SMC-Based Blind Detection for DS-CDMA ... - Semantic Scholar
IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 54, NO. 6, JUNE 2006
971
SMC-Based Blind Detection for DS-CDMA Systems Over Multipath Fading Channels Qian Yu, Guoan Bi, and Chunru Wan
II. SIGNAL MODEL
Abstract—This letter derives a computationally efficient sequential Monte Carlo solution for blind detection of direct-sequence code-division multiple-access systems over multipath fading channels by decomposing the observed data into a number of signal components. Then the parameters of each component can be estimated by the sequential importance sampling and Kalman filtering. In comparison with other similar receivers, simulation results demonstrate that the proposed solution achieves the desirable performance with a significantly reduced computational complexity.
Let us consider a DS-CDMA system that has active users whose signals are transmitted over multipath channels with additive white Gaussian noise (AWGN). Here we assume that the delay spread is small compared with the symbol interval, so that the intersymbol interference is negligible. The received signal vector is given by
Index Terms—Code-division multiple access (CDMA), expectation-maximization (EM), Kalman filtering, multipath fading, sequential importance sampling, sequential Monte Carlo (SMC).
(1)
I. INTRODUCTION EQUENTIAL Monte Carlo (SMC) methods have begun to show a great potential for solutions to a wide range of statistical inference problems [1], and been successfully applied to a few problems in communications, such as blind equalization and detection for fading-channel communication systems [2]–[8]. For applications to code-division multiple-access (CDMA) systems, the conventional Kalman filter and SMC technique were combined for the problem of single-user detection [3]. Although the solutions to the detection problem for multiuser systems were presented in [4] and [5], the required computational complexity of all these methods grows exponentially with the number of users. The work in [6]–[8] reported the method based on the output of a whitened matched filter to perform detection sequentially in order to achieve nonexponential complexity, but this algorithm was reported only for flat-fading channels. This paper presents an SMC-based formulation of multiuser detection for a direct-sequence (DS)-CDMA system over unknown multipath fading channels. First, the multiuser system is decomposed into separate single-user systems by the expectation-maximization (EM) decomposition algorithm. Then the sequential importance sampling (SIS) and the Kalman filtering (KF) are combined to perform the data detection and channel estimation for every single-user system. With the decomposition of the superimposed observation signals, the total computational complexity of the proposed method can be reduced to be linear with the number of the active users. In addition to the details of the proposed solution, simulation results are provided to show that the receiver performs well over multipath fading channels with a significantly reduced computational complexity.
S
where for user , the transmitted symbols are differentially modulated with the binary infor. Spreading matrix denotes a mation symbols matrix .. . .. . .. .
..
.
..
.
(2)
.. . .. .
where is the spreading gain and is the spreading sequence. The channel vector is expressed as (3) where is the total number of resolvable paths in the channel. The noise term in (1) is assumed to be an independent, identically distributed (i.i.d.) complex white , i.e., Gaussian vector with a zero mean and variance . III. EM-SMC RECEIVER Because conventional SMC methods make decisions for all users at a time, the complexity of prediction and update at each step is exponential to the number of users. In [9], the EM method suggests a specific way to decompose superimposed signals. Based on this method, the observation vector given in (1) can be decomposed into components, such that
Paper approved by X. Wang, the Editor for Wireless Spread Spectrum of the IEEE Communications Society. Manuscript received April 23, 2004; revised November 22, 2004; April 13, 2005; and November 25, 2005. The authors are with the School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore 639798 (e-mail: [email protected]. sg). Digital Object Identifier 10.1109/TCOMM.2006.876830 0090-6778/$20.00 © 2006 IEEE
(4)
(5)
972
IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 54, NO. 6, JUNE 2006
Fig. 1. Receiver framework.
