) systems, the multiple-access interference (MAI)often represents the main source of performance degradation.During the last decades, in order to counteract such a degra-dation, a great bulk of research activities has been devotedto multiuser detection (MUD) [2]. Among most recent MUDdevelopments, widely linear (WL) techniques [3]–[7] perform

Manuscript received May 28, 2008; revised November 18, 2008. First pub-lished December 31, 2008; current version published March 11, 2009. The as-sociate editor coordinating the review of this manuscript and approving it forpublication was Prof. Subhrakanti Dey. This work was supported in part by theItalian National Project: Wireless multiplatfOrm mimo active access netwoRksfor QoS-demanding muLtimedia Delivery (WORLD), by Grant 2007R989S.

The authors are with the Dipartimento di Ingegneria Biomedica, Elettronica edelle Telecomunicazioni (DIBET), Università degli Studi di Napoli Federico II,I-80125 Napoli, Italy (e-mail: [email protected]; [email protected];[email protected]; [email protected]).

Digital Object Identifier 10.1109/TSP.2008.2011828

MAI mitigation by exploiting the noncircular or improper [8]features of many digital modulation schemes, such as ASK,differential BPSK (DBPSK), offset QPSK (OQPSK), offsetQAM (OQAM), MSK, and its variant Gaussian MSK (GMSK).Unlike conventional L-MUD receivers [9]–[11], which processonly the complex envelope of the received signal, to ex-tract the information contained in its statistical autocorrelationfunction , WL-MUD receiversjointly elaborate and its complex-conjugate version, to takeadvantage also of the information contained in their statistical

cross-correlation function ,assuring thus potentially larger performance gains.

In a recent paper [1], the theoretical performance anal-ysis of the WL receiver designed according to the minimumoutput-energy (MOE) criterion [11], is considered in terms ofsignal-to-interference-plus-noise ratio (SINR). Implementationof the WL-MOE receiver requires knowledge of the (aug-mented) autocorrelation matrix (ACM) of the received signal,and of the received signature (possibly distorted by the channel)of each user to be demodulated. In particular, by applying afirst-order perturbative analysis, the performance degradationdue to finite-sample ACM estimation is evaluated in [1], withreference to two common implementations of the WL-MOEreceiver: the WL-SMI one, which is based on sample matrixinversion (SMI), and the WL-SUB one, which resorts to ACMsubspace (SUB) decomposition to improve robustness againstfinite-sample errors.1 The results of the analysis show that theWL-MOE receiver is more sensitive than its linear counterpartto finite sample-size effects associated to ACM estimation,and its subspace-based implementation is a viable strategy toachieve in practice the performance gains predicted by theory.

The analysis of [1] is exhaustive only when the receivedsignature is perfectly known at the receiver. When operatingover a multipath channel, however, due to the unknown channelresponse, the received signature is a distorted version of thetransmitted one, making channel estimation (CE) a necessarystep to implement both the L- and WL-MOE receivers. In sucha scenario, the performance of the receivers is affected notonly by imperfect ACM estimation, but also by inaccurate CE.Conventional CE methods are training-based, which might leadto significant waste of resources in mobile communicationsscenarios, especially when training must be frequently repeateddue to rapidly changing channel conditions. Therefore, the

1It should be mentioned that other robust techniques, suitably generalized tothe WL case, could be employed, such as those developed in [12]–[14], whichare in their turn based on the robust beamforming technique [15].

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CACCIAPUOTI et al.: WL VERSUS LINEAR BLIND MULTIUSER DETECTION 1427

past few decades witnessed many contributions in the areaof blind CE for DS-CDMA systems, which assume the onlyknowledge of the transmitted signature of the desired user.Blind CE approaches relying on second-order statistics (SOS)of the received data are particularly attractive, since they re-quire far fewer samples than methods based on higher-orderstatistics. SOS-based approaches for blind CE exploit thechannel information contained in by processing thereceived signal . Among existing SOS-based approaches,the subspace-based CE method, originally proposed in [16],is one of the most popular blind algorithms for DS-CDMAsystems, due to its closed-form expression and noise robustness[17], and its amenability to low-complexity and fast recur-sive implementation [18]. Moreover, channel identifiabilityconditions for subspace-based CE are well assessed [19] and,finally, subspace-based blind methods can be easily combinedwith existing training-based approaches (so-called semi-blindmethods [20]). Among the main drawbacks of subspace-basedCE are its performance degradation when the number of activeusers is comparable to the processing gain, and the need foraccurate rank estimation of the noise-free ACM. Under theassumption that the transmitted symbols are improper andthe noise is proper, the first drawback can be overcome byresorting to a generalized subspace-based CE method, whichallows one to enlarge the dimension of the observation space,by jointly processing both and its conjugate versionto exploit the channel information contained in bothand . Such an approach was originally proposed in[21]–[24] to improve channel identification in many applicationfields, including multicarrier CDMA [23] and single-carrierDS-CDMA systems [24]. To face up to the second drawback,one can use conventional rank estimation techniques, such asthe Akaike information criterion (AIC) [25] or the minimumdescription length (MDL) method [26], or, alternatively, asubspace tracking procedure with successive cancellationtechniques [27].

In this paper, the first-order perturbation analysis carriedout in [1] is extended to incorporate the effects of errors dueto subspace-based blind CE on the synthesis of the L- andWL-MOE receivers. It is worthwhile to note that, when thedesired channel vector has been estimated through the subspacemethod and, hence, the subspace decomposition of the ACMis already available, it is preferable from a computationalviewpoint to implement the L- and WL-SUB receivers ratherthan their SMI counterparts, since they do not require directACM inversion. Notwithstanding this, we have chosen to carryout also the performance analysis of the SMI versions of thereceivers with CE since, in this way, an interesting comparisonwith the SUB versions of the receivers, as well as with ourprevious results [1], can be established. A distinct advantageof our analysis is that it leads to easily interpretable formulas,clearly showing the fundamental relationships among the mainparameters (sample-size, processing gain, channel length, andnumber of users) of the DS-CDMA system. Although a similarperturbative performance analysis was addressed in [28]–[30]for the blind L-MMSE (minimum mean-square error) receiver,the analysis carried out in this paper for the L-MOE receiverwith blind CE allows a more direct and fruitful comparison

with the WL-MOE one and, moreover, leads to more easilyinterpretable results (albeit slightly less accurate) than thoseobtained in [28]–[30].

