The Role of Transmit Diversity on Wireless Communications Reverse Link Analysis With Partial Feedback

Transcription

1 2082 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL 50, NO 12, DECEMBER 2002 The Role of Transmit Diversity on Wireless Communications Reverse Link Analysis With Partial Feedback Vincent K N Lau, Senior Member, IEEE, Youjian Liu, Member, IEEE, and Tai-Ann Chen, Member, IEEE Abstract Transmit diversity through multiple transmit antennas is generally regarded as beneficial to the link-level performance In this paper, we shall investigate the role of transmit diversity with respect to the link level and the system-level performance We focus on the reverse link analysis, where mobile users are assumed to have independent fading, and they are equipped with multiple transmit antennas ( ) Each mobile station is assumed to have an average power constraint The base station is assumed to have receive antennas We consider two levels of partial feedback, namely, scalar feedback and per-antenna vector feedback In both cases, the transmitter does not have full knowledge of the channel matrix Based on the information theoretical analytical model, it is shown that transmit diversity could enhance link-level performance, but is harmful to the multiuser system performance when there is insufficient feedback information Index Terms Multiuser multiple-input multiple-output (MIMO) scheduling, space time scheduling, transmit diversity I INTRODUCTION AS COMPUTER and telecommunication services start to merge in the wired domain, there is a high demand for high-speed wireless multimedia service in the near future Traditionally, the achievable bit rate of a communication link is limited by the available bandwidth and power Given a fixed power budget, the only way to increase the bit rate is to increase the bandwidth of transmission Recently, another interesting approach to increasing the bit rate without increasing the bandwidth or power budget is to make use of multiple transmit and receive antennas This transforms the channel into a multiple-input multiple-output (MIMO) system It is shown [6], [8] that the channel capacity of MIMO systems is proportional to,, where is the number of transmit antennas and is the number of receive antennas For quasi-static fading channels, it is shown that diversity order is the key to link-level performance, because the asymptotic error rate is inversely proportional to, where is the diversity order achieved Hence, if we fixed the number of receive antennas so as to isolate the role of the transmit antennas, the link-level diversity order is enhanced by increasing the number of transmit antennas [4], [12] [15] A number of contributions are currently proposed to boost the data rate of universal mobile telecommunications Paper approved by J Wang, the Editor for Modulation Detection and Equalization of the IEEE Communications Society Manuscript received March 12, 2002; revised June 15, 2002 The authors are with Bell Laboratories, Lucent Technologies, Whippany, NJ USA (knlau@ieeeorg; eugeneliu@ieeeorg; tac@ieeeorg) Digital Object Identifier /TCOMM systems (UMTS) to 108 Mb/s utilizing the MIMO techniques based on transmit diversity techniques [7] On the other hand, since the source for high-speed data is usually very bursty, packet scheduling is an important component in the communication systems, in addition to the physical layer In the context of the data-oriented third-generation wireless systems and their evolution, the pursuit of high spectral efficiency and very high average throughput in the system level is the main theme Therefore, cross optimization between the scheduling layer and the link layer is very important to the system performance [1] [3] Traditional design methodologies followed the layering approach, where individual layers are optimized separately This results in suboptimal design, especially in the wireless domain, where the physical channel is time varying It has been shown in [17] and [18] that significant performance gain could be achieved by a joint design between the physical layer and the media access control (MAC) layer in wireless systems In fact, achieving physical layer optimization does not always imply achieving system-level optimization, especially for bursty traffic in time varying physical channels It is found that selection diversity among different users is the key to the performance gain in scheduling Intuitively, to have a large scheduling gain, we should have physical links with high fluctuations in channel conditions, so that individual scheduling decisions could exploit the time varying fading physical links In this paper, the role of transmit diversity in wireless multiuser communication systems is investigated We shall focus on the reverse link, which is the direction from mobile station to base station We assume mobile devices are equipped with transmit antennas, while the base station is equipped with receive antennas Mobile devices are under an average power constraint We consider two levels of partial feedback, namely scalar power feedback and per-antenna vector power feedback In both cases, the transmitter does not have full knowledge of the channel matrix Therefore, eigenbeamforming could not be performed Based on the information theoretical analysis, it is found that while transmit diversity could enhance the physical layer performance, increasing the number of transmit antennas is harmful to the system performance when feedback is not sufficient This paper is organized as follows In Section II, we shall outline the system model and the channel model We shall formulate the scheduling problem of the distributed multiuser MIMO system based on the information theoretical approach In Section III, the partial feedback model is introduced and the optimal system capacity is evaluated in terms of for various /03$ IEEE

