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Blind Detection of Synchronous CDMA in Non-Gaussian Channels Yingwei Yao, Member, IEEE, and H. Vincent Poor, Fellow, IEEE
Abstract—Most blind source separation algorithms assume the channel noise to be Gaussian. This paper considers the problem of noncooperative blind detection of synchronous direct-sequence code-division multiple-access communications (no knowledge of the spreading sequences or training data) in non-Gaussian channels. Three iterative algorithms with different performance and complexity tradeoffs are proposed. Simulation results show that they significantly outperform Gaussian-optimal blind source separation algorithms in non-Gaussian channels. The Cramér–Rao lower bound for this problem is computed, and the performance of the proposed algorithms is shown to approach this bound for moderate signal-to-noise ratios. Index Terms—Blind source separation, DS-CDMA, robust detection.
I. INTRODUCTION
I
N SYSTEMS adopting nonorthogonal multiple-access schemes such as direct-sequence code-division multiple-access (DS-CDMA), multiple-access interference (MAI) is the limiting factor of the systems’ capacity. To mitigate the MAI, various multiuser receivers have been proposed [1]. Most multiuser receivers require either the knowledge of the desired user’s spreading sequence or of a training sequence. When neither is available, the problem of extracting the transmitted data falls into the category of blind source separation (BSS). Typically, blind source separation algorithms make some assumptions about the information sources’ structure, for example, constant modulus [2] and source independence [3]. When dealing with digital communication systems, a promising approach is to take advantage of the finite alphabet (FA) property arising from the digital nature of the information sources. Several different algorithms that exploit this property have been proposed in recent years [4]–[8]. While most existing blind source separation algorithms assume the ambient noise to be additive white Gaussian noise (AWGN), in many physical channels, such as indoor radio channels [9] and underwater acoustic channels [10], the ambient noises are known to be non-Gaussian. Studies have shown that in such channels, maximum-likelihood multiuser detection can
Manuscript received May 20, 2002; revised November 1, 2002. This work was supported in part by the U.S. Office of Naval Research under Grant N00014-03-1-0102, and in part by the New Jersey Center for Wireless Telecommunications. The associate editor coordinating the review of this paper and approving it for publication was Prof. Nicholas D. Sidiropoulos. Y. Yao was with the Department of Electrical Engineering, Princeton University, Princeton, NJ 08544 USA. He is now with the University of Minnesota, Minneapolis, MN 55455 USA (e-mail:
[email protected]). H. V. Poor with the Department of Electrical Engineering, Princeton University, Princeton, NJ 08544 USA (e-mail:
[email protected]). Digital Object Identifier 10.1109/TSP.2003.820086
achieve significant performance gain over Gaussian-optimal techniques [11]. Suboptimal multiuser detectors have also been proposed for these channels [12], [13]. In this paper, we address the problem of blind source separation in non-Gaussian ambient noise. In particular, we consider the eavesdropping of user data in a synchronous DS-CDMA system with no multipath. No knowledge of the users’ spreading sequences or training data is assumed at the receiver. Several robust blind source separation algorithms are proposed. We demonstrate through analytical and numerical results that by taking the non-Gaussian nature of the channel noise into account, our algorithms significantly outperform the Gaussian optimum blind source separation algorithms. The rest of this paper is organized as follows. In Section II, we present the signal and noise models. Several different robust blind source separation algorithms are presented in Section III. We analyze the asymptotic performance of the maximum-likelihood algorithm in Section IV and present the simulation results in Section V. Finally, Section VI contains some concluding remarks. II. SIGNAL MODEL In a DS-CDMA system, the waveform received by a given terminal consists of superimposed data modulated signals in the ambient channel noise. Without loss of generality, we consider the baseband signal model (1) contains the useful signals, and is the channel where noise, which we assume to be white. Consider a -user system with a data frame size of symbols, in which case, can be written as (2) is the symbol interval and where , , , and denote the th user’s received amplitude, delay, symbol stream, and spreading waveform, respectively. We will assume that , with extension to more general cases being straightforward. The spreading waveform is of the following form: where
(3) where is the spreading gain, , spreading sequence of the th user, and ized chip waveform of duration
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is the is the normal. We consider
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the synchronous case, i.e., . After chipmatched filtering and chip-rate sampling, the received vector can be written as (4) where is the normalized spreading code of the th user, and is the channel noise vector during the th symbol interval. are assumed to be independent and identically distributed (i.i.d.) random variables with non-Gaussian distribution
thus impractical in most applications. In this section, we present several suboptimal approaches with much lower complexity. A. EM-Based Expected Maximum-Likelihood Algorithm (EM-EML) Assume without loss of generality that the transmitted data is binary and antipodally modulated with all symbol vectors being equally likely. With both and unknown, one approach is to average the probability density function over all possible and select that maximizes the resulting function . This is equivalent to modeling the received as a mixture with component densities: vector
