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Communications and Network, 2010, 2, 145-151 doi:10.4236/cn.2010.23022 Published Online August 2010 (http://www.SciRP.org/journal/cn) Copyright © 2010 SciRes. CN Downlink MBER Transmit Beamforming Design Based on Uplink MBER Receive Beamforming for TDD-SDMA Induced MIMO Systems Sheng Chen, Lie-Liang Yang School of Electronics and Computer Sciences, University of Southampton, Southampton, UK E-mail: {sqc, lly}@ecs.soton.ac.uk Received January 10, 2010; revised April 26, 2010; accepted May 30, 2010 Abstract The downlink minimum bit error rate (MBER) transmit beamforming is directly designed based on the uplink MBER receive beamforming solution for time division duplex (TDD) space-division multiple-access (SDMA) induced multiple-input multiple-output (MIMO) systems, where the base station (BS) is equipped with multiple antennas to support multiple single-antenna mobile terminals (MTs). It is shown that the dual relationship between multiuser detection and multiuser transmission can be extended to the rank-deficient system where the number of users supported is more than the number of transmit antennas available at the BS, if the MBER design is adopted. The proposed MBER transmit beamforming scheme is capable of achieving better performance over the standard minimum mean square error transmit beamforming solution with the support of low-complexity and high power-efficient MTs, particularly for rank-deficient TDD-SDMA MIMO systems. The robustness of the proposed MBER transmit beamforming design to the downlink and uplink noise or channel mismatch is investigated using simulation. Keywords: Minimum Bit Error Rate, Time Division Duplex, Multiple-Input Multiple-Output, Transmit Beamforming, Receive Beamforming, Space-Division Multiple-Access 1. Introduction Motivated by the demand for increasing throughput in wireless communication, antenna array assisted spatial processing techniques [1-7] have been developed in order to further improve the achievable spectral efficiency. In the uplink, the base station (BS) has the capacity to implement sophisticated receive (Rx) beamforming schemes to separate multiple user signals transmitted by mobile terminals (MTs). This provides a practical means of realising multiuser detection (MUD) for space-division multiple-access (SDMA) induced multiple-input multiple-output (MIMO) systems. Tradi- tionally, adaptive Rx beamforming is based on the mini- mum mean square error (MMSE) design [2,5,6,8,], which requires that the number of users supported is no more than the number of receive antenna elements. If this condition is not met, the system becomes rank- deficient. Recently, adaptive minimum bit error rate (MBER) Rx beamforming design has been proposed [9-12], which outperforms the adaptive MMSE Rx beamforming, particularly in hostile rank-deficient systems. In the downlink with non-cooperative MTs at the receive end, the mobile users are unable to perform sophisticated cooperative MUD. If the downlink's channel state information (CSI) is known at the BS, the BS can carry out transmit (Tx) preprocessing, leading to multiuser transmission (MUT). The assumption that the downlink channel impulse response (CIR) is known at the BS is valid for time division duplex (TDD) systems due to the channel reciprocity. However, for frequency division duplex systems, where the uplink and downlink channels are expected to be different, CIR feedback from the MT's receivers to the BS transmitter is necessary [13]. Many research efforts have been made to discover the equivalent relationship between the MUD and MUT [14-19]. Notably, Yang [20] has derived the exact equivalency between the MUD and MUT for TDD systems under the condition that the number of antennas at the BS is no less than the number of MTs supported1. 1All the existing works, such as [14-20], consider the designs of MUD and/or MUT using second-order statistics based criteria, which implies that the MIMO s y stem must have full r ank. S. CHEN ET AL. 146 According to the results of [20], the MUT can be obtained directly from MUD. Since the BS has to implement MUD, it can readily implement MUT based on its uplink MUD solution with no extra computational complexity cost. This is very attractive as this strategy enables the employment of low-complexity and high power-efficient MTs to achieve good downlink per- formance. In general, the BS can design MUT when the downlink CSI is available. The Tx beamforming design based on the MMSE criterion is popular owing to its appealing simplicity [13,21]. Since the bit error rate (BER) is the ultimate system performance indicator, research interests in MBER based Tx beamforming techniques