IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 55, NO. 7, JULY 2007 3325
Blind Subspace-Based Signature Estimation in
DS-CDM...
3326 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 55, NO. 7, JULY 2007
single-antenna receiver, but, in contrast to [22], ...
ZARIFI AND GERSHMAN: BLIND SUBSPACE-BASED SIGNATURE ESTIMATION 3327
where . According
to (4), if the spreading code of the...
3328 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 55, NO. 7, JULY 2007
where stands for the dimension of a subspace. If we...
ZARIFI AND GERSHMAN: BLIND SUBSPACE-BASED SIGNATURE ESTIMATION 3329
of these approximations will be validated in the simul...
3330 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 55, NO. 7, JULY 2007
from (49) and the centro-Hermitian property of we h...
ZARIFI AND GERSHMAN: BLIND SUBSPACE-BASED SIGNATURE ESTIMATION 3331
Fig. 3. MSEs of the estimated channel versus the numbe...
3332 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 55, NO. 7, JULY 2007
Fig. 6 shows the MSE curves for the proposed algori...
ZARIFI AND GERSHMAN: BLIND SUBSPACE-BASED SIGNATURE ESTIMATION 3333
where
(73)
Assuming that the received samples correspo...
3334 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 55, NO. 7, JULY 2007
Using (84) and (85) in (68) yields
(86)
Equations (...
ZARIFI AND GERSHMAN: BLIND SUBSPACE-BASED SIGNATURE ESTIMATION 3335
[24] K.-J. Wang, Z.-C. Zhou, and Y. Yao, “Performance ...
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Blind Subspace-Based Signature Estimation in DS-CDMA Systems With Unknown Wide-Sense Stationary Interference

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Blind Subspace-Based Signature Estimation in DS-CDMA Systems With Unknown Wide-Sense Stationary Interference

  1. 1. IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 55, NO. 7, JULY 2007 3325 Blind Subspace-Based Signature Estimation in DS-CDMA Systems With Unknown Wide-Sense Stationary Interference Keyvan Zarifi, Student Member, IEEE, and Alex B. Gershman, Fellow, IEEE Abstract—In this paper, we propose a new blind subspace-based signature waveform estimation technique for direct-sequence code division multiple-access (DS-CDMA) communication systems op- erating in the presence of unknown wide-sense stationary interfer- ence. Unlike the existing algorithms, the proposed technique re- quires just a single receive antenna and is applicable to the case of arbitrary transmitted symbol constellations. Necessary and suf- ficient conditions for identifiability of the proposed technique are derived. For practical scenarios where the data covariance matrix is estimated using a finite number of samples, a closed form expres- sion for the mean-squared error (MSE) of the estimated channel is obtained. For the high signal-to-interference ratio (SIR) regime and Gaussian interference, an approximation of the MSE expres- sion is also derived and the effects of different parameters on the performance of the proposed algorithm are analyzed. Numerical examples validate our analytical results. Index Terms—Blind multiuser CDMA receivers, perturbation analysis, subspace-based signature estimation. I. INTRODUCTION DURING the last two decades, multiuser detection has emerged as an attractive means to mitigate the inherent multiaccess interference (MAI) in direct-sequence code divi- sion multiple-access (DS-CDMA) communication systems due to its ability to provide performance improvement in terms of quality of service and user capacity [1], [2]. However, attaining the performance gains promised by multiuser detec- tion schemes depends critically on accurate knowledge of the signature waveform of the user-of-interest at the receiver side [3]–[6]. The main challenge is that due to the spread spectrum nature of the DS-CDMA signals, the transmission channel may be subject to frequency selective fading [7], [8]. This may cause a signature waveform distortion, and, consequently, an unknown mismatch between the presumed and the actual received signature waveforms. Therefore, signature waveform estimation is an important prerequisite of the multiuser detec- tion procedure. A classical approach to the signature waveform estimation problem boils down to transmission of a sequence of training symbols which is known at the receiver, and estimation Manuscript received September 28, 2005; revised July 30, 2006. The as- sociate editor coordinating the review of this manuscript and approving it for publication was Dr. Andrew C. Singer. This work was supported in part by the Wolfgang Paul Award Program of the Alexander von Humboldt Founda- tion and by German Ministry of Education and Research. The authors are with the Communication Systems Group, Darmstadt University of Technology, D-64283, Darmstadt, Germany (e-mail: zarifi@ nt.tu-darmstadt.de; gershman@ieee.org). Digital Object Identifier 10.1109/TSP.2007.893759 of the signature waveform at the receiver side by computing the correlation between the training sequence and the received data [9]. Since CDMA communication systems are typically employed in dynamic environments, a reliable estimation of the signature waveform requires frequent transmission of training sequences. This, in turn, reduces the information transmission rate and wastes the available channel bandwidth. To overcome this drawback, considerable efforts have been devoted to find blind signature estimation techniques that do not require any training symbols [4], [10]–[17]. Among numerous blind techniques, subspace-based methods [4], [10]–[14] constitute a prominent trend. These methods ex- ploit the facts that the user signals occupy a low-dimensional subspace in the observation space and that the signature wave- form of each particular user belongs to a subspace defined by its associated spreading sequence. The usual assumption made in the conventional subspace-based signature estimation tech- niques is that the additive ambient noise is temporally white, and, hence, the signal subspace can be obtained using eigende- composition of the received data covariance matrix. However, this assumption may be violated due to, for instance, the pres- ence of narrow-band interferers [18]–[32]. It is well known that in the latter case, the signal subspace cannot be identified from the subspace spanned by the eigenvectors associated with the largest eigenvalues of the received data covariance matrix, and, hence, some alternative means to identify the signal subspace should be sought. Assuming that the receiver is equipped with two well-sep- arated antennas