A Deep Learning-Based Low-Overhead Beam Tracking Scheme for Reconfigurable Intelligent Surface-Aided Multiple-Input and Single-Output Systems with Estimated Channels

Abstract

The next generation network requires not only an ultra-high data rate, global coverage, and connectivity, but also a reduction in network deployment costs and energy consumption. The emergence of reconfigurable intelligent surface (RIS) technology provides an effective way to improve efficiency and reduce cost, while the passive elements bring new challenges of channel estimation (CE) and beam tracking. For an RIS-aided multiple-input and single-output (MISO) system, in this paper, to obtain the channel state information (CSI), we propose a principle component analysis (PCA)-based staged channel estimation method. Based on the estimated channel, we propose a deep learning (DL)-based beam tracking scheme to realize low-complexity RIS reflection coefficient design, which effectively improves the signal-to-noise ratio (SNR) on the user side. The simulation results verified our proposed channel estimation scheme based on PCA, and the beam tracking scheme based on a deep neural network (DNN) for semi-active RIS-aided MISO systems can obtain approximate performances to traditional hand-crafted convex optimization-based algorithms like semi-definite relaxation (SDR) with much lower computational complexity.

1. Introduction

1.1. Motivation

In comparison to the widely utilized fifth-generation (5G) network, the beyond 5G and sixth-generation (6G) networks not only necessitate an ultra-high data rate, global coverage, connectivity, and extremely high reliability and low latency, but also prioritize reducing network deployment costs and energy consumption, while striving for the sustainable development of green mobile communications [1,2,3]. The introduction of RIS technology offers a viable solution for enhancing the efficiency and cutting down the expenses of 6G communications [4]. An RIS consists of a series of passive units that have the ability to independently adjust the amplitude and phase of incoming signals [5]. The enhancement is comparable to the improvement of the attributes of the wireless channel, and it works particularly well in situations where the initial quality of the channel is extremely low (for example, where there are barriers between transceivers) [6]. As a result, the use of an RIS has the potential to enhance wireless system propagation conditions and expand coverage. However, the implementation of an RIS also brings new challenges to the design of wireless communication systems [7]. Estimating and sharing CSI is a difficult task with current RIS architectures, which typically consist of passive elements. Additionally, designing the reflection coefficients of an RIS to achieve optimal beamforming requires significant overheads and computational complexity due to the large number of reflecting elements.

1.2. Related Work

In general, accurate CE is key to the optimization of the reflecting coefficients. Numerous research studies have been conducted on the estimation of channels in communication systems assisted by an RIS [8,9,10,11,12,13,14,15,16,17,18,19,20]. An RIS without RF chains can only achieve channel estimation by adjusting the reflection patterns of the RIS to transmit pilots from the transmitter (Tx) to the receiver (Rx). For instance, in [8,9], the authors utilized the compressive sensing (CS) technique to extract the individual channel of a Tx-RIS and RIS-Rx from the combined CSI. In [10,11,12], the authors proposed CE schemes based on the message passing technique with sparse matrix completion and the factorization method. In [13], with the hierarchical search codebook design, the author introduced a CE scheme with low complexity. In addition, a combined scheme of least-squares (LS) and linear minimum mean square error (LMMSE) channel estimation is suggested in [14]. A pilot training-based multi-round sub-channel estimation method [15] and a data-driven non-linear solution based on deep learning [16] can also be utilized to execute the CE in RIS-aided systems, which requires, however, large training periods. From the existing literature, there is a gap regarding practical channel estimation in a communication system with a passive RIS.

In contrast, for the RIS that is furnished with active components such as radio frequency (RF) chains [17,18,19,20], the estimation of Tx-RIS and RIS-Rx CSI can be achieved by receiving pilot signals from both the base station (BS) and users, allowing for the application of traditional CE schemes. For example, in [19], the researchers suggested a new design consisting of passive unit components and only one active RF chain (a semi-active RIS), providing a balance between the cost and effectiveness of CE.

For beam tracking, RIS are commonly used in wireless communication systems to dynamically adjust the reflection coefficient of the beams for improved signal reception. Therefore, the design of the reflection coefficient is generally aimed at maximizing the signal-to-noise ratio and the achievable rate. To maximize the downlink achievable rate of the user terminal (UT), an alternatively optimization algorithm is proposed in [21], which can obtain a suboptimal solution. Moreover, the performance of the MISO system assisted by an RIS has been compared with that of the traditional MISO system, and it has been demonstrated that a reasonable design of the reflection coefficient of the RIS can maximize the system capacity [22]. In [23], the optimization of beamforming for both the BS and RIS is jointly performed using SDR to minimize the transmission power of the BS, with the introduction of a discrete phase-shift constraint in [24]. Nevertheless, the computational complexity of these methods remains high and the optimization schemes need to be redesigned when considering the heterogeneous RIS systems.

Recently, the deep learning (DL) method has attracted wide interest in wireless communication [25,26]. Some researches have proved that introducing DL into beamforming design would reduce computing complexity and improve the performance of the communication system [27,28,29,30,31,32,33,34]. For example, in [29], the authors investigated the joint design of a transmit beamforming matrix at the base station and a phase shift matrix at the fully passive RIS with deep reinforcement learning (RL). Moreover, two deep RL schemes were developed to jointly optimize the beamforming vectors and phase shifts of the fully passive RIS elements with the channel uncertainties and quality of service constraints in a non-orthogonal multiple access (NOMA) system [31,32]. Furthermore, the authors in [33,34] proposed machine learning-based channel estimation techniques for the single-input-single-output (SISO) system with a nonfully passive RIS, i.e., an RIS with a few active sensing elements to help with channel estimation.

1.3. Our Contributions

In this paper, we consider both the CE and beam tracking of a semi-active RIS-aided MISO system in a practical manner. As far as we know, this scenario has not been investigated. The specific contributions of this paper are summarized as follows:

(1)

Utilizing a semi-active RIS architecture that consists of passive unit elements and only one active RF chain, we propose a PCA-based CE scheme to estimate the CSI of the RIS-BS, the RIS-UT, and the UT-BS. The proposed CE scheme allows a trade-off between cost and performance;

(2)

Based on the estimated channel, we propose a DNN to realize low-complexity beam tracking of a semi-active RIS-aided system, which effectively improves the SNR on the UT side. Our simulation results verified the accuracy of the proposed schemes on beam tracking for semi-active RIS-aided MISO systems.

The rest of this paper is organized as follows. Section 2 describes the semi-active RIS-aided MISO system model and formulates the constrained sum-rate maximization problem. The proposed PCA-based CE method is developed in Section 3, while the DL-based beam tracking scheme is presented in Section 4. In Section 5, we analyze the results of our simulation experiments, while Section 6 concludes the paper.

