IEEE COMMUNICATIONS LETTERS, VOL. 15, NO. 6, JUNE 2011
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A Linear Precoding Scheme for Downlink Multiuser MIMO Precoding Systems Hualei Wang, Lihua Li, Lei Song, and Xiangchuan Gao Abstract—In this letter, we propose a low complexity linear precoding scheme for downlink multiuser MIMO precoding systems where there is no limit on the number of multiple antennas employed at both the base station and the users. In the proposed algorithm, we can achieve the precoder in two steps. In the first step, we balance the multiuser interference (MUI) and noise by carrying out a novel channel extension approach. In the second step, we further optimize the system performance assuming parallel SU MIMO channels. Simulation results show that the proposed algorithm can achieve elaborate performance while offering lower computational complexity. Index Terms—Multiuser MIMO, precoding, MUI, block diagonalization (BD), channel extension.
I. Introduction
R
ECENTLY the research of the capacity region of the multiuser MIMO broadcast channels (BC) has been of concern. In multiuser MIMO systems, multiuser interference (MUI) is an important factor that affects system performance as well as noise. A zero-forcing channel inversion (ZF-CI) scheme [1] is proposed to eliminate MUI. However, the ZFCI scheme is scaled to the single receive antenna per user case, and its performance is poor due to suffering from a power penalty. Although a minimum mean-squared error channel inversion (MMSE-CI) method [1] overcomes the power penalty problem of ZF-CI, this scheme is still confined to a single receive antenna per user case. In the scenario where multiple antennas are located at both every user in the terminal and base station, [2] porposed a low-complexity block diagonalization (BD) method. But BD scheme only accommodates the scenario where the number of the transmit antennas at the base station is no smaller than the sum of the receive antennas of all user terminals. Moreover, the BD attempts to completely eliminate the MUI without any consideration of the noise. Therefore the performance of BD scheme is poor at low SNR regime. With the purpose of improving the performance of BD, a regularized BD (RBD) scheme [3] is proposed. However, the complexity of RBD is too high, which is difficult to implement in practice. Thus, in this letter, we propose a low complexity linear precoding scheme, which can obtain elaborate performance and be used for downlink multiuser MIMO systems with arbitrary system configuration in terms of the number of antennas.
Manuscript received March 1, 2011. The associate editor coordinating the review of this letter and approving it for publication was R. Nabar. This work is supported by the National Science Foundation of China (NSFC-AF 60910160), the Major National S&T Program (2009ZX03003001-01,2009ZX03002-003-04), and co-funded by Fujitsu. The authors are with the Key Lab. of Universal Wireless Communications, Beijing Univ. of Posts and Telecommunications (BUPT), Ministry of Education, Wireless Technology Innovation Institute (WTI), BUPT, China (e-mail:
[email protected],
[email protected]). Digital Object Identifier 10.1109/LCOMM.2011.040111.110449
The following notations are used throughout the letter. Boldface capitals and lowercases denote matrices and vectors. Superscripts (·)T ,(·)H , T r(·) denote the transpose, conjugate transpose operation, trace, respectively. I and 0 are identity matrix and zero matrix, respectively. II. System Model We consider a multiuser MIMO downlink precoding system with a base station communicating with K users simultaneously. The base station employs N transmit antennas and each user is also equipped with multiple antennas. Let Mi denote the number of receive antennas at the ith user. In this way, such system is expressed in the following as (M1 , · · · , MK ) × N. In this letter, xi (n) denotes the transmitted data for user i at time n, and Wi denotes the precoding matrix for user i, K which satisfies the power constraint T r( i=1 WiH Wi ) PT . The received vector of size Mi × 1 at the user i at the time n is given by yi (n) = Hi
K
Wk xk (n) +vi (n)
(1)
k=1
where vi (n) is the ith user’s noise vector, which is assumed to have independent complex Gaussian elements with variance σ2ω and zero mean. The elements of Hi are complex Gaussian variables with zero mean and unit variance. Also we assume perfect CSI available at the transmitter and the channel is quasistic. In this letter, for user i(i = 1, · · · , K), we denote matrix Hi as T Hi = HT1 , · · · , HTi−1 , HTi+1 , · · · , HTK (2) In other words, Hi is composed of all the channel matrixes in the system other than the user i own channel matrix. The total number of receive antennas MR can be expressed as MR = K Mi . Then similarly we denote Mi as M i = MR − Mi , which
i=1
is the number of all the receive antennas other than the user i own receive antennas. III. Proposed Precoding Scheme for Multiuser MIMO Systems
