Blind adaptive joint multiuser detection and ... - Semantic Scholar

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Blind Adaptive Joint Multiuser Detection and Equalization in Dispersive Differentially Encoded CDMA Channels Stefano Buzzi, Member, IEEE, Marco Lops, Senior Member, IEEE, and H. Vincent Poor, Fellow, IEEE

Abstract—The problem of blind adaptive joint multiuser detection and equalization in direct-sequence code division multiple access (DS/CDMA) systems operating over fading dispersive channels is considered. A blind and code-aided detection algorithm is proposed, i.e., the procedure requires knowledge of neither the interfering users’ parameters (spreading codes, timing offsets, and propagation channels), nor the timing and channel impulse response of the user of interest but only of its spreading code. The proposed structure is a two-stage one: The first stage is aimed at suppressing the multiuser interference, whereas the second-stage performs channel estimation and data detection. Special attention is paid to theoretical issues concerning the design of the interference blocking stage and, in particular, to the development of general conditions to prevent signal cancellation under vanishingly small noise. A statistical analysis of the proposed system is also presented, showing that it incurs a very limited loss with respect to the nonblind minimum mean square error detector, outperforms other previously known blind systems, and is near-far resistant. A major advantage of the new structure is that it admits an adaptive implementation with quadratic (in the processing gain) computational complexity. This adaptive algorithm, which couples a recursive-least-squares estimation of the blocking matrix and subspace tracking techniques, achieves effective steady-state performance. Index Terms—Blind multiuser detection, CDMA, channel estimation, fading channels, Moore–Penrose pseudo inversion, subspace approach.

I. INTRODUCTION

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IRECT-SEQUENCE code division multiple access (DS/CDMA) is the proposed basic technology for the realization of the air interface of most third-generation wireless cellular networks [1]–[3]. Indeed, when compared to other conventional multiple access techniques, such as those based on time and/or frequency division concepts, CDMA appears to Manuscript received April 25, 2001; revised December 30, 2002. This work was supported in part by the National Science Foundation under Grant ECS-9811095 and in part by the New Jersey Center for Wireless Telecommunications. This paper was partly presented at the SIAM Conference on Linear Algebra in Signals, Systems and Control, Boston MA, August 2001, and at the 2002 IEEE Wireless Communications and Networking Conference, Orlando FL, March 2002. The associate editor coordinating the review of this paper and approving it for publication was Dr. Alex C. Kot. S. Buzzi was with the Department of Electrical Engineering, Princeton University, NJ 08544 USA. He is now with DAEIMI, Università degli Studi di Cassino, Cassino, Italy (e-mail: [email protected]). M. Lops is with DAEIMI, Università degli Studi di Cassino, Cassino, Italy (e-mail: [email protected]). H. V. Poor is with the Department of Electrical Engineering, Princeton University, Princeton, NJ 08544 USA (e-mail: [email protected]). Digital Object Identifier 10.1109/TSP.2003.812732

be very advantageous, due to increased resistance to multipath distortion, higher immunity to co-channel and overlaid interfering sources, ability to enable a simple random access to the network with light signaling protocols, etc. The capacity of DS/CDMA systems can be enhanced through the use of multiuser detection, which addresses the key problem of mitigating the multiaccess interference (MAI) [4]. The most prominent results in this area include the development of several suboptimal multiuser detectors that overcome the exponential complexity of the optimum (in the minimum error probability sense) maximum likelihood (ML) multiuser receiver at the price of some performance degradation, as well as the application of well-known adaptive filtering algorithms, such as least-mean-squares (LMS), recursive-least-squares (RLS), and subspace-based methods [5]–[9]. More recently, research in this area has focused on the problem of multiuser detection in fading dispersive channels, due to the increased bandwidth and data-rates of emerging CDMA systems. In particular, there has been recent interest in the development of detection algorithms, possibly blind and adaptive, capable of coping with both MAI and intersymbol interference (ISI). If training data can be transmitted, the linear minimum mean square error (MMSE) receiver can be adaptively implemented with no knowledge of the channel realization or of the interfering users’ signatures and parameters [8]. However, the use of training data reduces the communication throughput, shortens the mobile transceivers battery life, and must be coupled with appropriate signaling protocols. In order to overcome these limitations, significant research effort has been directed toward the development of channel estimation algorithms [10], [11] and blind multiuser receivers requiring as little as possible a priori information on the interfering signals and on the fading channel realization. In [12], Tsatsanis proposes a blind receiver based on straightforward modification of the minimum mean-output-energy (MOE) blind multiuser detector, which was first proposed in [5] with reference to nonfading channels. For a multipath channel, he considers as a “signal of interest” just one (usually the one with the largest average energy) of several received signal replicas. The proposed procedure thus requires knowledge of the propagation delay for at least one replica of the signal of interest; moreover, it does not exploit the energy contained in the remaining replicas of the signal of interest. Such an approach is refined in [13], wherein, based on a min-max formulation, a receiver of the MOE family is introduced and analyzed. The proposed receiver, while exploiting all of the received

