Low-Feedback-Rate and Low-Complexity Downlink Multiuser MIMO Systems

Transcription

1 3640 IEEE TRANSACTIONS ON VEICULAR TECNOLOGY, VOL. 59, NO. 7, SEPTEMBER 200 [5] R. Jiang and B. Chen, Fusion of censored decisions in wireless sensor networks, IEEE Trans. Wireless Commun., vol. 4, no. 6, pp , Nov [6] C. Rago, P. Willett, and Y. Bar-Shalom, Censoring sensors: A lowcommunication-rate scheme for distributed detection, IEEE Trans. Aerosp. Electron. Syst., vol. 32, no. 2, pp , Apr [7] C. R. Berger, M. Guerriero, S. Zhou, and P. Willett, PAC vs. MAC for decentralized detection using noncoherent modulation, IEEE Trans. Signal Process., vol. 57, no. 9, pp , Sep [8] S. Yiu and R. Schober, Nonorthogonal transmission and noncoherent fusion of censored decisions, IEEE Trans. Veh. Technol., vol. 58, no., pp , Jan [9] A. Anandkumar and L. Tong, Type-based random access for distributed detection over multiaccess fading channels, IEEE Trans. Signal Process., vol. 55, no. 0, pp , Oct [0] S. M. Kay, Fundamentals of Statistical Signal Processing: Detection Theory. Englewood Cliffs, NJ: Prentice-all, 998. [] B. Picinbono, On deflection as a performance criterion in detection, IEEE Trans. Aerosp. Electron. Syst., vol. 3, no. 3, pp , Jul [2] Z. Quan, S. Cui, and A.. Sayed, Optimal linear cooperation for spectrum sensing in cognitive radio networks, IEEE J. Sel. Topics Signal Process., vol. 2, no., pp , Feb [3] J. Unnikrishnan and V. V. Veeravalli, Cooperative sensing for primary detection in cognitive radio, IEEE J. Sel. Topics Signal Process., vol. 2, no., pp. 8 27, Feb Low-Feedback-Rate and Low-Complexity Downlink Multiuser MIMO Systems youngjoo Lee, Illsoo Sohn, Student Member, IEEE,and Kwang Bok Lee, Senior Member, IEEE Abstract In this paper, we propose a practical downlink multiuser multiple-input multiple-output MU-MIMO) system. The proposed MU-MIMO system focuses on improving two limiting factors for practical implementations of MU-MIMO: ) feedback rate and 2) computational complexity. First, users efficiently feed their channel-state information back with low feedback rate based on the proposed channel-quantization method. Second, the beamforming matrix at the base station is easily computed using singular value decomposition SVD). Numerical results show that the proposed MU-MIMO system achieves a higher sum rate than conventional MU-MIMO systems, particularly at low feedback rate, while the computational complexity is kept reasonable. Index Terms Limited feedback, multiple-input multiple-output MIMO) broadcast channel, multiuser MIMO MU-MIMO), unitary beamforming. Manuscript received November 2, 2009; revised March 9, 200 and April 29, 200; accepted May 5, 200. Date of publication May 8, 200; date of current version September 7, 200. This work was supported by the National Research Foundation of Korea under Grant funded by the Korean government Ministry of Education, Science, and Technology). This paper was presented in part at the 5th Asia-Pacific Conference on Communications APCC 2009), Shanghai China, October 8 0, The review of this paper was coordinated by Prof..-F. Lu.. Lee and K. B. Lee are with the School of Electrical Engineering and Computer Science and Institute of New Media and Communications, Seoul National University, Seoul 0-799, Korea leehj@mobile.snu.ac.kr; klee@snu.ac.kr). I. Sohn was with the School of Electrical Engineering and Computer Science and Institute of New Media and Communications, Seoul National University, Seoul 0-799, Korea. e is now with Wireless Networking and Communications Groups, University of Texas at Austin, Austin, TX 7872 USA isohn@mail.utexas.edu). Color versions of one or more of the figures in this paper are available online at Digital Object Identifier 0.09/TVT I. INTRODUCTION In the last decade, linear beamforming techniques for multiuser multiple-input multiple-output MU-MIMO) systems have extensively been studied, such as zero-forcing beamforming ZFBF) [], block diagonalization BD) [3], and generalized multiuser orthogonal space-division multiplexing [4], assuming perfect channel-state information at the transmitter CSIT). Furthermore, the studies have been extended to a more important scenario of partial CSIT, such as limited feedback-based ZFBF [5], BD [6], and coordinated beamforming [7]. It has been verified that these MU-MIMO systems, which are based on interference elimination, are very sensitive to channel-state information CSI) accuracy [8], [9]. Thus, they suffer from severe performance degradation, particularly at low feedback rate [0]. To