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Teong Chee Chuah, Bayan S. Sharif, and Oliver R. Hinton. Abstract—Recently, a robust version of the linear decorrelating detector (LDD) based on the Huber's.

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Robust CDMA Multiuser Detection Using a Neural-Network Approach Teong Chee Chuah, Bayan S. Sharif, and Oliver R. Hinton

Abstract—Recently, a robust version of the linear decorrelating -estimation technique detector (LDD) based on the Huber’s has been proposed. In this paper, we first demonstrate the use of a three-layer recurrent neural network (RNN) to implement the LDD without requiring matrix inversion. The key idea is based on minimizing an appropriate computational energy function iteratively. Second, it will be shown that the -decorrelating detector (MDD) can be implemented by simply incorporating sigmoidal neurons in the first layer of the RNN. A proof of the redundancy of the matrix inversion process is provided and the computational saving in realistic network is highlighted. Third, we illustrate how further performance gain could be achieved for the subspace-based blind MDD by using robust estimates of the signal subspace components in the initial stage. The impulsive noise is modeled using non-Gaussian -stable distributions, which do not include a Gaussian component but facilitate the use of the recently proposed geometric signal-to-noise ratio (G-SNR). The characteristics and performance of the proposed neural-network detectors are investigated by computer simulation. Index Terms— -stable distributions, decorrelating detector, -estimation, non-Gaussian noise, recurrent neural network (RNN).

I. INTRODUCTION

D

URING the past two decades, there has been substantial progress in the development of multiuser detection techniques [1] for enhancing the performance of direct-sequence code-division multiple-access (DS-CDMA) systems. To date, a commonly made hypothesis in the pursuit of multiuser detection has been the Gaussian noise assumption, despite the presence of impulsive noise in many realistic channels [2]. As a result, research in robust multiuser detection has received little attention until recently [3]. The linear decorrelating detector (LDD) [4] is an important candidate among these Gaussian-based multiuser detectors, which provides least squares (LS) estimate for the transmitted data. However, the LS criterion has long been recognized in statistical literature to dramatically lack robustness against outliers [5]. Therefore, the performance of the LDD in realistic non-Gaussian channels becomes strongly questionable. Recently, a robust version of the LDD in impulsive noise -estimation method for based on the concept of Huber’s robust regression [5] has been proposed [6]. Analytical and simulation results show that the proposed detector offers significant performance gain over the LDD in non-Gaussian noise. Specifically, the estimate of the users’ data is solved

Manuscript received June 11, 2001; revised January 16, 2002 and June 22, 2002. The authors are with the Department of Electrical and Electronic Engineering, University of Newcastle upon Tyne, Newcastle upon Tyne NE1 7RU, U.K. (e-mail: [email protected]). Digital Object Identifier 10.1109/TNN.2002.804310

iteratively by a modified residual (MR) method commonly used in robust statistics [5]. With the advances in digital technology, artificial neural networks (ANNs) have been proposed for solving various optimization problems [7]. In particular, we are motivated by the recurrent neural network (RNN) proposed by Cichocki and Unbehauen [8] for solving a system of linear equations. The primary objective of this paper is threefold: to demonstrate the implementation of the LDD by using the RNN, which eliminates explicit matrix inversion. Second, it will be shown how a robust criterion can be easily incorporated to implement the -decorrelating detector (MDD) [6] with a lower complexity. Third, in the framework of subspace-based blind detection scheme, we illustrate with examples how further performance gain can be achieved with robust estimation of the signal subspace components. In this paper, the impulsive noise is modeled by the class of non-Gaussian -stable random variables [9], which do not include a Gaussian component. It should be noted that the absence of a Gaussian component from the -stable model is solely made to utilize the recently proposed geometric signal-to-noise ratio (G-SNR) [10], [11], and, therefore, does not reflect realistic channel conditions for all communication systems. II. SYSTEM DESCRIPTION For a coherent synchronous CDMA system employing BPSK signaling with periodic spreading waveforms, the received signal is first filtered by a chip-matched filter and then sampled at the chip rate 1 . The discrete-time signal corresponding to the th chip of the th symbol is given by (1) which can be expressed in vector form as (2) is the number of active users and for the th where denote the received ampliuser, tude and th information bit, respectively. The normalized signature sequence of the th user is denoted by , where is a signature sequence of 1s assigned to is the processing gain, and with the th user, define the symbol period. is the ambient noise sample vector at the th symbol interval, which is assumed to be temporarily white with a zero mean non-Gaussian distribution.