In the following, we outline the notations used. We use $\mathbb{C}$ to denote the complex domain. The small bold font is utilized to denote a vector, while the large bold font is employed to indicate a matrix. Let ${\left(\mathbf{X}\right)}^T$, ${\left(\mathbf{X}\right)}^{*}$, ${\left(\mathbf{X}\right)}^H$ denote the transpose, conjugate, and conjugate transpose of a matrix $\mathbf{X}$, respectively. $\mathit{a}$ represents a vector, $\mathbb{E}\{·\}$ represents the expectation, and diag$(·)$ denotes diagonalization. Let $Re(·)$ and $Im(·)$ denote the real and imaginary parts of complex numbers, respectively. $vec(·)$ denotes the vector mapping operation.

2. System Model and Problem Formulation

2.1. System Model

As illustrated in Figure 1, we consider an RIS-aided MISO system. In this system, the BS and the RIS are equipped with $\sqrt{M}\times \sqrt{M}$ antennas and $\sqrt{N}\times \sqrt{N}$ reflecting elements, respectively. The UT is equipped with a single antenna. Moreover, the distance between the adjacent antennas of the BS is $d_{BS}$. The reflecting element spacing of the RIS is $d_{RIS}$.

The received signal y of the UT is given by

$$y=({\mathit{H}}_{RU}\mathit{\varphi }{\mathit{H}}_{BR}+{\mathit{H}}_{BU})\mathit{b}x+w=\mathit{H}x+w,$$

(1)

where ${\mathbf{H}}_{BR}\in {\mathbb{C}}^{M\times N}$ denotes the channel between the BS and RIS, ${\mathbf{H}}_{RU}\in {\mathbb{C}}^{1\times M}$ is the channel between the RIS and UT, ${\mathbf{H}}_{BU}\in {\mathbb{C}}^{1\times N}$ is the channel between the BS and UT, $\mathit{b}$ is the beamforming vector with $N\times 1$ dimension, $\mathit{\varphi }$ denotes the reflection coefficient matrix with $M\times M$ dimensions, x is the complex data symbol intended for the UT, and $w\sim \mathcal{CN}(0,{\sigma }^2)$ is the additive white Gaussian noise (AWGN), which has zero mean and variance ${\sigma }^2$.

Then, based on the angular domain channel model, ${\mathbf{H}}_{BR}$ can be expressed as

$${\mathbf{H}}_{BR}=\sum _{p_{BR}=1}^{P_{BR}}{\beta }_{BR}^{p_{BR}}vec\left(\mathbf{a}\left(u_{RIS}^{p_{BR}}\right){\mathit{b}}^T\left(v_{RIS}^{p_{RB}}\right)\right)vec{\left(\mathbf{a}\left(u_{BS}^{p_{BR}}\right){\mathit{b}}^T\left(v_{BS}^{p_{BR}}\right)\right)}^T,$$

(2)

where $p_{BR}$ denotes the resolvable paths between the BS and RIS, ${\beta }_{BR}^{p_{BR}}$ is the path loss between the BS and RIS, $vec(·)$ denotes vectorization, and $\mathbf{a}\left(u_{BS}^{p_{BR}}\right){\mathit{b}}^T\left(v_{BS}^{p_{BR}}\right)$ and $\mathbf{a}\left(u_{RIS}^{p_{BR}}\right){\mathit{b}}^T\left(v_{RIS}^{p_{BR}}\right)$ denote the array manifold vector of the BS and RIS, respectively. Furthermore, these parameters can be expanded as follows:

$$\begin{array}{cc}\hfill & \mathbf{a}\left(u_{BS}^{p_{BR}}\right)={[1,exp\left\{j2\pi u_{BS}^{p_{BR}}\right\},\cdots ,exp\left\{j2\pi (\sqrt{N}-1)u_{BS}^{p_{BR}}\right\}]}^T,\hfill \end{array}$$

(3)

$$\begin{array}{cc}\hfill & {\mathit{b}}^T\left(v_{BS}^{p_{BR}}\right)={\left[1,exp\left\{j2\pi v_{BS}^{p_{BR}}\right\},\cdots ,exp\left\{j2\pi (\sqrt{N}-1)v_{BS}^{p_{BR}}\right\}\right]}^T,\hfill \end{array}$$

(4)

$$\begin{array}{cc}\hfill & \mathbf{a}\left(u_{RIS}^{p_{BR}}\right)={[1,exp\left\{j2\pi u_{BS}^{p_{BR}}\right\},\cdots ,exp\left\{j2\pi (\sqrt{M}-1)u_{BS}^{p_{BR}}\right\}]}^T,\hfill \end{array}$$

(5)

$$\begin{array}{cc}\hfill & {\mathit{b}}^T\left(v_{RIS}^{p_{BR}}\right)={\left[1,exp\left\{j2\pi v_{BS}^{p_{BR}}\right\},\cdots ,exp\left\{j2\pi (\sqrt{M}-1)v_{BS}^{p_{BR}}\right\}\right]}^T,\hfill \end{array}$$

(6)

where

$$\begin{array}{cc}\hfill & u_{BS}^{p_{BR}}={\displaystyle \frac{d_{BS}}{\lambda }}cos{\theta }_{BS}^{p_{BR}}\mathrm{and}{v}_{BS}^{p_{BR}}={\displaystyle \frac{d_{BS}}{\lambda }}sin{\theta }_{BS}^{p_{BR}}cos{\mathrm{\Phi }}_{BS}^{p_{BR}},\hfill \end{array}$$

(7)

$$\begin{array}{cc}\hfill & u_{RIS}^{p_{BR}}={\displaystyle \frac{d_{RIS}}{\lambda }}cos{\theta }_{RIS}^{p_{BR}}\mathrm{and}{v}_{RIS}^{p_{BR}}={\displaystyle \frac{d_{RIS}}{\lambda }}sin{\theta }_{RIS}^{p_{BR}}cos{\mathrm{\Phi }}_{RIS}^{p_{BR}}.\hfill \end{array}$$

(8)

Similarly, the channel ${\mathbf{H}}_{RU}$ and ${\mathbf{H}}_{BU}$ can be expressed as

$$\begin{array}{cc}\hfill & {\mathbf{H}}_{RU}=\sum _{p_{RU}=1}^{p_{RU}=P_{RU}}{\beta }_{RU}^{p_{RU}}vec\left(\mathbf{a}\left(u_{RU}^{p_{RU}}\right){\mathit{b}}^T\left(v_{RU}^{p_{RU}}\right)\right),\hfill \end{array}$$

(9)

$$\begin{array}{cc}\hfill & {\mathbf{H}}_{BU}=\sum _{p_{BU}=1}^{p_{BU}=P_{BU}}{\beta }_{BU}^{p_{BU}}vec\left(\mathbf{a}\left(u_{BU}^{p_{BU}}\right){\mathit{b}}^T\left(v_{BU}^{p_{BU}}\right)\right).\hfill \end{array}$$

(10)

From (2), (9), and (10), upon observation, it is evident that the RIS-aided MISO channel is closely linked to the angles of arrival and path loss. In the subsequent section, we present a staged approach for channel estimation.