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signal energy, still requires knowledge of the timing of the user of interest. Alternatively, in [14], the subspace approach is employed in order to develop a blind implementation of the linear MMSE receiver. Despite very good performance, the proposed procedure can be implemented only through batch processing of the observables with a computational complexity cubic in the processing gain, i.e., it does not lend itself to a recursive, quadratic-complexity implementation. Finally, other works on this subject include [15], wherein a receiver equipped with an antenna array is considered, and [16], in which a novel scheme based on the serial concatenation of a blind and a data-aided algorithm is proposed. In this paper, DS/CDMA systems operating on fading dispersive channels are considered. The contributions of this study may be summarized as follows. First of all, we introduce a new batch blind algorithm for joint multiuser detection and equalization. In particular, the algorithm that we introduce belongs to the family of the code-aided techniques, as it requires knowledge of only the spreading sequence of the user of interest; neither the propagation delays nor the channel realizations need to be estimated. The proposed receiver structure is a two-stage one, wherein a first stage, ensuring the system near–far resistance, accomplishes MAI and ISI suppression, whereas the second stage implicitly achieves channel estimation and signal-to-noise ratio (SNR) optimization. As a side result, it is also shown that the proposed algorithm provides, as a by-product, an estimate of the propagation channel impulse response for the user of interest. The second contribution of this work is the development of an analytical framework under which the “minimal” conditions required to prevent cancellation of the useful signal at high SNRs may be easily determined. This result provides useful insights into how the signal cancellation problem in MOE applications may be handled. Finally, the third contribution of this work is the development of a recursive, quadratic-complexity, adaptive implementation of the proposed receiver. In particular, this procedure relies on a coupling of RLS with the stochastic-gradient-based subspace tracking technique described in [17]. With regard to the performance analysis, we also present some simulation results, contrasting the bit-error-rate (BER) of the proposed receiver with that of previously derived blind structures and of the nonblind, nonadaptive, linear MMSE receiver. Some examples illustrating the convergence dynamics of the adaptive rule are also given. The rest of this paper is organized as follows. In Section II, we give some results from linear algebra that will be used throughout the paper. Section III is devoted to the problem formulation and to the signal model, whereas the structure of the proposed blind receiver is described in Section IV. In Section V, some simulation results are presented and discussed in order to give insight into the system performance, whereas in Section VI, the issue of recursive receiver implementation is addressed. Finally, concluding remarks are given in Section VII. , , and Notation: In the following, the superscripts denote conjugate, transpose, and conjugate transpose, denotes statistical expectation; the symbols respectively; and denote the Kronecker product and the Schur (i.e., denotes component-wise) matrix product, respectively; real part; column-vectors and matrices are indicated through

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boldface lowercase and uppercase letters, respectively. The symbol denotes an all-zero column vector of dimension , and denote the identity and all-zero whereas the terms square matrix of order , respectively; if is a square matrix, denotes a vector whose entries are the diagonal Diag elements of the matrix , whereas if is a vector, Diag denotes a diagonal matrix containing the entries of the vector on its diagonal. The set of all the -dimensional matrices . Finally, given a with complex entries is denoted by , denotes the (column) span of the matrix spanned by the matrix , namely, the vector subspace of columns of . II. SOME PRELIMINARY RESULTS FROM LINEAR ALGEBRA In this section, we present briefly some results and definitions from linear algebra that will be exploited in the rest of the paper. We start with the following Theorem 1 (Riedel’s theorem [20]): Let and be posi; ; ; tive integers with ; and . Assume that , , , , is is of full rank, and is of full rank. Assume invertible, , and let and also1 that . Then

(1) denotes the Moore–Penrose generalwhere the superscript ized inverse. Proof: See [20]. Riedel’s theorem represents a generalization that is valid under the conditions specified therein, of the (full-rank) matrix inversion lemma [19, p. 50]. with having Now, consider a matrix , and denote by the th column vector of , rank . It is well known that if , then the vectors form a set of linearly independent vectors, whereas , they are a set of linearly dependent vectors. In this if may be latter situation, one or more of the columns of obtained as a linear combination of the remaining columns of . Now, assume and consider the vector for some . The set can be partitioned in two and —with respect to sets—which we denote by according to the following procedure. If is the vector , then set linearly independent of and . If, instead, is linearly dependent , then start collecting in the on the vectors in vector plus all of the possible nondisjoint subsets of linearly and such that independent vectors taken from may be obtained as a linear combination of these vectors. Now, that are not in consider (if any) the remaining columns of

R

R

1As noted in [21], the hypothesis that (Y ) = (Z ) is never used in the proof given in [20], whence the said theorem is also true without this assumption.

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and put them in if they are linearly dependent on ; else, put them in . An example will the vectors in be helpful to clarify these definitions. columns in which Example: Consider a matrix with are linearly independent vectors, whereas and . Let us start by considering ; is linearly dependent on the sets and obviously, , whence we start with ; , we have since is independent of the vectors in . Consider now , which is linearly dependent on the and . As a consequence, we have sets , and . Following along this line, and for it is easily found that . Conversely, since is linearly independent and of all of the remaining vectors, we have . We now introduce the following concept. with Definition 1: Consider a matrix having rank , and denote by the th column vector of , . Let . The vectors and are said to be linearly dependent with respect to the matrix if or, equivalently, if . If, instead, [or, ], then the vectors and are equivalently, if said to be linearly independent with respect to the matrix . , which is deNow, consider the singular matrix fined as

that the channel coherence time is larger than the packet duration , i.e., we consider the case of a slowly varying fading , with the chip interval, the channel. Denoting by the spreading code of system processing gain, by a unit-energy rectangular waveform the th user, and by of duration , we also have

(3) where all of the (unknown) channel characteristics have been , which are defined as placed in the unknown functions (4) is the multipath delay spread, which is asNotice that if sumed to be equal for all of the channels, then the functions have compact support in , where the inclusion stems from the reasonable assumption that , which we henceforth adopt. As a consequence, assuming that the user “0” is the user of interest, the bit can be detected through the windowed observables

(5)

We are interested in studying the matrix product . We have the following result. with , and define Theorem 2: Let . Further, let , denote by its th entry, and, for any , denote by and the matrices containing on their columns the vectors and , respectively. Then, for any pair in the sets , if and are linearly independent with respect to . Proof: See Appendix A. The utility of this result will become clear in the sequel.

