Low-complexity signal detection networks based on Gauss-Seidel iterative method for massive MIMO systems

Yao, Haifeng; Li, Ting; Song, Yunchao; Ji, Wei; Liang, Yan; Li, Fei

doi:10.1186/s13634-022-00885-0

Research
Open Access
Published: 21 June 2022

Low-complexity signal detection networks based on Gauss-Seidel iterative method for massive MIMO systems

Haifeng Yao¹,
Ting Li¹,
Yunchao Song²,
Wei Ji¹,
Yan Liang¹ &
…
Fei Li ORCID: orcid.org/0000-0003-4114-9082¹

EURASIP Journal on Advances in Signal Processing volume 2022, Article number: 51 (2022) Cite this article

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Abstract

In massive multiple-input multiple-output (MIMO) systems with single- antenna user equipment (SAUE) or multiple-antenna user equipment (MAUE), with the increase of the number of received antennas at base station, the complexity of traditional detectors is also increasing. In order to reduce the high complexity of parallel running of the traditional Gauss-Seidel iterative method, this paper proposes a model-driven deep learning detector network, namely Block Gauss-Seidel Network (BGS-Net), which is based on the Gauss-Seidel iterative method. We reduce complexity by converting a large matrix inversion to small matrix inversions. In order to improve the symbol error ratio (SER) of BGS-Net under MAUE system, we propose Improved BGS-Net. The simulation results show that, compared with the existing model-driven algorithms, BGS-Net has lower complexity and similar the detection performance; good robustness, and its performance is less affected by changes in the number of antennas; Improved BGS-Net can improve the detection performance of BGS-Net.

Introduction

Beyond fifth generation (B5G) is a perfect process to solve some application scenarios and technologies of 5th generation mobile communication technology (5G) . Artificial intelligence technology is the engine of B5G. In recent years, machine learning algorithms have been used in various fields such as healthcare, transportation, energy, and self-driving cars. These algorithms are also being used in communication technologies to improve system performance in terms of spectrum utilization, latency and security. With the rapid development of machine learning techniques, especially deep learning techniques, it is crucial to consider symbol error ratio (SER) and complexity when applying algorithms [1]. Massive multi-input multi-output (MIMO) is a key technology in B5G, where tens, hundreds or thousands of antennas are equipped at the base station. This system makes the signal detection problem a big challenge, because the computational complexity of the detector increases with the number of antennas [2, 3]. Therefore, how to find a balance between detection accuracy and complexity in massive MIMO signal detection has become a hot topic of research for domestic and foreign scholars.

In conventional signal detection methods, maximum likelihood (ML) is a nonlinear maximum likelihood detector. However, its complexity increases exponentially as the number of transmitting antennas increases, hindering its implementation in practical MIMO systems [2]. The spherical decoding (SD) detector [3] and the k-best detector [4] are two variants of ML detectors that balance computational complexity and SER by controlling the number of nodes in each search phase. Unfortunately, QR decomposition in these nonlinear detectors leads to high computational complexity and low parallelism because of the inclusion of unfavorable matrix operations, such as element elimination. In contrast, suboptimal linear detectors, such as minimum mean square error (MMSE) [5] and zero forcing (ZF) [6], provide a better trade-off between SER and computational complexity, but their complexity still reaches three times the number of transmitting antennas .

In order to reduce the complexity of matrix inversion, in 2013, Wu et al. proposed an approximate inversion-based uplink detection algorithm in [7]. Over the next few years, a large number of MIMO detectors designed for specific massive MIMO systems continued to appear. The main idea of the proposed method is to use iterative methods to approximate the inverse of a matrix or to avoid the computation of exact matrices. For example, the neumann series method (NS) [8], the newton iterative method (NI) [9], the Gauss-Seidel method (GS) [10], the successive super-relaxation method (SOR) [11], the jacobi (JA) method [12], the richardson method (RI) [13], the conjugate gradient method (CG) [14], the lanczos method ( LA) [15], residual method [16], coordinate descent method (CD) [17], belief propagation method (BP) [18], etc., these algorithms successfully reduce the complexity to $O(\mathrm {M} ^2)$, but the SER is only close to MMSE.

In order to improve the performance of detection algorithms, [19,20,21] introduced deep learning into communication. These methods treat the functional blocks of wireless communication as black boxes and replace them with deep learning networks. The mapping relationship between input and output data is obtained by training a large amount of data in an offline training phase. However, deepening the network does not significantly improve the performance beyond a certain number of layers, for this reason [22] proposed a parallel detection network (PDN), which consists of several unconnected deep learning detection networks in parallel. By designing specific loss functions that reduce the similarity between detection networks, the PDN obtains considerable diversity results. These algorithms are pure black boxes and although they improve the performance of detection, they require a large amount of training data to learn a large number of parameters, the advantage of these algorithms is that they do not require the incorporation of communication knowledge. [23, 24] proposed a modern neural network structure suitable for this detection task, detection network(DetNet). The structure of DetNet is obtained by expanding the iterations of the projected gradient descent algorithm into the network. [25, 26] proposed the orthogonal approximate message passing (OAMPNet), a model-driven deep learning network for multiple-input multiple-output (MIMO) detection, and [27, 28] proposed MIMO detection network(MMNet), a deep learning MIMO detection scheme. The design of MMNet is based on the theory of iterative soft thresholding algorithms, which significantly outperforms existing methods on realistic channels of the same or lower computational complexity. These algorithms are purely white-box model iterative and have better performance than convolutional neural network (CNN), deep neural networks (DNN), but are not as applicable as its wide range. In addition to these, there are BP-Net [29], CG-Net [30], which are networks changed based on approximation methods. [31] proposed a data-driven implementation of an iterative soft interference cancellation (SIC) algorithm, called DeepSIC. This method significantly outperforms model-based methods in the presence of channel state information (CSI) uncertainty, but the network is more complex and this method is a combination of black-box and white-box methods. Therefore, we know that deep learning methods can improve the performance of detection. However, when the number of antennas is large, deep learning not only requires high hardware requirements, but also requires training, which in reality, can have a significant delay.

Therefore, we consider using approximate inversion methods to reduce complexity while using deep learning methods to improve SER. However, most of the above-mentioned work is directed to a single-antenna user equipment (SAUE) system, and it is assumed that the channel matrix is independent, identically distributed (i.i.d) and obeys gaussian distribution. Unfortunately, in practice, because a user maybe equipped with several antennas, the antennas from the same user equipment (UE) are not sufficiently separated [32], so their transmission vectors are usually related. The spatial correlation between antennas is a key factor affecting the performance of the massive MIMO (M-MIMO) system. Therefore, this paper also considers the multiple-antenna user equipment(MAUE) system while considering the SAUE system.

In this paper, we propose a model-driven deep learning detector network Block Gauss-Seidel Network (BGS-Net) to solve the high complexity caused by the parallel operation of traditional Gauss-Seidel [10], which is based on the Gauss-Seidel iterative method . We reduce the complexity by converting large matrix inversions $({\mathbf {D}} +{\mathbf {L}} )^{-1}$ to small matrix inversions and converting matrix-by-matrix to matrix-by-vector. This paper considers SAUE and MAUE systems [32, 33] under Rayleigh channels. In order to improve SER of BGS-Net under MAUE system, we improve the initial solution of BGS-Net by replacing ${\mathbf {x}}_0 ={\mathbf {D}} ^{-1}{\mathbf {H}} ^\mathrm {T} {\mathbf {y}}$ with ${\mathbf {x}}_0 ={\mathbf {A}} ^{-1}{\mathbf {H}} ^\mathrm {T} {\mathbf {y}}$ . For ${\mathbf {A}} ^{-1}$, we use block matrix approximation to reduce its complexity. Simulation results show that compared with existing model-driven algorithms, BGS-Net has lower complexity and similar SER; good robustness, its performance is less affected by changes in the number of antennas; SER is better than traditional Gauss-Seidel; Improved BGS-Net can improve the SER of BGS-Net.

This paper is organised as follows. Section 2 presents and analyses the channels required in this paper. Section 3 analyses the existing OAMPNet/MMNet-iid algorithms. Section 4 proposes the algorithm BGS-Net, and explains why BGS-Net is proposed. Section 5 analyses the problems that BGS-Net may encounter in MAUE systems and improve it, and proposes Improved BGS-Net. Section 6 analyses the complexity of BGS-Net and Improved BGS-Net in terms of their complexity. Experimental simulations and discussion are performed in Section 7. Section 8 concludes the full paper.

Notation

In this paper, lower-case and upper-case boldface letters are used to represent column vectors and matrices, respectively . ${\mathbf {I}} _n$ denotes a unit matrix of size n. For any matrix ${\mathbf {A}}$, ${\mathbf {A}}^\mathrm {T}$, ${\mathbf {A}}^\mathrm {H}$, $tr({\mathbf {A}} )$, and ${\mathbf {A}} ^+$ represent the transpose, conjugate transpose, trace, and pseudo-inverse of $\mathbf {A\mathrm {} }$. $N _C (s_i;r_i,\tau _t^2)$ denotes the univariate Gaussian distribution of a random variable $s_i$ with mean $r_i$ and variance $\tau _t^2$. The operator $\left\| \cdot \right\|$ denotes the vector/matrix parametrization. The notation $diag({\mathbf {x}} )$ creates a matrix with ${\mathbf {x}}$ in the diagonal and $diag({\mathbf {X}} )$ is the vector of the diagonal elements of ${\mathbf {X}}$.

Background

SAUE System

Consider an uplink massive MIMO system that uses $\mathrm {N_r}$ antennas at the BS to serve $\mathrm {N_t}$ single-antenna user terminals simultaneously, where $\mathrm {N_r\gg N_t}$. The SAUE system can be simply expressed as:

$${\tilde{\mathbf{y}}} = {\tilde{\mathbf{H}}}_{{\mathbf{S}}} {\tilde{\mathbf{x}}} + {\tilde{\mathbf{n}}}$$

(1)

where ${}\tilde{{\mathbf {y}} } \in \mathrm {{C}} ^{\mathrm {N_r\times 1} }$, ${\tilde{\mathbf {H}}_{\mathbf {S}}} \in \mathrm {{C}} ^{\mathrm {N_r\times N_t} }$, $\tilde{{\mathbf {x}} } \in \mathrm {{C}} ^{\mathrm {N_t\times 1} }$, and ${}\tilde{{\mathbf {n}} } \in \mathrm {{C}} ^{\mathrm {N_r\times 1} }$ are the receive symbol, channel response, transmit symbol, and system noise respectively. $\mathrm {N_r}$ and $\mathrm {N_t}$ are the numbers of receive and transmit antennas, respectively. ${}\tilde{{\mathbf {n}} }$ is distributed as $CN(0,\sigma ^2)$. For signal detection, the complex-valued system model (1) is converted to the corresponding real-valued system model as

$$\begin{aligned} \mathrm {\mathbf {y=H_{S}x+n} } \end{aligned}$$

(2)

where ${\mathbf {y}} =\left[ \Re ({}\tilde{{\mathbf {y}} } )^{\mathrm {T} } \quad \Im ({}\tilde{{\mathbf {y}} } )^{\mathrm {T} } \right] ^{\mathrm {T} } \in \mathrm {R} ^{2\mathrm {N_r} \times 1}$, ${\mathbf {x}} =\left[ \Re ({}\tilde{{\mathbf {x}} } )^{\mathrm {T} } \quad \Im ({}\tilde{{\mathbf {x}} } )^{\mathrm {T} } \right] ^{\mathrm {T} } \in \mathrm {R} ^{2\mathrm {N_t} \times 1}$, all ${\mathbf {x}}$ come from the discrete constellation diagram $\mathrm {{S} =\left\{ s_1,s_2,...,s_M \right\} }$,

${\mathbf {n}} =\left[ \Re ({}\tilde{{\mathbf {n}} } )^{\mathrm {T} } \quad \Im ({}\tilde{{\mathbf {n}} } )^{\mathrm {T} } \right] ^{\mathrm {T} }\in \mathrm {R ^{2N_r\times 1}}$, $\mathbf {H_{S}=\begin{bmatrix} \Re (\tilde{\mathbf {H_{S}} } ) &{}-\Im (\tilde{\mathbf {H_{S}} } ) \\ \Im (\tilde{\mathbf {H_{S}} } ) &{} \Re (\tilde{\mathbf {H_{S}} } ) \end{bmatrix}} \in \mathrm {R}^{\mathrm {2N_r\times 2N_t} }$, $\mathrm {N=2N_r}$, $\mathrm {M=}\mathrm {2N_t}$. $\mathrm {\mathbf {H_{S}} }$ denotes the flat Rayleigh fading channel matrix whose entries are assumed to be independently and identically distributed (i.i.d.) with zero mean and variance $\mathrm {(1/N){\mathbf {I}} }$. Since each user is a single antenna, the correlation between them is not considered.

