# Multi-Source HRRP Target Fusion Recognition Based on Time-Step Correlation
## 2 Proposed Model
### Feature Extraction Network  
In order to verify the effectiveness of the feature fusion method, seven feature extraction networks are selected in this paper, which are Recurrent Neural Networks (RNNs), LSTM [28], GRU [29], Simple Recurrent Units (SRUs) [30], encoder part in Transformer [31], R-Transformer [32] and R-DeLighT [32] models. This section introduces the R-DeLighT model, whose structure is shown in Figure 2. The model is mainly composed of a local RNN module and a Transformer-Encoder feature extraction module that introduces Group Linear Transformations (GLTs). Figure 3 shows the local RNN module structure when the window size of the sliding window is 3. For a more general explanation, we assume that the window size of the sliding window in the local RNN module is \(k\). A new input sequence \(\mathbf{X}^{\prime}\) is formed by adding \(k-1\) zero sequences of the same size in time steps to the left side of the original input \(\mathbf{x}\):  
\[\mathbf{X}^{\prime}=\left[0,\underbrace{...}_{k-1},0,\mathbf{x}_{1},\mathbf{x}_{2},..., \mathbf{x}_{T}\right] \tag{4}\]  
Figure 1: HRRP recognition model integrated into the TCFF module. The local RNN module consists of parallel RNN modules designed to take full advantage of RNN in extracting short sequence signal features. The structure of the local RNN module is shown in Figure 3[32]. The sliding window with a size of \(k\) is slid from left to right, and the sliding distance is 1. Then \(T\) short sequences are formed, and the hidden state vectors output by \(T\) RNN are spliced into new hidden state vectors. \[\mathbf{h_{t}}=RNN(\mathbf{x_{t-k+1}},\mathbf{x_{t-k+2}},...,\mathbf{x_{t}}) \tag{5}\]  
\[\mathbf{H}=[\mathbf{h_{1}},\mathbf{h_{2}},...,\mathbf{h_{T}}] \tag{6}\]  
where \(\mathbf{h_{t}}\) is the output of the hidden layer vector in the \(t\)-th RNN. \(RNN(\bullet)\) is a simplification of the RNN principal formula. The GLTs are introduced into the Transformer model encoder structure to improve the model recognition performance while reducing the model parameters. Comparison  
Figure 3: Structure diagram of the local RNN module with sliding window size of 3. Figure 2: Structure diagram of the R-DeLighT model. of standard Transformer model encoder structure (left) and Transformer model encoder structure combined with GLTs is shown in Figure 4[32]. The module first takes the hidden layer vector \(\mathbf{H}\) output by the local RNN module as the input of the model. Assuming there are N layers of GLTs in this module. The modules of GLTs have two stages: expansion stage and reduction stage. In the expansion stage, in \(\left\lceil N/2\right\rceil\) linear layer, the input of dimension \(d_{m}\) is projected onto a high-dimensional space of dimension \(d_{\max}\). where \(d_{\max}=w_{m}d_{m}\), and \(w_{m}\) is the expansion coefficients. Similarly, in the reduction stage, use the remaining \(N-\left\lceil N/2\right\rceil\) layers to project the \(d_{max}\) dimensional vector onto the \(d_{0}\) dimensional vector. \[g^{l}=\left\{ \begin{matrix}\min\left(2^{l-1},g_{\max}\right),1\leq l\leq\left \lceil N/2\right\rceil\\ g^{N-l},\textit{otherwise}\end{matrix} \right. \tag{7}\]  
where \(g_{\max}\) is the set maximum number of linear transformation layers per layer. \(g^{l}\) denotes the number of linear transformation layers contained in the \(l\)th layer. \[\mathbf{Y}^{l}=\left\{ \begin{matrix}\mathbf{F}\big{(}\mathbf{H},\mathbf{W}^{l},\mathbf{b}^{l},g^{ l}\big{)},l=1\\ \mathbf{F}\big{(}MIX\big{(}\mathbf{H},\mathbf{Y}^{l-1},\mathbf{W}^{l},\mathbf{b}^{l},g^{l}\big{)} \big{)},\textit{otherwise}\end{matrix} \right. \tag{8}\]  
where \(\mathbf{W}^{l}=\left[\mathbf{W}_{1}^{l},\mathbf{W}_{2}^{l},...,\mathbf{W}_{g}^{l}\right]\) and \(\mathbf{b}^{l}=\left[\mathbf{b}_{1}^{l},\mathbf{b}_{2}^{l},...,\mathbf{b}_{g}^{l}\right]\) are the weights and bias of the \(g^{l}\) group linear transformation function in the \(l\)th layer, respectively.