alleviate this problem, quantization-based combining QBC) [] and a maximum expected signal-to-interference-plus-noise ratio combiner MESC) [2], which improve the receiver combining technique to reduce the required feedback rate, were recently proposed. owever, the channel feedback rate is still a major limiting factor. On the contrary, the random orthogonal beamforming ROB) technique in [2] is designed to be appropriate for MU-MIMO systems with low feedback rate. It is shown that the ROB also achieves the sum-rate capacity only if the number of users goes to infinity. ence, a per-user unitary and rate control PU 2 RC) scheme [4], [5], which is a practical implementation of the ROB in codebook-based systems, has become one of the major candidates for MU-MIMO systems in next-generation wireless communication standards [6], [7]. owever, the major drawback of the PU 2 RC is that the choice of a beamforming matrix at the base station BS) is limited within the predefined codebook. Thus, the system becomes inefficient when there are a finite number of users whose channels are not well matched to the beamforming matrix in the predefined codebook. The solution for the problem has been searched in two ways. One approach is to improve the codebook to efficiently exploit the channel statistic [3]. The other approach is to adaptively compute the beamforming matrix. As an example of the latter, an enhanced unitary beamforming scheme that constructs the beamforming matrix without being constrained within the predefined codebook has been proposed [8]. Unfortunately, both works have not provided noticeable improvements. In particular, the enhanced unitary beamforming scheme in [8] includes an iterative algorithm for the construction of the appropriate beamforming matrix, which results in high computational complexity in return for the increased beamforming gain. In this paper, we propose a new MU-MIMO system that includes the improvements in both channel feedback of users and beamforming matrix construction at the BS. As the simulation results will show, the proposed MU-MIMO system provides improved sum-rate performance, particularly at low feedback rate, reducing the required computational complexity for the determination of the beamforming matrix. II. SYSTEM MODEL As shown in Fig., a downlink MU-MIMO system is considered, where the BS has antennas, and each of the K users has N R antennas. An independent identically distributed Rayleigh flat-fading channel is assumed. The elements of the user channel matrix are considered to be independent complex Gaussian random variables with zero mean and unit variance. For simulation and limited feedback codebook design purposes, the block-fading channel is also assumed to be static during a time slot and independently changing over time slots, which can practically be attained using multiple-input multipleoutput orthogonal frequency-division multiplexing MIMO-OFDM) /$ IEEE

2 IEEE TRANSACTIONS ON VEICULAR TECNOLOGY, VOL. 59, NO. 7, SEPTEMBER Finally, the BS broadcasts information about the set of selected users and the beamforming vector of each selected user through feedforward channels before it transmits user data through data channels. III. PROPOSED MULTI-USER MULTIPLE-INPUT-MULTIPLE-OUTPUT DOWNLINK SYSTEM Fig.. MU-MIMO system model. In this section, a new user feedback method that efficiently conveys information on both the channel state and the effect of interuser interference at low feedback rate is described. Two user-selection methods for the case when the number of users exceeds the number of transmitting antennas are presented and compared. Then, it is explained how a beamforming matrix can easily be constructed at the BS with simple manipulations using singular value decomposition SVD). Finally, receive antenna combining and data decoding at the selected users are described. Fig. 2. Timing diagram of system operation. The received signal at the kth user after receiver combining is given by y k = r k k Fx + n k ), k =, 2,...,K ) where k C N R is the channel matrix of the kth user, F C is the unitary beamforming matrix, x C is the transmitted symbol vector, r k C N R is the unit norm receiver combining vector for the kth user, and n k C NR is the additive white Gaussian noise vector at the receiver. ere, users are simultaneously supported after user selection, and each selected user is allocated a single data stream. Equal power allocation for each data stream is assumed to be E[xx ]=P/ )I NT,whereP denotes the total transmit power of the BS. Perfect CSI at the