Journal of Systems Engineering and Electronics Vol. 22, No. 1, February 2011, pp.170–174 Available online at www.jseepub.com
Codebook selected beamforming algorithm for multiuser MIMO systems Wengang Li1,* , Kechu Yi1 , Daixian Zhu1,2 , and Yongjun Sun1 1. State Key Laboratory of Integrated Service Networks, Xidian University, Xi’an 710071, P. R. China; 2. Communication and Information Engineering College, Xi’an University of Science and Technology, Xi’an 710054, P. R. China
Abstract: For reducing the inter-user interference in multi-user multiple-input multiple-output (MU-MIMO) wireless communication systems, e.g., MIMO-orthogonal frequency division multiplexing (MIMO-OFDM) systems, it is often desirable to the complex preprocessing at the transmitter. This paper proposes a multi-user beamforming algorithm with sub-codebook selection. Based on the minimal leakage criterion, the codebook selection, limited feed-forward and minimum mean square error (MMSE) detection are combined in the proposed algorithm. This avoids the complex channel matrix decomposition and inversion. Consequently, the computational complexity at the transmitter is significantly reduced. Simulation results show that the proposed algorithm performs better than existing beamforming algorithms. Keywords: multi-user multiple-input multiple-output (MU-MIMO), linear beamforming, co-channel interference (CCI), minimum mean square error (MMSE).
DOI: 10.3969/j.issn.1004-4132.2011.01.023
1. Introduction Multiple-input multiple-output (MIMO) systems, which use multiple antenna techniques at both transmitter and receiver, are characterized by the high data throughput. This technique has been adopted into WiMax and long term evolution (LTE). Comparing with single user MIMO (SUMIMO) systems, multi-user MIMO (MU-MIMO) systems attract more attention due to the significant data throughput improvement [1–3]. In a MU-MIMO system, all users share the same time slot and frequency spectrum. Therefore, it is necessary to resolve the co-channel interference (CCI) at receiver for each user [4]. Toward this end, zero-forcing (ZF) and block diagonalization (BD) schemes were proposed in [5–9]. Generally, optimal performance can not be achieved by the two methods. Another precoding scheme was proposed in Manuscript received August 7, 2009. *Corresponding author. This work was support by the National Natural Science Foundation of China (60702060) and the 111 Project.
[10,11], which measures how much signal power leaked from one user into other users and then chooses the available beamforming coefficients for maximizing the signalto-leakage and noise ratio (SLNR). However, this scheme can not effectively reduce the CCI for each user. In this paper, a MU-MIMO beamforming algorithm is proposed for time division duplex (TDD) systems. For every frame data, basing on the criterion of maximizing SLNR, the transmitter chooses one sub-codebook from the codebook as a beamforming vector, which is used to multiply the data for transmission. At the same time, the indices of the sub-codebooks are also transmitted via the control channel. After the signal is received at the receiver, the indices are recovered into the corresponding sub-codebooks, which are used for the minimum mean square error (MMSE) detection of the received signals. We call the algorithm as codebook selected beamforming algorithm (CSBF). The algorithm needs only a few control bits, but it has lower computational complexity than other conventional algorithms. Moreover, the CCI of users is significantly reduced, and it allows improving the performance of the system.
2. System model Consider an MIMO downlink system with Mt transmitter antennas and Nr receiver antennas. Supposing there are K users in the system, denoted as M S1 , M S2 , . . . , M SK . M Sk is the kth user with mk receiver antennas, then the K mk . total number of receiver antennas is Nr = k=1
The data of user k are denoted as sk , which is an mk × 1 scalar symbol and is normalized as 1. The Mt × mk beamforming matrix is denoted as Fk , and is normalized as Fk 2 = 1. sk needs to be multiplied by Fk prior to transmission. The received signal of user k is denoted as yk , which is formulated as
Wengang Li et al.: Codebook selected beamforming algorithm for multiuser MIMO systems
yk = Hk Fk sk +
K
Hk Fj sj + zk
(1)
j=1,j=k
where Hk is the channel matrix of mk × Mt , represented as ⎤ ⎡ k h1,1 · · · hk1,Mt ⎥ ⎢ .. .. .. Hk = ⎣ (2) ⎦ . . . k k hmk ,1 · · · hmk ,Mt and where zk is a complex Gaussian noise vector with variance σk2 and mean zero, i.e. E(zk zkH ) = σk2 I, where (·)H denotes the conjugate transpose operation.