In this equation, we have isolated the useful signal, i.e., the sig, from the internature multiplying the bit to be detected ference, which consists of noise, MAI, and ISI. In particular, the contains the contribution from the bits term and , . We now convert the received signal to discrete-time at a rate samples per chip by defining the projections of (6) Thus, the windowed signal (5) can be represented efficiently -dimensional vector through the

III. SIGNAL MODEL Consider a DS-CDMA system with asynchronous users, . The complex enwhose signatures we denote by velope of the observed signal is written as

(7) From (5) and (6), we have

(8)

(2) In this equation, is the transmitted packet is the th bit transmitted by the bit interval, denotes convolution, and impulse response of the th user’s channel.

length, th user, is the is the unknown We assume here

wherein samples analogously with to be given by

and in (6). The term

are defined can be easily shown

(9)

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where, in turn, we have (10) is supported in the interval , it follows that the samples are . As a consequence, on defining equal to zero -dimensional vector the Since the waveform

D

Fig. 1. Block-scheme of the linear receiver. is the interference-suppressing filter; represents the cascade of a whitening filter and a matched filter.

e

signaling interval is contained in the bits and Thus, the decision rule of a linear multiuser detector is

(11) it can be easily shown that the discrete-time observable (7) can be written as

in

. (14)

the incremental phase between where we have denoted by and and, by , a vector to be suitably designed. The detection structure that we propose is outlined in Fig. 1 and, overall, is a two-stage one. In the first stage, the observables undergo the linear transformation (15)

(12) , and are the discrete-time versions of the where , , waveforms , and , respectively, and is the matrix in (13), shown at where the bottom of the page. The representation (12) is extremely powerful. Indeed, with regard to the desired user contribution, it isolates the known (i.e., the matrix ) from the unknown, channel-dependent vector . Notice also that in the above derivation the channel delays have been assumed to be unknown, which explains the redundancy needed in the signal representation. Otherwise stated, the receiver knows that the useful signal is somewhere in the interval , which is thus to be spanned entirely. Of course, the availability of even incomplete information about the channel state (i.e., a rough estimate of the users’ delays) could allow spanning a reduced interval and would eventually lead to a re. This is, for example, the assumption duced-order matrix made in [13], [14], and [16]. IV. SYSTEM DESIGN The receiver family we focus on relies on neither channel state information nor pilot signals, whereby the modulation format cannot be a plain BPSK. We thus assume differential encoding and detection, implying that the information of the th

.. .

is a suitable rectangular matrix to be designed where to reduce, if not completely suppress, MAI and ISI. In the second stage, the purged observables undergo a further linear processing (through the vector ) aimed at bit-error rate (BER) in (14) is given by optimization. Obviously, the vector . Although the two linear blocks may easily collapse into a single linear transformation, the segmentation presented here is essential both to illustrate the design criteria and to develop adaptive implementations of the receiver. For the moment, let us leave aside the problem of designing the matrix and assume that this matrix is able to suppress all of the MAI plus ISI contribution so that the transformed observables can be written approximately as (16) The vector in Fig. 1 can be designed according to a number of optimization criteria. For the sake of simplicity and to enable a viable adaptive implementation of the receiver, we choose to maximize the SNR, i.e., as the cascade of a noise-whitening filter and a filter matched to the resulting useful signature. As to the whitening transformation, it is easily determined by noticing that the covariance matrix of the noise is now . Since this matrix is positive definite and Hermitian, the following Cholesky factorization applies: (17)

.. .

.. . (13)

.. .

.. .

.. .

.. .

.. .

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where is a nonsingular lower triangular matrix. The whitened observables are now given by (18) a known vector, the optimum processing would be at Were is unthis point an ordinary matched filter. However, since known to the receiver, a further step is necessary in the procedure in order to estimate the desired projection direction. To be more definite, consider the covariance matrix of the whitened observables:

(19) is the number of columns of the matrix .2 The where covariance matrix in (19) is thus the sum of a full-rank identity matrix and of a unit rank matrix, the latter admitting as its unique dominant eigenvector. Coneigenvalues of this matrix are sequently, coincident, whereas the largest eigenvalue corresponds to an . Thus, the matched eigenvector that is parallel to filter for the detection problem (18) is given by this eigenvector. Since the covariance matrix (19) is not actually known to the receiver, in practice, it can be replaced by its sample estimate, i.e., we consider the matrix (20)

is the -dimensional set indexing the estimation where epochs. The “estimated” matched filter for the detection problem (18) is given by the principal eigenvector, which we denote by , of the matrix (20). Summing up, the detector can be based on the following procedure. . a) Observe the vector . b) Evaluate . c) Perform the Cholesky factorization into the whitened vector d) Transform the vector . of the vector e) Evaluate the sample covariance matrix through (20). f) Determine the eigenvector corresponding to the maximum . eigenvalue of the matrix g) The vector to be used in the differential decision rule (14) is given by (21) Notice that the complexity of the above recipe is proportional , due to the computations required in steps and , to and, as shown in the following, to the computation of the matrix .

Q

Q =N

M

2So far, has not been specified; however, we anticipate here that in the following, we will show that ( + 1) .

A. Synthesis of the Blocking Matrix Let us now focus on the issue of designing the matrix . Looking at (12) and (13), it is seen that the useful signal is according to an a linear combination of the columns of and is thus a vector in . The supunknown vector matrix. Forcing pression matrix is a rectangular is thus a necessary condition to ensure that survive the first stage all of the principal directions of and that for no nonzero configuration of is the output useful ) nullified. This last signal signature (i.e., the vector be a full-rank square condition requires that the product , i.e., that its determinant be nonzero. matrix of order Several design criteria might be chosen at this point to determine . In the present paper, we consider a generalization of the minimum MOE (MMOE) criterion, which was already introduced with reference to known communication channels [5], as well as to fading dispersive channels [12], [13], and which has been shown to subsume the two key linear multiuser detectors, i.e., the decorrelating and MMSE receivers [9]. The matrix is thus chosen as the solution to the following problem: (22) denotes determinant, and is a proper vector. where Indeed, in what follows, we show that depending on the defini, the criterion (22) leads to either an MMSE-like or tion of to a decorrelating-like receiver. Notice that the usual Lagrangian techniques to determine a constrained minimum are not applicable to solve problem (22) in that the constraint is not linear in the unknowns. Indeed, rebe nonzero amounts to quiring that the determinant of eigenvalues of the said matrix be requiring that the nonzero. In order to circumvent this drawback, we start by obto have serving that a necessary condition for the matrix full rank is that it has no row or column identically zero. This is in turn ensured through the following. -dimensional maProposition 1: Given a given by (13), a necessary condition trix and the matrix to be nonsingular is that there exists a bi-injection for

