# An Improved Jacobi-Based Detector for Massive MIMO Systems
## 1 Introduction  
Massive multiple-input-multiple-output (MIMO) is an emerging technology for communication application which contributes a promising technology for the wireless sensor networks (WSNs) [1; 2; 3] and the fifth generation (5G) wireless communications [4]. In such systems, the base station (BS) was equipped with hundreds of antennas serving tens of single-antenna users in the same frequency band [5; 6; 7]. Benefit from the massive MIMO system is capable of achieving higher multiplexing and diversity gains compared with the conventional small-scale MIMO system [8; 9]. However, due to the increasing dimension, the signal detection in the uplink may become a challenge of the massive MIMO system [10]. According to [11], the maximum likelihood (ML) detector is an optimal algorithm. However, its substantially high computational complexity contributes the bottleneck of massive MIMO systems. Therefore, assorted detection algorithms have been proposed. Some nonlinear detection algorithms such as sphere decoder (SD) and its variants have been proposed to achieve near-optimal performance with low complexity [12; 13]. However, the complexity is stillunfavorable because of the large dimension of the systems or the high modulation order. Thus, one can resort to the linear detection algorithm, such as the minimum mean square error (MMSE) with the near-optimal detection accuracy [14]. However, the inverse of the large-dimensional covariance matrices required by MMSE would result in a prohibitively high complexity. Given this challenge, low-complexity approximate matrix inversion has drawn substantial attention. The typical algorithms can be divided into two categories. One typical category is based on the series expansion, such as the Neumann series expansion (NSE) detection algorithm [15]. The NSE algorithm utilizes the first few terms of the series expansion to approximate matrix inverse. However, its performance suffers from a significant loss with the scaling up massive MIMO and only marginal reduction in complexity can be achieved. Another category is based on the iterative algorithms derived from linear equations. To expand, it converts the matrix inversion process into solving linear equations, where the Gauss Seidel (GS) based iterative algorithm is one of the typical algorithms and utilizes the most up-to-date values at each iteration, leading to the faster convergence rate and lower complexity than NSE detection algorithm [16]. Furthermore, the symmetric successive over-relaxation (SSOR) algorithm [17] can also be applied, where different weights are put in the iteration structure. To enhance the robustness of the SSOR algorithm, the optimal parameter of weights is determined by exploiting channel characteristics, yielding a fast convergence and a high accuracy. Unfortunately, each iteration of the GS and SSOR calculations is serial in nature, where each component of the new iteration depends on all of the previously calculated components. Fortunately, the Jacobi algorithm is a simple iterative algorithm with high parallelism [18]. Some improved Jacobi-based algorithms have been proposed, one of which is the damped Jacobi (DJ) algorithm [19]. The algorithm has great convergence performance when the number antennas at the BS is far greater than the number of antennas of the users. However, in the uplink communications, the increase of the users will make the requirements of DJ algorithm no longer satisfied, resulting in a great performance degradation. Motivated by the above concerns, we focus our attention on further improving the convergence rate and calculation accuracy of the Jacobi algorithm with low-complexity calculation. Based on the Jacobi algorithm, we invoke the steepest descent (SD) method [20] and whole-correction method [21] to modify the iteration process. The major contributions of our work can be listed as below: (1) The SD method [20] is invoked in Jacobi algorithm to improve the convergence rate.