user terminal and an error-free and zero-delay feedback link are assumed. The codebook C = {c, c 2,...,c 2 B } is generated by the Grassmannian line packing method [9], where B denotes the number of feedback bits per user. For better understanding of the proposed MU-MIMO system, a timing diagram is shown in Fig. 2. First, each user estimates its own channel from pilot broadcasting. Then, each user quantizes its channel information into channel-quality information and channel-direction information and reports to the BS. Based on the feedback information of all users, the BS selects users to simultaneously transmit and constructs the unitary beamforming matrix for downlink transmission. A. User Feedback Signal-to-interference-plus-noise ratio SINR) is considered to capture the effects of interuser interference when determining mobile station MS) feedback. Theorem : If the beamforming matrix F =[f f 2 f NT ] is a unitary matrix, the SINR of the kth user after linear minimum mean square error LMMSE) combining η k depends only on both the corresponding beamforming vector f k and the instantaneous channel matrix k [2]. Proof: η k a) = b) = = [ ) ] F k kf + I NT k,k ) ] [F k k + I NT F f k k,k 2) k k + I NT fk where a) is from [20], b) follows from F F = FF = I NT,and denotes the signal-to-noise ratio SNR) at the kth user. According to the theorem, user k can find the best f k that maximizes η k, regardless of the beamforming vectors of other users. Since users should feedback information using the predefined codebook C, users determine their feedback as πk) =arg max. 3) i=,...,2 B c i k k + I NT ci ere, c πk) implies the preferred beamforming vector of the kth user. Now, the kth user reports the index πk) and the corresponding SINR value η k as feedback information. Conventional user-feedback methods in [5], [], and [8] only consider the CSI of individual users without interuser interference, i.e., user feedback is determined by computing inner products between own channel direction and codeword in the codebook. Thus, all processes for interference mitigation are performed at the BS, which becomes inefficient, particularly in a low-rate feedback channel. On Derivations in 2) are also used in [8] for a different application.

3 3642 IEEE TRANSACTIONS ON VEICULAR TECNOLOGY, VOL. 59, NO. 7, SEPTEMBER 200 the contrary, a key advantage of the proposed feedback method is that it considers interuser interference at the feedback stage. Thus, each user is able to effectively convey its CSI, combined with the effect of interuser interference, to the BS, even in a low-rate feedback channel. 2 Note that the user-feedback method of the PU 2 RC in [4] and [5] also considers both CSI and interuser interference. owever, the major weakness of the PU 2 RC codebook is that it should be implemented with the specific codebook structure, i.e., sets of -by- orthonormal matrices. One practical example of designing the PU 2 RC codebook is using the discrete Fourier transformation DFT) [6]. When the number of users is extremely large, there always exists a user whose channel is well matched to one of the codebooks. As shown in the previous literature, this codebook performs well when it is used for user feedback. owever, when the number of users is moderate, the PU 2 RC codebook becomes inefficient since the codewords are not designed to be isotropically distributed. On the contrary, the proposed user feedback method previously described can be implemented with any codebook, including Grassmannian, which is proven to be optimal [9], and DFT-based codebooks, which is the great advantage over the conventional user feedback methods. Later, in the numerical results, the Grassmannian codebook was adopted for simulations. B. User Selection When the number of users exceeds the number of transmit antennas, i.e., K, the BS selects up to users for simultaneous transmission based on the user feedback πk) and η k. Two user-selection schemes are presented in the following. ) Max-SINR User Selection: A simple and effective method to exploit multiuser diversity for the BS is to select the user with the largest η k in the user pool and repeat the selection process until all users have been selected. The method only considers the SINR values η k reported from the users, and it can be implemented through simple ordering of scalar values. ence, the required computational complexity is extremely low. 2) Semi-Orthogonal User Selection: The reported preferred beamforming feedback vectors c πk) and reported SINR values η k are also important