Robust Blind Multiuser Detection for Synchronous ... - IEEE Xplore

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Robust Blind Multiuser Detection for Synchronous CDMA Systems Using Worst-Case Performance Optimization Shahram Shahbazpanahi, Member, IEEE and Alex B. Gershman, Senior Member, IEEE

Abstract—The performance of blind multiuser detection methods is known to degrade severely in the presence of even small mismatches between the actual and the presumed desired user signatures. Such mismatches may occur in practical situations due to an imperfect knowledge of the channel impulse response. In this paper, we propose a new robust approach to blind multiuser detection in the presence of unknown arbitrary-type mismatches of the desired user signature. Two different formulations of a robust multiuser receiver are considered. The proposed formulations are based on the explicit modeling of uncertainties in the covariance matrix of the desired user signature and/or data covariance matrix and optimization of the worst-case performance. Simple closed-form solutions to the considered robust multiuser detection problems are derived. The proposed methods have a computational complexity comparable to that of the traditional blind multiuser detection algorithms, and, at the same time, offer an improved robustness and faster convergence rates. Index Terms—Blind linear receivers, multiuser detection, synchronous code-division multiple-access (CDMA).

I. INTRODUCTION

R

ECENTLY, code-division multiple-access (CDMA) schemes have been extensively studied as alternatives to traditional time-division multiple-access (TDMA) or frequency-division multiple-access (FDMA) approaches. Using direct sequence CDMA (DS-CDMA), different users can share the same time slot and/or the same bandwidth, and hence, the communication resources can be exploited in a more efficient way. However, since different users communicate through the same channel, complicated multiuser detection techniques are needed to extract the signal of each user in the presence of the interference from other users [which is hereafter referred to as the multiple access interference (MAI)]. Recently, multiuser detection has been a focus of an intensive study [1]–[15]. In [2], a maximum likelihood (ML) detector has been proposed. In the multiuser case, this approach can greatly Manuscript received November 13, 2002; revised May 6, 2003; accepted August 4, 2003. The editor coordinating the review of this paper and approving it for publication is J. Evans. The work of A. B. Gershman was supported by the Wolfgang Paul Award Program of the Alexander von Humboldt Foundation and German Ministry of Education and Research, Premier’s Research Excellence Award Program of the Ministry of Energy, Science, and Technology (MEST) of Ontario, and Discovery Grant Program of the Natural Sciences and Engineering Research Council (NSERC) of Canada. The authors are with the Department of Communication Systems, University of Duisburg-Essen, 47057 Duisburg, Germany, on leave from the Department of Electrical and Computer Engineering, McMaster University, Hamilton, ON, L8S 4K1 Canada. Digital Object Identifier 10.1109/TWC.2004.837641

outperform the matched filter receiver which is the standard technique to detect a single user while ignoring all other users [3]. However, the computational complexity of the ML multiuser detector increases exponentially with the number of users. In [4], a semidefinite relaxation based quasi-ML approach has been proposed which reduces the computational complexity of the ML multiuser detector while preserving its excellent performance. As suboptimal solutions, linear receivers have been widely considered in the literature [6]–[11]. Examples of such receivers are minimum mean-square error (MMSE) receiver [8] and [9], zero-forcing (ZF) receiver [10], and minimum output energy (MOE) receiver [11]. All these techniques require the exact knowledge of the channel impulse response or the desired user signature. Hence, such receivers have to use training, and this might not be consistent with the requirement of minimal cooperation among users. Furthermore, the performance of these receivers can degrade severely due to variations of the channel impulse response after the training period. This necessitates a frequent use of training and significantly reduces the information transmission rate. Among different approaches to multiuser detection, blind techniques (which do not require any training) are of great interest. These methods are entirely based on the spreading code of the desired user and are able to detect its symbols from the received data without any knowledge of the channel or spreading codes of other users. For example, the popular Capon estimator has been adopted in [14] to estimate the channel impulse response, and consequently, the desired user signature. However, the performance of the Capon multiuser detector degrades severely at low signal-to-noise ratios (SNRs) and short data lengths. In [11], a robust blind MOE multiuser receiver has been presented that minimizes the output power subject to linear and quadratic constraints. The linear constraint maintains the distortionless response of the MOE receiver while the quadratic constraint limits the norm of the difference between the receiver coefficient vector and the presumed desired user signature. In [13], another type of robust MOE multiuser receiver has been proposed where the receiver output power is minimized subject to a quadratic constraint (which limits the norm of the receiver coefficient vector), and a set of ad hoc linear constraints which are used to provide further robustness against signature mismatch and channel distortions. Interestingly, the robust receivers proposed in [11] and [13] are essentially similar, because both of them lead to diagonal loading of the covariance matrix of the

