Low-complexity RBD precoding method for downlink of multi-user MIMO system
where PT is the total transmit power. In the second step, Fb is used to decompose the effective block channel HiFai into parallel subchannels by applying the SVD operation:
X. Gao, L. Li, P. Zhang, L. Song, G. Zhang, Q. Wang and H. Tian
Hi Fai = Ui Si VH i
(4)
then Fbi ¼ Vi. Finally, the precoding matrix for user i is Many practical downlink multi-user multiple-input multiple-output (MIMO) linear precoding methods have been proposed recently. The block diagonalisation (BD) method has more advantages when each user is equipped with multiple receive antennas. However, BD has poor performance at the low and medium SNR regime with no consideration of noise. Hence, the regularised block diagonalisation (RBD) method has been proposed, which also has a high computational complexity for using singular value decomposition to suppress multi-user interference and noise. Proposed is a novel method of Cholesky factorisation for the RBD method, which can reduce computational complexity effectively without performance loss. Analysis and simulation results show the effectiveness of the proposed method.
Introduction: Multiple-input multiple-output (MIMO) systems have attracted much attention owing to large spectral efficiencies. Recently, it has been shown that the sum-capacity region is achieved by dirty paper coding (DPC). Motivated by DPC, several nonlinear precoding schemes have been developed but they require high complexity. On the other hand, linear precoding schemes have been proposed as practical approaches with low complexity. The most intuitive scheme for single antenna receivers is to use the inversion of the channel as the precoding matrix [1]. An extension for receivers with multiple antennas is the block diagonalisation (BD) of the channels [2, 3], which attempts to completely eliminate multi-user interference (MUI) without any consideration of noise and has a constraint on the number of transmit and receive antennas, otherwise the nullspace may not exist. Although the regularised block diagonalisation (RBD) method [4] overcomes the drawbacks of BD, it still has high computational complexity when using singular value decomposition (SVD) to suppress MUI and noise. In this Letter, we propose a novel method of Cholesky factorisation for the RBD method, which is a simple way to get the precoding matrix with lower computational complexity and is equivalent to the original RBD. Theoretical analysis and simulation results show there is no performance degradation. System model and conventional method: We consider a downlink MUMIMO system which has a single BS equipped with MT transmit antennas and K users each of which has ni receive antennas. The total number of receive antennas at all users is defined as MR = Ki=1 ni ; the channel matrix of the user i is denoted by Hi , which is an ni × MT complex Gaussian random matrix of zero-mean and unit-variance; and the overall channel matrix is denoted by H = [HT1 , HT2 · · · HTK ]T . Then we define Hi = [HTi · · · HTi−1 HTi+1 · · · HTK ]T . The received signal of the user i is: yi = H i
K
Fk sk + ni
Fi = bVi Dai Vi
Proposed Cholesky factorisation method: In this Section, we introduce a Cholesky factorisation method instead of SVD to simplify the solving process of the RBD. The original RBD optimisation criterion (2) can be expressed by (see [4], equation (27)]): K
MR s2n H tr FH H H + I Fai (6) Fa = min i M T ai i Fa PT i=1 Substituting the solution Fai = Vi Dai and (3) into the above equation, we can get (see [4], equations (29– 32))
K MR s2n H H H V S S I V D D V Fa = min tr VH i i MT i ai ai i i i Fa i=1 PT
K MR s2n (7) = min tr SH IMT D2ai i Si Fa i=1 PT = min
Fa = min Fa
i=1
Hi Fai 2F +
2
nF b2
(2)
The solution of the above criterion is Fai ¼ ViDai , which is obtained from the SVD of Hi for transmission with equal power distribution: ⎧ H ⎨ Hi = U i Si Vi
MR s2n ⎩ Dai = SH IMT i Si + PT
(3)
tr(IMT )
From the above result we can see that the solution of the optimisation criterion in (7) is to find a matrix Fai , such that
MR s2n H H H + I (8) Fai = tr(IMT ) tr FH i M T ai i PT MR s2n IMT in the above equation is Hermitian PT and positive definite, we can decompose this matrix using Cholesky factorisation as
Since the matrix HH i Hi +
HH i Hi +
MR s2n IMT = LH i Li PT
(9)
where Li is an MT × MT upper triangular matrix, and we choose Fai = L−1 i
(10)
substituting it into (8), we have
MR s2n H H H + I tr FH i MT Fai ai i PT =
k=1
K
K
Fa i=1
(1)
where sk is an ni × 1 matrix representing the transmitted signal of user i and Fk is an MT × ni precoding matrix; ni representing the complex Gaussian noise with each component having zero mean and variance s2n per dimension. Furthermore, we assume that the BS knows the channel matrix H, and user i only knows Hi. In [4], the RBD method generates the precoding matrix F in two steps: FaFb , where b is⎞the scaling factor, Fa ¼ [Fa1...Fai...FaK] and F ¼ b⎛ Fb1 · · · 0 ⎜ .. .. ⎟ Fb = ⎝ ... . . ⎠. In the first step, Fa is obtained by the follow0 · · · FbK ing optimisation criterion:
(5)
H −1 tr(L−H i Li Li Li )
(11)
= tr(IMT )
Now we can see that is equivalent to the Fai = Vi Dai , both of them can minimise the cost function in (2). Then the equivalent combined channel matrix of user i is HiL21 i , then use the same method as (4) to decompose this matrix into parallel subchannels: L21 i
H Hi L−1 i = Ui Si Vi
(12)
Finally, the precoding matrix for user i is Fi = bL−1 i Vi
(13)
Results: In this Section, we compare the computational complexity and the performance of systems employing the precoding techniques introduced in this Letter to the BD, RBD and the proposed methods. We employ the floating point operations (flops) as a measure to compare computational complexity. According to [5], the required flops of each matrix operation are described as follows: † Multiplication of an m × n matrix and n × p matrix: 2mnp. † SVD of an m × n ( m ≤ n ) matrix where only L and V are obtained: 4n 2m + l3m 3. † Cholesky factorisation of an m × m matrix: m 3/3. † Inversion of an m × m matrix using Gauss-Jordan elimination: 4m 3/3.
