2014 IEEE 8th Sensor Array and Multichannel Signal Processing Workshop (SAM)
Joint Beamforming and Transmit Design for the Non-Regenerative MIMO Broadcast Relay Channel Jens Steinwandt 1 , Sergiy A. Vorobyov 2,3 , and Martin Haardt 1 1
Communications Research Laboratory, Ilmenau University of Technology, P.O. Box 100565, 98684 Ilmenau, Germany 2 Department of Signal Processing and Acoustics, Aalto University, P.O. Box 13000, FI-0076 Aalto, Finland 3 Department of Electrical and Computer Engineering, University of Alberta, Edmonton, AB, T6G2V4, Canada Emails: {jens.steinwandt, martin.haardt}@tu-ilmenau.de,
[email protected]
Abstract—In this paper, we consider a multiple-input multiple-output (MIMO) broadcast relay channel (BRC), in which the communication of a multi-antenna base station (BS) with several multi-antenna mobile stations (MS) is assisted by a fixed half-duplex multi-antenna relay station (RS). Applying dirty paper coding (DPC) at the BS and beamforming at the RS, we jointly optimize the transmit covariance matrices at the BS and the beamforming matrix at the RS by maximizing the system sum rate, which is a nonconvex problem. To solve this problem, we resort to the more tractable sum rate maximization in the dual multiple access relay channel (MARC), which is still a nonconvex difference of convex functions (DC) programming problem. We develop an iterative algorithm, termed alternating matrix polynomial time DC (POTDC) algorithm, based on an alternating optimization of the beamforming matrix and the transmit covariance matrices. The resulting covariance matrices for the MARC are then mapped to the desired BRC covariance matrices. The sum rate performance of the proposed algorithm is demonstrated by simulations. Index Terms—Multiuser MIMO relaying, sum rate maximization, alternating optimization, difference of convex functions.
I. I NTRODUCTION In recent years, multiple-input multiple-output (MIMO) systems have been employed in numerous wireless communication systems, such as point-to-point multiple-antenna communications and cellular multi-user communications [1]. It is well known that multiple antennas at wireless terminals can achieve channel capacity enhancements and robustness against channel fading. Furthermore, with the deployment of fixed relay stations (RS) at the cell edges or severely blocked areas in cellular systems, the coverage area can be extended to enhance the throughput of cell-edge users [2]. Various relay strategies have been studied in the literature. The most prominent strategies are amplify-and-forward (AF) and decode-and-forward (DF). Due to its simplicity, the AF scheme is usually preferred in practice. In order to exploit the above-mentioned benefits, a promising compound scheme that incorporates MIMO technology into the fixed relay architecture with single antenna users was introduced in [3]. The joint source-relay optimization for an AF-based MIMO broadcast relay channel (BRC) with multi-antenna mobile stations (MSs) was considered in [4]. The maximization of the sum rate to jointly optimize the transmit covariance matrices at the base station (BS) and the beamforming matrix at the RS as in [4] is an intricate task. This is due to the nonconvex nature of the optimization problem, which is a difference of convex functions (DC) programming problem. The iterative algorithm developed in [4] is based on solving the same problem for the more tractable dual multiple access relay channel (MARC) by performing DC iterations and hence is suboptimal. Recently, a polynomial time DC (POTDC) method [5] for the class This work was partially supported by the International Graduate School on Mobile Communications (MOBICOM), Ilmenau, Germany.
978-1-4799-1481-4/14/$31.00 ©2014 IEEE
M1
Mr
Mb
M2
G2
H
BS
MS1
G1
MS2
RS GK
MK
MSK Fig. 1.
MIMO broadcast relay channel scenario with K terminals.
of DC problems with the optimization over a single variable was developed and its global optimality was proven in [6]. The authors of [7] and [8] have subsequently extended the POTDC algorithm to the optimization over a matrix. In this paper, we consider a MIMO BRC, where a BS simultaneously transmits to K MSs through a fixed half-duplex RS. All parties are equipped with multiple antennas and perfect CSI is assumed everywhere in the network [3], [4]. We jointly optimize the transmit covariance matrices at the BS and the beamforming matrix at the RS by maximizing the system sum rate. Similarly to [4], we resort to the more tractable sum rate maximization in the dual MARC to solve the original BRC problem. We develop an iterative algorithm, termed alternating matrix POTDC algorithm, based on an alternating optimization of the beamforming matrix and the transmit covariance matrices. The resulting covariance matrices for the MARC are then mapped to the desired BRC covariance matrices. Simulation results demonstrate that the proposed alternating POTDC algorithm outperforms the method in [4]. II. S YSTEM M ODEL AND P ROBLEM S TATEMENT A. MIMO Broadcast Relay Channel Consider the MIMO BRC scenario depicted in Fig. 1, where the BS serves K MSs simultaneously with the help of a single RS. We assume that there is no direct link between the BS and the K MSs. The BS, the RS, and the i-th MS are equipped with Mb , Mr , and Mi antennas, respectively. Moreover, we assume that all the channels are flat block-fading channels, i.e., they are constant over a block length and independent from block to block. Perfect channel state information (CSI) is considered at the BS, RS, and the MSs [3], [4]. The transmission takes place in two phases. In the first phase,Pthe BS employs dirty paper coding (DPC) K Mb and transmits x = is the i=1 xi to the RS, where xi ∈ C codeword transmitted to the i-th MS with the covariance matrix Σi = E{xi xH i }. Hence, the covariance matrix of x is given by