where weight coefficients are real-valued scalars satisfying , and chosen to be to achieve maximum convergence rate [10]. The EM framework assumes the following form for the th iteration: Expectation-Step: For , compute
probability of the information symbol as
can be estimated
(7)
(6) Maximization-Step: For , obtain the maximum-likelihood (ML) estimate of based on the . Then the SMC method is used to approach the optimum ML estimations for each user, respectively. After the EM decomposition, SMC estimation can be implemented with a parallel structure for each unknown user, respectively. This is the key point that leads to the better computational efficiency of the proposed SMC approach. The framework of this approach is presented in Fig. 1. The algorithm achieves improvement on subsequent parameter estimates by the feedback of current parameter estimates to more effectively decompose the observed data. Here, represents the estimates of unknown parameters. One important property of this structure is that for different users are estimated simultaneously in parallel, therefore, no error-propagation problem exists in the system. We now consider the estimation of unknown parameters based on each signal component . For simplicity of presentation, the hat and superscript of expressions hereafter are omitted, and the system described in (4) is considered. Denote . Let be a sample drawn by the SMC at time , and . For each value of , a set denote , which are properly of Monte Carlo samples, weighted with respect to the distribution , are obtained. For every symbol , the a posteriori
where samples
and is an indicator function. The are drawn from the trial sampling density (8)
and the importance weight can be updated according to
(9) By assuming the channel to be Gaussian, i.e., , the conditional distribution can be computed as
(10) where the a posteriori mean and covariance are updated recursively by the Kalman filter as
(11)
(12)
IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 54, NO. 6, JUNE 2006
973
TABLE I EM-SMC BLIND DETECTION ALGORITHM
Hence, the conditional density is given by
is about . However, domcomputation of inant computation for the proposed SMC needs one-step predictions for computing . Therefore, the entire computational complexity required is , where is the number of the EM iterations.
(13) (14) with the mean and the covariance
(15) Cov (16) Then,
in (9) can be computed as follows:
(17) Finally, the EM-SMC blind detector for the system is given in Table I. Computational complexity of conventional SMC method is on the order of , because the dominant
IV. SIMULATION RESULTS The section provides simulation results to illustrate the performance of the blind EM-SMC receiver for a DS-CDMA system in a multipath fading channel. The multipath length for each user is . The fading coefficients are generated according to , and the fading coefficients are assumed to remain constant over the block of symbols. All users’ spreading sequences are chosen as a short sequence with a processing gain and generated randomly. The number of users is , and the number of Monte Carlo samples is taken as . Let us first study the convergence of the proposed algorithm. The performance in terms of bit-error rate (BER) versus signal-to-noise ratio (SNR) for different numbers of EM iterations is shown in Fig. 2. It is observed that BER performance can be improved when the number of iterations increases. However, little gain can be obtained after four iterations, i.e., the receiver reaches the convergence with only several iterations. The desirable convergence ensures the superiority of the proposed EM-SMC to the conventional SMC method, in terms of computational complexity. The capability of near–far resistance is illustrated in Fig. 3. The near–far ratio is defined as the ratio between the powers of the interfering users and desired user. The results show that the proposed approach provides a reasonable near–far resistance,
974
Fig. 2. BER versus SNR for various numbers of iterations.
IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 54, NO. 6, JUNE 2006
Fig. 4. Comparisons on BER performance,
K = 8.
V. CONCLUSIONS In this letter, a blind multiuser receiver is developed based on incorporating SMC with an EM framework for a DS-CDMA system over multipath fading channel. With a substantially reduced computational complexity, the proposed EM-SMC receiver outperforms other reported methods and achieves a comparable performance to the conventional SMC detector. REFERENCES
Fig. 3. BER versus the near–far ratio, SNR = 15 dB.
compared with the conventional SMC which makes decisions for all users at the same time. Finally, comparisons are made among Gibbs sampler [11], QRD-M-EKF [12], particle filtering [6], the proposed EM-SMC receiver, and the conventional SMC receiver [4] in Fig. 4. The QRD-M-EKF receiver proposed in [12] employs the data-detection method based on QR decomposition combined with the algorithm and channel-estimation method based on the extended Kalman filter. The number of paths for QRD-M-EKF is selected to be 32. Gibbs sampler is performed for 100 iterations, with the first 50 iterations as the burning-in period. The number of Monte Carlo samples used for other methods is 50. Simulation results demonstrate that the proposed EM-SMC detector achieves comparable performance to the conventional SMC with a much lower computational complexity, and also outperforms the other detectors with a similar computational complexity.
