Design of Group Precoding for MU-MIMO Systems with Exponential Spatial Correlation Channel

In this paper, a low-complexity precoding algorithm is proposed to reduce the computational complexity and improve the performance for MU-MIMO systems under exponential spatial correlation channel conditions. The proposed precoders are designed consisting of two components: The first one minimizes the interference among neighboring user groups, while the second one improves the system performance. Numerical and simulation results show that the proposed precoders have remarkably lower computational complexities than their existing LC-RBD-LR-ZF and BD counterparts. Besides, BER performances of the proposed precoders are asymptotic to that of LC-RBD-LR-ZF precoder at the low SNR region and better than that of LC-RBD-LR-ZF precoder at the high SNR region. Simulation results also show that the performance of the proposed algorithms is significantly improved compared to the BD algorithm in an exponential spatial correlation channel.


Introduction
Multiple-Input Multiple-Output (MIMO) system has been widely studied in recent years and already applied in 4G mobile communication systems due to the fact that they can greatly increase the spectrum efficiency [1]. In order to utilize multiplexing gain, Multiuser MIMO (MU-MIMO) system has been proposed. In the MU-MIMO, each base station (BS) is equipped with multi antennas to simultaneously serve multi users using the same frequency resource. MU-MIMO not only inherits all the advantages of MIMO systems but also overcomes its limitation [2].
Unlike single user MIMO (SU-MIMO) systems, the received signals at the user sides of MU-MIMO systems not only suffer from noise and inter-antenna interference but also affected by the interference among neighboring users. In order to solve this problem, precoding techniques are applied at the BS side. Linear precoding algorithms with low-complexity such as Zero Forcing (ZF), Minimum Mean Square Error (MMSE) and Maximum Ratio Transmission (MRT) are suitable candidates [3], [4]. As shown in [2], when the number of antennas at the BS side is greater than the number of users, the simple linear precoders become nearly optimal. Obviously, nonlinear algorithms, such as Dirty Paper Coding (DPC) proposed in [5], can also be applied. However, the complexity of these algorithms becomes significantly large as the system dimensions grow due to the implementation of random nonlinear encoding and decoding [6], [7]. To improve the system performance, in [8], the authors combined Seysen's lattice reduction algorithm (SA) and linear precoding techniques for MU-MIMO systems. It is shown in [8] that the proposed algorithm gives better performance than the precoding algorithm with the Lenstra-Lenstra-Lovász (LLL) lattice reduction algorithm. In [9], Block Diagonalization (BD) precoding algorithm was proposed by combining QR decomposition and Pseudo-Inverse Block Diagonalization (PINV-BD) in [10]. In this proposal, LLL lattice reduction (LR) algorithm and Tomlinson-Halashima precoder (THP) are applied in each block to improve the quality of the system. In [11], the authors proposed the low-complexity lattice reduction-aided regularized Block Diagonalization using Zero Forcing precoding (e.g., LC-RBD-LR-ZF) and low-complexity lattice reduction-aided regularized Block Diagonalization using MMSE precoders (e.g., LC-RBD-LR-MMSE) for MU-MIMO systems. In this proposal, the first precoding matrix is obtained by using QR decomposition of the channel matrix. The second one is designed based on ZF or MMSE algorithm in combination with LLL lattice reduction algorithm. The precoders in [9] and [11] were shown to significantly improve system performance. However, their computational complexity is still very high due to the adoptions of QR decomposition and LR algorithms. In [12], the precoding algorithm is proposed based on system expansion. Besides, the precoding algorithm based on the principal component analysis technique (PCA) is proposed for Massive MIMO systems [13]. However, for the proposals in [12] and [13], the authors have not given the symbol error probability analysis expression of the system.
In this paper, we propose two linear group precoding algorithms, called BD-LR-ZF and BD-LR-MMSE precoders, that have low complexity for MU-MIMO systems working in the exponential correlation channel model. In our proposal, the channel matrix from the BS to all users is divided into two groups, each group consists of a number of rows of the channel matrix. Based on this grouping approach, the proposed precoders are designed consisting of two components. The first precoding matrix minimizes the interference from neighboring user groups by using traditional BD algorithm; the second one improves the BER performance of the system by combining the conventional linear precoders and the element-base lattice reduction shortest longest basis (ELR-SLB) technique. Performance evaluation by analyzing the so-called orthogonal deficiency (od) component is provided so that one can roughly estimate and compare the performances among precoders. Numerical results are also provided to show that the proposed precoders have remarkably lower computational complexities than both the LC-RBD-LR-ZF in [11] and the BD in [4]. Simulation results show that the BER performance of the proposed algorithm is asymptotic to that of the LC-RBD-LR-ZF algorithm and better than the BD algorithm. Moreover, the spatial correlation adversely affects the system performance no matter which precoder is adopted.
The rest of this paper is organized as follows. In Section II, we present MU-MIMO system model. The proposed algorithms in combination with ELR-SLB technique are presented in Section III. In Section IV, we present the simulation results. Finally, conclusions are drawn in Section V. is to round the real and imaginary parts of the complex number to the nearest integer. . In addition, the Channel State Information (CSI) is assumed to be perfectly known at the BS.

