# Efficient Massive MIMO Detection for M-QAM Symbols
## 1 Introduction  
Massive multiple-input multiple-output (MIMO) systems are well known as essential enablers for the future of wireless communication. An antenna array with many antennas enables time-frequency resources to be shared by all user terminals at the same time. Massive MIMO is better than traditional MIMO in many ways, such as making links more reliable and using less energy [1]. Quadrature amplitude modulation (QAM) provides extra benefits for data transmission. Unfortunately, interference from multiple antennas becomes much worse as the size of the system increases, making uplink data detection a tough problem. As a result, to estimate a signal, advanced signal processing techniques are needed. The MIMO data detection problem can be constructed as a linear system of equations. The maximum likelihood (ML) technique theoretically provides the optimal detection performance, but its computational cost is prohibitively high for massive MIMO systems. Therefore, suboptimal detectors with low complexity and strong detection performance are required in practical applications [2]. Sphere decoder (SD) can achieve near-ML performance while requiring less computational complexity than ML detectors. Its computational complexity, however, is inconsistent with the large-scale MIMO problem [3]. Belief propagation (BP) detectors provide the soft output in an iterative process via a factor graph and suffer less local minimum problems than other algorithms. However, BP-based algorithms show severe performance loss as the ratio of the number of receive antennas to the number of transmit antennas approaches one [4, 5]. Moreover, as the number of antennas increases, the complexity of a BP detector itself becomes a burden when the number of antennas becomes larger. Zero forcing (ZF) is simple to implement but provides poor detection performance. However, the linear system of equations has a channel matrix of ill-determined rank, and Tikhonov regularization is a prominent method for solving this problem. A ZF detector with Tikhonov regularization, referred to as a minimum mean-square error (MMSE) detector, obtains a better performance-complexity tradeoffthan nonlinear detection algorithms. In addition, linear MIMO receivers including ZF and MMSE detectors need to perform channel correlation inversion. Matrix inversion requires a complexity of \(O(N^{3})\) (where \(N\) is the system size), which is too high for large system implementation. To avoid the exact matrix inversion needed by linear detection in a MIMO uplink, various sophisticated techniques have been proposed, which can be generically divided into three categories: approximate matrix inversion methods, decomposition methods, and iterative methods [6]. Neumann series expansion is applied to approximate matrix inversion using a series of matrix-vector multiplications, which is simple to implement in hardware but converges slowly [7]. Newton iteration [8] and Chebyshev iteration [9] have been successively proposed to accelerate convergence. However, when the number of iterations exceeds one, their computational complexity exceeds that of exact matrix inversion [10]. The complexity of decomposition methods such as Cholesky decomposition, Gaussian elimination, and QR decomposition (QRD) is \(\mathcal{O}(N^{3})\). Iterative methods such as steepest descent (SD) and conjugate gradient (CG) achieve rapid convergence, but each iteration requires \(\mathcal{O}(N^{2})\) mathematical operations. Methods of coordinate descent (CD), such as Jacobi [11], Gauss-Seidel [12], and successive overrelaxation (SOR), have slower convergence but have only \(\mathcal{O}(N)\) operations per iteration. The computational complexity of these iterative techniques is proportional to the number of iterations executed. Only if the ratio of base station (BS) antennas to user antennas (\(\rho=\frac{K_{t}}{K_{t}}\)) is large enough can satisfactory performance and a reasonable level of complexity be reached.