Power-Efficient Transmission Design for MISO Broadcast Systems with QoS Constraints

Transcription

1 Power-Efficient ransmission Design for MISO Broadcast Systems with QoS Constraints Chunshan Liu Department of Engineering Macquarie University NSW 2109, Australia Min Li Department of Engineering Macquarie University NSW 2109, Australia Stephen V. Hanly Department of Engineering Macquarie University NSW 2109, Australia Abstract his paper considers a novel transmitter optimization problem for MISO broadcast systems, where a multi-antenna base station (BS serves a set of single-antenna mobile stations (MSs with Quality of Service (QoS targets formulated in terms of Symbol Error Probability (SEP: data inputs are drawn from a discrete alphabet, precoding is performed by the BS to generate appropriate signals transmitted over the channel, symbol-by-symbol detection is performed at each MS, and a minimum SEP target is imposed at each MS as the QoS target. Starting with the system adopting standard constellations and with focus on 16-QAM, the SEP constraints are formulated and characterized by a set of convex relaxations on the received signals. he resulting problem is convex, and can be efficiently solved by the primal-dual interior point method. Numerical results demonstrate that the proposed precoder provides significant transmission power reduction compared with the linear zeroforcing (ZF precoder under the same SEP targets. urning to the system with omlinson-harashima type constellations, a similar convex relaxation approach is applied and the corresponding convex optimization problem is presented. he proposed precoding scheme is shown to outperform both the previous scheme (based on regular constellations with convex relaxation and the standard omlinson-harashima precoding using zero forcing. I. INRODUCION Multiple-input and multiple-output (MIMO is known as an efficient technology to increase the data throughput of a wireless communication system without additional bandwidth or increased transmit power [1], at the cost of sophisticated processing techniques deployed at transmitters or/and at receivers. Precoding is one such technique in which the transmitter intelligently matches the input signals to the channel and is useful to suppress interference among parallel data streams in a multi-user MIMO system. In the literature, a vast amount of research has been carried out in designing MIMO precoders with respect to different metrics, such as system throughput, power consumption and quality of service (QoS at user terminals. Most relevant to the current work, various formulations with QoS requirements have been investigated by [2] [5]. In particular, when the QoS is characterized by a set of signal-to-noise-plus-interference ratio (SINR for different users, [2] and [3] studied the power-efficient linear precoding schemes subject to a set of SINR constraints. With the QoS formulated as a set of mean square error (MSE on the received signals, [4] and [5] instead designed the powerefficient linear and nonlinear precoders under the MSE targets. In this work, we take a different point of view of the QoS requirement and develop power-efficient precoding techniques under the QoS target considered. Specifically, we consider a downlink multi-user MIMO system where a BS with multiple antennas serves a number of single-antenna mobile stations (MSs. he data symbols intended for each MS are drawn from a discrete alphabet set, such as constellations of quadrature amplitude modulation (QAM. Precoding is performed at the BS and a simple symbol-by-symbol detection is performed at each MS. he QoS target corresponds to the minimum Symbol Error Probability (SEP for the detection at each MS. Starting from the system with standard constellations such as 16-QAM, the SEP constraints are represented as a set of convex relaxations on the received signals with respect to the linear Zero-Forcing (ZF precoding. he resulting optimization problem is convex and a primal-dual interior point method [6] is used to solve the problem efficiently. Furthermore, we apply the convex relaxation approach to the system with omlinson- Harashima (H type constellations and formulate another precoder optimization problem. Power reductions from the proposed schemes are demonstrated in comparison with linear ZF and standard ZF-based-H precoding schemes. It is noted that a similar convex relaxation approach was proposed in [7] for the MIMO system with focus on binary and 4-QAM signaling. Here, we integrate the SEP targets and generalize the relaxation approach to the system with higher order QAM such as 16-QAM. Moreover, the same approach is considered for the setup with H type constellation. In this context, instead of forcing the received signal to a specific lattice point as done by the vector perturbation technique in [8], the proposed scheme instead aligns the received signal to a pre-designed region associated with a lattice point at each MS. A preliminary work was carried out by the same authors in a distributed network MIMO system with standard constellations [9]. he rest of this paper is organised as follows. Section II presents the system model and a general problem formulation. Section III studies the system with standard constellations, while Section IV turns to the system with H-type constellations. Numerical results are provided in Section V in comparison with linear ZF and ZF-based-H precodings. Conclusions are drawn in Section VI.