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We model the ambient noise by zero-mean symmetric -stable (S S) random processes. The -stable distributions include Gaussian as a special case, and have been shown to accurately model impulsive noise processes under broad conditions [9] and provide a close approximation for the simple Middleton’s Class B noise [2], [9]. The impulsiveness of -stable family is controlled by the characteristic exponent, (0 2) and the dispersion ( 0) signifies the scale parameter [9]. Since -stable processes have no finite variance , we characterize the relative strength between signal for and noise by the G-SNR [10], [11]. For a detailed discussion and motivation behind the use of this novel framework, see [11].

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differentiate (7) with respect to

gives

(8) . Express the gradient of the energy funcwhere tion in vector form

(9) and the solution to (7) can be obtained by solving

III. MULTIUSER DETECTION AND LINEAR REGRESSION After dropping the symbol index and define can be written as

(10) , (1)

or the vector in (2) can be expressed as a linear regression model

denotes a -dimensional zero vector. Any estimator where defined by a minimum problem of the form given by (7) or (10) is called maximum likelihood type estimator or -estimator [5]. Note that the LS estimator corresponds to the -estimator in Gaussian noise by taking the penalty function and its derivative, respectively, as follows:

(4)

(11)

is the matrix whose columns where are the normalized spreading codes of all users. Subject to a is the unsuitable optimality criterion, known vector of user signals to be estimated. We assume that so that (4) constitutes a system of linear (overdetermined) equations. To formulate the above problem in terms of ANN parameters, the key step is to construct an appropriate Lyapunov enaccording to a specific optimality criterion ergy function and to minimize it iteratively so that the lowest energy state will yield the desired estimate . The derivation of the Lyapunov function enables us to transform the minimization problem into a set of ordinary differential or difference equations on the basis of which we can simulate and design ANN architectures with appropriate synaptic weights, input excitations and activation functions. To do so, the following energy function is defined:

, which is also the output of the LDD and the LS estimate [4], is given by the following unique solution [6]:

(3)

(12) is the cross-correlation matrix of all where normalized signature waveforms. To improve the robustness of the estimator in the presence of impulsive disturbance, other robust penalty functions that increase less rapidly than the LS can be used [5]. Two commonly used penalty functions are considered. For example, the robust MDD proposed in [6] is based on the Huber’s minimax function [5] for for

(13)

with derivative (5) where

for for

is the residual component due to the ambient noise (6)

represents suitably chosen penalty functions (usually and convex) [5], which correspond to different optimality criteria toward minimizing . The generic form of the estimate using a specific penalty can then be expressed as function (7)

(14)

is used to control the robustwhere the cutoff parameter ness of the estimator. Another penalty function of interest is the logistic function [3], [7] defined as (15) with derivative (16) Notice that (16) takes the form of sigmoidal function commonly used in ANNs, and can be considered as a smooth approxima-

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M

Fig. 1. General structure of the discrete-time ANN for implementing the decorrelating detectors. The sigmoidal neurons in the first layer for implementing the -detector are not shown explicitly.

tion to the clipper function in (14) and will be used in the neural detector studied in this paper. However, when such non-LS penalty functions are employed, no closed-form solutions exist for (10) and iterative procedures [12], such as the MR method [5], [6], must be used. In the next section, we demonstrate how this problem can be attacked easily by means of a simple ANN structure.

