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Lowcomplexity signal detection networks based on GaussSeidel iterative method for massive MIMO systems
EURASIP Journal on Advances in Signal Processing volumeÂ 2022, ArticleÂ number:Â 51 (2022)
Abstract
In massive multipleinput multipleoutput (MIMO) systems with single antenna user equipment (SAUE) or multipleantenna 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 GaussSeidel iterative method, this paper proposes a modeldriven deep learning detector network, namely Block GaussSeidel Network (BGSNet), which is based on the GaussSeidel 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 BGSNet under MAUE system, we propose Improved BGSNet. The simulation results show that, compared with the existing modeldriven algorithms, BGSNet has lower complexity and similar the detection performance; good robustness, and its performance is less affected by changes in the number of antennas; Improved BGSNet can improve the detection performance of BGSNet.
1 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 selfdriving 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 multiinput multioutput (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 kbest 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 tradeoff 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 inversionbased 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 GaussSeidel method (GS) [10], the successive superrelaxation 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 modeldriven deep learning network for multipleinput multipleoutput (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 whitebox 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 BPNet [29], CGNet [30], which are networks changed based on approximation methods. [31] proposed a datadriven implementation of an iterative soft interference cancellation (SIC) algorithm, called DeepSIC. This method significantly outperforms modelbased methods in the presence of channel state information (CSI) uncertainty, but the network is more complex and this method is a combination of blackbox and whitebox 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 abovementioned work is directed to a singleantenna 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 (MMIMO) system. Therefore, this paper also considers the multipleantenna user equipment(MAUE) system while considering the SAUE system.
In this paper, we propose a modeldriven deep learning detector network Block GaussSeidel Network (BGSNet) to solve the high complexity caused by the parallel operation of traditional GaussSeidel [10], which is based on the GaussSeidel iterative method . We reduce the complexity by converting large matrix inversions \(({\mathbf {D}} +{\mathbf {L}} )^{1}\) to small matrix inversions and converting matrixbymatrix to matrixbyvector. This paper considers SAUE and MAUE systems [32, 33] under Rayleigh channels. In order to improve SER of BGSNet under MAUE system, we improve the initial solution of BGSNet 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 modeldriven algorithms, BGSNet 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 GaussSeidel; Improved BGSNet can improve the SER of BGSNet.
This paper is organised as follows. Section 2 presents and analyses the channels required in this paper. Section 3 analyses the existing OAMPNet/MMNetiid algorithms. Section 4 proposes the algorithm BGSNet, and explains why BGSNet is proposed. Section 5 analyses the problems that BGSNet may encounter in MAUE systems and improve it, and proposes Improved BGSNet. Section 6 analyses the complexity of BGSNet and Improved BGSNet in terms of their complexity. Experimental simulations and discussion are performed in Section 7. Section 8 concludes the full paper.
1.1 Notation
In this paper, lowercase and uppercase 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 pseudoinverse 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 \({\mathbf {X}}\).
2 Background
2.1 SAUE System
Consider an uplink massive MIMO system that uses \(\mathrm {N_r}\) antennas at the BS to serve \(\mathrm {N_t}\) singleantenna user terminals simultaneously, where \(\mathrm {N_r\gg N_t}\). The SAUE system can be simply expressed as:
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 complexvalued system model (1) is converted to the corresponding realvalued system model as
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.
2.2 MAUE System
Consider an uplink massive MIMO with a multiantenna 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:
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 \({{\mathbf {T}} \in {R^{M\times M}} }\).
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].
2.3 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 largescale 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 nondiagonal 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 nondiagonal elements around the diagonal elements have a serious impact on the diagonal elements, and the approximation effect is very poor.
The MarchenkoPasture 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 offdiagonal 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
and
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.
3 Related work
The goal of the receiver is to calculate the maximum likelihood (ML) estimate \(\hat{{\mathbf {x}} }\) of \({\mathbf {x}}\) set as
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.
3.1 OAMPNet
OAMPNet is a modeldriven 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
here \({\mathbf {W}}_t\) is the optimal \({\mathbf {W}}\) in OAMP in [35]
In this way, the value of the linear estimate can be obtained
Step 2:We take the linear detection estimates as input and perform nonlinear detection
\(\tau _t ^2\) is the variance of the linear estimation error
\(p(x_i)\) is the prior probability
Bring \(\tau _t ^2\) and \(p(x_i)\) into \(E\left\{ {\mathbf {x}} {\mathbf {r}}_t ,\tau _t \right\}\)
In this way, the input value of the (t+1) layer can be obtained
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 pseudoinverse, which brings great complexity, not suitable for massive MIMO, suitable for mediumscale MIMO [25, 26].
3.2 MMNetiid