such that the interference is spatially white between them, Wang and Poor have proposed to identify the signal subspace from the cross-correlation between the re- ceived data of these antennas [21]. As deploying well-separated antennas in the current mobile transceivers may be practically infeasible, a single-antenna blind technique has been proposed in [22] to identify the signal subspace. This technique is based on the assumptions that the original radio-frequency (bandpass) interference is a wide-sense stationary random process and the user transmitted symbols are drawn from the binary phase-shift keying (BPSK) constellation. Note that most of the leading standard proposals for the third generation (3G) of wireless communication systems recommend symmetric constellation such as quadrature phase-shift keying (QPSK) [33], [34]. Therefore, practical applications of the latter technique may be rather limited. In this paper (see also [35]), we propose an alternative approach to the problem of blind subspace-based signature waveform estimation in the presence of unknown interference. Similar to [22], the proposed technique can be applied to a 1053-587X/$25.00 © 2007 IEEE
  2. 2. 3326 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 55, NO. 7, JULY 2007 single-antenna receiver, but, in contrast to [22], it is applicable to the case of an arbitrary transmitted symbol constellation. In our technique, it is only assumed that the unknown interference is wide-sense stationary. Note that this assumption has been frequently used in various interference rejection schemes for CDMA communication systems [18], [22], [23]–[25] and includes several more particular interference models such as multitonal and autoregressive (AR) interference models [20], [26]–[32]. Both these models are widely used in the literature, in particular, to represent overlaid narrow-band interference whose current value can be effectively estimated from its past values [19], [20]. Exploiting the idea presented in [37] in the context of array processing, we obtain a subspace which is orthogonal to the subspace spanned by the user signals. We then use the so-obtained subspace along with the known spreading sequence of the user-of-interest to identify the channel vector, and, subsequently, the signature vector of this user. We also derive the necessary and sufficient conditions which warrant the identifiability of the proposed technique. Similar to other subspace-based signature estimation techniques [4], [10]–[14], the proposed algorithm poses some restriction on the maximum admissible number of active users as well as the length of the channel vector. We show that using the temporal oversampling technique, our algorithm can also be effectively applied to the overloaded scenarios with lengthy channels. Note that the exact data covariance matrix can precisely identify the channel vector. However, this matrix is not exactly known in practice and is estimated from the data samples. For this case, we derive a closed-form analytical expression for the mean-squared error (MSE) of the estimated channel vector using the first-order perturbation theory [12], [38]–[40]. Assuming that the interference is Gaussian distributed and using additional mild and physically justifiable assumptions, a simplified version of this expression is also presented for the high signal-to-interference (SIR) regime. From the latter ex- pression, an impact of the key parameters (such as the number of data samples, the received power of the user of interest, and the received interference power) on the performance of the proposed algorithm is studied. The rest of our paper is organized as follows. In Section II, we present the signal model. Our technique is presented in Section III, where the necessary and sufficient identifiability conditions are also derived. The finite-sample performance of the proposed algorithm is analyzed in Section IV. In Section V, the temporally oversampled version of the proposed technique is presented. Section VI contains computer simulation results. Conclusions are drawn in Section VII. II. SIGNAL MODEL Consider a -user synchronous DS-CDMA system1 oper- ating in the presence of an external interference. The received continuous-time baseband signal can be modeled as [12] (1) 1The synchronous case is mainly considered for the sake of notational brevity. Extending our analysis to the asynchronous systems is direct [21]. where is the symbol period, , , and denote the received amplitude, the th data symbol, and the signature waveform of the th user, respectively, and is the interfer- ence term that also includes the white Gaussian ambient noise. We make use of the following common assumptions: A1) The chip sequence period is the same as the symbol pe- riod. This corresponds to the so-called short spreading code case [36]. A2) The user channels are quasi-static, i.e., the corre- sponding impulse channel responses are fixed during the observation period [12]. A3) For each user, the duration of the channel impulse re- sponse is much shorter than the symbol period , so that the effect of intersymbol-interference (ISI) can be neglected [12], [36]. A4) The transmitted symbols are zero-mean i.i.d. random variables with variance that are statistically indepen- dent with the interference [12]. The original radio-fre- quency (bandpass) interference is a zero-mean, wide- sense stationary process with an unknown arbitrary cor- relation function [18], [22] and, therefore, its baseband equivalent is a zero-mean, circular, and wide-sense stationary random process2 [19], [22], [41]. Let be the spreading factor and denote the spreading sequence associ- ated with the th user where stands for the transpose. According to assumptions A1 and A2, the signature waveform of this user is given by [12] (2) where is the channel impulse response of the th user, and is the chip period. Let us assume that has a finite support of , where and is a positive integer. From as- sumption A3, we have that . Sampling (1) in the in- terval corresponding to the th transmitted symbol of each user and neglecting the first ISI-contaminated samples, the ISI-free received sampled data vector can be written as [12] (3) where , , and . Using (2), the signature vector can be expressed as [12] ... ... ... (4) 2Note that the algorithm proposed in Section III requires only that i(t) is wide-sense stationary, while the circularity of i(t) is only used to simplify the analytical MSE expression of the estimated channel vector that is derived in Section IV.