2.2. Problem Formulation

Based on (1), the received signal of the UT can be written as

$$y=({\mathbf{H}}_{RU}\mathit{\varphi }{\mathbf{H}}_{BR}+{\mathbf{H}}_{BU})\mathit{b}x+w,$$

(11)

where $\mathit{\varphi }$ is a $M\times M$ diagonal matrix, and the i-th diagonal element of $\mathit{\varphi }$ is ${\varphi }_i=e^{j2\pi p_i},i\in [0,M-1]$, for any i, $p_i\in [0,1)$.

The SNR of the received signal can be expressed as

$$\epsilon ={\displaystyle \frac{|({\mathbf{H}}_{RU}\mathit{\varphi }{\mathbf{H}}_{BR}+{\mathbf{H}}_{BU}){\mathit{b}|}^2}{{\sigma }^2}}.$$

(12)

According to [35], we realize that for a fixed reflection coefficient matrix $\mathit{\varphi }$, the beamforming matrix $\mathit{b}$ that maximizes the received SNR is as follows

$${\mathit{b}}^{*}=\sqrt{P_{max}}{\displaystyle \frac{({\mathbf{H}}_{RU}\mathit{\varphi }{\mathbf{H}}_{BR}+{\mathbf{H}}_{BU})}{\left|\right|({\mathbf{H}}_{RU}\mathit{\varphi }{\mathbf{H}}_{BR}+{\mathbf{H}}_{BU})\left|\right|}},$$

(13)

where $\sqrt{P_{max}}$ denotes the maximum transmitting power of the base station.

Next, the problem of optimizing the reflection coefficient matrix $\mathit{\varphi }$ is given by

$$\begin{array}{c}\hfill \begin{array}{cc}& \underset{\mathit{\varphi }}{max}\left|\right|({\mathbf{H}}_{RU}\mathit{\varphi }{\mathbf{H}}_{BR}+{\mathbf{H}}_{BU}){\left|\right|}^2\hfill \\ & s.t.|{\varphi }_i|=1,\forall 0<=i<M.\hfill \end{array}\end{array}$$

(14)

The optimization problem referenced as (14) presents significant challenges in terms of a solution due to its classification as a typical NP-hard problem, compounded by the difficulty in estimating CSI. In the subsequent section, we will introduce a channel estimation scheme based on PCA to achieve precise CSI for the RIS-aided system.

3. The Proposed PCA-Based Channel Estimation Algorithm

3.1. Channel Estimation

As previously demonstrated, we are examining an RIS architecture that includes passive unit elements and a solitary active RF chain, as depicted in Figure 2. This lone RF chain facilitates baseband channel estimation using pilot signals and consists of a low noise amplifier, a mixer for downconverting the signal from RF to baseband, and an analog to digital converter.

We consider that all reflecting elements in the RIS are turned off. Therefore, (1) can be rewritten as

$$y={\mathbf{H}}_{BU}x+\mathbf{w}=\mathbf{H}x+\mathbf{w}.$$

(15)

According to the formula (15), we can find that the entire system is consistent with conventional massive MISO systems. A CE algorithm utilizing PCA is then introduced for the adaptive CE of ${\mathbf{H}}_{BU}$. Before formally introducing the channel estimation algorithm based on PCA, we first introduce the relevant knowledge of matrix decomposition as the precondition of the research.

3.2. SVD Operation

One common method for obtaining the channel matrix from the received signal is to use the singular value decomposition (SVD) operation to decompose the received signal matrix. This can be represented as:

$$\mathbf{Y}=\mathbf{S}\mathsf{\Sigma }{\mathbf{D}}^H,$$

(16)

where $\mathbf{S}$ and $\mathbf{D}$ are referred to as singular vectors, which are unitary matrices with $M\times M$ and $N\times N$ dimensions, respectively. $\mathrm{\Sigma }$ is called a singular value, which is a diagonal matrix with $M\times N$ dimensions. In addition, we assume that the $\mathrm{\Sigma }$ is sorted in descending order.

Based on the theory of array signal processing, matrices $\mathbf{S}$ and $\mathbf{D}$ are associated with the angles of arrival (DoA). However, the computational complexity of performing the SVD operation is quite high. As a result, a channel estimation algorithm based on PCA with low complexity is utilized for estimating matrices $\mathbf{S}$ and $\mathbf{D}$.

3.3. The PCA-Based Channel Estimation Algorithm

Based on the analysis in the previous section, estimating the DoA angles involves estimating matrices $\mathbf{S}$ and $\mathbf{D}$. The low complexity estimates of these matrices are primarily obtained through covariance matrix estimation. For simplicity, we will focus on recovering matrix $\mathbf{S}$ first. The correlation covariance matrix can be expressed as follows:

$$\begin{array}{c}\hfill {\mathbf{R}}_1=\mathbb{E}\left\{\mathbf{Y}{\mathbf{Y}}^H\right\}=\sum _{i=1}^P{\mathbf{H}}_i{\psi }_i{\mathbf{H}}_i^H+{\sigma }^2\mathbf{I},\end{array}$$

(17)

where ${\mathbf{R}}_1$ is covariance matrix, ${\psi }_i=\mathbb{E}\left\{x{x}^{*}\right\}$. Then, introducing EVD operations for the analysis, (17) can be rewritten as

$${\mathbf{R}}_1={\mathbf{S}}_P{\mathsf{\Lambda }}_s{\mathbf{S}}_P^H+{\mathbf{S}}_z{\mathsf{\Lambda }}_{z_1}{\mathbf{S}}_z^H,$$

(18)

where the matrix ${\mathbf{S}}_P$ consists of eigenvectors corresponding to eigenvalues, ${\lambda }_1,\dots ,{\lambda }_P$. The matrix ${\mathsf{\Lambda }}_s$ is a diagonal matrix with entries from ${\lambda }_1$ to ${\lambda }_P$. The matrix ${\mathbf{S}}_z$ is composed of eigenvectors corresponding to eigenvalues ${\lambda }_{P+1},\dots ,{\lambda }_M$ and the matrix ${\mathsf{\Lambda }}_{z_1}$ is a diagonal matrix with entries of eigenvalues from ${\lambda }_{P+1}$ to ${\lambda }_M$.