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received data. However, as shown in [15], the robustness of the diagonal loading based approach [11] may be insufficient. Motivated by this fact, the authors of [15] have proposed another promising approach to robust blind multiuser detection that explicitly models an arbitrary (but norm-bounded) uncertainty in the desired user signature and uses worst-case performance optimization to improve the robustness of the MOE receiver. This method is based on a convex optimization using second-order cone (SOC) programming. Although several efficient convex optimization software tools are currently available, the implementation of the SOC-based method can be quite consuming. Similar worst-case performance optimization ideas have recently emerged in array processing in application to minimum variance distortionless response (MVDR) beamformer, see [16]. In [17], a related idea has been used to develop another robust beamformer which, in contrast to the SOC-based approaches [15] and [16], yields a closed-form solution. In this paper, we apply the idea of worst-case performance optimization to blind multiuser detection to develop two new robust multiuser receivers in the presence of an arbitrary yet norm-bounded mismatch in the desired user signature. The first of them is based on the explicit modeling of uncertainty in the covariance matrix of the desired user signature, whereas the second one additionally models such an uncertainty in the covariance matrix of the received data. Both methods yield simple closed-form solutions which correspond to the maximum of the worst-case signal-to-interference-plus-noise ratio (SINR). Furthermore, unlike the existing blind multiuser receivers, the proposed methods can be applied to random time-varying channel scenarios where the channel impulse response, and consequently, the desired user signature are subject to substantial fluctuations during the observation interval. The paper is organized as follows. A background on the signal model and the MOE multiuser receiver is presented in Section II. In Section III, a new extended formulation for the MOE receiver is developed which is applicable to both cases of a fluctuating (fast fading) and a deterministic (quasi-static) channel. This formulation is then used to develop two robust multiuser receivers in Section IV. Section V presents our simulation examples. Conclusions are drawn in Section VI.

[7]. Furthermore, we assume that for each user, the data symbols are independent random variables which are equally likely drawn from a finite alphabet. We model the channel for each user as an FIR filter whose impulse response is much shorter than the symbol period , so that the effect of inter-symbol-interference (ISI) can be neglected [11]. However, the duration of the channel impulse response is assumed to be comparable to the chip period , so that there is a substantial inter-chip-interference (ICI) [13]. Using these assumptions, the signature waveform of the th user is given by [10] (2) where is the user spreading is its chip waveform convolved with the code vector, channel impulse response, is the spreading factor (the number is the chip period, and of chips per symbol), stands for the transpose. In an ICI-free scenario, spans only one chip, while in practice, due to the channel dispersion, can span several chips and this causes ICI. Using the assumption that there is no ISI, we obtain that for or . Then, sampling (1) at for and using vector notation, we have (3) where

, , and are the data vector, the signature vector of the th user, and the noise vector, respectively. It should be stressed that in the general case, the actual user signature is defined as the user spreading code distorted by the effect of the channel. In other words, each signature represents the result of convolution of the corresponding spreading code and the channel impulse response. Assuming without loss of generality that the first user is the desired one, let us rewrite (3) as

II. BACKGROUND A. Signal Model Consider a -user synchronous DS-CDMA system [6]. The received continuous-time baseband signal can be modeled as [15]

(1)

(4) where

contains the desired user data, and contains MAI.