(5) Here, we have adopted a commonly used two-term Gaussian mixture model that serves as an approximation to the more fundamental Middleton Class A noise model [10], [14] and has been used extensively to model additive non-Gaussian noise. Note that while we deal with the eavesdropping of a DS-CDMA system in this paper, the signal model (4) is general enough to include any instantaneous mixture of finite alphabet sources. The algorithms proposed here can hence be applied directly to other applications such as direction-of-arrival estimation for uncalibrated antenna arrays. III. ROBUST BLIND SOURCE SEPARATION diag and On denoting we can rewrite the received vector in the th symbol interval (4) as
,
(8) where is the set of parameters to be estimated. Here, is the th row of , and is the set of all possible transmitted vectors. The expected likelihood function of is (9)
(6) and the noise paOur objective is to estimate the matrix samples of the rameters , , and , given a block of . Denote received vectors and . The likelihood function of can be written as
that maximizes the likelihood We will call the parameter set function (9) the expected maximum likelihood estimate (MLE). Due to the complexity of the likelihood functions, it is not practical to solve this maximization problem directly. A popular approach for solving this kind of problem is the expectation–maximization (EM) algorithm [15]–[17], which obtains the approximate MLE iteratively. In particular, we will adopt the space-alternating generalized expectation-maximization (SAGE) algorithm proposed by Fessler and Hero [18]. To do this, we divide the parameter set into two subsets and . The update equations for estimating the parameters at iteration are presented as follows, whereas the details of the derivation have been relegated to the Appendix.
(7) (10) where is the th element of the vector , and is the th row of . Since the symbol vectors are unknown, the optimal algorithm for solving this problem is to obtain a maximum-likelihood estimate of these parameters for each possible transmitted sequence and select the data sequence and the corresponding parameter estimates that maximize the likelihood function (7). This algoand is rithm has a computational complexity exponential in
(11)
(12)
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and
and applying the SAGE algorithm, we obtain the following iterative algorithm. (An outline of the derivation of this algorithm can be found in the Appendix.) (13)
Here, the weight factors follows:
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and
sign
are defined as (23) (14)
and (15) (24)
where
(25) (16) and (17)
(26) and
B. EM-Based Algorithm With Projection (EM-P) While the complexity of the EM-based expected ML algorithm is much lower than that of the optimal algorithm, it is still exponential in the number of users . In this section, we present another EM-based algorithm whose complexity is polynomial in . Instead of modeling the received vectors as random variables component denhaving a probability density function with sities each corresponding to a possible transmitted vector, we will treat the unknown transmitted bits as parameters to be estimated as well. Designating the parameter set to be estimated as
(27) where we denote (28)
(29)
(18) the incomplete-data likelihood function is
(30) and (19) Dividing the parameter set into three subsets: (20) (21) and (22)
(31) Note that in obtaining the update formula (23) for the data vectors, we treat the transmitted vectors as continuous variables first and then project the estimation results onto discrete sets. With additional computational cost, it is possible to achieve improved performance by using enumeration or a slowest descent algo. rithm [13] to estimate
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C. Iterative M-Estimator With Projection (IM-P)
2) Update the matrix
A well-known approach to robust estimation is the M-estimation method proposed by Huber [19]. It has been used in [12] to construct a robust decorrelator for multiuser detection. In this section, we present an iterative M-estimator algorithm for robust blind source separation. In particular, we will use the approximate minimax M-estimator [12] since it requires only the knowledge of the total ambient noise variance. In Gaussian channels, the maximum-likelihood criterion often leads to the least-squares (LS) cost function. Since the LS estimate is sensitive to the tail behavior of the noise density, it suffers large performance loss when the channel noise is non-Gaussian. To improve estimation performance in impulsive channels, Huber proposed a class of estimators called M-estimators [19]. Instead of using the square of the residuals as a cost function, a nonlinear function of the residuals, which suppresses the impact of the outlier observations, is used. In our case, this leads to the following optimization problem: (32) To achieve the best performance, we need to select the function based on the noise parameters , , and . Typically, this knowledge is not available to the receivers, so we will use a suboptimal cost function as follows: (33) is the total noise variance, and . In [12], it is shown that approximate M-estimators defined by (33) achieve satisfactory performance. Since both and are unknown, we will update them iteratively. Each iteration of this algorithm consists of two steps. First, we decode the symbol vectors using an estimate of obtained in the previous iteration; then, assuming the data detection is correct, we update the estimate of . This procedure is similar to the iterative least-squares (ILS) algorithms [5] for Gaussian channels; therefore, this algorithm can be seen as a robustification of the ILS algorithms. The details of the updating procedure are as follows. by 1) Update the data vectors solving the following minimization problem: where
(34) sign
(35)
which in turn can be solved iteratively by the modified residual method [12]. On denoting the estimate at the th step as , it is updated as follows: (36) where is the step size, , and a vector sign
for
.
