Sum Rate Maximization and Transmit Power Minimization for Multi-User Orthogonal Space Division Multiplexing

Transcription

1 Sum Rate aximization and ransmit Power inimization for ulti-user Orthogonal Space Division ultiplexing Boon Chin Lim, Christian Schlegel and Witold A. rzymień *) Department of Electrical & Computer Engineering University of Alberta; *) also with RLabs Edmonton, Alberta, Canada {bclim, schlegel, Abstract We demonstrate that receive antenna selection (RAS) provides significant increase in the achievable sum rates for multi-user IO wireless downlinks that employ block diagonalization (BD) to achieve orthogonal space division multiplexing (OSD), where each user terminal has one or more antennas. Although dropping one or more receive antennas at a user terminal reduces its capacity and correspondingly, the system sum capacity, udicious RAS improves the proected channel spatial mode gains and provides aitional degrees of freedom to all other terminals within the BD-OSD context. In this way there is mutual benefit to be shared among users when RAS is applied to all users and numerical results show significant sum rate gains despite sum capacity loss due to RAS. In many cases, users with reduced array sizes also enoy increased channel rates. When proected virtual channels are used as a means of spatial mode allocation, this RAS concept is also beneficial and may be referred to as spatial mode selection (SS). RAS/SS is therefore a necessary first step in any resource allocation and power control exercise for BD-OSD. Further, the same RAS/SS algorithms for sum rate maximization also provide a systematic means of resource allocation and power control. o avoid exhaustive RAS search, which has exponential complexity, efficient RAS algorithms with linear complexity and near optimal performance are proposed. eywords- ulti-user IO, downlink beam-forming, antenna selection, spatial mode selection. I. INRODUCION For the downlink of a wireless base station equipped with multiple antennas where coordination is feasible among the transmit chains but not among the mobile user terminals, simultaneous transmissions to multiple users are possible when channel state information is available at the transmitter. he optimum approach to maximize the downlink sum rate is dirty paper coding [1] [3]. owever, dirty paper coding has very high computational complexity. A reduced complexity suboptimal alternative is beamforming where each user s stream is coded independently and multiplied by a beamforming weight vector for transmission via multiple antennas. Beamforming has been shown to achieve a large fraction of the capacity in multi-user systems when the number of users is large [4] [5], despite its reduced complexity. owever, the determination of optimal weight vectors is still a tedious non-convex optimization problem. A sub-optimal beamforming technique is zero-forcing beamforming (ZFBF) where the weight vectors are chosen to enforce zero co-channel interference (CCI) among all users. When users are equipped with single-antenna terminals, transmit zero-forcing beamforming can be implemented using channel inversion. When each user terminal has multiple antennas however, creating parallel channels with zero CCI at the same terminal is sub-optimal since each terminal is able to coordinate the processing of its receivers. In this case, techniques such as vertical Bell Laboratories layered space-time (V-BLAS) or singular value decomposition (SVD)-based techniques can be used to improve throughput. It would therefore be better to impose orthogonality between users only and not between antennas located at the same terminal. his is commonly referred to as block diagonalization (BD) and it is one way of achieving orthogonal space division multiplexing (OSD). Examples of BD-OSD schemes are found in [6] [8]. In this paper, we demonstrate that receive antenna selection (RAS) has significant impact on increasing the achievable sum rates and minimizing the average user transmit power for BD-OSD systems. his demonstration is focused on the non-iterative BD-OSD schemes from [6] and [7] and the iterative scheme from [8]. BD-OSD generally employs null space proection techniques to achieve orthogonality between user terminals. his creates parallel single-user IO channels with zero CCI among them. Optimal beamforming with SVD-based techniques can then be done at each user to maximize its channel rate. Despite this, udicious RAS yields antenna or spatial mode subsets that substantially increase the achievable sum rate even though sum capacity loss is present due to RAS. For those BD schemes in [6] and [7] that operate directly on the channel matrix, termed direct-bd (DBD) for convenience, it is interesting to note that users with reduced array sizes due to RAS enoy channel rate increase in many cases. he mechanisms behind this phenomenon are: (a) Judicious RAS reduces correlation among user terminals and improves the spatial mode gains during null space proection; (b) RAS at one terminal increases the degrees of freedom of the BD-OSD proection matrices of other terminals. Specifically, each receive antenna removal at one terminal provides an aitional degree of freedom to all other terminals, which has the effect of aing more transmission resources. In this way there is mutual benefit to be shared among users when RAS is applied to all users and numerical results show significant improvements over a BD-OSD system without RAS. Note that for single-antenna terminals, the RAS process corresponds to finding an active user subset, as described for example in [1]

2 and [9]. Next, the Coordinated ransmit-receive (CR) [6] and the iterative null space directed SVD (Nu-SVD) [8] schemes perform BD on proected virtual channels to provide the flexibility for service to more users via spatial mode allocation. his is generally done by means of appropriately dimensioned receive weight matrices that reflect the number of modes to be activated at each user terminal. In this way, no receive antennas are dropped during mode allocation and better performance results because diversity is preserved. he RAS concept for direct-bd can also be applied to the virtual channels in CR and Nu-SVD to provide substantial gains in the achievable sum rate. In this case, the RAS process is akin to spatial mode selection (SS). Since the RAS/SS process contributes to individual and sum rate increase, it is therefore a necessary first step in any resource allocation and power control exercise. Next, the same RAS/SS algorithms that are used for sum rate maximization also provide a systematic means of resource allocation and power control. his is due to their ability to identify the worst antennas or modes at the overall system level or individual terminal level. Such antennas or modes may then be eliminated with minimum impact on the individual user rates. Very importantly, this process frees resources that were originally committed to lower returns for the benefit of other terminals with better spatial mode gains. his helps raise the channel rates of those users associated with the remaining antennas or modes. ransmit power minimization is realized by lowering the powers to those users with excess rates. he savings may then be given to those users needing more power. he rest of the paper is organized as follows. In Section II, the system model is described. In Section III, the theory and algorithms for applying RAS/SS in BD-OSD are developed. In Section IV, the numerical results are presented and Section V contains conclusions. II. SYSE ODEL AND ASSUPIONS We focus on the multi-user IO downlink of a base station (BS) serving a group of geographically distributed users via spatial multiplexing that is achieved using linear preand post-processing at the transmitter and receiver. he BS has transmit- chains and antennas while each user has one or more antennas ( R ), each coupled with a receive-chain. he total number of receive antennas is R = Σ = 1 R. he overall ( R x ) channel matrix is while each user s R x channel sub-matrix is denoted as. Each data vector d of arbitrary dimension ( m x1) has complex entries and is precoded by a ( x m ) matrix to result in a ( x 1) transmission vector s = d. he overall ( x 1) transmission vector is s = Σ = 1d, the received ( R x1) signal vector y at user is given by (1), and the overall ( Σ = 1 R x 1) received vector y is given by () y = Es d, (1) i 1 i i + n = y= Es d+ n, () where = [ 1 ], = [ 1 ], d = [ d 1 d d ] and n = [ n 1 n n ], where [.] is matrix transposition. Using a post-processing (m x R ) matrix R, estimates of the transmitted data symbols at user are ( ) dˆ = R E d + n. (3) s i = 1 i i his paper assumes a quasi-static, flat fading Rayleigh channel that is constant over several transmission blocks. he entries of are zero mean ointly circular Gaussian with variances scaled by path loss and shadow fading and n is ( R x 1) with covariance E { nn } = N o IR. E s is the total average transmit energy per symbol and E s / is the average energy transmitted from each antenna per channel use. o constrain the total transmit power, R ss = E{ss } must satisfy tr(r ss ) =. Channel state information at the transmitter is assumed available, e.g., via time-division duplex. User scheduling is also assumed done and not covered in this paper. III. ROLE OF RECEIVE ANENNA SELECION IN BD-OSD A. Pertinent Points of