MAUE System

Consider an uplink massive MIMO with a multi-antenna user equipment (MAUE) system. A BS with $\mathrm {N_r}$ antennas communicates with m UEs, and each UE is equipped with $\mathrm {m_{UE}}$ antennas, as shown in Figure 1. The total number of antennas on the user side is $\mathrm {N_t=m\times m_{UE}}$. The transmission vector is expressed as ${\mathbf {x}} =\left[ {\mathbf {x}} _{1}, \cdot \cdot \cdot ,{\mathbf {x}} _{i} ,\cdot \cdot \cdot ,{\mathbf {x}} _{m} \right] ^\mathrm {T}$, where ${\mathbf {x}}_i =\left[ x_{i1},\cdot \cdot \cdot , x_{ij},\cdot \cdot \cdot , x_{im_{UE}} \right] \in \mathrm {R ^{1\times m_{UE}}}$, $E\left\{ \left| x_{ij}^{2} \right| \right\} =1$. $\mathrm {N=2N_r}$, $\mathrm {M=2N_t}$ , the vector ${\mathbf {y}} \in \mathrm {R ^{N\times 1}}$ received by the BS:

$$\begin{aligned} {\mathbf {y=H_{M}x+n} } \end{aligned}$$

(3)

where ${\mathbf {H_{M}}} =\left[ {\mathbf {H_{M}}} _1 ,\cdot \cdot \cdot ,{\mathbf {H_{M}}} _i ,\cdot \cdot \cdot , {\mathbf {H_{M}}} _m\right] \in \mathrm {R^{N\times M}}$, ${\mathbf {H_{M}}} _i=\left[ {\mathbf {H_{M}}} _{i1},\cdot \cdot \cdot \cdot , {\mathbf {H_{M}}} _{im_{UE}}\right]$. ${\mathbf {H_{M}}}_{ij} \in \mathrm {R} ^{\mathrm {N} \times 1}$, represents the uplink from the jth antenna of the ith UE to the BS, $\mathrm {{\mathbf {n}} \in R^{N\times 1}}$ is an additive white gaussian noise (AWGN) vector with a mean value of zero and a variance of $\sigma ^{2} /2$. The Kronecker channel model [34] is: $\mathbf {H_{M}} ={\mathbf {R}} ^{1/2}\mathbf {H_{S}} {\mathbf {T}} ^{1/2}$. Spatial correlation matrix ${{\mathbf {R}} \in {R^{N\times N}} }$ and ${{\mathbf {T}} \in {R^{M\times M}} }$.

$$\begin{aligned} R_{pq}={\left\{ \begin{array}{ll} \left( \xi _re^{j\theta } \right) ^{2\left( q-p \right) } &{} \text { if } p\le q\\ R_{qp}^{*} &{} \text { if } p> q \end{array}\right. } \end{aligned}$$

(4)

$$\begin{aligned} T_{pq}={\left\{ \begin{array}{ll} \left( \xi _te^{j\theta } \right) ^{2\left( q-p \right) } &{} \text { if } p\le q\\ T_{qp}^{*} &{} \text { if } p> q \end{array}\right. } \end{aligned}$$

(5)

where $\xi _r$ and $\xi _t$ are correlation coefficients, $R_{pq}$ is the value of the receive antenna correlation matrix ${\mathbf {R}}$, $T_{pq}$ is the value of the transmit antenna correlation matrix ${\mathbf {T}}$ for each user. It can be seen that the transmitting antennas from the same terminal usually have correlation. However, most of the current papers do not consider the correlation between antennas from the same terminal, which is impractical and inaccurate. Therefore, according to the actual situation of propagation, $\xi _r$ and $\xi _t$ are respectively defined as the correlation factors of the receiving antenna and the transmitting antenna from the same terminal. Note that the correlation between different terminal antennas is ignored [32, 33].

Channel characteristics

In this section, SAUE and MAUE system characteristics are analyzed. $\mathrm {N_t=4}$, $\mathrm {N_r=32}$ are used here. It can be seen from Figure 2 that when the SAUE system has a large-scale antenna and $\alpha =\mathrm {N_r/N_t}$ is large, the channel appears channel hardening, and $\mathbf {H_{S}^TH_{S}}$ is diagonally dominant, so the approximate inversion method is very suitable for this environment. Figure 3 shows that in the MAUE system environment with $\mathrm {\xi _r=0,\xi _t=0.2,m_{UE}=2}$, although $\mathbf {H_{M}^TH_{M}}$ is still dominant diagonally and exhibits blockiness, it can be seen from the color depth that the non-diagonal elements on both sides of the diagonal elements have begun to have an impact on the diagonal elements and cannot be ignored. Figure 4 shows that in the MAUE system environment with $\mathrm {\xi _r=0,\xi _t=0.4,m_{UE}=4}$, the two sides of the diagonal element have had a great impact on the diagonal element, and it is difficult to get a good approximation by the approximation method. Figure 5 shows that in the MAUE system environment with $\mathrm {\xi _r=0.2,\xi _t=0.4,m_{UE}=4}$, the non-diagonal elements around the diagonal elements have a serious impact on the diagonal elements, and the approximation effect is very poor.

The Marchenko-Pasture theorem of random matrix theory shows that when each element of the matrix channel ${\mathbf {H}}$ is independently and identically distributed at zero mean and the variance is 1/N, the number of rows N and the number of columns M tend to Infinity, that is, $\mathrm {M,N} \rightarrow \infty$, and the ratio of the two tends to a constant ($\mathrm {N/M} \rightarrow \beta$), the diagonal elements of the matrix ${\mathbf {H}} ^\mathrm {T} {\mathbf {H}}$ tend to a certain constant, and the off-diagonal elements tend to zero. The following analyzes the symmetry of ${\mathbf {H}} ^\mathrm {T} {\mathbf {H}}$ under SAUE and MAUE systems: Known from formulas (4) and (5): ${\mathbf {R}}$ and ${\mathbf {T}}$ are symmetric matrices, so suppose

$$\begin{aligned} \mathbf {R=\begin{bmatrix} {\mathbf {R}}_1 &{} {\mathbf {R}}_2 \\ {\mathbf {R}}_2^\mathrm {T} &{} {\mathbf {R}}_1 \end{bmatrix}} \end{aligned}$$

(6)

and

$$\begin{aligned} {\mathbf {T}} =\begin{bmatrix} {\mathbf {T}}_1 &{} {\mathbf {T}}_2 &{} \cdots &{} 0 &{} 0\\ {\mathbf {T}}_2^\mathrm {T} &{} {\mathbf {T}}_1 &{} \cdots &{} 0&{} 0\\ \vdots &{} \vdots &{} \ddots &{} \vdots &{} \vdots \\ 0&{} 0 &{} \cdots &{} {\mathbf {T}}_1 &{}{\mathbf {T}}_2 \\ 0&{} 0 &{} \cdots &{} {\mathbf {T}}_2^\mathrm {T} &{}{\mathbf {T}} _1 \end{bmatrix} \end{aligned}$$

(7)

where ${\mathbf {R}}_1,{\mathbf {R}}_2\in \mathrm {R^{N/2\times N/2}}$, $\begin{bmatrix} {\mathbf {T}}_1 &{} {\mathbf {T}}_2 \\ {\mathbf {T}}_2^\mathrm {T} &{}{\mathbf {T}}_1 \end{bmatrix}\in \mathrm {R^{m_{UE}\times m_{UE}}}$, ${\mathbf {T}}_1,{\mathbf {T}}_2\in \mathrm {R^{m_{UE}/2\times m_{UE}/2}}$.

