Multi-User MIMO Linear Precoding with Grassmannian Codebook. Fang Shu, Wu Gang, Xiao Yue, and Li Shao-qian. National key Lab of Communication.
2009 International Conference on Communications and Mobile Computing
Multi-User MIMO Linear Precoding with Grassmannian Codebook Fang Shu, Wu Gang, Xiao Yue, and Li Shao-qian National key Lab of Communication University of Electronic Science and Technology of China
Abstract
advance knowledge of the interference, it could design a code to compensate for it. An approach that was defined as dirty paper coding (DPC) [2] was developed. Although DPC can achieve the sum capacity of the broadcast MIMO channel [3], deploying DPC in real-time systems is impractical because of its complexity. The Tomlinson-Harashima Precoding (THP) [4] method is based on DPC theory. THP adopts QR decomposition of the MIMO channel to product of a lower triangular matrix and proceeding with an iterative precoding to cancel the interference from the previous users. However, THP is impractical due to its complexity, many suboptimal precoding MU-MIMO techniques emerged recently, typically as the channel inversion method [5-6], the block diagonalization (BD) method [7-8] and other precoding methods [912]. Channel inversion method just adopts MIMO detection such as the Zero Forcing (ZF) and Minimum Mean Squared Error (MMSE) criterions to the transmitter. It can suppress the CCI effectively but may amplify the noise because the precoding vectors are not normalized. BD method decomposes a multi-user MIMO channel into multiple single user MIMO channels in parallel to completely cancel the CCI by making use of the null space. The generated null space vectors are normalized vectors, which could suppress the noise efficiently. So BD method performs much better than channel inversion method. However, since BD method aims to cancel the CCI and suppress the noise, the precoding gain is not optimized. These MU-MIMO techniques require full channel information at the transmitter and depend on the spatial correlation of the MIMO channel. When the spatial correlation increases, these schemes will degrade rapidly, especially for channel inversion technique. The MU-MIMO scheme based on precoding under high correlated MIMO channel is for further study. Besides, there is another type of MU-MIMO precoding method that is called codebook based MU-MIMO. The codebook based MU-MIMO usually has fixed codebook both at the BS and the MS that is generated offline. The remarkable advantage of the codebook based MU-MIMO are low feedback and low complexity. In this paper, we propose MU-MIMO linear precoding
In this paper, multi-user MIMO (MU-MIMO) linear precoding schemes with Grassmannian codebook for downlink transmission are proposed. The proposed MU-MIMO schemes extend the point-to-point single-user Grassmannian precoding to point-to-multipoint multi-user Grassmannian precoding. We proposed two MU-MIMO schemes with the Grassmannian codebook. One scheme assumes the perfect channel information at the base station and the other scheme is under the limited feedback. These MU-MIMO linear precoding schemes with Grassmannian codebook provide significant system capacity enhancement compared with the single user MIMO (SU-MIMO) system. The results also show the performance gap between perfect channel information and limited feedback.
1. Introduction There has been a great deal of interest in MIMO wireless communications. When multiple antennas at the base station (BS) and also at the mobile station (MS) are employed, the space dimension can be exploited for scheduling multi-user transmission besides time and frequency dimension. Therefore, the traditional MIMO technique focused on point-to-point SU-MIMO has been extended to the point -to-multipoint MU-MIMO technique. MU-MIMO can realize spatial division multiple access (SDMA), where one BS communicates with several MS within the same time slot and the same frequency band. MU-MIMO improves system capacity taking advantage of multi-user diversity. However multi-user co-channel interference (CCI) becomes one of the main obstacles to improve MU-MIMO performance. Telatar generalized the Shannon capacity of the singleinput single-output channel to the MIMO case [1]. He showed that in the single-user environment, the optimal power allocation that maximizes the mutual information of the MIMO channel is achieved through water-filling over its eigenvalues. For the MU-MIMO channel, the problem is more complex. It was proven that when a transmitter has 978-0-7695-3501-2/09 $25.00 © 2009 IEEE DOI 10.1109/CMC.2009.188
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schemes with Grassmannian codebook for downlink transmission under spatial uncorrelated MIMO channel. The proposed schemes exploit efficient multi-user diversity gain and the precoding gain to enhance the system capacity. The adoption of Grassmannian precoding extends the point-topoint single-user Grassmannian precoding [13] to point-tomultipoint multi-user Grassmannian precoding. The proposed two schemes will perform scheduling and precoding simultaneously at the transmitter to maximize the MUMIMO system capacity. For the first scheme that perfect channel information is assumed at the based station, the scheduler chooses a set of users and the best precoding vectors for these users from the Grassmannian codebook according to the perfect channel information at the base station. For the second scheme that only limited feedback could obtained at the base station, the proposed scheme explores a function to measure the system capacity with the limited feedback. The aim of the comparison of the two extreme schemes is to obtained the performance gap between the perfect channel and the limited feedback situation. Simulation results show performance leakage of the limited feedback scheme compared with the perfect channel information scheme. This paper is organized into six parts. First, the system model of MU-MIMO is given in section 2. Then the Grassmannian line packing criterion is described in section 3. In section 4, the proposed multi-user MIMO linear precoding schemes with Grassmannian codebook for downlink transmission under perfect channel information and limited feedback are introduced. In section 5, the simulation results and comparisons are given. Conclusions are drawn in section 6.