Finally, it is noteworthy that the problem considered hereinexhibits interesting analogies with a well-studied topic in arrayprocessing, since the L-MOE-based multiuser detector is math-ematically equivalent to the linear minimum variance (L-MV)beamformer [31], where in the latter the role of the received sig-nature is played by the array steering vector (SV). Finite-sampleperformance analysis of the L-MV beamformer was carried outin [31]–[33] for the SMI version, and in [34] for the subspace-based implementation (so called projection method). Specifi-cally, in [32] only the effects of ACM estimation were consid-ered, whereas in [31] and [34] the effects of ACM estimationand SV perturbation were separately studied, and a completeanalysis of the joint effects of ACM estimation and SV pertur-bation was carried out only in [33]. However, the latter analysisdoes not explicitly account for the situation wherein the SV isblindly estimated from the received data and, consequently, theSV perturbation depends in its turn on the accuracy in ACM es-timation, which is exactly the case of the subspace-based CEalgorithms considered herein.

The rest of the paper is organized as follows. In Section II,the mathematical model of the DS-CDMA system is intro-duced, and the ideal WL-MOE and L-MOE receivers are brieflyreviewed. In Section III, two data-estimated versions of theWL-MOE receiver (SMI and SUB) with CE are presented,and their finite-sample performance analysis is developed. Thesame analysis is carried out for the SMI and SUB implemen-tations of the L-MOE receiver with CE in Section IV, where,moreover, a comparison between WL-MOE and L-MOE re-ceivers is assessed. The theoretical results of Sections III andIV are validated by computer simulations in Section V, whiletheir proofs are gathered in the Appendix 1. Finally, concludingremarks are given in Section VI.

A. Notations

The fields of complex, real, and integer numbers are de-noted with , and , respectively; matrices [vectors]are denoted with upper case [lower case] boldface letters(e.g., or ); the field of complex [real] ma-trices is denoted as , with used asa shorthand for ; the superscriptsand denote the conjugate, the transpose, the Hermitian(conjugate transpose), the inverse, and the Moore-Penrosegeneralized inverse [38] (pseudo-inverse) of a matrix, re-spectively; anddenote the null vector, the null matrix, and the identitymatrix, respectively; and represent thetrace and the rank; , , and denote thenull space, the range (column space), and the orthogonalcomplement of the column space of

in ; for any denotesthe Euclidean norm; is adiagonal matrix with elements on the main diagonal;

and denote ensemble averaging and variance,

respectively, and is the imaginary unit; for any



stationary discrete-time random vector process ,

we denote with and with

the ACM and the conjugatecorrelation matrix, respectively ( when

); throughout the paper, we occasionally use the

simplified notations ,

and .

II. PROBLEM FORMULATION AND IDEAL WL-MOE AND

L-MOE RECEIVERS

In this paper, we consider a synchronous DS-CDMA systemwith users, employing short spreading codes with

chips/symbol and transmitting over channels that intro-duce interchip interference and negligible intersymbol interfer-ence (ISI) [2]. After chip-matched filtering, perfect time syn-chronization and sampling with rate , the received vector

collecting the samples of the incoming signal inthe time interval , with , can be written[1], [2] as follows:

(1)

where, with reference to the th user, the real positive numberis the received amplitude (which is the product of the

transmitted amplitude and the channel gain), the matrixis Toeplitz having as first

row and as first column, with

the vector de-noting the unit-norm transmitted signature,2 the sequence

is the channel impulse response of length

, withbeing the corresponding unit-norm channel vector, and

is the transmitted symbol. Moreover, the matrix

embodies the effects of channelsand signatures, with

and , whereas

is the symbolvector, and accounts for additive thermal noise.

Throughout the paper, we will rely on the same assump-tions formulated in [1]: a1) is a binary real zero-meanrandom vector, whose entries are independent and identicallydistributed (i.i.d.) random variables assuming equiprobablevalues in , with and statisticallyindependent for ; a2) is a complex proper [8]zero-mean Gaussian random vector, independent of ,having and , with and

statistically independent of each other for . Asregards the considered signal model, it is noteworthy that theassumptions of synchronous transmissions and negligible ISIare taken only for the sake of simplicity, since our analysis canbe readily generalized to other scenarios (e.g., asynchronoususers and/or channels with ISI). For instance, an asynchronous

2The signature � accounts also for possible precoding phases, whose role indownlink is discussed in [1, Th. 1].

system with users can be described by the synchronousmodel (1) with equivalent users (see [10] for details).

The output of a WL receiver for user canbe expressed [1], [35] as

(2)

where andis the augmented received vector. According to (1), vectorcan be expressed as

(3)

with and. Moreover, the following condition is as-

sumed to hold: c1) when (underloaded systems),the matrix is full-column rank, i.e., . It canbe readily shown (see, e.g., [1]) that, in the downlink case,wherein all the user signals propagate through a common mul-tipath channel, the linear independence of the signatures

is a necessary and sufficient condition to ensure c1).It is noteworthy that, if is full-column rank, the augmentedmatrix is full-column rank, too, i.e., . In otherwords, in underloaded environments, condition c1) additionallyassures the full-column rank property of . However, thematrix can be full-column rank even when(overloaded systems), wherein is inherently rank-deficient.Thus, in addition to condition c1), we assume hereinafter that:c2) when , the matrix is full-column rank,i.e., . With reference to the downlink scenario,fulfillment of condition c2) is thoroughly discussed in [1, Th.1].

Accounting for (3), (2) can be equivalently written as

(4)

where , with being the

th column of , whereas ,with denoting the matrix that includes all thecolumns of except for the th column de-notes the vector that includes all the elements of except for

the th entry , and is theaugmented disturbance (interference-plus-noise) vector. Since,by virtue of assumption a1), is real-valued, an appropriateperformance measure for the th user is the output SINR [29]defined as

SINR (5)

Indeed, if the disturbance contribution in (4) can beapproximated as a Gaussian random variable, the error proba-

bility is well approximated as

SINR , where de-notes the function. Definition (5) of the SINR is quite gen-eral and allows for relatively simple calculations also whenand/or are estimated from data. As discussed in [1, Lemma 1],



any receiver maximizing (5) can be found, without loss of gen-erality, within the class of receivers possessing the conjugatesymmetry (CS) property, that is, in (2), which assuresthat the output given by (2) or (4) is real-valued. Belonging tothis class is the WL-MOE receiver [1], i.e.