factors for improving user selection performance. Since the final unitary beamforming matrix is constructed from the preferred beamforming feedback vectors of the selected users, as will be discussed in the next section, user selection without considering the direction information of the users may result in large discrepancies between the preferred beamforming feedback vectors c πk) and the final beamforming vectors f k. To minimize such discrepancies, it is effective to select users whose preferred vectors are as orthogonal as possible. The semi-orthogonal user selection in [], which is based on greedy user selection [25], may be used to consider both SINR values and preferred beamforming feedback vectors. Note that the greedy user selection applied here is different from the original form, in that we use the SINR value η k instead of the channel gain and the preferred beamforming vector c πk) instead of the channel-direction vector. Compared with the maximum-sinr user selection, this semiorthogonal user selection exploits better multiuser diversity gain, in return for increased computational complexity. 2 It is worth mentioning that the key idea of the proposed feedback method can be applied to different applications. In recent parallel works, a similar channel-feedback idea is adopted in ZFBF-based MU-MIMO systems [2], [3]. Normally, the beamforming matrix of the ZFBF system is not unitary. Assuming an infinite number of users, an ideal scheduler makes the beamforming matrix unitary in this case. Thus, the underlying system model becomes identical to our unitary beamforming-based MU-MIMO system. C. Construction of Unitary Beamforming Matrix The optimal unitary beamforming matrix can be found when the sum rate, which will be expressed in 0), is maximized. Unfortunately, this is a nonconvex problem, and there has been no general solution in the literature, to the best of our knowledge. In our earlier work [8], we proposed an iterative algorithm to search a suboptimal unitary beamforming matrix. In spite of the improved sum-rate performance, the proposed algorithm is not appropriate for practical implementations due to high computational complexity. Instead, we introduce an alternative method to construct the unitary beamforming matrix in this work. By modifying the sum-rate maximization problem as follows, we can determine the beamforming matrix with extremely low computational complexity. When each user reports its preferred beamforming vector at the feedback stage, the users believe that the reported preferred beamforming vectors would be served with them. owever, it is impossible to directly use them after aggregation as a beamforming matrix since they are not orthogonal to each other. ence, the BS needs to construct a new unitary beamforming matrix F such that each beamforming vector in F, i.e., f k, is as close to the reported preferred beamforming vector c πk) as possible. Computation of the appropriate unitary beamforming matrix F can be formulated as the following problem: N T maximize cπk), f k k= subject to F F = FF = I NT 4) where, denotes inner product operation of two vectors. The unitary beamforming matrix F maximizes the sum of the inner products between the preferred beamforming vector c πk) and the corresponding beamforming vector fk. In other words, the optimal beamforming vector fk is closely tuned to the preferred beamforming vector c πk) in terms of chordal distance. 3 To solve the optimization problem in 4), a well-known result of matrix computation [24] is used. The steps to construct the optimal beamforming matrix are described as follows: First, the preferred beamforming vectors of the selected users are aggregated as P = [ c π), c π2),...,c πnt )]. 5) Then, the aggregated preferred beamforming matrix is decomposed by SVD as P = UΣV 6) where U and V are unitary matrices, and Σ is a diagonal matrix. Finally, the optimal beamforming matrix is simply computed as F = UV 7) 3 In limited-feedback closed-loop multiantenna systems, which require efficient algorithms for the quantized information of beamforming vector and feedback vector, it is well known that chordal distance is an appropriate design criterion that improves system performance in terms of SNR maximization and outage minimization [9], [22]. Therefore, based on these results, in the literature [5], [], [23] and future wireless communication standards [6], such as LTE-Advanced and IEEE 802.6m, the design criterion using chordal distance has been adopted for the design of beamforming vector and feedback vector.