In order to realize the detector by digital neural networks, the key step is to convert the differential equations (18) into corresponding difference equations through first-order discretization [7], [8] (19) and where is a diagonal postypically , where itive-definite matrix whose entries to ensure stability of the algorithm [13], or in scalar form

IV. NEURAL-NETWORK IMPLEMENTATIONS OF THE DECORRELATORS There are two alternative implementations of ANNs [7], [8]. The first approach employs continuous-time computing elements to simulate appropriate differential equations. The second approach involves transforming the differential equations into corresponding (generally nonlinear) difference equations, which are simulated by discrete-time computing elements. Using the gradient approach to minimize the Lyapunov , the signal estimate can be mapped directly function to an initial value problem. This is described by a set of differential equations expressed in matrix form [7], [8] as

(17) is a positive-definite matrix that is often where diagonal. Substitute (9) into (17) yields

(18) The basic detection procedure involves iteratively computing starting form the initial point , which the trajectory of . The has the final estimate as the limiting point as proof of stability condition in [7] ensures that the system of differential equations (17) and (18) always has a stable asymptotic solution under the condition that the matrix is positive definite, and in the absence of round-off errors in the matrix .

(20) The corresponding discrete-time neural network for implementing the above difference equations is depicted in Fig. 1. In this setting, the decorrelating detectors can be viewed as a three-layer RNN followed by a bank of hard-limiters for making final decision after convergence. The synaptic weights are the coefficients of the matrices and . For the LDD, it is easy to see that the neurons in the first layer of the RNN . On the utilize linear transfer function, i.e., other hand, the MDD is easily obtained by incorporating the or into the first-layer neurons, which transfer function plays a critical role to compress excessively large outliers from exceeding the prescribed cutoff parameter , thereby achieving robustness. It should be noted that, although (19) has a complexity that is proportional to the number of users, as in the case of the MR-based MDD in (25) [6], it eliminates explicit inversion of . Although the matrix inverse needs only to be computed once for a given number of users and spreading codes, a change in one of these parameters will result in different and an update of its matrix inverse is required. For CDMA networks accommodating a very large number of users with high user entry/exit rate, it is not difficult to imagine that the matrix inverse could be computationally very costly.

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(a)

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(b)

Fig. 2. Convergence behavior of the RNN detectors and the MR-based MDD in (a) Gaussian channel and (b) impulsive noise ( 1.

x

=+

This investigation reveals that the step in calculating in [6] is redundant. Furthermore, in [6], the MDD takes the LS as its initial estimate, in fact this calculation is also redundant, and an arbitrary initial estimate such as the zero vector used in (19) will yield the same solution. A proof of these results is included in the Appendix. The saving in computational complexity of the RNN over the MR method depends on the particular algorithm used to perform the matrix operations and also on the relative sizes of and . For example, to implement the MDD using , the the MR method in (25) for a fully loaded system best known algorithm of Coppersmith and Winograd [14] takes to carry out each of the following operations: time and ; the matrix inverse the matrix multiplication between ; and the matrix multiplication between and . Therefore, in this case, the RNN-based MDD provides a for a given over the MR-based MDD in saving of [6]. Note that there could be many possible combinations and/or permutations of spreading codes giving rise to many different matrices in a dynamic network. Another approach (among others) to robust linear regression in non-Gaussian noise is the iterative procedures derived from the EM algorithm proposed by Kozick et al. [15]. In [16], similar detection methods achieved after a matched-filter bank have been studied, where it was pointed out that the LDD can be realized by linear multistage interference cancellation algorithms with ideally an infinite number of stages. More specifically, the linear successive interference cancellation and parallel interference cancellation schemes correspond to the Gauss–Seidel and Jacobi iteration [13], respectively, for approximating matrix inversion.