2 Notations: Capital and lower-case bold fonts denote matrices and vectors, respectively, e.g., A is a matrix and a is a vector; notation( denotes the matrix transpose and ( H the Hermitian transpose; R{ } and I{ } are the real and imaginary part of a complex value, respectively; I M denotes M M identity matrix. Finally, 1 = j. II. SYSEM MODEL AND PROBLEM FORMULAION We consider a MISO broadcast system, where a BS equipped with N antennas serves M single-antenna MSs. It is assumed that the number of transmitting antennas is no smaller than the number of MSs, i.e., N M. Let d m be the data symbol drawn from a set of discrete alphabets m and intended for the mth MS. At each time instant, a set of data symbols d = [d 1,...,d M ] are encoded and broadcast from the BS to the MSs. he signal received by the mth MS is: y m = N n=1 h m,nx n +v m, (1 where x n is the complex signal transmitted from the nth BS antenna, h m,n is the complex channel coefficient from the nth BS antenna to the mth MS, and additive noises v m are i.i.d circular-symmetric complex Gaussian with zero-mean and fixed variance2 2. With transmitted signals{x n } and received signals {y m } stacked into vectors as x = [x 1,...,x N ] and y = [y 1,...,y M ], the system input-output relationship can be represented in a more compact way as y = Hx+v, (2 where the noise vector v = [v 1,...,v M ] and channel matrix H = [h 1,...,h M ] with h m = [h m,1,...,h m,n ] denoting the channels from BS antennas to the mth MS (see Fig. 1. Under this system, we wish to construct an efficient mapping (precoding from the symbol-vector d to the transmit signal-vector x such that the BS power consumption is minimised and the set of SEP constraints imposed at MSs are satisfied. o this end, we first characterize the SEP constraints. In the formulation, a SEP requirement indicates that the signal y m received at MS m should potentially reside in the right decision region associated with d m. he probability of y m lying outside the region should be no larger than the SEP target, which can be described mathematically as Pr(y m = (h m x+v m A(d m Pe m, (3 where d m S m, A(d m corresponds to the decision region of symbol d m, and Pe m is the SEP target at MS m. hen the optimization problem is formulated as: P : min x P (x = x H x subject to Pr((h mx+v m A(d m Pe m, for a fixed set of {d m S m,m = 1,...,M}. III. SYSEM WIH SANDARD CONSELLAION INPUS In this section, we focus on the system with standard constellations: we characterise the constraints on the received signals and formulate the power optimization problem. (4 Fig. 1. x h 1 d d1, d2, d M... BS... h2... h M 1 MS1 MS2 MS M 1 h M MS M A MISO broadcast system with discrete input and QoS constraints. A. Characterizing SEP Constraints We mainly elaborate on the construction of SEP constraints for 16-QAM constellation points. Similar approach can be applied to the 4-QAM as well as any other higher-order QAM. Consider the gray-coded 16-QAM constellation as shown in Fig. 2. With a fixed noise variance, due to the SEP requirement at MS m, the constellation set S m = {s k : k = 1,...,16} is a scaled version of the standard one (whose inner points are the four corner-points of a unit-square by β m. he green dashed lines partition the complex plane into squared-shape decision regions centered on the constellation points{s k }; any received signals falling into the decision region centred on s k are mapped back ontos k. According to (3, we have to ensure that the probability of the received signal lying outside the right decision region should be no greater than Pe m. his equivalently imposes a set of constraints on the received signals. Define y m (r = R{h mx}, y(i m = I{h mx}, v(r m = R{v m }, v m (i = I{v m }, d m (r = R{d m } and d m (i = I{d m }. hen requirement (3 is equivalent to 1 d (r m y(r m +ρ(r + 2π d (r n y (r m ρ (r e ( v (r m dv (r m }{{} O (r 1 d (i m y(i m +ρ(i + 2π d m (i y m (i ρ (i e ( v (i m dv (i m }{{} where ρ (r + /ρ(i + O (i 1Pe m, (5 and ρ(r /ρ(i are parameters that characterize decision regions and define the upper and lower bounds for the integrals involved. able I specifies these parameters associated with different constellation points. Given a target Pe m, one can determine the constraint region where h mx should lie in by examining inequality (5 as defined. he boundary of the region may be defined by the equality O (r O (i = 1Pe m in (5. Since different constellation points are associated with different decision regions, the resulting constraints are different.

