Hybrid Diversity Maximization Precoding for the Multiuser MIMO Downlink

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1 his full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE ICC 0 proceedings ybrid Diversity Maximization Precoding for the Multiuser MIMO Downlin Farhan halid and Joachim Speidel Institute of elecommunications, University of Stuttgart Pfaffenwaldring 47, Stuttgart, Germany {halid, speidel}@inue.uni-stuttgart.de Abstract In this paper, we present a high-performance hybrid linear precoding scheme for the downlin of multiuser multipleinput multiple-output (MU-MIMO systems based on the combination of an iterative modified regularized bloc diagonalization (IMRBD precoding and minimum mean square error (MMSE precoding. he proposed hybrid diversity maximization (DM scheme aims to maximize the diversity gain by means of this dual-stage precoding strategy while maintaining relatively low complexity. he simulation results show that DM precoding can provide comparable or even better performance than other iterative precoding schemes, including some of the more complex ones. Index erms Diversity, linear precoding, minimum mean square error (MMSE, multiple-input multiple-output (MIMO, multiuser MIMO (MU-MIMO. I. IODUCION Multiuser MIMO (MU-MIMO constitutes an integral part of the fourth generation (4G mobile technologies and beyond due to its great potential for increasing the system capacity of cellular networs. he downlin transmission problem in MU- MIMO systems involves mitigating the multiuser interference (MUI using some linear or nonlinear precoding scheme at the base station (BS and optimizing the downlin transmit power allocation for each user subject to an average total power constraint. Simple linear techniques lie channel inversion and regularized channel inversion [] [4] are applicable if each user equipment (UE utilizes a single receive antenna though the performance is generally much lower than that of nonlinear techniques based on dirty paper coding [4], [5]. owever, linear MU-MIMO downlin techniques that allow the use of multiple receive antennas at the UE are of particular interest since they can provide higher diversity gain using single-stream transmission or alternatively, multi-stream transmission can be employed to obtain spatial multiplexing gain for the users. Bloc diagonalization (BD [6], successive minimum mean square error (SMMSE [7] and regularized bloc diagonalization (RBD [8] are examples of lowcomplexity precoding techniques for multi-antenna UEs which provide closed-form expressions for the precoding matrices. owever, this advantage comes at the cost of lower performance. Several linear transmission schemes based on iterative processing at the BS have also been proposed in literature (e.g., [8] [5]. Such schemes are capable of achieving higher performance gains at the expense of significantly increased complexity. otal-mmse (-MMSE [], the direct optimization scheme of [4], and modified MMSE (M- MMSE [5] are joint transmit-receive optimization techniques based on minimization of the sum of the mean square errors (MSEs for all simultaneous users. M-MMSE uses a modified total MMSE criterion resulting in better performance. In [] and [3], the uplin/downlin duality is exploited to obtain a convex objective function which converges to the exact MMSE solution. Iterative RBD (IRBD [8] is another interesting iterative scheme which allows the unused row subspace of a user s channel to be utilized for other users transmissions by iteratively performing RBD. Even though IRBD is generally outperformed by -MMSE, M-MMSE and the duality-based schemes, it is much simpler to implement and still provides good performance. In this paper, we present a new hybrid iterative MU- MIMO downlin transmission scheme referred to as hybrid diversity maximization (DM. DM combines a simple iterative modified RBD (IMRBD scheme and the minimum sum-mse criterion of [] and [4] to minimize the average bit-error rate (BER while maintaining reasonably low system complexity. We analyze the proposed DM scheme by using single-stream transmission for each multi-antenna UE in order to maximize the diversity gain. he paper is organized as follows: In Section II, we describe the system model for MU-MIMO downlin transmission. Section III presents a detailed description of the proposed downlin transmission scheme. he simulation results, comparing the performance of the proposed scheme with other techniques, are presented in Section IV. Section V finally concludes the paper. II. SYSEM MODEL he generalized bloc diagram of a MU-MIMO downlin transmission system is shown in Fig.. We consider a -user MU-MIMO system with N transmit antennas at the BS and N R receive antennas at UE, =,,,. he total number of receive antennas is denoted by //$ IEEE