An adaptive iterative method based on PCA is proposed in this section to estimate matrix $\mathbf{S}$ with low computational complexity. Initially, let us consider the estimation on ${\mathbf{S}}_P$ and assume that ${\mathbf{T}}_1$ represents the estimation on ${\mathbf{S}}_P$. Then, according to [36], we can define the objective function as follows:

$$\begin{array}{cc}\hfill \underset{{\mathbf{T}}_1}{min}J\left({\mathbf{T}}_1\right)& =tr\left({\mathbf{C}}_L\right)-2tr\left(\mathsf{\Omega }{\mathbf{T}}_1^H{\mathbf{C}}_L{\mathbf{T}}_1\right)\hfill \\ \hfill & +tr\left(\mathsf{\Omega }{\mathbf{T}}_1^H{\mathbf{C}}_L{\mathbf{T}}_1{\mathbf{T}}_1^H{\mathbf{T}}_1\right),\hfill \end{array}$$

(19)

where ${\mathbf{C}}_L=\mathbb{E}\left\{{\mathit{Y}}_k{\mathit{Y}}_k^H\right\}$, $\mathsf{\Omega }$ is a diagonal matrix of the singular values, denoted as $w_{i,i}$. Based on the analysis in [36], the estimated value of ${\mathbf{T}}_1$ can be close to ${\mathbf{S}}_P$ if the diagonal elements of $\mathsf{\Omega }$ satisfy $w_{1,1}>w_{2,2}>·>w_{P,P}>0$.

Problem (19) can be worked out by the gradient descent method; the update rule of ${\mathbf{T}}_1$ is given by

$${\mathbf{T}}_{1,j}={\mathbf{T}}_{1,j-1}-\eta {\displaystyle \frac{\partial J\left({\mathbf{T}}_1\right)}{\partial {\mathbf{T}}_1}}{|}_{{\mathbf{T}}_1={\mathbf{T}}_{1,j-1}}.$$

(20)

The gradient of $J\left({\mathbf{T}}_1\right)$ is as follows:

$$\begin{array}{cc}\hfill {\displaystyle \frac{\partial J\left({\mathbf{T}}_1\right)}{\partial {\mathbf{T}}_1}}& =-(4{\mathbf{C}}_L{\mathbf{T}}_1\mathsf{\Omega }-{\mathbf{C}}_L{\mathbf{T}}_1\mathsf{\Omega }{\mathbf{T}}_1^H{\mathbf{T}}_1-{\mathbf{T}}_1{\mathbf{T}}_1^H{\mathbf{C}}_L{\mathbf{T}}_1\mathsf{\Omega }\hfill \\ \hfill & -{\mathbf{C}}_L{\mathbf{T}}_1{\mathbf{T}}_1^H{\mathbf{T}}_1\mathsf{\Omega }-{\mathbf{T}}_1\mathsf{\Omega }{\mathbf{T}}_1^H{\mathbf{C}}_L{\mathbf{T}}_1).\hfill \end{array}$$

(21)

Then, (22) can be rewritten as based on (21):

$$\begin{array}{cc}\hfill & {\mathbf{T}}_{1,j}={\mathbf{T}}_{1,j-1}+\eta (4{\mathbf{C}}_L{\mathbf{T}}_{1,k-1}\mathsf{\Omega }-{\mathbf{C}}_L{\mathbf{T}}_{1,j-1}\mathsf{\Omega }{\mathbf{T}}_1^H{\mathbf{T}}_{1,j-1}\hfill \\ \hfill & -{\mathbf{T}}_{1,j-1}{\mathbf{T}}_{1,j-1}^H{\mathbf{C}}_L{\mathbf{T}}_{1,j-1}\mathsf{\Omega }-{\mathbf{C}}_L{\mathbf{T}}_{1,j-1}{\mathbf{T}}_{1,j-1}^H{\mathbf{T}}_{1,j-1}\mathsf{\Omega }\hfill \\ \hfill & -{\mathbf{T}}_{1,j-1}\mathsf{\Omega }{\mathbf{T}}_{1,j-1}^H{\mathbf{C}}_L{\mathbf{T}}_{1,j-1}).\hfill \end{array}$$

(22)

Ref. [36] proves that ${\mathbf{T}}_1$ converges to ${\mathbf{S}}_P$ when j tends to infinity; therefore, the estimate of ${\mathbf{S}}_P$ can be achieved based on (22). However, the covariance matrix ${\mathbf{C}}_L$ in (22) is essential for the update of ${\mathbf{T}}_1$. Since it is difficult to obtain the ${\mathbf{C}}_L$ in the actual process, a sampling covariance matrix proposed in [37] is used for approximate estimation, which is given by

$${\widehat{\mathbf{C}}}_{L,j}={\displaystyle \frac{1}{j}}\sum _{k=0}^j{\tau }^{j-i}{\mathbf{Y}}_k{\mathbf{Y}}_k^H=\tau {\displaystyle \frac{j-1}{j}}{\widehat{\mathbf{C}}}_{L,j-1}+{\displaystyle \frac{1}{j}}{\mathbf{Y}}_j{\mathbf{Y}}_j^H,$$

(23)

where $\tau \in [0,1]$ denotes the forgetting factor. The forgetting factor can provide better performance when dealing with adaptive signal processing problems in unstable environments. An adaptive forgetting factor balances the weight of the covariance matrices ${\mathbf{C}}_L$ and ${\mathbf{Y}}_{\mathbf{k}}{\mathbf{Y}}_{\mathbf{k}}^{\mathbf{H}}$. When the channel changes rapidly, the value of the forgetting factor is increased so that the earlier data take up a smaller proportion. On the contrary, when the channel changes slowly, the value of the forgetting factor is reduced to better retain the past information. Therefore, it is necessary to adjust the forgetting factor value adaptively according to the statistical characteristics of channel changes [37]. When $\tau =1$, the channel can be thought of as stationary, and (23) can be thought of as expectation of ${\mathbf{Y}}_{\mathbf{k}}{\mathbf{Y}}_{\mathbf{k}}^{\mathbf{H}}$. Then, combined with (23), the update rule of ${\mathbf{T}}_1$ can be rewritten as

$$\begin{array}{cc}\hfill & {\mathbf{T}}_{1,k}={\mathbf{T}}_{1,k-1}+\eta (4{\widehat{\mathbf{C}}}_{1,k-1}{\mathbf{T}}_{1,k-1}\mathsf{\Omega }-{\widehat{\mathbf{C}}}_{L,k-1}{\mathbf{T}}_{1,k-1}\mathsf{\Omega }{\mathbf{T}}_1^H{\mathbf{T}}_{1,k-1}\hfill \\ \hfill & -{\mathbf{T}}_{1,k-1}{\mathbf{T}}_{1,k-1}^H{\widehat{\mathbf{C}}}_{1,k-1}{\mathbf{T}}_{1,k-1}\mathsf{\Omega }-{\widehat{\mathbf{C}}}_{1,k-1}{\mathbf{T}}_{1,k-1}{\mathbf{T}}_{1,k-1}^H{\mathbf{T}}_{1,k-1}\mathsf{\Omega }\hfill \\ \hfill & -{\mathbf{T}}_{1,k-1}\mathsf{\Omega }{\mathbf{T}}_{1,k-1}^H{\widehat{\mathbf{C}}}_{1,k}{\mathbf{T}}_{1,k-1}).\hfill \end{array}$$

(24)

It is worth noting that the step size $\eta$ is a necessary condition to achieve fast convergence and a high accuracy in estimating ${\mathbf{T}}_1$. To speed up the convergence, we propose a method for calculating the optimal step size.