(a)
$\mathbf {H_{S}^TH_{S}}$ under SAUE system. From equation (2) we know that:
$$\begin{aligned} \begin{aligned} \mathbf {H_{S}^TH_{S}}&=\begin{bmatrix} \Re ({\tilde{\mathbf {H}}_{\mathbf {S}}} )^\mathrm {T} &{} \Im ({\tilde{\mathbf {H}}_{\mathbf {S}}} )^\mathrm {T} \\ -\Im ({\tilde{\mathbf {H}}_{\mathbf {S}}} )^\mathrm {T}&{}\Re ({\tilde{\mathbf {H}}_{\mathbf {S}}} )^\mathrm {T} \end{bmatrix}\begin{bmatrix} \Re ({\tilde{\mathbf {H}}_{\mathbf {S}}} ) &{} -\Im ({\tilde{\mathbf {H}}_{\mathbf {S}}} )\\ \Im ({\tilde{\mathbf {H}}_{\mathbf {S}}} ) &{}\Re ({\tilde{\mathbf {H}}_{\mathbf {S}}} ) \end{bmatrix}\\ {}&=\begin{bmatrix} \Re ({\tilde{\mathbf {H}}_{\mathbf {S}}} )^\mathrm {T} \Re ({\tilde{\mathbf {H}}_{\mathbf {S}}} )+ \Im ({\tilde{\mathbf {H}}_{\mathbf {S}}} )^\mathrm {T} \Im ({\tilde{\mathbf {H}}_{\mathbf {S}}} ) &{} -\Re ({\tilde{\mathbf {H}}_{\mathbf {S}}} )^\mathrm {T}\Im ({\tilde{\mathbf {H}}_{\mathbf {S}}} )+\Im ({\tilde{\mathbf {H}}_{\mathbf {S}}} )^\mathrm {T}\Re ({\tilde{\mathbf {H}}_{\mathbf {S}}} )\\ -\Im ({\tilde{\mathbf {H}}_{\mathbf {S}}} )^\mathrm {T}\Re ({\tilde{\mathbf {H}}_{\mathbf {S}}} )+ \Re ({\tilde{\mathbf {H}}_{\mathbf {S}}} )^\mathrm {T} \Im ({\tilde{\mathbf {H}}_{\mathbf {S}}} )&{}\Im ({\tilde{\mathbf {H}}_{\mathbf {S}}} )^\mathrm {T}\Im ({\tilde{\mathbf {H}}_{\mathbf {S}}} )+\Re ({\tilde{\mathbf {H}}_{\mathbf {S}}} )^\mathrm {T}\Re ({\tilde{\mathbf {H}}_{\mathbf {S}}} ) \end{bmatrix} \end{aligned} \end{aligned}$$
(8)
It can be seen from (8) that the upper left matrix and the lower right matrix of $\mathbf {H_{S}^TH_{S}}$ are the same, and the matrix is symmetric about the main diagonal.
(b)
$\mathbf {H_{M}^TH_{M}}$ under MAUE system (only ${\mathbf {T}} ^{1/2}$)
$$\begin{aligned} \mathbf {H_{M}^TH_{M}} =\left( \mathbf {H_{S}} {\mathbf {T}} ^{1/2} \right) ^\mathrm {T} \mathbf {H_{S}} {\mathbf {T}} ^{1/2} ={\mathbf {T}} ^{1/2}\mathbf {H_{S}^TH_{S}} {\mathbf {T}} ^{1/2} \end{aligned}$$
(9)
where ${\mathbf {H_{S}}} =\left[ {\mathbf { H_{S}}}_1 ,\dots ,{\mathbf {H_{S}}}_i ,\dots ,{\mathbf {H_{S}}}_m \right] \in \mathrm {R^{N\times M}}$. It can be seen from (8) that the upper left corner matrix and the lower right corner matrix of $\mathbf {H_{S}^TH_{S}}$ are the same, and the matrix is symmetric about the main diagonal. We rewrite (8) formula $\mathbf {H_{S}^TH_{S}} =\begin{bmatrix} {\mathbf {Q}}_1 &{}{\mathbf {Q}}_2 \\ {\mathbf {Q}}_2^\mathrm {T} &{}{\mathbf {Q}}_1 \end{bmatrix}$, and rewrite (7) formula ${\mathbf {T}}^{1/2} =\begin{bmatrix} {\mathbf {K}}_1 ^{1/2} &{}{\mathbf {0}} \\ {\mathbf {0}} &{}{\mathbf {K}}_1 ^{1/2} \end{bmatrix}$, where ${\mathbf {K}}_1 \in \mathrm {R^{M/2\times M/2}}$, ${\mathbf {Q}}_1 \in \mathrm {R^{M/2\times M/2}}$, ${\mathbf {Q}}_2 \in \mathrm {R^{M/2\times M/2}}$. Bringing ${\mathbf {T}}$ into equation (9), you can write equation (9) as
$$\begin{aligned} \begin{aligned} {\mathbf {T}} ^{1/2}\mathbf {H_{S}^TH_{S}} {\mathbf {T}} ^{1/2}&=\begin{bmatrix} {\mathbf {K}}_1^{1/2} &{} {\mathbf {0}} \\ {\mathbf {0}} &{}{\mathbf {K}}_1 ^{1/2} \end{bmatrix}\begin{bmatrix} {\mathbf {Q}}_1 &{}{\mathbf {Q}}_2 \\ {\mathbf {Q}}_2^\mathrm {T} &{}{\mathbf {Q}}_1 \end{bmatrix}\begin{bmatrix} {\mathbf {K}}_1 ^{1/2}&{} {\mathbf {0}} \\ {\mathbf {0}} &{}{\mathbf {K}}_1 ^{1/2} \end{bmatrix}\\ {}&=\begin{bmatrix} {\mathbf {K}}_1 ^{1/2}{\mathbf {Q}}_1 {\mathbf {K}}_1 ^{1/2} &{} {\mathbf {K}}_1 ^{1/2}{\mathbf {Q}}_2 {\mathbf {K}}_1 ^{1/2} \\ {\mathbf {K}}_1 ^{1/2}{\mathbf {Q}}_2 ^\mathrm {T} {\mathbf {K}}_1 ^{1/2} &{} {\mathbf {K}}_1 ^{1/2}{\mathbf {Q}}_1 {\mathbf {K}}_1 ^{1/2} \end{bmatrix} \end{aligned} \end{aligned}$$
(10)
Therefore, the upper left corner matrix of $\mathbf {H_{M}^TH_{M}}$ is the same as the lower right corner matrix, and the matrix is symmetric about the main diagonal.
(c)
$\mathbf {H_{M}^TH_{M}}$ under MAUE system (only ${\mathbf {T}} ^{1/2}$ and ${\mathbf {R}} ^{1/2}$)
$$\begin{aligned} \begin{aligned} \mathbf {H_{M}^TH_{M}}&=\left( {\mathbf {R}} ^{1/2}\mathbf {H_{S}} {\mathbf {T}} ^{1/2} \right) ^\mathrm {T} \left( {\mathbf {R}} ^{1/2}\mathbf {H_{S}} {\mathbf {T}} ^{1/2} \right) \\ {}&=\left( {\mathbf {T}} ^{1/2}\right) ^\mathrm {T} \left( {\mathbf {R}} ^{1/2}\mathbf {H_{S}} \right) ^\mathrm {T} {\mathbf {R}} ^{1/2}\mathbf {H_{S}} {\mathbf {T}} ^{1/2} \end{aligned} \end{aligned}$$
(11)
where
$${\mathbf{R}}^{{1/2}} {\mathbf{H}}_{{\mathbf{S}}} = \left[ {\begin{array}{*{20}c} {{\mathbf{R}}_{1}^{{1/2}} \Re (\tilde{\mathbf{{H}}}_{{\mathbf{S}}} )^{{\text{T}}} - {\mathbf{R}}_{2}^{{1/2}} \Im (\tilde{{\mathbf{{H}}}}_{{\mathbf{S}}} )^{{\text{T}}} } & {{\mathbf{R}}_{1}^{{1/2}} \Im (\tilde{\mathbf{{H}}}_{{\mathbf{S}}} )^{{\text{T}}} + {\mathbf{R}}_{2}^{{1/2}} \Re (\tilde{\mathbf{{H}}}_{{\mathbf{S}}} )^{{\text{T}}} } \\ {\left( {{\mathbf{R}}_{2}^{{\text{T}}} } \right)^{{1/2}} \Re (\tilde{\mathbf{{H}}}_{{\mathbf{S}}} )^{{\text{T}}} - {\mathbf{R}}_{1}^{{1/2}} \Im (\tilde{\mathbf{{H}}}_{{\mathbf{S}}} )^{{\text{T}}} } & {\left( {{\mathbf{R}}_{2}^{{\text{T}}} } \right)^{{1/2}} \Im (\tilde{\mathbf{{H}}}_{{\mathbf{S}}} )^{{\text{T}}} + {\mathbf{R}}_{1}^{{1/2}} \Re (\tilde{\mathbf{{H}}}_{{\mathbf{S}}} )^{{\text{T}}} } \\ \end{array} } \right]$$
(12)
After calculation, we can find that the matrix $\left( {\mathbf {R}} ^{1/2}\mathbf {H_{S}} \right) ^\mathrm {T} {\mathbf {R}} ^{1/2}\mathbf {H_{S}}$ is symmetric about the main diagonal, but the upper left matrix and the lower right matrix are not the same, we set
$$\begin{aligned} \begin{aligned} \left( {\mathbf {R}} ^{1/2}\mathbf {H_{S}} \right) ^\mathrm {T} {\mathbf {R}} ^{1/2}\mathbf {H_{S}} =\begin{bmatrix} {\mathbf {Z}}_{1} &{} {\mathbf {Z}}_2 \\ {\mathbf {Z}}_2^\mathrm {T} &{}{\mathbf {Z}}_3 \end{bmatrix} \end{aligned} \end{aligned}$$
(13)
where ${\mathbf {Z}}_1 \in \mathrm {R^{M/2\times M/2}}$, ${\mathbf {Z}}_2 \in \mathrm {R^{M/2\times M/2}}$, ${\mathbf {Z}}_3 \in \mathrm {R^{M/2\times M/2}}$. Substitute (13) into $\mathbf {H_{M}^TH_{M}}$, get
$$\begin{aligned} \begin{aligned} \mathbf {H_{M}^TH_{M}}&=\begin{bmatrix} {\mathbf {K}}_1^{1/2} &{}{\mathbf {0}} \\ {\mathbf {0}} &{}{\mathbf {K}}_1 ^{1/2} \end{bmatrix}\begin{bmatrix} {\mathbf {Z}}_1 &{} {\mathbf {Z}}_2 \\ {\mathbf {Z}}_2^\mathrm {T} &{}{\mathbf {Z}}_3 \end{bmatrix}\begin{bmatrix} {\mathbf {K}}_1^{1/2} &{} {\mathbf {0}} \\ {\mathbf {0}} &{} {\mathbf {K}}_1 ^{1/2} \end{bmatrix}\\ {}&=\begin{bmatrix} {\mathbf {K}}_1 ^{1/2}{\mathbf {Z}}_1 {\mathbf {K}}_1 ^{1/2} &{} {\mathbf {K}}_1 ^{1/2}{\mathbf {Z}}_2 {\mathbf {K}}_1 ^{1/2} \\ {\mathbf {K}}_1 ^{1/2}{\mathbf {Z}}_2 ^\mathrm {T} {\mathbf {K}}_1 ^{1/2} &{} {\mathbf {K}}_1 ^{1/2}{\mathbf {Z}}_3 {\mathbf {K}}_1 ^{1/2} \end{bmatrix} \end{aligned} \end{aligned}$$
(14)
It can be seen from formula (14) that the upper left corner matrix and the lower right corner matrix of $\mathbf {H_{M}^TH_{M}}$ are different, but the matrix is symmetric about the main diagonal.

Related work

The goal of the receiver is to calculate the maximum likelihood (ML) estimate $\hat{{\mathbf {x}} }$ of ${\mathbf {x}}$ set as

$$\begin{aligned} \begin{aligned} \hat{{\mathbf {x}} }=arg\min _{{\mathbf {x}} \in S} \left\| {\mathbf {y}} -\mathbf {Hx} \right\| _2 \end{aligned} \end{aligned}$$

(15)

However, its complexity is too high. In the past few decades, researchers have been studying various detectors to reduce their complexity while maintaining their SER.

OAMPNet

OAMPNet is a model-driven DL algorithm for MIMO detection derived from orthogonal approximate matching tracking (OAMP). Compared with approximate message passing (AMP), the advantage of OAMP is that it can be applied to unitary invariant matrices, while AMP is only applicable to Gaussian measurement matrices. OAMPNet has better performance than OAMP and can be adapted to various channel environments by using a number of learnable variables. The algorithm for OAMPNet is as follows.

Step 1:We need to design a linear detector. $v_t^2$ is the variance of the nonlinear estimation error

$$\begin{aligned} \begin{aligned} v_t^2=\frac{\left\| {\mathbf {y}} -{\mathbf {H}{\hat{\mathbf {x}}}}_t \right\| _2^2-\mathrm {N} \frac{\sigma ^2}{2} }{tr\left( {\mathbf {H}} ^\mathrm {T}{\mathbf {H}} \right) } \end{aligned} \end{aligned}$$

(16)

here ${\mathbf {W}}_t$ is the optimal ${\mathbf {W}}$ in OAMP in [35]

$$\begin{aligned} \begin{aligned} {\hat{\mathbf {W}}} _t =v_{t}^{2} {\mathbf {H}}^\mathrm {T} \left( v_{t}^{2}\mathbf {HH}^\mathrm {T} +\frac{\sigma ^{2} }{2} \right) ^{-1} \end{aligned} \end{aligned}$$

(17)

$$\begin{aligned} \begin{aligned} {\mathbf {W}}_t =\frac{\mathrm {M} }{tr\left( {\hat{\mathbf {W}}}_t{\mathbf {H}} \right) } {\hat{\mathbf {W}}} _t \end{aligned} \end{aligned}$$

(18)

In this way, the value of the linear estimate can be obtained

$$\begin{aligned} \begin{aligned} {\mathbf {r}}_t ={\hat{\mathbf {x}}}_t +\gamma _t{\mathbf {W}}_t \left( {\mathbf {y}} -{\mathbf {H}} {\hat{\mathbf {x}}} _t \right) \end{aligned} \end{aligned}$$

(19)

Step 2:We take the linear detection estimates as input and perform nonlinear detection

$$\begin{aligned} \begin{aligned} {\mathbf {C}}_t ={\mathbf {I}} -\theta _t^2{\mathbf {W}}_t{\mathbf {H}} \end{aligned} \end{aligned}$$