(37)
row by row: (38)
which is also done by the modified residual method: (39) where
is the
th row of .
, and
D. Computational Complexity Among the three algorithms we have discussed, the EM-based expected ML algorithm (EM-EML) has computational complexity that is exponential in the number of users while the other two algorithms’ complexities are polynomial in the number of users. We can expect that in a large system, EM-EML’s complexity is much higher than that of EM-P and IM-P. However, it is not clear how their computational costs compare with each other in systems with nominal number of users. Here, we investigate quantitatively the computational complexity of these algorithms to get a better understanding of their relative costs under different scenarios. We will use the number of multiplications in an algorithm to measure its complexity. Reciprocal operation can be implemented using Newton’s method and have complexity of about three multiplications [20]. The exponential function can be evaluated using table-based algorithm and needs about ten multiplications [21]. Counting the operations in these three algorithms, we obtain the following results: EM-EML: Each iteration takes multiplications. EM-P: In each iteration, the number of multiplications is . IM-P: Each iteration of the IM-P algorithm consists of two inner iterations for updating and . Each iteration of these inner loops costs (for ) and (for ) multiplications, respectively. In our simulations, these inner loops usually converge in fewer than four iterations; therefore, each iteration of IM-P takes multiabout plications. We can see that the complexity of these algorithms is basically and . The complexity of EM-EML is exponenlinear in tial in the number of users, whereas the other two algorithms’ complexity grows quadratically with . In Fig. 1, we show how the single iteration computational costs of these algorithms change with the number of users. We can see that even in a relatively small system (e.g., six users), the computational complexity of EM-EML is much higher than that of the other two algorithms. Comparing EM-P with IM-P, we see that the single iteration complexity of EM-P is only slightly higher than that of IM-P, although the former algorithm’s update equations seem to be much more complicated. The overall costs of these algorithms also depend on their convergence speed. In our simulations, we found that in low to moderate SNRs, the number of
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is a
where submatrix given by
10
EMEML EMP IMP
matrix
14
10
(42) Number of Multiplications
12
10
It can be shown that 10
10
8
10
6
(43)
10
is defined in (16), and
where 4
10
M
0
5
10
15 Number of Users
20
25
30
(44)
N
Fig. 1. Single iteration computational complexity of different algorithms = 200, = 31). (
If the SNRs are sufficiently high, we can show that iterations needed for these algorithms is comparable, whereas in high SNRs, the IM-P algorithm converges somewhat faster than the other two. In summary, the EM-EML algorithm is practical only in very small systems. The complexity difference between EM-P and IM-P is not dramatic; to determine which one is more suitable for a particular situation, we need to take into account their performance difference.