In this section, we first introduce the proposed channel extension approach, and then propose the precoding method for multiuser MIMO precoding systems. The idea of channel extension is always used in the singleuser MIMO detection. In this letter, a novel channel extension approach for multiuser MIMO precoding systems is proposed. Unlike conventional channel extension method, the proposed channel extension approach is to extend Hi rather than Hi in order to balance MUI and noise’s influence on system’s performance. And later it is proved that the proposed algorithm
c 2011 IEEE 1089-7798/11$25.00
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IEEE COMMUNICATIONS LETTERS, VOL. 15, NO. 6, JUNE 2011
indeed can balance MUI and noise well. On the other hand, the position of additional part is also different. In the proposed channel extension, the additional part is placed at the left not below of Hi , which can increase the number of column. Thus the proposed channel extension approach can overcome the dimension constraint for obtaining null space of Hi . The proposed precoder can be obtained in two steps. Step 1: By carrying out the proposed channel extension method, we can obtain the first precoding matrix Fai , which can balance MUI and noise effectively. In [3], RBD adopts SVD operation and inversion operation to obtain Fai and it can’t be replaced by some low complexity decomposition such as QR or LQ. But in the proposed algorithm, by carrying out the proposed channel extension, LQ decomposition can be used to obtain Fai . For user i(i = 1, · · · , K), by extending the matrix Hi , the matrix Hi can be obtained as Hi = ∂I, Hi (3) T = ∂I, HT1 , · · · , HTi−1 , HTi+1 , · · · , HTK
where I is a M i × M i identity matrix, the sizes of the matrices Qi,1 , Qi,2 , Qi,3 , Qi,4 are M i × M i ,M i × N , N × M i , N × Nrespectively, Li,1 is a lower M i × M i triangular matrix, and Li,0 is a M i × N zero matrix. From (6) and (7), we can obtain Fai = Qi,4 . And from (7), we can also obtain Hi × Qi,4 + ∂I × Qi,2 = Li,0 = 0 Due to the fact that the columns of [ unitary vectors, we can obtain
(8)
Qi,2 ] are orthonormal Qi,4
H H Qi,2 +Qi,4 Qi,4 = I Qi,2
(9)
From (8) and (9), we can achieve H
H Qi,4 (Hi Hi + ∂2 I)Qi,4 = ∂2 I
(10)
where ∂ is equal to MR × σ2 Ptotal and I is a M i × M i identity matrix. From (3), it can be clearly seen that the size of Hi is M i × (N+M i ), which guarantees the dimension require for obtaining null space of Hi . Thus, there is no additional requirement on the system’s configuration in terms of the number of antennas. Then, in order for the null space of Hi , we employ LQ based on Givens or Householder decomposition as
The expression (10) shows that the matrix Qi,4 is just the result of the optimal problem in [3], which aims to balance MUI and noise well. This concludes the proof of theorem 1. Step 2: By assuming parallel SU MIMO channels, we carry out SVD operation to obtain the second precoding matrix Fbi , which can further optimize the system’s performance. In this letter, we don’t consider power balance between users, and only balance per user own substream power by water-fillling algorithm. For user i , the SVD of Hi Fai is computed as
Hi = Li Qi
H Hi Fai = Ui Λi ViH = Ui Λi V1i , V0i
(4)
where Li is a lower triangular matrix, and Qi is a (N + M i ) × (N + M i ) unitary matrix. Thus we can obtain 1 H Qi Hi QiH = Hi = Li = L1i , L0i (5) 0 Qi where Q1i is composed of the first Mi rows of Qi , and Q0i is comprised of the last N rows of Qi . L1i is a M i × M i lower triangular matrix, which holds the first Mi columns of Li , and L0i is a Mi × N 0matrix, which holds the last N columns of Li . Thus, it follows Hi (Q0i )H = L0i = 0 . And (Q0i )H can be referred to as the null space of Hi . Then, the first precoding matrix Fai can be obtained as Fai = (Q0i )H (Mi + 1 : N + Mi , :)
(6)
where (Q0i )H (Mi +1 : N +Mi , :) denotes the submatrix of (Q0i )H with elements from the last rows 1, · · · , N. Theorem 1: By carrying out the proposed channel extension method, the obtained first precoding matrix Fai can balance the MUI and noise effectively. Proof: the expression (5) can be rewritten as Hi QiH = [∂I, Hi ][ = [ Li,1
Qi,1 Qi,3
Li,0 ]
Qi,2 ] Qi,4
(7)
(11)
where Ui and Vi are the left singular matrix and the right singular matrix of Hi Fai respectively. Λi is a diagonal matrix, which is composed of the singular values of Hi Fai . Here,li denotes the number of non-zero singular values of Hi Fai . V1i is composed of the first li columns of Vi and V0i holds the last N − li,1 columns of Vi . Then, the second precoding matrix Fbi can be obtained as Fbi = V1i Di
(12)
where Di is the power loading matrix, which is obtained by water-filling on the diagonal elements of Λi . [2] Shows that Fbi can maximize sum capacity for SU MIMO systems. Thus, the precoding matrix Wi for user i can be expressed as Wi = β ∗ Fai Fbi
(13)
where β represents a scaling number which is determined by K WiH Wi ) = Ptotal . T r( i=1 IV. Comparing with Existing Scheme In this section, we will prove that the proposed algorithm is mathematically equivalent to RBD. And complexity analysis is given.
