## 2 RIS-Assisted System and Channel Model
### Proposed Deep Learning Model
#### 2.4.2 Model Construction and Operation
The input state information \(X_{t}\) at the present instant and the hidden layer information \(H_{t-1}\) that is previously learned are used to compute the update \(U_{t}\) and reset gate \(R_{t}\) in the first step. The reset gate \(R_{t}\) is used in the second phase to count the number of new pieces of information that are stored in the node \(\widehat{H_{t}}\). The third step involves using the update gate \(U_{t}\) to determine the hidden layer output at the present time. The GRU computation procedure is explained using the formulas below:  
\[U_{t}= \sigma(W_{U}X_{t}+V_{U}H_{t-1}+b_{U}) \tag{14}\] \[R_{t}= \sigma(W_{R}X_{t}+V_{R}H_{t-1}+b_{R})\] (15) \[\widehat{H_{t}}= \tanh(W_{H}X_{t}+V_{H}(R_{t}\otimes H_{t-1})+b_{H})\] (16) \[H_{t}= (1-U_{t})\otimes H_{t-1}+U_{t}\otimes\widehat{H_{t}}, \tag{17}\]  
where the values for the bias are \(b_{U}\), \(b_{R}\), and \(b_{H}\); The gate activation function is calculated using the sigmoid function, which is denoted by \(\sigma(c)=(1+e^{-c})^{-1}\). The hyperbolic tangent function known as \((\tanh)\) is in charge of calculating the state activation function. \(W_{U}\), \(W_{R}\), \(W_{H}\), \(V_{U}\), \(V_{R}\), and \(V_{H}\) are weight matrices. While the input state \(X_{t}\) and the output of the hidden layer at the previous instant are merged to generate \(\widehat{H_{t}}\), the output of the hidden layer at the current instant is represented by \(H_{t}\). The Hadamard product is represented as \(\otimes\) for element-wise multiplication. _Fully connected layer (i = 4):_ The task of activation with the softmax is provided to the fully linked layer, which plays a significant role in classification, and the final classification is performed in this layer. Here, the model calculates the probability that each sample now belongs to one of the classes and then derives the expression of a feature (\(Y_{pred}\)), which is stated as follows:  
\[Y_{pred}(i)=f(L=l_{i}|S_{3};(W,b)), \tag{18}\]  
where the GRU features are \(S_{3}\) and the activation function for softmax is \(f(\cdot)\). The computation output for the \(i\)-th classes of the input data is represented by the variables \(l_{i}\), where \(W\) and \(b\) stand for the weight and bias values, respectively. The cross-entropy operation calculates the cross-entropy loss for single-label and multi-label classification tasks between network predictions and target values. Cross-entropy is frequently the best choice when output probability models are dealt with. Furthermore, \(L2\) regularization may be viewed as a successful compromise between the identification of tiny weights and the reduction of the cost function [51]. Therefore, overfitting is avoidedby using cross-entropy and \(L2\) regularization. The following formulation of the loss computation serves as the basis for the goal of model training:  
\[S_{i}=f(S_{i-1}\otimes w_{i}+b_{i}), \tag{12}\]0  
where \((Y^{(t)}(i)\) represents the prospect of the known goal, \((Y^{(t)}_{pred}(i))\) is the probability that the \(i\)-th sample belongs to the \(t\)-th class, the number of samples is \(n_{s}\), the number of the class is \(c\), and finally, the term \"\(\lambda\)\" defines the coefficient of the regularization of \(L2\). The Adam optimizing strategy is applied to reduce the loss [52].