2 LAU et al: THE ROLE OF TRANSMIT DIVERSITY ON WIRELESS COMMUNICATIONS 2083 perfect channel estimation at the receiver 1 At the receiver, there is a random space time decoder,, which maps the received signal vectors to the bit streams The bit rate of the MIMO link between the th user and the base station is given by bits per channel use, which is dependent on the channel matrix realization for the current frame A decoding error occurs if We assume that scalar-rate feedback information is available to the transmitter for rate adaptation, achieving ergodic capacity instead of outage capacity The rate-feedback information is computed at the receiver based on the instantaneous channel matrix realization For simplicity, we assume noiseless feedback, and this scalar-rate feedback channel is implicitly assumed for the rest of the paper Fig 1 Block diagrams of the system model feedback levels The link-level capacity is a special case of the multiuser capacity In Section IV, we present the numerical results and discuss the role of transmit antennas with respect to the link-level performance and system-level performance Finally, we conclude with a brief summary of results in Section V II DISTRIBUTED MIMO SYSTEM MODEL We consider a communication system with mobile users, each has transmit antennas All the mobile users communicate to a base station with receive antennas, as illustrated in Fig 1 For simplicity, we assume the channel between a mobile user and the base station is modeled by independent flat quasi-static fading, where the channel realization is constant over a frame of coded symbols, but independent between different frames of coded symbols At the mobile transmitter, we have a random space time encoder followed by a Gaussian constellation space time modulator The cascade of the space time encoder and the modulator forms a mapping, III EVALUATION OF OPTIMAL SYSTEM CAPACITY In this section, we shall quantify the link-level performance as well as the scheduling-level performance using the information theoretical approach In this paper, denotes a scalar, and denotes a matrix In the case of a random matrix, denotes a special realization, while denotes the random matrix itself denotes a concatenation of submatrices Given a set of channel matrices,, where is a realization of the channel matrix between user and the base station, the capacity region is well known to be [20] with some input distribution maximizing the mutual entropy where The capacity achieving distribution for is shown [8] to be circular symmetric complex Gaussian distribution with denoting the covariance Note that denotes the Hermitian transpose Therefore, we have (3) where is the frame length, is the message index, is the bit rate per channel use, and is the modulation symbol taken from the set of complex numbers The vector of received signals,, after matched filtering is given by (1) where (4) (2) where is the transmitted symbol vector for user with average transmitted power, and is the random channel matrix (linking user and the base station) with independent, identically distributed (iid) complex random variables of unit variance,, as the matrix elements is the noise vector of normalized variance of We assume and 1 Channel estimation is not a focus of this paper In practice, if the channel varies much more slowly than the data rate, channel estimation could be done accurately through preamble transmission before payload transmission The amount of overhead involved in preamble transmission is negligible compared to the payload size

3 2084 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL 50, NO 12, DECEMBER 2002 A Partial-Feedback Model In general, when there is full feedback available to the transmitter (client user), the optimal transmission strategy would be a cascade of beamforming matrix and temporal power water filling The input covariance matrix is given by average power constraint in (8) so that the ergodic capacities,, is maximized where Introducing the set of Lagrange multipliers,, the problem could be reformulated as (5) (9) where is the unitary beamforming matrix and is the transmitted power on the th eigenchannel Both and are functions of the channel matrix Therefore, in general, the feedback link capacity (in terms of number of quantities to feedback) to realize the above transmission strategy is not scalable with respect to,, and In this paper, we consider a partial-feedback model, scalar feedback, 2 as below By scalar feedback, we assume each