  3. 3. ZARIFI AND GERSHMAN: BLIND SUBSPACE-BASED SIGNATURE ESTIMATION 3327 where . According to (4), if the spreading code of the user-of-interest is known at the receiver, then, provided that the channel vector is es- timated, obtaining the signature vector is straightforward [12]. Hence, throughout this paper we consider the problem of channel vector estimation rather than the signature vector es- timation. For the sake of consistency, we also assume without any loss of generality that is a unit Euclidean norm vector [12], that is, the normalization factor is absorbed in . One can rewrite (3) in a more compact form as [12] (5) where , and . From assumption A4 and (5), it follows that: (6) where and stands for the Hermitian transpose. Since is wide-sense stationary, the entries of depend only on the difference between the observation times. Hence, is a Hermitian Toeplitz matrix, and, therefore, it is centro-Hermitian [37], [42], that is, (7) where is the permutation matrix with ones on the main anti-diagonal and zeros elsewhere, and denotes complex conjugate. III. THE PROPOSED TECHNIQUE Exploiting the idea presented in [37] for direction-of-arrival (DOA) estimation, let us use (7) to facilitate estimation of the user signature waveforms without knowing the correlation ma- trix . Let us form the covariance difference matrix [37] as follows: (8) From (6) and (7), it follows that (9) Note that depends on the user transmitted signals while it is independent from the unknown interference covariance matrix . We can rewrite (9) as [37] (10) where is the identity matrix. Since is an matrix, we have that if (11) then is rank-deficient. Considering hereafter such a case, we have that the matrix can be eigendecomposed as (12) where is the diagonal matrix whose diagonal el- ements are the nonzero eigenvalues of and is the matrix whose columns are the eigenvectors asso- ciated with these eigenvalues. In turn, is the matrix whose columns are the eigenvectors associated with the zero eigenvalues of . The following lemma is essential for our later analysis. Lemma 1: Assume that is an eigenvalue of and is its associated eigenvector. Then, is also an eigenvalue of this matrix and its associated eigenvector is . Moreover, there exists a unitary matrix such that (13) Proof: The first part of this lemma has been proved in [37]. See Appendix A for the proof of (13). From (10) and (12) it follows that: (14) Since all columns of are orthogonal to all vectors in , we have (15) Let us assume without any loss of generality that the first user is the user-of-interest. From (15), it follows that (16) (17) It can be shown that (16) and (17) are equivalent, i.e., from either of them the other can be obtained. For instance, from (16) we have that . Substituting (13) into the latter equation and applying conjugate operation, (17) follows. Using this fact, either (16) or (17) can be exploited to identify the channel vector . From (16) along with (4), it follows that , and, therefore, is a nontrivial solution to (18) where . It is easy to verify that (18) is a linear system with equations and unknowns. To have a unique nontrivial solution for (18), it is necessary that the number of equations is greater than or equal to the number of unknowns, that is (19) Obviously, (19) also implies (11) but not vice versa. Inequality (19) represents only the necessary condition for uniqueness of the solution of (18). The necessary and suffi- cient conditions for unique identifiability of can be obtained as follows. As is a nontrivial solution to (18), it follows that lies in both and . Therefore, is in the intersection between the two latter subspaces and can be uniquely identified from (18) if and only if (20)
  4. 4. 3328 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 55, NO. 7, JULY 2007 where stands for the dimension of a subspace. If we further assume that is a full column-rank matrix, then corresponds to a unique channel vector . Hence, can be uniquely identified from (18) if and only if (20) holds true and is full column-rank. In practice, is estimated using (21) and the proposed method can be formulated as follows. 1) Compute the eigendecomposition of (22) as (23) where the matrices , , and are the finite-sample estimates of the matrices , , and , respectively, and is the diagonal matrix whose diagonal elements are the eigenvalues of with the least absolute values. 2) Compute and find the least square (LS) esti- mate of the channel vector as (24) where stands for the normalized eigenvector associ- ated with the smallest (minor) eigenvalue. IV. PERFORMANCE ANALYSIS Using the first-order perturbation analysis, let us derive an approximate expression for the MSE of the channel vector esti- mate . The following theorem holds. Theorem 1: Assume that is estimated using (24). Then, the MSE of the estimation error is approximately given by (25) where is the pseudoinverse of and (26) (27) Note that for the sake of simplicity, the time index of has been omitted in (27). The proof of Theorem 1 is given in Appendix B. It should be noted that, as the interference power goes to zero, both and converge to zero matrices. In such a case, it can be observed from (25) that even for the finite number of data samples, the approximate MSE of the channel vector estimate tends to