have inten sified recently [22-25]. Th is MBER based MUT design invokes a constrained nonlinear optimi- sation [22-25], which is typically solved using the iterative gradient-based optimisation algorithm known as the sequential quadratic programming (SQP) [26]. However, the computational complexity of the SQP based MBER MUT solution can be excessive and may become impractical for high-rate systems [23]. This contribution adopts a very different approach to design the MBER Tx beamforming for TDD-SDMA induced MIMO systems, which does not suffer the above men- tioned difficulty of high complexity. We prove that Yang's results [20] are more general and the exact equivalency between the MUD and MUT is not restricted only to second-order statistics based designs. Therefore, we can apply the results of [20] to implement the MBER Tx beamforming scheme directly based on the MBER Rx beamforming solution already available at the BS with no computational cost at all, even for rank-deficient systems where the number of antennas at the BS is less than the number of MTs supported. The robustness of the proposed scheme is also investigated when the downlink and uplink noise powers or channels mismatch. 2. Multiuser Beamforming System The TDD-SDMA induced MIMO system considered is depicted in Figure 1, where the BS employs antennas to support L K single-antenna MTs. When the uplink is considered, the received signal vector at the BS is given by ,1 ,2, =[ ] UUU xx xxT UL =1 =n= n K UUkk k s xHsh U (1) where the channel matrix is given by LK 12 =[ ] K Hhhh, = kk Figure 1. Schematic diagram of the TDD-SDMA induced MIMO system employing transmit and receive beamfor- mings at the BS. The system employs L antennas at the BS to support K single-antenna MTs. dditive white Gaussian noise (AWGN) vaector with 2 [nn] = 2 H UU UL E I, and L I represents the LL ithout thloss of generalit assume the binary phase shift keying (BPSK) modulation. The result however can be extended to modulation schemes with multiple bits per symbol by adopting the minimum symbol-error-rate design [9]. The MUD at the BS consists of a bank of Rx beamformers ,=u, 1, H Ukk U ykKx identity matrix. We y, we (2) where is the Rx beamformer's uk an weight vector for user kd H denotes the Hermitian operator. The decisi variae vector ,1 ,2, =[ ] T UUU UK yy yy for the on bl K transmitted symbols ca == n HHH UU U yUxUHsU n be expressed as (3) with the LK Rx beamforming coeff y (4) The real part of is a sufficient s he BS employs the Tx beamforming for the do icient matrix expressed b 12 =[u uu]. K U U y tatistics for detecting s. T wnlink transmission to the K MTs, with the LK transmission preprocessing matx 12 =[d dd]D ri (5) where is the precoder's coef , K k dficient vector for preprocessing the symbol k s to be transmitted to the kth MT. Note that we u the same notation s to esent the downlink symbol vector, witout distinction from the uplink symbol vector for the purpose of notational simplification. Due to the reciprocity of the downlink and uplink channels, the received signal vector ,1 ,2, =[ ] T DDD DK yy yy, received by the se repr h K MTs, is k A hg with k A and denoting the channel coefficient and the steering vector for the kth user, respectively, k g 12 =[ T ] K s ss K s he uplink c hannel contains the data symbols transmitted by the MTs to the K BS, and nU denotes t expressed as ,=n T D D yHDs (6) where is the downlink AW n D GN vector with 2 [nn2] = H D DD E IK and denotes the transpose T Copyright © 2010 SciRes. CN S. CHEN ET AL.147 r. Thoperatoe real part of the decision variable , D k y is detecting the symbol k used by the th MT for k s transmitted from the BS to the kth MT. Under the condition LK, there exists an exact equivalency between the Tx beamforming preprocessi matrix D and the Rx beamfoing weng rm ight matrix U expressed by [2 0] * =, DU (7) where *denotes thejugate ope } conrator, 12 ,Λ,,={ K diag c r desi ne a d is for achieving the transmit power int, and U or D is assumed to have been based on some second-order statistic criterion. The exact relationship (7) is valid for 22 = onst g D U . A simple scheme to implement the transmit power constraint is to set =1/ kk u for 1kK relationship (7) is easy to understand. Under the condition of LK, H Und H) have the same full rank and the same second-order statistic properties, given (7). For rank-deficient systems where <LK, H UH and T HD no longer ha ve the full rank . Inde ed, both the [20]. The H of (3) a M TD of ormi (6 ngMSE Rx beamforming and MMSE Tf turn out to be deficient in this case, exhibiting a high BER floor. However, the MBER Rx beamforming scheme [10,12] has been shown to consistently outperform the MMSE design and is capable of operating in rank-deficient systems where the number of MTs is more than the number of receive antennas at the BS. We will show in the next section that the relationship (7) is not restricted only to second-order statistics based designs. Specifically, with the MBER MUD and MUT designs, (7) is valid and, moreover, the restriction LK is no longer required. This enables us to use (7) directly for designing the MBER Tx beamforming based on the available MBER Rx beamforming solution even for rank-deficient systems. 