The main idea of MMNetiid is to introduce an appropriate degree of flexibility in the linear and denoising components of the iterative framework while maintaining its linear plus nonlinear structure [27, 28].
Step 1: We need to design a linear detector to estimate \({\mathbf {z}}_t\).
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
\(\eta _t({\mathbf {z}}_t ;\sigma _t^2)\) is a nonlinear detection estimate
In this way, the input of the next layer can be obtained
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. MMNetiid 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.
4 The proposed BGSNet method
4.1 GaussSeidel 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.
The iterative formula of GS [36] is
For each layer, \(b_i\) is used to subtract the updated \({\textstyle \sum _{j=1}^{i1}a_{ij}x_j^{(k+1)}}\) and the not yet updated \({\textstyle \sum _{j=i+1}^{n}}a_{ij}x_j^{(k)}\), its matrix representation is
GS is used in communication, that is, the Hermitian positive semidefinite matrix \({\mathbf {A}}\) is decomposed into strictly lower triangular terms \({\mathbf {L}}\), strictly upper triangular terms \({\mathbf {U}}\) and diagonal terms \({\mathbf {D}}\)
Then the problem we solve is
Then solve a set of linear equations by calculating the solution of the iterative be havior [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 \({\mathbf {D}} ^{1}{\mathbf {H}} ^\mathrm {T} {\mathbf {y}}\). GaussSeidel 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 GaussSeidel for diagonally dominant or symmetric positive times, respectively [38].
Theorem 1
When \({\mathbf {A}}\) is diagonally dominant, GaussSeidel 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
Suppose \(\mathbf {B_G} =({\mathbf {D}} +{\mathbf {L}} )^{1}{\mathbf {U}}\), the characteristic value is \(\lambda\), then the characteristic equation is
When the determinant is zero, the equation has a nonzero solution. Use contradiction: suppose \(\left \lambda \right \ge 1\),
It is a strictly diagonally dominant matrix, so it is nonsingular, 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, GaussSeidel converges.
Theorem 2
When \({\mathbf {A}}\) is symmetrically positive, GaussSeidel 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
Because \({\mathbf {A}}\) is positive definite, so \(p={\mathbf {x}} ^\mathrm {T} \mathbf {Dx} > 0\), set \({\mathbf {x}} ^\mathrm {T} \mathbf {Ux} =a\), then
So \(\left \lambda \right < 1\) is \(\rho (\mathbf {B_G} )< 1\). When \({\mathbf {A}}\) is symmetric positive definite, GaussSeidel converges, and we divide the numerator and denominator of (44) by \(a^2\) at the same time, we get
From (45) we can see that the more diagonally dominant, the smaller the \(\lambda ^2\), the faster the convergence.
4.2 BGSNet architecture
In this section, a modeldriven DL detector network (called BGSNet) is proposed. The signal detector uses the GaussSeidel 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 MMNetiid need to be calculated once per layer because of the training parameters present in them. The structure of BGSNet 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 BGSNet 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 modeldriven and DNN is that many of the parameters of modeldriven are fixed values obtained from past experience, while the parameters of DNN are all variable values.
Algorithm1: BGSNet 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
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 \(4\rightarrow 5\), \(\tau _t^2\)â€™s dimension expansion to a vector \(\mathbf {\tau }_t^2\) .
4.3 Lowcomplexity 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 \({\hat{\mathbf {x}}}_t =({\mathbf {D}} +{\mathbf {L}} )^{1}[{\hat{\mathbf {x}}}_{MF} {\mathbf {U}}{\hat{\mathbf {x}}} _{t1} ]\) 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 \(\mathrm {M1}\) 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 BGSNet.
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:
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:
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 donâ€™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
4.4 Error analysis
In this section, we will study the reasons why BGSNet 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 \({\mathbf {e}}_{t1}^{den} ={\hat{\mathbf {x}}} _t {\mathbf {x}}\). We can rewrite the update equation of Algorithm 1 based on these two output errors as:
and
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 signaltonoise 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 \({\mathbf {e}}_{t1}^{den}\) 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 BGSNet has good performance on i.i.d Gaussian channels. Moreover, it is better than MMNetiidâ€™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:
Assuming that the true value \(x_{ti}\) is \(s_1\), then the above formula is equal to
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.
5 Improved BGSNet method
5.1 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 lowcomplexity method is used to approximate the solution of \({\mathbf {A}}^{1}\).
5.2 Improved BGSNet Design
In this section, BGSNet 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 \({\hat{\mathbf {x}}}_0 \leftarrow {\mathbf {A}} ^{1}{\mathbf {H}} ^\mathrm {T} {\mathbf {y}}\) .
5.2.1 Approximation of \({\mathbf {A}}^{1}\)
From Figures 3, 4 and 5, it can be seen that as the correlation coefficient increases, the channel is blocklike 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 donâ€™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 nondiagonal 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:
use \({\mathbf {E}}\) to approximate the offdiagonal block part of \({\mathbf {D}} _{(1,1)}\)
and use \({\mathbf {N}}\) to approximate \({\mathbf {D}}_{(1,1)} ^{1}\)
So we can approximate \({\mathbf {D}} _{(1,1)}^{1}\) by \(k_N\) times Neumann method.
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
In this way we can get the required \({\mathbf {D}}^{1}\). The same can be obtained:
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]
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.
5.2.2 Offline training algorithm
A relatively good initial solution is obtained from the approximation of \({\mathbf {A}}\) in 5.2.1.
Algorithm2: Improved BGSNet offline training  