V. ROBUST BLIND MULTIUSER DETECTION In [6], it is shown that the subspace approach for blind detection [17] can be applied to implement the MDD, in which the receiver only requires the knowledge of the spreading code of

= 1.5). G-SNR = 7 dB and

the desired user. In this section, we address the robustness issue of this blind detector. Specifically, if we let (21) , refer to the orthonormal eigenvectors where , , of the corresponding largest eigenvalues , (which span in decreasing order of the correlation matrix of the signal subspace). In reality, we have to estimate the signal , and subspace components , from finite samples of . Under the subspace-based blind approach, instead of jointly estimating the signals vector using the known spreading codes of all users, the estimate of the pa, denoted as , rameters . The blind signal estimate can be exare found by using pressed as [6] (22) are orthonormal, the update of the essince the columns of timate of is given as [6] (23) Since (23) has the same form as (19), the RNN in Fig. 1 can also be used to implement the blind MDD by replacing the for the synaptic weights with , and the final symbol estimate is obtained by taking the sign of (22). Under this special case, the RNN detector maintains identical complexity with the blind MDD in [6]. But it is important to note that in [6], the signal subspace components are obtained through, for example, after collecting received vector , batch eigenvalue a finite block of decomposition (ED) of the sample covariance matrix or batch singular value decomposition (SVD) of the sample matrix is performed. The projection approximation subspace tracking with deflation (PASTd) algorithm [18] can also be applied, whereby the signal subspace components are updated at symbol

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(a)

(b)

(c)

(d)

=

=

Fig. 3. BER of the first user in a 15-user system each using Gold code of length N 31. (a) Impulsive channel, 1.7, all users have equal powers. (b) Gaussian channel, all users have equal powers. (c) Gaussian channel, the power of each interferer is 3 dB above user 1. (d) Mild impulsive channel, 1.9, the power of each interferer is 3 dB above user 1.

rate using the received vector . Consequently, under the ED approach, the accuracy of the signal subspace components depends critically on the quality of the correlation matrix. is non-Gaussian, it has been widely When the noise in recognized that conventional covariance estimation is no longer optimal for inference of the subspace components, instead more robust procedures are required [19]–[21]. The same is also true is unbounded of the SVD and PASTd approaches since and an aberrant data may dramatically influence the signal subspace estimates. The weakness of this subspace detector was also independently reported in [22]. It can, therefore, be anticipated that further performance improvements can be gained by incorporating robust statistical procedure during the estimation of signal subspace components, as illustrated in the next section. VI. SIMULATION RESULTS In the following examples, Gold spreading sequences of are employed. We assume that both RNN-based length in MDD and the MR-based MDD use the logistic function for a given (16) with the threshold parameter set to impulsiveness . For the RNN detectors, all initial estimates start from zeros, while the MR-based MDD takes the LS in (26) as the initial estimates [6].

=

Fig. 2(a) and (b) depict the trajectory curves in a single-user Gaussian and impulsive channels ( 1.5), respectively. Two different values of learning gain are simulated, the unperturbed value of has a positive unit amplitude and the G-SNR is 7 dB. From Fig. 2, the following results are observed. • Both RNN- and MR-based MDDs produce similar results as the RNN-based LDD in Gaussian noise [Fig. 2(a)], but they exhibit robust performance in impulsive noise [Fig. 2(b)]. • Both RNN- and MR-based MDDs converge slower than the RNN-based LDD as a result of using less rapidly increasing penalty function, which is, however, more robust against outliers. • The converged estimates of all detectors are invariant with respect to changes in step gain and initial estimates, because there is only a unique solution for any convex penalty functions. As a result, we can speed up the convergence rate considerably by increasing . • The RNN-based MDD converges to the same final states as the MR-based MDD despite the removal of the matrix . inverse Fig. 3 shows the bit error rate (BER) of the first user in a CDMA network serving 15 users under various noise scenarios. , In Fig. 3(a), the channel noise is impulsive with where all users are assumed to have equal powers. As expected, the performance of both LDDs degrades significantly under

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Fig. 4. BER versus G-SNR of the blind MDDs using batch-ED technique with 1.5) supporting six different correlation structures in impulsive channel ( equal-power users.