(6)

whose corresponding maximum SINR can be expressed as

SINR SINR

(7)

Under assumptions a1) and a2), the matrix assumesthe form . Moreover, by virtue ofconditions c1) and c2), the matrix has only nonzeroeigenvalues . An equivalent form ofthe WL-MOE receiver can be obtained thus by exploiting theeigenvalue decomposition (EVD) of , which is given by

, where collectsthe eigenvectors associated with the largest eigenvalues of theACM , which span the signal subspace, i.e., the subspace

,, and, finally,

collects the eigenvectors associatedwith the eigenvalue , which span the noise subspace, i.e.,the subspace in . By substituting the EVD of

in (6) and exploiting the orthogonality between signal andnoise subspaces, one obtains the subspace-based form of theWL-MOE receiver as follows:

(8)

Turning to L-MUD receivers, it should be observed that (2) en-compasses, as a particular case, the linear input-output relation-ship , with , which can be ob-tained by setting and . It is worth noticingthat, since a linear receiver does not satisfy the CS property,i.e., , its output is not necessarily real-valued and,most important, a linear receiver does not generally belong to

the family of receivers maximizing (5). A popular linear MUDreceiver is the L-MOE one [11], i.e.

(9)

with being the dis-turbance vector, and the resulting SINR (5) is given by(10), shown at the bottom of the page (see [1]) where

. Clearly, sincethe WL-MOE (6) is a maximum-SINR receiver, it results thatSINR SINR . Similarly to the WL-MOEreceiver, under condition c1), the L-MOE one (9) can be equiv-alently represented in subspace-based form3

(11)

where the columns of coincide with the eigen-vectors corresponding to the largest eigenvalues of

, and, with being the nonzero

eigenvalues of .

III. PERFORMANCE ANALYSIS OF WL-MOE RECEIVERS WITH

CHANNEL ESTIMATION

Implementation of the WL-MOE receiver defined by (6) or(8) requires estimation from the received data of the ACMin (6) or its EVD in (8), and of (in both). Under mild con-ditions, a consistent estimate of is the sample ACMobtained as

(12)

where denotes the estimation sample size. Applying the EVDto , one obtains , where thematrices , ,and are estimates of , and

3The subspace receivers (8) and (11) are mathematically equivalent to theprojection method, proposed in [31] to improve the robustness of the L-MVbeamformer against ACM estimation and SV perturbation errors.

SINR SINR

(10)



, respectively. With regard to , we prelimi-narily observe that, according to (1), the th column of thematrix assumes the form

(13)

and, consequently, one has

(14)

where is a unitary matrix, i.e., .Assuming that the receiver has the only knowledge of the trans-mitted signature , the matrix in (14) is known, whereas,under conditions c1) and c2) and accounting for (14), blind es-timation of can be accomplished [16] by exploiting the or-thogonality between the signal subspace and the noisesubspace , that is

(15)

The unknown vector can be obtained as the solution of thelinear system (15), provided that this system uniquely charac-terizes the channel coefficients for each user, i.e., an arbitraryunit-norm vector (with corresponding ),satisfies (15) if and only if (iff) , withand . It is clear that (15) has a unique solu-tion (up to a scaling factor) iff the following condition is sat-isfied: c3) the null space of has dimension one or,equivalently,4 . A reformulation ofcondition c3) is given in [24]. It can be readily proven that,under c3), the following two statements are equivalent: i) theunit-norm vector is a solution of (15); ii) ,i.e., , with . In other words, differently fromconventional subspace-based multiuser CE [17], where the es-timated channel might differ from the true one by an unknownrotation , in generalized subspace-based CE based on (15)the residual channel ambiguity is limited to a possible sign in-version. It is important to observe that condition c3) necessarilyimposes that the number of rows of the matrix

be greater than or equal to its number of columns,i.e., . Thereby, it fol-lows that, from the point of view5 of the th user, the max-imum number of users supported bythe system is smaller than the maximum number of usersin the known-channel case [1]. We assume hereinafter that con-dition c3) is satisfied.

4The dimension of the null space of� �� is equal to�� . Moreover, since � is unitary and, hence, nonsin-gular, it results that �� .

5In order to meaningfully define the maximum number of users that can besupported by the system, we could consider the worst case, i.e., set � �� as the maximum channel length, obtaining � � �� .

In practice, however, (15) cannot be satisfied exactly when(and, hence, ) is estimated from a finite sample size.

In this case, a channel estimate can still beobtained by solving (15) in the least-squares sense

(16)

whose solution [36] is given by the eigenvector associatedwith the smallest eigenvalue of the matrix

, with . Bysubstituting in (6) and (8) the sample ACM (12) and its EVD,respectively, as well as the resulting estimateof the received signature [see (14)], the WL-SMI6 andWL-SUB receivers with CE are given by

(17)

(18)

Note that, while (6) and (8) are equivalent, their estimated coun-terparts (17) and (18) are different, even when . Afinal remark is in order about knowledge of the real scalarand of the sign inversion inherent to channel estimate , whichare needed to correctly build the estimated signature . Theseparameters cannot be estimated by means of SOS-based blindtechniques; in practice, they can be recovered by resorting toautomatic gain control and differential modulation or, more ro-bustly, by using a few training symbols. It should be noted,however, that their possible inaccurate knowledge merely in-troduce a real multiplicative factor in the expressions of theWL-SMI-CE and WL-SUB-CE receivers, which does not affectSINR calculation based on (5). Therefore, to simplify matters,we assume that they are both known exactly.

A. Finite-Sample SINR Analysis

In this section, we provide the finite-sample performanceanalysis of the WL receivers with CE (17) and (18). Tothis aim, we generalize the analysis carried out in [1] in theknown-channel case, by taking into account also the estima-tion errors in the received signature . We adopt as in [1] afirst-order perturbative approach [30], [37] to model all esti-mation errors, and derive closed-form expressions of the SINRdefined by (5) for the WL-SMI-CE and WL-SUB-CE receivers.In the following, in order to carry out the analysis in an unifiedframework, we denote with any data-estimated WL-MOEreceiver, i.e., or , andset , where is the ideal WL-MOE

6The subsequent performance analysis could be also extended to WL general-izations of the robust multiuser detectors proposed in [12]–[14], capitalizing onthe fact that, as recognized by the same authors, such receivers can be regardedas diagonally loaded versions of the SMI detector.



receiver given by (6) or (8). When is employed, accountingfor (4), it can be shown that (5) yields

SINR (19)

Since and exhibit the CS property, the real parts in(19) can be omitted, thus yielding

SINR (20)

According to the perturbative approach, the vectors andare expressed as and , respectively,where and are small (i.e., and )and zero-mean CS perturbation terms. Thus, we have

, since, from (6), . More-over, denoting with the average with respect to (w.r.t.)