(20)

$\tau _t ^2$ is the variance of the linear estimation error

$$\begin{aligned} \begin{aligned} \tau _t ^2=\frac{1}{\mathrm {M} }tr({\mathbf {C}} _t{\mathbf {C}} _t^\mathrm {T} )v_t^2+\frac{\theta _t^2\sigma ^2}{\mathrm {M} } tr({\mathbf {W}} _t{\mathbf {W}} _t^\mathrm {T} ) \end{aligned} \end{aligned}$$

(21)

$p(x_i)$ is the prior probability

$$\begin{aligned} \begin{aligned} p(x_i)= {\textstyle \sum _{j\in \mathrm {M} }\frac{1}{\sqrt{\mathrm {M} } } \delta (x_i-s_j)} \end{aligned} \end{aligned}$$

(22)

Bring $\tau _t ^2$ and $p(x_i)$ into $E\left\{ {\mathbf {x}} |{\mathbf {r}}_t ,\tau _t \right\}$

$$\begin{aligned} \begin{aligned} E\left\{ x_{ti} |r_{ti} ,\tau _t \right\} =\frac{ {\textstyle \sum _{s_i\in S}} s_iN_c(s_i;r_{ti},\tau _t^2)p(s_i)}{{\textstyle \sum _{s_i\in S}} N_c(s_i;r_{ti},\tau _t^2)p(s_i)} \end{aligned} \end{aligned}$$

(23)

In this way, the input value of the (t+1) layer can be obtained

$$\begin{aligned} \begin{aligned} {\hat{\mathbf {x}} }_{t+1} =E\left\{ {\mathbf {x}} \mid {\mathbf {r}}_t ,\tau _t \right\} \end{aligned} \end{aligned}$$

(24)

The performance of the OAMPNet detection algorithm is very good, and there are only two training parameters $(\gamma _t,\theta _t^2)$, but each layer of ${\hat{\mathbf {W}} }_t$ needs to be calculated once, and each calculation requires a pseudo-inverse, which brings great complexity, not suitable for massive MIMO, suitable for medium-scale MIMO [25, 26].

MMNet-iid

The main idea of MMNet-iid is to introduce an appropriate degree of flexibility in the linear and denoising components of the iterative framework while maintaining its linear plus non-linear structure [27, 28].

Step 1: We need to design a linear detector to estimate ${\mathbf {z}}_t$.

$$\begin{aligned} \begin{aligned} {\mathbf {z}}_t ={\hat{\mathbf {x}}}_t +\theta _t^{(1)}{\mathbf {H}}^\mathrm {T} ({\mathbf {y}} -{H{\hat{\mathbf {x}}}}_t ) \end{aligned} \end{aligned}$$

(25)

Step 2:We take ${\mathbf {z}}_t$ as input and perform nonlinear detection. $\sigma _t^2$ is the variance of the linear detection estimation error

$$\begin{aligned} \begin{aligned} \sigma _t^2=\frac{\theta _t^{(2)}}{\mathrm {M} } \left( \frac{\left\| {\mathbf {I}} -{\mathbf {A}}_t{\mathbf {H}} \right\| _F^2 }{\left\| {\mathbf {H}} \right\| _F^2 } \left[ \left\| {\mathbf {y}} -{H{\hat{\mathbf {x}}}}_t \right\| _2^2-\mathrm {N_r} \sigma ^2 \right] _++\frac{\left\| {\mathbf {A}}_t \right\| _F^2 }{\left\| {\mathbf {H}} \right\| _F^2 }\sigma ^2 \right) \end{aligned} \end{aligned}$$

(26)

$\eta _t({\mathbf {z}}_t ;\sigma _t^2)$ is a nonlinear detection estimate

$$\begin{aligned} \begin{aligned} \eta _t(z_{ti} ;\sigma _t^2) =\frac{1}{Z} {\textstyle \sum _{s_i\in S}s_iexp(-\frac{\left\| z_{ti} -s_i \right\| ^2 }{\sigma _t^2} )} \end{aligned} \end{aligned}$$

(27)

In this way, the input of the next layer can be obtained

$$\begin{aligned} \begin{aligned} {\hat{\mathbf {x}} }_{t+1} =\eta _t({\mathbf {z}}_t ;\sigma _t^2) \end{aligned} \end{aligned}$$

(28)

where $Z = {\textstyle \sum _{s_i\in S}exp(-\frac{\left\| z_{ti} -s_i \right\| ^2 }{\sigma _t^2} )}$, ${\mathbf {A}}_t =\theta _t^{(1)}{\mathbf {H}}^\mathrm {T}$. It has two training parameters $(\theta _t^{(1)},\theta _t^{(2)})$, and its complexity is much smaller than OAMPNet. MMNet-iid performs well when the number of antennas is large and the linear Gaussian channel is good, and has poor performance in correlated channels or when the number of antennas is low.

The proposed BGS-Net method

Gauss-Seidel iterative method

GS is one of the common iterative methods used to solve systems of linear equations. If a system of linear equations $\mathbf {Ax} ={\mathbf {b}}$ is required to be solved, it will be decomposed as follows.

$$\begin{aligned} \begin{aligned} a_{i1}x_1+a_{i2}x_2+\dots +a_{in}x_n=b_i\quad (i=1,2,\dots ,n) \end{aligned} \end{aligned}$$

(29)

The iterative formula of GS [36] is

$$\begin{aligned} \begin{aligned} x_i^{(k+1)}=(b_i- {\textstyle \sum _{j=1}^{i-1}a_{ij}x_j^{(k+1)}- {\textstyle \sum _{j=i+1}^{n}}a_{ij}x_j^{(k)})/a_{ii} }\\(i=1,2,\dots ,n;k=0,1,2,\dots ,t) \end{aligned} \end{aligned}$$

(30)

For each layer, $b_i$ is used to subtract the updated ${\textstyle \sum _{j=1}^{i-1}a_{ij}x_j^{(k+1)}}$ and the not yet updated ${\textstyle \sum _{j=i+1}^{n}}a_{ij}x_j^{(k)}$, its matrix representation is

$$\begin{aligned} \begin{aligned} {\mathbf {x}}^\mathrm {(k+1)} ={\mathbf {D}}^{-1} (-\mathbf {Lx}^\mathrm {(k+1)} -\mathbf {Ux}^\mathrm {(k)} +{\mathbf {b}} ) \end{aligned} \end{aligned}$$

(31)

$$\begin{aligned} \begin{aligned} ({\mathbf {D}} +{\mathbf {L}} ){\mathbf {x}}^\mathrm {(k+1)}=-\mathbf {Ux}^\mathrm {(k)} +{\mathbf {b}} \end{aligned} \end{aligned}$$

(32)

$$\begin{aligned} \begin{aligned} {\mathbf {x}}^\mathrm {(k+1)} =({\mathbf {D}} +{\mathbf {L}} )^{-1}(-{\mathbf {U}} {\mathbf {x}}^\mathrm {(k)} +{\mathbf {b}} ) \end{aligned} \end{aligned}$$

(33)

GS is used in communication, that is, the Hermitian positive semi-definite matrix ${\mathbf {A}}$ is decomposed into strictly lower triangular terms ${\mathbf {L}}$, strictly upper triangular terms ${\mathbf {U}}$ and diagonal terms ${\mathbf {D}}$

$$\begin{aligned} \begin{aligned} {\mathbf {A}} ={\mathbf {H}} ^\mathrm {T} {\mathbf {H}} +\frac{\sigma ^2 }{2} {\mathbf {I}} \end{aligned} \end{aligned}$$

(34)

$$\begin{aligned} \begin{aligned} {\mathbf {A}} ={\mathbf {D}} +{\mathbf {L}} +{\mathbf {U}} \end{aligned} \end{aligned}$$

(35)

Then the problem we solve is

$$\begin{aligned} \begin{aligned} \mathbf {Ax} ={\mathbf {H}} ^\mathrm {T} {\mathbf {y}} \end{aligned} \end{aligned}$$

(36)

Then solve a set of linear equations by calculating the solution of the iterative be havior [37].

$$\begin{aligned} \begin{aligned} {\hat{\mathbf {x}}}^{(\mathrm {n} )} =({\mathbf {D}} +{\mathbf {L}} )^{-1}[{\hat{\mathbf {x}}}_{MF} -{\mathbf {U}}{\hat{\mathbf {x}}}^{\mathrm {(n-1)} } ] \end{aligned} \end{aligned}$$

(37)

where ${\hat{\mathbf {x}}}^\mathrm {(n)}$ is the estimated signal, iteratively refined in each iteration and ${\hat{\mathbf {x}}}_{MF} ={\mathbf {H}} ^\mathrm {T}{\mathbf {y}}$, replacing ${\mathbf {b}}$ in (32). Here ${\hat{\mathbf {x}} }^{(0)}$ is initialised to ${\mathbf {D}} ^{-1}{\mathbf {H}} ^\mathrm {T} {\mathbf {y}}$. Gauss-Seidel has good convergence, and is guaranteed to converge when ${\mathbf {A}}$ is diagonally dominant or symmetrically positive definite. This is because in MAUE systems, ${\mathbf {A}}$ is not guaranteed to be diagonally dominant, but ${\mathbf {A}}$ is definitely symmetric positive definite. The following is a proof of convergence of Gauss-Seidel for diagonally dominant or symmetric positive times, respectively [38].

Theorem 1

When ${\mathbf {A}}$ is diagonally dominant, Gauss-Seidel can guarantee convergence.

Proof of Theorem 1. For strictly diagonally dominant matrix ${\mathbf {A}}$, its diagonal elements $a_{ii}\ne 0,i=1,2,\dots ,n$, so

$$\begin{aligned} \begin{aligned} \left| {\mathbf {D}} +{\mathbf {L}} \right| = {\textstyle \prod _{i=1}^{n}}a_{ii} \ne 0 \end{aligned} \end{aligned}$$

(38)

Suppose $\mathbf {B_G} =-({\mathbf {D}} +{\mathbf {L}} )^{-1}{\mathbf {U}}$, the characteristic value is $\lambda$, then the characteristic equation is

$$\begin{aligned} \begin{aligned} \left| \lambda {\mathbf {I}} -\mathbf {B_G} \right|&=\left| \lambda {\mathbf {I}} +({\mathbf {D}} +{\mathbf {L}} )^{-1}{\mathbf {U}} \right| =\left| ({\mathbf {D}} +{\mathbf {L}} )^{-1} \right| \left| \lambda ({\mathbf {D}} +{\mathbf {L}} )+{\mathbf {U}} \right| =0\\ {}&\Rightarrow \left| \lambda ({\mathbf {D}} +{\mathbf {L}} )+{\mathbf {U}} \right| =0 \end{aligned} \end{aligned}$$

(39)

When the determinant is zero, the equation has a non-zero solution. Use contradiction: suppose $\left| \lambda \right| \ge 1$,

$$\begin{aligned} \begin{aligned} \lambda ({\mathbf {D}} +{\mathbf {L}} )+{\mathbf {U}} =\begin{bmatrix} \lambda a_{11} &{} a_{12} &{} \dots &{} a_{1n}\\ \lambda a_{21} &{} \lambda a_{22} &{} \dots &{}a_{2n} \\ \vdots &{} \dots &{} \ddots &{}\vdots \\ \lambda a_{n1} &{} \lambda a_{n2} &{} \dots &{} \lambda a_{nn} \end{bmatrix} \end{aligned} \end{aligned}$$

(40)

It is a strictly diagonally dominant matrix, so it is non-singular, that is, $\left| \lambda ({\mathbf {D}} +{\mathbf {L}} )+{\mathbf {U}} \right| \ne 0$ contradicts the eigenvalue $\lambda$ satisfying $\left| \lambda ({\mathbf {D}} +{\mathbf {L}} )+{\mathbf {U}} \right| = 0$. So $\left| \lambda \right| < 1$ is $\rho (\mathbf {B_G} )< 1$ , when ${\mathbf {A}}$ is diagonally dominant, Gauss-Seidel converges.

