Efficient Detectors for Uplink Massive MIMO Systems

I. INTRODUCTION

AN ever-expanding growth in the demand for reliable, high data rate, ubiquitous, and high capacity wireless communication is driving the development of technologies and solutions to support beyond-fifth-generation (B5G) wireless systems [1]. These systems are expected to provide increasing mobile users with reliable, ultra-high data rates and ultralow latency connections to support a plethora of envisioned applications such as immersive reality and remote health care. They are also expected to effectively handle highly dense, highly heterogeneous networks that will emerge from massive device deployment in industrial, smart city, and smart home settings. Importantly, B5G systems are required to meet these demands while maintaining high energy and spectral efficiencies [2].

Massive multiple-input multiple-output (MIMO) systems [3] have been proposed as an effective solution to address many of the above challenges. By utilizing hundreds/thousands of antenna units at the base station (BS), these systems can host multiple users simultaneously and frequency resources, which can significantly enhance system efficiency [4]. The multitude of antennas creates a highly rich scattering environment, providing a considerable enhancement in the diversity and multiplexing gains compared to small-scale MIMO systems. The resulting systems can combine high reliability, high data rates, high energy efficiency, and low noise sensitivity [5].

While bringing about tangible improvements in system capacity, reliability, and energy and spectral efficiency, the transition to large-scale MIMO systems also brings forth a host of new challenges that must be addressed [6], [7]. The large number of antennas and radio frequency (RF) chains means more complex/costly hardware. Moreover, at the signal processing level, the large number of antennas and users directly impacts the dimensionality of the signal, which can drastically increase the computational complexity of basic receiver tasks such as channel estimation and data detection. Both tasks must be performed accurately and efficiently to bring to fruition the promised gains of massive MIMO systems. In the uplink, it is critical for the massive MIMO BS to accurately detect the simultaneous transmissions of a large number of users without incurring a substantial delay that would affect the system’s latency. Hence, uplink massive MIMO detection has emerged as a critical research problem, attracting substantial research efforts [8], [9].

A. Related Work

Optimal detection in the form of maximum-likelihood (ML) detection, while providing the highest accuracy, requires a high-dimensional exhaustive search, making it intractable for practical applications [ 10]. Hence, efforts have focused on developing low-complexity methods that provide the best tradeoff between performance and complexity. In classical (small-scale) MIMO, linear methods have emerged among the most promising solutions. Classical linear methods (minimum mean-squared error (MMSE) and zero-forcing (ZF) based detectors) apply an equalization matrix to the received signal to minimize the inter-link interference before performing detection on the equalized signal [ 11]. While MMSE has become the mainstay of linear detection methods for classical MIMO systems, it requires a large-dimensional matrix inversion, which is computationally costly and can potentially compromise the real-time implementation of the system [ 12]. More generally, the computation of the equalization matrix in linear methods requires the inversion of the Gram (Gramian) matrix. This inversion becomes more computationally demanding as the system size increases. Moreover, the system can be illconditioned if the Gramian matrix is singular [ 9].

Thus, Research has focused on developing approximate linear methods that approach the MMSE performance at significantly lower computational complexity [13]. The work in [13] provides a detailed overview of these approximate linear methods and the related performance-complexity tradeoffs of each method. One proposed approach has been to employ approximate matrix inversion methods to approximate the inverse of the Gramian matrix iteratively. Methods that exemplify this approach include the Neumann series (NS) method [14], [15] and the Newton iteration (NI) method [16], [17]. In [17], a hybrid method was proposed whereby a small number of NI iterations (two iterations) were applied to obtain the initial estimate for the RI method. The motivation was the high complexity of high-order NI. The result was a significant enhancement in performance and a reduction of the computational complexity by an order of magnitude. A drawback of this type of method, however, is that they generally involve multiple matrix multiplications per iteration, which has non-trivial computational complexity and makes them less hardware-friendly. In addition, this type of methods suffer from a severe performance loss when the number of transmitting users approaches the number of BS antennas.

Another proposed approach that has also received attention is solving the matrix inversion as a system of linear equations. These methods start with an initial estimate, and after a number of iterations yield an output that represents the solution to the linear system. Methods that exemplify this approach include the Richardson (RI) method [18], the Jacobi (JA) method [19], the successive over-relaxation (SOR) method [20], the conjugate gradient (CG) [21], the Gauss- Siedel (GS) [22] and the accelerated over-relaxation (AOR) method [23]. One drawback of the above methods is that they may require a large number of iterations to converge, especially if users’ numbers and BS antennas’ numbers are close [24]. It has been observed in multiple studies that the related performance, convergence rate, and complexity of these methods significantly depend on their initial solution [25].

