On the Second-Order Statistics of the Weighted Sample Covariance Matrix

Transcription

1 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 51, NO. 2, FEBRUARY On the Second-Order Statistics of the Weighted Sample Covariance Maix Zhengyuan Xu, Senior Member, IEEE Absact The second-order statistics of the sample covariance are encountered in many covariance based processing algorithms. This paper is to derive closed-form expressions for the covariance of the weighted sample covariance maix with an arbiary weight for both a real system and complex system. Given a system model, the results explicitly rely on the second-order and fourth-order statistics of the channel noise and inputs. They are shown to coincide with the existing results when the channel inputs and noise are Gaussian disibuted. Our results can be directly applied to analyze the statistical properties of subspace-based channel estimation methods for single-input multiple-output (SIMO) systems and code-division multiple access (CDMA) systems. Numerical examples are provided to further verify analyses. Index Terms Asymptotic analysis, covariance estimation, subspace decomposition. I. INTRODUCTION BLIND channel identification has received considerable attention in data communications. In a wireless communication system, digital signals are ansmitted through multipath channels that usually cause severe signal distortion. In order to reliably recover the input sequence, channel impulse responses can be first estimated based on the channel output. Then, receivers are designed. If aining symbols are not available, blind channel estimation methods are required. One of the most effective blind methods is the subspace technique [14], which relies on the second-order statistics of the received signal. By either singular value decomposition (SVD) on the collected data maix or eigenvalue decomposition (EVD) on the data covariance maix, both signal subspace and noise subspace can be obtained. Based on orthogonality property of subspaces, channel parameters are then estimated by minimizing projections of signature waveforms of input symbols onto the noise subspace or maximizing their projections onto the signal subspace under certain quadratic consaints. The method has also been successfully applied to estimate channel parameters for a direct-sequence (DS) code-division multiple access (CDMA) system [2], [13], [15], [16]. It is observed that the covariance of the received data plays an important role in determining the performance of subspace-based channel estimators. Since covariance maices of multivariate stationary signals show certain suctures, many Manuscript received March 12, 2002; revised September 27, This work was supported in part by the U.S. National Science Foundation under Grant NSF-CCR The associate editor coordinating the review of this paper and approving it for publication was Prof. Zhi Ding. The author is with the Department of Elecical Engineering, University of California, Riverside, CA USA ( dxu@ee.ucr.edu). Digital Object Identifier /TSP algorithms have been developed to estimate covariances [4], [5], [7], [8], [12], [17], [19], most of which are particularly designed for beamforming and array processing. In practice, it is much convenient to obtain their estimates from a finite number of observations, called sample covariance method. Some important results on the covariances of sample covariance maices have also been derived based on this estimation method. If Gaussian sources are assumed, the second-order statistics of the estimated covariance can be found in [3]. It has been applied to analyze the asymptotic property of eigenvectors of a covariance maix [6]. A statistical analysis of the subspace method [14] is carried out in [9] under some asymptotic approximations for some quantities. In the CDMA literature, channel estimation performance is also studied in [20], resulting in a general expression of covariance of the covariance estimator. Some aforementioned analyses are fulfilled based on the fact that errors are incurred in covariance estimation due to finite sample effect. For a given channel input/output model and some a priori statistical knowledge of the channel input and additive noise, it is desirable to obtain an accurate covariance of the sample covariance analytically for an arbiary number of available snapshots. Such results appear as prerequisites for further evaluation of performance of the associated channel estimators. If the received data obeys Gaussian disibution, the desired covariance can be directly obtained from ue data covariance [3]. For particular probability disibutions of the input signals and noise, one can follow the procedure described in [20] to achieve one s goal in principle. Although [20] presents a general guideline for such an endeavor, those analytical results for non-gaussian data samples appear rather complex and block further insight into the roles of each term in the analytical expression. Hence, explicit closed-form expressions are desired. They are also expected to be applied to analyze the performance of certain algorithms based on the second-order statistics