Theorem 2

When ${\mathbf {A}}$ is symmetrically positive, Gauss-Seidel can guarantee convergence.

Proof of Theorem 2. Suppose $\mathbf {B_G} =-({\mathbf {D}} +{\mathbf {L}} )^{-1}{\mathbf {U}}$, the eigenvalue is $\lambda$, and ${\mathbf {x}}$ is the eigenvector, then

$$\begin{aligned} & - ({\mathbf{D}} + {\mathbf{L}})^{{ - 1}} {\mathbf{Ux}} = \lambda {\mathbf{x}} \\ & \to - {\mathbf{Ux}} = \lambda ({\mathbf{D}} + {\mathbf{L}}){\mathbf{x}} \\ & \to - {\mathbf{x}}^{{\text{T}}} {\mathbf{Ux}} = \lambda {\mathbf{x}}^{{\text{T}}} ({\mathbf{D}} + {\mathbf{L}}){\mathbf{x}} \\ \end{aligned}$$

(41)

Because ${\mathbf {A}}$ is positive definite, so $p={\mathbf {x}} ^\mathrm {T} \mathbf {Dx} > 0$, set $-{\mathbf {x}} ^\mathrm {T} \mathbf {Ux} =a$, then

$$\begin{aligned} \begin{aligned} {\mathbf {x}} ^\mathrm {T} \mathbf {Ax} ={\mathbf {x}}^\mathrm {T} ({\mathbf {D}} +{\mathbf {L}} +{\mathbf {U}} ){\mathbf {x}} =p-2a>0 \end{aligned} \end{aligned}$$

(42)

$$\begin{aligned} \begin{aligned} \lambda =\frac{-{\mathbf {x}} ^\mathrm {T} \mathbf {Ux} }{{\mathbf {x}} ^\mathrm {T}({\mathbf {D}} +{\mathbf {L}} ){\mathbf {x}} }=\frac{a}{p-a} \end{aligned} \end{aligned}$$

(43)

$$\begin{aligned} \begin{aligned} \lambda ^2=\frac{a^2}{p^2-2pa+a^2}=\frac{a^2}{p(p-2a)+a^2} <1 \end{aligned} \end{aligned}$$

(44)

So $\left| \lambda \right| < 1$ is $\rho (\mathbf {B_G} )< 1$. When ${\mathbf {A}}$ is symmetric positive definite, Gauss-Seidel converges, and we divide the numerator and denominator of (44) by $a^2$ at the same time, we get

$$\begin{aligned} \begin{aligned} \lambda ^2=\frac{1}{(\frac{p}{a}-1 )^2} \end{aligned} \end{aligned}$$

(45)

From (45) we can see that the more diagonally dominant, the smaller the $\lambda ^2$, the faster the convergence.

BGS-Net architecture

In this section, a model-driven DL detector network (called BGS-Net) is proposed. The signal detector uses the Gauss-Seidel method and nonlinear activation to improve the detection performance. The only training parameters is $\mathbf {\Omega } =\mathbf {\gamma }_t$, $\mathbf {\gamma } _t\in \mathrm {R} ^{\mathrm {M} \times 1}$. In the algorithm $({\mathbf {D}} +{\mathbf {L}} )^{-1}$, ${\hat{\mathbf {x}} }_{MF}$, ${\mathbf {U}}$, $tr({\mathbf {H}} ^\mathrm {T} {\mathbf {H}} )$, $\frac{1}{\mathrm {M} } tr({\mathbf {C}} _t{\mathbf {C}} _t^\mathrm {T} )$, $\frac{\sigma ^2}{\mathrm {M} } tr({\mathbf {W}} _t{\mathbf {W}} _t^\mathrm {T} )$, all of which need to be computed only once and then reused at each layer. In contrast, ${\mathbf {W}}_t$ and ${\mathbf {A}}_t$ for OAMPNet and MMNet-iid need to be calculated once per layer because of the training parameters present in them. The structure of BGS-Net is shown in Figure 6, which is an improved algorithm by adding a learnable vector variable $\mathbf {\gamma } _{t}$ . The network consists of $L_{layer}$ cascaded layers, each of which has the same structure, including nonlinear estimator, error variance $\mathbf {\tau } _{t}^{2}$ , and tied weights. The input of the BGS-Net network is $\hat{{\mathbf {x}} }_{MF}$ and the initial value $\hat{{\mathbf {x}} }_{0}$, and the output is the final estimate of the signal $\hat{{\mathbf {x}} }_{Llayer}$ . To make it easier to see the deep learning structure, see Figure 7. We first calculate $\hat{{\mathbf {z}} } _{t}$ and scalar $\tau _{t}^{2}$ through GS detection block, plus the constellation map S as the input of the network. We introduce a vector variable $\mathbf {\gamma } _{t}$ in nonlinear detection, and finally output $\hat{{\mathbf {x}} } _{t+1}$ . The difference between model-driven and DNN is that many of the parameters of model-driven are fixed values obtained from past experience, while the parameters of DNN are all variable values.

Algorithm1: BGS-Net algorithm for MIMO detection
Input: Received signal ${\mathbf {y}}$, channel matrix ${\mathbf {H}}$, noise level $\sigma ^2/2$
Initialize: ${\hat{\mathbf {x}}}_0 \leftarrow {\mathbf {D}} ^{-1}{\mathbf {H}} ^\mathrm {T} {\mathbf {y}}$
$1.{\hat{\mathbf {z}}}_t =({\mathbf {D}} +{\mathbf {L}} )^{-1}[{\hat{\mathbf {x}}} _{MF} -{\mathbf {U}}{\hat{\mathbf {x}}}_t ]$
$2.v_t^2=\frac{\left\\| {\mathbf {y}} -{\mathbf {H}}{\hat{\mathbf {x}}}_t \right\\| _2^2-\mathrm {N} \frac{\sigma ^2}{2} }{tr({\mathbf {H}} ^\mathrm {T} {\mathbf {H}} )}$
$3.v_t^2=\max (v_t^2,10^{-9} )$
$4.\tau _t^2 =\frac{1}{\mathrm {M} }tr({\mathbf {C}} _t{\mathbf {C}} _t^\mathrm {T} ) v_t^2 +\frac{\sigma ^2}{\mathrm {M} } tr({\mathbf {W}} _t{\mathbf {W}} _t^\mathrm {T} )$
$5.\mathbf {\tau }_t^2 =\frac{\mathbf {\tau }_t^2 }{\mathbf {\gamma }_t }$
$6.{\hat{\mathbf {x}}}_{t+1} =E\left\{ {\mathbf {x}} \|{\hat{\mathbf {z}}}_t ,\mathbf {\tau }_t \right\}$

where

$$\begin{aligned} \begin{aligned} {\mathbf {W}}_t ={\mathbf {H}} ^\mathrm {T} \end{aligned} \end{aligned}$$

(46)

$$\begin{aligned} \begin{aligned} {\mathbf {C}}_t ={\mathbf {I}} -{\mathbf {W}} _t{\mathbf {H}} \end{aligned} \end{aligned}$$

(47)

$$\begin{aligned} \begin{aligned} E\left\{ x_{ti}|{\hat{z}}_{ti} ,\mathbf {\tau }_{ti} \right\}&= {\textstyle \sum _{s_j\in S}s_j\times p(s_j/{\hat{z}}_{ti} ,\tau _{ti} )}\\ {}&= {\textstyle \sum _{s_j\in S}s_j\times softmax(\frac{-\left\| {\hat{z}}_{ti}-s_j \right\| ^2 }{\tau _{ti}^2} )} \end{aligned} \end{aligned}$$

(48)

where $softmax(V_i)= \frac{exp^{V_i}}{ {\textstyle \sum _{j}exp^{V_j}} }$. It can be seen from Algorithm1 that we only have one training parameter per layer, and vector $\mathbf {\gamma } _t$ is used to adjust the variance $\mathbf {\tau } _t ^2$ which is an estimated value. Because $\frac{1}{\mathrm {M} } tr({\mathbf {C}} _t{\mathbf {C}} _t ^\mathrm {T})$ used is a constant value, it is multiplied by ${\mathbf {v}}_t^2$ each time in $\mathbf {\tau }_t ^2$, which saves a large amount of calculation. What we need to pay attention to is that when $4\rightarrow 5$, $\tau _t^2$’s dimension expansion to a vector $\mathbf {\tau }_t^2$ .

Low-complexity algorithm for $({\mathbf {D}} +{\mathbf {L}} )^{-1}$

In this section, the complexity of $({\mathbf {D}} +{\mathbf {L}} )^{-1}$ will be reduced. The complexity in calculating ${\hat{\mathbf {x}}}_t =({\mathbf {D}} +{\mathbf {L}} )^{-1}[{\hat{\mathbf {x}}}_{MF} -{\mathbf {U}}{\hat{\mathbf {x}}} _{t-1} ]$ is mainly concentrated in solving $({\mathbf {D}} +{\mathbf {L}} )^{-1}$. If the inverse is solved directly, the complexity of the algorithm will reach $O(\mathrm {M} ^3)$ , so a circular nesting method is proposed to reduce its complexity, as described below :

The first: From Eq. (30), the complexity of each row is $\mathrm {M-1}$ multiplication, 1 division, for a total of $\mathrm {M}$ rows, so the complexity of one iteration is $\mathrm {M^2}$. However, this method is not applicable to BGS-Net.

The second: Solve the lower triangular matrix in parallel, the structure is shown in Figure 8. For the inversion of the lower triangular matrix, we have the following properties:

$$\begin{aligned} \begin{aligned} {\mathbf {Y}} =\begin{bmatrix} {\mathbf {B}} &{} {\mathbf {0}} \\ {\mathbf {C}} &{}{\mathbf {F}} \end{bmatrix}\rightarrow {\mathbf {Y}}^{-1} =\begin{bmatrix} {\mathbf {B}} ^{-1}&{} {\mathbf {0}} \\ -{\mathbf {F}} ^{-1}{\mathbf {C}} {\mathbf {B}} ^{-1} &{}{\mathbf {F}} ^{-1} \end{bmatrix} \end{aligned} \end{aligned}$$

(49)

where ${\mathbf {B}}$, ${\mathbf {C}}$, ${\mathbf {F}}$ have the same size. The main complexity of (37) is to solve $({\mathbf {D}} +{\mathbf {L}} )^{-1}$, which we know to be a lower triangular matrix, and use the above property for its inverse. It can be known from (8),(10) and (14) that when the system is SAUE or MAUE (only ${\mathbf {T}}$), it has the following (50) properties; when MAUE (both ${\mathbf {T}}$ and ${\mathbf {R}}$), there is no (50) property:

$$\begin{aligned} \begin{aligned} \begin{bmatrix} a_{1,1} &{} 0 &{}\dots &{} 0 &{}0 \\ a_{2,1}&{} a_{2,2}&{} \dots &{} 0&{}0 \\ \vdots &{} \dots &{} \ddots &{} \dots &{} \vdots \\ a_{\frac{M}{2}-1 ,1} &{} a_{\frac{M}{2}-1 ,2} &{} \dots &{} a_{\frac{M}{2}-1 ,\frac{M}{2}-1} &{} 0\\ a_{\frac{M}{2} ,1} &{} a_{\frac{M}{2} ,2} &{} \dots &{} a_{\frac{M}{2} ,\frac{M}{2}-1} &{}a_{\frac{M}{2} ,\frac{M}{2}} \end{bmatrix}\\=\begin{bmatrix} a_{\frac{M}{2}+1 ,\frac{M}{2}+1}&{} 0 &{}\dots &{} 0 &{}0 \\ a_{\frac{M}{2}+2 ,\frac{M}{2}+1}&{} a_{\frac{M}{2}+2 ,\frac{M}{2}+2}&{} \dots &{} 0&{}0 \\ \vdots &{} \dots &{} \ddots &{} \dots &{} \vdots \\ a_{M-1,\frac{M}{2}+1} &{} a_{M-1,\frac{M}{2}+2} &{} \dots &{} a_{M-1 ,M-1} &{} 0\\ a_{M,\frac{M}{2}+1} &{} a_{M,\frac{M}{2}+2} &{} \dots &{} a_{M,M-1} &{}a_{M,M} \end{bmatrix} \end{aligned} \end{aligned}$$

(50)

We can see from Figure 8 that the specific steps of the loop nesting method are as follows.