2 his full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE ICC 0 proceedings x x x N Fig.. General bloc diagram of a multiuser MIMO downlin transmission system. N = R L =. x represents the transmit symbol vector for user L is the number of simultaneous data streams transmitted to user. We assume the elements of x to be zero-mean uncorrelated symbols with unit variance i.e., E xx = I L E[ ] denotes the expectation and ( represents the ermitian transpose. N = is the combined multiuser channel matrix consisting of the MIMO channel matrices N of all the users. We consider a Rayleigh flatfading channel model the elements of each are independent identically distributed (i.i.d. zero-mean complex Gaussian random variables with unit variance. his is a valid assumption in case of multicarrier transmission lie orthogonal frequency-division multiplexing (OFDM. We also assume complete channel state information (CSI of all UEs to be available at the BS while each UE only has nowledge of its own channel. he transmit precoding matrix for user and the corresponding receive matrix at the th UE are N L L represented by and R respectively. he estimated symbol vector after receive processing at the th UE is then given by N R N R N R n n n L ( j j j = R R R ˆx xˆ xˆ = R x + n ( N R n is the corresponding noise vector consisting of zero-mean additive white Gaussian noise samples with variance σ n. he combined output symbol vector for all users can be written as L ( xˆ = R x+ n ( R blocdiag( R R = N L = represent the combined multiuser ˆx L and receive (bloc-diagonal and transmit matrices respectively, L x = x x and n = n n are the concatenated transmit symbol and noise vectors respectively, and L = L = is the total number of transmitted data streams. III. YBRID DIVERSIY MAXIMIZAION (DM RANSMISSION SCEME A. Iterative Modified RBD (IMRBD Precoding IMRBD precoding is a modification to RBD and IRBD [8]. It is capable of providing a slight performance improvement over IRBD for equal number of iterations. Consequently, performance comparable to IRBD can be achieved with fewer iterations. Lie IRBD, IMRBD utilizes the unused row subspace of a user s channel matrix (corresponding to the unused singular values for the transmissions of other users. his requires L < ran( for at least one of the users in order to achieve any sort of performance gain. owever, in our paper the analysis is restricted to single-stream transmission only i.e., L =,. IMRBD precoding constitutes the first stage of DM. o start with, we calculate the precoding matrices of the users sequentially from user to user. owever, no particular ordering of the users is necessary. During the first iteration, we define the matrix as U ( L U ( L = + ( ( j N j R Lj N = which is a reduced channel matrix with the th user s channel eliminated along with the unused row subspaces of the preceding users channel matrices. Each matrix U j ( L j in (3 represents the first L j columns of the unitary matrix U j which contains the left singular vectors of the j th user s equivalent channel. he preliminary precoding matrix for user is then given by ( N N α N (3 = + I (4 α = N Rσn / P and P is the average total transmit power. Next, we perform the singular value decomposition (SVD of the th user s equivalent channel

3 his full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE ICC 0 proceedings = UΣ V (5 resulting in the unitary matrices U and V containing the left and the right singular vectors respectively, and the diagonal matrix Σ containing the singular values of the equivalent channel. he final precoding matrix of user is then given by = V ( L N L (6 only the first L columns of the product V retained. he precoding matrices of all the users are calculated in a similar fashion. owever, for all subsequent iterations, the matrix is redefined as U ( L U ( L = U+ ( L + + U ( L ( j= ( L j j N j each Uj ( L j represents the most recent version of the matrix, either computed in the current or the previous iteration. After the i th iteration, we obtain a set of transmit precoding matrices {,( i } and receive matrices { R,( i } = { U,( i ( L } for i =,, N I N I is the total number of iterations used for DM precoding. he number of iterations to be used depends on the desired performance complexity tradeoff, and an appropriate stopping criterion can be specified accordingly. B. Minimum Sum-MSE Precoding After a certain number of iterations, IMRBD does not provide any further performance improvement. herefore, the second and final stage of DM employs the minimum sum- MSE criterion of [] and [4] to obtain the final set of precoding matrices resulting in enhanced performance. his transmit precoding method constitutes the final (i.e., DM iteration and utilizes the set of receive matrices obtained in the last IMRBD iteration. he optimization problem can be written as s.t. min { } = tr = E ( = P are (7 N I th (8 E is the th user s mean square error and tr( represents the matrix trace operation. With the receive matrices fixed, the objective function in (8 becomes convex over the transmit matrices thus guaranteeing convergence to at least some local minimum. E is given by E = E xˆ x ( ( = tr R R + σ RR j j n j= R R + I L denotes the Euclidean vector norm. he Lagrange dual objective function for the problem can then be constructed as given in []. Solving the Lagrange dual problem by taing the partial derivate with respect to the transmit precoding matrices { } and equating to zero, we get the new precoding matrices as ( υ,( NI N,( NI (9 = A+ I R (0 A = j Rj,( NI Rj,( NI j ( j = and υ is the Lagrange multiplier which is simply given by υ = α/ = σn / P. his greatly simplifies the generation of the precoding matrices as compared to -MMSE which requires a numerical search to obtain the Lagrange multiplier. he precoding matrices are finally normalized and scaled so that the power constraint in (8 is fulfilled. he final DM precoding matrices are thus given by = j = P j,( NI F denotes the Frobenius norm. F,( NI ( he proposed DM scheme does not necessarily require the receive matrices { R } to be transmitted from the BS to the UEs. Instead, each UE can estimate its receive matrix locally which is simply a linear MMSE (LMMSE receiver given by (( α ( Rˆ = + I. (3 L he product in (3 represents the th user s equivalent channel.