, since is zero-mean and is a real-valued scalar, itturns out that and

(21)

and, therefore,, which substituted in (20) leads to

SINR

(22)

where denotes joint average w.r.t. and . Under

the simplifying and reasonable assumption [32] that is inde-pendent of

(23)

which, accounting for and leadsto . Bysubstituting such a result into (22), one obtains

SINR

(24)

Since, noting also that, according to (7),SINR we obtain the compact

expression

SINRSINR

SINR(25)

where only the average w.r.t. is left to be evaluated. To pro-ceed further, explicit expressions for the perturbation of theWL-SMI-CE and WL-SUB-CE receivers are needed.

Lemmma 1: Let denote first-order equality,7 the first-orderperturbation term of the WL-SMI-CE and WL-SUB-CE re-ceivers can be expressed as

(26)with

(27)

(28)

where isthe sample estimate of the cross-correlation betweenthe disturbance vector and the desired symbol

and are given by (29) and (30)

shown at the bottom of the page, with

denoting an oblique projection matrix [32] and, while

the diagonal matrixcollects the nonzero eigenvalues of .

Proof: See Appendix A.It should be noted that and represent the perturba-

tions due to estimation of and , respectively; indeed, acomparison shows that the expression of is the same as thatreported in [1, Lemma 2]. In order to characterize the perturba-tion term , it is necessary to evaluate the perturbationassociated with the subspace-based CE procedure given by (16).

7First-order equality means that, as the sample size � approaches infinity,we neglect all the summands that tend to zero faster than the norm of the corre-sponding perturbation term.

WL-SMI-CEWL-SUB-CE

(29)

WL-SMI-CE

WL-SUB-CE(30)



Lemma 2: Given the estimate of the sig-nature , where the channel estimate is the solution of (16),the perturbation can be expressed as

(31)

where, with and defined in Lemma 1, and

.Proof: See Appendix B.

Accounting for (31) and Lemma 1, the overall perturbationof the WL-SMI-CE and WL-SUB-CE weight vectors can beexpressed as a linear function of , as summarized by thefollowing Lemma.

Lemma 3: The first-order overall perturbation termof the WL-SMI-CE and WL-SUB-CE receivers

can be expressed in a unified manner as

(32)

where , withand given by

Lemma 1, whereas is defined in Lemma 2.It should be observed that Lemma 3 provides a compact char-

acterization of the overall perturbation , which is obtainedunder the simplifying assumption [32] that the error in esti-mating is mainly due to the term . Equipped with sucha nice result, we are now in the position to evaluate the average

at the denominator of (25). Dropping the sub-script in for notational simplicity, by accounting for(32) and using the trace identity, we have

(33)

where, moreover, by virtue of assumptions a1) and a2), it can beshown (see [1] for details) that .Therefore, by substituting such relation in (33), and the resultback in (25), we get

SINRSINR

SINR(34)

The final result is obtained by evaluating the trace term in (34),on the basis of the different expressions for given byLemmas 1–3. In order to do this, it is convenient to considerthe SMI and SUB cases separately. With reference to theWL-SMI-CE receiver, it is shown in Appendix C that

(35)

where .Instead, as regards the WL-SUB-CE receiver, it is shown inAppendix C that

(36)

The trace expressions (35) and (36) are still too complicated toallow for a simple discussion, but they can be considerably sim-plified in the high-SNR region, i.e., by studying their behavioras . Let us first examine the trace term, which is presentin both (35) and (36). One has

(37)

Therefore, for , observing that and usingalso the trace properties, one has

(38)

where we refer to Appendix C for a formal proof of the result. In addition, as , it can

be easily checked that and . Conse-quently, accounting for (34)–(36) and (38), the SINR behaviorin the high-SNR region of the WL-SMI-CE and WL-SUB-CEreceivers is (approximately) governed by

SINR SINRSINR

SINR(39)

SINR SINRSINR

SINR(40)

which are directly comparable8 to [1, eq. (44)] and [1, eq. (47)].Our simulation results show that (39) and (40) accurately predictthe SINR performances of the WL-SMI-CE and WL-SUB-CEreceivers not only in the high-SNR regime, but also for mod-erate values of the SNR, wherein many systems of practical in-terest are envisioned to operate. A first exam of the obtainedexpressions shows that, for , both receivers attain themaximum SINR equal to SINR . A more interestingcomparison is between (39)–(40) and the corresponding ones

8Equation (40), as well as the subsequent (50), is similar to [34, eq. (38)],which was derived for the L-MV projection beamformer by considering how-ever only the effects of ACM estimation.



derived in [1] in the known-channel case. For the WL-SUB re-ceiver, such a comparison shows that the SINR when the channelis estimated is the same as that obtained when the channel isknown, namely, for moderate-to-high values of the SNR, theWL-SUB-CE receiver (approximately) pays no penalty w.r.t.its counterpart employing the exact channel. Such a result in-directly shows the reliability of the considered subspace-basedCE procedure, which simultaneously exploits the channel infor-mation contained in both and by jointly processingthe received vector and its conjugate version . Sur-prisingly enough, the SINR of the WL-SMI-CE turns out tobe even better than that of the corresponding WL-SMI receiverwith known channel: as a matter of fact, this phenomenon iswell-known in the array processing literature (see, e.g., [31] and[34]), where it is sometimes referred to as signature mismatch,and its effects vanish only when . As a byproduct,(39) and (40), together with their corresponding ones derivedin [1] in the known-channel case, provide the SINR assessmentof the signature mismatch problem, thereby showing the sim-plicity and insightfulness of our SINR formulas. For a finitesample-size , indeed, accounting for [1, eq. (44)] and (39), theSINR degradation due to signature mismatch in the high-SNRregion is given by

SINRSINR

(41)

which increases with the channel length. Another interestingconclusion that can be drawn from (39) and (40) is that, notdifferently from the case [1] where the channel is known, thedata-estimated receivers exhibit a SINR saturation effect, forvanishingly small noise. Indeed, when and is full-column rank , it has been shown in [1, Subsec.IV-A] that SINR grows without bound. Thus, for

, accounting for (39) and (40), we get

SINR (42)

SINR (43)

which show that, in the high-SNR regime, the performance ofthe WL-SMI-CE receiver does not depend on the number ofusers , but it depends on the processing gain as well ason the channel length of user , whereas the performanceof the WL-SUB-CE receiver is independent of both the pro-cessing gain and the channel length , while depending onthe number of users .