of the channel output, such as subspace-based channel estimation methods. For a general discussion, we consider a weighted covariance of a sample covariance maix when the inputs and noise are independent random processes, which is ue in most communication applications. Since correlations of real random variables and complex random variables show different statistical properties [3] (as also discussed in Section III), two cases are differentiated: The system is real, and the system is complex. By appropriate manipulations in detailed procedures, closed-form expressions with an arbiary weighting maix are derived. It turns out that they depend on the second- as well as fourth-order statistics of both the inputs and channel noise. It is also shown that our analytical expressions degrade to the well-known results when the received data sequence is assumed X/03$ IEEE

2 528 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 51, NO. 2, FEBRUARY 2003 white Gaussian. Those derived results can be directly applied to obtain the channel mean-square-errors of the subspace-based channel estimation methods. Numerical examples are provided to further validate the proposed analyses. This paper is organized as follows. Section II describes the input/output model to be considered and assumptions explicitly imposed. In Section III, the asymptotic covariance maices of the sample covariance with arbiary but deterministic weighting maices are derived for real and complex systems, both in compact and closed forms. Section IV studies cases when the channel is driven by white Gaussian inputs and corrupted by white Gaussian noise. This section also shows consistence of the derived results with existing ones. Then, our results are applied to subspace-based channel estimation algorithms in Section V, followed by some numerical examples in Section VI. Finally, conclusions are made in Section VII. II. DATA MODEL AND ASSUMPTIONS Consider a vector form channel input/output data model where is a -dimensional received data vector at time with samples stacked in an ascending order of time, is a channel maix (could be suctured or not suctured), is a -dimensional input vector, and represents the additive noise. This model can describe a variety of communication systems. For example, in a single-input multiple-output (SIMO) system, includes inputs from time to emitted from the same source. When it is used to represent a multiple-input multiple-output (MIMO) system, has samples from either multiple antennas or upsamplers. Accordingly, contains symbols from different sources. In particular, the received signal in a DS/CDMA system can also fit into this model where becomes a signature waveform maix. Without loss of generality, we assume all enies in are mutually independent sequences. They are drawn from the same constellation with zero odd moments, equal variance, fourthorder absolute moment, and fourth-order cumulant. In the case when multiple sources have different ansmission powers, it is an easy task to cast some factors into columns of the channel maix. Additionally, we assume all enies in have zero odd moments, the same variance, fourth-order absolute moment, and fourth-order cumulant. They are also independent of input signals. Cumulant is related to moment and variance [11]. For a real input sequence,, whereas for a complex input sequence,. If a sequence is a Gaussian random process, then it has zero cumulant. This applies to the noise sequence as well. III. PERFORMANCE ANALYSIS OF COVARIANCE ESTIMATION According to (1), the covariance of has a form (1) (2) convention in our later discussion. In practice, is usually estimated from independent data vectors by sample average [20] to approximate the expectation. The approximation error is denoted by. It is noticed that is a Hermitian symmeic maix because both and are Hermitian symmeic. Here, we are interested in the performance of the covariance estimator in terms of a weighted covariance where is a deterministic weighting maix that may take different forms in different applications. We will proceed differently from [20]. Instead of performing a vec operation [10] on, first, we perform it later as necessary. Substituting by, we obtain. Since under independence assumption on,, where,wehave After substituting by (1) and imposing independence among all inputs in and noise, it can be found that only following terms survive in where denotes the ace of a maix, the first two terms result from ansposing some scalars ( ) in order to reorder corresponding quantities, and has been applied. The last two terms in (5) depend on the fourth-order moments of the source and the noise, respectively. All other terms are either already in evaluative forms or dependent on the second-order moments. As is known, for a random variable, the real or complex nature differentiates the quantity and.in order to obtain a closed-form expression for, we consider two scenarios separately; the communication system is either real or complex. A. Real System In this case, all quantities are assumed real valued. Under our assumptions on the statistics of the inputs and noise, (5) becomes (3) (4) (5) where represents Hermitian complex conjugate ( ) anspose ( ). For convenience, we have dropped the subscript for the identity maix to specify its dimensionality without incurring confusion in the context. We will continue to follow this (6)