It is noteworthy that the equalization matrix is diagonally dominant [26]. Hence, most detectors in existing literature mainly exploit the diagonal matrix in their design. However, in some cases, the diagonal matrix may not be used to converge. In [27], it is shown that the convergence rate can remarkably improve by using the stair matrix in massive MIMO detectors. In [28], a massive MIMO detector based on an iterative method using the stair matrix is proposed. It was demonstrated that the detectors based on the stair and diagonal matrices have the same computational complexity level. In [25], a stair matrix was employed to compute the initial solution for the NI, GS, SOR, and RI methods. This resulted in improved convergence, enhanced performance, and lower complexity. In [29], the banded matrix accelerates the GS, JA, and SOR’s convergence rate. The banded matrix is exploited in [30] to reduce the computational complexity of the likelihood ascent search (LAS) based detector. One drawback of the above methods is their performance deterioration in an imperfect CSI environment. They also suffer from a significant performance loss when the number of transmitting antennas approaches the number of receiving antennas.

B. Contribution and Organization

Inspired by the promising results achieved in [17], [25], and [28], in this paper, we aim to improve the initialization stage due to its significant impact on the detectors’ performance-complexity profile. In the MMSE, the equalization matrix is diagonally dominant. Hence, the majority of existing detectors have utilized diagonal matrix. However, it has been observed that, in some situations, the methods that use the diagonal matrix have convergence with a slow rate or no convergence [28], [31]. Hence, this paper proposes three different initialization methods for massive MIMO uplink systems based on the scaled identity matrix, the stair matrix, and the banded matrix accompanied by the first iteration of the NI method to approximate the initial vector. The scaling parameter depends on the lower and upper eigenvalues of the equalization matrix. The output of the proposed initialization stage will be an input to the detection stage of the AOR, the SOR, the GS, the JA, and the RI iterative methods. The contributions of this work can be summarized as follows:

· We exploit the channel hardening phenomenon to propose an efficient initialization based on a scaled identity matrix and the first iteration of the NI method. The relaxation parameter [TeX:] $$(\omega)$$ is selected based on the lower and upper eigenvalues of the equalization matrix. Then the initial vector is computed based on the first iteration of the NI method.

· We also propose an efficient initialization based on a stair matrix and the first iteration of the NI method. We first compute the stair matrix inversion, which has the same complexity as the diagonal matrix inversion. After that, we approximate the equalization matrix inversion using the NI method. Then the NI formula is employed to estimate the initial vector.

· We propose an efficient initialization based on a band matrix and the first iteration of the NI method. The inverse of the band matrix is first computed and then employed to initialize the massive MIMO detectors based on the NI method.

· We conduct extensive simulations to demonstrate the proposed detectors’ performance and computational complexity in different scenarios. Several modulation schemes (16QAM and 64QAM) and massive MIMO sizes are used. Furthermore, to avoid misleading conclusions, perfect and imperfect channel state information (CSI) are considered. We show that the proposed initializations for massive MIMO detectors achieve a significant performance improvement and remarkable complexity reduction in both perfect and imperfect CSI scenarios, especially when the user terminals’ number approaches the base station (BS) antennas’ number.

The rest of this article is organized as follows: The system model is described in Section II. In Section III, we demonstrate the stair and band matrices and their properties. The proposed initialization methods for several massive MIMO detectors are presented in Section IV. Section V presents the complexity profile for all the proposed detectors in terms of multiplications’ number. In Section VI, we present, discuss, and compare our simulation results with conventional detectors and other state-of-art methods. Finally, our conclusions are presented in Section VII.

II. SYSTEM MODEL

In this paper, a massive MIMO uplink system with N antennas at BS and K single antenna users is considered where [TeX:] $$N \gg K.$$ After the transmission of modulated symbols, the received signal at the BS is given by

where [TeX:] $$\mathbf{H} \text{ is the } N \times K$$ channel matrix, and [TeX:] $$\mathbf{w} \text { is } N \times 1$$ circularly symmetric complex additive white Gaussian noise (AWGN) vector with mean [TeX:] $$\mathbf{0}$$ and covariance matrix [TeX:] $$\sigma^2 \mathbf{I}_N.$$ In this paper, we also consider imperfect channel state information (CSI), such that the imperfect channel estimate [TeX:] $$\acute{\mathbf{H}}$$ is given as [32], [33]