zero. It is also noteworthy that, for any arbitrary wide- sense stationary interference, the MSE in (25) converges to zero with the rate . To further simplify the analysis of properties of the MSE ex- pression (25), let us assume for the remainder of this section that has a circular Gaussian distribution, and, therefore, the fourth-order moments of are representable in terms of the second-order moments [42], [43]. Note that circular Gaussian interference has been frequently considered in the literature on interference rejection for CDMA systems [20], [22], [30]–[32]. Before proceeding to obtain the MSE of the channel estimation error for a circular Gaussian interference, we need the following lemma. Lemma 2: Assume that is a zero-mean circular Gaussian random vector with correlation matrix and is a square ma- trix of a conformable dimension. Then (28) (29) Proof: See Appendix C. In the following theorem, we use Lemma 2 along with (25) to obtain an approximate expression for the MSE of the channel vector estimate in the presence of unknown correlated Gaussian interference. A high SIR approximation of the so-obtained MSE expression will also be derived. Theorem 2: Assume that is a circular Gaussian random vector and is estimated using (24). Then, the MSE of the estimation error is approximately given by (30) Moreover, if the following three conditions hold: (31) (32) (33) then (30) is simplified to (34) where stands for the maximum eigenvalue of a matrix, and is the Kronecker delta. Proof: See Appendix D. Note that although (31) and (32) do not perfectly hold, they appear to be reasonable practical approximations. In practice, CDMA spreading codes are deliberately designed so that even after passing through a frequency selective channel, the resulting spread-spectrum signature waveforms occupy a wide frequency band and behave as almost white pseudorandom signals (see, e.g., [19], [28], [36], [44], [45]). Hence, in most practical scenarios (31) and (32) should hold approximately because the signature vectors are the sampled versions of al- most white pseudorandom signature waveforms. The accuracy
  5. 5. ZARIFI AND GERSHMAN: BLIND SUBSPACE-BASED SIGNATURE ESTIMATION 3329 of these approximations will be validated in the simulation section. It should also be noted that the average received power of the user of interest is equal to , while the average interference power is lower bounded by the left-hand side of (33) because (35) Hence, if SIR is reasonably high, it is guaranteed that (33) holds. Based on this observation, one can consider (34) as a simple approximation of (30) in the high SIR regime that explicitly clarifies the MSE of the estimated channel vector in terms of the number of data samples, variance of the transmitted symbols, the received amplitude of the user of interest, and . Note that, according to (35), the latter quantity can be viewed as a weighted average interference power where the weighting factor depends on the matrix and the principal angles between and [12]. Assuming that the average interference power is equal to , i.e., , and is the unique nontrivial solution to (18), one can also obtain an upper-bound for the MSE in (34) as follows. As (18) has a unique nontrivial solution, it directly follows that . Let us denote the positive sin- gular values of as . Note that the positive eigenvalues of and those of are equal to . Since and are positive (semi-)definite matrices, we have [46] (36) It should be noted that if the largest eigenvalue of is unique, i.e., , then the first line of (36) holds with equality if and only if (37) where is the eigenvector of associated with . It follows from (36) that the MSE of the estimated channel vector can be- come very large if goes to zero. This is an expected result since if , then , and, therefore, (18) has a nontrivial solution other than . V. TEMPORALLY OVERSAMPLED VERSION OF THE PROPOSED ALGORITHM Condition (19) restricts both the maximum admissible number of active users and the channel length. However, if (19) does not hold, one can resort to the temporal oversampling tech- nique to facilitate identification of lengthier channels in more heavily loaded environments. Note that the latter technique requires higher computational complexity and more expensive hardware. Let us sample the continuous-time received signal (1) with the rate in the interval corresponding to the th transmitted symbol where the oversampling factor is an integer. The ISI-free part of the oversampled data vector is given by [13] (38) where (39) (40) (41) with (42) (43) (44) Using (2) and (40), it follows that [13] (45) where (46) (47) and stands for the Kronecker product. Let us write (38) in the following form (48) where . From as- sumption A4 and (48) it follows that (49) where . Note that is a matrix and . Introducing (50)
