Novel THP algorithms with minimum BER criterion for MIMO broadcast communications

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1 August 009, 6(4: he Journal of China Universities of Posts and elecommunications Novel P algorithms with minimum BER criterion for MIMO broadcast communications YUN Xiang, LÜ ie-jun ( School of Information and Communication Engineering, Beijing University of Posts and elecommunications, Beijing 00876, China Abstract In this article, two novel omlinson-arashima precoding (P algorithms are proposed at the downlin broadcast channel of multi-user multiple-input multiple-output (MIMO systems. he optimality criterion is the minimization of the bit error rate (BER at the receiver with a constraint on the overall transmitted power. In these algorithms, transmit power allocation (PA matrix is introduced to the traditional P algorithms as part of the precoding matrix. his leads to a power loading operation based on substreams signal to interference-plus-noise ratio (SINR and results in improvement of the error rate performance compared with traditional P algorithms. When the precoding loss factor is considered, further performance improvement is achieved by the enhanced algorithm. eywords MIMO system, P, minimum bit-error rate (MBER, broadcast communication Introduction he multi-user MIMO system has been considered as one of the most important techniques because of its significant advancement in capacity compared with the traditional single-input single-output (SISO wireless channel []. he non-linear vertical Bell Laboratories layered space-time (V-BLAS system [] and spatial decision feedbac equalization (DFE [] were proposed to eliminate the co-channel interference (CCI in MIMO systems. Ginis and Cioffi [4] have proved that the V-BLAS receiver processing is equivalent to the MIMO-DFE structure. When PA methods are introduced to the V-BLAS system, the performance improvement is remarable [5 6]. Although these algorithms achieve high spectral efficiency, receiver error propagation may still occur. he aforementioned methods in MIMO systems require channel state information (CSI at only the receiver. In some cases, if the communication environment is slowly time varying, the CSI at the transmitter is possible to be obtained by feedbac in frequency-division duplex (FDD systems or Received date: Corresponding author: LÜ ie-jun, lvtiejun@gmail.com DOI: 0.06/S ( the reciprocal principle in time-division duplex (DD systems. When the CSI is achieved at the transmitter, the downlin dirty paper coding (DPC precoding, which pre-eliminates the spatial CCI in MIMO channels and avoids the receiver error propagation, develops into the duality problem of the DFE system [7]. One suboptimal implementation of DPC is P. First introduced to MIMO channels by Fischer et al [8], it moves the feedbac filters of MIMO-DFE to the transmitter. his non-linear algorithm achieves in capacity the advantage of MIMO channels with reasonable complexity and avoids error propagation. For minimizing error rate performance, an iterative PA strategy was derived in Ref. [9] and a closed-form expression was obtained in Ref. [0] for MIMO-P systems. Both algorithms achieve better performance than MIMO-P system without PA [8]. owever, the feed-forward filters are still at the receiver, that is, the received signals must be processed coordinately. As a result, the methods in Refs. [8 0] are only applicable to the point-to-point MIMO systems. In Refs. [ ], the MIMO-P was extended to the downlin multi-user scenario. Both zero-forcing (ZF and minimum mean square error (MMSE based P algorithms were derived in Ref. []. An alternative MMSE P solution was developed in Ref. [], which constrained the feed-forward

2 7 he Journal of China Universities of Posts and elecommunications 009 matrix as a unitary matrix to maintain the transmit power. In Refs. [ ], ordering P algorithms were proposed to further improve the performance. owever, the modulo operators in the MIMO-P systems result in an increase of transmit power, which is quantified by the precoding loss factor and will lead to performance decrease when constellation sizes are small [8 ]. In this article, a P structure for the downlin multi-user MIMO-P system is proposed. During the derivation, unified expressions of the user SINR under ZF and MMSE criterions are achieved from the P structure. According to the unified expressions, a closed-form solution to PA matrix is determined with the minimum BER criterion under sum-power constraint. his solution is the minimum BER-P (MBER-P algorithm. Furthermore, an improved MBER-P (IMBER-P algorithm is proposed under the novel structure, which invoes the precise precoding loss factor for performance improvement. Finally, the IMBER-P algorithm is