B. Linear Receivers The output of a linear multiuser receiver is given by [6], [7]

is the received signal amplitude of the th user, where is the th data symbol of this user, is its signature waveis the symbol period, and is the zero-mean addiform, tive random noise process with the variance . In this paper, we consider the short spreading code case, i.e., it is assumed that the chip sequence period is the same as the symbol period

(5) where receiver coefficients, and pose.

is a complex vector of the stands for the Hermitian trans-

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The receiver output is used for symbol detection. For example, in binary phase shift keying (BPSK) systems, the symbol detection is made as [1]

is

In the finite-sample case, the data covariance matrix estimated as

(12) (6) and denote the sign function and the real where part, respectively. The system performance can be measured in terms of the bit-error rate (BER) or alternatively, the output SINR can be used [13]

where is the number of data vectors. Using (12), the finitesample versions of the MOE multiuser receivers (10) and (11) can be written as (13)

(7)

where

denotes the statistical expectation.

C. MOE Receiver In the MOE method [11], the receiver coefficient vector is designed to minimize the output power subject to one or more constraints, which ensures that the receiver response to the desired user is distortionless. The MOE detector will then minimize the power of the interference and noise only. The receiver output power is given by (8) is the covariance matrix of the where received data. In general, a set of linear constraints is used to guarantee that the desired user signal is not affected by ICI, ISI, or MAI [12], is known, the and [13]. If the desired user signature vector MOE multiuser receiver can be designed by solving the following minimization problem: (9) whose solution is given by [13] (10) where the subscript “c” is used to indicate that (10) is the clairvoyant MOE multiuser receiver (which uses an exact knowledge of the desired user signature ). In practice, the exact knowledge of the desired user signature is often unavailable. In this case, ignoring the effect of the unknown channel, one can use instead of [11], and the MOE multiuser receiver can be written as (11) It is well known that the performance of (11) can degrade severely in the presence of even a slight mismatch between and , see [11], [13], and [15].

(14) respectively. In scenarios with a short data length, the performance of the MOE receiver (14) can become severely degraded [13]. To provide robustness against the short data-length effects, one may use the so-called diagonal loading technique in which is replaced by where is the so-called loading factor. Using this approach, the diagonal loading based MOE receiver can be written as [11], [13]

(15) Note that the multiuser receiver (15) does not explicitly compensate for the mismatch between and . As a result, it can be quite sensitive to such mismatch and it is not clear how the loading factor should be chosen [15]. III. EXTENDED FORMULATION OF THE MOE RECEIVER In this section, we present a new, more general than (9), formulation of the MOE receiver which will help us to develop our robust receivers. This new formulation will also enable us to apply the MOE concept to random (time-varying) channels, while the conventional MOE receiver (10) can only be used for a deterministic (quasi-static) channel whose impulse response, and, consequently, the desired user signature remain invariant within the observation interval. Let us find that minimizes the output power subject to a constraint which requires that the power contribution of the desired user be a positive constant, i.e., (16) Note that

(17) where is the covariance matrix of the desired can have a rank higher user signature. We stress here that than one. Using (17) and noting that the value of the constant in

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(16) does not affect the output SINR and the probability of error at the output of the symbol detector, we can rewrite (16) as

is an unknown Hermitian error matrix which dewhere scribes the mismatch in the covariance matrix of the desired user signature. can be bounded Assume that the norm of the error matrix by some known constant

(18) Lemma 1: The solution to (18) is given by

(22) (19)

is the operator which yields the principal eigenwhere vector of a matrix (i.e., the eigenvector that corresponds to the in (18) maximal eigenvalue) and the constraint should be satisfied by a proper normalization of the eigenvector in (19). Proof: See Appendix A. Remark: Note that any eigenvector can be normalized in an arbitrary way. Hence, the resulting receiver coefficient vector can be easily normalized to satisfy the constraint in (18). However, it is obvious that the multiplication of the receiver coefficient vector by any positive constant does not affect the probability of error at the output of the symbol detector. Hence, such a normalization is immaterial. It is noteworthy that for a deterministic channel, we have , and hence, we can write (19) as

where denotes the Frobenius norm. In practice, can be determined by finding an upper bound on the ICI. Typically, some preliminary (coarse) knowledge about the wireless channel is available. This information can be obtained either from preliminary channel measurement campaigns [18], [19] or using channel simulators and/or numerical channel modeling. Examples of known channel parameters may include its approximate average delay spread, as well as the rough knowledge of the average form of its impulse response and the characteristics of fluctuations of this response. All these parameters can be used to obtain an upper bound on the ICI (for example, using numerical methods) and to determine a proper value of . To incorporate robustness against an arbitrary norm-bounded and , we modify the MOE problem (18) error between to guarantee that for all potential , the contribution of the desired user to the receiver output power is larger than a constant value, i.e., we require that