4 his full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE ICC 0 proceedings DM provides maximum diversity gain when singlestream transmission is employed per user. his high diversity gain is a consequence of the two-stage optimization process that constitutes DM. In the first stage, IMRBD precoding sequentially optimizes the utilization of the unused row subspaces of the users channel matrices by repeatedly applying the modified RBD precoding. Further performance improvement is then accomplished in the second stage by minimizing the sum-mse of all users for a given set of receive matrices. IV. SIMULAION RESULS We have used a quasi-static Rayleigh flat-fading channel model and quaternary phase-shift eying (QPS modulation for the simulations. A sufficiently large number of channel realizations are used for each simulated data point in the performance curves. erein, we use the notation N { ( L, ( L,, ( L } to represent a -user MU-MIMO system consisting of N BS antennas and receive antennas at the th UE supporting L data streams, for =,,. he performance comparison of various transmission schemes is provided in terms of the combined average BER of all the users versus the signal-tonoise ratio (S defined as P / N σ. R n Fig. shows the uncoded BER performance of the proposed DM scheme in comparison with other techniques for 4 {(,(} MU-MIMO configuration. Per-user SMMSE (PU-SMMSE proposed in [6] is a modified lowcomplexity version of SMMSE with similar performance. MMSE power loading (MMSE PL [8] is used to enhance the performance of PU-SMMSE but it still lags DM by a huge margin. IRBD with improved diversity (impd PL [8] and N I = 5 iterations performs quite well. owever, further increasing the number of iterations does not result in any significant performance improvement within the given S range and the performance gap between IRBD and DM widens as S increases. At higher S values, DM with N I = 5 iterations performs slightly better than -MMSE with 4 ε = 0 which is far more complex to implement and requires more iterations than DM. ere ε represents the threshold for the stopping criterion used for -MMSE in []. Using a smaller value for ε might improve the performance of -MMSE by allowing more iterations, with the obvious consequence of increasing the complexity even further. A major contributing factor to the complexity of the - MMSE scheme is the procedure needed to obtain the Lagrange multiplier. he Lagrange multiplier υ for -MMSE is calculated by numerically solving the equation P N λj = (4 j= ( λ + υ and selecting the value of υ which gives the minimum sum- MSE, E []. he λ j in (4 represent the singular = j Average BER 0 4 x {(, (} PU-SMMSE MMSE PL IRBDimpDPL,N I =5 -MMSE, ε =0 4 DM, N I = S (db Fig.. Performance comparison of the DM scheme with PU-SMMSE, IRBD and -MMSE for 4 {(, (} MU-MIMO configuration. Average BER 0 6 x {(, (, (} PU-SMMSE MMSE PL IRBD impd PL, N I =5 IRBD impd PL, N I =5,ρ= 0dB -MMSE, ε =0 4 DM, N I =5 DM, N I =5,ρ= 0dB S (db Fig. 3. Performance comparison of the DM scheme with PU-SMMSE, IRBD and -MMSE for 6 {(, (, (} MU-MIMO configuration. values of the matrix A in (. In the given analysis employing single-stream transmission per user, only the L largest singular values in (4 have been considered while discarding the rest. owever, the complexity still remains relatively high. Fig. 3 shows the uncoded BER performance comparison for 6 {(, (,(} MU-MIMO configuration. DM clearly outperforms the other techniques showing its ability to exploit the system diversity more effectively. It provides much higher diversity gain resulting in significant performance improvement over IRBD which has similar complexity. It even outperforms the significantly more complex -MMSE scheme. he performance of DM and IRBD with imperfect CSI at the BS is also shown in the figure. he relative strength of the channel estimate available at the BS is represented by N ρ = / E E is the channel estimation error matrix whose elements are i.i.d. zero-mean

