A Near-Optimal Iterative Linear Precoding With Low Complexity for Massive MIMO Systems

A Near-Optimal Iterative Linear Precoding With Low Complexity for Massive MIMO Systems

Abstract: The linear zero-forcing (ZF) precoding can achieve the near-optimal sum-rate performance when the favorable channel propagation is obtained in downlink massive multiple input multiple-output (MIMO) systems. However, it involves high complexity with the matrix inversion. To significantly reduce the complexity of ZF precoding, we propose a weighted two stage (WTS) precoding scheme with low complexity based on an iterative method. Specially, the proposed WTS precoding converts the complicated matrix inversion into two half iteration stages, and the result of each stage is weighted by a coefficient to further speed up the convergence and reduce the complexity. Theoretical analysis demonstrates that the proposed WTS precoding enjoys a fast convergence rate and low complexity.

Simulation results indicate that the proposed WTS precoding can achieve better bit-error-rate (BER) and sum-rate performance with a smaller number of iterations than the recently proposed schemes.

Existing system:

The proposed WTS precoding scheme can approximate the complicated matrix inversion required by the ZF precoding in an iterative manner, which results in faster convergence and lower complexity (as shown in Section III-C and Section III-D). Note that the existing GS, SOR, and NS precoding schemes exploit the similar iterative manner to circumvent the complicated matrix inversion. However, the proposed WTS precoding is different from these existing schemes.

Specifically, unlike the GS or SOR precoding, the proposed WTS precoding makes the iteration matrix symmetric without a relaxation parameter, which indicates that the proposed WTS precoding is more robust in practical applications. Compared with the NS precoding, the proposed WTS precoding exhibits a smaller approximation error by combining the two symmetric half iterations with weighted coefficients to obtain the signal vector.

Proposed system: The columns of H in (1) are asymptotically orthogonal in massive MIMO systems, which make W positive definite Hermitian. Although the computational complexity of the pre coded vector t is O , fortunately, the fact that W is positive definite Hermitian inspires us to efficiently realize t with much lower complexity.

To circumvent the complicated matrix inversion and speed up the convergence, we propose a WTS precoding scheme adopting two half iterations and combining the two half iterations with an weighted coefficient to iteratively solve t as follows: 1) Since W is a strictly diagonally dominant Hermitian matrix, and the more diagonally dominant W is, the more rapid convergence rate of the iterative method can be achieved, we decompose W as W= L + LH + D, where L and D denote the strict lower triangular and diagonal matrix of W, respectively.

Advantages: However, realizing the promising advantages of massive MIMO in practice faces the challenging interference problem. For mitigating multiuser interference and improving the quality of the signal transmission, precoding is usually required at the BS. The nonlinear precoding method can achieve the optimal sum-rate performance, but the complexity is too high to be implemented in practice. Since the channel matrix of massive MIMO systems is asymptotically orthogonal, the linear

capacity-approaching precoding schemes, e.g., zero forcing (ZF) precoding can achieve the near-optimal sum-rate performance.

Disadvantages: However, realizing the promising advantages of massive MIMO in practice faces the challenging interference problem. For mitigating multiuser interference and improving the quality of the signal transmission, precoding is usually required at the BS. The nonlinear precoding method can achieve the optimal sum-rate performance, but the complexity is too high to be implemented in practice. Since the channel matrix of massive MIMO systems is asymptotically orthogonal, the linear capacity-approaching precoding schemes, e.g., zero forcing (ZF) precoding can achieve the near-optimal sum-rate performance.

Modules: Zero forcing: MASSIVE multiple-input multiple-output (MIMO) is a key technology for 5G wireless communications. Compared with the traditional small-scale MIMO, hundreds of antennas are equipped at the base station (BS) to simultaneously serve many users in massive MIMO systems. Theoretical analysis shows that the use of moderately large number of antennas can significantly increase the energy and spectrum efficiency of communication systems. However, realizing the promising advantages of massive MIMO in practice faces the challenging interference problem. For mitigating multiuser interference and improving the quality of the signal transmission, precoding is usually required at the BS .The nonlinear precoding method can achieve the optimal sum-rate performance, but the complexity is too high to be implemented in practice. Since the channel matrix of massive MIMO systems is asymptotically orthogonal, the linear capacityapproaching precoding schemes, e.g., zero forcing (ZF) precoding can achieve the near-optimal sum-rate performance. Although the linear ZF precoding has lower complexity than nonlinear precoding schemes, it still involves the matrix inversion.