(20)

(23)

can be chosen as where the immaterial constant to satisfy the constraint . Therefore, in the case of a deterministic channel, the solution to the new MOE formulation in (18) simplifies to (10). Note that the traditional formulation in (9) assumes that the is a deterministic vector which does desired user signature not change during the observation time. However, in the timevarying channel case (where the channel impulse response is not fixed within the observation interval) the desired user signature vector can fluctuate. Hence, (9) is not applicable to scenarios with random channels where the new receiver (19) has to be used. In (18) and (19), the channel is modeled through the covariance matrix of the desired user signature rather than the desired user signature itself.

Comparing (23) with the constraint in (18), we see that the conventional MOE receiver uses a single equality constraint, while (23) sets an infinite number of inequality constraints. We stress that (23) guarantees that the desired user power will be not less than one for the worst-case mismatch which corresponds to the . Therefore, the proposed desmallest value of sign should improve the receiver robustness. in (18) by (23), we can Replacing the constraint write the robust formulation of the MOE problem as

IV. ROBUST MULTIUSER DETECTION

Proof: If (25) is not equivalent to (23) then the minis achieved when imum of the objective function . However, replacing with , one can decrease the objective function further while the constraint (23) will still be satisfied. This contradicts to the original statement that the objective function is minimized . Therefore, the minimum of the objective function when and this proves the equivalence of (23) and is achieved at (25). It is important to stress that the robust MOE problem (24) is equivalent to the problem of maximization of the worst-case SINR at the receiver output (i.e., the SINR which corresponds to the worst-case norm-bounded uncertainty in the covariance matrix of the desired user signature).

The generalized multiuser receiver in (19) assumes that the covariance matrix of the desired user signature and the data are exactly known. However, in practice, covariance matrix these covariance matrices may be known with some errors. In this section, we first present a multiuser receiver which is robust against an imperfect knowledge of . Then, we extend it to the and are considered. case where uncertainties both in In practice, due to an imperfect knowledge of the channel impulse response, there is always a certain mismatch between and its actual value . Therefore, we the presumed matrix have (21)

(24) Lemma 2: The constraint in (24) can be replaced by (25)

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Now, we use the following lemma to simplify (25): vector and Hermitian matrix Lemma 3: For any (26) where the matrix is constrained to belong to the class of Hermitian matrices. Proof: See Appendix B. Using Lemmas 2 and 3, the original robust multiuser detection problem (24) can be rewritten as (27) From (27), it is clear that an infinite number of inequality constraints [that have been used in the original problem (24)] is now replaced by the single quadratic equality (worst-case) constraint. Lemma 4: The solution to (27) exists only if is smaller than and is given by the maximal eigenvalue of (28)

takes into account all mismatches in the where the matrix data covariance matrix. is The practical significance to model the uncertainty in motivated by the fact that this matrix is always known with some is unavailable and the error. Indeed, in practice, the true sample covariance matrix is used instead of it. Furthermore, the data may be nonstationary and this may cause an additional un. certainty in Similar to (24), the problem (31) corresponds to maximization of the worst-case SINR, where the definition of the worst case is further extended by taking into account the additional in the data covariance matrix. uncertainty To solve (31), we use the following lemma: vector and Hermitian Lemma 5: For any fixed matrix (32) where the matrix is constrained to belong to the class of Hermitian matrices. Proof: See Appendix B. Using (26) and (32), the problem (31) can be rewritten in a much simpler equivalent form