5 his full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE ICC 0 proceedings complex Gaussian random variables and denotes the matrix norm. he performance of DM with ρ = 0dB is almost identical to that of IRBD with perfect CSI which demonstrates its robustness towards channel estimation errors. In fact at around 8dB, IRBD experiences an error floor and lags behind DM ( ρ = 0dB as a result. V. CONCLUSION In this paper, we have proposed a hybrid MU-MIMO downlin transmission scheme which effectively exploits the inherent diversity gain of the MIMO broadcast channel (MIMO-BC to significantly improve system performance, thus providing a higher sum-rate. As seen in the previous section, maximum diversity gain for a certain antenna configuration can be achieved by using single-stream transmission per user. his scheme provides a means of implementing high-performance MU-MIMO downlin transmission systems without any drastic increase in complexity. It can even outperform more complex techniques lie -MMSE despite the lower complexity. he proposed scheme has also shown robustness against channel estimation errors and is capable of maintaining good performance even when perfect CSI is not available at the BS. REFERENCES []. austein, C. V. elmolt, E. Jorswiec, V. Jungnicel, and V. Pohl, Performance of MIMO systems with channel inversion, in Proc. IEEE VC 00-Spring, 00, vol., pp []. austein, A. Forc,. Gäbler, C. V. elmolt, V. Jungnicel, and U. rüger, Implementation of adaptive channel inversion in a real-time MIMO system, in Proc. IEEE Int. Symp. Personal, Indoor Mobile Radio Communications (PIMRC 04, Barcelona, Spain, Sep. 004, pp [3] C. B. Peel, B. M. ochwald, and A. L. Swindlehurst, A vectorperturbation technique for near-capacity multiantenna multiuser communication Part I: Channel inversion and regularization, IEEE rans. Communications, vol. 53, no., pp. 95 0, Jan [4] Q.. Spencer, C. B. Peel, A. L. Swindlehurst, and M. aardt, An introduction to the multi-user MIMO downlin, IEEE Communications Magazine, vol. 4, no. 0, pp , Oct [5] C. B. Peel, B. M. ochwald, and A. Lee Swindlehurst, A vectorperturbation technique for near-capacity multiantenna multiuser communication Part II: Perturbation, IEEE rans. Commun., vol. 53, no. 3, pp , Mar [6] Q.. Spencer and M. aardt, Capacity and downlin transmission algorithms for a multi-user MIMO channel, in Proc. 36th Asilomar Conf. Signals, Systems, and Computers, Nov. 00, pp [7] V. Stanovic and M. aardt, Multi-user MIMO downlin precoding for users with multiple antennas, in Proc. th Wireless World Research Forum (WWRF, oronto, ON, Canada, Nov [8] V. Stanovic and M. aardt, Generalized design of multiuser MIMO precoding matrices, IEEE rans. Wireless Communications, vol. 7, no. 3, pp , Mar [9] Z. Pan,.. Wong, and. S. Ng, Generalized multiuser orthogonal space-division multiplexing, IEEE rans. Wireless Commun., vol. 3, pp , Nov [0] A. J. enenbaum and R. S. Adve, Joint multiuser transmit receive optimization using linear processing, in Proc. IEEE Int. Conf. Commun.(ICC 04, Paris, France, Jun. 004, pp [] M. Schubert, S. Shi, E. A. Jorswiec, and. Boche, Downlin sum- MSE transceiver optimization for linear multi-user MIMO systems, in Proc. 39th Asilomar Conf. Signals, Systems, and Computers, Pacific Grove, CA, USA, Oct [] J. Zhang, Y. Wu, S. Zhou, and J. Wang, Joint linear transmitter and receiver design for the downlin of multiuser MIMO systems, IEEE Commun. Lett., vol. 9, pp , Nov [3] A. Mezghani, M. Joham, R. unger, and W. Utschic, ransceiver design for multi-user MIMO systems, in Proc. IG/IEEE Worshop on Smart Antennas (WSA 006, Ulm, Germany, Mar [4] B. Bandemer, M. aardt, and S. Visuri, Linear MMSE multi-user MIMO downlin precoding for users with multiple antennas, in Proc. IEEE 7th Int. Symp. Personal, Indoor and Mobile Radio Comm. (PIMRC 06, 4 Sep. 006, pp. 5. [5] J. Joung and Y.. Lee, Regularized channel diagonalization for multiuser MIMO downlin using a modified MMSE criterion, IEEE rans. Signal Processing, vol. 55, no. 4, pp , Apr [6] M. Lee and S.. Oh, A per-user successive MMSE precoding technique in multiuser MIMO systems, in Proc. IEEE VC 007- Spring, 5 Apr. 007, pp