 Open Access
 Total Downloads : 575
 Authors : Abhishek Gupta, Garima Saini, Dr.Sbl Sachan
 Paper ID : IJERTV2IS100931
 Volume & Issue : Volume 02, Issue 10 (October 2013)
 Published (First Online): 25102013
 ISSN (Online) : 22780181
 Publisher Name : IJERT
 License: This work is licensed under a Creative Commons Attribution 4.0 International License
An Efficient Linear Precoding Scheme Based on Block Diagonalization for Multiuser MIMO Downlink System
Abhishek Gupta #, Garima Saini * Dr.SBL Sachan $
#ME Student, Department of ECE, NITTTR, Chandigarh
NITTTR, Chandigarh
*Assistant Professor, Department of ECE,
$Professor, Department of ECE,
Abstract
Block diagonalization (BD) based on singular value decomposition (SVD) is a wellknown precoding method that eliminates inter user interference in multiuser multiinput multioutput (MIMO) broadcast channels, but it is computationally inefficient. In this paper we examine the use of generalized singular value decomposition (GSVD) for coordinated beamforming in MIMO systems. GSVD facilitates joint decomposition of a class of matrices from source to destination MIMO broadcast scenarios. GSVD allows two channels of suitable dimensionality to be jointly diagonalized, through the use of jointly determined transmit precoding and receiver reconstruction matrices. Simulation results demonstrate that the proposed GSVD technique achieves considerable gains in terms of bit error rate (BER) over SVD technique with traditional interference aware block diagonalization (BD) scheme. Analysis and simulation results show that our proposal has lower complexity than the conventional BD method, with a significant increase there in sum throughput.

Introduction
Multipleinput multipleoutput communication systems have attracted attention in recent years. Such multiple antennas potentially allow higher throughput, increased diversity, and reduced interference as they communicate with multiple wireless users [1]. In the multiuser multipleinput multiple output (MIMO) downlink system, a base station with multiple antennas transmits data streams to multiple mobile users, each with one or more receive antennas. Due to the fact that the cochannel interference is hard to be managed by the receiver operation, and more importantly, capacity of interference channel can be made equivalent to capacity without interference, precancellation of the interference via the precoding at the transmitter has received much attention in recent years [2].
In [3][4], it was shown that the maximum sum rate in the multiuser MIMO broadcast channels (BC) can be achieved by dirty paper coding (DPC). However, the DPC is difficult to implement in practical systems due to its high computational complexity. As a suboptimal strategy of the DPC [5], the TomlinsonHarashima precoding, is still not practical due to its complexity, since this algorithm is based on nonlinear modulo operations.
In linear processing systems, several practical precoding techniques have been proposed, typically as the channel inversion method [6], the block diagonalization (BD) method [7]. A zeroforcing channel inversion (ZF) scheme
[6] can suppress CCI completely for the case where all users employ a single antenna. However its performance is rather poor due to a transmit power boost issue. Although a minimum meansquared error channel inversion (MMSE) method [6] overcomes the drawback of the ZF, this is still confined to a single receive antenna case. For the case where the users in the network have multiple antennas, the block diagonalization (BD) is wellknown precoding algorithm [7]. As a generalization of the ZF scheme, the BD can eliminate the multiuser interference (MUI) completely. Owing to the singular value decomposition (SVD) in the algorithm, the BD is not computationally efficient.In this paper, we present a new generalized singular value decomposition based on block diagonalization interference suppression precoding (GSVD) scheme for downlink multi stream MUMIMO channel. As a new approach of the conventional BDSVD scheme, BDGSVD shows a significant advantage in computational complexity. From the sum rate analysis as well as the bit error rate simulations in multiuser MIMO downlink, we show that the proposed BDGSVD brings substantial performance gain over existing multiuser MIMO algorithms scheme.
The rest of this paper is organized as follows. In Section II the system model and the summary of the conventional BD
SVD technique is described. In Section III, we present the proposed GSVDBD method. The complexity analysis is given in section IV. We present the simulation results in Section V and conclude in Section VI.

Multiuser MIMO Downlink
In this section, we present the system model for single cell multiuser MIMO downlink and then review the BD technique.