(23) , where is the th column of such that . the matrix Otherwise stated, the constraint in (22) requires forcing no scalar constraints. Notice also that upon less than proper permutation of the columns of the matrix , the condition of Proposition 1 can be recast into a constraint on the diag. onal of the matrix We thus start by considering the problem (24) Diag -dimensional vector with nonzero wherein is an entries, and we verify whether and under which conditions

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the minimal set of constraints in (24) is sufficient to ensure . Two situations are of theoretical interest here: a) The covariance matrix of

has full rank, and b) this matrix is singular. Situation a) occurs so that the cost function in (22) and (24) is the when classical one for minimum MOE. Situation b), instead, occurs so that (24) can be interpreted as a when constrained minimization of the output MAI-plus-ISI. Likewise, and the noise becomes vanishingly small, the if covariance matrix of the observables tends to singularity. Under situation a), application of standard Lagrangian techniques to (24) yields the solution Diag

(25)

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However, as a side remark, it is worth noticing that the netmay be increased through the oversamwork capacity pling factor .4 Under the above assumptions, it is shown in Appendix B that the solution to problem (22) is given by

Diag

(28)

As a reassuring checkpoint, solution (28) reduces, for the special case of nonfading channel and a priori known user “0” timing, to the linear decorrelating detector [9]. Thus, we refer to the matrix in (28) as a decorrelating-like filter. As already anticipated, evaluation of the matrix in (25) and (28) requires an computational burden. B. Receiver Properties

which is readily shown to fulfill the constraint in (22). Notice also that this solution subsumes, as the special case of nonfading channel with known timing for the user “0,” the classical MMOE solution, which is well-known to be equivalent to the linear MMSE one [5], [9]. Thus, we refer to solution (25) as an MMSE-like receiver. The unsuitability of (25) for the case of vanishingly small noise floor emerges as a consequence of the arguments of [13], where it is shown that

Several nice properties of the solutions (25) and (28) can be proven. In particular, we have the following. Property P1: The matrix in (25) and (28) may be replaced such that . by any other matrix Property P2: The choice of the elements of the vector is irrelevant, i.e., any vector whose entries are all nonzero leads to the same receiver. Property P3: The solutions (25) and (28) can be given the following unified form:

(26)

(29)

where is a matrix containing in its columns an orthonormal basis for the subspace

or for the MMSE-like with and decorrelating-like receivers, respectively. Property P4: If

Otherwise stated, since the matrix is the orthogonal projector onto the orthogonal complement of the , and since subspace belongs to such a subspace, for vanishingly small , , the columns of end up orthogonal to the useful signal which is thus completely nullified by the blocking matrix. In such a pathological situation, thus, the set of constraints in (24) is no longer sufficient. A hint on how to deal with this latter situation can be obtained by moving on to the solution of the MMOE problem with so that rank . Let us also assume that , which is a very mild assumption for DS/CDMA systems with carefully chosen spreading codes and with over,3 as well as that rank sampling factor rank rank . Notice that this last hypothesis may , i.e., it poses the be fulfilled only if following constraint on the number of users that may be supported by the network:

rank

rank

rank

(30)

, the receiver is near–far resistant in with the sense that it zero forces the MAI and ISI in the limiting situations of vanishingly small thermal noise and/or increasingly large other users’ signals energies. Proof: See Appendix C. C. Blind Channel Estimation In what follows, we illustrate briefly how the proposed procedure provides, as a by-product, an estimate of the discrete-time channel impulse response. Indeed, notice that the principal eigenvector of the covariance matrix of the whitened . As a consequence, given the observables (18) is principal eigenvector of the sample covariance matrix in (19), an estimate, say, of the vector can be obtained straightforwardly through the linear processing (31)

(27) 3Such an assumption is made here for the sake of simplicity, but what follows

may be easily modified to account for the existence of some i such that c 2 R (R ). However, as will be explained in the sequel, in such a situation, the resulting receiver is not near–far resistant.

In the following sections, we will provide a simulation result showing the effectiveness of this channel estimation technique. 4Another strategy to increase the network capacity is to consider a processing window possibly larger than 2T .

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V. NUMERICAL RESULTS Although some analytical results on the performance of the proposed system can be obtained, in terms, e.g., of system bit-error-rate and signal-to-interference-plus-noise-ratio, for the sake of simplicity and brevity, we give only some computer simulation results in order to corroborate the effectiveness of the proposed receiver and to contrast its performance with that of competing alternatives. In what follows, we consider a DS/CDMA system employing -sequences as spreading codes, with processing gain (unless otherwise specified). As to the channel, we consider the following multipath model:

(32)

generated from a complex Gaussian with the channel gains distribution and held constant for the entire frame length; the de, lays are generated from a uniform distribution in are generated from a uniform distribuwhereas the delays (this implies that the actual channel delay spread tion in ). Additionally, we set the oversampling factor and is . In Figs. 2 and 3, the system BER the number of users is versus the average received energy contrast shown for the following detectors: • the proposed MMSE-like detection algorithm (25) or its ; equivalent version (29) with • the detector proposed by Tsatsanis and Xu in [13] (in the design of such a system, the authors assumed that the propagation delay for the user of interest were known to the receiver. As a consequence, in their algorithm, the with a reduced number of authors have adopted a matrix tied to the channel multipath columns equal to delay spread); • a modified version of the detector proposed by Tsatsanis and Xu in [13], where the word “modified” stems from the fact that the timing information is no longer assumed available -dimensional and in the algorithm the , as expressed in (13), has been employed; matrix • the receiver proposed by Wang and Poor in [14], wherein, again, due to the assumption of known propagation delay, with a reduced number of columns has been a matrix considered; • a modified version of the receiver by Wang and Poor, wherein -dimensional matrix , as expressed the in (13), has been employed; • the nonblind, nonadaptive linear multiuser MMSE receiver [4]. All of the plots show an average over 100 independent random realizations of the delays and of the channel complex gains, and the received signal correlation matrix has been estisignaling intervals. mated through an average over have been In Fig. 2, the signals amplitudes assumed to be coincident, whereas Fig. 3 refers to a severe are near–far scenario where the amplitudes

M = 3, K = 6, N = 15, Q = 1500, power-controlled scenario. Fig. 2.