3. MBER Receive and Transmit x beam Beamforming The BER of detecting k s using th mforming with the e up be weior can lin be show k Rx n aght vectk u to be [10, 12] () () ,()[ ] 1 ()=, NqHq skk Rx kk sgn s PQ Hs u (8) =1q skU N u u where e usual Gaussian error function ()Q is th, [] denotes tal part, he re=2 K s N ate transmit sy is the number o equipritimmbol v f all th ectors, s e ()q obable leg for 1 s qN , and ()q k s denotes the k th element f ()q s. The MBER solutik is then defined as MBER, , =arg( ). min kRxkk k P u uu ( ) The optimisation (9) can be solved using a ient-based numerical optisation algorithm [1 o for 9 2]. BER is iario a ti als norlise e beamforming on nv way u im ant t ma gra No k u d te that the , and one canposi th ve scaling of weight vector to a unit length, yielding =1 k u, which significantly reduces the computational complexity of imisation. This is also useful for directly implementing the Tx beamforming design from the Rx beamforming design using the relationship (7), as the scalin matrix Λ can be chosen as the identity matrix in this case. Adaptive MBER Rx beamforming can be achieved using opt tha -mu g the stochastic gradient-based algorithm known as the least bit error rate [10, 12]. It is clear from (8) and (9) t the optimal MBER Rx beamforming solution to k u is self-centred without regarding the effect on the other users. This type of optimisation is referred to as the egocentric-optimisation (E-optimisation) and the resul ting solution is known to be egocentric-optomum (E-opti m) [20]. Th e average BER over a ll the K users for the Rx beamforming with the beamforming weight matrix U is obviously given by , =1 () () ()= ( )= ()[ 1 RxRx k k k NqHq Ksk PP K sgn s Q Uu uH (10) 1K =1 =1 ] , k kq skU KN s u and the MBER Rx beamforming solution U ply given by MBER =arg( ) min Rx P U U (11) is simMBERMBER,1 MBER,2MBER, =[ ] K Uuu u. u forSince all the Rx beamforming vectors MBER,k 1kK are optimum in some sen matrix MBER U is overall optimum se (E-optimum), the Rx beamforming (O-optimum) [20]. With the precoder coefficient matrix D, the BER of detecting k s by the kth MT is () () ,=1 Tx kq sD Q N (12) for 1kK ()[ ] qT skk sgn s hDs 1 ()= Nq P D . The avrage BER o K ever all the users eamforming with the precodeweight then given by for matrix the Tx b D is r's , =1 () () )= ()= ()[ ] 1 Tx k k NqTq Ks P K sgn s Q DD hDs (13) =1 =1 1 ( . K Tx kk kq sD P KN Copyright © 2010 SciRes. CN S. CHEN ET AL. 148 The MBER Tx beamforming solution for can be obtained by minimising subject to given transmit power constraint (14) r opt g the computatio Unlike the E-optimum of the MBER the optimal MBER soultion to a precoden D the() Tx PD MBER=arg( ) min Tx P D DD s.t.transmit power constraint ismet This constrained nonlineaimisation problem for example can be solved usinSQP algorithm [22, 23, 25], which is however nally expensive. Rx beamforming, r's colum vector k d is not self-centred, as it is clear from (12) to (14) that an optimal solution to k d of user k not only maximises the kth user's performance but also pays attention on mitigating its effect on the other 1K users. This type of optimisation is referred to as the altruistic-optimisation (A-optimisation) and the resulting solution is known to be altruistic-optimum (A-optimum) [20]. Denote M RMBER,1MBER,2MBER, =[ ] k Ddd d. Since all the columns MBER,k d for 1kK are optimu some sense (A-optimum), the precoding matrix MBER D is also O-optimum [20]. BE m in applying the computationally ex ectly derive it as (15) cost at all, provided that the more general. We start by examining the probabilit Instread ofpensive SQP algorithm to find the MBER Tx beamforming solution, we can dir * MBER MBER =,DU with no computational relation (7) is not restricted to second-order statistics based designs and it is also valid for the MBER design. We now show that (7) is indeed much y density functions (PDFs) of [] D y and [] u y, the real part of D y in (6) and the real part of U y in (3), respectively. Without loss of generality, we assume that =1 H kk uu , 1kK , as the BER is invariant to the length of k u [12]. Denote = R I jH, wHH ith th part =[] RHHthe imaginary part =[ IHH and 2=1j. Similarly , let = e real ] , R I jUU U and = UU U R I jnn n []=( )(). TTT T URRII RUIU . Then R I UyUHUHsUn n (16) [] U y, denoted as ( ) UK pw , is obviously Gause mean [[ ]) U I y s The PDF of sian with th (17) and the covariance matrix H On the other hand, with the notation 12 , ,,w w ]=(TT RRI EUHUH 22 12 [] U H UU uu 2 1 222 21 2 12 [[ ]]= [ ] [] [ [] [] U K H UUU UK UKU Cov y uu uu u uu uu 2] . H K u (18) 22HH = D DD R I j nn n and given , . * =DU []=( ) TT DRRIID R y HUHU sn The PDF of (19) [] D y, denoted as 12 (, ,,) D K pww w, is Gaussian with the mean s (20) nce [[ ]]=(TT DR and the covaria matrix ) R II EyHUHU 2 [[ ]]=. D DK Cov yI by the (21) The BER (10) is determined K marginal PDFs of [] U y, ,() Uk pw k nl ar forhich are Gausd are speci eleovariance gin 1kK fied ments of t al PDFs , w he c of sian distributed aby the mean vector (17) and the diagona matrix (18). The K m[] D y, ,() D kk pw for 1kK , are als of the co o G atri aussian distributed and are specified by the mean vector ( and the diagonal evariance m (21). These two sets of the marginal PDFs are almost ``identical'', given 22 = UD 20) xlements . The difference is that ,() Uk k pw only depends on the kcolumn vector of U while ,() th D kk pw depends on the entire matrix * U, leading to the BER expressions (10) and (13). We quote the following result from [20]. Proposition 1 An E-optimum solution in a MUD is equivalent an A-optimum solu the corresponding MUT et MBER U be the MBER Rxeamforming solution of (11). That is, the column vectors of MBER U are E-optimum in the c totion in . Now l B ontext of the MBER Rx linear aK use The ngles of arrival (departure) for the four users were Beamforming. Then * MBER U is the MBER Tx Beamforming solution of (1 4), and the co lumn vectors of * MBER are Aum in the context of the MBER Tx Beamforming. Note that we do not require LK 4. Simulation Study Full-rank system. The BS employed a four-element ntenna array with half-wavelength element spacing to support four single-antenna BPSrs. U-optim . a 2 , plink were 16, 15 and 30, respectively, and the u elffor the four users chann coeficients = 0.70710.7071 k Aj , 14k . The full uplink CSI was assumed to be known at the BS and was used by the BS to design the uplink Rx beamforming. Figure 2 compares the average BER performance, defined in (, nk MM and MBER Rx beamforming schemes. The BS then directly implemented the down- 10) of the upliSE Copyright © 2010 SciRes. CN S. CHEN ET AL.149 Figure 2. Average BER performance comparison of the uplink Rx and downlink Tx beamforming schemes with both the MMSE and MBER designs for the TDD-SDMA induced MIMO system which consists of a four-antenna BS to support four single-antenna BPSK MTs. link Tx beamforming from the resulting uplink Rx beamforming according to (7). Assuming the exact reciprocity of the uplink and downlink channels as well as 22 = UD , the average BERs, as defined in (13), of both the MMSE and MBER Tx beamforming schemes so esigned are also plotted in Figure 2, in comparison with h that of d the average BERs of the MBER and MMSE Rx beamforming designs. As expected, the performance of the proposed downlink Tx beamforming design agreed witthe uplink Rx beamforming design. The robustness of the proposed Tx beamforming design was next investigated when the downlink and uplink noise powers or channels were mismatched. In the case of noise mismatching, the downlink noise power was 3 dB more than the uplink noise power. The average BERs of the MBER and MMSE Tx beamforming desi- gns under this uplink and downlink noise power mismatching are plotted in Figure 3, in comparison with the case of equal uplink and downlink noise powers. It can be seen that the 3 dB noise-power mismatching had little influence on the performance of the MBER Tx beamforming scheme but it had some influence on the performance of the MMSE Tx beamforming design. This was expected as the MMSE design is explicitly influen- ced by the noise power while the BER calculation is relatively insensitive to the noise variance estimate used. In the case of channel mismatching, the uplink channel coefficients were = 0.70710.7071 k Aj for 14k , but the downlink channel coefficients were =0.6 0.8 k A j for 14k. The average BER performance of the MBER and MMSE Tx beamforming designs under this uplink and downlink channel mis- matching are plotted in Figure 4, in comparison with the Figure 3. Average BER performance of the downlink Tx beamforming schemes with both the MMSE and MBER designs for the TDD-SDMA induced MIMO system which consists of a fou r-antenna BS to support four single-antenna BPSK MTs, when the dow nlink and uplink noise powers mismatch. Figure 