(45) where
(46) If
, then we have
IV. ASYMPTOTIC PERFORMANCE ANALYSIS In this section, we examine the asymptotic performance of the expected maximum likelihood algorithm. In particular, we will investigate the estimation error of the mixing matrix . Denote the set of parameters to be estimated as , is the th row of . To simplify the analysis, we aswhere sume perfect knowledge of , , . The probability distribuis tion of the observed vector given
(47) Summarizing the above arguments, we obtain the following lemma. Lemma 1: Assuming the number of observed samples is sufficiently large and the signal-to-noise ratios are sufficiently of the parameter should satisfy high, the MLE (48)
(40) In [8], using results of Redner and Walker [17], we have shown that when the additive noise is white Gaussian noise, given a sufficiently large number of observed samples, is asymptotically normally distributed with mean zero and covariance matrix , where is the true value of the pais the maximum-likelihood estimate of given rameter, observed samples, and is the Fisher information matrix. It is straightforward to show that this result also holds here. Since the detailed proof is lengthy and tedious, we will not include it in this paper. Using the definition of [22], the Fisher information matrix can be written as
If we let is Gaussian,
in (46), we have that when the channel noise . V. NUMERICAL RESULTS
We simulate a system with six users each assigned a different Gold sequence of length 31. The received powers of all users are and the same, and the noise distribution parameters are . The number of received samples we use is 200, and the results are obtained by averaging over 1000 simula, tion runs. The initial estimates of and are obtained using the analytical constant modulus algorithm (ACMA) [6], whereas the noise parameters are initialized as follows [23]: (49) median
(41)
(50)
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2
0
limit (non-Gaussian) limit (Gaussian) EM-EML EM-P IM-P ACMA
10
10
10
10
1
Bit Error Rate
Mean Square Estimation Error
10
EM-EML EM-P IM-P ACMA
0
10
10
-1
-2
10 0
5
10
-1
-2
-3
-4
15
0
5
10
Fig. 2. Mean square estimation error versus SNR ( = 0:01 and = = 100, where all users have the same power).
15
SNR (dB)
SNR (dB)
Fig. 3. BER performance of different algorithms ( = 0:01 and = =
100, where all users have the same power). 10
1
and
limit (non-Gaussian) limit (Gaussian) EM-EML EMP IM-P ACMA
(51)
(52) when the SNRs are high. For comparison, we also plot the asymptotic performance of the MLEs in the Gaussian noise case. We can see the performance gain over the case of Gaussian noise. The performance of the three robust blind source separation algorithms is quite close and is accurately predicted by the asymptotic limit (52) for high SNRs. We can also see that the CRB for the Gaussian channel well approximates the performance of the ACMA in the impulsive channel when the SNR is high. This can be explained by the asymptotic consistency and asymptotic normality of the Gaussian optimal parameter estimator in non-Gaussian noises [22]. The bit error rates (BERs) of the different algorithms are presented in Fig. 3. We can see that the optimal algorithm achieves the best performance, whereas the performance of EM-P and IM-P is very close to it. All three algorithms significantly outperform the ACMA algorithm. The performance of different algorithms in a very impulsive channel is presented in Figs. 4 and 5. In this simulation, and . The interfering users’ rewe set ceived powers are 10 dB larger than that of the desired user. It can be seen that while the iterative M-estimator-type receiver achieves substantial performance gain over the ACMA, the performance gap between it and the two EM-based algorithms is also significant. We also examine the performance of the proposed algorithms in Gaussian noise. The mean-square estimation errors and BER’s of different algorithms are plotted in Figs. 6 and 7,
Mean Square Estimation Error
, according
10
10
10
0
-1
-2
-3
0
1
2
3
4
5 SNR (dB)
6
7
8
9
10
Fig. 4. Mean square estimation error versus SNR ( = 0:1, = = 100, and the powers of the interferer are 10 dB larger than that of the desired user). 0
10
EM-EML EM-P IM-P ACMA -1
10
-2
10 Bit Error Rate
Fig. 2 shows the mean square estimation error of different algorithms. Since we have set to Lemma 1 , it should be close to
10
-3
10
-4
10
-5
10
0
1
2
3
4
5 SNR (dB)
6
7
8
9
10
Fig. 5. BER performance of different algorithms ( = 0:1, = = 100, and the powers of the interferer are 10 dB larger than that of the desired user).