where [TeX:] $$\tilde{\mathbf{E}}$$ is the error matrix whose i.i.d. elements are modeled as complex Gaussian with mean 0 and variance 1, and [TeX:] $$0 \leq \zeta \leq 1.$$ The vectors [TeX:] $$\mathbf{x}=\left[x_1, x_2, \cdots, x_K\right]^T$$ and [TeX:] $$\mathbf{y}=\left[y_1, y_2, \cdots, y_N\right]^T$$ denote the vector of data symbols transmitted by the K users and the corresponding received signal vector at the BS, respectively. The main objective of the massive MIMO detector is to estimate [TeX:] $$\mathbf{x}.$$ While many massive MIMO detection techniques exist in the literature, this work focuses on linear detectors due to their simplicity and low complexity.

A. Linear MMSE Data Detection

The massive MIMO detector based on the MMSE is given as

where [TeX:] $$\mathbf{b}=\mathbf{H}^H \mathbf{y}$$ is the matched-filter output and [TeX:] $$\mathbf{W}=\mathbf{H}^H \mathbf{H}+\sigma^2 \mathbf{I}_K$$ represents the MMSE equalization matrix. On the other hand, in the zero-forcing (ZF) based detector, the noise effects are ignored, and the signal is estimated as

where [TeX:] $$\mathbf{G}=\mathbf{H}^H \mathbf{H}$$ is the Gram matrix or Gramian. Unlike the ZF detector, the MMSE equalization matrix considers the noise effects and achieves higher performance gains. Notably, the Gramian matrix is invertible in massive MIMO [26]. As obvious from (3) and (4), both the ZF and MMSE based detectors include a matrix inversion, which is not desirable in hardware implementations, particularly for large N and K. Therefore, iterative methods have been proposed to approximate or avoid matrix inversion. Although these methods achieve a reasonable performance when [TeX:] $$N \gg K,$$ their performance tends to deteriorate as K approaches N. In addition, a large number of iterations is often required, which leads to increased computational complexity.

B. Newton Iteration Method

In the Newton iteration (NI) method, we obtain an approximate estimate of the matrix inverse [TeX:] $$\mathbf{W}^{-1}$$ through n iterations. If [TeX:] $$\mathbf{X}^{(0)}$$ is the initial estimate of [TeX:] $$\mathbf{W}^{-1},$$ then the nth iteration estimate is

It should be noted that the selection of [TeX:] $$\mathbf{X}^{(0)}$$ plays a crucial role in the convergence rate and the computational complexity of the NI method. A common choice is [TeX:] $$\mathbf{X}^{(0)}=\mathbf{D}^{-1}$$ where [TeX:] $$\mathbf{D}$$ is the diagonal matrix [16]. Equation (5) converges quadratically to [TeX:] $$\mathbf{W}^{-1}$$ if

The signal in (3) can be estimated as

where [TeX:] $$\mathbf{E}$$ consists of the off-diagonal entries of [TeX:] $$\mathbf{W}$$.

C. Data Detection based on Iterative Methods

The NI based detector has a high complexity due to the large number of iterations needed to converge. Hence, alternative methods such as the AOR, the SOR, the GS, the JA, and the RI have been proposed to detect the signal without the explicit computation of [TeX:] $$\mathbf{W}^{-1}$$.

The AOR is a stationary iterative method for solving linear systems where the signal can be estimated as

where [TeX:] $$\mathbf{U} \text { and } \mathbf{L}$$. are the strictly upper diagonal matrix, and strictly lower diagonal matrix, respectively. Moreover, ω is the relaxation parameter, and γ is the acceleration parameter, and both are related to the eigenvalues of equalization matrix [34], [35]. Furthermore, based on [TeX:] $$\omega \text { and } \gamma$$, the AOR method is reduced to the JA, GS, and SOR methods as:

When [TeX:] $$\gamma = \omega$$, a detector based on the SOR method can estimate the signal as

When [TeX:] $$\omega = 1,$$ a detector based on the GS method can estimate the signal as

When [TeX:] $$\gamma=0, \omega=1 \text {, }$$ a detector based on the JA method can estimate the signal as

which holds if

In (12), the condition is realized with very high probability in massive MIMO systems [36]. In parallel computing platforms, the JA method can be easily implemented [37]. However, the JA method is neither robust nor as fast as the GS and SOR methods in sequential computing platforms.