in (27) should be satiswhere the constraint fied by a proper normalization of the eigenvector in (28). Proof: See Appendix C. Remark 1: Similar to (19), the normalization of the eigenvector in (28) is immaterial because the multiplication of the receiver coefficient vector by any positive constant does not affect the probability of error at the output of the symbol detector. Remark 2: According to Lemma 4, we have to guarantee that the chosen value of is smaller than the maximal eigenvalue of . From Lemma 4, we see that the robust multiuser receiver problem (24) has a simple closed-form solution given by (28). should be used instead of , and In the finite-sample case, we can write the sample version of our robust multiuser receiver as

where the “tilde” sign is used to distinguish between the multiuser receivers (28) and (34). It follows from (34) that the solution to (31) naturally combines two different types of diagonal loading, where the positive diagonal load is applied to the ma, while the negative load is applied to the matrix . trix by Considering the finite-sample case and replacing in (34), we obtain the sample version of this multiuser receiver

(29)

(35)

Note that that the multiuser receiver in (29) applies a negative diagonal load to the presumed covariance matrix of the desired user signature. In the case when the channel is unknown, one can use instead of in (29) and it can be rewritten as (30)

A. Extended Formulation In the previous formulation of our robust multiuser receiver, we considered a mismatch in the desired user signature covari. Now, let us consider mismatches both in ance matrix and . In this case, (24) can be extended as

(31)

(33) Similar to the problem (27), the solution to (33) can be expressed in a closed form as (34)

In the case of unknown channel, (35) can be written as (36) instead of . where we use The parameter can be chosen based on some a priori coarse knowledge about the amount of uncertainty in the data covariance matrix. A reliable way of choosing in practical applications is to make preliminary Monte-Carlo runs with the explicit modeling of all uncertainties that are expected in practical scenarios and then to apply the so-obtained value of to real data. V. SIMULATION EXAMPLES In our simulation examples, we consider a 7-user DS-CDMA system which uses Gold codes of length as user spreading codes. The users are all synchronized and have BPSK modulation. The interferers are assumed to have an

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Fig. 1.

BER versus the SNR for

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N = 100 and INR = 20 dB; first example.

Fig. 2. Output SINR versus the SNR for

N = 100 and INR = 20 dB; first example.

interference-to-noise ratio (INR) equal to 20 dB. For each scenario, an average of 100 independent simulation runs is used. The performances of the following techniques are compared in terms of the BER or the output SINR: 1) the clairvoyant MOE algorithm (13) which corresponds to the ideal case when the desired user signature is known exactly (this algorithm does not correspond to any practical situation but is considered for comparison reasons only); 2) the conventional and diagonal loading based MOE receivers (14) and (15); and 3) the robust algorithms (30) and (36). It is important to stress that the clairvoyant MOE receiver should converge to the optimal SINR at large number of sam-

ples, while the other receivers tested may not converge to this value even if the number of samples is large. Except in a few cases where the parameter is varied, we have which gives nearly the best performance for chosen the robust multiuser receivers tested. Furthermore, the parameter in the diagonal loading receiver (15) and in our robust multiuser receiver (35) is also kept fixed throughout the simula. Note that this choice provides nearly the tions and is equal best performance of the diagonal loading based MOE receiver throughout the examples tested. Moreover, the choice of the diagonal loading factor around 10–15 dB above the noise level is a commonly accepted ad hoc choice of this parameter which is recommended by several authors, see [16], [20], [21].

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BER versus the data length

N for SNR = 5 dB and INR = 20 dB; first example.

Fig. 4. Output SINR versus the data length

N for SNR = 5 dB and INR = 20 dB; first example.

A. Example 1: Deterministic Channel In the first example, for each user, we assume that its spreading code is distorted by a multipath deterministic , , and . The (quasi-static) channel with three taps is assumed to be equal to one, while the values of value of and are uniformly drawn from the interval [ 0.8, 0.8] in each simulation run and for each user (however, note that these values remain fixed within each simulation run). Figs. 1 and 2 show, respectively, the BER and the output SINR of the multiuser detectors tested versus the SNR of the desired symbols are used to obtain user. In these figures, . As follows from Fig. 1, the sample covariance matrix our receivers provide excellent BER performances while the

