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Blind Adaptive Multiuser Detection Based on Kalman Filtering Xian-Da Zhang, Senior Member, IEEE, and Wei Wei
Abstract—Although several Kalman filtering algorithms have been presented for adaptive multiuser detection, none is “blind” due to requiring training data sequences and/or more knowledge than the spreading waveform and delay of the desired user. This paper proposes a novel blind adaptive multiuser detector based on Kalman filtering and compares it with previously published LMS and RLS algorithms for blind adaptive multiuser detection. It is shown that the steady-state excess output energy of the Kalman filtering algorithm is identically zero for stationary environment. Simulation results show the effectiveness of the new Kalman filtering algorithm. Index Terms—Blind multiuser detection, code-division multiple access, Kalman filter, minimum mean-square error detection, multiple-access interference, Rayleigh fading channel.
I. INTRODUCTION
D
IRECT-sequence code-division multiple-access (DSCDMA) has been widely studied in the literature. Recently, adaptive interference suppression techniques based on multiuser detection have been considered as powerful methods for increasing the quality, capacity, and coverage of CDMA systems. The mitigation of multiple access interference (MAI) in CDMA systems is a problem of continuing interest since MAI is the dominant impairment for CDMA systems. It is widely recognized that MAI exists even in perfect power-controlled CDMA systems, and multiuser detectors perform better than the conventional detector under all power distributions, except in pathological cases, such as the decorrelating detector in extremely low SNR. Therefore, multiuser detection is not only a solution to the near-far problem but is also useful even with power control. In order to successfully eliminate the MAI and detect the desired user’s information bits, one or more of the following is usually required at the receiver end: 1) spreading waveform of the desired user; 2) spreading waveforms of the interfering users; 3) propagation delay (timing) of the desired user; Manuscript received October 24, 2000; revised October 3, 2001. This work was supported by the National Natural Sciences Foundation of China under Grant 60072043. The associate editor coordinating the review of this paper and approving it for publication was Dr. Rick S. Blum. X.-D. Zhang is with the Key Laboratory for Radar Signal Processing, Xidian University, Xi’an, China. He is also with the Department of Automation, Tsinghua University, Beijing, China (e-mail: [email protected]; [email protected]). W. Wei was with the Department of Automation, Tsinghua University, Beijing, China. He is now with the Department of Electrical Engineering and Computer Science, University of California, Berkeley, CA 94720 USA. Publisher Item Identifier S 1053-587X(02)00404-X.
4) propagation delays of the interfering users; 5) received amplitudes of the interfering users (relative to that of the desired user); 6) training data sequences for every active user. The optimal multiuser detector proposed by Verdu [4] is a major theoretical milestone but is disadvantaged by its exponential computational complexity in the number of users and its requiring knowledge of all delays, amplitudes, and modulation waveforms of the desired user and the interfering users. Decorrelating detectors [1], the minimum mean square error (MMSE), and the equivalent minimum output energy (MOE) detectors (see, e.g., [2] and [3]) are near-far resistant and have much reduced complexity than the optimal multiuser detector in [4]. The adaptive decorrelating detectors in [7] and adaptive MMSE detectors in [8] need to have training data sequences for every active user. In comparison, the “blind” adaptive multiuser detectors require only the knowledge of (1) and (3), that is, the same knowledge as the conventional receiver. Previous work on blind adaptive multiuser detection dates back to a 1995 paper by Honig et al. [2], who established a canonical representation for blind multiuser detectors and used stochastic gradient algorithms such as least mean squares (LMS) to implement the blind adaptive MOE detector. As elegantly shown by Roy [5], the blind adaptive MOE detector has a smaller eigenvalue spread than the training-based adaptive LMS detector; hence, the blind LMS algorithm always provides (nominally) faster convergence than the training driven LMS-MMSE receiver but at the cost of increased tap-weight fluctuation or misadjustment. It is well-known that the recursive least squares (RLS) algorithm and the Kalman filtering algorithm are better than the LMS algorithm in convergence rate and tracking capability. Using the exponentially weighted sum of error squares cost function, Chen and Roy [6] proposed an RLS algorithm that requires the knowledge of (1)–(4) and, thus, is not a blind multiuser detector. Later, Poor and Wang [9] proposed an exponentially windowed RLS algorithm for blind multiuser detection requiring only the knowledge of (1) and (3). On the other hand, the RLS is a special case of the Kalman filter [10], [11], whereas the Kalman filter is known to be a linear minimum variance state estimator [10], [12], [13]. Motivated probably by these two facts, some attention has been focused recently on Kalman filter-based adaptive multiuser detection [14]–[18]. In particular, it is shown [16] that when applied in an asynchronous CDMA system, the RLS algorithm performs more poorly than the more general Kalman filter algorithm. The objective of this paper is to develop a blind adaptive multiuser detector based on Kalman filtering.
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This paper is organized as follows. A novel blind adaptive multiuser detector based on Kalman filtering is developed in Section II. This is followed by comparison of the new Kalman filtering algorithm with previous published Kalman filtering and RLS algorithms for multiuser CDMA detection in Section III. Further, the steady-state excess output energy of the new multiuser detector is deduced in Section IV. Simulation results are presented in Section V to demonstrate the performance of the new detector and make a comparison with the blind multiuser detector based on the LMS and RLS algorithms. Finally, the paper is concluded in Section VI. II. NOVEL BLIND ADAPTIVE MULTIUSER DETECTOR Consider an antipodal -user synchronous DS-CDMA system signaling through an additive white Gaussian noise channel. By passing through a chip-matched filter, followed by a chip-rate sampler, the discrete-time output of the receiver during one symbol interval can be modeled as
(1) where ambient channel noise; number of users; received amplitude of the th user; information symbol sequence from the th user, ; chosen independently and equally from th signature waveform that is assumed to have unit . energy is supported only on the interval It is assumed that , where symbol interval; chip interval; processing gain. Defining
(2)
A. State-Space Model Consider a dynamical system defined by process or state equation (5) and measurement equation (6) and are an state vector and an obwhere is a known servation vector at time , respectively; state transition matrix relating the states of the system at and , is a known measurement matrix, times vector is called the process noise, whereas and the vector represents a measurement noise. It is asthe is uncorrelated with both sumed that the initial state vector and for , and the noise vectors and are statistically independent, i.e., for all and . To develop a blind adaptive multiuser detector based on Kalman filtering, we need to devise a novel linear first-order state-space model for the blind multiuser detection problem that is a perfect match for the application of Kalman filter theory in blind scenario. By Honig et al. [2], the canonical representation for any linear detector for user 1 is defined by Type I: subject to (8) is orthogonal to the signature wavenamely, the vector form vector of user 1. The constraint (8) can be equivalently written as (9) We refer to (7) as a Type I canonical representation for a blind linear multiuser detector since there is another form, as shown in the following. The LMS algorithm is given by [2] (10)
we can express (1) in vector form (3) is the code sewhere quence assigned to the th user. For convenience, we will assume that the desired user is . It is well known that any linear multiuser detector for user 1 such that can be characterized by the tap-weight vector during the th symbol interval is given by the decision on
is the output of the conventional where single-user matched filter, and is the output of the linear detector. “Beamforming” is an efficient technique in blind multiuser detection [19]–[21]. Following the framework of the well-known generalized sidelobe canceler [22] and by conto present a unit response straining the linear detector to user 1’s signature sequence, Kapoor et al. [21] presented another canonical representation for linear detectors for user 1: Type II:
sgn
(7)
sgn
(11)
(4)
In the following, we consider how to establish the state-space model for Kalman filtering algorithm for blind adaptive mul. tiuser detector
is the adaptive part of , and the columns of the matrix span the null space of , i.e., . It is easy to see that Type I and II canonical representations are identical.