Another iterative method to achieve the MMSE performance is the RI method. Although it has a low complexity, the performance-complexity profile of the RI iterative method is very sensitive to the value of the relaxation parameter [TeX:] $$(\omega)\text {. }$$ The signal in the RI method is estimated as

For the iterative methods, if the spectral radius [TeX:] $$\rho\left(\mathbf{I}-\mathbf{D}^{-1} \mathbf{G}\right)\lt 1$$ is satisfied, the method is convergent for all initial vectors. However, a smaller spectral radius leads to faster convergence. Therefore, the selection of the initial estimate [TeX:] $$\left(\hat{\mathbf{x}}^{(0)}\right)$$ impacts the number of iterations needed to detect the signal, and accordingly, the computational complexity. In most iterative methods, it is common to set the initial estimate [16] as

III. STAIR AND BAND MATRICES AND THEIR PROPERTIES

As the equalization matrix in MMSE is diagonally dominant, the diagonal matrix has conventionally played a key role in formulating most approximate inversion methods and iterative methods for massive MIMO detection. Several recent works, however, have established the merit of other formulations, such as the band matrix [30] and the stair matrix [28]. Both types of matrices will play essential roles in the efficient initialization techniques proposed in our work. It is thus worth introducing these types of matrices and their properties in this section.

A. Stair Matrix and Its Properties

Definition 1: A stair matrix [TeX:] $$(\mathbf{S})$$ is a special tri-diagonal matrix where one of conditions below is fulfilled:

- Type I: [TeX:] $$\mathbf{S}_{(i, i-1)}=0, \mathbf{S}_{(i, i+1)}=0,$$ where [TeX:] $$i=1,3, \cdots, 2\left\lfloor\frac{K-1}{2}\right\rfloor+1,$$

- Type II: [TeX:] $$\mathbf{S}_{(i, i-1)}=0, \mathbf{S}_{(i, i+1)}=0,$$ where [TeX:] $$i=2,4, \cdots, 2\left\lfloor\frac{K}{2}\right\rfloor .$$

In other words, [TeX:] $$\mathbf{S}$$ is a tri-diagonal matrix where the offdiagonal elements on either odd or even rows are zeros [28]. In this paper, the stair matrix is denoted by [TeX:] $$\mathbf{S}=\left(s_{i, i-1}, s_{i i}, s_{i, i+1}\right).$$

Examples of Type I and Type II of the stair matrix are shown as:

or

It is also noted that if [TeX:] $$\mathbf{S}$$ is a stair matrix, then [TeX:] $$\mathbf{S}^H \text { and } \mathbf{S}^{-1}$$ are also stair matrices [28]. If we have a linear system [TeX:] $$\mathbf{S x}=\mathbf{d} \text {, }$$ the solution can be immediately obtained by computing [TeX:] $$\mathbf{S}^{-1} \mathbf{d}$$ [38]. Algorithm (1) solves the stair linear system where [TeX:] $$d_i=0$$ if [TeX:] $$i\lt 1 \text { or } i\gt K.$$

It is also worth noting that the [TeX:] $$K \times K$$ stair matrix is nonsingular only in the case that the diagonal elements of [TeX:] $$\mathbf{S}$$ are nonsingular. In addition, if [TeX:] $$\mathbf{S}$$ is nonsingular, then

B. Band Matrix and Its Properties

We begin with the following definition:

Definition 2: A band matrix refers to a square matrix in which zero elements are located at a distance of p above and below the main diagonal. Here, p represents a value smaller than the matrix’s size. Therefore, for a matrix of size [TeX:] $$K \times K$$, it holds that [TeX:] $$p\lt K .$$ Let [TeX:] $$\mathbf{W}=\left(W_{i j}\right)$$ denote a [TeX:] $$K \times K$$ matrix, and [TeX:] $$\mathbf{T}=\left(T_{i j}\right)$$ represent a banded matrix with a bandwidth of [TeX:] $$2 p+1,$$ as defined by the following expression:

where p is called the matrix bandwidth or the band parameter. In the banded matrix, non-zero elements are only restricted to the diagonal band, which includes the main and secondary diagonals. An example of a banded matrix is shown below.

According to [39], LU decomposition is usually used to find [TeX:] $$\mathbf{T}_p^{-1}$$ where the jth column of [TeX:] $$\mathbf{T}_p^{-1}$$ can be calculated by solving the following linear equations:

where [TeX:] $$\mathbf{e}_j$$ is the jth column of the identity matrix, [TeX:] $$\mathbf{V}_j$$ is an intermediate vector, and [TeX:] $$T_j$$ is the jth column of the inverse matrix, respectively. In [39], the authors proposed several lowcomplexity methods for obtaining the inverse of the banded matrix.