BD-OSD We begin by highlighting the pertinent points of block diagonalized orthogonal space division multiplexing (BD- OSD). Due to space constraints, the non-iterative BD scheme from [6] that operates directly on the channel matrix will be used to illustrate the core concepts. We will refer to the scheme simply as direct block diagonalization (DBD). o eliminate cochannel interference (CCI) between users, BD imposes i = 0 for i. he channel rate for such a system with a power constraint is [10] C = max log det I + AR BD tr( Rss ) =, R, R, i = 0, i = 1 I + A Rd tr,, d Rss = = R R, i = 0, i max log det, (4) where A = E s / N o and [.] indicates ermitian transpose. Next, define = [ ], (5) which is actually equal to, except for the absence of. he zero CCI constraint forces to lie in the null space of and one way of finding is via the SVD of (1) = U Σ V V x x x [ ] 1, Ri i i i 1, i Ri = = i= 1, i Ri. (6) Since V forms an orthonormal basis for the row or left null space of and its column vectors can therefore be used as part of the pre-coding matrix of user, i.e., = V P, where P is the other part of the pre-coding matrix to be determined. his form of makes (4) realizable because = diag ( 1V 1 P1, V P,, V P). (7) Note that each pre-coded channel of the form V may be thought of as a proected channel with dimensions R i= 1, i Ri x ( ). (8) he block orthogonalization has created single-user IO channels and the optimal solution for P is then clear via [11], i.e., using SVD( V (1) ), set P = V, where Σ 0 V = U V V 0 0 (1) P [ ] (9)

3 his will then allow direct access to the spatial modes of the proected channels and waterfilling can be done to maximize each user s throughput. ence C = max log det I + AΣ R, (10) BD tr( Rss ) =, R where Σ= diag( Σ1, Σ,, Σ ). Next, the Coordinated ransmit-receive (CR) [6] and the iterative null space directed SVD (Nu-SVD) [8] schemes perform BD on a proected virtual channel e, which is defined as e [ [ R 1 1] [ R ] ]. he post-processing matrices R are appropriately dimensioned according to the desired number of spatial modes to be activated for a user. For CR, the R matrices are labeled as W in [6]. B. Impact of Receive Antenna Selection (RAS) on BD-OSD Judicious implementation of RAS for DBD improves the spatial mode gains of the proected channels P in two ways. First, the removal of antennas with high inter-terminal correlation increases the orthogonality among the user channel sub-matrices. Since DBD achieves zero inter-terminal interference by proecting each into the corresponding null space of, improving the orthogonality among users has the effect of decreasing the degree of orthogonality between the spaces spanned by and null( ). his is advantageous when improved spatial mode gains in the proected channels P = V of the other users result in rate gains that outweigh the rate loss for the user affected by RAS. Second, each receive antenna removal at a particular terminal provides an aitional degree of freedom to all other terminals. For example if one antenna is removed from a user k, then the dimension of the proected channels P of any other user is R R i= 1, i Ri x + 1. (11) he number of columns in P is increased by one and this has the effect of aing more transmission resources to all users other than k. his is in contrast to [7] where increasing is mentioned as a means of aing more resources. o minimize the loss at user k, RAS may be optimally done by exhaustive search over Rk row vectors incurring an exponential search complexity of ( Rk 1). his can be reduced to linear search complexity with Rk steps using RAS algorithms such as [9] and [1]. Noting that the capacity loss for user k arising from the removal of an antenna with high intra-terminal correlation is low in percentage terms, weeding out such antennas throughout the system can result in higher overall sum rates due to (11). When RAS is applied to all terminals, there is mutual benefit to be shared among users and this translates to sum rate increase. Expanding on the second point, we will first examine the impact of performing RAS on one user, both on itself and on the other users. Let the number of antennas at user k be reduced by one. he resulting channel sub-matrix k is then ( R k x ), where Rk = Rk 1. his means rank( k) = Rk 1 and user k s capacity is consequently reduced. he system sum capacity is also correspondingly reduced. If RAS is not performed on any other user, then k = k from (5), which leads to V k = V using (6). k ence the dimensions of user k s proected channel Pk = kv k are ( Rk 1) x ( Σ i= 1, i kri ). he number of rows reduces by one while the number of columns remains the same. Let the singular values of Pk be σ max ( Pk) σ ( Pk) σ min ( Pk ). Since rows( Pk ) columns( Pk ), then [13] σ max ( P k) σ max ( Pk) σ min ( Pk) σ min ( Pk). (1) he singular values of Pk lie between those of the original Pk and hence the total channel power gain Pk F < Pk F, where P tr( P P ) Rk k F k k = Σi= 1 λ i and λ i are the eigenvalues of P P k k. For any other user, the row dimension of is reduced by one and hence the dimensions of P = V are shown in (11). ence, by virtue of a oneantenna reduction in user k, the column dimension of the proected channels of all other users is increased by one. Since rows( P ) columns( P ), then [13] σ ( ) σ ( ) σ ( ) σ ( ). (13) max P max P min P min P Equation (13) shows that the singular values in P may be greater than the original singular values of P. Given that tr ( P P ) > tr ( P P) because of the increased column dimension in P and that rank( P ) = rank( P ), this ensures that σ i( P ) > σ i( P ) will be true for some values of i in (13). In turn, this creates the potential for higher total channel power gain, i.e., P F > P F and hence the potential for a higher system sum rate despite sum capacity loss due to user k. In this respect, aing more columns to a proected channel is like providing more transmission resources for its associated user. When RAS is done on more than one user, it is important to note that (13) also applies to those users with reduced antenna array sizes due to RAS. Let ε k and ε represent the total number of receive antennas eliminated from user k and user respectively. Let { R : = 1,, } be the total number of antennas at each user prior to RAS and hence ε {0,1,, R}. he dimensions of the proected channel = V for user k is then more generally expressed as Pk k k ( = 1, ) ( R k ) ( R ) ε x ε. (14) k k From (14), user k s proected channel will have its row dimension reduced by ε k after RAS is applied on it, while its column dimension may be increased by the amount βk = Σ = 1, k ( ε ) when RAS is performed on other users as well. ence the singular values of those users with reduced array size may still be increased by virtue of (13). In this way there is mutual benefit to be obtained when each user performs RAS to remove those receive-antennas with high correlation. It can be shown that the above analysis extends readily to the CR and Nu-SVD schemes where applying RAS to the proected virtual channel is equivalent to spatial mode selection (SS). An example to illustrate the benefits of RAS/SS is given using DBD and Nu-SVD with a particular channel realization. A system with 8 users, each equipped with 4 antennas is used. able 1 shows the individual and overall sum channel rates (in bits/sec/z) with and without RAS/SS. A RAS algorithm known as JWFAS from [9] is used in this example to perform both RAS and SS. As shown, RAS/SS has substantial impact on the system sum rates. It is interesting to note that in many cases, users with reduced antenna array sizes or reduced spatial mode sets enoy rate increase. Note also that the rate loss for Users # and #5 in the DBD scheme is

4 able 1. utual benefit arising from RAS/SS User #1 # #3 #4 #5 #6 #7 #8 otal DBD without RAS #Ants Rate DBD with RAS #Ants Rate Nu-SVD without RAS #Ants Rate Nu-SVD with RAS #Ants Rate not large even though antennas were removed. his demonstrates the mutual benefit effect when udicious RAS/SS is performed across the system and that it is done in a way that minimizes rate loss to those users that are dropping antennas or modes. Note that RAS/SS may be done at a global system level or at the local user level. Better solutions arise from global search since oint maximization is done rather than localized maximization. Note also that Nu-SVD is the same as DBD when all modes are activated without RAS/SS. As expected, Nu-SVD with SS performs better than DBD with RAS since all receive antennas are utilized. C. Receive Antenna & Spatial ode Selection Algorithms his section considers the possible RAS/SS algorithms for DBD, CR and Nu-SVD. Beginning with DBD, we note that a one-to-one correspondence between the spatial modes of each user and its receive antennas is not apparent. ence, even though one could associate the weakest overall spatial mode with a particular user, the choice of antenna de-selection is unclear. he use of antenna selection algorithms developed for single-user IO systems can be considered for this purpose. In the BD-OSD context, these algorithms can operate on the composite channel matrix, which is formed by appending all user channel sub-matrices RAS algorithms perform better than incremental ones because it approaches the selection problem globally. he decremental RAS algorithm in [1] has fairly high computational complexity at O( 5.0 ). It is applicable