3 ABLE I INEGRAL PARAMEERS ASSOCIAED WIH GREY-CODED 16-QAM s 3 s 4 s 7 s 8 s 15 s 16 s 11 s 12 ρ (i /ρ(i + + /β m β m/β m β m/+ ρ (r /ρ(r + + /β m s 1 s 2, s 4 s 3 β m/β m s 5, s 13 s 6, s 8, s 14, s 16 s 7, s 15 β m/+ s 9 s 10, s 12 s 11 s 2 s 1 m s 6 s 5 s 14 s 13 s 10 s 9 Fig QAM: β m is a scaling factor; s 1,s 2,...,s 16 are constellation points; decision regions are separated by green dashed lines; linear constraints on h mx are marked by shadow areas. In the following, we explain how to determine a set of critical points on the boundary of the corresponding constraint regions for a group of representative constellation points {s 11,s 12,s 16 }. For the corner constellation point s 11, the critical boundary points can be determined according to three different combinations of (O (i,o (r : (1,1Pe m : y (i m = +,y (r m = 2β m Q 1 (1Pe m ; (1Pe m,1:y (r m = +,y (i m = 2β m Q 1 (1Pe m ; ( 1Pe m, 1Pe m : y (i m = y (r m = 2β m Q 1 ( 1Pe m, where Q 1 (. denotes the inverse of the standard Q-function. For the centre constellation point s 16, the critical boundary points can be found by examining the following combinations of (O (i,o (r : : y (i m = β m, y (r m = β m ±δ 1 ;,P: y (r m = β m, y (i m = β m ±δ 1 ; ( 1Pe m, 1Pe m: y (i m = β m ±δ 0, y (r (, 1Pem ( 1Pem where = 1 βm 2π β m e 2 and δ 1 are chosen to satisfy: Q( δ0β m Q( δ1β m 2 m = β m ±δ 0, v 2 dv 1Pe m, parameters δ 0 Q( δ0+β m Q( δ1+β m = 1Pe m, = 1Pem. For the side constellation point s 12, the critical boundary points can be found similarly by examining the following combinations of (O (i,o (r : : y (i m = β m, y (r m = β n Q 1( 1Pe m ; (1Pe m,1: y (r m = 2β m Q 1 ( 1Pe m, y (i (, 1Pem β m ±δ 0 ; ( 1Pe m, 1Pe m: y (i m = β m ±δ 2, y (r m = +, where parameter δ 2 is chosen to satisfy: ( δ2 β m Q Q (6 m = ( δ2 +β m = 1Pe m. (7 In principle, the curved-shape boundary of the constraint regions can be found by examining all possible combinations of O (i and O (r. However, there is an infinite number of combinations. A practical approach is to find a polytype contained in the constraint region, i.e., we conservatively approximate the region using the area bounded by line segments between a finite number of points on or within the boundary. In the simplest case, a set of constraints can be constructed as follows, resulting in linear constraints on h x. For s 11, the resulting constraints are represented as: y m (i 2β m Q 1 ( 1Pe m and y m (r 2β m Q 1 ( 1Pe m ; for s 12, the resulting constraints are represented as: β m δ 0 y m (i β m +δ 0 and y m (r 2β m Q 1 ( 1Pe m ; while for s 16, the corresponding constraints are: β m δ 0 y (r m β m +δ 0 and β m δ 0 y m (i β m +δ 0. he constraint regions constructed in this way is illustrated in Fig. 2 (shadow areas, along with the true constraint region obtained numerically by examining a number of possible combinations of O (i and O (r (red-curve regions. B. Discussion on Choice of Scaling Factor β m In the formulation, given a noise variance, there exists a minimum requirement on the scaling factor β m βm such that each SEP target can be satisfied. he lower bound βm can be found by considering only the four centre constellation points, i.e., {s 6,s 8,s 14,s 16 }, as for these four points, the integral limits in (5 are more restricted than others, yielding the worst case among the constellation points. Specifically, by noticing that the length of integral intervals for both the real and imaginary part is equal to 2β m and that the noise is zero-mean, we have the following: ( 2 1 βm e v 2 2 dv 2 O (r O (i 1Pe m. (8 2π β m he minimum scaling factor βm corresponds to the solution of (8 when equality holds and can be calculated by ( 1 βm = Q Pem. (9 2 he minimum scaling factor indicates extreme constraint on the received signal when one of the four centre constellation points is transmitted: h m x = s k, k {6,8,14,15}. (10 With the same scaling factor βm, the constraints of one of the side points (e.g., s 12 becomes: y m (i = βm and y(r m