1
2 User 1
User 2
. . .
User X
1
User 1
re lu de hc S
User 2
. ..
. ..
re odc er P
User K
User 1
N
2 2
.. .
1 User 2
. . .
N M
2 N
1 .. .
User K
Figure 1. The configuration of MU-MIMO system
The MIMO channel matrix at the k-th MS is ⎡ ⎤ h1,1 h1,2 ... h1,M ⎢ h2,1 h2,2 ... h2,M ⎥ ⎥ Huk = ⎢ ⎣ ... ... ... ... ⎦ hN,1 hN,2 ... hN,M
(1)
Where hi,j indicates the channel impulse response coupling the j-th transmit antenna to the i-th receive antenna. Its amplitude obeys independent and identically Rayleighdistribution. The received signal at the k-th MS can be denoted as yuk = Huk ws + nuk s=
√
pu1 su1
√
pu2 su2
K
2. System mode of MU-MIMO
w=
...
√
puK suK
T
puk = p0 vu2
...
(3) (4)
k=1
vu1
(2)
vuK
(5)
Where yuk is the received vector of the k-th user, the additive noise nk has elements with distribution CN (0, N0 ) and is spatially and temporarily white. puk is the transmitted signal power of the k-th data stream, p0 is the transmitted power of all data streams. s is the transmitted symbol vector with K data streams and unit power. w is the M ×K MIMO precoding matrix, which contains K precoding vectors. vui is the unit precoding vector for user i with vuHi vui = 1 and [·]T denotes the matrix transposition.
We consider a MU-MIMO system with M transmit antennas at the base station, N receive antennas at each mobile station as shown in figure 1. There are totally X users to be scheduled at the BS and the scheduler will determine (K ≤ M ) users to be transmitted according to the channel information and the scheduling criterion. The scheduled K users will be precoded by the precoding matrix w before transmission. Because MU-MIMO technique aims to transmit data streams of multiple-users at the same time and the same frequency resources, we discuss the algorithm at single-carrier case in this paper. And for each sub-carrier of the multicarrier system, it is processed as same as the single-carrier case. On the other hand, the orthogonal frequency division multiplexing (OFDM) technique has become one of the brilliant techniques in the next generation of wireless communications. Since OFDM technique deals the frequency selective fading as flat fading, we model the channel as the flat fading MIMO channel.
3. Grassmannian line packing criterion The Grassmannian line packing problem is the problem of optimally packing one-dimensional subspaces, which results in finding the set or packing of Nt lines in CM that has maximum minimum distance between any pair of lines [13]. Grassmannian line packing provides the optimal codebook design for average received signal to noise ratio (SNR) under spatial uncorrelated MIMO channel.
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M We represent in C by a P × Nt a packing of Nt lines matrix V = v1 v2 ... vNt whose column space vi is the i-th line in the packing. vi is a unit vector with viH vi = 1 and viH vj = 1 (i = j) . It is proved that maximizing the transmitting precoding gain Gvi (H) = 2 arg max Hvi 2 with a finite codebook vectors corre-
For MU-MIMO system, the channel matrix for the k-th MS after precoding is ˜ uk = Huk w H
The linear MMSE decoding matrix of the k-th MS for the MU-MIMO correspondingly is
1≤i≤Nt
sponds to maximizing the minimum distance between any pair of lines spanned by the codebook vectors. Defining a distance function d(v1 , v2 ) by letting the distance between the two lines generated from unit vectors v1 and v2 be the sine of the angle θ1,2 between the two lines. The distance is 2 d(v1 , v2 ) = sin(θ1,2 ) = 1 − v1H v2 (6)
KN0 ˜H ˜ uk = h ˜ ˜H G IM )−1 uk (Huk Huk + p0
˜ u = Hu vu = H ˜ uk = [Huk w] h k k k k k
(13) (14)
Where vuk is the precoding vector for the k-th MS, [·]k denotes the k-th column of the matrix. Then the detected data stream sˆuk for the k-th MS is
The minimum distance of a packing is the sine of the smallest angle between any pair of lines. This is written as 2 1 − vkH vl (7) δ(V) = min
˜ uk yuk sˆuk = G
(15)
The detected SINR Suk for the k-th MS with the linear detection is 2 ˜ puk G uk Huk vuk Suk = K (16) 2 ˜ ˜ 2 pui Guk Huk vui + Guk N0
1≤k