for both cases where R and < R. Whenever R, a decremental RAS algorithm based on [9] with a lower complexity of O( 3.8 ) and better performance than [1] can be used. he algorithm is known as JWFAS and was developed for user selection in transmit zero-forcing beamforming systems with singleantenna terminals. Its complexity can be further lowered to O( 3.1 ) by means of partitioned matrix inversion identities together with the fact that switching a pair of rows in corresponds to switching a pair of rows and columns in. In general, decremental 1 with the same corresponding indices. For sum rate maximization, the RAS algorithm operating on a global basis requires a maximum of R iterations instead of ( R 1) iterations. A DBD process is needed for rate evaluation at each iteration. In our simulations, we break the search whenever the next antenna elimination results in a lower sum rate. Numerical results show that JWFAS provide near optimal performance in DBD when compared to exhaustive search. For CR, the RAS process operates on the composite proected virtual channel e and is equivalent to spatial mode selection (SS). In CR, there is a one-to-one correspondence between the spatial mode gains and the columns of the postprocessing matrices defined as W in [6], which are dimensioned according to the desired number of spatial modes for each user. It is therefore possible to implement a simple SS algorithm that proceeds by removing the column in W associated with spatial mode to be eliminated. For convenience, we will refer to such a SS algorithm as poorest spatial mode elimination (or PSE). he CR process is repeated after each column-elimination and the elimination process is stopped whenever the next iteration results in a lower sum rate. As shown later in the numerical results, the JWFAS algorithm provides better performance for CR than the PSE approach. Note that for CR, it is possible to use JWFAS whenever N, where N = Σ = 1N is the total number of activated spatial modes. For Nu-SVD, there is also a one-to-one correspondence between the spatial mode gains and the post-processing matrices defined as R. ence, the PSE approach is possible and it involves removing the column in R associated with spatial mode to be eliminated. he Nu-SVD process is repeated after each column-elimination and the elimination process is stopped whenever the next iteration results in a lower sum rate. Numerical results show that the PSE and JWFAS algorithms have identical performance. D. Resource Allocation and Power Control for BD-OSD As discussed in Section III-B, the removal of antennas or modes with low contribution improves the BD-OSD spatial mode gains of the remaining users and also provides aitional transmission resources for them. As a result, these remaining users experience higher channel rates and the overall sum rate is increased. It is clear therefore that a sum rate maximization process should precede any resource allocation or power control exercise. he same RAS/SS algorithms used for sum rate maximization in BD-OSD can also be used to provide a systematic mechanism for resource allocation and power control. his is due to their ability to rank the antennas or spatial modes in an order that represent their contribution to the overall sum rate. Removing an antenna or mode with low contribution will result in a low rate loss to the affected user and a low loss to the overall sum rate. his mechanism is useful when reducing the rates of those users with excess rate in order to aid those that are lacking. One may proceed by dividing the user pool into groups, viz., those with excess rates (Group #1) and those who are in deficit (Group #). he rate allocation process may then proceed by eliminating the worst antenna or mode within Group #1. If the elimination causes a user to go from Group #1 to Group #, undo the elimination and go for the next worse antenna or mode in Group #1. Repeat this process until all individual user rates are satisfied. Note that a solution may not be found and other allocation policies may then be invoked, e.g., serving the higher priority users. In this case the RAS/SS algorithms are of help again as it can identify the worst antennas and modes to be eliminated so that the overall rate loss impact is minimized. Power control can proceed after rate allocation is done according to the procedures described above. his will help achieve transmit power minimization as the poorer antennas