IV. PROPOSED METHODS

In this section, we present the proposed hybrid massive MIMO detectors to enhance the BER performance. We consider five well-known iterative methods, namely, the AOR, the SOR, the GS, the JA, and the RI methods. Due to its high impact on the convergence rate and the detectors’ performance, the initial solution should be carefully chosen.

A. Initialization based on NI and Scaled Identity Matrix (Detector 1)

In this subsection, we propose an efficient initialization of several massive MIMO detectors based on the NI and the scaled identity matrix. The main idea behind the proposed initialization is that utilization of the identity matrix instead of the diagonal matrix [TeX:] $$(\mathbf{D})$$ in a detector increases the convergence rate. For instance, the iteration matrix in the RI method is [TeX:] $$\varphi_{R I}=\mathbf{I}-\omega \mathbf{W}$$ while it is [TeX:] $$\varphi_{J A}=\mathbf{I}-\mathbf{D}^{-1} \mathbf{W}$$ in the JA method. However, the convergence rate of the RI based detector [TeX:] $$\frac{1}{\omega} \mathbf{I}$$ is faster than that in the JA based detector. Therefore, the convergence rate of [TeX:] $$\frac{1}{\omega} \mathbf{I}$$ in the NI based detector should be faster than that of [TeX:] $$\mathbf{X}^{(0)}=\mathbf{D} .$$ Due to channel hardening phenomena [ 40], we also can assume that [TeX:] $$\mathbf{D}=\mathbf{W} \simeq N \mathbf{I}.$$ If N and K grow to infinity, the smallest and largest values of eigenvalues of [TeX:] $$\mathbf{W}$$ would be stable [ 41] and can be presented as

In this paper, we use the optimum ω as

Therefore, matrix approximation after the first iteration of the NI method can be expressed as

and the signal can be initially estimated as

However, based on the NI method, [TeX:] $$\hat{\mathbf{x}}^{(0)}$$ can be presented as

In the AOR based detector, the corresponding estimation of the signal after the first iteration is presented as

In the SOR based detector, the corresponding estimation of the signal after the first iteration is given by

In the GS based detector, the corresponding estimation of the signal after the first iteration is given by

In the JA based detector, the corresponding estimation of the signal after the first iteration is given by

In the RI based detector, the corresponding estimation of the signal after the first iteration is given by

Algorithm (2) presents the proposed initialization of massive MIMO detectors with the corresponding iterative methods.

B. Initialization based on the NI and the Stair Matrix (Detector 2)

According to [25], [37], computation of [TeX:] $$\mathbf{S}^{-1}$$ incurs the same complexity order as the computation of [TeX:] $$\mathbf{D}^{-1}.$$ In Detector 2, we use the stair matrix to initialize all detectors based on iterative methods as

where [TeX:] $$\mathbf{S}^{-1}$$ can be easily computed as shown in Algorithm (3). Therefore, the first iteration of the AOR based detector is

The SOR based detector’s first iteration is

The GS based detector’s first iteration is

The JA based detector’s first iteration is

The RI based detector’s first iteration is

In order to accelerate the convergence rate and hence, reduce the computational complexity, we propose to utilize the stair matrix [TeX:] $$(\mathbf{S})$$ to approximate [TeX:] $$\mathbf{W}^{-1}$$ based on the NI formula. Then, the approximated matrix [TeX:] $$(\mathbf{W}^{-1})$$ will be used to find the initial solution of the iterative methods. We propose to approximate the equalization matrix using the first iteration of the NI method, where [TeX:] $$\mathbf{D}$$

Therefore, [TeX:] $$\bar{\mathbf{W}}^{-1}$$ is calculated based on matrix-vector multiplications instead of matrix-matrix multiplications. The initial estimation [TeX:] $$\left(\bar{\mathbf{x}}^{(0)}\right)$$ based on the first NI iteration can be expressed as

Therefore, the AOR based detector’s first iteration is

The SOR based detector’s first iteration is

The GS based detector’s first iteration is

The JA based detector’s first iteration is

The RI based detector’s first iteration is

In order to further accelerate the convergence rate, we use the Newton-Schultz [30] and (49) to estimate the initial vector [TeX:] $$\left(\hat{\mathbf{x}}^{(0)}\right)$$ as

Therefore, the AOR-based detector’s first iteration is

The SOR based detector’s first iteration is

The GS based detector’s first iteration is

The JA based detector’s first iteration is

The RI based detector’s first iteration is

Algorithm (3) describes the proposed massive MIMO detectors based on the stair matrix and the NI formula.