Step 1: Find the reciprocals of $a_{1,1},a_{2,2},a_{3,3},\dots ,,a_{\frac{\mathrm {M}}{2} ,\frac{\mathrm {M}}{2}}$ and assign them to ${\mathbf {B}}_{1,t}^{-1}$ and ${\mathbf {F}}_{1,t}^{-1}$ , $t\in (1,\frac{\mathrm {M}}{4} )$ respectively.

Step 2: Bring the resulting ${\mathbf {B}}_{i,t}^{-1}$ and ${\mathbf {F}}_{i,t}^{-1}$, and the corresponding ${\mathbf {C}}_{i,t}$, $t\in (1,\frac{\mathrm {M}}{2^{i+1}} )$ into (49), we can get ${\mathbf {B}}_{i+1,t} ^{-1}$ and ${\mathbf {F}}_{i+1,t}^{-1}$, $t\in (1,\frac{\mathrm {M}}{2^{i+2}} )$. If ${\mathbf {B}}_{i+1,t}^{-1}$ is an $\mathrm {\frac{M}{2} \times \frac{M}{2}}$ matrix, then the next step, otherwise loop the second step.

Step 3:In the case of Section 2.3 (a) (b), assign ${\mathbf {B}}^{-1}={\mathbf {B}}_{i+1,t}^{-1}$ to ${\mathbf {F}}^{-1}$; otherwise, the same method as for ${\mathbf {B}}^{-1}$, solve ${\mathbf {F}}^{-1}$. In this way, we can obtain $({\mathbf {D}} +{\mathbf {L}} )^{-1}=\begin{bmatrix} {\mathbf {B}}^{-1} &{}{\mathbf {0}} \\ -{\mathbf {F}} ^{-1}{\mathbf {C}} {\mathbf {B}} ^{-1} &{}{\mathbf {F}} ^{-1} \end{bmatrix}$

Note that we don’t need to find the value of $({\mathbf {D}} +{\mathbf {L}} )^{-1}$, just take the following formula to solve the linear detection term. (Because ${\mathbf {P}}_1 \in \mathrm {R}^{\mathrm {P}\times \mathrm {Q}}$, ${\mathbf {P}}_2 \in \mathrm {R} ^{\mathrm {Q}\times \mathrm {K}}$, ${\mathbf {b}} \in \mathrm {R} ^{\mathrm {K}\times 1}$, we know $({\mathbf {P}} _1{\mathbf {P}} _2 ){\mathbf {b}} ={\mathbf {P}}_1 ({\mathbf {P}} _2{\mathbf {b}} )$. Let ${\mathbf {b}} ={\hat{\mathbf {x}}}_{MF} -{\mathbf {U}}{\hat{\mathbf {x}}}_t =\begin{bmatrix} {\mathbf {c}}_1 \\ {\mathbf {c}}_2 \end{bmatrix}$, where ${\mathbf {c}}_1 ,{\mathbf {c}}_2 \in \mathrm {R} ^{\mathrm {\frac{M}{2}} \times 1 }$. So

$$\begin{aligned} \begin{aligned} ({\mathbf {D}} +{\mathbf {L}} )^{-1}[{\hat{\mathbf {x}}}_{MF} -{\mathbf {U}}{\hat{\mathbf {x}}}_t ]&=\begin{bmatrix} {\mathbf {B}} ^{-1} &{} {\mathbf {0}} \\ -{\mathbf {F}} ^{-1}\mathbf {CB} ^{-1} &{}{\mathbf {F}} ^{-1} \end{bmatrix}\begin{bmatrix} {\mathbf {c}}_1 \\ {\mathbf {c}}_2 \end{bmatrix}\\ {}&=\begin{bmatrix} {\mathbf {B}} ^{-1}{\mathbf {c}}_1 \\ -{\mathbf {F}} ^{-1}\mathbf {CB} ^{-1}{\mathbf {c}}_1 +{\mathbf {F}} ^{-1}{\mathbf {c}}_2 \end{bmatrix} \end{aligned} \end{aligned}$$

(51)

Error analysis

In this section, we will study the reasons why BGS-Net can improve performance. The analysis of the error (${\hat{\mathbf {x}}}_t -{\mathbf {x}}$) is as follows:

Define the output error ${\mathbf {e}}_t^{lin} ={\mathbf {z}}_t -{\mathbf {x}}$ for the linear phase at iteration t and the output error at the previous iteration $t -1$ as ${\mathbf {e}}_{t-1}^{den} ={\hat{\mathbf {x}}} _t -{\mathbf {x}}$. We can rewrite the update equation of Algorithm 1 based on these two output errors as:

$$\begin{aligned} \begin{aligned} {\mathbf {e}}_t^{lin}&=({\mathbf {D}} +{\mathbf {L}} )^{-1}{\mathbf {H}} ^\mathrm {T} {\mathbf {y}} -({\mathbf {D}} +{\mathbf {L}} )^{-1}{\mathbf {U}}{\hat{\mathbf {x}}}_t -{\mathbf {x}} \\ {}&=({\mathbf {D}} +{\mathbf {L}} )^{-1}{\mathbf {H}}^\mathrm {T} (\mathbf {Hx} +{\mathbf {n}} )-({\mathbf {D}} +{\mathbf {L}} )^{-1} {\mathbf {U}}{\hat{\mathbf {x}}}_t -{\mathbf {x}} \\ {}&=({\mathbf {D}} +{\mathbf {L}} )^{-1}{\mathbf {H}}^\mathrm {T} (\mathbf {Hx} +{\mathbf {n}} )-({\mathbf {D}} +{\mathbf {L}} )^{-1}({\mathbf {A}} -{\mathbf {D}} -{\mathbf {L}} ){\hat{\mathbf {x}}}_t -{\mathbf {x}}\\ {}&={\mathbf {e}}_{t-1}^{den} +({\mathbf {D}} +{\mathbf {L}} )^{-1}{\mathbf {H}}^\mathrm {T} (\mathbf {Hx} +{\mathbf {n}} )-({\mathbf {D}} +{\mathbf {L}} )^{-1}({\mathbf {H}} ^\mathrm {T}{\mathbf {H}} +\frac{\sigma ^2}{2}{\mathbf {I}} ){\hat{\mathbf {x}}}_t\\ {}&=({\mathbf {I}} -({\mathbf {D}} +{\mathbf {L}} )^{-1}{\mathbf {H}} ^\mathrm {T} {\mathbf {H}} ){\mathbf {e}}_{t-1}^{den} +({\mathbf {D}} +{\mathbf {L}} )^{-1}({\mathbf {H}} ^\mathrm {T} {\mathbf {n}} -\frac{\sigma ^2}{2}{\hat{\mathbf {x}} }_t )\\ {}&=({\mathbf {I}} -({\mathbf {D}} +{\mathbf {L}} )^{-1}({\mathbf {H}} ^\mathrm {T} {\mathbf {H}} +\frac{\sigma ^2}{2} {\mathbf {I}} )){\mathbf {e}}_{t-1}^{den} +({\mathbf {D}} +{\mathbf {L}} )^{-1}({\mathbf {H}} ^\mathrm {T} {\mathbf {n}} -\frac{\sigma ^2}{2} {\mathbf {x}} ) \end{aligned} \end{aligned}$$

(52)

and

$$\begin{aligned} \begin{aligned} {\mathbf {e}}_t^{den} =E\left\{ {\mathbf {x}} |{\hat{\mathbf {z}} }_t ,\mathbf {\tau }_t \right\} -{\mathbf {x}} \end{aligned} \end{aligned}$$

(53)

From Figure 2, we know that under channel hardening conditions, the first term of equation (52) $({\mathbf {I}} -({\mathbf {D}} +{\mathbf {L}} )^{-1}({\mathbf {H}} ^\mathrm {T} {\mathbf {H}} +\frac{\sigma ^2}{2}{\mathbf {I}} ))$ tends to 0; the second term is divided into the effect of ${\mathbf {n}}$ and the effect of ${\mathbf {x}}$, for the noise, where ${\mathbf {n}} =\sqrt{\sigma ^2/2}*N(0,1)$, so when the signal-to-noise ratio(SNR) is small, ${\mathbf {H}} ^\mathrm {T} {\mathbf {n}}$ becomes larger, $\frac{\sigma ^2}{2} {\mathbf {x}}$ also becomes larger; when the SNR is large, ${\mathbf {H}} ^\mathrm {T} {\mathbf {n}}$ becomes smaller and $\frac{\sigma ^2}{2} {\mathbf {x}}$ also becomes smaller. And $({\mathbf {D}} +{\mathbf {L}} )^{-1}({\mathbf {H}} ^\mathrm {T} {\mathbf {n}} -\frac{\sigma ^2}{2}{\mathbf {x}} )$ can in turn be reduced to $({\mathbf {D}} +{\mathbf {L}} )^{-1}{\mathbf {H}} ^\mathrm {T} {\mathbf {y}} -({\mathbf {D}} +{\mathbf {L}} )^{-1}({\mathbf {H}} ^\mathrm {T} \mathbf {Hx} +\frac{\sigma ^2}{2}{\mathbf {x}} )=({\mathbf {D}} +{\mathbf {L}} )^{-1}{\mathbf {H}} ^\mathrm {T} {\mathbf {y}} -({\mathbf {D}} +{\mathbf {L}} )^{-1}({\mathbf {H}} ^\mathrm {T} {\mathbf {H}} +\frac{\sigma ^2}{2}{\mathbf {I}} ){\mathbf {x}}$, which is approximated under good channel hardening as ${\mathbf {x}}_{MMSE} -{\mathbf {x}}$, and as far as we know, the gap between ${\mathbf {x}}_{MMSE} -{\mathbf {x}}$ decreases as the channel hardens more. Under channel hardening conditions the error ${\mathbf {e}}_{t-1}^{den}$ from the previous stage, suppressed by $({\mathbf {I}} -({\mathbf {D}} +{\mathbf {L}} )^{-1}({\mathbf {H}} ^\mathrm {T} {\mathbf {H}} +\frac{\sigma ^2}{2} {\mathbf {I}} ))$, is significantly attenuated. These calculations explain why BGS-Net has good performance on i.i.d Gaussian channels. Moreover, it is better than MMNet-iid’s ${\mathbf {I}} -\theta _t^{(1)}{\mathbf {H}} ^\mathrm {T} {\mathbf {H}}$ on correlated channels, where channel hardening disappears when the channel is correlated and there is no way for ${\mathbf {I}} -\theta _t^{(1)}{\mathbf {H}} ^\mathrm {T} {\mathbf {H}}$ to converge to ${\mathbf {0}}$ as the number of antennas increases, while ${\mathbf {I}} -({\mathbf {D}} +{\mathbf {L}} )^{-1}({\mathbf {H}} ^\mathrm {T} {\mathbf {H}} +\frac{\sigma ^2}{2} {\mathbf {I}} )$, since ${\mathbf {A}}$ is symmetric and $\mathbf {D+L}$ itself contains all the information in ${\mathbf {A}}$. When the number of antennas increases, it can be approximated as ${\mathbf {I}} -{\mathbf {A}} ^{-1}{\mathbf {A}}$, tending to ${\mathbf {0}}$ but not to ${\mathbf {0}}$.