  6. 6. 3330 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 55, NO. 7, JULY 2007 from (49) and the centro-Hermitian property of we have (51) We can eigendecompose as (52) where is a matrix which contains the eigenvectors associated with the zero eigenvalue of . Similar to our discussion in Section III, it can be easily shown that is a nontrivial solution to (53) where is a ma- trix. To have a unique nontrivial solution for (53), it is required that (54) Comparing (54) with (19), it follows that using the oversam- pling factor of , the maximum admissible number of users is increased by times. It can also be observed from (54) that for a fixed number of active users, the oversampling technique con- siderably increases the maximum admissible channel length. VI. SIMULATIONS Computer simulations have been conducted to evaluate the performance of the proposed algorithm and validate the ob- tained theoretical results. In all numerical examples, and the spreading sequence associated with each user has been randomly drawn from the binary set of and then fixed throughout all examples. The entries of the channel vectors of the length have been randomly and independently drawn from a zero-mean complex Gaussian process and then have been normalized so that and fixed throughout all examples. In all but the last example, the trans- mitted symbols have been drawn from the QPSK constellation with the variance . Three different models have been used to simulate the interference vector: • Model 1: The interference vector is a circular Gaussian random vector such that the th entry of the correlation matrix is [25]. • Model 2: Interference is a second-order Gaussian autore- gressive process [22], [30] with the poles at and , that is (55) where is the th entry of a vector and is the vector of white complex Gaussian noise. • Model 3: Interference is tonal (harmonic) with (56) where describes the normalized interference fre- quency offset from the carrier frequency, and is a random phase uniformly distributed in the interval [18], [29]. Fig. 1. MSEs of the estimated channel versus the number of data samples N for the first interference model. Fig. 2. MSEs of the estimated channel versus SIR for the first interference model. In Figs. 1 through 3, the first interference model has been used, while in Figs. 4 and 5 the second interference model has been applied. In Fig. 6, the third interference model is employed. Throughout our simulations, we assume that all the users have identical average powers and is taken in all the figures but Fig. 3. Each point of the simulation curves is the result of averaging over 1000 Monte Carlo realizations of interference, noise, and the transmitted data sequences. Fig. 1 shows the experimental MSE of the proposed algo- rithm as well as the theoretical MSEs (30) and (34) versus for . For the sake of comparison, the MSE curve of the conventional (i.e., white noise assumption-based) Liu and Xu (LX) algorithm [12] is also drawn. It can be observed from Fig. 1 that the analytical MSE curves obtained from (30) and (34) follow the experimental MSE curve with quite a good accu- racy. As predicted in Section IV, these curves converge to zero with the rate . From Fig. 1, it also follows that the MSE of the estimated channel using the LX algorithm is constantly
  7. 7. ZARIFI AND GERSHMAN: BLIND SUBSPACE-BASED SIGNATURE ESTIMATION 3331 Fig. 3. MSEs of the estimated channel versus the number of users for the first interference model. Fig. 4. MSEs of the estimated channel versus the number of data samples N for the second interference model. high and does not converge to zero. Note that the LX algorithm is based on the mismatched assumption that the signal subspace is spanned by the eigenvectors associated with the largest eigenvalues of . Hence, even if becomes arbitrarily large and converges to , the LX algorithm does not offer better estimation performance. Fig. 2 shows the experimental and analytical MSEs versus SIR for . A substantial performance improvement can be observed from this figure with respect to the LX algorithm. Note that the effect of interference is negligible in high SIRs where, as Fig. 2 demonstrates, the conventional LX algorithm can also be used to obtain a reliable channel vector estimate. Note also that the MSE expressions (30) and (34) are derived using the first-order perturbation theory, and, hence, their va- lidity requires a small perturbation assumption. Therefore, as shown in Fig. 2, (30) and (34) cannot accurately predict the MSE values at very low SIRs, where the MSE of the channel vector estimate is quite large. Fig. 5. MSEs of the estimated channel versus SIR for the second interference model. Fig. 6. MSEs of the estimated channel versus SIR for the third interference model. Fig. 3 displays the MSE of the proposed technique versus for and in the case of oversampling. Three different oversampling factors of (no oversam- pling), , and are considered. If no oversampling is used, then, as follows from (19), the proposed algorithm fails to estimate the channel vector for . This theoretical re- sult is validated by Fig. 3. At the same time, it can be observed from Fig. 3 that using the oversampled version of the proposed algorithm with the oversampling factors of and , the channel vector can be reliably estimated in the presence of up to and users, respectively. Figs. 4 and 5 show the MSE curves versus and SIR, re- spectively, where (55) is used to model interference. Except the interference model, the simulation setup in Figs. 4 and 5 is iden- tical to that of Figs. 1 and 2, respectively. As Figs. 4 and 5 show, the proposed technique outperforms the conventional LX algo- rithm, and the analytical MSE curves coincide with the experi- mental MSE curves with a good precision.