4 IEEE TRANSACTIONS ON VEICULAR TECNOLOGY, VOL. 59, NO. 7, SEPTEMBER where the optimal beamforming vectors fk are the column vectors of F. The details of proof can be found in [24]. Note that the derived solution F is optimal for the alternative problem in 4), which implies that it does not guarantee its optimality in terms of sum rate. Nonetheless, the solution F is closely related to the optimal beamforming matrix, which maximizes the sum rate since the beamforming vectors tend to align with user feedback if they are optimal. Perhaps surprisingly, the beamforming matrix F indeed provides better sum-rate performance than the iterative algorithm [8], as will be shown in numerical results. Furthermore, the beauty of this solution is that it can be obtained with only simple matrix computations. One SVD and one matrix multiplication are all required computations, which can be implemented within the computational complexity of ON 3 T ). Thus, the solution enables extremely low-complexity construction of the beamforming matrix at the BS. D. User Data Decoding For coherent data decoding, the selected users require the beamforming vectors f k. As presented in the system model, the information can be conveyed in the feedforward channel or broadcast by a dedicated pilot. Given the information of the beamforming vector, the LMMSE receiver of the kth user is 4 f k r k = f k k k + I NT k ). 8) ρ k k k + I NT k The postprocessing SINR of the kth user after receive combining is expressed as SINR k = f k Finally,thesumrateR can be computed as N T R = log + SINR k ) k= N T = log fk k=. 9) k k + I NT fk ρk k k + I NT fk. 0) IV. NUMERICAL RESULTS In this section, the average sum-rate performance of the proposed unitary beamforming system is evaluated and compared with conventional beamforming schemes. Simulation results are averaged over 000 independent channel realizations. In Fig. 3, the average sum-rate performance versus SNR is plotted for =4BS antennas, N R =2 MS antennas, K =4users, and B =4bits for user feedback. The proposed unitary beamforming system outperforms previous schemes, such as PU 2 RC with unitary codebook [4], PU 2 RC with improved codebook [3], ZFBF-QBC [], ZFBF-MESC [2], and the enhanced unitary beamforming [8] scheme in all SNR regions. Since the user Fig. 3. Average sum rate versus SNR, where =4, N R =2, K =4, and B =4. pool is not large enough to efficiently match users to predefined beamforming vectors, PU 2 RC shows low sum-rate performance. Although PU 2 RC with an improved codebook increases the sum-rate performance over PU 2 RC with a unitary codebook, PU 2 RC with an improved codebook still suffers from ineffective matching between the users and the predefined beamforming vectors in a small userpool environment. On the contrary, the proposed unitary beamforming system achieves high performance gain by freely constructing a unitary beamforming matrix based on MS feedback, which results in higher sum-rate performance than PU 2 RC. ZFBF combined with the QBC technique [] or the MESC technique [2] alleviates performance degradation resulting from channel-quantization error by effectively using multiple receiving antennas. owever, ZFBF shows low sumrate performance due to channel-quantization error, particularly in a low-rate feedback. In addition, ZFBF becomes inefficient in the low-snr region by its nature. Compared with the enhanced unitary beamforming scheme [8], the proposed unitary beamforming system provides higher average sum rate, particularly in the high-snr region. This is because the effect of interuser interference at the MS is not considered in [8]. In the proposed unitary beamforming system, information on both the user channel state and the effect of interuser interference is compressed and reported through a low-rate feedback channel. Therefore, the BS can effectively mitigate interuser interference, even with limited feedback information. Fig. 4 shows the average sum-rate variations, along with the number of users when the number of users exceeds the number of transmit antennas. The proposed unitary beamforming systems that adopted maximum-sinr user selection and greedy user selection are compared with PU 2 RC with a unitary codebook [4], PU 2 RC with an improved codebook [3], ZFBF-QBC [], 5 and ZFBF-MESC [2]. In both cases, the proposed systems outperform the PU 2 RC, ZFBF-QBC, and ZFBF-MESC schemes, particularly in the presence of small and medium numbers of users. In detail, the proposed system adopting the greedy user-selection method outperforms the method using maximum-sinr user selection. This is achieved because the greedy user-selection process jointly utilizes both the preferred beamforming 4 From [20], the LMMSE combining receiver of the kth user can be derived as the kth row vector of / )F k kf + I NT F k. Since F F = FF = I NT, the LMMSE combining receiver is transformed as 8). 5 The user-selection method for ZFBF-QBC [] has not explicitly been discussed. ence, we also use the conventional greedy user-selection algorithm in [25] for user selection.