WANG et al.: A LINEAR PRECODING SCHEME FOR DOWNLINK MULTIUSER MIMO PRECODING SYSTEMS
TABLE I The number of floating point operations of the proposed algorithm and RBD algorithm The expression of complexity
RBD The proposed algorithm
⎛ 2 ⎞ ⎜⎜⎜ N (4M + 6M + 2) + N M ⎟⎟⎟ ⎟⎟⎟ K ⎜⎜⎜⎝ 3 ⎠ 3 +13M + 13M + 4M 3 2 4 2 3 K 8N M + 3 M + 13M + 2N M
6168 2871
where Fb,RBD is from SVD of Hi Fa,RBD as eq.(11)and eq.(12). i i a,RBD a,RBD And Fi is expressed as Fi = Vi Di where Di is equal to (Σi Σi + ∂2 I)−1/2 . H Then (Fa,RBD )H (Hi Hi + ∂2 I)Fa,RBD can be induced as i i H
= DiH (Σi Σi + ∂2 I)Di =I
(15)
From (10) and (15), we can obtain that Fai = ∂Fa,RBD . Aci cording to Lemma 1 and [2], the expression Fbi = 1∂ Fb,RBD can i can be be obtained. Thus Wi can be shown as Fbi = ∂1 Fb,RBD i RBD W . obtained. Thus Wi can be shown as Wi = β∗Fai Fbi = β∂ i β1 RBD satisfy the same constraint, we Then because Wi and Wi can obtain β1 = β. Further we can obtain that Wi = WRBD i
20
Instantiation (2,2,2)x6
A. Mathematical analysis Lemma 1: If matrix A is k times of matrix B, then matrix A and matrix B have the same singular vectors [4]. Theorem 2: The proposed algorithm is mathematically equivalent to RBD. Proof: Let us define the SVD of Hi as Hi = Ui Σi ViH . From [3], the precoding matrix for user i of RBD can be expressed as = β1 Fa,RBD Fb,RBD (14) WRBD i i i
(Fa,RBD )H (Hi Hi + ∂2 I)Fa,RBD i i
Capacity for (2,2,2)x6 25
(16)
Theorem 2 has been proven. B. Complexity analysis In this part, we will give the comparison of the complexity of the proposed precoding algorith and RBD as shown in Table I where we assume Mi = M for all i , and M = (K − 1) × M.The computation complexity is measured as the number of flops. A flop is a floating point operation. According to [4], the required flops of each matrix operation are described as follows : • Multiplication of an m × n matrix and an n × p matrix: 2mnp . • SVD of an m×n(m n) where only Λ and V are obtained: 4n2 m + 13m3 . • LQ decomposition of an m × n(m n) : 2m2 (n − m/3) As the instantiation in Table I shows, the proposed algorithm has lower complexity than RBD. And the complexity advantage grows as N and K increase. This is because that performing the SVD operation needs higher computational complexity than LQ decomposition. And RBD must adopt to balance MUI and noise. But in SVD to obtain Fa,RBD i the proposed algorithm, by carrying out proposed channel extension, LQ decomposition can be used to obtain Fai to balance MUI and noise.
Capacity[bits/Hz]
The algorithm
655
BD RBD No interference Proposed
15
10
5
0 −5
0
5
10
15
20
SNR(dB)
Fig. 1. Comparison of the sum rate as a function of SNR for multiuser MIMO systems.
V. Simulation Results In this section, we compare the performance of the proposed algorithm, the BD algorithm [2], RBD algorithm [3] and the no-interference case (a hypothetical scenario that refers to a single-user environment). For all simulations, spatially uncorrelated MIMO channels with the elements complex Gaussian elements with zero mean and unit variance are adopted. And all simulations are constructed using a QPSK transmit constellation. For the purpose of the simplicity, here, we consider an equal power allocated case and each user is transmitted one single data stream. In Fig. 1, we compare the system’s capacity of the four methods with respect to the SNR in dB for the (2,2,2)x6 system. The figure shows that the proposed algorithm outperforms BD algorithm. This is because that the proposed algorithm balances MUI and noise effectively, compared with BD algorithm especially in the low SNR regime, where the noise influences the system’s performance severely. The figure 1 further shows that the proposed algorithm is equivalent to RBD algorithm in terms of performance. VI. Conclusion In this letter, a novel linear MU-MIMO precoding scheme for DL MIMO systems is proved to balance MUI and noise effectively with low complexity. Simulation results show that the proposed algorithm can effectively improve the system’s capacity performance with low complexity, compared with several existing methods. References [1] C. B. Peel, B. M. Hochwald, and A. L. Swindelhurst, “A vector-perturbation technique for near-capacity multiantenna multiuser communication–part I: channel inversion and regularization,” IEEE Trans. Commun., vol. 53, pp. 195-202, Jan. 2005. [2] Q. H. Spencer, A. L. Swindelhurst, and M. Haardt, “Zero-forcing methods for downlink spatial multiplexing in multiuser MIMO channels,” IEEE Trans. Signal Process., vol. 52, pp. 461-471, Feb. 2004. [3] V. Stankovic and M. Haardt, “Generalized design of multiuser MIMO precoding matrices,” IEEE Trans. Wireless Commun., vol. 7, pp. 953961, Mar. 2008. [4] G. H. Golub and C. F. V. Loan, Matrix Computations, 3rd edition. The Johns Hopkins University Press, 1989.