4 Fig. 3. Constraint regions of h mx with different scaling factors at Pe m = 10 3 : (1 Blue curves: β m = 1.05β m; (2 Red curves: β m = 1.20β m. 2βm Q 1 ( 1Pe m. his is equivalent to doing a ZF for the imaginary part while having a relaxed constraint on the real part. Choosing a larger scaling factor β m above βm may or may not improve the performance. Fig. 3 plots two instances of the constraint regions for Pe m = 10 3 when the constellation is scaled up with β m = 1.05βm (blue curves and β m = 1.20βm (red curves. It can be seen that with a larger β m above βm, the constraints are relaxed for the centre points. he corresponding constraint region on h mx gets enlarged and subsumes the region with smaller β m, hence potential benefit can be accrued. However, the corresponding constraint regions for the corner points always shrink as the constellation is scaled up. In this case, potential power efficiency loss may be induced because the choice of h mx becomes more restricted. When one of the side points is transmitted, it is unclear how the performance reacts as the constraint regions with larger β m partially overlaps with that for a smaller β m. C. Problem Solving In Section III-A, we have characterized the constraint regions corresponding to SEP targets for 16-QAM constellation points (constraint regions for 4-QAM can be constructed similarly to the corner points of 16-QAM and conservatively approximate them by linear convex relaxation. Different input data symbols corresponds to different types of constraints, equality or/and inequality. herefore, by defining x = [ R{x },I{x } ], the resulting optimization problem is generally in the form of: min P ( x = x x x R 2M 1 P 1 : s. t. U xb 0, (11 A xc = 0, b R M1 1,c R M2 1, where U = [u 1,...,u M1 ], A = [a 1,...,a M2 ], b = [b 1,...,b M1 ] and c = [c 1,...,c M2 ]. Parameters M 1 and M 2 depends on the data symbols as well as the scaling factors. E.g., assuming β m = βm is used, if the data symbols only corresponds to the inner four points, thenm 1 = 0 andm 2 = 2M; if the data symbols only corresponds to the corner points, then M 1 = 2M and M 2 = 0. Matrices U and A depend on the channel matrix and the data symbols to be transmitted. Vectors b and c depend on the data symbols, the scaling factor as well as the SEP targets. For instance, assuming the minimum scaling factor βm is used and d 1 = s 12 is the desired data symbol at the 1st MS in a 16-QAM system, then the inequality constraint is specialised to: u 1 = [ R{h 1 },I{h 1 }] and b 1 = 2β1 Q1 ( 1Pe m ; the equality constraint is specialised to: a 1 = [ I{h 1 },R{h 1 }] and c1 = β1. In (11, the objective function is quadratic and the constraints are linear, thus the problem is strictly convex and the unique solution can be found efficiently using standard numerical optimisation algorithms such as the prima-dual interior point method [6]. IV. SYSEM WIH REPLICA INPUS In the last section, we have focused on the system with standard QAM constellation points for which convex constraints are constructed according to the SEP targets. While power reduction may be gained by our approach with the standard constellations as inputs, room for improvement still exists if we break the asymmetric property by replicating the constellation points and introducing box-like constraint regions for every replica point. his idea is mainly inspired by the capacity-achieving dirty-paper coding (DPC and its simplified suboptimal variant omlinson-harashima Precoding (HP with replica inputs for the MIMO broadcast system. A. omlinson-harashima Precoding We first briefly review some basic concepts related to HP. o form the replica constellation, each point in the standard one is copied and placed at a regular manner in the complex plane. For instance, the replica constellation points of 16-QAM can be represented as: dm = d m + 4(km r + jkm, i where d m corresponds to the standard 16-QAM points, and km r and km i are integer numbers. he HP is done in a successive manner in which interference created by previous transmissions will be pre-cancelled for the current transmission at each stage. he encoding is accommodated by the replica constellation and modulo-operation at the transmitter [10]. Specifically, let the channel matrix be represented as H H = FR as a result of QR factorization, where F is a unitary matrix and R is an upper triangular matrix. hen B = HF = R H is a lower triangular matrix. he successive precoding operates as x m = 1 B m,m β m dm m1 l=1 B m,l x l, (12 for each MS m, where β m is an appropriate scaling factor chosen according to the SEP constraints and is the modulo operation [10]. he transmit signal is then formed by multipling F with x, i.e., x = F x. In this way, at the