To obtain the optimal $\eta$, we consider the expression of the optimization function $J\left({\mathbf{T}}_1\right)$ at the $(k+1)th$ iteration

$$\begin{array}{cc}\hfill J\left({\mathbf{T}}_{1,k+1}\right)& =tr\left({\mathbf{R}}_{1,k+1}\right)-2tr\left(\mathsf{\Omega }{\mathbf{T}}_{1,k+1}^H{\mathbf{R}}_{1,k+1}{\mathbf{T}}_{1,k+1}\right)\hfill \\ \hfill & +tr\left(\mathsf{\Omega }{\mathbf{T}}_{1,k+1}^H{\mathbf{R}}_{1,k+1}{\mathbf{T}}_{1,k+1}{\mathbf{T}}_{1,k+1}^H{\mathbf{T}}_{1,k+1}\right).\hfill \end{array}$$

(25)

To simplify subsequent formulas, we use ∇ to replace ${\displaystyle \frac{\partial J\left({\mathbf{T}}_1\right)}{\partial {\mathbf{T}}_1}}$. Then, by combining (22) with (25), (25) can be rewritten as

$$\begin{array}{cc}\hfill & J({\mathbf{T}}_{1,k+1},\eta )=\hfill \\ \hfill & tr\left({\widehat{\mathbf{C}}}_{L,k+1}\right)-2tr\left[\mathsf{\Omega }{({\mathbf{T}}_{1,k}-\eta \nabla )}^H{\widehat{\mathbf{C}}}_{L,k+1}({\mathbf{T}}_{1,k}-\eta \nabla )\right]+\hfill \\ \hfill & tr\left[\mathsf{\Omega }{({\mathbf{T}}_{1,k}-\eta \nabla )}^H{\widehat{\mathbf{C}}}_{L,k+1}({\mathbf{T}}_{1,k}-\eta \nabla ){({\mathbf{T}}_{1,k}-\eta \nabla )}^H({\mathbf{T}}_{1,k}-\eta \nabla )\right].\hfill \end{array}$$

(26)

Considering that ${\mathbf{T}}_{1,k+1}^H{\mathbf{T}}_{1,k+1}={\mathbf{U}}^H{\mathbf{S}}_P^H{\mathbf{S}}_P\mathbf{U}=\mathbf{I}$, (26) can be rewritten as

$$\begin{array}{cc}\hfill J({\mathbf{T}}_{1,k+1},\eta )& =tr\left({\widehat{\mathbf{C}}}_{L,k+1}\right)-tr\left[\mathsf{\Omega }\right({\mathbf{T}}_{1,k}-\hfill \\ \hfill & {\eta \nabla )}^H{\widehat{\mathbf{C}}}_{L,k+1}({\mathbf{T}}_{1,k}-\eta \nabla )].\hfill \end{array}$$

(27)

We find that $J\left({\mathbf{T}}_{1,k+1}\right)$ obtains the global maximum when step size is ${\eta }_{opt}$. Then, the partial derivative of $J\left({\mathbf{T}}_{1,k+1}\right)$ with respect to ¦Ç can be written as

$$\begin{array}{cc}\hfill & {\displaystyle \frac{\partial J\left({\mathbf{T}}_{1,k+1},\eta \right)}{\partial \eta }}={\displaystyle \frac{\partial }{\partial \eta }}\{-2tr\left[\mathsf{\Omega }{({\mathbf{T}}_{1,k}-\eta \nabla )}^H{\widehat{\mathbf{C}}}_{L,k+1}({\mathbf{T}}_{1,k}-\eta \nabla )\right]+\hfill \\ \hfill & tr\left[\mathsf{\Omega }{({\mathbf{T}}_{1,k}-\eta \nabla )}^H{\widehat{\mathbf{C}}}_{L,k+1}({\mathbf{T}}_{1,k}-\eta \nabla ){({\mathbf{T}}_{1,k}-\eta \nabla )}^H({\mathbf{T}}_{1,k}-\eta \nabla )\right]\}\hfill \\ \hfill & =-2tr\left[\mathsf{\Omega }{(-\nabla )}^H{\widehat{\mathbf{C}}}_{1,k+1}({\mathbf{T}}_{1,k}-\eta \nabla )\right]\hfill \\ \hfill & -2tr\left[\mathsf{\Omega }{({\mathbf{T}}_{1,k}-\nabla )}^H{\widehat{\mathbf{C}}}_{L,k+1}(-\eta \nabla )\right]\hfill \\ \hfill & =2tr\left[\mathsf{\Omega }{\nabla }^H{\widehat{\mathbf{C}}}_{L,k+1}{\mathbf{T}}_{1,k}\right]-2\eta tr[\mathsf{\Omega }{\nabla }^H{\widehat{\mathbf{C}}}_{L,k+1}\nabla ]\hfill \\ \hfill & +2tr[\mathsf{\Omega }{\mathbf{T}}_{1,k}^H{\widehat{\mathbf{C}}}_{L,k+1}\nabla ]-tr[\mathsf{\Omega }{\nabla }^H{\widehat{\mathbf{C}}}_{L,k+1}\nabla ].\hfill \end{array}$$

(28)

   When ${\displaystyle \frac{\partial J\left({\mathbf{T}}_{1,k+1}\right)}{\partial \eta }}=0$, $J\left({\mathbf{T}}_{1,k+1}\right)$ obtains the global maximum. Then, ${\eta }_{opt}$ is given by

$${\eta }_{opt}={\displaystyle \frac{tr\left[\mathsf{\Omega }{\nabla }^H{\widehat{\mathbf{C}}}_{L,k+1}{\mathbf{T}}_{1,k}\right]+tr[\mathsf{\Omega }{\mathbf{T}}_{1,k}^H{\widehat{\mathbf{C}}}_{L,k+1}\nabla ]}{2tr[\mathsf{\Omega }{\nabla }^H{\widehat{\mathbf{C}}}_{L,k+1}\nabla ]}}.$$

(29)

Based on Equation (29), it is possible to obtain the best update rule for estimating the target matrix ${\mathbf{S}}_P$. Using a similar approach, it is also feasible to retrieve ${\mathbf{D}}_P$.