IV. COMPARISON BETWEEN L- AND WL-MOE RECEIVERS

WITH CHANNEL ESTIMATION

As done in [1] in the case of known channel, it is interestingto compare the SINR performances of the data-estimatedWL-MOE receivers with CE based on (16) against the data-es-timated L-MOE receivers with CE based on the algorithmof [17]. Similarly to the WL-MOE one, the synthesis of theL-MOE receiver given by (9) or (11) involves estimation from

the received data of in (9) or its EVD in (8), as well as anaccurate estimate of in both cases. Under mild conditions, aconsistent estimate of is given by

(44)

Let collect the eigenvectors associated withthe eigenvalue of , under condition c1), channel estima-tion can be blindly carried out [17] by exploiting the orthogo-nality between the signal subspace and the noise subspace

, thus obtaining

(45)

where we have also used (13). In this case, (45) uniquely char-acterizes the channel coefficients for each user iff the followingcondition holds: c4) the null space of has dimensionone or, equivalently,9 . A discussionabout condition c4) is made in [17]. If condition c4) is satisfied,then an arbitrary unit-norm vector satisfies (45) iff

, with and . Itis noteworthy that fulfillment of condition c4) requires that thenumber of rows of the matrix be greaterthan or equal to its number of columns, i.e.,

, and, hence, from the point of view10 of the thuser, the maximum number of users that canbe supported by the system is smaller than the number ofusers when the channel is assumed to be perfectly known. Ob-serve that the maximum number of allowable users for the linearcase is exactly one-half of the corresponding number for the WLcase. In practice, when both blind L and WL re-ceivers can be utilized, whereas foronly the blind WL receivers can work (note that the above lim-itations are mainly due to the considered blind channel identifi-cation procedure). Hereinafter, we assume that condition c4) issatisfied. When (and, hence, ) is estimated from a finitesample size, a channel estimate can be obtained by solving(45) in the least-squares sense

(46)

where the matrix is the sample estimate of. The solution [36] of (46) is the eigenvector associated with

the smallest eigenvalue of the matrix. By substituting in (9) and (11) the sample ACM (44)

and its EVD, respectively, as well as the resulting estimate

9The dimension of the null space of � � � is equal to� � �� .

10Following footnote 5, the maximum number of users that can be supportedby the system is given by � � �� .



of the signature [see (13)] in both, we obtain theL-SMI-CE and L-SUB-CE receivers defined as

(47)

(48)

As for WL receivers, while (9) and (11) are perfectly equivalent,their estimated counterparts (47) and (48) are different, evenwhen . The performance analysis of the L-SMI-CE andL-SUB-CE receivers is complicated from the fact that the SINR(19) must again be evaluated but, differently from the WL ones,linear receivers do not exhibit the CS property, sincein (2). Such an analysis is similar in principle to the one carriedout in [28]–[30], but the approach adopted here leads to moreeasily interpretable results, which are directly comparable withthose obtained in the WL case, at the cost of a minimal loss inaccuracy. To avoid a burdensome treatment, we will report onlythe final results and defer their synthetic proofs to Appendix D.By assuming that both and are known exactly, it turns outthat, in the high-SNR regime, the output SINR of the L-SMI-CEand L-SUB-CE receivers can be approximately written as

SINR SINRSINR

SINR(49)

SINR SINRSINR

SINR(50)

Our simulation results show that the SINR performances of theL-SMI-CE and L-SUB-CE receivers are accurately described by(49) and (50) even when the SNR assumes moderate values. Dueto the similarity between the SINR expressions obtained for Land WL receivers, most observations of Section III-A regardingthe comparison between receivers with or without CE apply alsoin this case. Summarizing, the SINR of the L-SUB-CE receiverturns out to be (approximately) equal to that of the L-SUB one.Moreover, due to the mentioned signature mismatch problem[31], [34], the SINR of the L-SMI receiver with known channelis worse than that of the corresponding L-SMI-CE receiver: in-deed, for a finite sample size , in the high-SNR regime, it re-sults that

SINRSINR

(51)

Additionally, similarly to the WL case, the data-estimated linearreceivers exhibit a SINR saturation effect, for . In thiscase, if is full-column rank , it is readily verifiedthat SINR . Henceforth, for ,accounting for (49) and (50), one obtains

SINR (52)

SINR (53)

which show that, at high SNR, the performance of theL-SMI-CE receiver depends on the processing gain and thenumber of users , as well as on the channel length of theth user, whereas the performance of the L-SUB-CE receiver

is independent of both the processing gain and the channellength , while depending on the number of users .

At this point, we are able to establish a direct compar-ison between L- and WL-MOE receivers with CE, focusingour attention to the case , whereinboth L- and WL-MOE receivers with CE can work [noteindeed that the WL-MOE with CE can accommodate upto users]. By comparing (40)and (50) for the subspace-based receivers, it turns out thatSINR SINR for any value of

. Instead, for the SMI-based receivers [see (39) and (49)],it results that SINR SINR onlywhen , where

SINR SINR(54)

is a threshold sample size, that is, similarly to the known channelcase described in [1], the WL-SMI-CE receiver assures a per-formance advantage only by processing a sufficient number ofsamples.11 Finally, for , as regards the compar-ison between the saturation SINRs (i.e., the SINRs for )of the L- and WL-MOE receivers with CE, it can be observedfrom (42)–(43) and (52)–(53) that the value for the L-SMI-CEreceiver is better than the corresponding value for WL-SMI-CE,whereas the saturation SINRs for the subspace-based receiversare exactly coincident.

V. NUMERICAL EXAMPLES

In this section, Monte Carlo simulations are presented, aimedat validating and extending our performance analysis. We con-sider a DS-CDMA system withand . The users employ unit-norm (i.e., )random signatures , whose entries are i.i.d. random vari-ables assuming equiprobable values in the complex set

, with and statistically indepen-dent of each other for . The channellengths are , i.e., they are equal forall the users, and, as in [24], [30], the entries of the unit-normchannel vectors are randomly and independently drawn withequal power from a zero-mean complex circular (or proper)Gaussian process. The symbol and noise sequences are gen-erated according to assumptions a1) and a2), and the SNR isdefined as . In each simulation, we carry out indepen-dent Monte Carlo runs, with each run employing a different setof spreading sequences, channel vectors, symbol sequences andnoise. In all simulations, we assume that the users have identicalpowers, i.e., there is perfect power control, and, without loss ofgenerality, that the desired user is the first one, i.e., . Note

11A comparison with [1, eq. (53)] shows that, in the estimated-channel case,the value of � is slightly lower.