System Model
In the multiuser multipleinput multipleoutput (MIMO) downlink system, a base station with multiple antennas transmits data streams to multiple mobile users, each with one or more receive antennas [8]. Consider a singlecell multiuser MIMO downlink system with independent users and a base station. A base station is equipped with T antennas transmits information to mobile users. Each mobile user has R,k receive antennas

Review of conventional BDSVD method
The BD algorithm is an extension of zeroforcing method for multiuser MIMO systems where each user has multiple antennas. Each users linear precoder and receive filter can be obtained by twice SVD operations. In the first SVD operation, we get precoding matrix which can eliminate the other users interference. In the second SVD operation, each users block channel is decomposed into parallel subchannels in order to optimize the power allocation for each user Nk The key idea of the BD algorithm is to employ the precoding matrix W to suppress the MUI completely. To eliminate all MUI, the following constraint is improved.
Hk Wk = 0 (2)
Let the SVD of Hk be
Hk = Uk k[ Vk,1 Vk,0 ] H (3)
Where Uk and k are matrices of the left singular vectors
and thus the total number of receive antennas is NR = R, k .
In the sequel, the notation (
,,
)Ã—
=1
to d e the
and the ordered singular values of Hk , respectively, and Vk,1
R,1
R,K
T escrib
& Vk,0 are right singular matrices corresponding to non zero
multiuser system MIMO downlink system. The downlink
channel between the base station and user is described by the matrix H = [H T H T … H T]T is perfectly known at the base
singular values and zero singular values, respectively. We choose Vk,0 as an orthonormal basis for null space of Hk. Hk
1 2
station. The downlink
K
channel between the base station and user
N Ã— N
Vk,0 is the equivalent channel for user k after elimination
MUI, these equivalent channel has the same properties as a
is described by the matrix Hk k T, In the MIMO BC, the signal received at user k can be written as
conventional SUMIMO channel. In order to maximize the achieve sum rate of the BD, the water filling algorithm can be additionally incorporated. Define the SVD of Hk Vk,0 is
] H (4)
] H (4)
H Vk,0 = Uk k [ Vk,1 Vk,0
Thus we define the precoding matrix as
WBDSVD = [V1,0 V1,1 V2,0 V2,1 . Vk,0 Vk,1 ] 1/2 (5)
where is a diagonal matrix whose element i scale the power transmitted into each of columns of WBDSVD. With WBDSVD chosen in (5), the capacity of the BDSVD is [9].
Figure1. Multiuser MIMO Downlink Chanel
CBDSVD
= max log2
 I + 2
2
 (6)
yk = Hk Wk xk + k Wj xj + nk (1)
=1
=1
=1
Where xk represents information data of user k, Wk is an NT Ã— 1 precoding vector assoiated with user k and denotes the complex valued additive white Gaussian noise at user k, which has a variance of 2. Note that the precoding vectors are normalized to be unity, i.e., Wk 2 = 1 for k=1,, K. Furthermore, the power allocated to user k is given by Pk = E(x xH) and a sum power constraint is applied at the BS, corresponding to k P , the second term on the right hand side represents the cochannel
interference (CCI), which is desirable to be eliminated by designing appropriate precoding algorithms.
Where = diag(1 ,,,, k ) the optimal power loading coefficient in are determined by using waterfilling on the diagonal elements of under the power constraint.