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1
10
putational costs have been proposed. Analytical and numerical results show that these algorithms achieve significant performance gain over Gaussian optimal algorithms in impulsive channels. We have adopted an instantaneous multiple-input multiple-output (MIMO) channel model in this paper. In systems with multipath, blind channel equalization must be performed first to mitigate the intersymbol interference (ISI). Since most blind channel estimation/equalization algorithms only perform well under high SNRs and are designed for Gaussian channels, eavesdropping in impulsive channels with large ISI could be a challenge.
limit EM-EML EM-P IM-P ACMA
Mean Square Estimation Error
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0
10
-1
10
APPENDIX A DERIVATION OF UPDATE (10)–(13) -2
10
0
5
10
15
SNR (dB)
Define the set of complete data as
, where
Fig. 6. Mean square estimation error versus SNR (Gaussian channel noise; all users have the same power).
(53) 0
10
EM-EML EM-P IM-p ACMA
in which indicates which data vector is transindicates mitted in the th symbol interval, whereas whether the additive noise during the th chip of the th symbol or . We will denote is and
-1
Bit Error Rate
10
-2
10
-3
10
(54)
-4
10
0
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6
8
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14
SNR (dB)
Fig. 7. BER performance of different algorithms (Gaussian channel noise; all users have the same power).
respectively. It can be seen that the robust algorithms’ performance is also close to optimal (CRB) in Gaussian channels. In this case, the performance gains of these algorithms over the ACMA are minimal. Summarizing the above simulation results, the proposed algorithms significantly outperform the ACMA in non-Gaussian channels, whereas in Gaussian channels, their performance is similar to that of the ACMA. The complexity of the EM-EML is prohibitively high, but the complexities of the EM-P and and , IM-P are much more reasonable [ respectively]. Considering the fact that the complexity of the [6], applying the EM-P or IM-P to the ACMA is output of the ACMA can bring substantial performance gain with only a modest increase of computational cost.
First, we update is
. The complete-data log-likelihood function
(55) We can show that
VI. CONCLUSION In this paper, we have considered the problem of robust blind source separation. Three algorithms with different com-
(56)
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Substituting (54) into (56), we have that
(57) (65) Denote (58) Solving (57), we obtain the following updating formula for
:
(66) (59)
,
where ,
, and .
Denote (60)
APPENDIX B DERIVATION OF EM ALGORITHM WITH PROJECTION We set the complete data as Denote
where
(61) We can show that (67)
(62) We have Second, we update
. Similarly, we can show that
(68) Maximizing
with respect to
, we have
(63) The maximization step yields the iteration formulas for as follows:
(64)
(69)
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The update formula (23) is obtained by first treating , as continuous vectors and then projecting the esis timates onto the discrete set. The weight factor defined in (28). For updating , we have
(70) The new estimate for
should satisfy
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[10] D. Middleton, “Channel modeling and threshold signal processing in underwater acoustics: An analytical overview,” IEEE J. Ocean. Eng., vol. 12, pp. 4–28, Jan. 1987. [11] H. V. Poor and M. Tanda, “Multiuser detection in impulsive channels,” Ann. Telecommun., vol. 54, pp. 392–400, July–Aug. 1999. [12] X. Wang and H. V. Poor, “Robust multiuser detection in non-Gaussian channels,” IEEE Trans. Signal Processing, vol. 47, pp. 289–305, Feb. 1999. [13] P. Spasojevic´ and X. Wang, “Improved robust multiuser detection in nonGaussian channels,” IEEE Signal Processing Lett., vol. 8, pp. 83–86, Mar. 2001. [14] S. M. Zabin and H. V. Poor, “Efficient estimation of the class a parameters via the EM algorithm,” IEEE Trans. Inform. Theory, vol. 37, pp. 60–72, Jan. 1991. [15] A. Dempster, N. Laird, and D. Rubin, “Maximum likelihood estimation from incomplete data via the EM algorithm (with discussion),” J. R. Stat. Soc. B, vol. 39, no. 1, pp. 1–38, 1977. [16] G. J. McLachlan and T. Krishnan, The EM Algorithm and Extensions. New York: Wiley, 1997. [17] R. A. Redner and H. F. Walker, “Mixture densities, maximum likelihood and the EM algorithm,” SIAM Rev., vol. 26, no. 2, pp. 195–239, 1984. [18] J. A. Fessler and A. O. Hero, “Space-alternating generalized expectation-maximization algorithm,” IEEE Trans. Signal Processing, vol. 42, pp. 2664–2677, Oct. 1994. [19] P. J. Huber, Robust Statistics. New York: Wiley, 1981. [20] R. P. Brent, “Multiple-precision zero-finding methods and the complexity of elementary function evaluation,” in Analytic Computational Complexity, J. F. Traub, Ed. New York: Academic, 1975, pp. 151–176. [21] D. Defour, F. de Dinechin, and J.-M. Muller, “Correctly rounded exponential function in double precision arithmetic,” Inst. Nat. Recherche Inform. Automat., Res. Rep. RR-4231, 2001. [22] H. V. Poor, An Introduction to Signal Detection and Estimation. New York: Springer-Verlag, 1994. [23] L. Ljung, System Identification: Theory for the User. Englewood Cliffs, NJ: Prentice-Hall, 1987.