=

the influence of impulsive noise, whereas the performance improvement achieved by both MDDs is very significant. Fig. 3(b) shows the BER performance under Gaussian noise with perfect power control. In this case, all detectors have comparable performance, though both MDDs only incur negligible performance loss relative to both LS detectors. Fig. 3(c) plots the BER in Gaussian noise when the power of each interferer is 3 dB above the first user, and virtually the same results as in Fig. 3(b) are observed. This is because decorrelating-type detectors are near-far resistant [1]. Fig. 3(d) 1.9 with plots the BER under mild impulsive noise having the same multiuser interference as in Fig. 3(c). In this case, the sensitivity of the LS detectors to outliers is obvious and both MDDs continue to exhibit robust performance. We can conclude from Fig. 3 that the performance of both LDD and its RNN version are virtually indistinguishable from each other. The same is also true of both MDDs, while implementing the MDD using the RNN reduces system complexity. Next, we show two examples in which the performance of the blind MDD [6] could be further improved by using robust estimates of the signal subspace components. We first illustrate the BER of the blind MDD based on the batch ED ap. For simplicity, proach with a data block chosen as we follow two simple methods used in [23] when constructing the sample correlation matrix. In the first approach, covariation matrix, which uses fractional lower order moments (FLOM) [19] instead of a covariance matrix, is used. The norm param. In the second method, the received eter is chosen as is first processed by data adaptive zero memory nonvector linearity (ZMNL) [24], e.g., a soft-clipper function like (14), which limits the influence of impulsive noise while adapting the cutoff parameter to avoid clipping the useful signals. The preprocessed data resulting from the ZMNL are then used to form a covariance matrix. The result is plotted in Fig. 4, where there are six equal-power users and the channel noise is im). As anticipated, a further substantial perforpulsive ( mance gain is achieved. It is seen that the performance offered by ZMNL is better than that of the FLOM estimator, as was also observed in [25], [23]. This is due to the fact that ZMNL is

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Fig. 5. BER versus characteristic exponent of the blind MDDs using linear and nonlinear PASTd algorithms in impulsive channel ( 1.5), G-SNR 8 dB with six equal-power users.

=

=

more effective in removing outliers while preserving useful signals, on the other hand, FLOM estimator reduces the influence of impulsive noise by imposing the same weights on both the noise and useful signals, thereby distorting the useful signals to some extent. Finally, we study the performance improvement of a blind MDD using a nonlinear PASTd algorithm that is achieved by preceding the linear PASTd algorithm by a soft-clipper mentioned above. In this case, we plot the BER against the characteristic exponent in Fig. 5, and it is observed that the nonlinear PASTd algorithm also achieves a considerable gain over its linear counterpart, which again validates the motivation of using robust signal subspace estimation [26]. It should be noted that other robust covariance estimation techniques, e.g., [27], could also be used to achieve better performance but at the expense of higher complexity.

VII. CONCLUDING REMARKS In this paper, we demonstrate how an RNN can be used to implement both the LDD and MDD. In this scenario, the only difference between the two boils down to the transfer function in the first-layer neurons; the LDD is achieved by using linear transfer functions, whereas the MDD is easily realized by incorporating sigmoidal functions. The desirable feature of the RNN-based MDD lies in the elimination of explicit matrix inverse which results in computational saving and, thus, presents a significant advantage to realistic networks with large user population having high user entry/exit rate. Simulation results demonstrated that the RNN-based LDD achieves equivalent performance to that of the original LDD. The same is also true of the RNN-based MDD and the MR-based MDD. We have also shown that the blind MDD can be implemented using the RNN structure and suggested how to improve robustness through two simple techniques for estimating the signal subspace components. As a final remark, it is interesting to draw a comparison between the robust MDD with other robust detection techniques, such as the ones proposed in [22] and [28]. The latter approach