where -by-
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Our problem here is how to update adaptively using is given and is precomputed offline (11) after via one of many orthogonalizing procedures (such as the Gram–Schmidt orthogonalization and so on) [22]. To this end, we define the mean output energy (MOE) and the (scaled) mean-square-error (MSE) as MOE
(12)
MSE
(13)
Defining
is an identity matrix, and the process noise is a zero vector; b) the observation vector becomes the scalar , and the measurement matrix reduces to the . vector B. Kalman Filtering Algorithm From the corresponding relationship between the conventional dynamical system model and the dynamical system model of user 1, and using the standard Kalman filtering algorithm, we have the Kalman filtering algorithm for blind adaptive multiuser detection in stationary synchronous CDMA:
(14) (20)
has zero mean and variance [2], [3, p. 317]
then
MSE
cov
(15)
For a time-invariant CDMA system, an important fact is that is timeits optimum linear detector or tap-weight vector . Letting invariant as well, namely be the weight vector corresponding to in Type II canonical representation (11), we then have the following state : equation on (16) On the other hand, substituting (11) into (14) yields (17) and . If achieves , then (17) can be rewritten as the following measurement equation:
Put
(18) Both the state equation (16) and the measurement equation (18) construct the dynamical system model of user 1, which is the basis for Kalman filtering. The Kalman filtering problem in the blind multiuser detection may now be formally stated as fol, use the observed lows: Given the measurement matrix to find for each the minimum mean-square esdata . timates of the components of the state vector Using (15), we have cov
(19)
MSE denotes the MMSE where is optimum, and hence, when the tap-weight vector MOE is the minimum mean output energy of the dynamical system of user 1. Moreover, for synchronous model (1), it is easily shown that for due to for . Therefore, is a white noise with zero mean and variance in the synchronous case. As compared with the conventional dynamical system model specified by (5) and (6), the new linear first-order state-space model specified by (16) and (18) has the following characteris, the state-transition matrix tics: a) The state vector is
(21) (22) (23) , , , and are where vectors, and is the matrix having a length of . For the due to the state vector Kalman filter to be optimal, it is required that the initial state is Gaussian. Let its mean be zero and variance vector be one. By [10, p. 320], we may choose the initial predicted and its correlation estimate as matrix . Equation (22) is of fundamental significance. It shows that the correction term equals the innovation process premultiplied by the Kalman gain vector . Although the computation of in (20) requires that be known or estimated, this requirement is the parameter is only a not crucial. Note that the Kalman gain vector . By Honig et al. time-varying “step size” of updating [2], the signal-to-interference radio (SIR) of the desired user’s output after the MAI suppression is defined by SIR dB
(24)
where we have used the constraint (9) in Type I canonical representation. Since the maximum SIR of the desired user’s output is is usually expected to be higher than 10 dB, the MMSE usually smaller than 0.1. Note that since the received amplitude of the desired user during the th symbol interval is usually , we can directly take in larger such that . (19) as the estimate of the unknown parameter By Haykin [10, p. 702], an unknown dynamic system can be modeled as a transversal filter whose tap-weight vector undergoes a first-order Markov process, namely (25) is the where is a fixed parameter of the model, and process noise vector assumed to be of zero mean and correla-
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tion matrix Hence, (21) and (22) in the Kalman filtering algorithm should be replaced by
(26) (27) For a slowly time-varying CDMA system, it can be assumed that the value of parameter is very close to 1, and every element takes a very small value. Therefore, of its correlation matrix although, in a slowly time-varying environment, the proposed Kalman filtering algorithm is conditioned on perfect knowledge , , and , even with imperfect knowledge of of , and (null matrix), the new algorithm still works well. See Section V for simulation results. III. COMPARISON WITH OTHER RELATED ALGORITHMS The Kalman filtering algorithm established in Section II is related to many existing algorithms for adaptive multiuser detection. Now, let us compare its differences from other prior algorithms. In recent years, several Kalman filtering algorithms have been developed for adaptive multiuser detection [14]–[18]. However, due to the requirement of more knowledge than the desired user’s signature vector and timing, none of the previously published Kalman filtering algorithms can be considered blind. Since Kalman filtering algorithms and RLS algorithms are similar, it is necessary to compare the Kalman filtering algorithm of this paper with the previously published RLS algorithms for multiuser CDMA detection. Two typical RLS algorithms have been proposed for multiuser CDMA detection [6], [9]. The RLS algorithm in [6] requires spreading waveforms and propagation delays of all users, and it is not a blind multiuser detector. In addition, this RLS algorithm is a chip-rate algorithm, whereas our Kalman filtering algorithm is a symbol-rate one. Using the minimum output energy criterion, Poor and Wang [9] proposed a RLS algorithm for blind multiuser detection. The exponentially windowed RLS algorithm selects the weight vector to minimize the sum of exponentially weighted output energy, namely minimize
(28)
subject to
(29)
is the forgetting factor. The solution to this where constrained optimization problem yields the linear MMSE detector, which is given by [2], [9] (30) where (31)
A recursive procedure for updating can be implemented by the RLS algorithm [9]. The following are the computation complexities for updating per symbol interval of the three the tap-weight vector blind adaptive multiuser detectors: multiplications and adds; LMS algorithm [2]: multiplications and RLS algorithm [9]: adds; mulKalman filtering algorithm in this paper: adds. tiplications and IV. STEADY-STATE EXCESS OUTPUT ENERGY In this section, we analyze steady-state excess output energy of the Kalman filtering algorithm (20)–(23). A. Convergence Properties of the LMS and RLS Algorithms For convenience of comparison with the LMS algorithm [2] and the RLS algorithm [9], we first recall briefly the convergence properties of these two algorithms. Let MSE MOE
(32) (33)
represent the mean square error and mean output energy denote the minat iteration , respectively, and let , where imum mean square error with and is the minimum mean output energy, whereas is called the excess output energy (EOE) due to adaptation at time . Define (34) as the steady-state or asymptotic EOE; then, it is known by [2] that the steady-state excess MSE due to adaptation is equal to the steady-state EOE, i.e., (35) and for the LMS algorithm tr tr
(36)
where (37)
tr
that is the cross-correlation with being the eigenvalue of and , namely, matrix of . Note that the approximation (37) , , and (36) is one of becomes exact as several representations of the equivalence between the MMSE criterion and minimum output energy criterion. Upon convergence of the RLS algorithm, by Poor and Wang [9], for large tr
(38)
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where
is the average excess MSE at time , is the weight error correlation matrix at time , and the steady-state excess MSE or the steady-state EOE is given by (39)
and hence, the steady-state EOE in stationary situation is identically zero, i.e., (47) For the sake of comparison, we rewrite (35), (39), and (47) as an equation: tr tr
B. Steady-State EOE of the Kalman Filtering Algorithm
for RLS for Kalman.
