## 3 MIMO System Model and the Implementation Steps of Channel Estimation
### Basic Attention LSTM  
Attention-based LSTM [35] consists of two main parts: the attention model and LSTM. The attention network in LSTM selects only the useful information at a time sequence from the input signal, and their weights are assigned to reduce the overload in the signals present in its hidden state. LSTM in the network uses sequential data as input and stores information in the memory for future use. Thus, the LSTM provides an association between the preceding and succeeding data. In this wireless communication network, LSTM has the potential to forecast future data using previous information. Managing and tracking the information in LSTM is carried out using three gates, namely, a \"forget gate, input gate, and output gate\".  
The input data consist of related and unrelated information. Unrelated information in the network creates signal loss during communication. The signal loss is reduced by using the forget gate in LSTM, which helps to eradicate this unrelated information and the processing. The forget gate is defined by means of Equation (8).  
\[G(y)=\beta\Big{(}E_{g,y}\vec{e}_{y}+E_{g,y}\vec{j}_{y-1}+n_{g,y}\Big{)} \tag{8}\]  
In Equation (8), the term \(\beta\) represents the sigmoid function, and the biasing term of the forget gate is denoted by \(n_{g,y}\). If the forget gate produces output \(1\), then the unrelated information is dropped from the input data. Only the useful information from the input gate is fed to the next update gate.  
The update gate restores the present state of the data using the previous state. The computation of the update gate is given by Equation (9).  
\[\overline{V}_{t}=\tan\Big{(}E_{vZ_{y}}+E_{vZ_{y-1}}+\overline{n}_{g,y}\Big{)} \tag{9}\]  
\[V_{y}=g_{y}\cdot V_{y-1}+o_{y}\cdot\overline{V}_{y} \tag{10}\]  
\[Out=\sigma\big{(}E_{p}h\big{(}j_{y-1},c_{y}\big{)}+n_{p}\big{)} \tag{11}\]  
In Equations (9)-(11), the terms \(V_{y}\) and \(V_{y-1}\) denote the current and past status of the memory and \(n_{p}\) indicates the biased term. Finally, the output is generated using the tan function and threshold value to achieve the essential operation. However, autoencoders are highly sensitive to input errors. Hence, an autoencoder with a serial cascaded structure is used in this hybrid network to reduce the repetition of the input signal and minimize the number of errors in the received information. The attention mechanism used in the autoencoder to estimate the components and assign them based on their weights. A linear transformation is performed in the attention module to initiate the input signal, and this input signal is resized using a fully connected layer in the network. The input signal is split into three sub-signals: query, key, and value. These sub-signals are represented in Equation (12).  
\[L,U,B\in T\Big{(}\frac{mg}{2}\Big{)}\times 2\times M_{head} \tag{12}\]  
In Equation (12), the term \(M_{head}=B_{pilot}\) represents the size of the head. A scaled dot-product attention mechanism is performed, and the integrated result is conveyed to the next fully connected layer. Finally, the attention module output is generated using the softmax function, and it is determined using Equation (13).  
\[Atten=Sof\left(\frac{WL^{y}}{\sqrt{\frac{K_{g}}{2}}}\right)B \tag{13}\]  
The adaptability between the query and the key vector is computed using a feed-forward network, and their comparability is measured using the dot-scale attention mechanism. Long input sequence data are handled by using an attention network and the interpretability of the model is improved by selecting the relevant part from the input using the attention mechanism.