applicable to the ordering MIMO-P system for maximizing all users SNR, and achieves better performance than conventional ordering MIMO-P systems. he remainder of this article is organized as follows. he downlin multi-user MIMO-P broadcast communication systems are described in Sect.. In Sect., the MBER-P and the IMBER-P schemes are derived for the downlin multi-user MIMO-P structures in perfect CSI case. Simulation results are given in Sect. 4 and conclusions are drawn in Sect. 5. he following notations are used. he boldface is used to denote matrices and vectors. Let tr S, S, S and S denote the matrix trace, transpose, conjugate transpose and inverse for the matrix S, respectively. [ S denotes the entry at the ith row and jth column of S. diag[ S ] is the diagonal matrix with the diagonal entries of S, [ x x x ] is ] ij diag,,..., the diagonal matrix with diagonal entries x, x,..., x and E[ ] is the expectation operator. Downlin multi-user MIMO-P broadcast communication system. System model and assumptions A downlin multi-user MIMO system is considered with N transmit antennas and N R mobile stations (each with one antenna in a flat fading channel. Let x x, x,..., x N and y y, y,..., y NR represent the transmitted and the received signals, respectively. he system model can be written as y x n ( where is the N R N channel gain matrix. he channel gain from the transmit antenna j to the receive antenna i is denoted by h ij [ ] ij, which is assumed to be a zero-mean, unit-variance, circularly symmetric and complex Gaussian random variable. Assume that the noise vector n is a zero-mean, circularly symmetric complex Gaussian (CSCG vector with E[ nn ] σ n I and n is independent of x. For simplicity, both sides of the system are supposed to have equal antennas number (i.e. N NR.. Downlin multi-user P structure First, the authors review the proposed downlin MIMO-P structure. he overall system structure is illustrated as the bloc diagram in Fig.. he pre-processing matrix U EP are composed of PA matrix E and permutation matrix P. E diag E, E,..., E is a diagonal matrix with allocated power E on the th substream. P ee μ is the permutation matrix for user ordering, where e is the th column of identity matrix I while { μ, μ,..., μ } is a suitable permutation of the set {,,..., }. Without loss of generality, now set P I, which refers to the system without ordering. It is worth noting that the proposed algorithms will be extended to the ordering system in Sect.. For other matrices, B is unit lower triangular matrix (lower left triangular matrix with ones on the main diagonal and F is unitary matrix. G diag [ g, g,..., g ] is diagonal matrix. a [ a ], a,..., a is the data vector. Each user data a,,,..., uses the same M-ary square constellation (M is constellation size and has unity power (i.e., σa σ,,,...,. a a Ua Ea is the pre-possessed data vector and the variance of each component is σa,,,...,. Eσa E he modulo operator is introduced to constrain the power of the precoded vector a. t [ t, t,..., t ] is the modulo-bound vector determined by the allocated power and constellation size, t 6 M ( M E,,,...,. he output of i.e., ( Mod t ( i (modulo t operator is bounded on

3 Issue 4 YUN Xiang, et al. / Novel P algorithms with minimum BER criterion for MIMO broadcast communications 7 [ t, t j[ t, t and assumed to be uniform distribution for,,...,. For a complex variable c, the modulo operator is defined as Re( c Im( c Mod t ( c c t j t t t where the floor operator i rounds the argument to the nearest integer towards minus infinity. From the modulo operator, the precoded symbols a can be calculated as [ I B ] a a a d ;,,..., l l l where d tn and N is a complex-value variable whose real and imaginary parts are integers that constrain a to the bounded region. his indicates that the modulo operator can be represented as a linear model [] and generates an effective data vector v a d, where d [ d ], d,..., d is add to a to constrain the value of a and can be eliminated at the receivers by another modulo operator. considered for the system illustrated by Fig. should be the distortion between the effective data vector v and the data vector r [ r ], r,..., r which will enter the receiver modulo operators, i.e., r v. he covariance matrix of error is R E[ ]. By minimizing the MSE tr R, the QR-factorization of ( ς I is performed to obtain the filter matrices, i.e. ( ς I QR G diag r, r,..., r (4 B GR F Q where ς σn σa σn, R [ r ij ] is an upper triangular matrix and Q is a unitary matrix. According to the criterion of the MMSE, the detected signal â is expressed as aˆ a and decoupled into parallel substreams aˆ E a ;,,..., (5 where the variance of each error element is [] σ R σ σ (6 and 4 [ ] g n n h j Ej j h j. j. Unified SINR expressions