After the estimation of ${\mathbf{S}}_P$ and ${\mathbf{D}}_P$ is carried out, the MUSIC method is suggested for estimating $u_{BU}$ and $v_{BU}$ in (10). Then, the pseudospectrum with respect to $u_{BU}$ based on the MUSIC method is given by

$$f\left(u\right)={\displaystyle \frac{1}{{\displaystyle \sum _{k=P+1}^M}|\mathbf{a}{\left(u_i\right)}^H{\mathbf{s}}_{z,k}{|}^2}}={\displaystyle \frac{1}{{\mathbf{p}}^H\left(\zeta \right){\mathbf{S}}_z{\mathbf{S}}_z^H\mathbf{p}\left(\zeta \right)}},$$

(30)

where $\mathbf{p}\left(\zeta \right)={\left[1,\zeta ,\dots ,{\zeta }^{(M-1)}\right]}^T$. Furthermore, ${\mathbf{S}}_z{\mathbf{S}}_z^H$ can be obtained by ${\mathbf{S}}_P$, which is given by

$${\mathbf{S}}_z{\mathbf{S}}_z^H=\mathbf{I}-{\mathbf{S}}_P{\mathbf{S}}_P^H.$$

(31)

Substituting (31) into (30), the estimation of ${\mathbf{u}}_{BU}$, $\forall 1\le i\le P$ can be obtained by identifying the peaks of the pseuospectrum in (30). Similarly, the same approach can be utilized to estimate $v_{BU}$ from ${\mathbf{D}}_P$.

After estimating $u_{BU}$ and $v_{BU}$ from ${\mathbf{S}}_P$ and ${\mathbf{D}}_P$, respectively, it is necessary to match the estimated $u_{BU}$ and $v_{BU}$. Therefore, we first define projection operation as

$$P_r=<\mathbf{J}|\mathbf{K}>=trace\left({\mathbf{J}}^H\mathbf{K}\right).$$

(32)

Let J and K represent two matrices. Using the estimated values of $u_{BU}$ and $v_{BU}$, we can express the projection of the received data matrix $\mathbf{Y}$ onto the steering matrix ${\mathbf{A}}_i(u_i,v_j)$.

$$\begin{array}{ccc}\hfill P_r& =& <{\mathbf{A}}_i(u_i,v_j)|\mathbf{Y}>\hfill \\ & =& trace\left({\mathbf{A}}_i^H(u_i,v_j)\mathbf{Y}\right)\hfill \\ & =& trace\left(\mathit{b}{\left(v_j\right)}^{*}{\mathbf{a}}^H\left(u_i\right)\mathbf{Y}\right)\hfill \\ & =& {\mathbf{a}}^H\left(u_i\right)\mathbf{Y}\mathit{b}{\left(v_j\right)}^{*},\hfill \end{array}$$

(33)

where $i,j=1,2,3\dots P$. It is evident that the maximum value of Pr will only be reached when $j=i$, as the received signal at the BS primarily travels through the P resolvable paths. In other words, the contribution of received signals from P paths has the greatest impact on the energy received at the BS. Therefore, by identifying the maximum value of Pr in (34), $u_{BU}$ can be matched with $v_{BU}$. Following the estimation and matching of $u_{BU}$ and $v_{BU}$, the corresponding DoA angles can be determined using the following formula:

$$\begin{array}{ccc}& & {\theta }_i=arccos\left({\displaystyle \frac{u_i}{\tau }}\right),\forall 1\le i\le P,\hfill \end{array}$$

(34)

$$\begin{array}{ccc}& & {\varphi }_i=arccos\left({\displaystyle \frac{v_i}{\tau \sqrt{1-{\left(\frac{u_i}{\tau }\right)}^2}}}\right),\forall 1\le i\le P,\hfill \end{array}$$

(35)

where $\tau ={\displaystyle \frac{d}{\lambda }}$.

3.4. Staged Channel Estimation Scheme

Utilizing the PCA algorithm introduced in the preceding section, channel estimation between the UT and the base station in the first stage can be achieved. In the second stage, we consider turning on the RIS, and the UT side sends a signal to the RIS. Because of the reciprocity of the TDD system channel, the received signal on the RIS can be expressed as

$$\begin{array}{cc}\hfill {\mathbf{Y}}_{RIS}& ={\mathbf{H}}_{RU}^{T}x+\mathbf{W}\hfill \\ \hfill & =\sum _{p_{RU}=1}^{p_{RU}=P_{UR}}{\beta }_{RU}^{p_{RU}}vec{\left(\mathbf{a}\left(u_{RU}^{p_{RU}}\right){\mathit{b}}^T\left(v_{RU}^{p_{RU}}\right)\right)}^{T}x+\mathbf{W}.\hfill \end{array}$$

(36)

According to (36), it is not difficult to find that the channel estimation between the UT and the RIS on the RIS side is similar to the channel estimation between the UT and the base station on the base station side. Therefore, the PCA-based channel estimation algorithm proposed in previous section can be used to recover the channel ${\mathbf{H}}_{RU}$ by estimating the DoA angles.

During the third phase, we take into account the transmission of a signal from the BS to the RIS. The signal received by the RIS can be expressed as follows:

$$\begin{array}{cc}\hfill & {\mathbf{Y}}_{RISB}={\mathbf{H}}_{BR}x+\mathbf{W}=W+\hfill \\ \hfill & \sum _{p_{BR}=1}^{p_{BR}=P_{BR}}{\beta }_{BR}^{p_{BR}}vec\left(\mathbf{a}\left(u_{BS}^{p_{BR}}\right){\mathit{b}}^T\left(v_{BS}^{p_{BR}}\right)\right)vec{\left(\mathbf{a}\left(u_{RIS}^{p_{BR}}\right){\mathit{b}}^T\left(v_{RIS}^{p_{RB}}\right)\right)}^T.\hfill \end{array}$$

(37)

According to (37) and (3)–(6), it can be known that the recovery of the channel ${\mathbf{H}}_{RB}$ is based on the estimation of the DoA angles. As a result of receiving signals from three neighboring antennas on the base station, the signal received on the RIS side can be represented as

$$\begin{array}{cc}\hfill {\mathbf{Y}}_{RISB}^{a_1}& ={\mathbf{H}}_{BR}x+\mathbf{W}\hfill \\ \hfill & =\sum _{p_{BR}=1}^{p_{BR}=P_{BR}}{\beta }_{BR-a1}^{p_{BR}}vec\left(\mathbf{a}\left(u_{RIS}^{p_{BR}}\right){\mathit{b}}^T\left(v_{RIS}^{p_{RB}}\right)\right)+W,\hfill \end{array}$$

(38)

where $a_1$ represents the signal transmitted by the first antenna on the base station. According to (38), we can obtain the estimate of $u_{RIS}$ and $v_{RIS}$. Then, selecting the antennas $a_2$ and $a_3$ adjacent to the first antenna in the horizontal and vertical directions, the path gain between BS and RIS has the following relationship:

$$\begin{array}{ccc}\hfill {\beta }_{BR-a2}& =& {\beta }_{BR-a1}exp\left\{j2\pi u_{BS}\right\},\hfill \end{array}$$

(39)

$$\begin{array}{ccc}\hfill {\beta }_{BR-a3}& =& {\beta }_{BR-a1}exp\left\{j2\pi v_{BS}\right\}.\hfill \end{array}$$

(40)

We can achieve the estimate of $u_{BS}$ and $v_{BS}$ based on (39) and (7). Then, the channel $H_{BR}$ can be recovered based on $u_{BS}$ and $v_{BS}$.