client user is supplied with a scalar information from the base station, where is a real and continuous mapping from to a real number 3 The transmission strategy of a client user is given by a power control matrix Moreover, the entries of the power control matrix,, are constrained to be a linear function of the scalar feedback That is, where is independent of the feedback (6) IV PERFORMANCE ANALYSIS In this section, we shall derive the optimal system capacity and link-level capacity as a special case at two feedback levels We shall consider the cases of scalar power feedback and per-antenna power feedback These two cases are regarded as partial feedback, because, in general, the transmit does not have full knowledge of the channel matrix Hence, eigenbeamforming could not be performed at the transmitter In this paper, we shall assume the total number of client users for system-level performance A Scalar Power Feedback We first consider a case when the th mobile device has a scalar power feedback channel,, from the base station as illustrated in Section III-A The mobile transmitter has antennas, while the base station receiver has antennas, and there are such users The concatenated channel matrix is given by (10) B Problem Formulation The average capacity region,, is then characterized by if, where the expectation is over the random channel matrices, Since mobile station has an average power constraint of,wehave To obtain the optimal multiuser capacity, we have to solve the following optimization problem Problem 1: Given any channel matrices realization,,we have to find the optimal power control policy, satisfying the 2 To simplify the analysis, the feedback capacity requirement is expressed in terms of differential entropy (rather than the absolute capacity) That means although the absolute capacity of the feedback channel (required to transmit one real number) is still infinite, there is a relative capacity associated with the feedback In other words, the amount of relative feedback capacity required for the scalar feedback is smaller than that for the vector feedback 3 Although one could theoretically feed back the quantized channel matrix through such a real number channel, our intention is to constrain the feedback on a single parameter only Hence, feeding back the quantized matrix is regarded as feeding back 2n n parameters instead of the single parameter This is not allowed in our partial-feedback constraint (7) (8) where is an matrix representing the channel states from all the transmit antennas of user to all the receive antennas at the base station The concatenation of the transmitted symbol, which is a -dimension matrix Since users do not have interactions, the conditional covariance matrix of the aggregate input distribution is given by (11) where is the conditional variance of the input distribution of user, satisfying the average power constraint, and is the scalar power feedback The capacity-achieving input distribution is circular symmetric Gaussian with covariance matrix and is given by the following lemma Lemma 1: The optimal conditional covariance matrix of user in the presence of a scalar feedback channel defined in Section III-A,, is given by p D (12)

4 LAU et al: THE ROLE OF TRANSMIT DIVERSITY ON WIRELESS COMMUNICATIONS 2085 where for normalization and is a function of the scalar power feedback, satisfying the constraint Hence, the optimization problem of the system is to find the optimal power-control policy,, so that is optimized 4 (13) This could be proved by expanding as, where and are independent of the feedback Note that could be interpreted as the transmit antenna power allocation configuration Furthermore, asymmetric transmit antenna power allocation is equivalent to symmetric transmit antenna power allocation with a transformed channel matrix given by h h h D Complete optimization, therefore, requires optimization with respect to the set of power control policies,as well as the power allocation coefficient across antennas As a result of the above, and the fact that the optimization procedure below does not depend on the distribution of, the general optimization problem is first solved with respect to in terms of, based on the symmetric power allocation on the input covariance matrix with a transformed channel matrix, The results will be further optimized with respect to in Lemma 1 Given a realization of the concatenated channel matrix, the capacity region of the system in (7) is characterized by (15) where is chosen to satisfy the average power constraint, The solution is summarized in the following algorithm Theorem 1: Please refer to Appendix A for proof Given a channel matrix realization,, let be the set of user indices with nonzero power allocation We have the cardinality of the set, The algorithm to obtain the set is based on some nonlinear optimization techniques and is outlined below Step 1) Let and Set Step 2) Set Set Step 3) If argmax and, then (16) and where and is given by the solution of the equations below for and is given by (17) (18) (14) where is the rank of, with the nonzero eigenvalues of, and is the corresponding eigenvector matrix which is partitioned as follows: Repeat Step 3 If there is no such, procedure ends with obtained above and the optimal power allocation is given by in (17) If, the optimal power allocation is Therefore, the optimal power allocation strategy is to perform water filling across the spatial and temporal domain At any instant, at most users should be allocated nonzero power For illustration, assume In this case,, and we have the rank of the matrix equals one (ie, ) Furthermore, the nonzero eigenvalue,, is given by and is the norm of the vector Considering users together, and dropping the subscript of to simplify notation The rank of the matrix is, and hence, there are only nonzero eigenvalues There are, at most, nonzero terms in the summation of (14), and without loss of generality, assume The corresponding normalized eigenvector,, is given by 4 The optimization problem belongs to the class of max-determinant problems, where there is no analytical solution, in general In this paper, we approximate the solution with a reasonably tight bound in (14) This bound is asymptotically tight as n!1

5 2086 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL 50, NO 12, DECEMBER 2002 Hence, the optimal system capacity (per user) after power water filling is given by (19) where, and is the distribution of given by (20) where and are the cumulative distribution function (cdf) and the probability density function (pdf) of the channel matrix, respectively, and is the Lagrange multiplier chosen to satisfy the average power constraint of user, We found that in case of insufficient feedback, the optimal power allocation strategy for the system-level perspective is different from that for the link-level perspective This is summarized in the lemma below (proved in Appendix B) Lemma 2: For sufficiently large SNR and large, the optimal power allocation across the transmit antennas when we have scalar power feedback is given by (a) for link-level performance, for system-level performance, Therefore, from a link-level perspective, transmit diversity enhances link capacity by transforming the fading statistics to additive white Gaussian noise (AWGN) statistics and therefore, the optimal power allocation strategy is to uniformly allocate transmit power across all antennas On the other hand, from a system-level perspective, multiuser selection diversity is of prime importance, and therefore, the optimal power allocation strategy is to focus transmit power on a single transmit antenna only Fig 2(a) illustrates the optimal system capacity versus the number of transmit antennas, It is shown that the system performance degrades as increases This is attributed to the fact that as increases, the randomness of in reduces, and in the limit as,wehave by law of large numbers Since the random variable is given by the maximum of independent components of, the distribution of skews toward a delta function at a lower mean of ] as increases, as illustrated in Fig 2(b) This illustrates that transmit diversity actually hurts the system performance in the presence of scalar power feedback channel, because it reduces the randomness of the channel with respect to each user, and therefore, suppresses the selection diversity among users or the scheduling gain On the other hand, if we consider the link-level performance, we see that the link-level capacity improves as the transmit diversity increases This is illustrated in Fig 3 In general, for, similar observations on transmit diversity apply Say we have (client) and (base station) From the results of this paper, it is more effective Fig 2 (b) Role of transmit diversity at the system level with scalar power feedback Note that although the optimal power allocation strategy is to focus power on a single transmit antenna only, for illustration, the label n = 8 refers to uniform power sharing across eight transmit antennas to transmit independent data streams from multiple users (four users at a time) than to transmit an independent data stream from multiple antennas of the same user (due to the scalar feedback constraint) This is because for the former strategy, one could enjoy the multiuser diversity by selecting good users to form the distributed 4 4 configurations For the latter case, one does not have a choice over the antennas of a client user (due to insufficient feedback), and therefore, do not enjoy the multiuser diversity If we follow the former strategy, four good users are selected and a distributed 16 4 configuration is formed Obviously, only four spatial channels are available (contributed by the four independent users), and additional transmit antennas within a user only contributed to the transmit diversity That is why one should focus power on one single transmit antenna of the client user only, to reduce the transmit diversity so as to enhance the randomness for multiuser selection diversity The result is illustrated in Fig 4