C. Initialization based on the NI and the Band Matrix (Detector 3)

According to [29], if the equalization matrix is diagonally dominant, then for any [TeX:] $$K \leq N$$ the iterative methods are convergent for any initial vector. In the case of the band matrix, the band parameter (p) plays a crucial role in achieving a fast convergence rate. In Detector 3, we employ the band matrix defined in Section III-B and the NI method to initialize the massive MIMO detectors. We first use the band matrix to initialize all detectors based on iterative methods as

where [TeX:] $$\mathbf{F}$$ presents the band matrix extracted from [TeX:] $$\mathbf{W}$$, and [TeX:] $$\mathbf{F}^{-1}$$ can be easily computed as shown in Algorithm (4). In order to achieve a fast convergence rate and hence, reduce the computational complexity, we also propose to use the band matrix [TeX:] $$(\mathbf{F})$$ to approximate the equalization matrix inverse [TeX:] $$\mathbf{W}^{-1}$$ based on the first iteration of the NI formula as

It is clear that [TeX:] $$\bar{\mathbf{W}}^{-1}$$ is calculated based on matrix-vector multiplications instead of matrix-matrix multiplications. The initial estimation [TeX:] $$\left(\bar{\mathbf{x}}^{(0)}\right)$$ based on the first NI iteration can be expressed as

Therefore, [TeX:] $$\bar{\mathbf{x}}^{(0)}$$ can be presented as

Therefore, the AOR based detector’s first iteration is written as

The SOR based detector’s first iteration is

The GS based detector’s first iteration is

The JA based detector’s first iteration is

The RI based detector’s first iteration is

Algorithm (4) shows the details of initialization and the proposed massive MIMO detectors based on the band matrix and the NI method.

V. COMPLEXITY ANALYSIS

The fact that the hardware complexity is primarily determined by the number of multiplications is widely recognized [42]. Consequently, this paper focuses on assessing the computational complexity by examining the necessary number of multiplications. The computational complexity of the proposed massive MIMO data detection techniques is split into two stages; preparation & initialization and iteration. For the preparation & initialization stage, the computation of the matrix inversion, the matched filter output [TeX:] $$({\mathbf{b}})$$, and the first iteration of the NI method are considered. All proposed detectors require the computation of [TeX:] $$\mathbf{b}$$, where 4NK multiplications are needed. The computation of [TeX:] $$\mathbf{D}^{-1}$$, requires K multiplications. The number of multiplications required to compute [TeX:] $$\hat{\mathbf{x}}^{(0)}$$ based on the NI method is 8(KN +K). In addition, the computation of [TeX:] $$\mathbf{F}^{-1}$$ requires [TeX:] $$2 p K^2+(3 p+5) p K$$ multiplications [39] while [TeX:] $$\mathbf{S}^{-1}$$ needs 3(K−1) multiplications [28]. Multiplications’ number in the iteration stage in all iterative methods (conventional detectors) is presented in Table I. Clearly, the number of iterations (n) significantly impacts the number of multiplications.

The total number of multiplications required for each of the proposed detectors (preparation & initialization and iteration) is presented in Table II. Detector 1, detector 2, and detector 3 correspond to the proposed detectors in Algorithm (2), Algorithm (3), and Algorithm (4), respectively. As shown in Table II, the MMSE complexity [TeX:] $$O\left(K^3\right)}$$ is reduced to [TeX:] $$O\left(K^2\right)$$ using the proposed detectors. However, the complexity of the detector in [17] is presented as [TeX:] $$O\left(NK\right)$$ where two iterations of the NI method are required to initialize the RI based detector. In Section VI, it is shown the proposed detectors converge faster than the detector in [17].

VI. NUMERICAL RESULTS

In this section, we use simulation results to investigate the performance of our proposed methods and compare them with conventional techniques. Our results are obtained by averaging over 10,000 instances of the channel matrix H, whose elements are independent complex Gaussian with mean zero and variance 1. We consider QPSK, 16QAM and 64QAM modulations in our simulations. We also consider several massive MIMO size, including 20×160, 30×160, 40×160. For Detector 3, the band parameter is set as p = 5. While the majority of our results assume perfect CSI, the impact of imperfect CSI is also considered. The performance of the classical MMSE detector is shown as a benchmark in all our simulation results. In addition, it is noteworthy that not every iteration could improve the performance. However, every extra iteration could increase the computational complexity. In this paper, we are using the smallest number of iterations to attain the MMSE performance.