4. Average BER performance of the downlink Tx beamforming schemes with both the MMSE and MBER designs for the TDD-SDMA induced MIMO system which consists of a four-antenna BS to support four single-antenna BPSK MTs, when the downlink and uplink channels mismatch. SI. case of the exact reciprocity of uplink and downlink channels. From Figure 4, it can be seen that this imperfect CSI had little influence on the MMSE Tx beamforming scheme. It is also seen that the MBER Tx beamforming design was not overly sensitive to the mperfect CiRank-deficient system. The BS again employed a four-element linear antenna array with half-wavelength element spacing but the number of single-antenna BPSK users was increased to six. The angles of arrival (depar- ture) for the six users were 2 , 15, 10, 30, 25 Copyright © 2010 SciRes. CN S. CHEN ET AL. 150 Figure 5. Average BER performance comparison of the uplink Rx and downlink Tx beamforming schemes with both the MMSE and MBER designs for the TDD-SDMA induced MIMO system which consists of a four-antenna BS to support six single-antenna BPSK MTs. and respectively, and the uplink channel co-effi- 36, cients for the six users were = 0.70710.7071 k Aj, 16k. Given the exact reciprocity of the uplink and downlink channels as well as 22 = UD , the average BERs of both the MBER and MMSE Tx beamforming esigns, implemented directly from the corresponding eam e average BERs of th rank-defi d simila d Rx bforming schemes according to (7), are plotted in Figure 5, in comparison with the MBER and MMSE Rx beamforming designs. For this cient system, both the MMSE Rx and Tx beamforming solutions exhibiter high BER floors while both the MBER Rx and Tx beamforming solutions achieved similarly adequate performance. The robustness of the proposed Tx beamforming design was then investigated again under the scenarios of mismatched downlink and uplink noise powers or channels. In the case of the downlink channel having 3 dB more noise power than the uplink channel, the average BERs of the MBER and MMSE Tx beamforming designs, implemented directly from the corresponding Rx beamforming solutions according to the relation (7), are plotted in Figure 6, in comparison with the case of 22 = UD . It can be seen that the 3 dB noise-power mismatching had little influence on performance. In the case of channel mismatching, the uplink channel coefficients were = 0.70710.7071 k Aj for 16 k, but the downlink channel coefficients were =0.6 0.8 k A j for 16k. The average BER performance of th and MMSE Tx beamforming designs under this uplink and downlink channel mismatching are plotted in Figure 7, in comparison with the case of an identical uplink and downlink channel. Figure 6. Average BER performance of the downlink Tx beamforming schemes with both the MMSE and MBER designs for the TDD-SDMA induced MIMO system which consists of a four-antenna BS to support six single-antenna BPSK MTs, when the downlink and uplink noise powers mismatch. Figure 7. Average BER performance of the downlink Tx beamforming schemes with both the MMSE and MBER designs for the TDD-SDMA induced MIMO system which consists of a four-antenna BS to support six single-antenna BPSK MTs, when the downlink and uplink channels mismatch. From Figure 7, it can be seen that the MBER Tx nk MBER transmit beamforming solution has where the number of MTs supported is beamforming design was not overly sensitive to the imperfect CSI. 5. Conclusions he downliT been derived directly based on the uplink MBER receive beamforming design for TDD-SDMA induced MIMO systems. It has been shown that even for rank-deficient TDD systems, e MBER Copyright © 2010 SciRes. CN S. CHEN ET AL. Copyright © 2010 SciRes. CN 151 me than the number of transmit antennas available at lent relationship between the MUD still valid, if the MBER ime Wireless Communications,” Cambridge ss, Cambridge, 2003. . Viswanath, “Fundamentals of Wireless munications, Vol. 5, 1998, pp. 23-27. “Adap- [10] ro c e e d i n g s o f International or the BS, the equiva and the corresponding MUT is design is adopted. The proposed MBER transmit beam- forming design imposes no computational cost at all at the BS and is capable of achieving good downlink BER performance with the support of low-complexity and high power-efficient MTs. The robustness of this transmit beamforming scheme to the downlink and uplink noise or channel mismatching has been investigated by simulation. 6. References [1] A. J. Paulraj, D. A. Gore, R. U. Nabar and H. Bölcskei, “An Overview of MIMO Communications - A Key to Gigabit Wireless,” Proceedings of IEEE, Vol. 92, No. 2, 2004, pp. 198-218. [2] A. Paulraj, R. Nabar and D. 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