BER versus the average received energy contrast. System parameters:

Fig. 3.

BER versus the average received energy contrast. System parameters:

M = 3, K = 6, N = 15, Q = 1500, severe near–far scenario.

15 dB above .5 It is seen that the proposed receiver largely outperforms the previously derived receiver by Tsatsanis and Xu; in particular, at 10 BER, the performance gap between the proposed receiver and the one by Tsatsanis and Xu, which does exploit the timing information, is about 5 dB, whereas the gap with respect to its modified version, which relies on the same prior information as the newly proposed receiver, is more than 10 dB. Additionally, the performance loss with respect to the nonblind, nonadaptive MMSE receiver is less BER, and this loss is further reduced in than 2 dB at 10 the near–far situation represented in Fig. 3. It is also seen that the severe near–far situation only slightly affects the system performance, thus confirming, at an experimental level, the above cited receiver immunity to the near–far effect. With regard, instead, to the comparison with the Wang and Poor receiver, it is seen that this receiver slightly outperforms the 5The power advantage of the interfering signals with respect to the signal of interest is referred to as interference-to-signal ratio (ISR).

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Fig. 4. BER versus the users number. System parameters: dB, N = 15, Q = 1000, power-controlled scenario.

M = 1,

= 22

proposed approach in the power-controlled scenario, whereas the two receivers achieve the same performance in the near–far situation depicted in Fig. 3. However, since the Wang and Poor receiver is based on the subspace approach, its performance suddenly degrades in moderately loaded networks. Indeed, in Fig. 4, the system BER for all of the aforementioned receivers dB, , is shown versus the users number for . It is seen that for , the and with proposed receiver outperforms the blind receivers by Tsatsanis , instead, the Tsatsanis and Xu receiver and Xu. For achieves better performance than the new receiver, which, in turn, outperforms the modified version of the receiver. The Wang and Poor receiver, instead, exhibits the best performance for low values of , whereas its performance degrades for increasing users number. In order to test the receiver performance for larger values of users number and system processing gain, in Fig. 5, the BER versus is reported for a system with users, ISR dB, and . It is here assumed , the oversampling factor that the channel delay spread is , and . In addition, in this case, it is seen is that the proposed receiver achieves satisfactory performance. Overall, results show that the proposed receiver achieves very satisfactory performance both for power-controlled and near–far scenario, as well as for moderately loaded networks. As previously commented, a side result of the proposed detection algorithm is a novel channel estimation algorithm. In order to give an insight into the performance of such an estimator, in Fig. 6, we report the following normalized correlation coefficient

versus . In particular, we consider a system with , , ISR dB, and with and oversampling factor , the proposed algorithm is 14. The results show that for outperformed by the Wang and Poor receiver, whereas the oppo: a situation in which the site behavior is observed for

Fig. 5.

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BER versus the average received energy contrast. System parameters:

M = 2, K = 25, N = 63, Q = 3500, ISR = 5 dB.

Fig. 6. Correlation coefficient versus the average received energy contrast for the channel estimation algorithm. System parameters: M = 2, N = 15, Q = 1500, ISR = 0 dB, K = 7; 14.

latter receiver’s performance exhibits a dramatic performance degradation. On the other hand, the estimation algorithm proposed in [13] is outperformed by the proposed algorithm in both cases. In summary, simulation results show that the proposed receiver exhibits very satisfactory performance and is effective both as a symbol detection algorithm and as a channel estimation procedure. VI. RECURSIVE BLIND JOINT EQUALIZATION AND MULTIUSER DETECTION The proposed batch estimation procedure is applicable only when the scenario is stationary in the long term, i.e., when the covariance properties of the observables may be considered constant for sufficiently long time intervals. In many situations of

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practical interest, this is not the case in that the cell may be either entered by new users or abandoned by former users, which requires a symbol-by-symbol update of the covariance matrix. Additionally, it may also be the case that the propagation channels’ impulse responses vary with time as the packet duration exceeds the channels’ coherence time. In these situations, it is customary to resort to recursive algorithms, which enable a symbol-by-symbol updating of the receiver. In this study, we consider only the adaptive implementation of the MMOE receiver (25), leaving to future developments the issue of adaptive implementation of the decorrelating-like receivers. Implicit in our choice is the decision of disregarding the pathological situation of vanishingly small noise floor and cancellation of the useful signal. This decision relies on the experimental evidence that the numerical algorithms presented here are effective even for extremely large SNRs, well beyond the values of practical interest. The first step to derive an adaptive algorithm is to devise a tracking procedure for the blocking matrix . Starting on a set say, an estimate of the of observables, can be obtained through the following exblocking matrix ponentially weighted counterpart of the minimization problem (24):

[see (68)], the cascade of the blocking matrix and of the , where whitening filter amounts to multiplying the data by is the matrix containing on its columns the dom. On the other hand, these eigenvecinant eigenvectors of , may be effitors, which form an orthonormal basis for ciently tracked by generating data whose covariance matrix is and passing them on to a subspace tracking algorithm. Since the matrix is estimated on a symbol-by-symbol basis through the procedure (34), the auxiliary observables to be generated at epoch are (37) is an -dimensional vector whose entries where . We are independent binary variates taking on values in say, of , whereby the are thus left with an estimate, whitened data

can be formed and finally forwarded to a subspace tracking alsay, of their cogorithm to track the principal eigenvector, variance matrix. The estimate of the vector , which is to be used in (14), can thus be obtained as (38)

(33) Diag where is a (close-to-one) forgetting factor. Applying, again, standard Lagrangian techniques, we obtain the updating algorithm