Consider the stationary situation. By [12, p. 139], we have the following information matrix propagation equation: (40) 1, 2 in (40), we get
given the state equation (16). Taking
Mimicking this process for general form:
, we have the following
(41) , and denote
Put
and (42) into (41), we (43)
We remark that for the slowly time-varying situation in which is close to the correlation matrix of the process noise vector a null matrix, (40), (41), and (43) become approximate expressions, respectively. yields the estimate of the optimum weight Since , vector associated with the optimum tap-weight vector , we let i.e., MOE
(44)
represent the mean output energy at iteration . Theorem 1: For stationary CDMA systems, when is sufof the Kalman filficiently large, the mean output energy tering algorithm is given by (45) where is the processing gain of the DS-CDMA system. Proof: See the Appendix. From (45), we get the excess mean output energy at time given by (46)
(48)
when Remarks: From (45), we know that , and when the iteration number . This implies that the output energy of the Kalman filtering algorithm rapidly tends to the minimum mean as increases. On the convergence of the output energy , we have the following. mean output energy a) The Kalman filtering algorithm depends only on the processing gain and is independent of the data autocorrelation matrix , as shown in (45). b) The LMS algorithm depends on the eigenvalue distribution of , as is well known. c) The RLS algorithm depends on the trace of the matrix , as shown in (38). product Finally, we point out that no steady-state EOE analysis was given for Kalman filter-based (nonblind) adaptive multiuser detection algorithms in [14]–[18]. V. SIMULATION RESULTS
(42) Substituting the initial value have
for LMS
In this section, we present several simulation results to compare the new algorithm with other algorithms for blind multiuser detection. For each run, the LMS algorithm in [2], the RLS algorithm in [9], and the Kalman filtering algorithm in this paper are applied at the same time. Following (70) in [2], when implementing the LMS algorithm, the step-size must satisfy the stability condition of convergence of output MSE: (49) As a figure of merit for assessing the MAI suppression capability of the blind LMS, RLS, and Kalman filtering algorithms, the time-averaged SIR (in decibels) at the th iteration is given by [2] SIR
(50)
is the number of independent runs, and the subscript where indicates that the associated variable depends on the particular run. All signal energies are given in decibels relative to the background noise variance , i.e., the SNR of user is defined , where is the bit energy by SNR of user . In all simulations, user 1 is assumed to be the desired and an SNR of 20 dB (i.e., user that has the unit energy ), and the processing gain . In the following, the data in each plot are the average over 500 independent runs.
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Fig. 3. Time-averaged measured EOE versus time for 500 runs when using the three algorithms to a synchronous CDMA system with processing gain = 31.
N
Fig. 1. Time-averaged SIR versus time for 500 runs when using the three = 31. algorithms to a synchronous CDMA system with processing gain
N
Fig. 4. Time-averaged SIR versus time for 500 runs when using the three algorithms to a synchronous CDMA system with processing gain = 31 and the time-varying number of users.
N
Fig. 2. Time-averaged SIR versus time for 500 runs when using the three algorithms to an asynchronous CDMA system with processing gain = 31.
N
1) Convergence Rate Comparison: In Example 1, DS-CDMA systems in a Gaussian channel are simulated, and there are nine multiple-access interfering users among which five users have an SNR of 30 dB each, three users have SNR of 40 dB each, and another , user has an SNR of 50 dB, i.e., , and . Then, from (49), it follows that the step size should satisfy , and thus, was used in the LMS algorithm. When applying the RLS algorithm, takes , and the the initial value is taken. forgetting factor Fig. 1 depicts a plot of the time-averaged SIR versus iteration number (time) for the three algorithms applied in a synchronous CDMA system. Fig. 2 shows the timeaveraged SIR versus time for an asynchronous CDMA system with the time delays of the interferers , , where is the chip interval. Fig. 3 depicts the time-averaged measured EOE versus time for a synchronous CDMA system.
2) Tracking Dynamical Environment: In Example 2, we compare the tracking capabilities of the LMS, RLS, and Kalman filtering algorithms in a dynamical environment with a time-varying number of users for DS-CDMA , the systems in a Gaussian channel. When , configuration is the same as Example 1. At three interfering users with SNR of 40 dB are added to , four the CDMA system at the same time. At interfering users with SNR of 40 dB and one interfering user with SNR of 50 dB are removed from the system. Figs. 4 and 5 show the tracking behaviors of the three blind adaptive algorithms in a synchronous system and an asynchronous one, respectively. 3) Slowly Time-Varying Environment: Example 3 is a synchronous DS-CDMA system in a Rayleigh fading channel realized via harmonic decomposition technique [23], which undergoes a positive Doppler shift of about is used 22 Hz. The channel information of slot as an imperfect channel information of slot . The user configuration is the same as Example 1. Figs. 6 and 7 show the plots of the time-averaged SIR versus time and
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Fig. 5. Time-averaged SIR versus time for 500 runs when using the three = 31 algorithms to an asynchronous CDMA system with processing gain and the time-varying number of users.