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incorporates applying some nonlinearity to the received signal after chip-matched filtering. Under certain conditions, such as in the context of locally optimum detection, the nonlinearity arises naturally. On the other hand, the MDD cast the robust multiuser detection problem coherently in a multiple regression framework and introduces the nonlinearity on the noise residuals, rather than on the received signal itself. APPENDIX

Assume that the penalty function is convex and bounded from below, the same is also true of the Lyapunov function in (5) with a unique minimum . During the minimization will finally reach a process under the gradient approach, stable asymptotic solution. Therefore, from (31) we achieve as

(33)

for the MR-based MDD, and from (32)

For the MR-based MDD, the update of the estimate is given by [6] (24) (25) . The MR-based MDD take where we have assumed the LS solution as the initial estimate (26) . We denote the th row of the matrix by To provide a fair comparison, we adopt the same technique used in [5] by defining the following function:

as

(34)

for the RNN-based MDD. Since, from (10), we know that at , this implies that the vector vanishes along the trajectories of as . More importantly, assume that all spreading codes are linin (33) or the matrix early independent, then the matrix in (34) is constant and always finite, and, . hence, will not affect the final solution as In other words, the left-hand sides of (33) and (34) approach vanishes and this ensures that as zero as the vector we have , which is the only unique solution regardless of the step gain , as long as it satisfies the stability conditions. This completes the proof.

(27)

ACKNOWLEDGMENT

(28)

The authors would like to thank Dr. D. Coppersmith of IBM T. J. Watson Research Center for discussions on the complexity of matrix inversion and for pointing out [14]. The constructive critique of the anonymous reviewers and the Associate Editor is gratefully acknowledged.

Note that the following are true for (27) [6]

REFERENCES

(29)

(30) where (30) is achieved under the assumption that with signifying the matrix – is positive semidefinite. From elementary calculus, it can be concluded from (28)–(30) for any . The MR-based MDD uses that in (27)

(31) Now we observe the effect of removing the matrix inverse by substituting into (27)

(32)

[1] S. Verdu, Multiuser Detection. Cambridge, U.K.: Cambridge Univ. Press, 1998. [2] D. Middleton, “Non-Gaussian noise models in signal processing for telecommunications: New methods and results for class A and class B noise models,” IEEE Trans. Inform. Theory, vol. 45, pp. 1129–1149, Oct. 1999. [3] H. V. Poor and M. Tanda, “Multiuser detection in impulsive channels,” Ann. Telecommun., vol. 54, pp. 392–400, 1999. [4] R. Lupas and S. Verdu, “Linear multiuser detectors for synchronous code-division multiple-access channels,” IEEE Trans. Inform. Theory, vol. 35, pp. 123–136, Feb. 1989. [5] P. J. Huber, Robust Statistics. New York: Wiley, 1981. [6] X. Wang and H. V. Poor, “Robust multiuser detection in non-Gaussian channels,” IEEE Trans. Signal Processing, vol. 47, pp. 289–305, Feb. 1999. [7] A. Cichocki and R. Unbehauen, Neural Networks for Optimization and Signal Processing. London, U.K.: Wiley, 1993. [8] , “Neural networks for solving systems of linear equations and related problems,” IEEE Trans. Circuits Syst. I, vol. 39, pp. 124–138, 1992. [9] C. L. Nikias and M. Shao, Signal Processing With Alpha-Stable Distributions and Applications. New York: Wiley, 1995. [10] J. G. Gonzalez, “Robust techniques for wireless communications in nonGaussian environments,” Ph.D. dissertation, Dept. Elect. Comput. Eng., Univ. Delaware, Newark, 1997. [11] T. C. Chuah, B. S. Sharif, and O. R. Hinton, “Robust adaptive spreadspectrum receiver with neural-net preprocessing in non-Gaussian noise,” IEEE Trans. Neural Networks, vol. 12, pp. 546–558, May 2001. [12] D. I. Clark and M. R. Osborne, “Finite algorithms for Huber -estimator,” SIAM J. Sci. Stat. Comput., vol. 7, pp. 72–85, 1986. [13] G. H. Golub and C. F. Van Loan, Matrix Computation. Oxford, U.K.: North Oxford Academic, 1989.