diagonal loading based receiver breaks down at high values of SNR. Moreover, both the conventional and clairvoyant MOE receivers perform poor at all values of SNR. Fig. 2 shows that the output SINRs of our robust receivers are higher than the SINRs of other methods tested (although the performance improvements in terms of BER are much more , even the clairvoyant pronouncing). Interestingly, for MOE receiver does not provide satisfactory performance in terms of BER and output SINR. This is apparently an effect of the short data length (small number of data vectors). Figs. 3 and 4 show, respectively, the BER and the output SINR of the multiuser receivers tested versus the data length . In . It is clear from these figures that the both figures, proposed robust receivers converge much faster than the clair-

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Fig. 5.

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BER versus the SNR for  = 0:2, N = 100 and INR = 20 dB; second example.

Fig. 6. Output SINR versus the SNR for  = 0:2, N = 100 and INR = 20 dB; second example.

voyant and conventional MOE receivers (both in terms of BER and output SINR). Also, from Figs. 3 and 4, it can be seen that the convergence of our receiver (30) in terms of BER and output SINR is faster than that of the diagonal loading based multiuser receiver. However, the convergence rates and BER performances of the receiver (36) and the diagonal loading based receiver are nearly identical in this example. B. Example 2: Deterministic Channel In the second example, we have chosen another way of modeling the effect of ICI. Following [15], we distort the spreading codes of the users at the receiver by an additive random Gaussian . For each user, such a random vector drawn from vector is added to the spreading code vector to simulate the ef-

fect of the ICI. The parameter is the variance of an ICI-related distortion of the user spreading code. In this example, we consider the deterministic channel case again, i.e., the random Gaussian distortions (which simulate the effect of the ICI) are fixed in each simulation run. We assume and use symbols to obtain the sample cothat variance matrix . Figs. 5 and 6 show, respectively, the BER and the output SINR of the multiuser detectors tested versus the SNR of the desired user. Fig. 5 demonstrates that with increasing the SNR, the BERs of our robust receivers decrease much faster than the BERs of the other receivers tested. The BER of the diagonal loading based multiuser receiver does not decrease monotonically because it is severely affected by spatial signature errors at high values of SNR. Fig. 6 shows that, similar to the previous example, the output SINR performances of

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Fig. 7. BER versus the data length

Fig. 8.

N for  = 0:2, SNR = 5 dB and INR = 20 dB; second example.

Output SINR versus the data length N for  = 0:2, SNR = 5 dB and INR = 20 dB; second example.

our robust receivers are higher than those of the other methods tested, although, as before, the performance improvements in terms of BER are much more pronouncing. Figs. 7 and 8 display, respectively, the BER and the output SINR of the multiuser receivers tested versus the data length for and . From these figures, we see that the proposed robust receivers provide much faster convergence than the clairvoyant and conventional MOE receivers (both in terms of the BER and SINR). This observation is consistent with the previous example. Also, from Fig. 7 one can see that the convergence of our receivers in terms of BER is faster than that of the diagonal loading based multiuser receiver. To study the sensitivity of our robust techniques to the choice of , we display the BER and output SINR of (30) and (36) versus in Figs. 9 and 10, respectively. In these figures,

, and different values of have been examined. As it can be seen from these figures, the robust multiuser receiver (36) is somewhat less sensitive to the parameter as compared to the multiuser receiver (30). From Figs. 9 and 10, we see that there is an abrupt performance breakdown of the proposed multiuser receivers around . This phenomenon can be explained by the fact that in this case, the value of is approaching the principal eigenvalue (which is equal to ). of the matrix C. Example 3: Random Channel In the third example, we model the effect of ICI in the same way as in the previous example. However, we now consider a scenario with a random (fast fading) channel. In this example, the additive random Gaussian distortions (which simulate the

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Fig. 9. BER of the proposed robust receivers versus " for N = 100, SNR = 5 dB, INR = 20 dB and different values of  ; second example.

Fig. 10.