A detailed recipe for this algorithm is given in 07Table I. In order to test the learning capabilities of the proposed tracking procedure, we consider again a system with and users. In Fig. 7, the normalized correlation coefficient

(34) where

is the following Kalman gain: (35)

is the exponentially weighted sample covariance maand trix of the observables (36) The next step is to track the whitening filter, which would in . principle require a Cholesky decomposition of Unfortunately, however, such a matrix cannot be recursively computed through rank-one updates; therefore, known algorithms for recursive Cholesky decomposition cannot be applied. This implies that such a decomposition should be performed from scratch at each bit-interval, thus leading to a procedure computational complexity. with In order to avoid this problem, we propose a different approach, which relies on coupling the above RLS procedure with subspace tracking techniques, so that the overall complexity is still quadratic. In particular, we consider the stochastic-gradientbased subspace tracking algorithm developed in [17]. To illustrate further, we recall that based on the results of Appendix C

is shown versus the number of symbol-intervals used in adaptation. The average received energy contrast is set to 16 dB, , and two different values (i.e., the oversampling factor 0 and 6 dB) of the ISR are considered. The plots are the result of an average over 50 independent random realizations of the propagation channels and delays for all of the users. In order , the estimate to speed up convergence, at iteration has been obof the principal directions of the subspace tained by applying an SVD to the first 130 auxiliary observables . It is seen that even though a somewhat slow convergence may be noticed, the proposed algorithm achieves a satisfactory steady-state performance. VII. CONCLUSIONS The problem of joint multiuser detection and equalization for DS/CDMA systems operating in fading channels has been considered in this paper. A new detection structure has been introduced based on the key point that in a dispersive environment the received signature waveforms may be written as the product of a code-matrix, containing properly shifted versions of the spreading codes, times an unknown vector, in which the effect of the unknown timing and channel impulse response is lumped. The proposed receiver has a two-stage architecture: The first stage is a noninvertible linear transformation aimed at MAI and

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TABLE I RECURSIVE RECEIVER IMPLEMENTATION

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aimed at SNR optimization, i.e., it performs a whitening transformation and matched filtering. A useful by-product of this approach is the introduction of a new blind channel estimation technique. We have also considered the problem of recursive adaptive receiver implementation and have proposed a learning algorithm based on a coupling of RLS with subspace tracking. Overall, simulation results confirm that the proposed receiver compares favorably with competing alternatives and that its performance is quite close to that of the nonblind, nonadaptive, linear MMSE multiuser receiver. Finally, the learning capabilities of the proposed adaptive receiver implementation have been shown to be satisfactory. APPENDIX A PROOF OF THEOREM 2 of the matrix , and build the Consider the first column and . Denote by and by two sets and matrices containing on their columns the vectors in , respectively. We have

(39) the cardinality of the set and by Now, denoting by the rank of , let us assume, with no loss columns of are linof generality, that the first columns early independent, so that the remaining can be obtained through a linear combination of the first columns. The matrix can thus be written as

(40) the first (linIn the above equation, we have denoted by columns of and expressed as a early independent) the remaining linear combination of the columns of columns of . Notice that . Substituting (40) into (39), we have

(41) the rank of , and consider the folNow, denote by lowing “economy-size” singular value decomposition (SVD):

M =

Fig. 7. Convergence dynamics of the proposed adaptive receiver implementation. System parameters: 2, = 16 dB, N = 15, K = 5.

ISI suppression. This transformation has been designed based on the minimum MOE criterion. In particular, analytical results have been presented in order to define, and understand how to prevent, the pathological situations that may lead to the cancellation of the useful signal. The second receiver stage, instead, is

(42) is an -dimensional matrix containing The matrix most dominant eigenvectors of on its columns the , which represent an orthonormal basis for the , whereas is a diagonal subspace on its matrix with the nonzero eigenvalues of and the orthogonal diagonal. Denote now by

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projections of the matrix along the subspace and its orthogonal complement, respectively. That is and Since , the matrix

(43) , if we let in (41) can be written as

wherein the fact that has is the first column been exploited. Notice also that since , and since of , it follows that . Accordingly, we have

(50) (44) , and , it is readily checked that the hypotheses of Riedel’s theorem are fulfilled. In particular, notice that coincides with the sum of the ranks since the rank of and , the matrix is of full rank [21, of Prop. 5]. As a consequence, letting, for notational simplicity, , the Moore–Penrose generalized inverse of the matrix is written as

As to the second block, premultiplying both sides of the above , we have relation by

Since

(51) Finally, consider the third block. Since , we have

(52)

(45) From (42), it follows that shown that

is the first column of has been wherein the fact that such that exploited. From the last relation, it is thus seen that , the th element of is equal to zero. In a simand ilar way, i.e., considering the partitions , the theorem follows. APPENDIX B PROOF OF SOLUTION (28)

, whence it is easily

(46) and denote the first column of the matrices wherein and , respectively, the second equality in (46) , and the third equality follows from the fact that is orthogonal to the columns of follows from the fact that . Likewise, it is easily checked that

Based on Proposition 1, instead of considering the original MOE problem (22), we consider the following constrained minimization: (53) Diag Solving the above problem entails minimizing the Lagrangian cost function

(47) Consider now the following block matrix:

(54) (48)

is the set of the Lagrange multiwhere pliers. It is easily seen that minimizing the above quantity is tantamount to minimizing the following cost functions:

Substituting (45) into the above expression, and taking into account (46) and (47), we have, for the first block in (48) (55) which, in turn, are associated with the following constrained minimization problems:

(56) (49)

BUZZI et al.: BLIND ADAPTIVE JOINT MULTIUSER DETECTION AND EQUALIZATION

It thus follows that problems (53) and (56) are equivalent. Minimizing the cost functions (55) entails solving the equations

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whence the transformation can be compactly written as in (28), which we rewrite here as

(57) , the above equations have no solution. Since However, observe that the problems in (56) are equivalent to the problems