3 XU: SECOND-ORDER STATISTICS OF THE WEIGHTED SAMPLE COVARIANCE MATRIX 529 To evaluate the second term to the last, we perform vec and then the reverse unvec operations where has been applied. Observe that. Substituting (12) and (13) in (6) first and then (6) in (4), we obtain unvec vec (7) where properties of vec and the Kronecker product have been applied [10]. After some saightforward algebra, it can be verified that [18] vec (8) (14) where has been used, is a block diagonal maix Using once more, (14) becomes diag and has been partitioned into sub-blocks with the (, )th sub-block being. Substituting (8) in (7), we obtain where unvec vec unvec vec It is observed that has two typical terms unvec vec, and unvec vec. Express maix explicitly by columns as whose th diagonal element is. Then, according to the definition of, wehave unvec vec (9) diag (10) where represents element-wise multiplication. Then, only keeps the diagonal elements of. According to the definition of, the th subvector of vector vec is, which can be written as. Since, it becomes. Therefore unvec vec (11) With (10) and (11), (9) becomes Similar to (12), the last term in (6) is simplified to (12) (13) (15) It can be observed that depends on not only the second-order statistics of the inputs and noise but also on their high-order statistics. B. Complex System In this case, the real parts and imaginary parts of inputs are assumed to be independent and have the same disibutions. Similar assumptions are made on the noise. Then,, and. Equation (5) becomes In parallel with (7) and (8), we can obtain the following: unvec vec (16) (17) vec (18) where has been substituted by. Notice that (18) is different from (8) because of complex inputs. Substituting (18) in (17) and noticing (10), we obtain Similarly, noticing (19), in this case, we obtain (20) Observe that. Substituting (19) and (20) in (16) first and then (16) in (4), we obtain (21)

4 530 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 51, NO. 2, FEBRUARY 2003 Using once more, (21) becomes In fact, (29) is also a direct application of that in [3] or [6]. After applying (29), we have (22) Results in (15) or (22) can be simplified if inputs and/or channel noise are Gaussian disibuted. IV. SPECIAL CASE GAUSSIAN INPUTS AND NOISE In this section, we derive the covariance of the estimated covariance for Gaussian inputs with additive white Gaussian noise (AWGN) in the system directly from the probability theory. Then, we compare those results with our results in the previous section in such a scenario and show that they are consistent although derived differently. A. Real System To easily derive, we consider first, where and are arbiary deterministic vectors. After substituting by and observing (3), the following can be expanded and verified: (23), we can sim- For a zero-mean real Gaussian random process plify the first term [3], [11] Then (24) (25) In order to apply this result to evaluate, we perform SVD on, where are singular values (non-negative), and and are left and right singular vectors. Therefore (26) It is interesting to compare (26) with (15). For Gaussian processes,, and. Therefore, (15) degrades to (26). However, for a given system, only is required in (26), which is different from our general result (15) with non-gaussian variables, where the channel maix is explicitly involved. B. Complex System Similar to (23) and (24), we can obtain Then (27) (28) (29) (30) Consistency between (30) and (22) is clearly observed after noting and for Gaussian processes. Again, (22) is more general but requires channel maix. Next, we will show how to apply our results to analyze subspace-based channel estimators. V. APPLICATIONS IN SUBSPACE-BASED CHANNEL ESTIMATION ALGORITHMS There are different approaches to estimate channel parameters. One of the widely used methods is the subspace method [14]. It can be applied to estimate multiple channels in a SIMO system. It has also been successfully applied to CDMA systems [2], [16]. Before we study the asymptotic performance of the corresponding estimators based on our previous results, let us first review the method [14]. A. Review of SIMO Subspace Approach Assume there are subchannels, each of which has order. Stack all unknown channel coefficients in a vector. Collect inputs from the same source in a vector and outputs in a vector whose enies are sorted based on time. Then, the data model to be considered in an AWGN environment obeys (31) where is a block Toeplitz maix consucted from. All columns in can be obtained by shifting the column vector up/down by multiples of positions. It can be well represented by the operation for..., where For notational convenience, we have defined and. As is well known, the subspace method employs either the noise subspace or the signal subspace. Both subspaces can be obtained from EVD of the covariance maix, which yields diag (32)