Output SINR of the proposed robust receivers versus " for N = 100, SNR = 5 dB, INR = 20 dB and different values of  ; second example.

effect of the ICI) change from one data vector to another. The other conditions are the same as in the second example. In Figs. 11 and 12, respectively, the BER and output SINR of the multiuser receivers tested are shown versus the SNR for and . As it can be seen from these figures, in the random channel case, the proposed robust receivers substantially outperform the other receivers tested. These improvements are especially substantial at high SNRs. In Figs. 13 and 14, respectively, the BER and output SINR and are displayed versus the data length for . From these plots, we observe that our robust techniques have faster convergence rates than that of the clairvoyant and conventional MOE receivers, as well as the diagonal loading based method.

In summary, our simulation examples demonstrated that the proposed blind multiuser receivers consistently enjoy better performance (both in terms of BER and SINR) as compared to the MOE receiver and the robust diagonal loading based receiver. These improvements are significant, both in the cases of a deterministic (quasi-static) channel and random (time-varying) channel. VI. CONCLUSION In this paper, a novel approach to robust blind multiuser detection in synchronous DS-CDMA systems has been proposed. Our new multiuser receivers are based on the explicit modeling of arbitrary (yet norm-bounded) uncertainties in the covariance matricesof thedesired usersignature and of the receiveddata, and use

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Fig. 11.

BER versus SNR for  = 0:2, N = 100 and INR = 20 dB; third example.

Fig. 12.

Output SINR versus SNR for  = 0:2, N = 100 and INR = 20 dB; third example.

worst-caseperformanceoptimizationtoyieldsimpleclosed-form solutions. In contrast to the existing blind multiuser receivers, the proposed techniques can be applied to random time-varying scenarioswherethechannelimpulseresponse,and,consequently,the desiredusersignaturearesubjecttosubstantialfluctuationsduring the observation interval. APPENDIX A PROOF OF LEMMA 1 Using the Lagrange multipliers method, we should minimize the function (37)

Taking the gradient of (37) with respect to and equating it to zero, we obtain that the solution to (18) satisfies the following generalized eigenvalue problem: (38) where the Lagrange multiplier can be viewed as a generalized eigenvalue of the matrix pencil . It is easy to show that all generalized eigenvalues in (38) are nonnegative real numbers. Indeed, multiplying (38) by , we get (39) and are positive and, since the covariance matrices semidefinite, we obtain that all s are real and nonnegative.

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N for  = 0:2, SNR = 5 dB and INR = 20 dB; third example.

Fig. 13.

BER versus the data length

Fig. 14.

Output SINR versus the data length N for  = 0:2, SNR = 5 dB and INR = 20 dB; third example.

Using (39), we obtain that the solution to the optimization problem (18) is the generalized eigenvector corresponding to the . smallest generalized eigenvalue of the matrix pencil , we can rewrite it as Multiplying (38) by

APPENDIX B PROOF OF LEMMAS 3 AND 5 We have to solve the following constrained optimization problems: (41)

(40)

(42)

. which is the characteristic equation for the matrix Since all s are nonnegative, the minimum generalized eigenin (38) corresponds to the maximum eigenvalue value in (40). Using this fact, the optimal vector of the receiver coefficients can be explicitly written as (19).

and are Hermitian matrices, and is a where positive constant. We observe here that the objective function in (41) and (42) is a linear function of the optimization variable . Hence, the inequality constraint in (41) and (42) can be replaced by the equality constraint

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. Therefore, the solutions to (41) and (42) can be obtained using Lagrange multipliers method, by means of minimizing/maximizing the function

The receiver coefficient vector that minimizes (52) can be found by solving the following generalized eigenvalue problem: (53)

(43) Multiplying (53) by

The gradient of (43) is given by

, we obtain (54)

(44) where is the trace of a matrix and we take into account that is Hermitian. Computing this gradient and equating the result to zero yields (45) Inserting this equation into the constraint, we have that and, therefore, the gradient of (43) is zero at (46) Substituting (46) in

, we obtain

is positive definite, and are realSince valued and have the same sign. However, if , . then it is impossible to satisfy the constraint which implies that . As a result, Hence, the solution to the optimization problem (27) can only involve the generalized eigenvectors corresponding to the positive genin eralized eigenvalues. Using the constraint (54), we obtain that the solution to (27) is the generalized eigenvector corresponding to the smallest positive generalized eigen. Note that if value of the matrix pencil is negative definite (that is, if is larger than the maximal eigen), then (27) does not have any solution. Therefore, value of when choosing the parameter , we have to guarantee that is smaller than the maximal eigenvalue of . Note that (53) can be written as (55)

(47) Since is positive, then implies that

which

which is the characteristic equation for the matrix . Since the smallest positive corresponds to the maximal value of , then the explicit solution to (27) is given by the and can be principal eigenvector of the matrix written as (28).