The downlink channel model in MU-MIMO systems
In real applications, both the BS side and user side do not have large spaces to arrange the antenna elements distant enough. Therefore, spatial correlations always exist among transmit and receive antennas, resulting in performance degradation. In order to take into account the effect of spatial correlation, the channel model is given by the following equation [14]: H is the uncorrelated channel matrix, whose entries, ij h , are complex Gaussian random variables with zero mean and unit variance. In this paper, we investigate MU-MIMO systems in exponentially correlated channel model [15]. In this model, the components of Applying the same steps to 1 H , we are able to get the next precoding matrix 2 BD W . The precoding matrix BD W is designed to have the following form: 12 .
After getting the first weight matrix BD W , we define the effective channel matrix for the first group as follows: 1 11 . BD H H W (9) The channel matrix 1 H is then transposed and converted into the matrix 1 LR H in the LR domain by using ELR-SLB algorithm in [16] to give:  (15) In order to make sure that the transmit power is unchanged after having precoded the transmit signals, the normalized power factor is computed as follows: The proposed BD-LR-ZF and BD-LR-MMSE algorithms are summarized in Algorithm 1.  (11) and (12).

Repeat
Step 7 to Step 9 for the second user group.
11. Generate the matrix LP W as in (13) and (14) (17) The received signal y is then quantized to the nearest constellation symbols to give the recovered signals for all users.

Performance Analysis
From (17), the estimated signal vector of all users is given by:

Computational Complexity Analysis
In this sub section, we evaluate the computational complexity of the proposed precoders and compare them with those of LC-RBD-LR-ZF algorithm in [11] and of BD algorithm in [4]. The complexities are evaluated by counting the necessary floating point operations (flops).
We assume that each real operation (such as an addition, a multiplication or a division) is counted as a flop. Hence, a complex multiplication and a division require 6 flops and 11 flops, respectively. According to [20], SVD operation of an mn complex matrix with mn requires The number of flops for SVD operations is given by: F is calculated to be: Since ELR-SLB algorithm is adopted, 3 F is given by: The complexities all of the precoders under consideration are summarized in Table I.

Simulation Results
In this Section, we compare both the computational complexities and the BER performances of the proposed algorithms with those of LC-RBD-LR-ZF algorithm in [11] and BD algorithm in [4]. In all simulation results, the channel from BS to all users are assumed to be quasistatic Rayleigh fading channel.  6( )( However, at sufficiently high SNRs, they provide better system performance than LC-RBD-LR-ZF precoder.
More importantly, in all scenarios, the proposed precoders outperform the BD one in the entire SNR region.  Fig. 7 also show that the correlation coefficient has an adverse effect on the system performance no matter which precoder is employed.

Conclusions
In this paper, we propose the BD-LR-ZF and BD-LR-