5 XU: SECOND-ORDER STATISTICS OF THE WEIGHTED SAMPLE COVARIANCE MATRIX 531 All vectors are orthogonal to, which leads to. Therefore, the noise subspace-based method becomes [14] (33) where takes integers from to. For simplicity, we have dropped and will continue to drop the limits in the summation. Under certain identifiability conditions [14], optimization of (33) guarantees a unique eigenvector of the following objective maix (34) corresponding to its null eigenvalue, which is the channel vector up to a multiplicative scalar. We will study the performance of this channel estimator next when the noise subspace is obtained from the estimated data covariance maix. B. Asymptotic Performance of the Channel Estimator Assume is estimated from data sample vectors according to (3). The finite number of data samples will determine the accuracy of the subspace estimate, thus affecting the performance of the channel estimator. For notational convenience, let be the noise-free data covariance maix has the fol- Then, the perturbation of the noise subspace of lowing form [6]: diag (35) (36) where denotes the pseudo-inverse. If a perturbation occurs in estimating, then an error will be ansferred to our channel estimate. We have shown that is a unique eigenvector corresponding to the null eigenvalue of. Assume the channel estimation error is and the perturbation of is. Then, has the following form [1], [6]: According to (34), is given by (37) (38) Substituting (36) in (38) and then (38) in (37) and noticing the orthogonality property, we obtain The covariance of channel estimate becomes (39) (40) To evaluate this quantity, we may set and apply either (15) or (22) to compute. It is noted that can be first simplified in two cases, respectively. However, since both and are orthogonal to, direct application will cause to have the same form. It can be easily verified that, and Then, (40) becomes (41) (42) where is given by (41). The mean-square-error (MSE) of the estimated channel vector is then the ace of maix (43) According to (43), it can be observed that the MSE is proportional to. Meanwhile, it is determined by projections of onto the noise subspace according to the quantity inside the ace operator. If we examine the scalar, we can conclude that the MSE is approximately inversely proportional to the power of the ansmitted signal because of the term. According to subspace decomposition of in (35), also depends on projections of signature vectors of various bits onto the signal subspace. More interestingly, the MSE does not depend on the fourth-order statistics of the input sequence. C. Performance Analysis of Channel Estimation for a CDMA System We may proceed in a much similar way to analyze the statistical property of the subspace-based channel estimator for a multiuser CDMA system. As an example, we only resict our attention to a synchronous CDMA system. It is easy to generalize the results to an asynchronous system. Consider a user system. User is assigned spreading codes. Its channel is assumed to have order, and corresponding coefficients are stacked in a vector. Then, in one symbol interval, the received data vector has a form [13], [18], [20] (44) where is a code filtering maix obtained from the spreading codes of user. Now, the signature waveform maix is given by (45) Without loss of generality, assume user 1 is the desired user. We still use to denote the data covariance maix, the noise-free data covariance maix, and the noise subspace. Since, under some identifiability conditions, can be estimated from [13] (46)

6 532 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 51, NO. 2, FEBRUARY 2003 within a scalar ambiguity. following objective maix: is then a unique eigenvector of the (47) corresponding to its null eigenvalue. It can be observed from (34) and (47) that the results in the previous subsection can be applied by dropping the summation indexed by. Therefore, we can obtain the following: where The MSE of the estimated channel vector has a form (48) (49) (50) (51) According to (43), similar conclusions can be made on the MSE as in [1]. This time, the MSE is still proportional to and does not depend on the fourth-order statistics of the input sequence. It depends on projections of the code maix onto the signal and noise subspaces. It is approximately inversely proportional to the power of the ansmitted signal. Fig. 1. Covariance estimation error for a real system with real inputs. VI. NUMERICAL EXAMPLES Several experiments have been conducted to verify our analyses in the previous sections. Since the expression of the covariance of the estimated data covariance depends on whether the communication system is real or complex, we differentiate these cases in the simulation. We adopt the normalized squared Frobenius norm of the difference between and its estimate as the performance measure and test the asymptotic performance with respect to the number of independent snapshots ( ). Here, we take only those data vectors independent to each other to make simulations consistent with our analysis, which is similar to [20]. Each time, we collect ten data samples to form a data vector from four inputs