5 receiver side, no user experiences interference because of the pre-cancellation operations done at the transmitter. wo remarks are made here. First, a natural choice of the scaling factor β m is the minimum one β m as given by (9. Secondly, since HP is performed in a successive way, different encoding orders may lead to different performance. o find the optimal ordering, one needs to do an exhaustive search over all possible combinations, which is generally infeasible. In this work we simply adopt the suboptimal V- BLAS (VB ordering [11]. B. Convex Relaxation With the minimum scaling factor βm in the HP as described, the received symbols h mx is forced to reside exactly at one of the replica points d m and hence the choice of input signals is very restricted. When we increase the scaling factor, the ZF-type constraints can be relaxed such that the input signals is chosen in a more flexible way as done for the standard constellation in Section III. In particular, with the replica constellation scaled up, the resulting constraint regions become boxes centered at each replica point as constructed and approximated similarly for the inner four points of the 16-QAM constellation (see Fig. 2. he power optimization problem is then formulated as P 2 : minp (x = x H x x s.t. β m d(r m δ 0 y (r m β m d(r β m d(i m δ 0 y (i m β m d(i m = 1,...,M, m +δ 0, m +δ 0, (13 where β m > βm, parameter δ 0 determines the size of the constraint box and is calculated according to (6, and { d m (r (i, d m } are the real and imaginary parts of a particular replica d m. We can simply use the same set of { d m } determined in the standard HP encoding procedure. Problem P 2 is convex and thus can be efficiently solved using standard algorithms such as the primal-dual interior point method. V. SIMULAION RESULS In this section, we present numerical results for the proposed schemes. In the simulation, the number of antennas at the BS is equal to the number of MSs with M = N = 10. Flat fading is assumed and the entries of the complex channel matrix are i.i.d. Rayleigh distributed with unit variance. he noise variance in each signal dimension is set to unity, i.e. 2 = 1. All the MSs share the same input alphabet and the same SEP target. In these symmetric simulation settings, the scaling factors for the MSs are identical: β 1 =... = β M = β and β 1 =... = β M = β. Given a specific target, the power consumption at the BS is calculated based on averaging over 1000 random realizations, where random data and channel coefficients are generated. Both 4-QAM and 16-QAM inputs are considered in the simulation. A. Choice of β for the Standard 16-QAM As described in Section III-B, a minimum scaling factor β for the standard 16-QAM input is required so that the SEP target can be met. o address the question whether choosing a scaling factor β larger than β is of benefit or not, we plot the transmit power as a function of the SEP target in Fig. 4(a, with three different scaling factors considered: β = {1.01β,1.10β,1.20β }. It can be seen that when the scaling factor is too large, e.g., β = 1.20β, the transmit power is increased compared with the case using the minimum scaling factor. his is due to the fact that although scaling up may bring power reduction when the inner four points are transmitted, the power increase corresponding to the transmission of the corner points dominates. In contrast, a small enlargement of the scaling factor from the minimum can potentially save some powers, especially for low SEP targets as observed from Fig. 4(a where β = 1.01β. But since the gain is quite marginal, fixing the scaling factor at the minimum one will not incur much loss in optimality. B. Choice of β for the H-type Constellation For the case with replica constellation, the scaling factor β should be larger than the minimum one β in order to enable the convex relaxation as discussed in Section IV-B. Although a larger scaling factor naturally increases the averaged power of the replica constellations, it leads to larger constraint regions on received signals and hence admits more choices of transmit signals. herefore, potential power saving may still be accrued by scaling up the constellation. Choosing a balanced scaling factor is quite a challenge and we select the appropriate one by numerical experiments. Fig. 4(b shows the power consumption versus target SEPs for different scaling factors for the H-type 16-QAM constellation. It can be seen that performance gain in term of power reduction indeed can be harvested by scaling up the constellation. In particular, as the scaling factor increases, the power reduction first increases and then decreases and even goes negative after crossing a threshold. he best one is β = 1.06β and is used afterward in the simulation for 16-QAM. For 4-QAM systems, we adopt β = 1.08β via the same method. C. Power Consumption We now present the power consumption performance for the proposed schemes in comparison with the standard H precoder (12 and the linear ZF precoder in form of x ZF = H ( HH 1( β I M d. Fig. 5 and Fig. 6 plot the transmit power versus different SEP targets for the 4-QAM and 16-QAM signalling, respectively. As shown, significant power reduction is attained by our convex relaxation approach for systems with both standard and H-type constellations. In system with standard constellations, a power reduction of about 8 db for 16-QAM (13 db for 4- QAM is obtained over the linear ZF precoder. However, the power consumption is still higher than that for the system with H-type constellation. It is also worth noticing that the