  8. 8. 3332 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 55, NO. 7, JULY 2007 Fig. 6 shows the MSE curves for the proposed algorithm and for the Buzzi and Poor (BP) algorithm [22] versus the SIR for both the cases of BPSK and QPSK transmitted symbols. The analytical MSE curve of the proposed algorithm is also shown in this figure. Note that, as interference is not Gaussian, the latter curve is obtained from (25). As it can be observed from Fig. 6, the proposed algorithm performs equally well for both the BPSK and QPSK transmission schemes. Moreover, both the experimental MSE curves are quite close to the analytical curve (25). It can also be seen from Fig. 6 that, although the perfor- mance of the BP algorithm for the BPSK transmission scheme is comparable to that of the proposed algorithm, the BP algorithm becomes completely unreliable in the case of QPSK symbols. This is, however, an expected result since the BP algorithm is exclusively designed for the BPSK symbol case [22]. VII. CONCLUSION In this paper, we have proposed a new blind subspace-based signature waveform estimation technique for DS-CDMA com- munication systems operating in the presence of unknown wide- sense stationary interference. Using the centro-Hermitian prop- erty of the interference covariance matrix along with the idea of covariance differencing [37], we derive a new algorithm for blind identification of user signatures. In contrast to the ex- isting algorithms [21], [22], the proposed technique can be im- plemented using a single receiving antenna and is applicable to arbitrary constellations of transmitted symbols. Necessary and sufficient conditions for identifiability of the proposed technique have also been derived. Using the first-order perturbation theory, closed-form expressions for the mean-squared error of the esti- mated channel vector have been obtained and the effects of dif- ferent parameters on the performance of the proposed algorithm have been studied. It has been shown that the temporally over- sampled version of the proposed technique can be applied to overloaded communication systems with lengthy channels. Nu- merical examples have verified the advantages of the proposed technique and validate its error analysis. APPENDIX A It follows from (8) that (57) To prove (13), note that (58) Left-multiplying (58) with , using (57), and applying conju- gate operation, we obtain (59) As columns of span the null-space of , all of the columns of are in , or, equivalently, there exists a matrix such that (60) Applying Hermitian operation to both sides of (60), we have (61) Left-multiplying each side of (61) with the corresponding side of (60) and using the fact that , we obtain (62) Since is a square matrix, from (62) it follows that is unitary. This completes the proof. APPENDIX B First, note that from (15) and (26), we have (63) It is also obvious that (64) Let us denote (65) (66) Using the perturbation theory, the first-order approximation of can be written as [12], [38]–[40] (67) where stands for the “equality up to the first order,” and (68) As , it follows that: (69) Inserting (67) in (69) and applying the expectation operation to the squared norm of the resulting expression, we obtain (70) Hence, to find the MSE, we need to obtain . Note that (71) where the last line of (71) follows from (64). Using (22) in the right-hand side of (71), we have (72)
  9. 9. ZARIFI AND GERSHMAN: BLIND SUBSPACE-BASED SIGNATURE ESTIMATION 3333 where (73) Assuming that the received samples corresponding to two dif- ferent sampling times are independent from each other, the ex- pression inside the expectation operation in the last line of (72) is equal to , which, according to (64), is identically zero. Hence, we have (74) where (75) (76) and for the sake of simplicity, the time index in has been omitted. Inserting (5) in (75) and (76), and using (63) along with the fact that and are zero-mean independent random vectors, we have (77) (78) Computing the first term of , we have (79) Note also that, as the original bandpass interference is a wide- sense stationary random process, is a circular random vector [41], and, therefore, the first term in the right-hand side of (78) is equal to zero. Using the so-obtained values of and in (74), after straightforward manipulations it can be shown that (80) where is defined in (27). Substituting (80) in (70), (25) di- rectly follows. This completes our proof of the Theorem. APPENDIX C Assume that is a vector with the length . To prove (28), we note that (81) where is the th entry of a matrix. Hence, we have (82) where the second line of (82) holds due to the fact that is a circular Gaussian random vector (for example, see [42, p. 68]). As is the covariance matrix of , we have which directly proves (28). To prove (29), using derivations sim- ilar to (81) and (82), we obtain which proves (29). APPENDIX D Using (28) and (29) in (27), it can be readily shown that for any circular Gaussian interference vector , we have (83) Substituting (83) into (25) and using (6), the approximation (30) follows. To prove (34), note that if (31) and (32) hold true, then, according to (14), is an orthogonal matrix whose columns span , and, hence, the columns of are normalized versions of the corresponding columns of . Therefore, we have (84) (85)