In Figs. 1(a), 1(b), and 1(c), we show the BER performances of the three proposed detectors combined with the selected iterative methods and compare them with the performances of the conventional iterative methods that do not use the proposed initialization. This is done using 16QAM, a 20×160 massive MIMO system, and only a single iteration (n = 1). We also compare it with the performance of the detector developed in [17].

It is clear from Fig. 1(a) that detector 1 with all the proposed methods yields performance that overlaps with MMSE for the whole SNR range. The performance is also approximately 0.5 dB better than the detector proposed in [17] at BER of [TeX:] $$10^{-4}.$$ For each method, there are substantial gains for using the proposed initialization compared to the conventional (diagonal-based) initialization. For instance, at BER of [TeX:] $$10^{-3},$$ the proposed GS exhibits a gain of approximately 1 dB, the proposed JA a gain of approximately 3.2 dB, while the RI, SOR, and AOR all exhibit very high gains (more than 6 dB). Similar trends are also observed for detectors 2 and 3 in Figs. 1(b) and 1(c), respectively, with approximately similar gains. These figures confirm the advantage of our proposed initialization, especially that MMSE performance is achieved with a single iteration n = 1.

In Figs. 2(a), 2(b), and 2(c) we repeat the same experiments as Figs. 1(a), 1(b), and 1(c), but using 64QAM. In Fig. 2(a) the performance of the proposed AOR, almost overlaps with the MMSE, while those of the GS, RI, and SOR are very close to MMSE. However, the proposed JA diverges from the MMSE at high SNR. In particular, it diverges by more than 2 dB at BER of [TeX:] $$10^{-6}.$$ At BER of [TeX:] $$10^{-3},$$ all the proposed detectors yield a gain of approximately 2 dB compared to the detector in [17]. The gains compared to the conventional detectors are higher than those obtained at 16QAM. For instance, the gain of the GS is higher than 6 dB at BER of [TeX:] $$10^{-3},$$ while the gains of the other method are much higher since the conventional methods seem to exhibit error floors at high SNR, unlike the proposed methods. For detector 2, we can see in Fig. 2-b that the proposed GS and SOR overlap with the MMSE. The RI and AOR diverge from the MMSE starting from SNR of 16 dB, with a performance loss of more than 2 dB at BER of [TeX:] $$10^{-6}.$$ The JA, however, diverges from the MMSE earlier, at approximately 13 dB, with a performance loss of approximately 3 dB at BER of [TeX:] $$10^{-4}.$$ Needless to say, all the proposed methods outperform the method [17] and significantly outperform their conventional counterparts. For detector 3, we can see in Fig. 2(c) that all the proposed methods overlap with MMSE. Hence, among the three proposed detectors, detector 3 seems to offer the best performance as the modulation order increases. [TeX:] $$10^{-4},$$ though the gap increases to about 2.5 dB at BER of [TeX:] $$10^{-3}.$$ For the GS, the gap is approximately 1.5 dB at BER of [TeX:] $$10^{-4}$$ and approximately 3 dB at BER of [TeX:] $$10^{-5}.$$ For the SOR, the gap is approximately 6 dB at BER of [TeX:] $$10^{-4}.$$ For the RI, the gap is approximately 6.8 dB at BER of [TeX:] $$10^{-3}.$$ In general, the proposed AOR and GS are the closest to the MMSE, while the JA is the farthest. Still, with the proposed detector we achieve viable performance with AOR, GS, and SOR, compared to the conventional methods whose performance degrades significantly with the increase in the size of the system.

Fig. 3(b) shows the performance of Detector 2 for the 40× 160 massive MIMO system with n = 3. In this case, both the AOR and the GS almost overlap with the MMSE, while the SOR is close to the MMSE, slightly diverging at high SNR. While the proposed RI performs significantly better than the conventional RI (more than 10 dB better at BER of [TeX:] $$4 \times 10^{-2}$$), it is still significantly far from MMSE performance and seems to encounter an error floor. Moreover, there is no remarkable difference between the performance of the conventional and proposed JA, both of which are significantly degraded. It is also noticed that the AOR, GS, SOR, and RI all outperform the detector proposed in [17].