Proposed BDGSVD Scheme
Generalized singular value decomposition is a joint matrix decomposition technique. Consider two matrices, Hk mÃ—n
and Hk+1 pÃ—n for m>n. The GSVD of the matrices is given by pair of factorizations.
Hk = Uk 1 VT Hk+1 = Uk+1 2 VT (7)
The matrices in this factorization have following properties: Uk is an (m Ã— m) orthogonal matrix, Uk+1 is a (p Ã— p) orthogonal matrix, and V is an (n Ã— n) non singular matrix. The columns of V are called the right singular vectors. The columns of U are called the left singular vectors. 1 is an (m
Ã— r) and 2 is a (p Ã— r) real nonnegative and diagonal matrix.
1 = diag[1 , .. r] (8)
2 = diag[1 , .. r] (9)
The ratios Âµr = r / r are called the generalized singular values of the matrix pair Hk, Hk+1[10]. In our GSVD calculation we are only using V vector for both the matrix
i.e. common orthogonal basis for null spaces for both the matrix and the reason for using this matrix is that only V satisfy the condition with Hk ,Hk+1 and all other arithmetic operations. The Vr (the columns of V) are referred to as
generalized singular vectors. Thus we define the precoding matrix as
method is shown in this section. For the sake of the simplicity, all users are assumed to have the same number of the receive antennas NR and KNRNT. The complexities of the methods are usually compared by the number of floating point operations (flops) [11]. A flop is defined as real floating operations, e.g. a real addition, multiplication or division. One complex addition and multiplication involve 2 and 6 flops respectively
The complexity of some utilized operations in these for an mÃ—n complexvalued matrix A with mn, its multiplication with another nÃ—p complexvalued matrix B requires 8mnp flops. But when B has a special form the complexity reduces. For instance, when B= AH, the complexity reduces to 4nm(m+1) flops.
k
k
For the conventional SVDBD method, obtaining the orthogonal complementary basis Vk,0 requires K times SVD operations in step 1.a. In step 1.b calculating all HkVk,0 requires K matrix multiplications while in step 1.c obtaining the singular vectors Vk,1 and the singular values k,1 requires another K SVD operations. In step 2, one water filling is needed to find Pk. In step 3, the square root of the real valued diagonal matrixP 1/2 needs to be first calculated and then multiplied by Vk,1 and Vk,0 ,respectively. Those operations repeat K times as well. By treating all operations
WBDGSVD
= V1
(10)
as multiplications, the SVD operation on A takes about 24m n2 + 48m2 n + 54m3.
where is chosen to satisfy the power constraint. The capacity for BDGSVD is given by
CBDGSVD = max log2  I + 2  (11)
where is a diagonal matrix corresponding to the singular values Âµr. As an example, suppose that Hk and Hk+1 above represents the channel matrices of MIMO system having two
users. Assume perfect channelstateinformation (CSI) for
For the conventional GSVDBD method, obtaining the orthogonal Complementary basis V requires K times GSVD operations in step 1.a. If we only want to know V it is not computationally efficient. In step 1.bcalculating all HkV for precoding matrix W requires K matrix multiplications while in step 1.c obtaining the singular vectors and the singular values requires another K GSVD operations. Here no water filling is needed to find Pk because all diagonal elements are same. In step 2, the square root of the realvalued diagonal
matrix P 1/2 needs to be first calculated and then multiplied.
Hk and Hk+1. With a transmit precoding matrix V1, and
k
Those operations repeat K times as well. By treating all
receiver reconstruction matrices Uk, Uk+1. The invertible factor V in facilitates joint precoding for the MIMO subsystems. Diagonal elements of , represents the gains of these virtual channels. Since V is nonunitary, precoding would cause the instantaneous transmit power to fluctuate. This is a drawback not present in SVDbased beamforming. Transmit signal should be normalized to maintain the average total transmit power at the desired level. This is the essence of GSVDbased beamforming for a single source and two destinations.

Complexity Analysis
The mathematical analysis for computational complexity reduction of proposed BDGSVD method over BDSVD
operation as multiplication, the GSVD operation of A & B takes 6(5m+5mp2+9m2p+9m3).

Simulation Results
Computer simulations are performed to evaluate the performance of GSVDISP approach. In all simulations, each users data are modulated by quadrature phase shift keying (QPSK) signaling. We assume the channel matrix and the interference channel matrix are perfectly known both at the transmitter and the receiver.
These results compare the sumrate of proposed BDGSVD scheme and BDSVD scheme for multiuser MIMO downlink system having various antenna configurations both at transmitter and receiver side. Parameters and values for single cell multiuser MIMO downlink channel are considered with number of frames per packet as 10 with
number of packets as 100, number of bits per QPSK symbol
= 2, number of bits per packet = 40 and total number of bits transmitted = 4000.
Fig.1.1, Fig.1.2 compares the sum rates of proposed BD GSVD scheme against SNR with the perfect channel information in the cases of {2,2}Ã—4,{3,3,3}Ã—10. It is observed from the figures that BDGSVD has more sumrate compared to BDSVD[12][13] and outperforms the BD SVD scheme.
Fig.2 compares the simulated bit error rate (BER) or single cell multiuser MIMO downlink system. The bit error rate analysis at different signal to noise ratio (SNR) is done by considering number of transmitting antennas (NT = 4) and number of receiving antennas (NR = 2) for K=2 users. From the figure 2 shown below it can be seen that the BER decreases as SNR increases by using the proposed BD GSVD method as compared to conventional BDSVD method. From the BER curves versus the transmit SNR, it can be seen that BDGSVD achieves less BER.
Fig. 3.1(a), we consider the case that =KR. We set R =2 and express the computation cost as a function of T (or equivalently ). Later we consider the case that k < T.
We fix T = 64 and NR = 2 while express the computation cost as a function of K. The result is shown in Fig. 3.1(b). Obviously, BDGSVD method shows a clear advantage in both comparisons. The larger T or R, the more it shows the significant difference.