  10. 10. 3334 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 55, NO. 7, JULY 2007 Using (84) and (85) in (68) yields (86) Equations (31) and (32) can be used along with (86) to obtain the following equalities: (87) (88) (89) Using (87)–(89) in (30), it follows that: (90) where (91) Note that for any vector and positive (semi-)definite matrix , we have (92) Moreover (93) where the third line of (93) follows from the facts that and is Hermitian. From (92) and (93) it follows that (94) where stands for the absolute value. Assuming that (33) holds, (34) immediately follows from inserting (94) into (90). This completes the proof. REFERENCES [1] S. Verdú, Multiuser Detection. Cambridge, U.K.: Cambridge Univ. Press, 1998. [2] D. N. C. Tse and S. V. Hanly, “Linear multiuser receivers: Effective interference, effective bandwidth and user capacity,” IEEE Trans. Inf. Theory, vol. 45, pp. 641–657, Mar. 1999. [3] M. Honig, U. Madhow, and S. Verdú, “Blind adaptive multiuser detec- tion,” IEEE Trans. Inf. Theory, vol. 41, pp. 944–960, Jul. 1995. [4] X. Wang and H. V. Poor, “Blind multiuser detection: A subspace ap- proach,” IEEE Trans. Inf. Theory, vol. 44, pp. 677–690, Mar. 1998. [5] S. L. Miller, M. L. Honig, and L. B. Milstein, “Performance analysis of MMSE receivers for DS-CDMA in frequency-selective fading chan- nels,” IEEE Trans. Commun., vol. 48, pp. 1919–1929, Nov. 2000. [6] K. Zarifi, S. Shahbazpanahi, A. B. Gershman, and Z.-Q. Luo, “Robust blind multiuser detection based on the worst-case performance opti- mization of the MMSE receiver,” IEEE Trans. Signal Process., vol. 53, pp. 295–305, Jan. 2005. [7] S. Kondo and B. Milstein, “Performance of multicarrier DS CDMA systems,” IEEE Trans. Commun., vol. 44, pp. 238–246, Feb. 1996. [8] E. A. Sourour and M. Nakagawa, “Performance of orthogonal multi- carrier CDMA in a multipath fading channel,” IEEE Trans. Commun., vol. 44, pp. 356–367, Mar. 1996. [9] U. Madhow and M. L. Honig, “MMSE interference suppression for direct-sequence spread spectrum CDMA,” IEEE Trans. Commun., vol. 42, pp. 3178–3188, Dec. 1994. [10] S. E. Bensley and B. Aazhang, “Subspace-based channel estimation for code-division multiple-access communication systems,” IEEE Trans. Commun., vol. 44, pp. 1009–1020, Aug. 1996. [11] M. Torlak and G. Xu, “Blind multiuser channel estimation in asyn- chronous CDMA systems,” IEEE Trans. Signal Process., vol. 45, pp. 137–147, Jan. 1997. [12] H. Liu and G. Xu, “A subspace method for signature waveform esti- mation in synchronous CDMA systems,” IEEE Trans. Commun., vol. 44, pp. 1346–1354, Oct. 1996. [13] 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. [14] Z. Xu, P. Liu, and X. Wang, “Blind multiuser detection: From MOE to subspace methods,” IEEE Trans. Signal Process., vol. 52, pp. 510–524, Feb. 2004. [15] M. K. Tsatsanis and Z. Xu, “Performance analysis of minimum variance CDMA receivers,” IEEE Trans. Signal Process., vol. 46, pp. 3014–3022, Nov. 1998. [16] Z. Xu and M. K. Tsatsanis, “Blind adaptive algorithms for minimum variance CDMA receivers,” IEEE Trans. Commun., vol. 49, pp. 180–194, Jan. 2001. [17] Q. Li, C. N. Georghiades, and X. Wang, “Blind multiuser detection in uplink CDMA with multipath fading: A sequential EM approach,” IEEE Trans. Commun., vol. 52, pp. 71–81, Jan. 2004. [18] H. V. Poor and X. Wang, “Code-aided interference suppression for DS/CDMA communications—Part I: Interference suppression capa- bility,” IEEE Trans. Commun., vol. 45, pp. 1101–1111, Sep. 1997. [19] S. Buzzi, M. Lops, and H. V. Poor, “Code-aided interference sup- pression for DS/CDMA overlay systems,” Proc. IEEE, vol. 90, pp. 394–435, Mar. 2002. [20] J.-T. Yuan and J.-N. Lee, “Narrow-band interference rejection in DS/CDMA systems using adaptive (QRD-LSL)-based nonlinear ACM interpolators,” IEEE Trans. Veh. Technol., vol. 52, pp. 374–379, Mar. 2003. [21] X. Wang and H. V. Poor, “Blind joint equalization and multiuser detec- tion for DS-CDMA in unknown correlated noise,” IEEE Trans. Circuits Syst. II, Analog Digit. Signal Process., vol. 46, pp. 886–895, Jul. 1999. [22] S. Buzzi and H. V. Poor, “A single-antenna blind receiver for multiuser detection in unknown correlated noise,” IEEE Trans. Veh. Technol., vol. 51, pp. 209–215, Jan. 2002. [23] E. Masry, “Closed-form analytical results for the rejection of narrow-band interference in PN spread-spectrum systems—Part I: Linear prediction filters,” IEEE Trans. Commun., vol. 32, pp. 888–896, Aug. 1984.