link channels Hk (k = {1, . . . , K}) are known at transmitter before beamforming. We pre-set a codebook C = {Ci |i ∈ (1, . . . , N )} based on householder transformation presented in [12]. Each sub-codebook Ci is unitary. The number N equals 2B , where B is the bit size of sub-codebook index (binary bits). This codebook is known to both the transmitter and the receiver prior to transmission. Each codebook, Ci for example, can be used as the beamforming vector. Therefore, (4) can be rewritten as
3.1 SLNR criterion When user k receives the signal yk which is formulated in (1), we expect the signal-to-interference-plus-noise rate (SINR) at the input of the receiver is maximized. The expression of SINR is given by δSINRk =
For user k, since Hk (k ∈ (1, . . . , K)) is known to the transmitter, δSINRck should be maximized by selecting an optimal sub-codebook from C. In other words, the optimal sub-codebook Cko can be selected through maximizing δSINRck as described in (6). Cko = arg max δSINRck = Ci ∈C
arg max
(3)
Ci ∈C
2
Hk Ci = K 2 mk σk2 + Hj Ci j=1,j=k
j=1,j=k
where mk σk2 is the power of Gaussian white noise, Hk Fk 2 is the signal power of user k, and K 2 Hk Fj is the power of signal leaked to user j=1,j=k
k from other users. Instead of solving the maximization problem in (3), δSINRk can be corresponding maximized if the power of signal leaked to other users from user k (k ∈ {1, . . . , K}) is minimized. This consideration motivates us to introduce a new criterion in terms of an SLNR [10,11], which is formulated as δSINRk =
2
Hk Fk K 2 mk σk2 + Hj Fk
(4)
arg max
Ck ∈C
where
WkH =
The downlink channel can be perfectly estimated by its corresponding uplink channel because of channel reciprocity in a TDD system. Thus, we assume that the down-
K i=1
j=1,j=k
3.2 Proposed CSBF algorithm
(6)
˜ k = [H1 . . . Hk−1 Hk+1 . . . HK ]H . where H After user k selected its optimal sub-codebook Cko from C, user data are multiplied by Cko and then transmitted. The indices of selected sub-codebook need to be transmitted over the control channel. When user k receives the control information, all the selected sub-codebook will be recovered. Then the received symbols can be easily recovered by using the MMSE detection algorithm. The linear MMSE filter of user k is given as
Hj Fk 2 is the signal power leaked to other
users from user k.
CiH HkH Hk Ci ˜ k )Ci ˜ HH CiH (mk σk2 + H k i ∈ (1, . . . , N )
j=1,j=k
K
(5)
j=1,j=i
2
Hk Fk K 2 mk σk2 + Hk Fj
2
Hk Ci K mk σk2 + Hj Ci 2
δSINRck =
3. CSBF algorithm and analysis In this section, we detail the proposed beamforming algorithm and its deployment. For simplicity, the proposed algorithm is denoted as CSBF hereinafter.