4. The Proposed RIS Beam Tracking Scheme

For the proposed optimization problem, in this section, we will introduce two RIS beam tracking schemes, i.e., the SDR-based algorithm and the DL-based scheme. As an RIS usually consists of a large number of elements, the design of reflection coefficients is usually hard and complex. To avoid the problem that the performance of traditional high complexity algorithms cannot be guaranteed during the solution process, a design scheme of the reflection coefficient based on DL is proposed in this section.

4.1. The SDR-Based Algorithm

We introduce the auxiliary variables $\mathbf{\Theta }=diag\left({\mathbf{H}}_{RU}\right){\mathbf{H}}_{BR}$, $\mathbf{v}={[v_1,\dots ,v_i,\dots ,v_M]}^H$, and $v_n=e^{j2\pi p_i},n\in [1,M]$. Then, we can find that ${\mathbf{H}}_{RU}\mathit{\varphi }{\mathbf{H}}_{BR}={\mathbf{v}}^H\mathbf{\Theta }$. Therefore, the optimization problem (14) can be rewritten as

$$\begin{array}{c}\hfill \begin{array}{cc}& \underset{\mathbf{v}}{max}{\mathbf{v}}^H\mathbf{\Theta }{\mathbf{\Theta }}^H\mathbf{v}+{\mathbf{v}}^H\mathbf{\Theta }{\mathbf{H}}_{BU}+{\mathbf{H}}_{BU}^H{\mathbf{\Theta }}^H\mathbf{v},\hfill \\ & s.t.|v_i|=1,\forall 0<=i<M.\hfill \end{array}\end{array}$$

(41)

It is evident that the problem (41) involves a non-convex quadratic constrained quadratic programming (QCQP) problem, which can be transformed into a homogeneous QCQP problem. By introducing an auxiliary variable t, the original problem (41) can be reformulated as follows:

$$\begin{array}{c}\hfill \begin{array}{cc}& \underset{\overline{\mathbf{v}}}{max}{\overline{\mathbf{v}}}^H\mathbf{R}\overline{\mathbf{v}},\hfill \\ & s.t.|{\overline{v}}_i|=1,\forall 0<=i<=M,\hfill \end{array}\end{array}$$

(42)

where

$$\mathbf{R}=\left[\begin{array}{cc}{\displaystyle \mathbf{\Theta }{\mathbf{\Theta }}^H}& \mathbf{\Theta }{\mathbf{H}}_{BU}\\ {\displaystyle {\mathbf{H}}_{BU}^H{\mathbf{\Theta }}^H}& 0\end{array}\right],\overline{\mathbf{v}}=\left[\begin{array}{c}{\displaystyle \mathbf{v}}\\ {\displaystyle t}\end{array}\right].$$

(43)

We define $\mathbf{V}=\overline{\mathbf{v}}{\overline{\mathbf{v}}}^H$, which needs to satisfy $\mathbf{V}⪰0$ and $rank\left(\mathbf{V}\right)=1$. Since the constraint of rank 1 is non-convex, the SDR is used to relax the constraint, so (42) can be rewritten as

$$\begin{array}{c}\hfill \begin{array}{cc}& \underset{\mathbf{V}}{max}tr\left(\mathbf{R}\mathbf{V}\right),\hfill \\ \hfill s.t.& {\mathbf{V}}_{i,i}=1,\forall 0<=i<=M,\hfill \\ & \mathbf{V}⪰0.\hfill \end{array}\end{array}$$

(44)

It is evident that the problem described in Equation (44) conforms to the standard convex semi-definite form, and can therefore be effectively addressed using established convex optimization solvers such as CVX. Finally, we can obtain a suboptimal solution for v:

$${\mathbf{v}}^{*}=e^{jarg\left({\left[{\displaystyle \frac{\overline{\mathbf{v}}}{{\overline{\mathbf{v}}}_M}}\right]}_{(0:M-1)}\right)},$$

(45)

where $arg(·)$ denotes the magnitude of a complex number. $\left[a\right]$ denotes the first M elements of the vector.

4.2. The Proposed Beam Tracking Scheme Based on DL

Recommending a DL-based framework to improve the efficiency and stability of the optimal reflection coefficient of the RIS due to the high computational complexity and inconsistent performance of SDR-based algorithms is our focus in this section. The structure diagram of the DNN designed in this paper, as shown in Figure 3, consists of seven hidden layers, one input layer, and one output layer.

As shown in Figure 3, $\mathit{H}$ is the input to the network, which consists of channels ${\mathbf{H}}_{BR}$, ${\mathbf{H}}_{RU}$, and ${\mathbf{H}}_{BU}$. However, considering that the channel $\mathit{H}$ is a complex matrix, it cannot be directly input to the neural network. Therefore, the imaginary and real parts need to be combined as the input, which is given by

$$\begin{array}{cc}\hfill & \mathbf{H}=[Re\left(vec\left({\mathbf{H}}_{BR}\right)\right),Im\left(vec\left({\mathbf{H}}_{BR}\right)\right),\hfill \\ & Re\left(H_{RU}\right),Im\left(H_{RU}\right),Re\left(H_{BU}\right),Im\left(H_{BU}\right){]}^T,\hfill \end{array}$$

(46)

where $Re(·)$ and $Im(·)$ denote the real and imaginary parts of complex elements, respectively. Then, the dimension of the input vector $\mathbf{H}$ is $1\times (2\ast M\ast N+M+N)$. We selected seven hidden layers after repeated experiments. To ensure the scalability of the system, the numbers of neurons in each layer were set to $64N,32N,16N,8N,4N,N$, which are proportional to the number of antennas on the base station. In addition, after comparing the network effects under various activation functions, we finally decided to use the Rectified Linear Unit (ReLu) as the activation function for the hidden layer, and a linear function for the output layer. Moreover, to prevent the overfitting of the network, we chose to add the BatchNorm layer by comparing the actual effects of the BatchNorm(BN) layer and the Dropout layer.

After setting up the structure of the network, we used negative (14) as the loss function to train the network. Considering that the output value of the network is a real number, and the reflection coefficient of the RIS should be a complex number, the Lambda layer was designed to convert the output value of the real number into a complex number, which is expressed as

$$\mathbf{v}=cos\left(\mathbf{o}\right)+j·sin\left(\mathbf{o}\right)=e^{j·\mathbf{o}},$$

(47)

where $\mathbf{o}$ is the output of the network.

