Nowadays, fifth generation (5G) wireless communications systems have been introduced by several mobile companies to meet user demands for high data rates (up to 10 Gbps) and quality of service (QoS). The main target is to obtain an enhanced mobile broadband (eMBB), massive machine type communications (mMTC), and ultra-reliable and low-latency communications. In 5G, several technologies are utilized such as the millimeter-wave (mmWave), the Internet of Things (IoT), the visible light communication (VLC), and the massive multiple-input multiple-output (M-MIMO) [

1]. In mmWave, a wide spectral resource can be exploited to support a wide signal bandwidth and high data throughput. The electromagnetic wave at mmWave band is highly attenuated because of the path loss and shadowing. However, high antenna gain can be obtained when the mmWave band is utilized [

2]. In the IoT, a large number of devices are connected to process huge data rates, make classifications and decisions, and solve problems. It would exploit the artificial intelligence and linked to cloud networks [

3]. In the VLC, a light emitting diode (LED) is utilized in an indoor scenario for simultaneous wireless communications. In addition, the VLC provides high energy efficiency, high security, and free of interference [

4]. The M-MIMO system is an extension of a small scale MIMO where a large number of antennas at the base-station (BS) are deployed to serve multiple user terminals [

5]. It contributes positively in spectrum efficiency (SE) and energy efficiency (EE), increases the degree of freedom, and reduces the latency. However, a large number of antennas requires advanced signal processing techniques to equalize and estimate the signal. Although a detector based on the maximum likelihood (ML) obtains the optimum performance, it is banned in implementation because of the exponential computational complexity [

6]. Sphere decoding (SD) is a possible solution to achieve a quasi-optimum performance. However, it depends on the sphere radius selection and the complexity increases as the radius increased [

7]. Other detectors are also proposed such as detectors based on successive interference cancellation (SIC) [

8], local search [

9], and belief propagation (BP) [

10]. The computational complexity of the SIC based detector is high. The local search based detector depends on the size of the neighborhood. In the BP, the damping factor is required, and it is very hard to find it. In addition, the convergence is not guaranteed. However, a large number of antennas at the BS leads to an interesting phenomena called channel hardening where the columns of the channel matrix are being orthogonal or nearly orthogonal. Thus, a simple linear detector can achieve a satisfactory performance. Linear minimum mean square error (MMSE) is a possible solution, but it sustains a significant performance loss in highly loaded systems [

11]. It also contains an exact matrix inversion that increases the computational complexity. In order to avoid the exact matrix inversion, a plethora of iterative linear M-MIMO detection algorithms have been proposed such as the Neumann series (NS) [

12], Newton iteration (NI) [

13], successive over relaxation (SOR) [

14], Gauss–Seidel (GS) [

15], Jacobi (JA) [

16], and Richardson (RI) [

17]. A detector based on the NS and NI methods approximates the matrix inversion of the

Gram matrix instead of computing it. A detector based on the SOR, GS, JA, and RI methods estimates the signal (

$\widehat{\mathbf{x}}$) by avoid a matrix inversion. However, the behavior of a detector based on iterative methods is greatly influenced by selection of the initial solution (

${\widehat{\mathbf{x}}}_{0}$) where an inappropriate initial solution could increase the number of iterations, and hence increases the complexity. In contrast, good initialization reduces the number of iterations and hence reduces the computational complexity.

Most of the existing iterative linear detectors are utilizing the diagonal matrix (

$\mathbf{D}$) in estimating the initial vector because the equalization matrix is diagonally dominant. In some cases, the diagonal matrix has a slow convergence, or it may not converge. In [

18], a stair matrix (

$\mathbf{S}$) is proposed as an alternative solution to replace

$\mathbf{D}$ in initializing iterative linear detectors based on the NS and JA methods. This paper aims to study the impact of

$\mathbf{S}$ and

$\mathbf{D}$ in initialization of iterative linear M-MIMO UL detectors where iterative methods are utilized such as the SOR, GS, JA, and RI methods. In addition, a comparison between the performance-complexity profiles of

$\mathbf{S}$ and

$\mathbf{D}$ will be presented. Therefore, thousands of random channel matrices have been generated to examine the convergence rate using both

$\mathbf{S}$ and

$\mathbf{D}$.

This paper is organized as follows:

Section 2 presents the system model and the definition of a stair matrix and a diagonal matrix.

Section 3 presents the NS, NI, SOR, GS, JA, and RI methods with the initialization using stair and diagonal matrices.

Section 4 provides a comprehensive analysis of the computational complexity required by each detector. In

Section 5, results and discussion are presented.

Section 6 concludes the paper.