Input: Received signal \({\mathbf {y}}\), channel matrix \({\mathbf {H}}\), noise level \(\sigma ^2/2\)  
Initialize: \({\hat{\mathbf {x}}}_0 \leftarrow \tilde{{\mathbf {A}} } ^{1}{\mathbf {H}} ^\mathrm {T} {\mathbf {y}}\)  
\(1.{\hat{\mathbf {z}}}_t =({\mathbf {D}} +{\mathbf {L}} )^{1}[{\hat{\mathbf {x}}}_{MF} {\mathbf {U}}{\hat{\mathbf {x}} }_t ]\)  
\(2.v_t^2=\frac{\left\ {\mathbf {y}} {\mathbf {H}}{\hat{\mathbf {x}}}_t \right\ _2^2\mathrm {N} \frac{\sigma ^2}{2} }{tr({\mathbf {H}} ^\mathrm {T}{\mathbf {H}} )}\)  
\(3.v_t^2=\max (v_t^2,10^{9} )\)  
\(4.\tau _t^2 =\frac{1}{\mathrm {M} }tr({\mathbf {C}} _t{\mathbf {C}} _t^\mathrm {T} ) v_t^2 +\frac{\sigma ^2}{\mathrm {M} } tr({\mathbf {W}} _t{\mathbf {W}} _t^\mathrm {T} )\)  
\(5.\mathbf {\tau }_t^2 =\frac{\mathbf {\tau } _t^2 }{\mathbf {\gamma } _t }\)  
\(6.{\hat{\mathbf {x}}}_{t+1} =E\left\{ {\mathbf {x}} {\hat{\mathbf {z}}}_t ,\mathbf {\tau }_t \right\}\) 
6 Complexity analysis
6.1 Complexity of \(({\mathbf {D}} +{\mathbf {L}} )^{1}\)
In this section, the complexity of (37) is analysed. \({\hat{\mathbf {x}}}_t =({\mathbf {D}} +{\mathbf {L}} )^{1} [{\hat{\mathbf {x}}}_{MF} {\mathbf {U}}{\hat{\mathbf {x}} }_{t1} ]\) 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 \(\mathrm {\frac{1}{12} M^3+\frac{3}{4} tM^2+(\frac{2}{3}+t )M}\).
6.2 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 BGSNet and BGSNet 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 \(\mathrm {\frac{1}{4} M^3+(k_F+\frac{1}{2} )M^2+MN}\).