(58) solving since it may be easily checked that any vector problem (56) also solves problem (58), and vice versa. The vectors s solving the problems (58) can be written as (59) to be determined so that the with the Lagrange multiplier is fulfilled. Now, if we denote by constraint the matrix

(60)

Diag

(64)

The next step now is to verify that (64) actually solves problem (22), i.e., that it minimizes the output energy subject to the constraint that is nonzero. To this end, let us denote by the -dimensional matrix, which is obtained by in (60). Since, we deleting the first column in the matrix , we have recall,

(65) Since we have assumed that rank rank rank , i.e., that rank

rank rank

and since the columns of are linearly independent of the with respect to the matrix columns of the matrix , by virtue of Theorem 2, we have that the matrix

are defined in Section III, and if we where the vectors , it is easily checked that let solution (59) is also written as (61) , we have Since we have assumed that , is linearly independent of the useful that with respect to the matrix , whence, in light signature , i.e., the useful signal of Theorem 2, we have is again totally nullified by the blocking matrix . In order to avoid signal cancellation for all nonzero realizations of , we have to properly modify the solution (59). Since the signal canand are linearly cellation is caused from the fact that , the only means to independent with respect to the matrix a set of rank-one prevent this effect is to add to the matrix and end up linearly dependent with matrices so that . Since is unknown at the receiver, respect to the matrix the “minimal” set of such matrices is (62) We thus come up with the following solution:

has all-zero entries. On the other hand, notice that the noiseless consists of a linear combination (whose observable coefficients are the random information bits) of the columns of , whence all of the vectors in the matrix (which represent the MAI and ISI) are totally nullified by the matrix , i.e., we have (66) A straightforward application of Theorem 2 and, in particular, of (51) leads to the following result:

Diag

(67)

, is always nonzero. As a consewhich, for any quence, since the filter in (28) nullifies all of the interference and ensures that the useful signature, whatever the channel realization and the signal propagation delay are, is not canceled, it solves problem (22). APPENDIX C PROOFS OF P1–P4 Proof of P1: Consider the following “economy-size” SVD

(63)

where

, , and

contains the

dominant

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eigenvectors of the matrix . Since the covariance matrix of the thermal noise contribution at the output of the filter is , an alternate expression of the . As a consequence, the vector whitening filter is in (18) can be also expressed as

wherein the fact that for nonsingular matrices the Moore–Penrose generalized inverse reduces to classical matrix inversion has been exploited. Now, in the light of Proposition 2, we have just to prove that coincides with the column span of the in (73). Applying twice the matrix inversion lemma matrix [18], the matrix in (73) is written as

(68) has been exploited. wherein the fact that The proposed receiver has to evaluate the dominant eigenvector , i.e., the solution to the equation of the matrix (69) so that the final decision variable to be forwarded to the differential decision rule is (70) is replaced by Assume now that the blocking matrix such that a new blocking matrix . Proceeding as in the previous situation, we end up with the decision statistic

(74)

As a consequence, since the matrix can be expressed as the times a nonsingular square matrix, it product of the matrix coincide. follows that the column spans of and Proof of P4: In Appendix B, it has already been shown that the decorrelating-like receiver zero-forces the MAI and the ISI and that a similar operation is performed by the MMSE-like . It thus remains to prove that for nonvanishing filter as thermal noise, the MMSE-like filter in (73) zero forces the users whose signals have an increasingly large energy. To this is written as end, let us assume that the observable

(71) (75) where

solves the eigenvalue equation (72)

Premultiplying both sides of (69) and (72) by respectively, and exploiting the fact that follows that

and

, , it

where is an irrelevant scalar constant. Substituting the above equation into (70) yields

Since the differential rule is not influenced by , P1 is proven. Proof of P2: Since the column span of the matrix is not affected by the actual values of the diagonal matrix Diag (provided, of course, that all of the entries of are nonzero), P2 is an obvious corollary of P1. Proof of P3: The fact that solution (28) is equivalent to (29) may be trivially proven by letting

the signatures wherein we have denoted by , of the users whose users have an increasingly large ampli), whereas contains the tude (which we denoted by remaining contributions from MAI, ISI, and thermal noise. the -dimensional maIn addition, denote by . trix having on its columns the signatures , , with a suitable Since nonsingular matrix, the contribution from the signatures , at the output of the filter is written as

(76) As

, the matrix , whence we have

(77) is a matrix having all-zero where entries. It is thus seen that the matrix zero forces those signatures having an increasingly large amplitude.

With regard, instead, to solution (25), notice that if we let

REFERENCES then it reduces to the simplified form . In addition, notice that solution (29), for the case at hand, can be written as (73)

[1] E. Dahlman et al., “WCDMA—The radio interface for future mobile multimedia communications,” IEEE Trans. Veh. Technol., vol. 47, pp. 1105–1118, Nov. 1998. [2] F. Adachi, M. Sawahashi, and H. Suda, “Wideband DS-CDMA for nextgeneration mobile communication systems,” IEEE Pers. Commun. Magazine, vol. 36, pp. 56–69, Sept. 1998.