5 and spatial modes are eliminated while meeting the individual user rates. Further adustments to the final transmission rate and powers may be done via power scaling. IV. NUERICAL RESULS he presence of spatial fading correlation in is 1/ 1/ captured by modeling the channel as = Rr wr t, where w is the i.i.d. spatially white channel and R r and R t are positive definite ermitian matrices that specify the receive and transmit correlations respectively. We assume that the base station antennas are well spaced enough to allow Rt = I and the users are well separated enough to consider only the intraterminal antenna correlation. An exponential correlation model is used where each element r i in R r is r i = ρ i, where ρ is the maximum correlation between two antennas at each user terminal. Fig. 1 shows the ergodic sum rates of DBD and Nu- SVD with and without RAS for a 4-user system each with antennas. he RAS is done via exhaustive search and JWFAS. As shown, RAS provides substantial sum rate gain and the JWFAS algorithm is near optimal. Also shown are the upperbounded sum capacities with and without RAS. he upperbounding is done via single-user IO channel capacities. It is seen that the sum capacity loss due to RAS is accompanied by increase in the achievable sum rate. Fig. compares the performance of DBD, CR and Nu-SVD using the JWFAS and PSE algorithms for RAS/SS. As shown, all three BD- OSD schemes benefited from RAS/SS. JWFAS performs better than PSE for CR but both provide the same performance for Nu-SVD. As expected, Nu-SVD provides the best performance among the three schemes. owever it is computationally more expensive as it is an iterative algorithm. CR is attractive in that it provides a means of mode selection at a computational cost that is practically the same as DBD. V. CONCLUSION We have shown that udicious receive antenna selection (RAS) has significant impact on increasing the achievable sum rate and minimizing the average transmitted power per user in multi-user IO wireless downlinks that employ block diagonalization to achieve orthogonal space division multiplexing (BD-OSD). For BD-OSD schemes that use proected virtual channels for spatial mode allocation, the RAS procedure is akin to spatial mode selection (SS) and is also beneficial. Algorithms for RAS/SS are proposed and numerical results have shown significant sum rate gains when they are incorporated. Given this, RAS/SS is a necessary first step to be taken prior to any resource allocation or power control exercise. he same RAS/SS algorithms for sum rate maximization can be used to provide a systematic mechanism to aress user rate allocation and power control in BD- OSD. ACNOWLEDGEN he authors gratefully acknowledge the funding for this work provided by DSO National Laboratories (Singapore), Natural Sciences and Engineering Research Council (NSERC) of Canada, Alberta Informatics Circle of Research Excellence (icore), Rohit Sharma Professorship, and RLabs. REFERENCES [1] G. Caire and S. Shamai, On the achievable throughput of a multiantenna Gaussian broadcast channel, IEEE rans. Inform. heory, vol.49, pp , Jul []. Costa, Writing on dirty paper, IEEE rans. Inform. heory, vol.9, pp , ay [3] W. Yu and J.. Cioffi, Sum capacity of Gaussian vector broadcast channels, IEEE rans. Inform. heory, vol.50, pp , Sep [4] B. ochwald and S. Vishwanath, Space-time multiple access: Linear growth in the sum rate, Proc. 40th annual Allerton conf. Communications, control and computing, Allerton IL, Oct. 00. [5]. Viswanathan, S. Venkatesan and. uang, Downlink capacity evaluation of cellular networks with known-interference cancellation, IEEE Journal on Selected Areas in Communications, vol.1, no.5, Jun [6] Q.. Spencer, A.L. Swindlehurst and.. aardt, Zero-forcing methods for downlink spatial multiplexing in multiuser IO channels, IEEE rans. Sig. Proc., vol.5, no., pp , Feb [7] L. Choi and R.D. urch, A transmit preprocessing technique for multiuser IO systems using a decomposition approach, IEEE rans. on Wireless Comms., vol.3, no.1, pp , Jan.004. [8] Z. Pan,.. Wong and.s. Ng, Generalized multiuser orthogonal space division multiplexing, IEEE rans. on Wireless Comms., vol.3, no.6, pp , Nov [9] B.C. Lim, C. Schlegel, W.A. rzymień, Efficient receive antenna selection algorithms and framework for transmit zero-forcing beamforming, in Proc. VC-06 Spring, ay 006. [10] I. elatar, Capacity of multi-antenna Gaussian channels, Eur. rans. el., vol.10, no.6, pp , Nov/Dec [11] G.G. Raleigh and J.. Cioffi, Spatio-temporal coding for wireless communication, IEEE rans. Commun., vol.46, pp , ar [1] A. Gorokhov, D. Gore and A. Paulra, Receive antenna selection for IO spatial multiplexing: theory and algorithms, IEEE rans. Sig. Proc., vol. 51, no. 11, pp , Nov [13]. Lütkepohl, andbook of atrices, John Wiley & Sons, Chichester, Fig. 1. DBD and Nu-SVD sum rates with and without JWFAS Fig.. Comparison of DBD, CR and Nu-SVD with different RAS algorithms (JWFAS and PSE)