5 3644 IEEE TRANSACTIONS ON VEICULAR TECNOLOGY, VOL. 59, NO. 7, SEPTEMBER 200 Fig. 4. Average sum rate versus the number of users, where =4, N R =2, B =4, and SNR =0dB. Fig. 6. Average sum rate versus the number of users, where =4, N R =2, B =8, and SNR =0dB. systems. ZFBF combined with the QBC technique or the MESC technique tends to achieve a higher sum rate in high SNR regions than the proposed scheme as the feedback bits increase. This behavior of the proposed SVD-based unitary beamforming scheme results from the constraint that the beamforming matrix should be a unitary matrix and is consistent with that of general unitary beamforming schemes in [2], [4], and [8]. In Fig. 6, we plot the simulation results for a system with =4 BS antennas, N R =2MS antennas, and B =8bits for user feedback. In this case, we assume that the SNR is equal to 0 db. The proposed unitary beamforming scheme outperforms PU 2 RC, ZFBF-QBC, and ZFBF-MESC in the presence of small numbers of users. In high-rate feedback, ZFBF-MESC achieves a higher sum rate than the proposed scheme as the number of users increases, because it is reasonable to apply assumptions in [2] that the selected users are almost orthogonal to each other. Fig. 5. Average sum rate versus SNR, where =4, N R =2, K =4, and B =8. feedback vector information c πk) and the SINR information η k at the cost of increased computational complexity when selecting users. In addition, Fig. 4 indicates that PU 2 RC with an improved codebook [3] rapidly boosts the average sum rate as the number of users increases because PU 2 RC with an improved codebook can efficiently compensate for performance degradation from ineffective matching between users and predefined beamforming vectors in the medium or large user pool. In Figs. 5 and 6, the average sum-rate performance is evaluated when a high-rate feedback is employed at a multiuser MIMO system. In Fig. 5, a system with =4BS antennas, N R =2MS antennas, K =4users, and B =8bits for user feedback is considered. Note that the feedback bits of PU 2 RC are fixed for 4 bits because, in highrate feedback, PU 2 RC suffers from considerable performance loss due to ineffective matching between users and predefined beamforming vectors in a small user-pool environment, which results in significant decreasing multiplexing gain. The proposed SVD-based unitary beamforming scheme achieves higher sum-rate performance than the enhanced unitary beamforming scheme. In addition, the proposed scheme requires extremely low computational complexity, compared with the enhanced unitary beamforming scheme. As a result, the proposed scheme is highly appropriate for practical multiuser MIMO V. C ONCLUSION In this paper, a new low-feedback-rate and low-complexity system for downlink MU-MIMO has been proposed. The main features of the proposed system are that both the user channel state and interuser interference effect information are effectively conveyed, even in a low feedback rate, and the beamforming matrix is computed with extremely low computational complexity by using SVD. Our numerical results have verified that the proposed system achieves higher sum-rate performance than previous MU-MIMO systems, such as the PU 2 RC, ZFBF-QBC, ZFBF-MESC, and enhanced unitary preprocessing schemes. Based on the aforementioned features and the numerical results, the proposed system appears highly appropriate for practical MU-MIMO systems. REFERENCES [] T. Yoo and A. Goldsmith, On the optimality of multiantenna broadcast scheduling using zero-forcing beamforming, IEEE J. Sel. Areas Commun., vol. 24, no. 3, pp , Mar [2] M. Sharif and B. assibi, On the capacity of MIMO broadcast channels with partial side information, IEEE Trans. Inf. Theory, vol. 5, no. 2, pp , Feb [3] Q.. Spencer, A. L. Swindlehurst, and M. aardt, Zero-forcing methods for downlink spatial multiplexing in multiuser MIMO channels, IEEE Trans. Signal Process., vol. 52, no. 2, pp , Feb [4] Z. Pan, K. K. Wong, and T. S. Ng, Generalized multiuser orthogonal space-division multiplexing, IEEE Trans. Wireless Commun., vol. 3, no. 6, pp , Nov