and process. The weighting maix is set to be an identity maix. In total, 100 independent Monte Carlo runs are performed to obtain the average estimation error. Channel coefficients are randomly selected from a zero-mean white Gaussian process with unit variance and fixed in all realizations. Channel noise is Gaussian disibuted giving the system a 10 db signal-tonoise ratio (SNR). Both inputs and noise are random in each realization. First, we consider a system with real inputs, channel, and noise. These inputs are drawn from real constellations such as BPSK, PAM, and Gaussian. The ue maix is obtained according to (15) or (26). The estimation errors are plotted in Fig. 1. The solid line is for the BPSK source, the dashed-dotted line for the 8PAM source, and the dashed line for the Gaussian source. Differences are observed with different inputs, which implies that the sample covariance estimation method may Fig. 2. Covariance estimation error for a complex system with complex inputs. perform differently, as predicted by our analysis. With BPSK inputs, the estimation error is smaller than other two kinds of inputs. The Gaussian inputs produce the largest estimation error with any given. A similar conclusion can be made when the system is complex with complex channel, Gaussian noise, and inputs in the corresponding types 8PSK, 16QAM, and Gaussian, as shown by Fig. 2. Now, the ue covariance is computed from (22) or (30). However, differences caused by different inputs decrease compared with Fig. 1. With 1000 independent data vectors, the estimation error reduces to a similar level at 10. The weighting maix may take forms different from an identity maix in some applications. According to our analytical results, it also affects the estimation performance. Although it is impossible to test all possible cases, we set it equal to an identity maix, which is a random maix whose enies are generated as zero-mean unit-variance Gaussian processes in different realizations and, respectively, to gain some insights. The system to be considered is the same as for the first figure. The average results are presented in Fig. 3. We can see that the estimation

7 XU: SECOND-ORDER STATISTICS OF THE WEIGHTED SAMPLE COVARIANCE MATRIX 533 Fig. 3. Covariance estimation error with different weighting maices. Fig. 5. Data length effect on channel estimation error for a synchronous CDMA system. Fig. 4. Channel estimation error for a SIMO system. error highly depends on the weighting maix, with the smallest error when and the worst performance when is randomly generated. When an identity maix is used, the error is in between, but closer to, the Gaussian case. After we verify those analytical results, we apply them to channel estimation in different scenarios. First, a SIMO system with BPSK source is assumed. Two subchannels of order 3 each are randomly generated. To guarantee channel identifiability, six samples from each subchannel are collected each time to form our 12 1 data vector. SNR is set to be 20 db. These subchannels are estimated based on the subspace technique [14]. The experimental result is compared against our analytical result given by (43) for different number of independent data vectors in Fig. 4. The solid line represents the experimental channel estimation error, whereas the dashed line denotes the analytical result. Each independent data vector is, in fact, conibuted by nine input symbols since the channel maix is Therefore, the observation window for independent data vectors spans 9 symbol intervals. It is observed that the simulation agrees with our analysis when is large. For small, the covariance estimation method (3) does not yield a good estimate for. Therefore, the perturbation technique generates a larger discrepancy. We also verify our analyses by applying the derived result to channel estimation for a synchronous CDMA system [13]. We assume ten equally powered users in a system with 15 db AWGN. Each user is assigned a Gold sequence of length 31 to spread its data seams [18]. Channel order is set to be 3 for each user. We collected 28 chip-rate samples, corresponding to each symbol to eliminate intersymbol interference. Again, our experimental result is compared with the analytical result for different number of bit periods and shown in Fig. 5. High consistency between two lines can be observed for longer observation time. Up to 60 symbol periods, which is a fairly good estimate for the data covariance, is reflected by small channel estimation error in the figure. Since applicability of perturbation analysis for channel estimation requires not only large number of data samples but small noise in the system as well, we then test the noise effect on the estimation performance. The results after 200 symbols are ansmitted are presented for a large range of SNRs in Fig. 6. It is clear that when SNR is greater than 0 db, a significantly well match between two curves can be observed. However, when the system is much noisy (low SNR), the subspace-based method does not provide a reliable channel estimate, and the perturbation analysis cannot predict the actual performance of the channel estimator. The near far effect is also tested in a 15 db noise environment. The channel estimation error versus signal to interference ratio (SIR) is plotted in Fig. 7. For different SIR levels, the experimental data varies closely around the analytical curve. VII. CONCLUSIONS We have derived the covariance of a data covariance estimator in closed and compact forms for both real and complex systems with arbiarily disibuted inputs. They are shown to agree with the existing approaches when the channel input and

