# Multi-Source HRRP Target Fusion Recognition Based on Time-Step Correlation
## 2 Proposed Model
### Time-Step Correlation-Based Feature Fusion (TCFF) Module  
The process of the TCFF module is shown in Figure 6. The method is mainly composed of three parts: calculating the time-step correlation matrix, calculating the time-step weight vector and obtaining the fusion feature. It is assumed that the original features to be fused are \(F_{1},F_{2}\in\mathbb{R}^{H\times T}\), where \(H\) and \(T\) are the dimensions and time steps of the features. The calculation steps are as follows. Figure 5: Characteristic mixing process in dimension-up phase of GLTs module. For the two feature vectors \(\mathbf{F}_{1},\mathbf{F}_{2}\in\mathbb{R}^{H\times T}\) of the input module: the correlation of the corresponding time steps is calculated, and two eigenvector covariance matrices \(\mathbf{TC}\in\mathbb{R}^{T\times T}\) are obtained. The calculation process is as follows. \[\mathbf{F}_{1}^{\prime}=\mathbf{F}_{1}^{\prime}-\mathbf{F}_{1}^{\prime} \tag{12}\]  
where \(\mathbf{F}_{1}^{\prime}\) represents the \(r\)-th time step of feature vector \(\mathbf{F}_{1},\mathbf{F}_{1}^{\prime}\) represents the matrix consisting of H copies of the mean of feature vector \(\mathbf{F}_{1}\) at the \(i\)-th time step, \(\mathbf{F}_{1}^{\prime},\mathbf{F}_{1}^{\prime}\in\mathbb{R}^{H\times 1}\). The procedure for calculating the feature vector \(\mathbf{F}_{2}\) is the same as that of \(\mathbf{F}_{1}\). The calculation results of each time step \(\mathbf{F}_{1}^{\prime},\mathbf{F}_{2}^{\prime}\) are spliced to obtain vector \(\mathbf{F}_{1}^{\prime},\mathbf{F}_{2}^{\prime}\in\mathbb{R}^{H\times T}\). Transpose \(\mathbf{F}_{1}^{\prime}\) and carry out matrix multiplication with \(\mathbf{F}_{2}^{\prime}\) to obtain the time-step correlation matrix. \[\mathbf{TC}=\mathbf{F}_{1}^{\prime\ T}\otimes\mathbf{F}_{2}^{\prime} \tag{13}\]  
where \(\otimes\) denotes matrix multiplication. The time-step correlation matrix \(\mathbf{TC}\in\mathbb{R}^{T\times T}\), \(TC_{ij}\) represents the correlation between \(\mathbf{F}_{1}\) the \(i\)-th time step and \(\mathbf{F}_{2}\) the \(j\)-th time step. The matrix \(\mathbf{TC}\) allows the statistical correlation between \(\mathbf{F}_{1}\) and \(\mathbf{F}_{2}\) to be analyzed for subsequent more efficient fusion of features. After the time-step correlation matrix is obtained, the rows and columns are summed, respectively, and two column vectors \(\mathbf{TC}_{row},\mathbf{TC}_{col}\in\mathbb{R}^{T\times 1}\) are obtained, which represent the correlation degree between the \(i\) th time step of \(\mathbf{F}_{1}\) and \(\mathbf{F}_{2}\), and the correlation degree between the \(j\) th time step of \(\mathbf{F}_{2}\) and \(\mathbf{F}_{1}\). According to the magnitude of the summation: the larger the value, the stronger the correlation. For the feature vectors extracted from the same target: the stronger the correlation, the more representative the features are under the current time step, and the features under the current time step should be given a higher weight value. On the contrary, the weaker the correlation, the feature that represents the current time step is not important for the target, and a smaller weight value should be given to avoid information redundancy and to reduce recognition performance. In order to limit the weight value between 0 and 1, we use the Sigmoid function to calculate the two column vectors \(\mathbf{TC}_{row},\mathbf{TC}_{col}\), the corresponding weight size. The Sigmoid function is a monotonically increasing function, which can convert the values greater than 0 in the column vector into an interval of 0.5-1 and convert the values less than 0 into 0-0.5. The corresponding weight calculation formula of each feature is as follows:  
\[\mathbf{W}_{\mathbf{F}_{1}}=\sigma(\mathbf{TC}_{row}) \tag{14}\]  
Figure 6: Flow chart of TCFF module. \[\mathbf{W_{F_{2}}}=\sigma(\mathbf{TC_{col}}) \tag{15}\]  
where \(\sigma\) is the Sigmoid function. The weight vector \(\mathbf{W_{F_{1}}}\), \(\mathbf{W_{F_{2}}}\) is copied \(H\) times and spliced to obtain the new weight vector.