6 LAU et al: THE ROLE OF TRANSMIT DIVERSITY ON WIRELESS COMMUNICATIONS 2087 Fig 3 Role of transmit diversity in link-level performance with scalar power feedback (a) (b) Fig 4 Effect of transmit diversity (n ) in system performance with scalar power feedback (n =2) B Per-Antenna Power Partial Feedback Next, we shall consider the case when the mobile user has scalar power feedback channels from the base station This is regarded as partial feedback, because, in general (when ), the mobile user does not have full knowledge of the channel matrix Similarly, the transmission strategy of user is given by (6), where the entries of the power matrix,, is a function of the -dimension feedback Following a similar argument as in Lemma 1, the equivalent input covariance matrix for user is given by where is chosen to satisfy the average power constraint Fig 5 System-level and link-level performance versus transmit diversity with per-antenna power feedback Hence, when we have per-transmit antenna power feedback, the optimal resource allocation strategy in the MIMO system would be to split the user into virtual users and allocate power on the virtual user basis based on the algorithm in Theorem 1 Similarly, consider as an example for illustration The optimal system capacity is given by (19), with given by (21) Figs 5(a) and (b) show the system- and link-level performance of a MIMO multiuser system with per-antenna power feedback Observe that as increases, the scheduling performance actually enhances This is because with per-antenna power feedback, each mobile user could perform water filling across different antennas Hence, the randomness of the component in is not suppressed when increases, and therefore, the scheduling diversity is not compromised when we have per-antenna power feedback

7 2088 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL 50, NO 12, DECEMBER 2002 V CONCLUSION In this paper, we have investigated the role of transmit diversity in the reverse link of multiuser systems with multiple transmit and receive antennas We found that with scalar power feedback, the transmit diversity enhances the link-level capacity, but hurts the scheduling-level capacity This is due to the fact that the transmit diversity suppresses fluctuation or randomness of channels per user This is good from a link-level perspective, but bad from a scheduling point of view, because the key gain in scheduling is attributed to the multiuser selection diversity On the other hand, at per-antenna power feedback, the transmit diversity becomes beneficial to both the link-level performance and the system-level performance This is attributed to the fact that transmit diversity in the presence of per-antenna feedback enhances the signal-to-noise ratio (SNR) of the received signal without suppressing the randomness or fluctuation in a statistical sense Hence, the scheduling gain is not harmed (23) (24) Consider the th subproblem in (24) Let us define the index set for the th subproblem,, as the set of user indices with the minimum APPENDIX A PROOF OF THE OPTIMAL MULTIUSER MIMO SCHEDULER We shall first establish the first part of the theorem by proving Observe that the original capacity region consists of superposition of terms in the summation of (14) However, the maximization problem could not be split into subproblems, because of the coupling between the subproblems through the common arguments, We first separate the original problem into subproblems by introducing additional variables as follows: Let, the capacity region, becomes The optimal received power vector for the th subproblem has nonzero entry only when Note that the set is a function of the Lagrange multipliers In other words, for However, from (22), we have the constraint that for all, the solution is consistent iff is selected such that This is equivalent to the condition for all, and for all This is again equivalent to the following set of equations: Splitting as, where is the th capacity subregion Hence, the original capacity region,, could be split into independent subregions, where the th subregion depends only on the variables The additional constraint relating the interaction between the eigenchannels is given by (22) This introduces additional Lagrange multipliers into the optimization problem Finally, for the average power constraint term on, we split into Hence, the original optimization problem is written as and (25) (26) The number of unknowns in (25) and (26) is given by, and the number of equations in the system is given by Note that with probability one, the set of equations in (25) will be inconsistent when the number of unknowns is larger than the number of equations Hence, for consistent solution of, we must have, which implies Therefore, the number of nonzero optimal is, at most, Going back to the original problem in terms of, this implies that the maximum cardinality of the set of user indices with nonzero,, is, at most,