8 534 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 51, NO. 2, FEBRUARY 2003 Fig. 6. system. Noise effect on channel estimation error for a synchronous CDMA [5] S. Degerine, Maximum likelihood estimation of autocovariance maices from replicated short time series, IEEE Trans. Inform. Theory, vol. 44, pp , Mar [6] B. Friedlander and A. Weiss, On the second-order statistics of the eigenvectors of sample covariance maices, IEEE Trans. Signal Processing, vol. 46, pp , Nov [7] D. R. Fuhrmann, Application of toeplitz covariance estimation to adaptive beamforming and detection, IEEE Trans. Signal Processing, vol. 39, pp , Oct [8] M. Jansson and B. Ottersten, Suctured covariance maix estimation: A parameic approach, Proc. IEEE Int. Conf. Acoust., Speech, Signal Process., vol. 5, pp , June [9] M. Kristensson and B. Ottersten, Statistical analysis of a subspace method for blind channel identification, Proc. IEEE Int. Conf. Acoust., Speech, Signal Process., vol. 5, pp , May [10] P. Lancaster and M. Tismenetsky, The Theory of Maices, 2nd ed. San Diego, CA: Academic, [11] A. Leon-Garcia, Probability and Random Processes for Elecical Engineering, 2nd ed. Reading, MA: Addison-Wesley, [12] H. Li, P. Stoica, and J. Li, Computationally efficient maximum likelihood estimation of suctured covariance maices, IEEE Trans. Signal Processing, vol. 47, pp , May [13] H. Liu and G. Xu, A subspace method for signature waveform estimation in synchronous CDMA systems, IEEE Trans. Commun., vol. 44, pp , Oct [14] E. Moulines, P. Duhamel, J.-F. Cardoso, and S. Mayrargue, Subspace methods for the blind identification of multichannel FIR filters, IEEE Trans. Signal Processing, vol. 43, pp , Feb [15] X. Wang and H. V. Poor, Blind multiuser detection: A subspace approach, IEEE Trans. Inform. Theory, vol. 44, pp , Mar [16] X. Wang and H. Poor, Blind equalization and multiuser detection in dispersive CDMA channels, IEEE Trans. Commun., vol. 46, pp , Jan [17] D. B. Williams and D. H. Johnson, Robust estimation of suctured covariance maices, IEEE Trans. Signal Processing, vol. 41, pp , Sept [18] Z. Xu, Asymptotically near-optimal blind estimation of multipath CDMA channels, IEEE Trans. Signal Processing, vol. 49, pp , Sept [19] C. Yang and H. Van Trees, Blind beamforming with suctured covariance, in Proc. Fifth Int. Conf. Signal Process., vol. 1, Beijing, China, Aug , 2000, pp [20] N. Yuen and B. Friedlander, Asymptotic performance analysis for signature waveform estimation in synchronous CDMA systems, IEEE Trans. Signal Processing, vol. 46, pp , June Fig. 7. Near-far effect on channel estimation error for a synchronous CDMA system. noise are Gaussian disibuted. Different channel inputs are used to validate the analytical results. In addition, those results are also applied to subspace-based channel estimation algorithms for SIMO and CDMA systems. REFERENCES [1] E. Aktas and U. Mia, Complexity reduction in subspace-based blind channel identification for DS/CDMA systems, IEEE Trans. Commun., vol. 48, pp , Aug [2] S. Bensley and B. Aazhang, Subspace-based channel estimation for code division multiple access communication systems, IEEE Trans. Commun., pp , Aug [3] D. R. Brillinger, Time Series Data Analysis and Theory. San Francisco, CA: Holden-Day, [4] J. P. Burg, D. G. Luenberger, and D. L. Wenger, Estimation of suctured covariance maices, Proc. IEEE, vol. 70, pp , Sept Zhengyuan Xu (S 97 M 99 SM 02) received the B.S. and M.S. degrees in eleconic engineering from Tsinghua University, Beijing, China, in 1989 and 1991, respectively, and the Ph.D. degree in elecical engineering from Stevens Institute of Technology, Hoboken, NJ, in From 1991 to 1996, he worked as an engineer and department manager at the Tsinghua Unisplendour Group Corp., Tsinghua University. From 1996 to 1999, he was a research assistant and research associate at Stevens Institute of Technology, working on signal processing for wireless communications, especially for multiuser CDMA systems. In 1999, he joined the Department of Elecical Engineering, University of California, Riverside, as an assistant professor. His current research interests include advanced signal processing, multirate communication, multiuser detection, and system identification. Dr. Xu received the Outstanding Student Award and the Motorola Scholarship from Tsinghua University and the Peskin Award from Stevens Institute of Technology. He also received the Academic Senate Research Award and the Regents Faculty Award from University of California, Riverside. He is Associate Editor for the IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY and the IEEE COMMUNICATIONS LETTERS.