Based on the above analysis and research, the algorithm for tracking beams in RIS-aided MISO systems is summarized in Algorithm 1.

Algorithm 1 The proposed beam tracking scheme for RIS-aided MISO system

1: Disable the RIS and utilize the PCA-based channel estimation technique to calculate the channel between the base station and the UT. 2: The UT transmits a pilot sequence to the RIS, and the channel between the UT and the RIS is estimated by the RIS using the PCA algorithm proposed in this study. 3: By utilizing the channel reciprocity of the TDD system, it is possible to acquire the channel between the RIS and the UT. 4: The pilot sequence is transmitted from the base station to the RIS, and the channel between the base station and the RIS is estimated based on the correlation between neighboring antennas and using the PCA algorithm. 5: Obtain the reflection coefficient by substituting the estimated channel into the trained DNN.

5. Simulation Results

To assess the effectiveness of the suggested algorithms, numerical outcomes were acquired through computer-based simulations. The system’s setup was determined by a specific set of parameters: the antenna space and reflecting elements space were set to $d_s=\lambda /2$. In addition, all channels in this paper were modeled using Rayleigh small-scale fading, and the path loss in dB is expressed as $20lo{g}_{10}(d/d_{ref})$, where d and $d_{ref}=1$ m denote the distance and reference distance, respectively. Finally, the transmit power was 5 dBm and the noise power was −70 dBm.

5.1. The Estimation Accuracy

In this section, we will initially validate the accuracy of CE in the proposed scheme. The normalized mean square error (MSE) on DoA angle estimation is presented to illustrate the CE performance under various SNRs. The MSE is defined as

$$\begin{array}{cc}\hfill & {\mathbf{MSE}}_{\mathrm{\Phi }}={\displaystyle \frac{1}{p_{BR}}}\sum _{p_{BR}=1}^{p_{BR}=P_{BR}}\left({\displaystyle \frac{{\mathrm{\Phi }}_{BS}^{{\overline{p}}_{BR}}-{\mathrm{\Phi }}_{BS}^{p_{BR}}}{{\mathrm{\Phi }}_{BS}^{p_{BR}}}}\right),\hfill \end{array}$$

(48)

$$\begin{array}{cc}\hfill & {\mathbf{MSE}}_{\theta }={\displaystyle \frac{1}{p_{BR}}}\sum _{p_{BR}=1}^{p_{BR}=P_{BR}}\left({\displaystyle \frac{{\theta }_{BS}^{{\overline{p}}_{BR}}-{\theta }_{BS}^{p_{BR}}}{{\theta }_{BS}^{p_{BR}}}}\right).\hfill \end{array}$$

(49)

In Figure 4, we illustrate the MSE performance for DoA angle estimation, showing that the estimation MSE decreases as the received SNR at the BS increases. The proposed scheme was compared with the backtracking line search (BLS) method [38] in terms of DoA angle estimation MSE, and it is evident that the proposed CE scheme has a better accuracy than the BLS algorithm. Additionally, due to both the estimated $u_{BS}^{p_{BR}}$ and $v_{BS}^{p_{BR}}$ affecting azimuth DoA angle estimation, elevation DoA angle estimation has a higher accuracy than azimuth DoA angle estimation.

5.2. Convergence Analysis under Different Network Structures

Firstly, we hypothesized the existence of three unique routes connecting the BS and the UT, the BS and the RIS, and the RIS and the UT. Moreover, each path has a corresponding azimuth and elevation angle. To assess the effectiveness of the network structure suggested in this paper, we replaced the BN layer in the designed DNN with the Dropout layer for comparison, both of which are used to avoid overfitting. Then, according to $M=16,N=64$, $M=16,N=16$, and $M=64,N=16$, i.e., three different base station antenna and RIS reflector unit configurations, we randomly generated 100,000 sets of channel data for DNN training. The Adam optimizer was initialized with a learning rate of 0.001, and the model was trained for 50 epochs with a batch size of 500.

The convergence performance under different network structures is shown in Figure 5. To compare the loss function values under various system configurations intuitively, we normalized and changed the value while retaining the original convergence trend. As shown in Figure 5, the networks based on the BN layer can converge very well, while the networks based on the Dropout layer can hardly converge. Therefore, this demonstrates the applicability of the BN layer in this model. Both the BN layer and the Dropout layer are used to avoid overfitting, but for the seven-layer network architecture proposed in this paper, subtle parameter changes will be infinitely amplified. Then, the input distribution will change greatly, which makes the network difficult to adapt to changes and fail to converge. Therefore, the normalization of the BN layer can reduce the influence of differences in the distribution of different batches of training data by changing the distribution of the data, reinforcing the network convergence speed, and enhancing the overall adaptability of the network.

5.3. Performance Analysis under Different Configurations

In this experiment, we generated 1000 sets for prediction based on the network model trained in the previous section. Then, the effects of different configurations of antennas and reflection elements on system performance were analyzed. Among them, the number of base station antennas in Figure 6 is 16, and the number of RIS reflection elements in Figure 7 is 64. According to Figure 6 and Figure 7, it can be observed that the received SNR increases with a larger number of antennas and reflection elements. Additionally, it is evident that the performance of the DL-based reflection coefficient estimation algorithm proposed in this paper closely resembles that of the SDR algorithm.

5.4. Complexity Analysis

Based on [23], the complexity of the SDR approach is $\mathbb{O}\left({(N+1)}^6\right)$, while the complexity of the proposed DL-based scheme is $\mathbb{O}\left((2788+128M)N^2\right)$. To compare the performance of these two different algorithms, the running times of the two schemes under different RIS configurations are shown in

Table 1. Running time comparison.

Configuration	Run Time
	SDR	DNN
$M=16,N=16$	49.8893s	0.1816s
$M=16,N=64$	107.0052s	0.2233s
$M=64,N=16$	121.5610s	0.2613s

. Both algorithms ran under the configuration of Core i5-9400F and RTX2060s for consistency. Furthermore, both algorithms made predictions on the same 100 sets of data. The SDR-based algorithm spent much more time than the DL-based scheme, which verifies the effectiveness of our proposed DL-based beam tracking algorithm.

6. Conclusions

In this paper, we present a channel estimation and beam tracking method for RIS-assisted MISO systems. Initially, the channel estimation of the system was divided into three stages based on the PCA algorithm. Subsequently, we achieved channel estimation between the base station and the RIS, as well as between the RIS and the UT, through a specific semi-passive RIS device. Additionally, a DNN was developed to estimate the RIS reflection coefficient. Ultimately, the simulation results demonstrate the superiority of our proposed scheme in terms of computational complexity compared to the traditional SDR algorithm.