171
(Hk Cko )H Hk Cio F
(7) 2
+ mk σ I
where the downlink channel Hk can be estimated from the transmitted pilots. That’s all for the proposed algorithm. Since the user can easily compute the power leakage K Hk Cio from other users at the receiver, the CCI i=1,i=k
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can be cancelled for each user. It is the most important different and advantage of the proposed algorithm compared with the ZF and BD algorithm, which can not do similarly operation to remove the CCI at the receiver. A summary of the proposed CSBF algorithm is given below. • At transmitter (i) Initially: let C = (C1 C2 . . . CN ) and Fk = C0 , estimate Hk (k ∈ (1, . . . , K)) by uplink channel of each user. (ii) For k = 1 : K ˜ k = [H1 . . . Hk−1 Hk+1 . . . a) For each user k, let H HK ]. b) For i = 1 : N , let Fk = Ci , compute δSLNRik = FkH HkH Hk Fk . ˜ k )Fk ˜ HH FkH (mk σk2 I + H k c) Search δSLNRmax = max(δSLNR1k , δSLNR2k , . . . , k δSLNRN ), let Fkoptimal = Ci (δSLNRmax ). k k
d) Use Fkoptimal to beamform the transmit signal and record codebook index of user k. (iii) Sent all users transmit signals and sub-codebook indices. • At receiver For k = 1 : K (i) For each user k, estimate the Hk by pilots and recover the codebook form control information. (ii) Compute detector filter weight WkH by using MMSE detection algorithm. (iii) Recover the transmit signal.
4. Algorithm complexity analysis and comparison The aim of the proposed algorithm is to reduce the computation complexity of MU-MIMO beamforming, and also to improve the performance of systems. Thus the computation complexity analysis and comparison between the proposed algorithm and other conventional algorithms is described in detail as follows. 4.1 Complexity of typical matrix operation We firstly define the computation complexity as ψ, and define a real floating point operation as f . Thus the complexity ψ is counted as the number of f . Although ψ can not characterize the true computational complexity, it captures the order of the computation load, so suffices for the purpose of the complexity analysis in this paper. For some simplicity computation, we directly give the complexity. For example, a real addition, multiplication, or division is counted as one f , a complex addition and multiplication have two f and six f , respectively.
For an m × n complex-valued matrix H ∈ C m×n , there are some correlative computation complexity listed as follows: 2 (i) Frobenius norm HF takes 2mn real multiplications and 2mn real additions, hence the complexity in total is 4mnf . (ii) The complexity count for SVD(H) of real-valued m × n matrices is (4m2 n + 8mn2 + 9n2 )f . For complexvalued m × n matrices is (48m2 n + 24mn2 + 54m3 )f . (iii) HF denotes the matrix multiplication, its’ complexity is 8mn2 f . 4.2 Algorithm complexity comparison Based on the analysis of matrix computation complexity [13], the BD algorithm and the proposed CSBF algorithm complexity are derived as follows. We focus on the computation complexity of transmitter beamforming. Because the important difference between two algorithms is beamforming methods at the transmitter. Besides, the other processing is similar, so we can ignore them for comparation. For the BD algorithm, in beamforming processing, user k (k = 1, . . . , K) needs to select the channel matrix H H H ˜ k = [H H . . . H H Hk+1 . . . HK ] zero H 1 k−1 subspace vectors as the beamfroming vectors. Hence the computation complexity ψBD of user k is ψBD ≈ [48(Nr − mk )2 Mt + 32(Nr − mk )Mt2 + 54(Nr − mk )3 + 4(Nr − mk )Mt ]f
(8)
For the proposed algorithm, compared with the conventional BD algorithm, the matrix decomposition operation at the transmitter is avoided. Basing on the SLNR criterion, we use the codebook selection instead of the matrix decomposition. As a result, the computation complexity can be reduced by CSBF. The complexity of the proposed algorithm ψCSBF is N ψCSBF ≈ [8Nr Mt2 + 4Nr Mt ]f (9) i=1
where N is the number of sub-codebook of the preset codebook. We can see the computation complexity is proportional to N . When it is small, and the transmitter/receiver antennas are ascertained, comparing with (9), ψCSBF is less than ψBD . In other words, the computation complexity of the proposed algorithm is less than the conventional BD algorithm’s. It is difficult to know which has higher complexity between (8) and (9). To further show the complexity in intuitive, we characterize the complexity with curves in Fig. 1. Fig. 1 clearly shows the computational complexity comparison between the CSBF algorithm and the BD algorithm. In this comparison, we assume the number of transmitter antennas is four, and there are two or four users in
Wengang Li et al.: Codebook selected beamforming algorithm for multiuser MIMO systems
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the systems. Hence, the number of receiver antennas is two or one, respectively. The ‘mk’ denotes the number of receiver antennas for each user. Where the B = log2 N , it denotes the feedforward bits.