N
the time-averaged measured EOE versus time for the RLS and Kalman filtering algorithms, respectively. In simulations, we used, respectively, the estimates , , and in (20), but Kalman curves in Figs. 1–7 have few changes. From Figs. 1–7, we can see the following facts. a) The RLS and Kalman filtering algorithms have faster convergence and better tracking capability than the LMS algorithm in both the synchronous and asynchronous systems. b) The Kalman filtering algorithm achieves nearly the con, in contrast to the LMS and RLS vergence when algorithms. c) The measured EOE in Kalman filtering algorithm is close in to the “ideal” EOE value of zero when both stationary and slowly time-varying environments, in . contrast to the LMS algorithm and RLS one with The first fact is well known. The second fact is not surprising since the performance analysis in Section IV told us , one has that when the iteration number that is already close to . The third fact , the steady-state shows that with imperfect knowledge of EOE in the Kalman filtering algorithm is not zero but is close to zero.
Fig. 6. Time-averaged SIR versus time for 500 runs when using Kalman and RLS algorithms with = 0:997 to a synchronous CDMA system in Rayleigh fading channel; the processing gain is N = 31.
Fig. 7. Time-averaged measured EOE versus time for 500 runs when using Kalman and RLS algorithms with = 0:997 to a synchronous CDMA system in Rayleigh fading channel; the processing gain is N = 31.
Although only the single-path delay channel was considered in this paper, the proposed algorithm, as well as the adaptive MOE detector, can be applied to multipath channels with the understanding that represents the desired user’s channel-distorted spreading code, i.e., the spreading waveform after passing through the channel. With perfect channel estimates, it would be possible to have this waveform at the receiver.
VI. CONCLUSION In this paper, we have used the Kalman filter to propose a novel blind adaptive multiuser detector in both a stationary environment and a slowly time-varying one. It is shown in theory that the steady-state EOE of the Kalman filtering algorithm is identically zero for the stationary environment, and simulations have shown that its measured EOE is close to zero in either stationary or slowly time-varying environments. Although the Kalman filtering algorithm is conditioned on perfect knowledge of some system parameters, simulations show that even when such knowledge is not easily available, the new algorithm still works well and has performance advantages over the LMS and RLS algorithms.
APPENDIX PROOF OF THEOREM 1 From (18) and (44), it follows that
(51) By the orthogonality principle, we have immediately that or
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. Using this result, we expand
as
ACKNOWLEDGMENT
follows:
(52)
The authors would like to thank Associate Editor Dr. R. S. Blum and the anonymous reviewers for their constructive and valuable comments and suggestions that helped to greatly improve the quality and clarity of the presentation. They also wish to thank the anonymous reviewers for bringing their attention to [5], [17], [19], and [20].
which may be equivalently rewritten in the following form: tr
REFERENCES (53)
and its predicted Since the state vector at time is , the predicted state-error vector estimate is is defined by (54) and the predicted state-error correlation matrix defined by
Since equation can be written as
is
in the stationary case, the above
(55) Substituting (55) and have tr
into (53), we
(56)
Since (43) holds strictly for the stationary environment, after is replaced by (43), for sufficiently large , we , and (56) may be approximated as follows: have tr Note that tr tr and we have that for sufficiently large tr which may be written in a compact form as that is (45). This completes the proof of Theorem 1.
[1] S. Verdu, “Multiuser detection,” in Advances in Statistical Signals Processing. Greenwich, CT: JAI, 1993, vol. 2, pp. 369–409. [2] M. L. Honig, U. Madhow, and S. Verdu, “Blind adaptive multiuser detection,” IEEE Trans. Inform. Theory, vol. 41, pp. 944–960, July 1995. [3] S. Verdu, Multiuser Detection. Cambridge, U.K.: Cambridge Univ. Press, 1998. , “Minimum probability of error for asynchronous Gaussian [4] multiple-access channels,” IEEE Trans. Inform. Theory, vol. IT-32, pp. 85–96, 1986. [5] S. Roy, “Subspace blind adaptive multiuser detection for CDMA,” IEEE Trans. Commun., vol. 48, pp. 169–175, Jan. 2000. [6] D. S. Chen and S. Roy, “An adaptive multiuser receiver for CDMA systems,” IEEE J. Select. Areas Commun., vol. 12, pp. 808–816, June 1994. [7] G. Woodward and B. S. Vucetic, “Adaptive detection for DS-CDMA,” Proc. IEEE, vol. 86, pp. 1413–1434, July 1998. [8] U. Madhow and M. Honig, “MMSE interference suppression for directsequence spread spectrum CDMA,” IEEE Trans. Commun., vol. 42, pp. 3178–3188, Dec. 1994. [9] H. V. Poor and X. Wang, “Code-aided interference suppression for DS/CDMA communications, Part II: Parallel blind adaptive implementations,” IEEE Trans. Commun., vol. 45, pp. 1112–1122, Sept. 1997. [10] S. Haykin, Adaptive Filter Theory, 3rd ed. Englewood Cliffs, NJ: Prentice-Hall, 1996. [11] A. H. Sayed and T. Kailath, “A state-space approach to adaptive RLS filtering,” IEEE Signal Processing Mag., vol. 11, pp. 18–60, July 1994. [12] B. D. O. Anderson and J. B. Moore, Linear Optimal Control. Englewood Cliffs, NJ: Prentice-Hall, 1979. [13] G. C. Goodwin and K. S. Sin, Adaptive Filtering, Prediction and Control. Englewood Cliffs, NJ: Prentice-Hall, 1984. [14] R. Iltis and L. Mailander, “An adaptive multiuser detector with joint amplitude and delay estimation,” IEEE J. Select. Areas Commun., vol. 12, pp. 774–785, June 1994. [15] S. H. Isabelle and G. W. Wornell, “Efficient multiuser detectors for intersymbol interference channels,” in Proc. SPAWC, 1997, pp. 277–280. [16] T. J. Lim, L. K. Rasmussen, and H. Sugimoto, “An asynchronous multiuser CDMA detector based on the Kalman filter,” IEEE J. Select. Areas Commun., vol. 16, pp. 1711–1722, Sept. 1998. [17] T. J. Lim and Y. Ma, “The Kalman filter as the optimum linear minimum mean squared error multiuser CDMA detector,” IEEE Trans. Inform. Theory, vol. 46, pp. 2561–2566, July 2000. [18] P.-W. Fu and K. C. Chen, “Linear-complexity equalized multiuser receivers for wideband CDMA in time-varying channels,” in Proc. Veh. Technol. Conf., 1999, pp. 2338–2342. [19] J. B. Schodorf and D. B. Williams, “A constrained optimization approach to multiuser detection,” IEEE Trans. Signal Processing, vol. 45, pp. 258–262, Jan. 1997. [20] , “Array processing techniques for multiuser detection,” IEEE Trans. Commun., vol. 45, pp. 1375–1378, Nov. 1997. [21] S. Kapoor, S. Gollamudi, S. Nagaraj, and Y. -F. Huang, “Adaptive multiuser detection and beamforming for interference suppression in CDMA mobile radio systems,” IEEE Trans. Veh. Technol., vol. 48, pp. 1341–1355, Sept. 1999. [22] B. D. Van Veen and K. M. Buckley, “Beamforming: A versatile approach to spatial filtering,” IEEE Acoust., Speech, Signal Processing Mag., vol. 5, pp. 4–24, Apr. 1988. [23] P. M. Crespo and J. Jimenez, “Computer simulation of radio channels using a harmonic decomposition technique,” IEEE Trans. Veh. Technol., vol. 44, pp. 414–419, Aug. 1995.