(48) REFERENCES (49) Replacing we obtain

,

, and

in (48) by

,

, and , respectively,

(50) which completes the proof of Lemma 3. Replacing in (49) by , and , respectively, yields

, , and

(51) and the proof of Lemma 5 is completed. APPENDIX C PROOF OF LEMMA 4 The optimal solution to (27) can be obtained from the minimum of the Lagrangian function defined as (52)

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Alex B. Gershman (M’97–SM’98) received the M.Sc. Diploma and the Ph.D. degree in radiophysics from the Nizhny Novgorod State University, Russia, in 1984 and 1990, respectively. From 1984 to 1989, he was with the Radiotechnical and Radiophysical Institutes, Nizhny Novgorod. From 1989 to 1997, he was with the Institute of Applied Physics of Russian Academy of Science, Nizhny Novgorod, as a Senior Research Scientist. From the summer of 1994 until the beginning of 1995, he was a Visiting Research Fellow at the Swiss Federal Institute of Technology, Lausanne. From 1995 to 1997, he was Alexander von Humboldt Fellow at Ruhr University, Bochum, Germany. From 1997 to 1999, he was a Research Associate at the Department of Electrical Engineering, Ruhr University. In 1999, he joined the Department of Electrical and Computer Engineering, McMaster University, Hamilton, ON, Canada where he is now a Professor. Currently, he also holds a Visiting Professorship at the Department of Communication Systems, Gerhard-Mercator University, Duisburg, Germany. His research interests are in the area of signal processing and wireless communications and include statistical signal and array processing, adaptive beamforming and smart antennas, spatial diversity in wireless communications, space-time coding and MIMO systems, parameter estimation and detection, and spectral analysis. Dr. Gershman was a recipient of the 1993 International Union of Radio Science (URSI) Young Scientist Award, the 1994 Outstanding Young Scientist Presidential Fellowship (Russia), the 1994 Swiss Academy of Engineering Science and Branco Weiss Fellowships (Switzerland), and the 1995–1996 Alexander von Humboldt Fellowship (Germany). He received the 2000 Premier’s Research Excellence Award of Ontario, Canada, and 2001 Wolfgang Paul Award from the Alexander von Humboldt Foundation, Germany. He is also a recipient of the 2002 Young Explorers Prize from the Canadian Institute for Advanced Research (CIAR) which has honored Canada’s top twenty researchers aged forty or under. He is on editorial boards of IEEE TRANSACTIONS ON SIGNAL PROCESSING and EURASIP Journal on Wireless Communications and Networking and serves as a Member of the Sensor Array and Multichannel (SAM) Signal Processing Technical Committee of the IEEE Signal Processing Society.

Shahram Shahbazpanahi (M’02) was born in Sanandaj, Kurdistan, Iran. He received the B.Sc., M.Sc., and Ph.D. degrees in electrical engineering from Sharif University of Technology, Tehran, Iran, in 1992, 1994, and 2001, respectively. From September 1994 to September 1996, he was a faculty member with the Department of Electrical Engineering, Razi University, Kermanshah, Iran. Since July 2001, he has been conducting research as a Postdoctoral Fellow at the Department of Electrical and Computer Engineering, McMaster University, Hamilton, ON, Canada. Since March 2002, he has been a summer Visiting Researcher at the Department of Communication Systems, Gerhard-Mercator University, Duisburg, Germany. His research interests include statistical and array signal processing, space-time adaptive processing, detection and estimation, smart antennas, spread spectrum techniques, as well as DSP programming and hardware/real-time software design for telecommunication systems.