Fig. 1. ASINR versus SNR for WL-MOE receivers (� � �� users and � �

�� symbols).

Fig. 2. ASINR versus SNR for L-MOE receivers (� � �� users and� � ��

symbols).

that, in the considered scenario, the maximum number of usersthat can be accommodated by the receivers with CE is equalto for the L-MOE receivers andfor the WL-MOE receivers. To extensively compare WL-MOEand L-MOE receivers, we assume that the number of userssatisfies the first, more stringent condition, exception made forthe second experiment, where we evaluate the performances asa function of .

Example 1: In this experiment, we evaluate the averageSINR (ASINR) as a function of SNR for the WL-MOE (Fig. 1)and L-MOE (Fig. 2) receivers (both with and without CE), for

users and a sample size equal to symbols.For the sake of comparison, we also report the ASINR of theexact (i.e., data-independent) WL-MOE and L-MOE receiversgiven by (6) and (9), respectively. All the curves show a goodagreement between simulation and analytical results. Lookingin detail at Fig. 1, the simulation results confirm the theoreticalprediction that the two subspace versions of the WL-MOE

receiver (with or without CE) exhibit practically the same per-formances, whereas the WL-SMI-CE receiver performs slightlybetter than the WL-SMI one (with known channel), since thelatter is penalized by the signature mismatch phenomenon;in particular, the asymptotic (for ) differencebetween the ASINR curves of the WL-SMI-CE and WL-SMIreceivers is about 1.5 dB, which is in good agreement with thevalue theoretically predicted by (41). Similar considerationsapply to Fig. 2, where the asymptotic gain of the L-SMI-CEreceiver over the L-SMI one (with known channel) is about1 dB, as correctly predicted by (51). As regards the comparisonbetween WL-MOE and L-MOE receivers, results of Figs. 1 and2 allow us to extend an important conclusion of our previouswork [1], relative to the underloaded case (i.e., ): al-though the exact WL-MOE receiver generally exhibits a SINRgain over the L-MOE one also when , in practice,due to SINR saturation effects, the subspace implementationsof the WL-MOE and L-MOE receivers exhibit the same perfor-mances, whereas the L-SMI receivers (both with and withoutCE) outperform their WL-SMI counterparts.

Example 2: In this experiment, we evaluate the ASINR asa function of the number of users for the WL-MOE (Fig. 3)and L-MOE (Fig. 4) receivers (both with and without CE), fora sample size equal to symbols and dB.Since the subspace-based CE procedure poses a strict limit of

users for the WL-MOE receivers andfor the L-MOE receivers with CE, the performances of the re-

ceivers with CE are not reported (i.e., the corresponding curvesare truncated) for values of exceeding these limits. Besidesconfirming again a good agreement between simulation and an-alytical results, the curves for the WL-MOE receivers (Fig. 3)show that the performance advantage of the WL-SUB receiverover the WL-SMI one (both with and without CE) progressivelydecreases as increases, becoming negligible in correspon-dence of about users for the receivers with CE, and

users for the receivers with known channel. It is worthwhileto observe, moreover, that when approaches the upper limit

for CE, the performances of the WL-MOE re-ceivers with CE degrade rapidly, suffering from a clear thresholdeffect. Similar considerations apply to Fig. 4, where, however,the ASINR curves of the L-MOE receivers are more closelyspaced and the performance advantage of the L-SUB receiverover the L-SMI one becomes negligible in correspondence ofabout users for the receivers with CE, and usersfor the receivers with known channel. A careful comparisonbetween the performances of WL-MOE and L-MOE receiversshows again that the largest advantage in using WL-MOE re-ceivers is obtained in the “overloaded” region, i.e., when

for the receivers with CE (where the L-MOE receiverscannot operate at all), and when for the receiverswith known channel (where the L-MOE receivers, although ca-pable of operating, exhibit poor performances).

Example 3: In this last experiment, we report the ASINRas a function of the sample size for the WL-MOE (Fig. 5)and L-MOE (Fig. 6) receivers (both with and without CE), for

users and dB. The ASINR values of theexact (i.e., data-independent) WL-MOE and L-MOE receivers,in this scenario, are equal to 21.5 and 19.2 dB, respectively, and



Fig. 3. ASINR versus number of users � for WL-MOE receivers (��

dB and � � �� symbols).

Fig. 4. ASINR versus number of users � for L-MOE receivers (��

dB and � � �� symbols).

obviously do not depend on . The simulation and analyticalresults are again in good agreement, and, as expected, the ac-curacy of the formulas (39)–(40) and (49)–(50) improves asincreases. In particular, Fig. 5 shows that the two versions ofthe WL-SUB receivers (with or without CE) exhibit almost thesame performances, outperforming the WL-SMI-CE receiverby about 2 dB, and the WL-SMI one (with known channel) byabout 3 dB, for all considered values of . Instead, the ASINRcurves of the L-MOE receivers (see Fig. 6) are more closelyspaced, exhibiting only marginal differences in performancesbetween the various receivers. By comparing Figs. 5 and 6, itcan be seen that the two WL-SUB receivers (with or withoutCE) outperform the corresponding L-SUB ones, for all the con-sidered values of . In contrast, the WL-SMI receiver (withknown channel) again performs worse than its linear counterpartfor all values of (in this case the threshold sample size eval-uated as in [1] is , thus larger than the maximumvalue of considered in the simulations), whereas the

Fig. 5. ASINR versus sample size � for WL-MOE receivers (� � �� usersand �� dB).

Fig. 6. ASINR versus sample size� for L-MOE receivers (� � �� users and�� dB).

performances of the WL-SMI-CE receiver approaches those ofthe L-SMI-CE one for approaching 2500, which agrees verywell with the value predicted by (54).