8 LAU et al: THE ROLE OF TRANSMIT DIVERSITY ON WIRELESS COMMUNICATIONS 2089 APPENDIX B OPTIMAL POWER ALLOCATION ACROSS ANTENNAS When, the closed-form solution is given by, where is given by the maximum selection of random variables Observe that the mean of the random variables within the maximizing function is always, irrespective of On the other hand, the variance of the th random variable is given by For link-level performance, and there is no selection diversity in In that case, the link capacity is maximized when the variance is minimized, which is achieved on uniform power allocation On the other hand, for system-level performance with, wehave selection diversity in and therefore, the system capacity is maximized when the variance is maximized, which is achieved by focusing power on a single antenna When 1, no closed-form solution is available However, observe that the capacity region is determined by the matrix w h h, where h h D Hence w h h We notice that the mean of all diagonal terms of is equal to one, and the mean of all off-diagonal terms of for all Furthermore, the variance of all off-diagonal terms for all In other words, only the variance of the diagonal terms of depends on in the same way as above Therefore, the system performance, with respect to power allocation across transmit antennas, could be characterized by a variance of the diagonal terms of It is shown numerically that for, minimum variance gives the maximum link capacity because of the absence of selection diversity When, maximum variance gives the maximum system capacity due to selection diversity in REFERENCES [1] A Naguib, N Seshadri, and A R Calderbank, Increasing data rates over wireless channels, IEEE Signal Processing Mag, pp 76 92, May 2000 [2] D Tse and S V Hanly, Multiaccess fading channels Part I: Polymatroid structure, optimal resource allocation and throughput capacities, IEEE Trans Inform Theory, vol 44, pp , Nov 1998 [3] R Knopp and P A Humblet, Information capacity 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Digital Modulation and Coding, 1st ed Englewood Cliffs, NJ: Prentice-Hall, 1996 [55] John G Proakis, Digital Communications, 3rd ed New York: Mc- Graw-Hill, 1995 [56] Wm C Jakes, Mobile Radio Systems, 1st ed New York: McGraw- Hill, 1974 Vincent K N Lau (M 97 SM 01) was born in Hong Kong He received the undergraduate degree in electrical and electronics engineering from the University of Hong Kong with distinction (first-class honors) in 1992, and the PhD degree from Cambridge University, Cambridge, UK in 1997 He joined Lucent Technologies, Whippany, NJ, as a Member of Technical Staff in 1997 He left Lucent and joined the University of Hong Kong in 1999 as an Assistant Professor, and was later appointed the Codirector of Information Engineering Programmes as well as the Codirector of the 3G Wireless Technology Center He left the University and rejoined Lucent in July 2001 His research interests include digital transceiver ASIC design, communication theory with feedback, space time scheduling, iterative decoding and equalization, space time multiuser detection, power control and CREST factor control algorithms, and jointly adaptive multiple-access protocols, as well as short-range wireless ad-hoc networking (Wireless LAN and Bluetooth) His other interests include FPGA and ASIC prototyping, digital hardware design, digital signal processing, and microcontroller firmware development over various platforms He has published more than 60 papers in IEEE conference proceedings and journals, and is the author of a book chapter on wideband CDMA technologies In addition, he has five US patents pending Dr Lau was the invited Session Chair of the IEEE WCNC 2000 International Conference, the IEEE CAS 3G Workshop 2000, and the SCT 2001 International Conference He received two Best Paper Awards from the Institution of Electrical Engineers and the Hong Kong Institute of Engineers (HKIE) in 1999 Youjian (Eugene) Liu (S 98 M 01) was born in Kunming, Yunnan, China He received the MS and PhD degrees in electrical engineering from Ohio State University, Columbus, in 1998 and 2001, respectively, the MS degree in electronics from Beijing University, Beijing, China, in 1996, and the BE degree in electrical engineering from Beijing University of Aeronautics and Astronautics, Beijing, China, in 1993 Since January 2001, he has been a Member of the Technical Staff in CDMA System Analysis and Algorithms Group, Wireless Advanced Technology Laboratory, Lucent Technologies, Whippany, NJ His research interests include communication, coding theory, and information theory Tai-Ann Chen (S 96 M 98) received the BSEE and BSIE double-major degrees with the highest honor from National Tsing Hua University, Hsinchu, Taiwan, in 1989, and the MSEE and PhD degrees from the School of Electrical and Computer Engineering, Purdue University, West Lafayette, IN, in 1994 and 1998, respectively He has been with the Wireless Advanced Technology Labs, Lucent Technologies, Whippany, NJ, since 1998 as a Member of Technical Staff He is currently working on the design and analysis of 3G UMTS/CDMA2000 systems His research interests include space time fading channel characteristics, multiple-antenna techniques, physical layer communication theory, and information theory In addition to various paper publications, he has also contributed to three US patents Dr Chen was the recipient of the IEEE Leonard G Abraham Prize Paper Award in 2001 as being the primary author of the selected best paper in the IEEE JOURNAL OF SELECTED AREAS IN COMMUNICATIONS He is a member of the IEEE Communication Society and the IEEE Information Theory Society, and is a Planning Committee Member of the Wireless and Optical Conference (WOCC)