For the effect of the nonlinear activation function, $E\left\{ {\mathbf {x}} |{\hat{\mathbf {z}}}_t ,\mathbf {\tau }_t \right\} -{\mathbf {x}}$ in (53) reduces the difference of ${\hat{\mathbf {x}}}_{t+1} -{\mathbf {x}}$. The proof is as follows:

$$\begin{aligned} \begin{aligned} E\left\{ x_{ti} |{\hat{z}}_{ti} ,\tau _{ti} \right\} -x_{ti} = {\textstyle \sum _{s_j\in S}s_j\times softmax(\frac{-\left\| {\hat{z}}_{ti}-s_j \right\| ^2 }{\tau _{ti}^2} )-x_{ti}} \end{aligned} \end{aligned}$$

(54)

Assuming that the true value $x_{ti}$ is $s_1$, then the above formula is equal to

$$\begin{aligned} \begin{aligned} {\textstyle \sum _{s_j\in S}s_j\times p(s_j/ {\hat{z}}_{ti},\tau _{ti} )-s_1} \end{aligned} \end{aligned}$$

(55)

We know that the softmax soft decision uses an exponent that allows judgments with larger probabilities to become larger and smaller probabilities to become smaller, but the total probability is still 1. It can be seen that as the probability of judging $s_1$ increases, the first term of equation (55) gets closer and closer to $s_1$, so using this activation function can further reduce the error.

Improved BGS-Net method

Analysis of the problem

Under the MAUE system, ${\mathbf {A}}$ cannot be approximated as a diagonal matrix ${\mathbf {D}}$, which has a great influence on ${\hat{\mathbf {x}}}_0 \leftarrow {\mathbf {D}} ^{-1}{\mathbf {H}} ^\mathrm {T}{\mathbf {y}}$. ${\hat{\mathbf {x}}}_0$ is a initial solution. If ${\hat{\mathbf {x}}}_0$ is given well, then the number of iterations required is small. In the SAUE system, the value of ${\hat{\mathbf {x}}}_0 \leftarrow {\mathbf {D}} ^{-1}{\mathbf {H}} ^\mathrm {T} {\mathbf {y}}$ is approximately equal to $({\mathbf {H}} ^\mathrm {T} {\mathbf {H}} +\frac{\sigma ^2}{2} {\mathbf {I}} )^{-1}{\mathbf {H}} ^\mathrm {T}{\mathbf {y}}$, so the number of iterations is small, which can also explain why fast iterations converge under channel hardening conditions. In the MAUE system, ${\mathbf {H}} ^\mathrm {T} {\mathbf {H}}$ loses the diagonal dominance, and the sum of the other elements in the same line is no longer much smaller than the diagonal elements. ${\mathbf {D}} ^{-1}{\mathbf {H}} ^\mathrm {T} {\mathbf {y}}$ cannot approach the real solution ${\mathbf {x}}$, here ${\hat{\mathbf {x}}}_0 \leftarrow {\mathbf {A}}^ {-1}{\mathbf {H}}^ \mathrm {T} {\mathbf {y}}$ replaces ${\hat{\mathbf {x}}}_0 \leftarrow {\mathbf {D}} ^{-1}{\mathbf {H}} ^\mathrm {T} {\mathbf {y}}$, so that no matter how the channel changes, a good initial solution can be extracted.

However, calculating ${\mathbf {A}}^{-1}$ has a high complexity, so the low-complexity method is used to approximate the solution of ${\mathbf {A}}^{-1}$.

Improved BGS-Net Design

In this section, BGS-Net is improved to adapt to the MAUE system by replacing ${\hat{\mathbf {x}}}_0 \leftarrow {\mathbf {D}} ^{-1}{\mathbf {H}} ^\mathrm {T} {\mathbf {y}}$ with ${\hat{\mathbf {x}}}_0 \leftarrow {\mathbf {A}} ^{-1}{\mathbf {H}} ^\mathrm {T} {\mathbf {y}}$ .

Approximation of ${\mathbf {A}}^{-1}$

From Figures 3, 4 and 5, it can be seen that as the correlation coefficient increases, the channel is block-like in character. To approximate ${\mathbf {A}}^{-1}$, as in Figure 9, divide the diagonal of matrix ${\mathbf {A}}$ into $\mathrm {\frac{M}{m_{UE}} }$ small matrices, the size of the small matrice is $\mathrm {m_{UE}} \times \mathrm {m_{UE}}$, each small matrice in order is set to ${\mathbf {D}}_{(2,t)}\in \mathrm {R^{\mathrm {m_{UE}} \times \mathrm {m_{UE}}}}, (t\in [1:\mathrm {T} ])$ respectively. The matrix ${\mathbf {A}}$ is again divided into 4 matrices, with the upper left and lower right matrices set to ${\mathbf {D}} _{(1,1)}, {\mathbf {D}} _{(1,2)} \in \mathrm {R^{M/2\times M/2}}$ respectively. To ensure convergence of the Neumann Series, here, unlike in [32], the parameter $\alpha _{opt}=1+\eta$, $\eta =\mathrm {M/N}$ was introduced [39]. We do the following for all small matrices ${\mathbf {D}}_{(2,t)}$ as ${\mathbf {D}}_{(2,t)}={\mathbf {D}}_{(2,t)} \times \alpha _{opt}$ and find the inverse of each small matrix ${\mathbf {D}}_{(2,t)} ^{-1}$. In order to get ${\mathbf {D}}_{(1,1)} ^{-1}$, we don’t want to find it directly, because its complexity is $O(\mathrm {M^3} /4)$, at this time ${\mathbf {D}}_{(1,1)} ^{-1}$ has both diagonal elements and non-diagonal elements, unlike only diagonal elements under channel hardening conditions. In this paper, the Neumann Series is used to approximate it [32, 40] by the following method:

$$\begin{aligned} \begin{aligned} {\mathbf {N}} =diag[{\mathbf {D}}_{2,1}^{-1} ,{\mathbf {D}}_{2,2}^{-1} ,\dots ,{\mathbf {D}}_{2,\frac{\mathrm {T}}{2} }^{-1} ] \end{aligned} \end{aligned}$$

(56)

$$\begin{aligned} \begin{aligned} {\tilde{\mathbf {D}} }_{1,1} =diag[{\mathbf {D}}_{2,1} ,{\mathbf {D}}_{2,2} ,\dots ,{\mathbf {D}}_{2,\frac{\mathrm {T}}{2} } ] \end{aligned} \end{aligned}$$

(57)

use ${\mathbf {E}}$ to approximate the off-diagonal block part of ${\mathbf {D}} _{(1,1)}$

$$\begin{aligned} \begin{aligned} {\mathbf {E}} ={\tilde{\mathbf {D}}} _{1,1}-{\mathbf {D}} _{(1,1)} \end{aligned} \end{aligned}$$

(58)

and use ${\mathbf {N}}$ to approximate ${\mathbf {D}}_{(1,1)} ^{-1}$

$$\begin{aligned} \begin{aligned} {\mathbf {D}}_{(1,1)} ^{-1}={\mathbf {N}} \end{aligned} \end{aligned}$$

(59)

So we can approximate ${\mathbf {D}} _{(1,1)}^{-1}$ by $k_N$ times Neumann method.

$$\begin{aligned} \begin{aligned} k_N \quad times:{\mathbf {D}} _{(1,1)}^{-1} =\mathbf {NED} _{(1,1)}^{-1} +{\mathbf {N}} \end{aligned} \end{aligned}$$

(60)

As above, we can obtain ${\mathbf {D}}_{(1,2)} ^{-1}$, and then splice ${\mathbf {D}}_{(1,1)} ^{-1}$ and ${\mathbf {D}}_{(1,2)} ^{-1}$ together

$$\begin{aligned} \begin{aligned} {\mathbf {D}}^{-1} =diag[{\mathbf {D}}_{(1,1)}^{-1} ,{\mathbf {D}}_{(1,2)}^{-1} ] \end{aligned} \end{aligned}$$

(61)

In this way we can get the required ${\mathbf {D}}^{-1}$. The same can be obtained:

$$\begin{aligned} \begin{aligned} {\mathbf {E}} ={\mathbf {D}}^{-1} -{\mathbf {A}} \end{aligned} \end{aligned}$$

(62)

$$\begin{aligned} \begin{aligned} {\tilde{\mathbf {A}} }^{-1} ={\mathbf {D}} ^{-1} \end{aligned} \end{aligned}$$

(63)

$$\begin{aligned} \begin{aligned} k_F\quad times:\tilde{{\mathbf {A}} }^{-1} ={\mathbf {D}} ^{-1}{\mathbf {E}} \tilde{{\mathbf {A}} }^{-1} +{\mathbf {D}} ^{-1} \end{aligned} \end{aligned}$$

(64)

In this way, we get ${\tilde{\mathbf {A}} }^{-1}$. Considering the high complexity of Eq. (64), we rewrite Eq. (64) without changing its principle [41]

$$\begin{aligned} \begin{aligned} {\mathbf {S}}_t =\mathbf {\vartheta S}_{t-1} +{\mathbf {S}}_0 \quad (k_F\ge t\ge 1) \end{aligned} \end{aligned}$$

(65)

where ${\mathbf {S}}_0 ={\mathbf {D}}^{-1} ({\mathbf {H}} ^\mathrm {T} {\mathbf {y}} )$, $\mathbf {\vartheta } ={\mathbf {D}} ^{-1}{\mathbf {E}}$. In this way, we can bypass the high complexity of solving Eq. (64) and directly obtain ${\hat{\mathbf {x}}}_0 \leftarrow {\mathbf {A}} ^{-1}{\mathbf {H}} ^\mathrm {T}{\mathbf {y}}$ with low complexity.

Offline training algorithm

A relatively good initial solution is obtained from the approximation of ${\mathbf {A}}$ in 5.2.1.

Algorithm2: Improved BGS-Net offline training
Input: Received signal ${\mathbf {y}}$, channel matrix ${\mathbf {H}}$, noise level $\sigma ^2/2$
Initialize: ${\hat{\mathbf {x}}}_0 \leftarrow \tilde{{\mathbf {A}} } ^{-1}{\mathbf {H}} ^\mathrm {T} {\mathbf {y}}$
$1.{\hat{\mathbf {z}}}_t =({\mathbf {D}} +{\mathbf {L}} )^{-1}[{\hat{\mathbf {x}}}_{MF} -{\mathbf {U}}{\hat{\mathbf {x}} }_t ]$
$2.v_t^2=\frac{\left\\| {\mathbf {y}} -{\mathbf {H}}{\hat{\mathbf {x}}}_t \right\\| _2^2-\mathrm {N} \frac{\sigma ^2}{2} }{tr({\mathbf {H}} ^\mathrm {T}{\mathbf {H}} )}$
$3.v_t^2=\max (v_t^2,10^{-9} )$
$4.\tau _t^2 =\frac{1}{\mathrm {M} }tr({\mathbf {C}} _t{\mathbf {C}} _t^\mathrm {T} ) v_t^2 +\frac{\sigma ^2}{\mathrm {M} } tr({\mathbf {W}} _t{\mathbf {W}} _t^\mathrm {T} )$
$5.\mathbf {\tau }_t^2 =\frac{\mathbf {\tau } _t^2 }{\mathbf {\gamma } _t }$
$6.{\hat{\mathbf {x}}}_{t+1} =E\left\{ {\mathbf {x}} \|{\hat{\mathbf {z}}}_t ,\mathbf {\tau }_t \right\}$

Complexity analysis

Complexity of $({\mathbf {D}} +{\mathbf {L}} )^{-1}$

In this section, the complexity of (37) is analysed. ${\hat{\mathbf {x}}}_t =({\mathbf {D}} +{\mathbf {L}} )^{-1} [{\hat{\mathbf {x}}}_{MF} -{\mathbf {U}}{\hat{\mathbf {x}} }_{t-1} ]$ in which the complexity is mainly solving $({\mathbf {D}} +{\mathbf {L}} )^{-1}$. If the inverse is solved directly, the complexity of the algorithm will reach $O(\mathrm {M^3} )$, so we use a nested loop method to reduce its complexity.