(71) Solving the above minimization problem, it is straightforward to obtain the update (24) for . Following similar procedures, we as given by (25)–(27). can obtain the update equations for
Yingwei Yao (S’98–M’03) received the Ph.D. degree in electrical engineering from Princeton University, Princeton, NJ, in 2002. He is currently a postdoctoral researcher with University of Minnesota, Minneapolis. His research interests include multiuser communications theory and statistical signal processing for wireless communications.
REFERENCES [1] S. Verdú, Multiuser Detection. Cambridge, U.K.: Cambridge Univ. Press, 1998. [2] R. Johnson, P. Schniter, T. J. Endres, J. D. Behm, D. R. Brown, and R. A. Casas, “Blind equalization using the constant modulus criterion: A review,” Proc. IEEE, vol. 86, pp. 1927–1950, Oct. 1998. [3] J. F. Cardoso, “Blind signal separation: Statistical principles,” Proc. IEEE, vol. 86, pp. 2009–2025, Oct. 1998. [4] K. Anand and V. U. Reddy, “Maximum likelihood estimation of constellation vectors for blind separation of co-channel BPSK signals and its performance analysis,” IEEE Trans. Signal Processing, vol. 45, pp. 1736–1741, June 1997. [5] S. Talwar, M. Viberg, and A. Paulraj, “Blind separation of synchronous co-channel digital signals using an antenna array—Part I: Algorithms,” IEEE Trans. Signal Processing, vol. 44, pp. 1184–1197, May 1996. [6] A. J. van der Veen, “Analytical method for blind binary signal separation,” IEEE Trans. Signal Processing, vol. 45, pp. 1078–1082, Apr. 1997. [7] T. Li and N. D. Sidiropoulos, “Blind digital signal separation using successive interference cancellation iterative least squares,” IEEE Trans. Signal Processing, vol. 48, pp. 3146–3152, Nov. 2000. [8] Y. Yao and H. V. Poor, “Eavesdropping in the synchronous CDMA channel: An EM-based approach,” IEEE Trans. Signal Processing, vol. 49, pp. 1748–1756, Aug. 2001. [9] K. L. Blackard, T. S. Rappaport, and C. W. Bostian, “Measurements and models of radio frequency impulsive noise for indoor wireless communications,” IEEE J. Select. Areas Commun., vol. 11, pp. 991–1001, Sept. 1993.
H. Vincent Poor (S’72–M’77–SM’82–F’87) received the Ph.D. degree in electrical engineering and computer sciemce in 1977 from Princeton University, Princeton, NJ, where he is currently the George Van Ness Lothrop Professor of electrical engineering. From 1977 until he joined the Princeton faculty in 1990, he was a faculty member at the University of Illinois at Urbana-Champaign. He has also held visiting and summer appointments at several universities and research organizations in the United States, Britain, and Australia. His research interests are primarily in the area of statistical signal processing, with applications in wireless communications and related areas. Among his publications in this area is the recent book Wireless Communication Systems: Advanced Techniques for Signal Reception. Dr. Poor is a member of the National Academy of Engineering and is a Fellow of the Institute of Mathematical Statistics, the Optical Society of America and other organizations. His IEEE activities include serving as the President of the IEEE Information Theory Society in 1990 and as a member of the IEEE Board of Directors from 1991 to 1992. Among his recent honors are an IEEE Third Millennium Medal (2000), the IEEE Graduate Teaching Award (2001), the Joint Paper Award of the IEEE Communications and Information Theory Societies (2001), the NSF Director’s Award for Distinguished Teaching Scholars (2002), and a Guggenheim Fellowship (2002–2003).