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[14] D. Coppersmith and S. Winograd, “Matrix multiplication via arithmetic progressions,” J. Symbolic Comput., vol. 9, pp. 251–280, 1990. [15] R. J. Kozick, R. S. Blum, and B. M. Sadler, “Signal processing in nonGaussian noise using mixture distributions and the EM algorithm,” in Proc. Asilomar Conf. Signals, Syst., Computers, vol. 1, Pacific Grove, CA, 1998, pp. 438–442. [16] H. Elders-Boll, H. D. Schotten, and A. Busboom, “Efficient implementation of linear multiuser detectors for asynchronous CDMA systems by linear interference cancellation,” Eur. Trans. Telecomm., vol. 9, pp. 427–437, 1998. [17] X. Wang and H. V. Poor, “Blind multiuser detection: A subspace approach,” IEEE Trans. Inform. Theory, vol. 44, pp. 677–690, June 1998. [18] B. Yang, “Projection approximation subspace tracking,” IEEE Trans. Signal Processing, vol. 44, pp. 95–107, Jan. 1995. [19] P. Tsakalides and C. L. Nikias, “The robust covariation-based MUSIC (ROC-MUSIC) algorithm for bearing estimation in impulsive noise environments,” IEEE Trans. Signal Processing, vol. 44, pp. 1623–1633, Sept. 1996. [20] D. B. Williams and D. H. Johnson, “Robust estimation of structured covariance matrices,” IEEE Trans. Signal Processing, vol. 41, pp. 2891–2906, Dec. 1993. [21] A. Moghaddamjoo, “Eigenstructure variability of the multiple-source multiple-sensor covariance matrix with contaminated Gaussian data,” IEEE Trans. Acoust., Speech, Signal Processing, vol. 36, pp. 153–167, Jan. 1988. [22] S. N. Batalama, M. J. Medley, and D. A. Pados, “Robust adaptive recovery of spread-spectrum signals with short data records,” IEEE Trans. Commun., vol. 48, pp. 1725–1731, Oct. 2000. [23] B. M. Sadler, R. J. Kozick, and T. Moore, “Performance analysis for direction finding in non-Gaussian noise,” in Proc. IEEE Int. Conf. Acoust., Speech, Signal Processing, vol. 5, Phoenix, AZ, 1999, pp. 2857–2860. [24] B. M. Sadler, “Detection in correlated impulsive noise using fourth-order cumulants,” IEEE Trans. Signal Processing, vol. 44, pp. 2793–2800, Dec. 1996. [25] R. J. Kozick and B. M. Sadler, “Robust subspace estimation in nonGaussian noise,” in Proc. IEEE Int. Conf. Acoust., Speech, Signal Processing, Istanbul, Turkey, 2000, pp. 4192–4195. [26] A. Swami and B. Sadler, “TDE, DOA and related parameter estimation problems in impulsive noise,” in Proc. IEEE Signal Processing Workshop on Higher Order Statistics, Banff, AB, Canada, July 21–23, 1997, pp. 273–277. [27] R. J. Kozick and B. M. Sadler, “Maximum-likelihood array processing in non-Gaussian noise with Gaussian mixtures,” IEEE Trans. Signal Processing, vol. 48, pp. 3520–3535, Dec. 2000. [28] S. N. Batalama, M. J. Medley, and I. N. Psaromiligkos, “Adaptive robust spread-spectrum receivers,” IEEE Trans. Commun., vol. 47, pp. 905–917, Apr. 1999.

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