ZHANG AND WEI: BLIND ADAPTIVE MULTIUSER DETECTION
Xian-Da Zhang (SM’93) was born in Jiangxi, China, in 1946. He received the B.S. degree in radar engineering from Xidian University, Xi’an, China, in 1969, the M.S. degree in instrument engineering from Harbin Institute of Technology, Harbin, China, in 1982, and the Ph.D. degree in electrical engineering from Tohoku University, Sendai, Japan, in 1987. From August 1990 to August 1991, he was a Postdoctoral Researcher with the Department of Electrical and Computer Engineering, University of California at San Diego, La Jolla. Since 1992, he has been with the Department of Automation, Tsinghua University, Beijing, China, as a Professor and is currently with the Key Laboratory for Radar Signal Processing, Xidian University, Xi’an, China, as a Specially Appointed Professor awarded by the Minister of Education of China and the Cheung Kong Scholars Programme. His current research interests are signal processing with applications in radar and communications and intelligent signal processing. He has published over 20 papers in several IEEE Transactions and is the author of five books (all in Chinese). He holds one patent on radar target recognition.
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Wei Wei was born in Shanxi, China, in 1977. He received the B.S. and M.S. degrees from the Department of Automation, Tsinghua University, Beijing, China, in 1999 and 2001, respectively. He is currently pursuing the Ph.D. degree at the Department of Electrical Engineering and Computer Science, University of California, Berkeley.
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Blind Adaptive Multiuser Detection Based on Kalman Filtering Xian-Da Zhang, Senior Member, IEEE, and Wei Wei
Abstract—Although several Kalman filtering algorithms have been presented for adaptive multiuser detection, none is “blind” due to requiring training data sequences and/or more knowledge than the spreading waveform and delay of the desired user. This paper proposes a novel blind adaptive multiuser detector based on Kalman filtering and compares it with previously published LMS and RLS algorithms for blind adaptive multiuser detection. It is shown that the steady-state excess output energy of the Kalman filtering algorithm is identically zero for stationary environment. Simulation results show the effectiveness of the new Kalman filtering algorithm. Index Terms—Blind multiuser detection, code-division multiple access, Kalman filter, minimum mean-square error detection, multiple-access interference, Rayleigh fading channel.
I. INTRODUCTION
D
IRECT-sequence code-division multiple-access (DSCDMA) has been widely studied in the literature. Recently, adaptive interference suppression techniques based on multiuser detection have been considered as powerful methods for increasing the quality, capacity, and coverage of CDMA systems. The mitigation of multiple access interference (MAI) in CDMA systems is a problem of continuing interest since MAI is the dominant impairment for CDMA systems. It is widely recognized that MAI exists even in perfect power-controlled CDMA systems, and multiuser detectors perform better than the conventional detector under all power distributions, except in pathological cases, such as the decorrelating detector in extremely low SNR. Therefore, multiuser detection is not only a solution to the near-far problem but is also useful even with power control. In order to successfully eliminate the MAI and detect the desired user’s information bits, one or more of the following is usually required at the receiver end: 1) spreading waveform of the desired user; 2) spreading waveforms of the interfering users; 3) propagation delay (timing) of the desired user; Manuscript received October 24, 2000; revised October 3, 2001. This work was supported by the National Natural Sciences Foundation of China under Grant 60072043. The associate editor coordinating the review of this paper and approving it for publication was Dr. Rick S. Blum. X.-D. Zhang is with the Key Laboratory for Radar Signal Processing, Xidian University, Xi’an, China. He is also with the Department of Automation, Tsinghua University, Beijing, China (e-mail: [email protected]; [email protected]). W. Wei was with the Department of Automation, Tsinghua University, Beijing, China. He is now with the Department of Electrical Engineering and Computer Science, University of California, Berkeley, CA 94720 USA. Publisher Item Identifier S 1053-587X(02)00404-X.
4) propagation delays of the interfering users; 5) received amplitudes of the interfering users (relative to that of the desired user); 6) training data sequences for every active user. The optimal multiuser detector proposed by Verdu [4] is a major theoretical milestone but is disadvantaged by its exponential computational complexity in the number of users and its requiring knowledge of all delays, amplitudes, and modulation waveforms of the desired user and the interfering users. Decorrelating detectors [1], the minimum mean square error (MMSE), and the equivalent minimum output energy (MOE) detectors (see, e.g., [2] and [3]) are near-far resistant and have much reduced complexity than the optimal multiuser detector in [4]. The adaptive decorrelating detectors in [7] and adaptive MMSE detectors in [8] need to have training data sequences for every active user. In comparison, the “blind” adaptive multiuser detectors require only the knowledge of (1) and (3), that is, the same knowledge as the conventional receiver. Previous work on blind adaptive multiuser detection dates back to a 1995 paper by Honig et al. [2], who established a canonical representation for blind multiuser detectors and used stochastic gradient algorithms such as least mean squares (LMS) to implement the blind adaptive MOE detector. As elegantly shown by Roy [5], the blind adaptive MOE detector has a smaller eigenvalue spread than the training-based adaptive LMS detector; hence, the blind LMS algorithm always provides (nominally) faster convergence than the training driven LMS-MMSE receiver but at the cost of increased tap-weight fluctuation or misadjustment. It is well-known that the recursive least squares (RLS) algorithm and the Kalman filtering algorithm are better than the LMS algorithm in convergence rate and tracking capability. Using the exponentially weighted sum of error squares cost function, Chen and Roy [6] proposed an RLS algorithm that requires the knowledge of (1)–(4) and, thus, is not a blind multiuser detector. Later, Poor and Wang [9] proposed an exponentially windowed RLS algorithm for blind multiuser detection requiring only the knowledge of (1) and (3). On the other hand, the RLS is a special case of the Kalman filter [10], [11], whereas the Kalman filter is known to be a linear minimum variance state estimator [10], [12], [13]. Motivated probably by these two facts, some attention has been focused recently on Kalman filter-based adaptive multiuser detection [14]–[18]. In particular, it is shown [16] that when applied in an asynchronous CDMA system, the RLS algorithm performs more poorly than the more general Kalman filter algorithm. The objective of this paper is to develop a blind adaptive multiuser detector based on Kalman filtering.