6 otal ransmit Power (db otal ransmit Power (db (a 10 3 (b β = β β = 1.01β β = 1.10β β = 1.20β HP HP with relaxation β=1.01β HP with relaxation β=1.06β HP with relaxation β=1.10β HP with relaxation β=1.20β otal ransmit Power (db Zero Forcing (ZF Standard constellation with relaxation HP HP with relaxation β=1.06β HP with relaxation β=1.40β Fig. 4. Effect of different scaling factors: (a the standard 16-QAM with convex relaxation; (b the H-type 16-QAM with convex relaxation. Fig. 6. ransmit power versus target SEP with 16-QAM inputs: standard constellation and H-type constellation. otal ransmit Power (db Zero Forcing (ZF Standard constellation with relaxation HP HP with relaxation β = 1.08 β at each MS. A convex relaxation approach is proposed to construct the SEP constraints for the systems with standard constellations and H type replica constellations. he resulting power optimization problem is convex and can be solved efficiently. Numerical results demonstrate that the proposed convex relaxation approach leads to significant transmit power reduction for systems with either standard constellations or H-type constellations. For a high order QAM-based system, the proposed approach with H-type constellations is preferable to that with standard constellations. 20 ACKNOWLEDGMEN Supported by the CSIRO Macquarie University Chair in Wireless Communications. his chair has been established with funding provided by the Science and Industry Endowment Fund. Fig. 5. ransmit power versus target SEP with 4-QAM inputs: standard constellation and H-type constellation. benefit introduced by our relaxation approach on the standard 16-QAM constellation is not as significant as that for the 4- QAM based system. his is mainly because convex relaxation benefits only when the side and outer points are transmitted as pointed out in Section III-B. herefore, for higher order inputs such as 16-QAM, H-type constellation with convex relaxation is desirable. VI. CONCLUSIONS In this paper, we have formulated a novel transmission power optimization problem for the MISO broadcast system with QoS targets to meet at MSs. Minimum symbol error probability (SEP has been introduced as the QoS metric REFERENCES [1] D. Gesbert, M. Shafi, D. Shiu, P. J. Smith, and A. Naguib, From theory to practice: an overview of MIMO space-time coded wireless systems, IEEE Journal on Selected Areas in Communications, vol. 21, no. 3, pp , [2] A. Wiesel, Y. Eldar, and S. Shamai, Linear precoding via conic optimization for fixed MIMO receivers, IEEE ransactions on Signal Processing, vol. 54, no. 1, pp , [3] W. Yu and. Lan, ransmitter optimization for the multi-antenna downlink with per-antenna power constraints, IEEE ransactions on Signal Processing, vol. 55, no. 6, pp , [4] L. Sanguinetti and M. Morelli, Non-linear pre-coding for multipleantenna multi-user downlink transmissions with different QoS requirements, IEEE ransactions on Wireless Communications, vol. 6, no. 3, pp , [5] M. B. Shenouda and. N. Davidson, Nonlinear and linear broadcasting with QoS requirements: ractable approaches for bounded channel uncertainties, IEEE ransactions on Signal Processing, vol. 57, no. 5, pp , [6] S. Boyd and L. Vandenberghe, Convex optimization. Cambridge University Press, 2004.

7 [7] R. R. Muller, D. Guo, and A. L. Moustakas, Vector precoding for wireless MIMO systems and its replica analysis, IEEE Journal on Selected Areas in Communications, vol. 26, no. 3, pp , [8] B. M. Hochwald, C. B. Peel, and A. L. Swindlehurst, A vector-perturbation technique for near-capacity multiantenna multiuser communication-part II: Perturbation, IEEE ransactions on Communications, vol. 53, no. 3, pp , [9] M. Li, C. Liu, S. V. Hanly, and I. B. Collings, ransmitter optimization for the network MIMO downlink with finite-alphabet and QoS constraints, in Australian Communications heory Workshop, [10] C. Windpassinger, R. F. H. Fischer,. Vencel, and J. B. Huber, Precoding in multiantenna and multiuser communications, IEEE ransactions on Wireless Communications, vol. 3, no. 4, pp , [11] G. D. Golden, C. J. Foschini, R. A. Valenzuela, and P. W. Wolniansky, Detection algorithm and initial laboratory results using V-BLAS space-time communication architecture, Electronics letters, vol. 35, no. 1, pp , 1999.