Fig. 3(c) shows the performance of Detector 3 for the 40×160 massive MIMO system with n = 2. Again as before, it is observed that all proposed detectors outperform their conventional counterparts and outperform the Detector of [17]. Moreover, the GS, the proposed AOR, GS, and the SOR all overlap with MMSE performance. As with Detector 1 and Detector 2, the RI and the JA are the most affected by the increase in the size of the system. However, both of them perform much better with Detector 3 than with Detectors 1 and 2, with the RI achieving BERs below [TeX:] $$4 \times 10^{-3}$$ at high SNR. While all the proposed detectors outperform their conventional counterparts, it seems that Detector 3 with GS, AOR, and SOR offers the best performance for large-sized systems.

While the previous results all assume perfect CSI, in Fig. 4, we investigate the impact of imperfect CSI on the performance of the proposed estimators. In order to avoid any misleading conclusions, we present the performances of the proposed detectors in case of perfect and imperfect CSI. In order to model imperfect CSI, we assume that the estimated channel is related to the true channel by (2) where ζ is set to 0.9. We use QPSK with a system size is 30 × 160 and n = 1 iteration for all detectors. In case of imperfect CSI, in general, for all iterative methods, Detector 3 seems to offer the best performance with imperfect CSI, followed by Detector 1 and then Detector 2. Moreover, all the proposed detectors, except JA based Detector 2, perform better than the detector proposed in [17] for imperfect CSI. The detector in [17] diverge from the proposed detectors starting from SNR of 4 dB, with a performance loss of more than 3 dB at BER of [TeX:] $$10^{-4}.$$ It is also noticed that for RI and JA, the performances of both Detector 1 and Detector 2 are affected by imperfect CSI. While for the AOR, GS and SOR, all detectors perform close to the MMSE. The proposed JA based detector 2 suffers from performance loss at high SNR. In general, it is observed that Detector 3 is the most robust to channel errors.

We next investigate the computational complexity of the proposed estimators. Fig. 5 illustrates all the proposed detectors as well as the benchmark MMSE in terms of the multiplications versus the number of transmitting antennas. The number of received antennas is fixed at 160, and the number of iterations is set to n = 1. From Fig. 5(a), it is clear that the proposed detectors provide a huge reduction in complexity compared to the MMSE. In fact, compared to the MMSE, the proposed detectors require approximately 10−20× less multiplications. Due to the large gap between the proposed detectors and the MMSE, we show the complexity of the proposed detectors without the MMSE in Fig. 5(b), for better resolution. It is observed that Detectors 1 and 2 have almost identical complexity for all the methods, while Detector 3 has higher complexity than both. Moreover, the gap between Detectors 1, 2, and Detector 3 increases with the number of transmitting antennas. However, Detector 3 does not seem to exceed twice the number of computations of Detectors 1 and 2.

In Fig. 6, we compare the computational complexity of the proposed estimators in terms of the number of multiplications needed to achieve a BER of [TeX:] $$10^{-4}$$ for a 40×160 massive MIMO system and 64QAM. For reference, we also provide the complexity of the detector in [17]. It is again observed that there is a huge reduction in complexity compared to the MMSE detector.For all proposed detectors, the AOR has the lowest complexity to achieve the target performance. However, the JA method based on Detectors 1 & 2 failed to attain BER=[TeX:] $$10^{-4}$$. However, it works well when the ratio between the number of transmitting and receiving antennas is very small.

VII. CONCLUSION

In this paper, we considered the problem of data detection in massive MIMO systems. Three different initialization methods for massive MIMO detectors were developed based on the NI method, the scaled identity matrix, the stair matrix, and the band matrix. The proposed initialization methods were combined with the AOR, the SOR, the GS, the JA, and the RI methods. Using simulation results, the proposed detectors achieved a good performance with a significant complexity reduction in both perfect and imperfect CSI scenarios, under different modulation schemes, and when the number of users approaches the number of BS antennas. An attractive feature of the proposed detectors is that a large number of iterations is not required to attain MMSE performance when the number of receiving antennas is much larger than the number of transmitting antennas. We also showed that the proposed detectors achieved a significant improvement in BER performance with a significant complexity reduction compared to the conventional detectors that employ the diagonal matrix in the initialization stage. Moreover, the proposed AOR and SOR based detectors achieved the best performance and lowest complexity in the various considered scenarios. Finally, many of the steps in the implementation of the proposed massive MIMO detectors lend themselves to a real-time implementation in the presence of appropriate computational resources.