Conclusion
In this paper, we proposed BDGSVD method to obtain the
precoding matrix for multiuser MIMO downlink systems where each user has more than one antenna. We have proved that BDGSVD scheme has more sumcapacity than the BD SVD scheme. Through complexity analysis, we show that the new method has lower complexity than the conventional BD method and efficiency improvement becomes significant
when there have a large number of transmit antennas and users.

References

G. Caire and S. Shammai, On the achievable throughput of a multiantenna Gaussian broadcast channel, IEEE Trans. Inform.Theory, vol. 3, no. 7, pp.16911706, July 2003.

I.E. Telatar, Capacity of multiantenna Gaussian channels, European
Trans. Telecommun., vol. 10, pp. 585595, Nov. 1999.

H. Weingarten, Y. Steinberg, and S. Shamai, The capacity region of the Gussian MIMO broadcast channel, IEEE Trans. Inform. Theory, vol. 52, no. 9, pp. 39363964, Sept. 2006.

M. H. M. Costa, Writing on dirty paper, IEEE Trans. Inform. Theory, vol. 29, pp. 439441, May 1983.

C. Windpassinger, R. F. H. Fischer, T. Vencel, and J. B. Huber, Precoding in multiantenna and multiuser communications, IEEE Trans Wireless Commun., vol. 3, no.4, pp. 13051315, July 2004.

C. B. Peel, B. M. Hochwald, and A. L. Swindelhurst, A vector perturbation technique for nearcapacity multiantenna multiuser communication—part I: channel inversion and regularization, IEEE Trans. Commun, vol. 53, 195 202, Jan. 2005.

Q. H. Spencer, A. L. Swindelhurst, and M. Haardt, Zero forcing method for downlink spatial multiplexing in multiuser MIMO channels, IEEE Trans. Signal Processing, vol. 52, no. 2,pp. 461471, Feb. 2004.

M. Rim, Multiuser downlink beamforming with multiple transmit and
receive antennas,"Electron. Lett., vol. 38, no. 25, pp. 1725
1726, 2002.

Wei Li,Member, Matti Latvaaho, An Efficient Channel Block Diagonalization Method for Generalized Zero Forcing Assisted MIMO Broadcasting Systems ,IEEE Transactions on wireless communications, vol. 10, no. 3,pp. 739744, March 2011.

Q. H. Spencer and M. Haardt, Capacity and downlink transmission algorithms for a multiuser MIMO channel," in Proc. 36th Asilomar Conf. Signals, Syst. Comput., vol. 2, pp. 13841388, Pacific Grove, CA,
USA, Nov. 2002.

Z. Shen, R. Chen, J. G. Andrews, R. W. Heath Jr., and B. L. Evans, Sum capacity of multiuser MIMO broadcast channels with block diagonalization, IEEE Trans. Wireless Comm.,vol.6, no.6,pp2040 2045, Jun. 2007.

Golub, G.H., V, C.F., and Loan, : Matrix computations 3rd edn., The Johns Hopkins University Press, Baltimore and London, 1996, pp. 87470, 1996.

Jungyong Park, Byungju Lee, and Byonghyo Shim, A MMSE Vector Precoding with Block Diagonalization for Multiuser MIMO Downlink,
IEEE Transactions on communications, Vol. 60, No. 2,pp. 569577, Feb 2012.