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Signal Process., vol. 53, no. 0, pp , Oct On Discrete-Time Modeling of Time-Varying WSSUS Fading Channels Christian Sgraja, Jun Tao, Student Member, IEEE,and Chengshan Xiao, Fellow, IEEE Abstract The general property of noncommutativity of linear timevarying LTV) systems has to be considered when a digital transmission system composed of LTV channel and transmit/receive filtering shall be represented in the equivalent discrete-time domain. In this correspondence, we analyze the implications of serially concatenated LTV systems with respect to the discrete-time representation of wide-sense stationary uncorrelated scattering WSSUS) fading channels. We provide a detailed discussion of the largest tolerable channel Doppler spread in relation to the system bandwidth) for which the concatenation is commutative in good approximation. In this case, the discrete-time description facilitates an efficient simulation model. The approximation accuracy is quantified by a detailed error analysis and further verified by numerical simulation adopting Universal Mobile Telecommunications System UMTS) parameters. Index Terms Approximation methods, discrete-time systems, fading channels, system modeling, time-varying systems. I. INTRODUCTION As to compactly represent and efficiently simulate a digital transmission over linear time-varying LTV) wide-sense stationary uncorrelated scattering WSSUS) fading channels, one can take advantage of focusing on the discrete-time digital) domain. Discrete-time simulation models for both single-input single-output SISO) [] [4] and multiple-input multiple-output MIMO) channels [5] [7] have previously been proposed, together with corresponding methods for an efficient generation of the Rayleigh fading process. The discretetime model integrates the LTV physical channel, transmit and receive filtering, digital/analog conversion, and receiver-side sampling into an equivalent discrete-time channel response, which, in general, loses the property of uncorrelated scattering US) due to the inclusion of transmit and receive filtering []. In contrast, the wide-sense stationarity WSS) of the channel is preserved as also verified in this paper). Throughout, we further adopt the common assumption that the 2-D channel scattering function is decomposable into the Doppler spectrum and power delay profile [3]. In this paper, we highlight and discuss the fact that, similar to the U.S. property, the separability of the channel scattering function is, in general, also not preserved in the associated equivalent discrete-time model. owever, in cases where the so-called jointly underspread condition [8] is fulfilled, which translates into the requirement that the channel is virtually constant during receive filtering or, equivalently, the Doppler spread is small, compared with the system bandwidth), the discrete-time system response admits a simplified representation. The Manuscript received October 2, 2009; revised March 5, 200; accepted May 5, 200. Date of publication June 28, 200; date of current version September 7, 200. The work of J. Tao and C. Xiao was supported in part by the National Science Foundation under Grant CCF The review of this paper was coordinated by Prof. M. D. Yacoub. C. Sgraja is with Qualcomm CDMA Technologies Gmb, 904 Nuremberg, Germany csgraja@qualcomm.com). J. Tao is with the Department of Electrical and Computer Engineering, University of Missouri, Columbia, MO 652 USA jtb84@mail.missouri.edu). C. Xiao is with the Department of Electrical and Computer Engineering, Missouri University of Science and Technology, Rolla, MO USA xiaoc@mst.edu). Digital Object Identifier 0.09/TVT /$ IEEE