[1] J. S. Liu and R. Chen, “Sequential Monte Carlo methods for dynamic systems,” J. Amer. Statist. Assoc., vol. 93, pp. 1032–1044, 1998. [2] R. Chen, X. Wang, and J. S. Liu, “Adaptive joint detection and decoding in flat-fading channels via mixture Kalman filtering,” IEEE Trans. Inf. Theory, vol. 46, no. 6, pp. 2079–2094, Jun. 2000. [3] R. A. Iltis, “A sequential Monte Carlo filter for joint linear/nonlinear state estimation with application to DS-CDMA,” IEEE Trans. Signal Process., vol. 51, no. 2, pp. 417–426, Feb. 2003. [4] D. Guo and X. Wang, “Blind detection in MIMO systems via sequential Monte Carlo,” IEEE J. Sel. Areas Commun., vol. 21, no. 3, pp. 464–473, Mar. 2003. [5] E. Punskaya, C. Andrieu, A. Doucet, and W. J. Fitzgerald, “Particle filtering for multiuser detection in fading CDMA channels,” in Proc. 11th IEEE Signal Process. Workshop Statist. Signal Process., Aug. 2001, pp. 38–41. [6] P. M. Djuric, J. H. Kotecha, J. Zhang, Y. Huang, T. Ghirmai, M. F. Bugallo, and J. Miguez, “Particle filtering,” IEEE Signal Process. Mag., vol. 20, no. 5, pp. 19–38, 2003. [7] J. Zhang, Y. Huang, and P. M. Djuric, “Multiuser detection with particle filtering,” in Proc. 11th Eur. Signal Process. Conf., Toulouse, France, Sep. 2002, vol. 2, pp. 307–310. [8] Y. Huang, J. Zhang, I. T. Luna, P. M. Djuric, and D. P. R. Padillo, “Adaptive blind multiuser detection over flat fast fading channels using particle filtering,” EUJASP [Online]. Available: http://www.hindawi.co.uk/open-access/wcn/forthcoming/S1687147204410032.pdf [9] M. Feder and E. Weinstein, “Parameter estimation of superimposed signals using the EM algorithm,” IEEE Trans. Acoust., Speech, Signal Process., vol. 36, no. 4, pp. 477–489, Apr. 1988. [10] J. A. Fessler and A. O. Hero, “Complete-data spaces and generalized EM algorithm,” in Proc. IEEE Int. Conf. Acoust., Speech, Signal Process., Apr. 1993, vol. 4, pp. 2535–2538. [11] Z. Yang and X. Wang, “Blind turbo multiuser detection for long-code multipath CDMA,” IEEE Trans. Commun., vol. 50, no. 1, pp. 112–125, Jan. 2002. [12] K. J. Kim and R. A. Iltis, “Joint detection and channel estimation algorithms for QS-CDMA signals over time-varying channels,” IEEE Trans. Commun., vol. 50, no. 5, pp. 845–855, May 2002.
971
SMC-Based Blind Detection for DS-CDMA Systems Over Multipath Fading Channels Qian Yu, Guoan Bi, and Chunru Wan
II. SIGNAL MODEL
Abstract—This letter derives a computationally efficient sequential Monte Carlo solution for blind detection of direct-sequence code-division multiple-access systems over multipath fading channels by decomposing the observed data into a number of signal components. Then the parameters of each component can be estimated by the sequential importance sampling and Kalman filtering. In comparison with other similar receivers, simulation results demonstrate that the proposed solution achieves the desirable performance with a significantly reduced computational complexity.
Let us consider a DS-CDMA system that has active users whose signals are transmitted over multipath channels with additive white Gaussian noise (AWGN). Here we assume that the delay spread is small compared with the symbol interval, so that the intersymbol interference is negligible. The received signal vector is given by
Index Terms—Code-division multiple access (CDMA), expectation-maximization (EM), Kalman filtering, multipath fading, sequential importance sampling, sequential Monte Carlo (SMC).
(1)
I. INTRODUCTION EQUENTIAL Monte Carlo (SMC) methods have begun to show a great potential for solutions to a wide range of statistical inference problems [1], and been successfully applied to a few problems in communications, such as blind equalization and detection for fading-channel communication systems [2]–[8]. For applications to code-division multiple-access (CDMA) systems, the conventional Kalman filter and SMC technique were combined for the problem of single-user detection [3]. Although the solutions to the detection problem for multiuser systems were presented in [4] and [5], the required computational complexity of all these methods grows exponentially with the number of users. The work in [6]–[8] reported the method based on the output of a whitened matched filter to perform detection sequentially in order to achieve nonexponential complexity, but this algorithm was reported only for flat-fading channels. This paper presents an SMC-based formulation of multiuser detection for a direct-sequence (DS)-CDMA system over unknown multipath fading channels. First, the multiuser system is decomposed into separate single-user systems by the expectation-maximization (EM) decomposition algorithm. Then the sequential importance sampling (SIS) and the Kalman filtering (KF) are combined to perform the data detection and channel estimation for every single-user system. With the decomposition of the superimposed observation signals, the total computational complexity of the proposed method can be reduced to be linear with the number of the active users. In addition to the details of the proposed solution, simulation results are provided to show that the receiver performs well over multipath fading channels with a significantly reduced computational complexity.