Since the complexity of matrix inversion is cubic with respect to the number of

users the computational complexity of ZF precoding is still high for massive MIMO. Gauss – seidel: To this end, many iteration-based precoding schemes without the complicated matrix inversion, such as Gauss-Seidel (GS), successive over relaxation (SOR), Neumann-Series (NS), and joint steepest descent and Jacobi iteration (SDJC) have been proposed. The key idea of these methods is to use a series of low-complexity matrix multiplications and additions instead of matrix inversion. However, a small number of iterations will result in an unsatisfactory performance. Moreover, a larger number of iterations may even cause a higher computational complexity than the traditional ZF precoding. In this letter, we propose a near-optimal linear ZF precoding without the complicated matrix inversion based on an iterative method.

Weighted two – stage: Unlike GS and SOR schemes, the proposed weighted two-stage (WTS) precoding consists of two symmetric half iterations. To prohibit the spectral radius of the iterative matrix close to unity, the update is slipped by the two half iterations.

Then, the two half iterations are merged together by a weighted coefficient to obtain the iterative solution of the proposed WTS, which can reduce the complexity and acquire a faster convergence. Therefore, the proposed WTS precoding can mitigate multiuser interference by using low-complexity two half iterations. Furthermore, the weighted combination of the two half iterations can further speeds up the convergence and achieves the satisfactory sum-rate performance. Theoretical analysis shows that the proposed WTS precoding can reduce the complexity and improve the convergence. We also provide the simulation results to demonstrate that the proposed WTS precoding can achieve better bit-error-rate (BER) and sum rate performance with a smaller number of iterations than the recently proposed schemes.

Proposed WTS precoding: The columns of H in are asymptotically orthogonal in massive a MIMO system, which makes W positive definite Hermitian. Although the computational

complexity of the pre coded vector t is O, fortunately, the fact that W is positive definite Hermitian inspires us to efficiently realize t with much lower complexity.

To circumvent the complicated matrix inversion and speed up the convergence, we propose a WTS precoding scheme adopting two half iterations and combining the two half iterations with an weighted coefficient to iteratively solve t as follows: Since W is a strictly diagonally dominant Hermitian matrix, and the more diagonally dominant W is, the more rapid convergence rate of the iterative method can be achieved , we decompose W as W= L + LH + D, where L and D denote the strict lower triangular and diagonal matrix of W, respectively.

WTS precoding: The BER performance of the proposed WTS precoding and other precoding schemes is shown. It is clear that the ZF precoding enjoys the best BER performance. The BER performance of the proposed WTS precoding is superior to the GS precoding and SOR precoding when the number of iterations is the same.

For example, when i = 3, the proposed WTS precoding requires the SNR of 23 dB to achieve the BER of 10 3, while the GS precoding and SOR precoding require the SNR of 23.5 dB, SDJC precoding requires the SNR of 24 dB, and NS precoding requires the SNR of 25 dB. Moreover, when WTS precoding, GS precoding, and SOR precoding have the same complexity, for example, the iteration number of the proposed WTS precoding is 2 and the iteration number of the GS and SOR precoding is 4, the GS and SOR precoding is lightly inferior to the proposed WTS precoding. Compares the convergence against the number of iterations in the massive MIMO systems. When i = 1 the capacity of the proposed WTS precoding is 133.2 bps/Hz, while the capacity of the NS, GS, and SOR precoding is 94.1 bps/Hz, 132.1 bps/Hz, and 128.8 bps/Hz when i = 2, which means that the proposed WTS precoding can ultilize the two symmetric half iterations to achieve better performance with the same complexity.