(a)
When the system is SAUE or MAUE (only \({\mathbf {T}}\)): \(\mathrm {\frac{k_N+1}{8} M^3+(\frac{k_N2}{4}m_{UE}+k_F+\frac{1}{2} )M^2+Mm_{UE}^2+MN}\)

(b)
When MAUE (both \({\mathbf {T}}\) and \({\mathbf {R}}\)): \(\mathrm {\frac{k_N}{4} M^3+(\frac{k_N2}{2}m_{UE}+k_F+\frac{1}{2} )M^2+2Mm_{UE}^2+MN}\)

(a)
6.3 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 BGSNet is about \(16\%\) of MMSE, \(0.01\%\) of OAMPNet, \(30\%\) of TLBDINSA , and \(20\%\) of MMNetiid; the complexity of Improved BGSNet is about \(38\%\) of MMSE, \(0.02\%\) of OAMPNet, \(42\%\) of TLBDINSA, and \(65\%\) of MMNetiid. In the environment of b (when MAUE (with both \({\mathbf {T}}\) and \({\mathbf {R}}\))), the complexity of BGSNet is approximately \(19\%\) of MMSE, \(0.01\%\) of OAMPNet, \(32\%\) of TLBDINSA, and \(21\%\) of MMNetiid; the complexity of Improved BGSNet is about \(46\%\) of MMSE, \(0.02\%\) of OAMPNet, \(80\%\) of TLBDINSA, and \(52\%\) of MMNetiid. So our algorithm complexity is very low. From [28] we can know that the complexity of MMNetiid nonlinear detection is \(O(\mathrm {M^2} )\), while the complexity of BGSNetâ€™s nonlinear detection is much smaller than that of MMNetiid nonlinear detection, and the complexity of nonlinear detection of MMNetiid is much smaller than that of linear detection, so we ignore its complexity.
7 Numerical results and discussion
In this section we give simulation results for MIMO detection of BGSNet and Improved BGSNet, and evaluate the performance by the symbol error rate(SER) for different signaltonoise ratios(SNR). The SNR of the system, defined as
7.1 Experimental description
BGSNet and Improved BGSNet 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.
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 \({\hat{\mathbf {x}}} =({\mathbf {H}} ^\mathrm {T} {\mathbf {H}} +\frac{\sigma ^2}{2}{\mathbf {I}} ) ^{1}{\mathbf {H}} ^\mathrm {T} {\mathbf {y}}\).

GaussSeidel: 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 twolevel block diagonal.

OAMPNet: It is a DLbased 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.

MMNetiid: MMNetiid is specially designed for i.i.d. Gaussian channels. In our simulation, the number of layers of MMNetiid is set to 4, and each layer has 2 learnable variables.

GSNet: The structure of BGSNet is shown in Section 4. It has 4 layers and each layer has 1 learnable vector variable.