VI. CONCLUSION

We presented a comprehensive performance comparisonbetween different versions of the L- and WL-MOE receiverswith blind CE, when both the ACM and the channel im-pulse response of the desired user are estimated from a finitesample-size. The analysis extends our previous study [1] andthe obtained formulas are fully supported by computer simu-lation results. The answers to the two questions put forward inthe abstract are the following ones. With reference to their sub-space-based implementations, for moderate-to-high values ofthe SNR, errors in estimating the L-SUB-CE and WL-SUB-CEreceivers are essentially due to ACM estimation. The same isnot true for the L-SMI-CE and WL-SMI-CE receivers, imple-mented by using the sample ACM directly, for which CE errors



undesirably combine with ACM errors (signature mismatchphenomenon); however, compared with the known-channelcase, CE errors adversely affect the SINR performances ofL-SMI-CE and WL-SMI-CE receivers in a similar way. In con-clusion, when considering finite sample-size implementation,the more sophisticated subspace-based implementation is aneffective method to assure that the WL-MOE receiver (withor without CE) significantly outperform (for low-to-moderatevalues of the SNR) its linear counterpart. In this case, for agiven channel length, the WL-MOE receiver allows one towork with an increased number of users , which makes it aviable choice in heavily congested DS-CDMA networks. Afuture interesting development is the extension of our analysisto other robust multiuser detectors, e.g., those belonging to thefamily of diagonal loading methods.

APPENDIX

PROOFS

A. Proof of Lemma 1

It is shown in [1, Proof of Lemma 2] that, for mod-erate-to-high values of the sample size, i.e., , thesample ACM (12) can be decomposed as ,

where , with

. Consequently, admitsthe first-order approximation .

First, let us consider the SMI-CE implementation (17) ofthe WL-MOE receiver. Substituting the previous approxi-mation of and in (17), after somealgebraic manipulations, one obtains the first-order approxi-mation of the weight vector (55), shown at the bottom of the

page, with. Observe

that, by virtue of (6), the matrix can be equiva-lently expressed as . Substituting

the expression of in , and observingthat and

, one has

(56)

Since both and exhibit theCS property, the scalar is realand, thus,

. Consequently

(57)

At this point, we focus attention on the SUB-CE implemen-tation (18) of the WL-MOE receiver. When the EVD is appliedto the sample ACM given by (12), for a sufficiently largesample size , the matrices and can be decomposed[30], [37] as and , whereand represent the resulting perturbation in the estimatedsignal subspace, whose norm is of the order of . More-over, it results [30], [37] that ,

with , and. Consequently, we can write

(58)

Observe that, since , one has .Hence, using (58), accounting for the first-order perturbations of

and , and remembering that , one obtains

(59)

Substituting (58) and (59) in (18), after some tedious butstraightforward algebra, the first-order approximation of theweight vector can be concisely written as

(60)

where

(61)

(55)



(62)

Then, substituting the expression of the perturbationin (61), remembering again that

and, one gets

(63)

with .Moreover, using again the fact that

and observing that, by virtue of the EVD properties,, the perturbation term (62) can be

rewritten as in (64) shown at the bottom of the page.

B. Proof of Lemma 2

For a sufficiently large sample size , when the EVD is

applied to , where

, with ,the matrix can be decomposed [30], [37] as

and the perturbation in the estimated noise subspacehas the following form , with

. By substituting theexpression of and noticing that , oneobtains

(65)

The perturbation induces an error in the channel estimategiven by (16), which assumes the form ,

where represents the CE error. Remembering thatis the estimate of the signature

, one easily gets . Accordingto (15), the channel vector is the unique eigenvector corre-sponding to the null eigenvalue of ,

with . The sample es-timate of matrix can be de-composed as where, accounting for(65), the perturbation has the form

(66)

Based on (15), one has

, which implies that, whose min-

imal-norm least-squares solution [38] is given by

(67)

since is unitary. Substituting (66) in (67) and observing that,due to (15), , one has

(68)

from which we finally have

(69)

C. Evaluation of

Initially, we will proceed in a unified manner bytreating the SMI and SUB cases jointly. Since

(see Lemma 3), using the linearityproperty of the trace operator and observing that isHermitian (see Lemma 1), we can write

(70)

(64)



By invoking the properties of the trace operator, it follows that:

(71)

which shows that the third summand in (70) is the con-jugate version of the second one. Moreover, rememberingthat and , andaccounting for the expressions of (see Lemma 1)and (see Lemma 2), it can be directly verifiedthat and

. Thus, the first summand in (70) becomes, whereas the fourth one

reduces to ,where we have also used the properties of the trace operatoragain. This last trace can be further expanded by replacing

with its expression given in Lemma 2: in particular,using , remembering that

and , and observingthat, on the basis of the Moore-Penrose conditions [38],

, one has

. Consequently, sub-stituting also the expression of in the second summandof (70), we get

(72)

At this point, we have to consider the SMIand SUB cases separately. Let us start from theSMI case, for which and

. Following [1], it canbe shown that .As regards the second summand in (72), we observe that

, where the secondequality follows by noticing that and

. Henceforth, observing thatand using the trace properties, the second summand in (72)becomes

(73)

with ,where the last equality comes from the fact thatis the orthogonal projector onto the subspace

(see the Moore definition of the generalized inverse[38]) and, hence,12

, while by virtueof condition c3). Considering the third summand in (72),we note that

which, usingthe facts that

and , endsup to . Con-sequently, the third summand in (72) assumes the form

. Thus, we have proven(35).

Let us consider now the SUB case, wherein

(74)

(75)

with .Following [1], it can be shown that

(76)

As to the second summand in (72), since, with

and , we obtain thatand

.Consequently, we get

(77)

where we have used the facts that

and . Therefore, observing again thatand using the trace

properties, the second summand in (72) simplifies to

(78)

With reference to the third summand in (72), we notethat

which,exploiting the EVD

12If � is an eigenvalue of the orthogonal projector �� , then � �� .



and its related properties (as done for the second sum-mand), and using the facts that

and , boils down to.

Consequently, the third summand in (72) assumes the form

(79)

Thus, we have also proven (36).

D. Performance Analysis of the L-SMI-CE and L-SUB-CEReceivers

Let us denote with any data-estimated L-MOE receiver,i.e., L-SMI-CE or L-SUB-CE, and set

for simplicity. When a linear data-estimatedreceiver is employed [i.e., and in (2)],accounting for (4), (5) assumes the form

SINR

(80)

It is important to observe that, differently from the WL case, thereal parts in (80) cannot be omitted, since andare in general complex-valued quantities. This fact significantlycomplicates the analysis with respect to the WL case. Assumethat , where is a small (i.e.,

) zero-mean perturbation term, and let be the averagew.r.t. . Thus, one has

(81)

and

(82)

with and

(83)

since by (9), is zero-mean by assumption, and. Consequently, it

follows:

(84)

Similarly to the WL case, we assume that the weight vectoris independent from the data vector . Let denotethe joint average w.r.t. and , using again the identity

, performing the av-erage w.r.t to , and recalling that, due to assumptions a1)and a2), the vector is zero-mean, one obtains

(85)

At this point, noticing that and, collecting all the previously obtained results

and substituting in (80), we, thus, get (86) shown at the bottomof the page, where we have also accounted for (10) and used theproperties of the trace operator. The following Lemma gives afirst-order characterization of the perturbation vector .