for P sub-channel model Fig. Bloc diagram of modified downlin multi-user MIMO-P for PA If ZF criterion is used, the filter matrices G, B and F can be obtained by performing the QR-factorization to [], i.e., QR G diag r, r,..., r ( B GR F Q where R [ r ij ] is an upper triangular matrix with real diagonal entries and Q is a unitary matrix. he detected signal vector â is given as aˆ a n and can be decoupled into parallel substreams aˆ Ea n ;,,..., ( where n gn,,,...,. On an alternative criterion, the MMSE based solution has also been given in Ref. []. he error that needs to be he equivalent sub-channel model under ZF and MMSE criterions is described in Eqs. ( and (5, respectively. When ZF filter is used, the signal to interference plus noise ratio (SINR of the th user can be calculated as a function of E from Eq. (, i.e., ( σ a E ρ ZF E (7 SINR σ σ g where [ ] n n g G is the scaling coefficients at the th user. When MMSE filter is used, the SINR of the th user can be obtained from Eq. (5 as σ a E ρ MMSE ( E (8 SINR σ 4 g σn σn h j Ej j owever, MMSE case is not as simple as the ZF case. For the clarity of analysis, the approximation of σ is completed by E j E, where E is the sum transmit power. Without loss of generality, the sum power is set as E for comparison with equal power allocation; Eq. (8 is then rewritten as

4 74 he Journal of China Universities of Posts and elecommunications 009 ρ MMSE SINR E ( E (9 4 g σn σn h j j As a result, the MIMO-P sub-channel SINR in Eqs. (7 and (9 can be combined into a unified form as: ρsinr E SINR ( ρ I ; ZF σ n g (0 E ; MMSE 4 σn σn h j g j It is noteworthy that the th user SINR of both ZF and MMSE filters are expressed as a product of their allocated transmit power E and the SINR obtained from E I. hus, the PA matrix E can be calculated in the same manner for both ZF and MMSE cases. Solution to transmit power allocation matrix his article first proposes the MBER-P algorithm. Subsequently, an IMBER-P scheme is obtained, which taes the precoding loss factor into account during the derivation. Finally, the proper ordering scheme of different users is described.. MBER-P for MIMO systems o derive the MBER-P scheme, the BER of each sub-channel is depicted as a function of the transmit power E. he PA matrix E is achieved by means of minimizing the average BER. From the equivalent sub-channel model discussed in Sect.., the average BER P b is expressed as an arithmetic mean of the BER for every sub-channel Pb Pb( ρsinr ( where Pb( ρ SINR is the BER of the th sub-channel. Assume that the constellation sizes of all sub-channels are the same. herefore, the MBER-P scheme under sum-power constraint is given by { E} arg min P b P, P,..., P ( s.t. σ x E where σ x is the power of the th transmit antenna. owever, the expression of P b ( ρ SINR is often complex for square QAM modulation. For simplicity, it can be tightly approximated as an exponential function of the th SINR [4], ρsinr Pb( ρsinr exp ( 5 ( M hus, the average BER can be expressed as: ρsinr b exp P α exp( βp 5 ( M (4 where α (5 and β ρsinr ( I (( M. hen the sum-power constraint is obtained. As illustrated in Sect., the precoded symbol a is assumed to have uniform distribution bounded on. he variance of the precoded symbol can be approximated as σ a t 6 E, which ignores the precoding loss of the P systems. Because the feed forward matrix F in Fig. is a unitary matrix, the power of the th transmit antenna is the same with the variance of precoded symbol a, i.e. σx σ a E. hus, the problem of Eq. ( can be rewritten as an optimization problem { E} arg min P b arg min α exp( βe P, P,..., P P, P,..., P (5 s.t. σ x E E he problem in Eq. (5 can be solved by Lagrange multiplier method [5] and obtain a closed form solution as ln( αβ ln λ E ;,,..., (6 β where ( x max( x,0, and the Lagrange multiplier λ is set to satisfy the sum-power constraint of Eq. (5, i.e., ln( αβ β λ exp (7 β It should be noted that the result has the same form as the classical inverse water filling solutions in MIMO-DFE systems [6]. his means that when the precoding loss of modulo operator is ignored, the MIMO-P system and MIMO-DFE system has strict dual property [7]. When the channel condition is bad, the small constellation modulation (BPS or 4QAM is preferred. In this case, precoding loss factor needs to be considered to improve the system performance.. IMBER-P scheme for MIMO systems In this section, more complicated cases considered. As explicated in Sect.., the precoded symbols a are assumed to be uniform distributions bounded on an enlarged