Improved BGSNet: The structure of Improved BGSNet 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 \(\mathrm {N_t=16 \ or\ 32}\).
7.2 SAUE system
The performance of BGSNet was tested on the SAUE system using QPSK modulation; the SNR was 3dB during training.
7.2.1 Convergence analysis
As shown in Figure 10, the convergence speed of BGSNet 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 3layer BGSNet has converged, while MMNetiid needs at least 7 layers to converge and OAMPNet needs 4 layers which indicates that BGSNet converges fastest in the SAUE system environment. The SER of BGSNet is much better than that of MMNetiid in the SAUE system, but there is still a gap between the SER of BGSNet and that of OAMPNet.
7.2.2 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, BGSNet can approximate the performance of OAMPNet with much lower complexity. When \(\mathrm {N_r} =24\), the gap between BGSNet and OAMPNet is \(9\times 10^{5}\); when \(\mathrm {N_r} =40\), the gap between BGSNet and OAMPNet is \(3\times 10^{7}\).
7.2.3 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 MMNetiid changes the most, and the performance of BGSNet changes little. Affected by the number of antennas is small, which shows that BGSNet is very robust. At the same time, BGSNet has always been better than GaussSeidel, which shows that the nonlinear activation function improves the performance of GaussSeidel.
7.2.4 Effect of modulation order
This section analyzes the impact of modulation methods on algorithm performance. We compare the performance of MMSE, GaussSeidel, and BGSNet under QPSK and 16QAM. When the test SNR is set to 59dB, 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 BGSNet is always better than GaussSeidel, and the performance of GaussSeidel is close to that of MMSE.
7.3 MAMU system
The MAUE system uses QPSK modulation, and the SNR during training is 4dB.
7.3.1 Convergence analysis
In order to study the performance of the algorithm, what is the difference between the MAUE system and the SAUE system. As shown in Figure 17, we tested the convergence speed of BGSNet and Improved BGSNet at different network layers under the same SNR of 4dB and the same number of antennas \(\mathrm {N_t} =4,\mathrm {N_r} =32\), and found that the 3layer Improved BGSNet has converged, and the BGSNet and OAMPNet require 4layer can converge, while MMNetiid needs 7layer network to converge. The performance of MMNetiid is much lower than that of other algorithms, while the performance of BGSNet and Improved BGSNet maintains a slight performance gap with OAMPNet, and the performance of Improved BGSNet is better than BGSNet.
7.3.2 Impact of ratio \(\alpha\)
This section analyzes the effect of \(\alpha\) on the performance of the algorithm under \(\xi _r=0\), \(\xi _t=0.2\), we set \(\mathrm {N_t} =4\), SNR=4dB, and compare the performance of the algorithm when \(\mathrm {N_r} =32,40\) respectively. As shown in Figure 18, it is found that as \(\alpha\) increases, the performance of Improved BGSNet is closer to that of OAMPNet. As shown in Figure 19, when the antenna ratio \(\alpha =11\), the performance gap between Improved BGSNet and OAMPNet is \(2.5\times 10^{6}\). This shows that as long as \(\alpha\) is large enough, the performance of Improved BGSNet can approach OAMPNet with low complexity.
7.3.3 Impact of the number of antennas
This section analyzes the influence of the number of antennas on the performance of the algorithm, we set \(\alpha\) to a constant value, i.e. \(\alpha =8\). The performance of the algorithm is compared for \(\mathrm {N_t=4,N_r=32/N_t=8,N_r=64/N_t=16,N_r=128}\) respectively, as shown in Figures. 20, 21, and 22. It is found that the performance gap between Improved BGSNet and BGSNet decreases as the number of antennas increases in the \(\xi _r=0\), \(\xi _t=0.2\) environment, while at the same time the MMNetiid performance improves much faster than the others, suggesting that the impact brought by correlation can be improved by increasing the number of antennas in this environment. The fact that our proposed algorithm is consistently better than \(\mathrm {TL\_BD\_INSA}\) suggests that Improved BGSNet does improve the performance of the algorithm based on using \(\mathrm {TL\_BD\_INSA}\) as the initial solution.
7.3.4 Effect of modulation order
This section analyzes the effect of modulation on the performance of the algorithm. We compare the performance of MMSE, GaussSeidel, BGSNet, and Improved BGSNet under QPSK and 16QAM. As shown in Figure 23, it is found that the larger the modulation order, the lower the performance of the algorithm, and the gap between Improved BGSNet and BGSNet has increased a bit. Under QPSK, BGSNet coincides with Improved BGSNet at 5dB; under 16QAM, Improved BGSNet has always been better than BGSNet.
7.3.5 Effect of transmit correlation
To explore the effect of the correlation between the userâ€™s multiple antennas on the performance of the algorithm, we made two sets of comparisons, one for the performance of the algorithm with \(\mathrm {N_t=8,N_r=64,\xi _r=0,\xi _t=0.2 \ or \ 0.4}\), as shown in Figure 24. one for the performance of the algorithm with \(\mathrm {N_t=16,N_r=128,\xi _r=0,}\)
\(\mathrm {\xi _t=0.2 \ or \ 0.4}\), as shown in Figure 25. it was found that when the greater the correlation between multiple antennas, the lower the performance of the algorithm and the greater the difference between Improved BGSNet and BGSNet, which suggests that our improvements can make BGSNet better adapted to the MAUE system environment.
7.3.6 Effect of receive correlation
This section analyzes the impact of the correlation between the multiple antennas of the BS on the performance of the algorithm. We have made two comparisons, one is the performance of the algorithm under \(\mathrm {N_t=16,N_r=128,\xi _r=0 \ or \ 0.2,\xi _t=0.4}\) , As shown in Figure 26. One group is the performance of the algorithm under \(\mathrm {N_t=32,N_r=256,\xi _r=0 \ or \ 0.2,\xi _t=0.4}\), as shown in Figure 27. It is found that the greater the correlation between the multiple antennas of the BS, the lower the performance of the algorithm, and the greater the gap between Improved BGSNet and BGSNet. And in this environment, Improved BGSNet and MMSE are very close, so our algorithm can only be applied to low and medium correlations.
7.4 Other performance analysis
7.4.1 Comprehensive analysis of complexity and performance
From Figure 28, Table 2, and Table 3, we can see that when the number of antennas is the same, as the correlation degree increases, the proposed algorithm requires more layers to converge; To achieve the same performance, although BGSNet and Improved BGSNet require more layers than OAMPNet, their required complexity is much lower than OAMPNet. When the number of antennas is increased, the performance of the algorithm should be better, but since the number of individual terminals is changed from 2 to 4, more layers are required to converge.
7.4.2 SER performance with channel estimation error
In the presence of channel estimation errors, the performance of the proposed algorithm in uplink multiuser massive MIMO systems is investigated. The estimated channel gain matrix is given by
where \(\Delta \tilde{{\mathbf {H}} } \in \mathrm {C} ^{\mathrm {N_{r}} \times \mathrm {N_{t}} }\) is the error matrix of the complex Gaussian terms of iid with zero mean and variance \(\sigma _{\epsilon }^{2}\) .
As shown in Figures 29 and 30, when there is a channel estimation error, the Improved BGSNet performance is very close to OAMPNet; With the increase of channel estimation error, the performance of all algorithms decreases, but the proposed detection algorithm still has good SER performance and is more robust to channel estimation error.
7.4.3 SER performance with noise uncertainty
Next, we investigate the effect of noise variance uncertainty on the performance of different DL detectors. It is assumed that the noise variance is unknown in both training and testing phases. Therefore, when evaluating performance on test data, the noise variance is not the same as when training. Suppose the estimated noise variance is \({\hat{\sigma }} ^2=\eta \sigma ^2\). We also define the noise uncertainty factor (NUF) as \(\mathrm {NUF} = 10\mathrm {log} _{10}\eta\).
As can be seen in Figure 31, both MMNetiid and OAMPNet incur considerable performance losses when the estimated noise variance deviates from the true variance.When the estimation of noise variance is inaccurate, the performance gap between OAMPNet and BGSNet, Improved BGSNet becomes more obvious. At the same time, BGSNet and Improved BGSNet are hardly affected by inaccurate estimation of noise variance and have good robustness.
8 Conclusion
We propose a new modeldriven deep learning network for MIMO detection, BGSNet, and build on it with Improved BGSNet. The network is based on GaussSeidel, coupled with a nonlinear activation function, and exhibits excellent performance. The network needs to be optimised with few adjustable parameters and the training process is simple and fast. In this paper, singleantenna user equipment (SAUE) and multipleantenna user equipment (MAUE) systems are considered under Rayleigh channels. Simulation results show that the performance of BGSNet is significantly better than that of the GaussSeidel algorithm; the proposed scheme is suitable for massive MIMO with low complexity, and the performance can be improved by increasing the ratio between the receiving and transmitting antennas; the robustness of BGSNet is good, and the performance is little affected by the variation of the number of antennas; under the MAUE system, the performance of Improved BGSNet is better than that of BGSNet, and both are suitable for lowand mediumcorrelation MAUE systems.
Availability of data and materials
Mostly, I got the writing material from different journals as presented in the references. A python tool has been used to simulate my concept.
Abbreviations
 BS::