Lemma 4: Given the estimate ofthe signature , where the channel estimate is the solutionof (46) and is a small (i.e., ) zero-mean pertur-bation term, the first-order perturbation term of the L-SMI-CEand L-SUB-CE receivers can be expressed as

(87)

with

(88)

(89)

where , with

(90)

and the random vectorbeing the sample estimate of the cross-correlation between

the disturbance vector and the desired symbol ,whereas

L-SMI-CEL-SUB-CE

(91)

SINRSINR

SINR(86)



L-SMI-CE

L-SUB-CE

(92)

and , with

being an oblique projection matrix [32] and

,

while the diagonal matrixcollects the nonzero eigenvalues of , and, finally,

.Proof: The proof can be conducted along the same lines

of Appendices A and B, with the additional complication that,contrary to , the scalar product is complex ratherthan real.

By virtue of Lemma 4, the overall perturbation of theL-SMI-CE and L-SUB-CE weight vectors can be expressed,similarly to the WL case, as a linear function of

, that is, ,

where , with

and . Therefore, the SINRin (86) assumes the form shown in (93) at the bottom ofthe page, where, by virtue of assumptions a1) and a2), wehave used (see [1] for details) the fact that

and , with

. The matrixhas a particular block structure where the lower-right block

is the conjugate of the upper-left one , and thelower-left block is the conjugate of the upper-right

one . Moreover, since

and , onehas and

and, hence, and

. By exploiting the block structure ofand , it follows that

(94)

Using the expressions of and , after simple manipu-lations, we get

(95)

where . There-fore, proceeding similarly to the WL case (see Appendix C) andaccounting for [1, eq. (79)], it can be verified that, with referenceto the L-SMI-CE receiver, the first trace term in (93) is given by

(96)

whereas, for the L-SUB-CE receiver, one obtains

(97)

As regards the other trace term in (93), proceeding as done forthe first one, it can be shown that

(98)

Since, in addition to , the fact that

also implies that, by resorting to the properties of the trace oper-

ator, one has

(99)

SINRSINR

SINR(93)



where it is verified that and. Moreover, observing that

is symmetric, substituting the expression ofand [see (9)], one obtains

(100)

where we have also observed that, with reference to bothL-SMI-CE and L-SUB-CE receivers,

, since and more-over, that

. The evaluation of the trace terms at the lasthand of (99) and (100) are complicated and, to obtain manage-able expressions, it is convenient to consider their asymptoticvalues as . Using the limit formula for the generalizedinverse [38], one gets

(101)

with , where we haveobserved that . Similarly, it can be verified that

and . Hence-forth, noticing that the trace term at the last hand of (99) hasbeen evaluated in [1, Appendix E], accounting for (101), it canbe shown that, with reference to both L-SMI-CE and L-SUB-CEreceivers

(102)

(103)

Finally, for , as in the WL case (see Section III-A), onehas and

. Thus, it follows from (96) and (97) that

L-SMI-CEL-SUB-CE

(104)

By substituting (103) and (104) into (93), (49) and (50) areeasily obtained.

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Angela Sara Cacciapuoti was born in Pozzuoli,Italy, on August 19, 1980. She received the Dr. Eng.degree summa cum laude in telecommunications en-gineering in 2005 and the Ph.D. degree in electronicand telecommunications engineering in 2009, bothfrom the University of Napoli Federico II.

She is currently an Assistant Researcher with theDepartment of Biomedical, Electronic and Telecom-munication Engineering, University of Napoli Fed-erico II. Her research activities lie in the area of statis-tical signal processing, digital communications, and

communication systems. In particular, her current interests are focused on equal-ization, channel identification, and multiuser detection.

Giacinto Gelli was born in Napoli, Italy, on July29, 1964. He received the Dr. Eng. degree summacum laude in electronic engineering in 1990, andthe Ph.D. degree in computer science and electronicengineering in 1994, both from the University ofNapoli Federico II.

From 1994 to 1998, he was an Assistant Professorwith the Department of Information Engineering,Second University of Napoli. Since 1998, he hasbeen with the Department of Biomedical, Electronicand Telecommunication Engineering, University of

Napoli Federico II, first as an Associate Professor, and since November 2006,as a Full Professor of Telecommunications. He also held teaching positionswith the University Parthenope of Napoli. His research interests are in the fieldsof statistical signal processing, array processing, image processing, and mobilecommunications, with current emphasis on code-division multiple-accesssystems and multicarrier modulation.

Luigi Paura was born in Napoli, Italy, on February20, 1950. He received the Dr. Eng. degree summacum laude in electronic engineering in 1974 from theUniversity of Napoli Federico II.

From 1979 to 1984 he was with the Department ofBiomedical, Electronic and Telecommunication En-gineering, University of Napoli, first as an AssistantProfessor and then as an Associate Professor. Since1994, he has been a Full Professor of Telecommuni-cations: first, with the Department of Mathematics,University of Lecce, Italy; then, with the Department

of Information Engineering, Second University of Napoli; and, finally, since1998, he has been with the Department of Biomedical, Electronic and Telecom-munication Engineering, University of Napoli Federico II. He also held teachingpositions with the University of Salerno, Italy, the University of Sannio, Italy,and the University Parthenope of Napoli. During 1985–1986 and 1991, he was aVisiting Researcher with the Signal and Image Processing Laboratory, Univer-sity of California, Davis. At the present time, his research activities are mainlyconcerned with statistical signal processing, digital communication systems,and medium access control in wireless networks.

Francesco Verde was born in Santa Maria CapuaVetere, Italy, on June 12, 1974. He received the Dr.Eng. degree summa cum laude in electronic engi-neering in 1998 from the Second University of Napoli,and the Ph.D. degree in information engineering in2002 from the University of Napoli Federico II.

Since 2002, he has been an Assistant Professor ofSignal Theory with the Department of Biomedical,Electronic and Telecommunication Engineering,University of Napoli Federico II. He also heldteaching positions with the Second University of

Napoli. His research activities lie in the broad area of statistical signal pro-cessing, digital communications, and communication systems. In particular,his current interests include cyclostationarity-based techniques for blind identi-fication, equalization and interference suppression for narrowband modulationsystems, code-division multiple-access systems, multicarrier modulationsystems, and space-time processing for cooperative communications systems.


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