5 Issue 4 YUN Xiang, et al. / Novel P algorithms with minimum BER criterion for MIMO broadcast communications 75 range. herefore, the transmit power will be increased and is quantified by the precoding loss, which is defined as γ E a E a M ( M,,,..., for square QAM constellations [8 ]. owever, because B is a unit lower triangular matrix, the first entry a of precoding data vector a is directly passed through modulo operator and will not be interfered by other precoding data components. Consequently, the distribution of a is the same with a rather than the uniform distribution bounded on. hus, the precoding loss is redefined as ; γ M (8 γ ;,,..., M Because the feed forward matrix F is unitary, the exact power of x is given by ; σ a E σx σ a γ σa γσ a γ E ;,,..., (9 hen, the problem of Eq. ( can be modified from Eq. (5 as an optimization problem { E} arg min P b arg min Pb( ρsinr P, P,..., P,,..., P P P (0 s.t. σx γ E E In the Appendix A it is shown that the solution to the optimization problem in Eq. (0 is given by αβ ln ln λ γ E ;,,..., ( β where the Lagrange multiplier λ is set to satisfy the sum-power constraint Eq. (0, i.e., γ αβ ln β γ λ exp ( γ β By applying Eq. ( to Eq. (, the closed-form expression of IMBER-P algorithm is obtained. Because the precoding loss factor is involved during the derivation, the IMBER-P algorithm has more accurate PA matrix and obtains global optimal solution to MIMO-P systems. Note that the IMBER-PA structure of the MIMO-P system illustrated in Fig. can be considered as a generalized MIMO-P system. By setting the PA matrix P I, the IMBER-PA structure degenerates to the conventional MIMO-P structure in Ref. [].. Ordering strategy of different users It is well nown that properly ordering different users during the precoding process may achieve better performance for P systems. An interesting best-first ordering strategy was proposed for P in Refs. [ ]. his method aims to minimax the noise variance for the ZF-P or the error variance for the MMSE-P to exchange different rows of channel matrix (channel of different users. From Ref. [], the feasible order of different users { μ, μ,..., μ } is given. he corresponding variance matrix of noise and error can be calculated. According to the corresponding SINR, the IMBER-P algorithm can also be used in the ordering MIMO-P system in Ref. [] to maximize the SNR of all users. he enhanced performance of the IMBER-P algorithm will be shown in Sect Simulation results Computer simulations are performed to evaluate the uncoded BER performance in terms of signal to noise ratio (SNR. Square 4QAM with Gray encoding are employed [5] during the simulation. Assume that the downlin scenario has 4 antennas at the base station (BS and 4 decentralized MS users (i.e., N NR 4. A new channel matrix is generated at each simulation. Its entries are modeled as independent Gaussian random variables with zero mean and unit variance. Perfect CSI is assumed at the transmitter. he simulation results are shown in Figs. and that illustrate the accuracy of the SINR approximation by Eq. (9 and BER approximation by Eq. (, respectively. Without loss of generality, the results are acquired from the first user and the last user sub-channels in the MMSE-P system. As shown in these figures, both approximations are tight and Fig. SINR of the first and last users in MMSE MIMO-P system

6 76 he Journal of China Universities of Posts and elecommunications 009 Fig. 5 illustrates average uncoded BER performances in ordering MIMO-P systems. he ordering strategy employs the best-first strategy proposed in Ref. []. Fig. 5 shows that the ordering IMBER-P algorithm effectively improves the BER performance of ordering MIMO-P [] under both ZF and MMSE scenarios. If ordering ZF-P systems are employed, the SNR gain is about 4.8 db at BER of 0. In the case of MMSE, the ordering IMBER-P algorithm provides.4 db SNR gain at BER of 0. Fig. BER of the first and last users in MMSE MIMO-P system reasonable for use in the proposed algorithms. Fig. 4 shows the performance comparison of the MIMO-P systems under ZF and MMSE criterions. he three inds of curves in this figure denote the conventional MIMO-P systems in Ref. [], the MBER-P in Eq. (6 and the IMBER-P in Eq. (, respectively. It is obvious that the two proposed P schemes achieve better performance than conventional MIMO-P systems. More importantly, when the precoding loss factor is considered during the derivation, the IMBER-P further improves the BER performance of the MBER-P. In ZF cases, it is observed that at BER of 0, the IMBER-P achieves nearly 0.7 db and 4.0 db SNR gains over the MBER-P and the conventional ZF MIMO-P, respectively. When MMSE criterion is employed, the IMBER-P has nearly 0.8 db and 4. db SNR gains over the MBER-P and the conventional MMSE MIMO-P at BER of 0, respectively. his is because the introduced PA matrix leads to power loading operations to optimize the system performance. Fig. 5 Uncoded BER performance comparison in ordered downlin