Base station
 BGSNet::

Block GaussSeidel network
 MAUE::

Multipleantenna user equipment
 DL::

Deep learning;
 SAUE::

Singleantenna user equipment
 MMSE::

Minimum mean square error
 ZF::

Zero forcing
 MIMO::

Multipleinput multipleout
 B5G::

Beyond 5G
 ML::

Maximum likelihood
 SD::

Spherical decoding
 NS::

Neumann series
 NI::

Newton iterative
 GS::

GaussSeidel
 SOR::

Successive superrelaxation
 JA::

Jacobi
 RI::

Richardson
 CG::

Conjugate gradient
 LA::

Lanczos
 CD::

Coordinate descent
 BP::

Belief propagation
 PDN::

Parallel detection network
 SIC::

Soft interference cancellation
 UE::

User equipment
 MMIMO::

Massive multipleinput multipleout
 QR::

Normal orthogonal matrix Q and upper triangular matrix R
 SER::

Symbol error rate
 SNR::

Signal to noise ratio
 OAMP::

Orthogonal approximate matching tracking
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Acknowledgements
Thanks to the help of the intelligent information processing team and This work is supported by National Natural Science Foundation of China under Grants No.61871238 and No.61771254.
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Haifeng Yao conceived and designed the methods. Haifeng Yao performed the experiments and wrote the paper. Ting Li analyzed the simulation data. Fei Li and Wei Ji gave valuable suggestions on the structure of the paper. Yan Liang and Yunchao Song revised the original manuscript. All authors read and agreed the manuscript.
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Yao, H., Li, T., Song, Y. et al. Lowcomplexity signal detection networks based on GaussSeidel iterative method for massive MIMO systems. EURASIP J. Adv. Signal Process. 2022, 51 (2022). https://doi.org/10.1186/s13634022008850
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DOI: https://doi.org/10.1186/s13634022008850