S
where for user , the transmitted symbols are differentially modulated with the binary infor. Spreading matrix denotes a mation symbols matrix .. . .. . .. .
..
.
..
.
(2)
.. . .. .
where is the spreading gain and is the spreading sequence. The channel vector is expressed as (3) where is the total number of resolvable paths in the channel. The noise term in (1) is assumed to be an independent, identically distributed (i.i.d.) complex white , i.e., Gaussian vector with a zero mean and variance . III. EM-SMC RECEIVER Because conventional SMC methods make decisions for all users at a time, the complexity of prediction and update at each step is exponential to the number of users. In [9], the EM method suggests a specific way to decompose superimposed signals. Based on this method, the observation vector given in (1) can be decomposed into components, such that
Paper approved by X. Wang, the Editor for Wireless Spread Spectrum of the IEEE Communications Society. Manuscript received April 23, 2004; revised November 22, 2004; April 13, 2005; and November 25, 2005. The authors are with the School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore 639798 (e-mail: [email protected]. sg). Digital Object Identifier 10.1109/TCOMM.2006.876830 0090-6778/$20.00 © 2006 IEEE
(4)
(5)
972
IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 54, NO. 6, JUNE 2006
Fig. 1. Receiver framework.
where weight coefficients are real-valued scalars satisfying , and chosen to be to achieve maximum convergence rate [10]. The EM framework assumes the following form for the th iteration: Expectation-Step: For , compute
probability of the information symbol as
can be estimated
(7)
(6) Maximization-Step: For , obtain the maximum-likelihood (ML) estimate of based on the . Then the SMC method is used to approach the optimum ML estimations for each user, respectively. After the EM decomposition, SMC estimation can be implemented with a parallel structure for each unknown user, respectively. This is the key point that leads to the better computational efficiency of the proposed SMC approach. The framework of this approach is presented in Fig. 1. The algorithm achieves improvement on subsequent parameter estimates by the feedback of current parameter estimates to more effectively decompose the observed data. Here, represents the estimates of unknown parameters. One important property of this structure is that for different users are estimated simultaneously in parallel, therefore, no error-propagation problem exists in the system. We now consider the estimation of unknown parameters based on each signal component . For simplicity of presentation, the hat and superscript of expressions hereafter are omitted, and the system described in (4) is considered. Denote . Let be a sample drawn by the SMC at time , and . For each value of , a set denote , which are properly of Monte Carlo samples, weighted with respect to the distribution , are obtained. For every symbol , the a posteriori
where samples
and is an indicator function. The are drawn from the trial sampling density (8)
and the importance weight can be updated according to
(9) By assuming the channel to be Gaussian, i.e., , the conditional distribution can be computed as
(10) where the a posteriori mean and covariance are updated recursively by the Kalman filter as
(11)
(12)
IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 54, NO. 6, JUNE 2006
973
TABLE I EM-SMC BLIND DETECTION ALGORITHM
Hence, the conditional density is given by
is about . However, domcomputation of inant computation for the proposed SMC needs one-step predictions for computing . Therefore, the entire computational complexity required is , where is the number of the EM iterations.
(13) (14) with the mean and the covariance
(15) Cov (16) Then,
in (9) can be computed as follows:
(17) Finally, the EM-SMC blind detector for the system is given in Table I. Computational complexity of conventional SMC method is on the order of , because the dominant
IV. SIMULATION RESULTS The section provides simulation results to illustrate the performance of the blind EM-SMC receiver for a DS-CDMA system in a multipath fading channel. The multipath length for each user is . The fading coefficients are generated according to , and the fading coefficients are assumed to remain constant over the block of symbols. All users’ spreading sequences are chosen as a short sequence with a processing gain and generated randomly. The number of users is , and the number of Monte Carlo samples is taken as . Let us first study the convergence of the proposed algorithm. The performance in terms of bit-error rate (BER) versus signal-to-noise ratio (SNR) for different numbers of EM iterations is shown in Fig. 2. It is observed that BER performance can be improved when the number of iterations increases. However, little gain can be obtained after four iterations, i.e., the receiver reaches the convergence with only several iterations. The desirable convergence ensures the superiority of the proposed EM-SMC to the conventional SMC method, in terms of computational complexity. The capability of near–far resistance is illustrated in Fig. 3. The near–far ratio is defined as the ratio between the powers of the interfering users and desired user. The results show that the proposed approach provides a reasonable near–far resistance,
974
Fig. 2. BER versus SNR for various numbers of iterations.
IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 54, NO. 6, JUNE 2006
Fig. 4. Comparisons on BER performance,
K = 8.
V. CONCLUSIONS In this letter, a blind multiuser receiver is developed based on incorporating SMC with an EM framework for a DS-CDMA system over multipath fading channel. With a substantially reduced computational complexity, the proposed EM-SMC receiver outperforms other reported methods and achieves a comparable performance to the conventional SMC detector. REFERENCES
Fig. 3. BER versus the near–far ratio, SNR = 15 dB.
compared with the conventional SMC which makes decisions for all users at the same time. Finally, comparisons are made among Gibbs sampler [11], QRD-M-EKF [12], particle filtering [6], the proposed EM-SMC receiver, and the conventional SMC receiver [4] in Fig. 4. The QRD-M-EKF receiver proposed in [12] employs the data-detection method based on QR decomposition combined with the algorithm and channel-estimation method based on the extended Kalman filter. The number of paths for QRD-M-EKF is selected to be 32. Gibbs sampler is performed for 100 iterations, with the first 50 iterations as the burning-in period. The number of Monte Carlo samples used for other methods is 50. Simulation results demonstrate that the proposed EM-SMC detector achieves comparable performance to the conventional SMC with a much lower computational complexity, and also outperforms the other detectors with a similar computational complexity.
[1] J. S. Liu and R. Chen, “Sequential Monte Carlo methods for dynamic systems,” J. Amer. Statist. Assoc., vol. 93, pp. 1032–1044, 1998. [2] R. Chen, X. Wang, and J. S. Liu, “Adaptive joint detection and decoding in flat-fading channels via mixture Kalman filtering,” IEEE Trans. Inf. Theory, vol. 46, no. 6, pp. 2079–2094, Jun. 2000. [3] R. A. Iltis, “A sequential Monte Carlo filter for joint linear/nonlinear state estimation with application to DS-CDMA,” IEEE Trans. Signal Process., vol. 51, no. 2, pp. 417–426, Feb. 2003. [4] D. Guo and X. Wang, “Blind detection in MIMO systems via sequential Monte Carlo,” IEEE J. Sel. Areas Commun., vol. 21, no. 3, pp. 464–473, Mar. 2003. [5] E. Punskaya, C. Andrieu, A. Doucet, and W. J. Fitzgerald, “Particle filtering for multiuser detection in fading CDMA channels,” in Proc. 11th IEEE Signal Process. Workshop Statist. Signal Process., Aug. 2001, pp. 38–41. [6] P. M. Djuric, J. H. Kotecha, J. Zhang, Y. Huang, T. Ghirmai, M. F. Bugallo, and J. Miguez, “Particle filtering,” IEEE Signal Process. Mag., vol. 20, no. 5, pp. 19–38, 2003. [7] J. Zhang, Y. Huang, and P. M. Djuric, “Multiuser detection with particle filtering,” in Proc. 11th Eur. Signal Process. Conf., Toulouse, France, Sep. 2002, vol. 2, pp. 307–310. [8] Y. Huang, J. Zhang, I. T. Luna, P. M. Djuric, and D. P. R. Padillo, “Adaptive blind multiuser detection over flat fast fading channels using particle filtering,” EUJASP [Online]. Available: http://www.hindawi.co.uk/open-access/wcn/forthcoming/S1687147204410032.pdf [9] M. Feder and E. Weinstein, “Parameter estimation of superimposed signals using the EM algorithm,” IEEE Trans. Acoust., Speech, Signal Process., vol. 36, no. 4, pp. 477–489, Apr. 1988. [10] J. A. Fessler and A. O. Hero, “Complete-data spaces and generalized EM algorithm,” in Proc. IEEE Int. Conf. Acoust., Speech, Signal Process., Apr. 1993, vol. 4, pp. 2535–2538. [11] Z. Yang and X. Wang, “Blind turbo multiuser detection for long-code multipath CDMA,” IEEE Trans. Commun., vol. 50, no. 1, pp. 112–125, Jan. 2002. [12] K. J. Kim and R. A. Iltis, “Joint detection and channel estimation algorithms for QS-CDMA signals over time-varying channels,” IEEE Trans. Commun., vol. 50, no. 5, pp. 845–855, May 2002.
