## 3 CE Methods and System Model
### Proposed Method  
This section presents a DL-based peephole LSTM model for channel state estimation. Another popular LSTM alternative with an associated connection, namely the peephole connection that the cell requires to control the gates. As the gates of the LSTM cells do not have direct connections from the cell state, which has only been possible through an open output gate, the network's performance is affected by a lack of essential information. [PERSON] and [PERSON] have connected LSTM memory cells with peepholes from cell to gates. Also, peephole connections allow gates to use the previous internal and hidden states (which the LSTM cell is limited to). This allows peephole LSTM to learn more precise timings than traditional LSTM cells. Peephole connections have several advantages over other types of connections in that they can connect the associated gates of the current information of the memory cell at any time since they can observe the state of the memory cell before its modulation by the output gate. Peephole connections boost LSTM performance greatly, leading to their adoption as a typical application technique [27; 28; 29]. The peephole LSTM functions similarly to the standard LSTM, except for the computations for the peephole connections. Figure 2 demonstrates a peephole LSTM structure implemented as follows: There are several variants of the basic LSTM architecture. By making peephole connections, the gates can rely not only on the input layer \(x_{t}\) and the past hidden state \(h_{t-1}\), but also on the preceding internal state \(c_{t-1}\), which adds another term to the gate equations that also comes back from the cell \(c_{t}\), and the forget gate \(f_{t}\) is generated by the activation function \(\delta_{g}\). The forgetting gate's output, which represents the capability for forgetting to hide the state of the cell in the top layer, is evaluated using the sigmodal's output, which has values between 0 and 1, for which the mathematical formulation is given below:  
Figure 1: General channel estimation. The mathematical expression is:  
\[f_{t}=\delta_{g}\left(W_{f}\ x_{t}+V_{f}h_{t-1}+U_{f}c_{t-1}+b_{f}\right) \tag{1}\]  
The actual series position's input is processed by the input gate, which has three elements; the cell state, input data, and peephole connection value. When the sigmoid function is activated, the output is \(i_{t}\):  
\[i_{t}=\delta_{g}(W_{i}x_{t}+V_{i}h_{t-1}+U_{i}c_{t-1}+b_{i}) \tag{2}\]  
Next, combine two elements to update the cell state. The first element is generated by multiplying \(c_{t-1}\) and the forget gate's output, while the second part is generated by the activation function's output of \(x_{t}\), \(c_{t-1}\), \(h_{t-1}\) by an activation function. In this way, the updated cell state is given:  
\[c_{t}=i_{t}\delta_{c}\left(W_{c}x_{t}+U_{f}c_{t-1}+V_{f}h_{t-1}+b_{i}\right)+ c_{t-1}f_{t} \tag{3}\]  
The following information affects the output gate's decision: input value, most recent cell state, and peephole value. Memory also processes the new cell state. The sigmoid function determines the linked cell state and output gate value. \[o_{t}=\delta_{g}(W_{O}x_{t}+U_{O}c_{t-1}+V_{O}h_{t-1}+b_{O}) \tag{4}\]  
To update the hidden state by multiplying the output gate and the cell state activated by the function. \[h_{t}=o_{t}\delta_{h}(c_{t}) \tag{5}\]  
The output of the entire LSTM model is as follows:  
\[y_{t}=k\left(W_{h}h_{t}+b_{y}\right) \tag{6}\]  
where \(W_{f}\), \(W_{i}\), \(W_{o}\), \(V_{f}\), \(V_{i}\), \(V_{o}\), and \(u_{f},u_{i},u_{o}\) are weight matrices, \(W_{h}\) represents the hidden output weight matrix, and \(b_{f},b_{i},b_{o},b_{y}\) are bias vectors. The DL peephole LSTM model was used to estimate the channel constructed from five layers and is presented below. The input data properties have a 256-bit serial input layer. The subsequent layer is a peephole LSTM layer having 16 hidden units that produce  
Figure 2: Peephole LSTM architecture. the final element of the sequence.