# Computationally Efficient Channel Estimation in 5G Massive Multiple-Input Multiple-output Systems
## 1 Introduction  
Multiple-input multiple-output (MIMO) technology is becoming more and more mature, especially when combined with orthogonal frequency division multiplexing (OFDM) [1; 2; 3; 4; 5], which has been successfully applied in multiple wireless communications fields such as Long-Term Evolution (LTE) and LTE-Advanced. However, traditional MIMO technology can only achieve a \(4~{}\times~{}4\) or \(8~{}\times~{}8\) scale system [6], which makes it difficult to meet the explosive growth in mobile data services. Therefore, in recent years, massive MIMO has been proposed based on traditional MIMO technology [7]. Massive MIMO systems configure up to hundreds of antenna arrays at the base station to serve multiple single-antenna end-users simultaneously [8], which can improve spectrum utilization and power utilization in wireless communications systems by two to three orders of magnitude [9; 10; 11]. This has become one of the most promising enabling technologies and one of the hottest research directions in 5G [12]. The maximum likelihood (ML) algorithm is the optimal algorithm in massive MIMO detection algorithms, but its computational complexity increases exponentially with the number of system antennas and the modulation order of baseband signals. It is difficult for it to be fast, effective, and realized in practical applications. Linear detection methods, such as the zero-forcing (ZF) algorithm and minimum mean square error (MMSE) algorithm, can achieve near-optimal detection performance in massive MIMO systems. The complexity in this kind of detection algorithm is greatly reduced, compared with the complexity of the ML algorithm, but introduces a complex high-dimensional matrix inversion operation, so a low-cost and efficient engineering implementation is still a problem to be solved. Aimed at this problem, many simplified algorithms based on the MMSE detectionscheme have been proposed in recent years, and can be roughly divided into three types: The series expansion class-approximation method [13; 14], the iterative class-approximation method [15; 16] and a gradient-based search for an approximate solution [17; 18; 19; 20]. The authors in [13] proposed a method of using Neumann series expansion to approximate the inverse MMSE filter matrix, but when the number of expansion stages was gradually increased \((i~{}>~{}2)\), the computational complexity was still high, even equal to or exceeding the exact MMSE. The complexity of the filter matrix inversion algorithm also loses a large degree of detection performance. The authors in [14] applied the Newton algorithm derived from the first-order Taylor series expansion (similar to Neumann series expansion) to massive MIMO signal detection, and used the iterative method to improve the estimation accuracy of the MMSE filter matrix inversion. However, from the aspects of detection performance and computational complexity, the algorithm based on the Newton iteration was not dominant. Different from the two series expansion-based algorithms above, it is necessary to estimate the signal vector sent by the user by inverting the approximate matrix. Some iterative algorithms based on solving linear equations, such as the Richardson iterative (RI) algorithm [15] and the successive over-relaxation (SOR) algorithm [16], use the special properties of the MMSE filter's symmetric positive definite matrix. Through the method of solving linear equations, they directly estimate the transmission vector, thus avoiding the inversion of high-dimensional matrices. The RI and SOR algorithms mentioned above have lower computational complexity at a fixed number of iterations, but RI convergence is slower and requires a higher number of iterations to achieve certain detection performance requirements. In SOR, although the detection performance is close to excellent, its internal iterative structure means the algorithm cannot be implemented in parallel in practical applications.