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Vaccaro, “Performance analysis for DOA estimation algorithms: Unification, simplification, and observations,” IEEE Trans. Aerosp. Electron. Syst., vol. 29, pp. 1170–1184, Oct. 1993. [39] Z. Xu, “Perturbation analysis for subspace decomposition with appli- cations in subspace based algorithms,” IEEE Trans. Signal Process., vol. 50, no. 11, pp. 2820–2830, Nov. 2002. [40] ——, “On the second-order statistics of the weighted sample covari- ance matrix,” IEEE Trans. Signal Process., vol. 51, no. 2, pp. 527–534, Feb. 2003. [41] F. D. Neeser and J. L. Massey, “Proper complex random processes with applications to information theory,” IEEE Trans. Inf. Theory, vol. 39, pp. 1293–1302, Jul. 1993. [42] S. Haykin, Adaptive Filter Theory, 4th ed. Upper Saddle River, NJ: Prentice-Hall, 2002. [43] T. W. Anderson, An Introduction to Multivariate Statistical Analysis, 2nd ed. New York: Wiley , 1984. [44] M. Lops, G. Ricci, and A. M. Tulino, “Narrow-band-interference sup- pression in multiuser CDMA systems,” IEEE Trans. Commun., vol. 46, pp. 1163–1175, Sep. 1998. [45] O. C. Ugweje and S. A. Matin, “Performance of CDMA overlay sys- tems using filtering and diversity,” IEEE Trans. Veh. Technol., vol. 52, pp. 456–462, Mar. 2003. [46] R. A. Horn and C. R. Johnson, Matrix Analysis. Cambridge, U.K.: Cambridge Univ. Press, 1999. Keyvan Zarifi (S’04) received the B.Sc. and M.Sc. degrees, both in electrical engineering, from the Uni- versity of Tehran, Tehran, Iran, in 1997 and 2000, respectively. He is currently pursuing the Ph.D. degree at the Department of Communication Systems, Darmstadt University of Technology, Darmstadt, Germany. From January 2002 until March 2005, he was with the Department of Communication Systems, University of Duisburg-Essen, Duisburg, Germany. Since April 2005, he has been with the Darmstadt University of Technology. From September 2002 until March 2003, he was a visiting scholar with the Department of Electrical and Computer Engineering, McMaster University, Hamilton, ON, Canada. His research interests include statistical signal processing, multiuser detection, blind estimation techniques, and MIMO communications. Mr. Zarifi was student contest finalist in the IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP), Toulouse, France, May 2006. Alex B. Gershman (M’97–SM’98–F’06) received the Diploma and Ph.D. degrees in radiophysics and electronics from the Nizhny Novgorod State University, Russia, in 1984 and 1990, respectively. From 1984 to 1999, he held several full-time and visiting research appointments in Russia, Switzerland, and Germany. In 1999, he joined the Department of Electrical and Computer Engineering, McMaster University, Hamilton, ON, Canada, where he became a Professor in 2002. He held visiting pro- fessorships with the University of Duisburg-Essen, Duisburg, Germany (2003–2005) and the Munich University of Technology, Munich, Germany (2004). Since April 2005, he has been with the Darmstadt University of Technology, Darmstadt, Germany, as a professor of Communi- cation Systems. His research interests are in the area of signal processing and communications with the primary emphasis on array processing, beamforming, multiantenna and multiuser communications, and estimation and detection theory. He has coedited two books and (co-)authored several book chapters and more than 100 journal and 150 conference papers on these subjects. Dr. Gershman is the recipient of several awards including the 2004 IEEE Signal Processing Society Best Paper Award, the IEEE Aerospace and Elec- tronic Systems Society Barry Carlton Award for the best paper published in 2004, the 2002 Young Explorers Prize from the Canadian Institute for Ad- vanced Research (CIAR), the 2001 Wolfgang Paul Award from the Alexander von Humboldt Foundation, Germany, and the 2000 Premier’s Research Excel- lence Award, Ontario, Canada. He also coauthored the paper that received the 2005 IEEE Signal Processing Society Young Author Best Paper Award. He is currently Editor-in-Chief of the IEEE SIGNAL PROCESSING LETTERS and is on the Editorial Boards of the EURASIP Journal on Wireless Communications and Networking and the EURASIP Signal Processing Journal. He was Associate Editor for the IEEE TRANSACTIONS ON SIGNAL PROCESSING during 1999–2005. He is a member of both the Sensor Array and Multichannel (SAM) and the Signal Processing Theory and Methods (SPTM) Technical Committees of the IEEE Signal Processing Society and is the current Chair of the SAM Technical Committee. He was Technical Co-Chair of the IEEE International Symposium on Signal Processing and Information Technology (ISSPIT), Darmstadt, December 2003; Co-Chair of the IEEE/ITG International Workshop on Smart Antennas, Duisburg, April 2005; General Co-Chair of the First IEEE Workshop on Computational Advances in Multi-Sensor Adaptive Processing (CAMSAP), Puerto Vallarta, Mexico, December 2005; Technical Co-Chair of the Fourth IEEE Sensor Array and Multichannel Signal Processing Workshop, Waltham, MA, June 2006; and Tutorial Chair of EUSIPCO, Florence, Italy, September 2006.

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