## 3 The Proposed Scheme
### Network Structure  
The architecture of the proposed model is depicted in Figure 2, which is composed by subnet \(\mathcal{S}_{1}\) with \(\mathcal{L}_{1}\) convolution layers, subnet \(\mathcal{S}_{2}\) with \(\mathcal{L}_{2}\) LSTM layers, and subnet \(\mathcal{S}_{3}\) with \(\mathcal{L}_{3}\) dense layers. The samples \(\overline{\mathcal{Y}}\in\mathbb{R}^{(N-2m+2)\times m\times m}\) are then fed into \(\mathcal{S}_{1}\) for feature extraction. The structure of \(\mathcal{S}_{1}\) is composed of a convolution layer, pooling layer, flatten layer, and dense layer. The convolution layer employs a series of two-dimensional convolution filters (2D-Conv) to \(\overline{\mathcal{Y}}\), so as to extract different feature maps of received signals. Let \(\mathcal{K}_{l}\) and \(\mathcal{C}_{l}\) denote the size of the convolutional kernel and the number of filters of the \(l\)-th convolutional layer, respectively. For simplicity, the stride was fixed to \(1\), and the Relu function and the same padding were employed in 2D-Conv. After the convolution calculation, the max-pooling layer is used to extract the invariant features with the non-linear downsampling, which will eliminate the non-maximal values. The same signal processing is implemented in the next 2D-Conv and max-pooling layer. After that, the flatten layer is employed to reshape the data size, and then, the dense layer is linked behind accordingly. The CNN outputs should be firstly transformed as a three-dimensional vector \(\overline{\mathcal{Y}}_{1}\in\mathbb{R}^{(N-2m+2)\times m\times\mathcal{D}_{1 }^{S2}}\) and fed into \(\mathcal{S}_{2}\), where \(\mathcal{D}_{1}^{S2}\) denotes the cell number of the first layer of \(\mathcal{S}_{2}\). The structure of \(\mathcal{S}_{2}\) is made up by cascaded sub-layer blocks composed by multiple LSTM cells. In addition, different amounts of LSTM cells can be deployed in different sub-layers. The internal structure of a single LSTM cell, as shown in Figure 2, contains three Sigmoid gates in terms of the forget gate, input gate, and output gate. These gates can selectively influence the model state at each time step. The forget gate is also the core of a single LSTM cell, since it determines the information that should be retained or  
Figure 2: Schematic of the proposed DL scheme with the Volterra spatial feature. discarded according to the current input \(y_{t}\) and the previous output \(H_{t-1}\). The output of forget gate can be expressed as  
\[f_{t}=\sigma\big{(}\mathbf{W}_{f}\cdot[y_{t},H_{t-1}]+\mathbf{b}_{f}\big{)}, \tag{11}\]  
where \(\sigma\) is the Sigmoid function and \(\mathbf{W}_{f}\) and \(\mathbf{b}_{f}\) represent the parameter matrix and bias matrix of the forget gate, respectively. After forgetting part of the previous state, the input gate picks up some new information and then adds it into the former state \(\mathrm{C}_{t-1}\). Therefore, the new cell state is formed as  
\[C_{t}=f_{t}\mathrm{C}_{t-1}+i_{t}z_{t}, \tag{12}\]  
where \(z_{t}\) denotes the temporary cell state and \(i_{t}\) is the output of the input gate. Furthermore, the expression of \(z_{t}\) is given as  
\[z_{t}=\tanh(\mathbf{W}_{z}\cdot[y_{t},H_{t-1}]+\mathbf{b}_{z}), \tag{13}\]  
and the output \(i_{t}\) can be expressed as  
\[i_{t}=\sigma(\mathbf{W}_{i}\cdot[y_{t},H_{t-1}]+\mathbf{b}_{i}), \tag{14}\]  
where \(\mathbf{W}_{z}\), \(\mathbf{W}_{i}\), \(\mathbf{b}_{z}\), and \(\mathbf{b}_{i}\) denote the parameter matrix and bias matrix, respectively. Therefore, the value \(f_{t}\) and \(i_{t}\) between 0 and 1 also indicates the proportion of the important information in \(\mathrm{C}_{t-1}\) and \(z_{t}\), thereby determining which information is to be updated. Then, the output of the LSTM cell can be calculated by  
\[H_{t}=\tanh(C_{t})\sigma(\mathbf{W}_{o}\cdot[y_{t},H_{t-1}]+\mathbf{b}_{o}). \tag{15}\]  
As a result, the outputs of this layer are fed into the corresponding LSTM cell of the next layer.