So the complexity calculation formula of $({\mathbf {D}} +{\mathbf {L}} )^{-1}$ is:

(a)
When the system is SAUE or MAUE (only ${\mathbf {T}}$)
$$\begin{aligned} \begin{aligned} \frac{\mathrm {M} }{2} +\mathrm {(2\times (2^0)^3\times \frac{M}{8} \times 2+2\times (2^1)^3\times \frac{M}{16}\times 2+\dots }\\+ 2\times (2^{\log _{2}{\mathrm {M} }-3 })^3\times \frac{\mathrm {M} }{2^{\log _{2}{\mathrm {M} }}} \times 2 ) +\frac{\mathrm {M} ^3}{32}=\frac{1}{24} \mathrm {M} ^3+\frac{1}{3} \mathrm {M} \end{aligned} \end{aligned}$$
(66)
Because $({\mathbf {D}} +{\mathbf {L}} )^{-1}$ only needs to be calculated once, the complexity of formula (37) is $\mathrm {\frac{1}{24} M^3+\frac{3}{4} tM^2+(\frac{1}{3}+t )M}$.
(b)
When MAUE (both ${\mathbf {T}}$ and ${\mathbf {R}}$)
$$\begin{aligned} \begin{aligned} 2\times [\frac{\mathrm {M} }{2} +\mathrm {(2\times (2^0)^3\times \frac{M}{8} \times 2+2\times (2^1)^3\times \frac{M}{16}\times 2+\dots }\\+ 2\times (2^{\log _{2}{\mathrm {M} }-3 })^3\times \frac{\mathrm {M} }{2^{\log _{2}{\mathrm {M} }}} \times 2 ) +\frac{\mathrm {M} ^3}{32}]=\frac{1}{12} \mathrm {M} ^3+\frac{2}{3} \mathrm {M} \end{aligned} \end{aligned}$$
(67)
So the complexity of formula (37) is $\mathrm {\frac{1}{12} M^3+\frac{3}{4} tM^2+(\frac{2}{3}+t )M}$.

Complexity of ${\hat{\mathbf {x}}_0 }$

In this section, the complexity analysis of ${\hat{\mathbf {x}}_0 }$ is carried out. The initial solutions of Improved BGS-Net and BGS-Net are different:

(1)
The complexity of ${\hat{\mathbf {x}}}_0 \leftarrow {\mathbf {D}} ^{-1}{\mathbf {H}} ^\mathrm {T} {\mathbf {y}}$ is: $\mathrm {MN+M}$ .
(2)
The complexity of ${\hat{\mathbf {x}}}_0 \leftarrow {\mathbf {A}} ^{-1}{\mathbf {H}} ^\mathrm {T} {\mathbf {y}}$ is: From [32] we can see that the complexity of (60) is $\frac{\mathrm {k_N}-1}{8} \mathrm {M} ^3+\frac{\mathrm {k_N}-2}{4} \mathrm {m_{UE}M^2} +\mathrm {Mm_{UE}^2}$, and the complexity of $\mathbf {\vartheta }$ is $\mathrm {M^2\frac{M}{2}-M(\frac{M}{2} ) ^2=\frac{1}{4} M^3}$, then the complexity of equation (64) becomes $\mathrm {\frac{1}{4} M^3+(k_F+\frac{1}{2} )M^2+MN}$.
1. (a)
  When the system is SAUE or MAUE (only ${\mathbf {T}}$): $\mathrm {\frac{k_N+1}{8} M^3+(\frac{k_N-2}{4}m_{UE}+k_F+\frac{1}{2} )M^2+Mm_{UE}^2+MN}$
2. (b)
  When MAUE (both ${\mathbf {T}}$ and ${\mathbf {R}}$): $\mathrm {\frac{k_N}{4} M^3+(\frac{k_N-2}{2}m_{UE}+k_F+\frac{1}{2} )M^2+2Mm_{UE}^2+MN}$

Complexity comparison

This section will analyze the computational complexity according to the number of multiplications of different algorithms. Here $\mathrm {N_r} =256$, $\mathrm {N_t} =32$, $\mathrm {k_N} =\mathrm {k_F} =2$, $\mathrm {m_{UE}} =4$. From Table 1, we can know that in the environment of a (when the system is SAUE or MAUE (only ${\mathbf {T}}$)), the complexity of BGS-Net is about $16\%$ of MMSE, $0.01\%$ of OAMPNet, $30\%$ of TL-BD-INSA , and $20\%$ of MMNet-iid; the complexity of Improved BGS-Net is about $38\%$ of MMSE, $0.02\%$ of OAMPNet, $42\%$ of TL-BD-INSA, and $65\%$ of MMNet-iid. In the environment of b (when MAUE (with both ${\mathbf {T}}$ and ${\mathbf {R}}$)), the complexity of BGS-Net is approximately $19\%$ of MMSE, $0.01\%$ of OAMPNet, $32\%$ of TL-BD-INSA, and $21\%$ of MMNet-iid; the complexity of Improved BGS-Net is about $46\%$ of MMSE, $0.02\%$ of OAMPNet, $80\%$ of TL-BD-INSA, and $52\%$ of MMNet-iid. So our algorithm complexity is very low. From [28] we can know that the complexity of MMNet-iid nonlinear detection is $O(\mathrm {M^2} )$, while the complexity of BGS-Net’s nonlinear detection is much smaller than that of MMNet-iid nonlinear detection, and the complexity of nonlinear detection of MMNet-iid is much smaller than that of linear detection, so we ignore its complexity.

Table 1 Computational complexity comparison

Full size table

Numerical results and discussion

In this section we give simulation results for MIMO detection of BGS-Net and Improved BGS-Net, and evaluate the performance by the symbol error rate(SER) for different signal-to-noise ratios(SNR). The SNR of the system, defined as

$$\begin{aligned} \begin{aligned} \mathrm {SNR} =\frac{E\left\| \mathbf {Hx} \right\| _2^2 }{E\left\| {\mathbf {n}} \right\| _2^2 } \end{aligned} \end{aligned}$$

(68)

Experimental description

BGS-Net and Improved BGS-Net simulations were implemented using Tensorflow. The number of layers T was set to 4. The training data consisted of a number of randomly generated pairs $({\mathbf {x}} ,{\mathbf {y}} )$. where the data ${\mathbf {x}}$ is generated by QPSK modulation symbols. We trained the network for 1000 iterations using stochastic gradient descent and the Adam optimiser. The learning rate was set to 0.001. $\mathrm {k_N} =\mathrm {k_F} =2$ and in the experimental setup we chose $l_{2loss}$ as the cost function.

$$\begin{aligned} \begin{aligned} l_{2loss}=\frac{1}{L_{layer}} {\textstyle \sum _{t=1}^{L_{layer}}}\left\| {\hat{\mathbf {x}}}_t -{\mathbf {x}} \right\| _2^2 \end{aligned} \end{aligned}$$

(69)

The different detectors are described in detail below, and in order to reduce the high latency of the deep learning method, all networks and iterations are set to 4 layers.

MMSE: MMSE detector uses ${\hat{\mathbf {x}}} =({\mathbf {H}} ^\mathrm {T} {\mathbf {H}} +\frac{\sigma ^2}{2}{\mathbf {I}} ) ^{-1}{\mathbf {H}} ^\mathrm {T} {\mathbf {y}}$.
Gauss-Seidel: The maximum number of iterations is set to layer=4.
${{TL\_BD\_INSA} }$ [32]: $\mathrm {TL\_BD\_INSA}$ is an improved Neumann series approximation algorithm based on two-level block diagonal.
OAMPNet: It is a DL-based detector that develops the OAMP detector. In our simulation, the number of layers of OAMPNet is set to 4 layers, and each layer has 2 learnable variables.
MMNet-iid: MMNet-iid is specially designed for i.i.d. Gaussian channels. In our simulation, the number of layers of MMNet-iid is set to 4, and each layer has 2 learnable variables.
GS-Net: The structure of BGS-Net is shown in Section 4. It has 4 layers and each layer has 1 learnable vector variable.
Improved BGS-Net: The structure of Improved BGS-Net is shown in Section 5, it has 4 layers with 1 learnable vector variable per layer. $\mathrm {m_{UE}} =2$ when $\mathrm {N_t=4 \ or\ 8}$; $\mathrm {m_{UE}} =4$ when $\mathrm {N_t=16 \ or\ 32}$.

SAUE system

The performance of BGS-Net was tested on the SAUE system using QPSK modulation; the SNR was 3dB during training.

Convergence analysis

As shown in Figure 10, the convergence speed of BGS-Net was tested for different network layers with the same SNR of 3dB and the same number of antennas $\mathrm {N_r} =32$, $\mathrm {N_r} =4$. It can be observed that the 3-layer BGS-Net has converged, while MMNet-iid needs at least 7 layers to converge and OAMPNet needs 4 layers which indicates that BGS-Net converges fastest in the SAUE system environment. The SER of BGS-Net is much better than that of MMNet-iid in the SAUE system, but there is still a gap between the SER of BGS-Net and that of OAMPNet.

Impact of ratio $\alpha$

This section analyzes the effect of the ratio $\alpha$ of the receiving antenna and the transmitting antenna on the performance of the algorithm. We set $\mathrm {N_t} =4$, SNR=3dB, and compare the performance of the algorithm when $\mathrm {N_r} =24,32,40$, respectively. As shown in Figure 11, it is found that as the ratio $\alpha$ increases, the lower the SER, the smaller the gap between the algorithms. Studying $\mathrm {N_t} =4, \mathrm {N_r} =40$ separately, as shown in Figure 12, BGS-Net can approximate the performance of OAMPNet with much lower complexity. When $\mathrm {N_r} =24$, the gap between BGS-Net and OAMPNet is $9\times 10^{-5}$; when $\mathrm {N_r} =40$, the gap between BGS-Net and OAMPNet is $3\times 10^{-7}$.

Impact of the number of antennas

This section analyzes the influence of the number of antennas on the performance of the algorithm, and sets the ratio $\alpha$ as a fixed value, that is, $\alpha =8$. As shown in Figure 13, Figure 14, and Figure 15, when the number of antennas increases, the performance of all algorithms is improving, the performance of MMNet-iid changes the most, and the performance of BGS-Net changes little. Affected by the number of antennas is small, which shows that BGS-Net is very robust. At the same time, BGS-Net has always been better than Gauss-Seidel, which shows that the nonlinear activation function improves the performance of Gauss-Seidel.

Effect of modulation order

This section analyzes the impact of modulation methods on algorithm performance. We compare the performance of MMSE, Gauss-Seidel, and BGS-Net under QPSK and 16QAM. When the test SNR is set to 5-9dB, the training test ratio is set to 7dB. As shown in Figure 16, it is found that when the modulation order increases, the performance of the algorithm decreases, but the performance of BGS-Net is always better than Gauss-Seidel, and the performance of Gauss-Seidel is close to that of MMSE.