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[3] P. Taaghol et al., “Satellite UMTS/IMT2000 W-CDMA air interfaces,” IEEE Commun. Mag., vol. 37, pp. 116–126, Sept. 1999. [4] S. Verdú, Multiuser Detection. Cambridge, U.K.: Cambridge Univ. Press, 1998. [5] M. Honig, U. Madhow, and S. Verdù, “Blind adaptive multiuser detection,” IEEE Trans. Inform. Theory, vol. 41, pp. 944–960, July 1995. [6] H. V. Poor and X. Wang, “Code-aided interference suppression for DS/CDMA communications—Part II: Parallel blind adaptive implementations,” IEEE Trans. Commun., vol. 45, pp. 1112–1122, Sept. 1997. [7] U. Madhow, “Blind adaptive interference suppression for CDMA,” Proc. IEEE, vol. 86, pp. 2049–2069, Oct. 1998. [8] P. B. Rapajic and B. S. Vucetic, “Adaptive receiver structures for asynchronous CDMA systems,” IEEE J. Selected Areas Commun., vol. 12, pp. 685–697, May 1994. [9] X. Wang and H. V. Poor, “Blind multiuser detection: A subspace approach,” IEEE Trans. Inform. Theory, vol. 44, pp. 677–690, Mar. 1998. [10] M. Torlak and G. Xu, “Blind multiuser channel estimation in asynchronous CDMA systems,” IEEE Trans. Signal Processing, vol. 45, pp. 137–147, Jan. 1997. [11] E. Aktas and U. Mitra, “Complexity reduction in subspace-based blind channel identification for DS/CDMA systems,” IEEE Trans. Commun., vol. 48, pp. 1392–1404, Aug. 2000. [12] M. K. Tsatsanis, “Inverse filtering criteria for CDMA systems,” IEEE Trans. Signal Processing, vol. 45, pp. 137–147, Jan. 1997. [13] M. K. Tsatsanis and Z. (D.) Xu, “Performance analysis of minimum variance CDMA receivers,” IEEE Trans. Signal Processing, vol. 46, pp. 3014–3022, Nov. 1998. [14] X. Wang and H. V. Poor, “Blind equalization and multiuser detection in dispersive CDMA channels,” IEEE Trans. Commun., vol. 46, pp. 91–103, Jan. 1998. [15] D. Gesbert et al., “Blind multiuser MMSE detector for CDMA signals in ISI channels,” IEEE Commun. Lett., vol. 3, pp. 233–235, Aug. 1999. [16] G. Caire, “Two-stage nondata-aided adaptive linear receivers for DS/CDMA,” IEEE Trans. Commun., vol. 48, pp. 1712–1724, Oct. 2000. [17] B. Yang, “Projection approximation subspace tracking,” IEEE Trans. Signal Processing, vol. 43, pp. 95–107, Jan. 1995. [18] R. A. Horn and C. R. Johnson, Matrix Analysis. Cambridge, MA: Cambridge Univ. Press, 1985. [19] G. H. Golub and C. F. Van Loan, Matrix Computations, 3rd ed. Baltimore, MD: John Hopkins Univ. Press, 1996. [20] K. S. Riedel, “A Sherman–Morrison–Woodbury identity for rank-augmenting matrices with application to centering,” SIAM J. Matrix Anal. Applicat., vol. 13, pp. 659–662, 1992. [21] J. A. Fill and D. E. Fishkind, “The Moore–Penrose generalized inverse for sums of matrices,” SIAM J. Matrix Anal. Applicat., vol. 21, pp. 629–635, 1999.

Stefano Buzzi (M’98) was born in Piano di Sorrento, Italy, on December 10, 1970. He received the Dr.Eng. degree with honors in 1994 and the Ph.D. degree in electronic engineering and computer science in 1999, both from the University of Naples “Federico II,” Naples, Italy. In 1996, he spent six months at the Centro Studi e Laboratori Telecomunicazioni, Turin, Italy, while from November 1999 through December 2001, he spent eight months with the Department of Electrical Engineering, Princeton University, Princeton, NJ, as a Visiting Research Fellow. He is currently an Associate Professor with the University of Cassino, Cassino, Italy. His current research and study interests lie in the area of statistical signal processing, with emphasis on signal detection in non-Gaussian noise and multiple access communications. Dr. Buzzi received, from the Associazione Elettrotecnica ed Elettronica Italiana, the “G. Oglietti” scholarship in 1996 and was the recipient of a NATO/CNR advanced fellowship in 1999 and of a CNR short-term mobility grant in 2000 and 2001.

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Marco Lops (M’96–SM’01) was born in Naples, Italy, on March 16, 1961. He received the Dr.Eng. degree in electronic engineering from the University of Naples in 1986 and the Ph.D. degree in electronic engineering from the University of Naples in 1992. From 1986 to 1987, he was with Selenia, Roma, Italy, as an engineer in the Air Traffic Control Systems Group. From 1991 to 2000, was an Associate Professor of radar theory and digital transmission theory at the University of Naples, and since March 2000, he has been a Full Professor with the University of Cassino, Cassino, Italy, where he has been engaged in research in the field of statistical signal processing, with emphasis on radar processing and spread spectrum multiuser communications. He also held teaching positions at the University of Lecce, Lecce, Italy, and during 1991, 1998, and 2000, he was on sabbatical leaves at the University of Connecticut, Storrs; Rice University, Houston, TX; and Princeton University, Princeton, NJ, respectively.

H. Vincent Poor (S’72–M’77–SM’82–F’87) received the Ph.D. degree in electrical engineering and computer science in 1977 from Princeton University, Princeton, NJ, where he is currently a Professor of electrical engineering. He is also affiliated with Princeton’s Department of Operations Research and Financial Engineering and with its Program in Applied and Computational Mathematics. From 1977 until he joined the Princeton faculty in 1990, he was a faculty member at the University of Illinois at Urbana-Champaign. He has also held visiting and summer appointments at several universities and research organizations in the United States, Britain, and Australia. His research interests are in the area of statistical signal processing and its applications, primarily in wireless multiple-access communication networks. His publications in this area include the forthcoming book, Wireless Communication Systems: Advanced Techniques for Signal Reception. Dr. Poor is a member of the National Academy of Engineering and is a Fellow of the Acoustical Society of America, the American Association for the Advancement of Science, the Institute of Mathematical Statistics, and the Optical Society of America. He has been involved in a number of IEEE activities, including serving as President of the IEEE Information Theory Society in 1990 and as a member of the IEEE Board of Directors in 1991 and 1992. Among his recent honors are an IEEE Third Millennium Medal in 2000, the IEEE Graduate Teaching Award in 2001, the Joint Paper Award of the IEEE Communications and Information Theory Societies in 2001, the NSF Director’s Award for Distinguished Teaching Scholars in 2002, and a Guggenheim Fellowship from 2002 to 2003.