multi-user MIMO-P systems for square 4QAM ( N N 4 5 Conclusions R MBER-P and IMBER-P algorithms for downlin multi-user MIMO-P systems are proposed under ZF and MMSE criterions. Both schemes achieve closed-form solutions by minimizing the average BER under sum-power constraint. he proposed structure performs power loading operation on sub-channel SINR. As a result, the simulation results show that all of the proposed P algorithms outperform the corresponding traditional MIMO-P in Ref. []. When the precoding loss factor is involved, the IMBER-P algorithm results in more accurate PA matrix and performs better than the MBER-P scheme in cases of lower constellation. When properly ordering strategy is employed, further performance improvement can be achieved by IMBER-P algorithm. Acnowledgements Fig. 4 Uncoded BER performance comparison in downlin multi-user MIMO-P systems for square 4QAM (N N R 4 his wor was supported by the Program for New Century Excellent alents in University (NCE

7 Issue 4 YUN Xiang, et al. / Novel P algorithms with minimum BER criterion for MIMO broadcast communications 77 Appendix A his Appendix highlights the major steps leading to Eq. (. he goal is to find the solution to the optimization problem in Eq. (0. he cost function can be expressed as L ( E, E,..., E, λ Pb( ρsinr λ γe (A. o satisfy the condition, the authors obtain a set of equations from L E 0 P b ( ρsinr E λ γ ;,,..., (A. By substituting Eq. (4 into Eq. (A., and solving equations composed of Eq. (A. and the sum-power constraint in Eq. (0, the closed-form solution of the optimal transmit power is αβ ln ln λ γ E ;,,..., (A. β where the Lagrange multiplier λ is set to satisfy the sum-power constraint Eq. (0, i.e., γ αβ ln β γ λ exp (A.4 γ β By applying Eq. (A. into Eq. (A.4, the optimization problem Eq. (0 has been solved and the PA matrix of IMBER-P algorithm is obtained. References. Foschini G J, Gans M J. On limits of wireless communications in a fading environment when using multiple antennas. Wireless Personal Communications, 998, 6(: 5. Wolniansy P W, Foschini G J, Golden G D, et al. V-BLAS: an architecture for realizing very high data rates over the rich-scattering wireless channel. Proceedings of the 998 URSI International Symposium on Signals, Systems and Electronics (ISSSE 98, Sep 9 Oct, 998, Pise, Italy. Piscataway, NJ, USA: IEEE, 998: Cioffi J M, Forney G D. Generalized decision-feedbac equalization for pacet transmission with ISI and Gaussian noise. Communications, Computation, Control and Signal Processing : a ribute to homas ailath Norwell, MA, USA: luwer Academic Publishers, 997: Ginis G, Cioffi J M. On the relation between V-BLAS and the GDFE. IEEE Communications Letters, 00, 5(9: Nam S, Shin O S, Lee B. ransmit power allocation for a modified V-BLAS system. IEEE ransactions on Communications, 004, 5(7: Wang N, Blostein S D. Minimum BER power allocation for MIMO spatial multiplexing systems. IEEE ransactions on Communications, 007, 55(: Vishwanath S, Jindal N, Goldsmith A. Duality, achievable rates, and sum-rate capacity of Gaussian MIMO broadcast channels. IEEE ransactions on Information heory, 00, 49(0: Fischer R F, Windpassinger C, Lampe A, et al. Space-time transmission using omlinson-arashima precoding. Proceedings of the 4th International IG Conference on Source and Channel Coding (SCC 0, Jan 8 0, Berlin, Germany. Piscataway, NJ, USA: IEEE, 00: Bizai, Falahati A. Power loading by minimizing the average symbol error rate on MIMO P systems. Proceedings of the IEEE 9th International Conference on Advanced Communication echnology (ICAC 07: Vol, Feb 4, 007, Phoenix Par, orea. Piscataway, NJ, USA: IEEE, 007: 6 0. Yun X, Lv J, Gong P, et al. Approximate minimum BER power allocation for MIMO-P System. Proceedings of the IEEE International Conference on Wireless Communications and Mobile Computing Conference 008 (IWCMC 08, Aug 6 8, 008, Crete Island, Greece. Piscataway, NJ, USA: IEEE, 008: Fischer R F. Precoding and signal shaping for digital transmission. New Yor, NY, USA: John Wiley & Sons, 00. Liu J, rzymien W A. Improved omlinson-arashima precoding for the downlin of multiple antenna multi-user systems. Proceedings of the 005 IEEE Wireless Communications and Networing Conference (WCNC 05: Vol, Mar 7, 005, New Orleans, LA, USA. Piscataway, NJ, USA: IEEE, 005: usume, Joham M, Utschic W, et al. Cholesy factorization with symmetric permutation applied to detecting and precoding spatially multiplexed data streams. IEEE ransactions on Signal Processing, 007, 55(6: Zhou S L, Giannais G B. Adaptive modulation for multi-antenna transmissions with channel mean feedbac. IEEE ransactions on Wireless Communications, 004, 5(: Proais J. Digital communications. 4th edition. New Yor, NY, USA: McGraw-ill, 00 (Editor: WANG Xu-ying