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2014 IEEE 8th Sensor Array and Multichannel Signal Processing Workshop (SAM)
PK Σ = E{xxH } = i=1 Σi . The received signal at the RS is given by yr = Hx + nr , where H ∈ CMr ×Mb is the BS-to-RS channel and nr ∈ CMr contains the i.i.d. noise samples at the RS 2 with nr ∼ CN (0 PMKr , σnr IMr ). The transmission power at the BS is constrained by i=1 Tr{Σi } = PBt , where PBt is the maximum allowable transmit power and the subscript B stands for BRC. In the second transmission phase, the RS processes the received signal yr by the beamforming matrix DB ∈ CMr ×Mr and transmits the vector xr = DB yr = DB Hx + DB nr to the K MSs. The received signal at the i-th MS can be expressed as X X yi = Gi DB Hxi + Gi DB Hxj + Gi DB Hxj ji
+ G i DB nR + ni ,
(1)
Mi ×Mr
is the channel from the RS to the i-th MS and where Gi ∈ C ni ∈ CMi is the vector of i.i.d. noise samples at the i-th MS with ni ∼ CN (0Mi , σn2 i P IMi ). The transmit power constraint at the RS H 2 H is given by Tr{DB ( K i=1 HΣi H + σnr IMr )DB } = PBr , where PBr is the maximum relay transmit power. Applying the concept of DPC, the BS chooses codeword xi for the i-th MS by taking into account the knowledge of the interference caused by the previously chosen codewords xj for j < i. Thus, the P BS pre-cancels the interference term j 0, ǫ2 > 0, Q(0) = (n−1) RM |
10
IKM ;
9
while − > ǫ1 do Set l = 1; ˜ (n,l) − R ˜ (n,l−1) | > ǫ2 do while |R M M (n,l) Solve (14) to obtain ΛD ; (n,l) (n,l) ΛD,c = ΛD ; l = l + 1; end while (n) (n,l) ΛD = ΛD ; Solve (11) to obtain Q(n) ; n = n + 1; end while 1/2 Compute D ∗ = UH ΛD VGH ; ∗ Solve (7) to obtain Q ; Output: Q∗ , D ∗ .
1 −1/2 1/2 1/2 −1/2 M Ui ViH Bi Qi Bi Vi UiH Mi , c2 i
−1/2
7 6 5 4 3 2 1 0
as the respective interference-plus-noise covariance matrices seen by the i-th MS in the BRC and the dual MARC. Then, the Σi can be generated from the Qi as Σi =
DC−Iter AM−POTDC
8 Sum rate (bps/Hz)
2: 3: 4: 5: 6: 7: 8: 9: 10: 11: 12: 13: 14: 15:
(n) |RM
PT KM
(15)
−1/2
where Mi H H DGH = Ui Λi ViH is the SVD of the i Bi effective channel matrix experienced by the i-th MS. Note that H H 2 2 B1 = G1 DD H GH 1 + σn1 IM1 and MK = H DD H + σnb IMb as MS1 in the BRC and MSK see no interference from the other users. Moreover, Σi only depends on Σj , j < i and thus, the Σi can be computed sequentially from the Qi in an ascending order.
5
10
15 SNR (dB)
20
25
30
Fig. 2. Sum rate versus the SNR for Mb = Mr = 5, K = 2 users with M = 5 and PBt = PBr = PMt = PMr = 1.
the alternating matrix POTDC algorithm, which is based on an alternating optimization of the beamforming matrix and the transmit covariance matrices. The resulting covariance matrices for the MARC are then mapped to the desired BRC covariance matrices. Simulation results have demonstrated the superior performance of the proposed alternating matrix POTDC algorithm compared to the method in [4].
IV. S IMULATION R ESULTS In this section, we present simulation results to demonstrate the performance of the proposed alternating matrix POTDC algorithm “AM-POTDC” for solving the sum rate maximization problem for the MIMO BRC. For comparison purposes, we include the recently proposed method “DC-Iter” in [4] into our evaluation. In the simulations, the channels H and Gi , i = 1, . . . , K, are randomly generated and drawn from an i.i.d. complex Gaussian distribution with zero mean and unit variance. Moreover, we assume that the K MSs are equipped with the same number of antennas, i.e., Mi = M . For the proposed algorithm and the method from [4], the initialization P Mt Q(0) = KM IKM is chosen such that equal power is allocated across all MSs. The thresholds ǫ1 and ǫ2 are both set to 10−5 and the results are obtained by averaging over 1000 independent Monte Carlo trials. In Fig. 2, we illustrate the sum rate as a function of the signal-tonoise ratio (SNR). We have Mb = Mr = 5, and K = 2 users with M = 5. The transmit power limits are given by PBt = PBr = PMt = PMr = 1. It can be seen that the proposed AM-POTDC algorithm outperforms the DC-Iter method and provides a higher sum rate. V. C ONCLUSION In this paper, we have considered the nonconvex sum rate maximization problem for the MIMO broadcast relay channel (BRC). In this scenario, a multi-antenna base station communicates with several multi-antenna mobile stations with the help of a fixed halfduplex multi-antenna relay station. We jointly optimize the transmit covariance matrices at the BS and the beamforming matrix at the RS by maximizing the sum rate. To solve this problem, we resort to the more tractable sum rate maximization in the dual multiple access relay channel (MARC), which is still a nonconvex difference of convex functions (DC) programming problem. We have developed
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