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This paper is organized as follows. A novel blind adaptive multiuser detector based on Kalman filtering is developed in Section II. This is followed by comparison of the new Kalman filtering algorithm with previous published Kalman filtering and RLS algorithms for multiuser CDMA detection in Section III. Further, the steady-state excess output energy of the new multiuser detector is deduced in Section IV. Simulation results are presented in Section V to demonstrate the performance of the new detector and make a comparison with the blind multiuser detector based on the LMS and RLS algorithms. Finally, the paper is concluded in Section VI. II. NOVEL BLIND ADAPTIVE MULTIUSER DETECTOR Consider an antipodal -user synchronous DS-CDMA system signaling through an additive white Gaussian noise channel. By passing through a chip-matched filter, followed by a chip-rate sampler, the discrete-time output of the receiver during one symbol interval can be modeled as
(1) where ambient channel noise; number of users; received amplitude of the th user; information symbol sequence from the th user, ; chosen independently and equally from th signature waveform that is assumed to have unit . energy is supported only on the interval It is assumed that , where symbol interval; chip interval; processing gain. Defining
(2)
A. State-Space Model Consider a dynamical system defined by process or state equation (5) and measurement equation (6) and are an state vector and an obwhere is a known servation vector at time , respectively; state transition matrix relating the states of the system at and , is a known measurement matrix, times vector is called the process noise, whereas and the vector represents a measurement noise. It is asthe is uncorrelated with both sumed that the initial state vector and for , and the noise vectors and are statistically independent, i.e., for all and . To develop a blind adaptive multiuser detector based on Kalman filtering, we need to devise a novel linear first-order state-space model for the blind multiuser detection problem that is a perfect match for the application of Kalman filter theory in blind scenario. By Honig et al. [2], the canonical representation for any linear detector for user 1 is defined by Type I: subject to (8) is orthogonal to the signature wavenamely, the vector form vector of user 1. The constraint (8) can be equivalently written as (9) We refer to (7) as a Type I canonical representation for a blind linear multiuser detector since there is another form, as shown in the following. The LMS algorithm is given by [2] (10)
we can express (1) in vector form (3) is the code sewhere quence assigned to the th user. For convenience, we will assume that the desired user is . It is well known that any linear multiuser detector for user 1 such that can be characterized by the tap-weight vector during the th symbol interval is given by the decision on
is the output of the conventional where single-user matched filter, and is the output of the linear detector. “Beamforming” is an efficient technique in blind multiuser detection [19]–[21]. Following the framework of the well-known generalized sidelobe canceler [22] and by conto present a unit response straining the linear detector to user 1’s signature sequence, Kapoor et al. [21] presented another canonical representation for linear detectors for user 1: Type II:
sgn
(7)
sgn
(11)
(4)
In the following, we consider how to establish the state-space model for Kalman filtering algorithm for blind adaptive mul. tiuser detector
is the adaptive part of , and the columns of the matrix span the null space of , i.e., . It is easy to see that Type I and II canonical representations are identical.
where -by-
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Our problem here is how to update adaptively using is given and is precomputed offline (11) after via one of many orthogonalizing procedures (such as the Gram–Schmidt orthogonalization and so on) [22]. To this end, we define the mean output energy (MOE) and the (scaled) mean-square-error (MSE) as MOE
(12)
MSE
(13)
Defining
is an identity matrix, and the process noise is a zero vector; b) the observation vector becomes the scalar , and the measurement matrix reduces to the . vector B. Kalman Filtering Algorithm From the corresponding relationship between the conventional dynamical system model and the dynamical system model of user 1, and using the standard Kalman filtering algorithm, we have the Kalman filtering algorithm for blind adaptive multiuser detection in stationary synchronous CDMA:
(14) (20)
has zero mean and variance [2], [3, p. 317]
then
MSE
cov
(15)
For a time-invariant CDMA system, an important fact is that is timeits optimum linear detector or tap-weight vector . Letting invariant as well, namely be the weight vector corresponding to in Type II canonical representation (11), we then have the following state : equation on (16) On the other hand, substituting (11) into (14) yields (17) and . If achieves , then (17) can be rewritten as the following measurement equation:
Put
(18) Both the state equation (16) and the measurement equation (18) construct the dynamical system model of user 1, which is the basis for Kalman filtering. The Kalman filtering problem in the blind multiuser detection may now be formally stated as fol, use the observed lows: Given the measurement matrix to find for each the minimum mean-square esdata . timates of the components of the state vector Using (15), we have cov
(19)
MSE denotes the MMSE where is optimum, and hence, when the tap-weight vector MOE is the minimum mean output energy of the dynamical system of user 1. Moreover, for synchronous model (1), it is easily shown that for due to for . Therefore, is a white noise with zero mean and variance in the synchronous case. As compared with the conventional dynamical system model specified by (5) and (6), the new linear first-order state-space model specified by (16) and (18) has the following characteris, the state-transition matrix tics: a) The state vector is
(21) (22) (23) , , , and are where vectors, and is the matrix having a length of . For the due to the state vector Kalman filter to be optimal, it is required that the initial state is Gaussian. Let its mean be zero and variance vector be one. By [10, p. 320], we may choose the initial predicted and its correlation estimate as matrix . Equation (22) is of fundamental significance. It shows that the correction term equals the innovation process premultiplied by the Kalman gain vector . Although the computation of in (20) requires that be known or estimated, this requirement is the parameter is only a not crucial. Note that the Kalman gain vector . By Honig et al. time-varying “step size” of updating [2], the signal-to-interference radio (SIR) of the desired user’s output after the MAI suppression is defined by SIR dB
(24)
where we have used the constraint (9) in Type I canonical representation. Since the maximum SIR of the desired user’s output is is usually expected to be higher than 10 dB, the MMSE usually smaller than 0.1. Note that since the received amplitude of the desired user during the th symbol interval is usually , we can directly take in larger such that . (19) as the estimate of the unknown parameter By Haykin [10, p. 702], an unknown dynamic system can be modeled as a transversal filter whose tap-weight vector undergoes a first-order Markov process, namely (25) is the where is a fixed parameter of the model, and process noise vector assumed to be of zero mean and correla-
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tion matrix Hence, (21) and (22) in the Kalman filtering algorithm should be replaced by