ELECTRONICS LETTERS 11th November 2010 Vol. 46 No. 23
Note that, when the results of multiplication and inversion are m × m Hermitian matrices or a triangular matrix, it is possible to reduce the complexity proportional to (m + l)/2m. We compare the required flops of each method in Table 1 where we assume ni ¼ n for all i, and MT ¼ MR ¼ Kn. For instance, in the case of a {2,2,2,2} × 8 system, the required flops of the BD, RBD and the proposed methods are counted as 18432, 22464, 6856, respectively. The number of flops is plotted against the number of users K in Fig. 1. It is obvious that the proposed method exhibits lower complexity than BD and RBD.
Conclusion: We propose the Cholesky factorisation method to replace the original SVD-based RBD method. This method effectively reduces computational complexity without any performance loss. Both analysis and simulation results validate the effectiveness of the proposed method. Acknowledgment: This work is jointly supported by the National Science Foundation of China (NSFC 60702051, NSFC-AF 60910160), the Major National S&T Program (2009ZX03003-001-01), and the National 973 Program (2009CB320400), NCET-08-0735 and SRFDP 20070013028.
(x 106) 4.0 n = 2, original-BD n = 2, original-RBD n = 2, proposed n = 3, original-BD n = 3, original-RBD n = 3, proposed
3.5 3.0
# The Institution of Engineering and Technology 2010 3 September 2010 doi: 10.1049/el.2010.2470 One or more of the Figures in this Letter are available in colour online.
flops
2.5 2.0 1.5 1.0 0.5 0 2
3
4
5 6 7 number of users K
8
9
10
E-mail:
[email protected]
Fig. 1 Comparison of average flops against number of users
References
Table 1: Comparison of computational complexity SVD-based BD K(17MT3 − 43nMT2 + 43n2 MT + 4n3 ) SVD-based RBD K(17MT3 − (35n − 2)MT2 + (39n2 + 3)MT ) K(2MT3 + (7n + 2)MT2 + nMT + 13n3 )
Proposed method
X. Gao, L. Li, P. Zhang, L. Song, G. Zhang, Q. Wang and H. Tian (Key Laboratory of Universal Wireless Communications, Beijing University of Posts and Telecommunications, Ministry of Education, Wireless Technology Innovation Institute (WTI), P. O. Box 92, No. 10 Xi Tu Cheng Road, Hai Dian District, Beijing 100876, People’s Republic of China)
Fig. 2 shows the sum rates of the three methods against SNR with the perfect channel information in the cases of {2,2,2,2} × 8 and {3,3,3,3} × 12. We present the performance of each method through Monte Carlo simulations. For all simulations, spatially uncorrelated MIMO channels with elements generated by I.I.D. complex Gaussian random variables with zero-mean and unit-variance are used. For simplicity, we assume no power distribution strategy is specified and consider an equal power allocation case. Simulation results show that the performance of our proposed method is the same as the SVD-based RBD method, there being no performance degradation.
1 Gershman, A.B., and Sidiropoulos, N.D.: ‘Space-time processing for MIMO communications’ (John Wiley & Sons Ltd, Chichester, UK, June 2005), pp. 209– 236 (ISBN: 978-0-470-01002-0) 2 Spencer, Q.H., Swindelhurst, A.L., and Haardt, M.: ‘Zero-forcing methods for downlink spatial multiplexing in multiuser MIMO channels’, IEEE Trans. Signal Process., 2004, 52, pp. 461–471 3 Choi, L.U., and Murch, R.D.: ‘A transmit preprocessing techinique for multiuser MIMO systems using a decomposition approach’, IEEE Trans. Wirel. Commun., 2004, 3, (1), pp. 20– 24 4 Stankovic, V., and Haardt, M.: ‘Generalized design of multiuser MIMO precoding matrices’, IEEE Trans. Wirel. Commun., 2008, 7, pp. 953–961 5 Golub, G.H., V, C.F., and Loan, : ‘Matrix computations’ 3rd edn., (The Johns Hopkins University Press, Baltimore and London, 1996), pp. 87–470
60 Mt = 8, K = 4, n = 2; original-BD Mt = 8, K = 4, n = 2; original-RBD Mt = 8, K = 4, n = 2; proposed Mt = 12, K = 4, n = 3; original-BD Mt = 12, K = 4, n = 3; original-RBD Mt = 12, K = 4, n = 3; proposed
sum rate, bits/Hz
50
40
30
20
10
0 –5
0
5
10
15
20
SNR, db
Fig. 2 Comparison of sum rate against SNR
ELECTRONICS LETTERS 11th November 2010 Vol. 46 No. 23