Fig. 2
Fig. 1
BER of one user in MIMO system with two users
Complexity comparison with the feedforward bits increasing
Obviously, compared with the conventional BD algorithm, the matrix decomposition operation at the transmitter is avoided by using the codebook selection in the proposed algorithm. We can see from Fig. 1 that, when B is small ( 5), computational complexity can be significantly reduced by CSBF than BD algorithm, especially for the system with a large number of users.
5. Results and analysis The proposed algorithm is incorporated into the MIMOorthogonal frequency division multiplexing (MIMOOFDM) system platform for simulation, and the urban channel model is employed [12]. System parameters are listed in Table 1. The channel estimation is assumed perfectly, and each user transmits a stream at a slot. Table 1
Simulation parameters
Parameter Number of transmitter antennas Number of receiver antennas Number of subcarriers Number of FFT points Coding rate Detection algorithm
Specification 4 2 300 512 1/3 MMSE
The proposed CSBF algorithm is compared with the conventional ZF and BD beamforming algorithms for evaluation. The bits error rate (BER) performance of three beamforming algorithms in the two-user and the four-user MIMO system are shown in Fig. 2 and Fig. 3, respectively. In Fig. 2, it is clear that the conventional ZF or BD beamforming algorithm has worse BER, while the CSBF algorithm has better performance in BER. The BER in Fig. 3 shows that the proposed algorithm has also remarkable improvement compared with the ZF and BD beamforming algorithms.
Fig. 3 BER of one user in MIMO system with four users
The proposed algorithm has better performance because it has approximately known all the beamforming vectors from control information before symbol is detected, meanwhile the downlink channel can be estimated from transmitted pilots. Based on that, the user can accurately and easily estimate the CCI, and completely eliminate it for each user. However, in the same condition, the interference is only restrained at transmitter for the ZF and BD beamforming algorithms. In other words, even if the users have similar channel state information at transmitter, the CCI is still difficult to be cancelled at receiver, which must degrade the performance of each user. The proposed algorithm can significantly improve the BER performance of a system, which is the most important advantage of the CSBF algorithm. Since the interference power is getting stronger with the number of users increasing, the BER performance improvement appears to be more remarkable in Fig. 3 than Fig. 2 thanks to the CCI canceling.
6. Conclusion In this paper, we investigate the interference cancellation
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algorithms for the downlink MU-MIMO systems. Combined the conventional SLNR criterion with codebook selection mechanism, a new beamforming algorithm is proposed. Instead of using the channel matrix decomposition as the BD algorithm, the CSBF algorithm selects subcodebook from the corresponding codebook for each user and transmits their indices to the users. Consquently, it not only improves the system performance, but also reduces the computational complexity compared with the conventional BD beamforming algorithm. However, the channel state information must be known at transmitter. Hence, it is suitable to use in the TDD system by its channel reciprocity.
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Biographies Wengang Li was born in 1980. He is a Ph.D. in Xidian University. His research interests are wireless communication and signal processing. E-mail:
[email protected];
[email protected]
Kechu Yi was born in 1942. He is a professor in Xidian University. His research interests are communication signal processing and navigation position technology. E-mail:
[email protected]
Daixian Zhu was born in 1970. He is a lecturer in Xi’an University of Science and Technology. He is currently pursuing the Ph.D. degree in communication and information systems at Xidian University. His research interests are short distance wireless communications technology and their applications. E-mail:
[email protected] Yongjun Sun was born in 1973. He is a lecturer in Xidian University. His research interests are wireless communication and navigation position technology.