(26) (27) For a slowly time-varying CDMA system, it can be assumed that the value of parameter is very close to 1, and every element takes a very small value. Therefore, of its correlation matrix although, in a slowly time-varying environment, the proposed Kalman filtering algorithm is conditioned on perfect knowledge , , and , even with imperfect knowledge of of , and (null matrix), the new algorithm still works well. See Section V for simulation results. III. COMPARISON WITH OTHER RELATED ALGORITHMS The Kalman filtering algorithm established in Section II is related to many existing algorithms for adaptive multiuser detection. Now, let us compare its differences from other prior algorithms. In recent years, several Kalman filtering algorithms have been developed for adaptive multiuser detection [14]–[18]. However, due to the requirement of more knowledge than the desired user’s signature vector and timing, none of the previously published Kalman filtering algorithms can be considered blind. Since Kalman filtering algorithms and RLS algorithms are similar, it is necessary to compare the Kalman filtering algorithm of this paper with the previously published RLS algorithms for multiuser CDMA detection. Two typical RLS algorithms have been proposed for multiuser CDMA detection [6], [9]. The RLS algorithm in [6] requires spreading waveforms and propagation delays of all users, and it is not a blind multiuser detector. In addition, this RLS algorithm is a chip-rate algorithm, whereas our Kalman filtering algorithm is a symbol-rate one. Using the minimum output energy criterion, Poor and Wang [9] proposed a RLS algorithm for blind multiuser detection. The exponentially windowed RLS algorithm selects the weight vector to minimize the sum of exponentially weighted output energy, namely minimize
(28)
subject to
(29)
is the forgetting factor. The solution to this where constrained optimization problem yields the linear MMSE detector, which is given by [2], [9] (30) where (31)
A recursive procedure for updating can be implemented by the RLS algorithm [9]. The following are the computation complexities for updating per symbol interval of the three the tap-weight vector blind adaptive multiuser detectors: multiplications and adds; LMS algorithm [2]: multiplications and RLS algorithm [9]: adds; mulKalman filtering algorithm in this paper: adds. tiplications and IV. STEADY-STATE EXCESS OUTPUT ENERGY In this section, we analyze steady-state excess output energy of the Kalman filtering algorithm (20)–(23). A. Convergence Properties of the LMS and RLS Algorithms For convenience of comparison with the LMS algorithm [2] and the RLS algorithm [9], we first recall briefly the convergence properties of these two algorithms. Let MSE MOE
(32) (33)
represent the mean square error and mean output energy denote the minat iteration , respectively, and let , where imum mean square error with and is the minimum mean output energy, whereas is called the excess output energy (EOE) due to adaptation at time . Define (34) as the steady-state or asymptotic EOE; then, it is known by [2] that the steady-state excess MSE due to adaptation is equal to the steady-state EOE, i.e., (35) and for the LMS algorithm tr tr
(36)
where (37)
tr
that is the cross-correlation with being the eigenvalue of and , namely, matrix of . Note that the approximation (37) , , and (36) is one of becomes exact as several representations of the equivalence between the MMSE criterion and minimum output energy criterion. Upon convergence of the RLS algorithm, by Poor and Wang [9], for large tr
(38)
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where
is the average excess MSE at time , is the weight error correlation matrix at time , and the steady-state excess MSE or the steady-state EOE is given by (39)
and hence, the steady-state EOE in stationary situation is identically zero, i.e., (47) For the sake of comparison, we rewrite (35), (39), and (47) as an equation: tr tr
B. Steady-State EOE of the Kalman Filtering Algorithm
for RLS for Kalman.
Consider the stationary situation. By [12, p. 139], we have the following information matrix propagation equation: (40) 1, 2 in (40), we get
given the state equation (16). Taking
Mimicking this process for general form:
, we have the following
(41) , and denote
Put
and (42) into (41), we (43)
We remark that for the slowly time-varying situation in which is close to the correlation matrix of the process noise vector a null matrix, (40), (41), and (43) become approximate expressions, respectively. yields the estimate of the optimum weight Since , vector associated with the optimum tap-weight vector , we let i.e., MOE
(44)
represent the mean output energy at iteration . Theorem 1: For stationary CDMA systems, when is sufof the Kalman filficiently large, the mean output energy tering algorithm is given by (45) where is the processing gain of the DS-CDMA system. Proof: See the Appendix. From (45), we get the excess mean output energy at time given by (46)
(48)
when Remarks: From (45), we know that , and when the iteration number . This implies that the output energy of the Kalman filtering algorithm rapidly tends to the minimum mean as increases. On the convergence of the output energy , we have the following. mean output energy a) The Kalman filtering algorithm depends only on the processing gain and is independent of the data autocorrelation matrix , as shown in (45). b) The LMS algorithm depends on the eigenvalue distribution of , as is well known. c) The RLS algorithm depends on the trace of the matrix , as shown in (38). product Finally, we point out that no steady-state EOE analysis was given for Kalman filter-based (nonblind) adaptive multiuser detection algorithms in [14]–[18]. V. SIMULATION RESULTS
(42) Substituting the initial value have
for LMS
In this section, we present several simulation results to compare the new algorithm with other algorithms for blind multiuser detection. For each run, the LMS algorithm in [2], the RLS algorithm in [9], and the Kalman filtering algorithm in this paper are applied at the same time. Following (70) in [2], when implementing the LMS algorithm, the step-size must satisfy the stability condition of convergence of output MSE: (49) As a figure of merit for assessing the MAI suppression capability of the blind LMS, RLS, and Kalman filtering algorithms, the time-averaged SIR (in decibels) at the th iteration is given by [2] SIR
(50)
is the number of independent runs, and the subscript where indicates that the associated variable depends on the particular run. All signal energies are given in decibels relative to the background noise variance , i.e., the SNR of user is defined , where is the bit energy by SNR of user . In all simulations, user 1 is assumed to be the desired and an SNR of 20 dB (i.e., user that has the unit energy ), and the processing gain . In the following, the data in each plot are the average over 500 independent runs.
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Fig. 3. Time-averaged measured EOE versus time for 500 runs when using the three algorithms to a synchronous CDMA system with processing gain = 31.
N
Fig. 1. Time-averaged SIR versus time for 500 runs when using the three = 31. algorithms to a synchronous CDMA system with processing gain
N
Fig. 4. Time-averaged SIR versus time for 500 runs when using the three algorithms to a synchronous CDMA system with processing gain = 31 and the time-varying number of users.
N
Fig. 2. Time-averaged SIR versus time for 500 runs when using the three algorithms to an asynchronous CDMA system with processing gain = 31.
N
1) Convergence Rate Comparison: In Example 1, DS-CDMA systems in a Gaussian channel are simulated, and there are nine multiple-access interfering users among which five users have an SNR of 30 dB each, three users have SNR of 40 dB each, and another , user has an SNR of 50 dB, i.e., , and . Then, from (49), it follows that the step size should satisfy , and thus, was used in the LMS algorithm. When applying the RLS algorithm, takes , and the the initial value is taken. forgetting factor Fig. 1 depicts a plot of the time-averaged SIR versus iteration number (time) for the three algorithms applied in a synchronous CDMA system. Fig. 2 shows the timeaveraged SIR versus time for an asynchronous CDMA system with the time delays of the interferers , , where is the chip interval. Fig. 3 depicts the time-averaged measured EOE versus time for a synchronous CDMA system.
2) Tracking Dynamical Environment: In Example 2, we compare the tracking capabilities of the LMS, RLS, and Kalman filtering algorithms in a dynamical environment with a time-varying number of users for DS-CDMA , the systems in a Gaussian channel. When , configuration is the same as Example 1. At three interfering users with SNR of 40 dB are added to , four the CDMA system at the same time. At interfering users with SNR of 40 dB and one interfering user with SNR of 50 dB are removed from the system. Figs. 4 and 5 show the tracking behaviors of the three blind adaptive algorithms in a synchronous system and an asynchronous one, respectively. 3) Slowly Time-Varying Environment: Example 3 is a synchronous DS-CDMA system in a Rayleigh fading channel realized via harmonic decomposition technique [23], which undergoes a positive Doppler shift of about is used 22 Hz. The channel information of slot as an imperfect channel information of slot . The user configuration is the same as Example 1. Figs. 6 and 7 show the plots of the time-averaged SIR versus time and
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Fig. 5. Time-averaged SIR versus time for 500 runs when using the three = 31 algorithms to an asynchronous CDMA system with processing gain and the time-varying number of users.
N
the time-averaged measured EOE versus time for the RLS and Kalman filtering algorithms, respectively. In simulations, we used, respectively, the estimates , , and in (20), but Kalman curves in Figs. 1–7 have few changes. From Figs. 1–7, we can see the following facts. a) The RLS and Kalman filtering algorithms have faster convergence and better tracking capability than the LMS algorithm in both the synchronous and asynchronous systems. b) The Kalman filtering algorithm achieves nearly the con, in contrast to the LMS and RLS vergence when algorithms. c) The measured EOE in Kalman filtering algorithm is close in to the “ideal” EOE value of zero when both stationary and slowly time-varying environments, in . contrast to the LMS algorithm and RLS one with The first fact is well known. The second fact is not surprising since the performance analysis in Section IV told us , one has that when the iteration number that is already close to . The third fact , the steady-state shows that with imperfect knowledge of EOE in the Kalman filtering algorithm is not zero but is close to zero.
Fig. 6. Time-averaged SIR versus time for 500 runs when using Kalman and RLS algorithms with = 0:997 to a synchronous CDMA system in Rayleigh fading channel; the processing gain is N = 31.
Fig. 7. Time-averaged measured EOE versus time for 500 runs when using Kalman and RLS algorithms with = 0:997 to a synchronous CDMA system in Rayleigh fading channel; the processing gain is N = 31.
Although only the single-path delay channel was considered in this paper, the proposed algorithm, as well as the adaptive MOE detector, can be applied to multipath channels with the understanding that represents the desired user’s channel-distorted spreading code, i.e., the spreading waveform after passing through the channel. With perfect channel estimates, it would be possible to have this waveform at the receiver.
VI. CONCLUSION In this paper, we have used the Kalman filter to propose a novel blind adaptive multiuser detector in both a stationary environment and a slowly time-varying one. It is shown in theory that the steady-state EOE of the Kalman filtering algorithm is identically zero for the stationary environment, and simulations have shown that its measured EOE is close to zero in either stationary or slowly time-varying environments. Although the Kalman filtering algorithm is conditioned on perfect knowledge of some system parameters, simulations show that even when such knowledge is not easily available, the new algorithm still works well and has performance advantages over the LMS and RLS algorithms.
APPENDIX PROOF OF THEOREM 1 From (18) and (44), it follows that
(51) By the orthogonality principle, we have immediately that or
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. Using this result, we expand
as
ACKNOWLEDGMENT
follows:
(52)
The authors would like to thank Associate Editor Dr. R. S. Blum and the anonymous reviewers for their constructive and valuable comments and suggestions that helped to greatly improve the quality and clarity of the presentation. They also wish to thank the anonymous reviewers for bringing their attention to [5], [17], [19], and [20].
which may be equivalently rewritten in the following form: tr
REFERENCES (53)
and its predicted Since the state vector at time is , the predicted state-error vector estimate is is defined by (54) and the predicted state-error correlation matrix defined by
Since equation can be written as
is
in the stationary case, the above
(55) Substituting (55) and have tr
into (53), we
(56)
Since (43) holds strictly for the stationary environment, after is replaced by (43), for sufficiently large , we , and (56) may be approximated as follows: have tr Note that tr tr and we have that for sufficiently large tr which may be written in a compact form as that is (45). This completes the proof of Theorem 1.
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ZHANG AND WEI: BLIND ADAPTIVE MULTIUSER DETECTION
Xian-Da Zhang (SM’93) was born in Jiangxi, China, in 1946. He received the B.S. degree in radar engineering from Xidian University, Xi’an, China, in 1969, the M.S. degree in instrument engineering from Harbin Institute of Technology, Harbin, China, in 1982, and the Ph.D. degree in electrical engineering from Tohoku University, Sendai, Japan, in 1987. From August 1990 to August 1991, he was a Postdoctoral Researcher with the Department of Electrical and Computer Engineering, University of California at San Diego, La Jolla. Since 1992, he has been with the Department of Automation, Tsinghua University, Beijing, China, as a Professor and is currently with the Key Laboratory for Radar Signal Processing, Xidian University, Xi’an, China, as a Specially Appointed Professor awarded by the Minister of Education of China and the Cheung Kong Scholars Programme. His current research interests are signal processing with applications in radar and communications and intelligent signal processing. He has published over 20 papers in several IEEE Transactions and is the author of five books (all in Chinese). He holds one patent on radar target recognition.
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Wei Wei was born in Shanxi, China, in 1977. He received the B.S. and M.S. degrees from the Department of Automation, Tsinghua University, Beijing, China, in 1999 and 2001, respectively. He is currently pursuing the Ph.D. degree at the Department of Electrical Engineering and Computer Science, University of California, Berkeley.
