# A Long Time Span-Specific Emitter Identification Method Based on Unsupervised Domain Adaptation
## 3 Methodology
### Classifier
The second part will control the distance between different class centers to enhance the interclass separation of features. \(L_{CL}\) can be expressed as  
\[L_{CL}=\frac{1}{2}(\sum\limits_{i=1}^{K}\left\|\mathbf{f}_{S}^{i}-\mathbf{c}_{y_ {i}}\right\|_{2}^{2}+\sum\limits_{i,j=1,j}^{N}\max(0,\sigma-\left\|\mathbf{c}_{y _{i}}-\mathbf{c}_{y_{j}}\right\|_{2}^{2})) \tag{9}\]  
where \(K\) represents the number of small batch samples, \(\mathbf{f}_{S}^{i}\) represents the feature vector extracted from the source domain sample \(X_{i}\) by the last fully connected layer, and \(\mathbf{c}_{y_{i}}\) represents the class center corresponding to the real label of \(X_{i}\).\(\left\|\cdot\right\|_{2}^{2}\) is used to calculate the square Euclidean distance. \(\sigma\) represents a variable parameter that controls the distance between different class centers. After each iteration, the class centers of the entire training set need to be updated because all training sample features need to be recalculated. However, this is cumbersome and impractical because the model uses only a small batch of samples to participate in training each time. Therefore, we use the mean of small batch sample features to approximate the global class centers. Batch class center \(\mathbf{c}_{y_{i}}\) can be described as  
\[\mathbf{c}_{y_{i}}=\frac{1}{b}\sum\limits_{i=1}^{b}\mathbf{f}_{y_{i}} \tag{10}\]  
where \(\mathbf{f}_{y_{i}}\) and \(b\) represent the features and their numbers of the same category as \(X_{i}\), respectively. Since \(\mathbf{c}_{y_{i}}\) does not train with the network parameters, we introduce \(\gamma\in[0,1]\) to control the learning rate of \(\mathbf{c}_{y_{i}}\) to prevent singular samples in the training set from causing considerable fluctuation for class centers in the feature space. The update method of \(\mathbf{c}_{y_{i}}\) is designed as follows:  
\[\mathbf{c}_{y_{i}}^{t+1}=\mathbf{c}_{y_{i}}^{t}-\gamma\cdot\Delta\mathbf{c}_{y _{i}} \tag{11}\]  
\[f_{c}:F_{i}\in\mathbb{R}^{1\times fi}\overset{\theta_{c}}{\rightarrow}\textbf {f}_{i}\in\mathbb{R}^{1\times N} \tag{5}\]0  
where \(\mathbf{c}_{y_{i}}^{t+1}\) represents the class center updated by class \(y_{i}\) at time \(t+1\), \(\mathbf{c}_{y_{i}}^{t}\) represents the class center updated by class \(y_{i}\) at time \(t\), and \(\delta(\cdot)\) represents the impulse function. When the condition in the parentheses holds, \(\delta(\cdot)=1\). Otherwise, \(\delta(\cdot)=0\). 3. HOMM3 Loss: We added a new constraint to the deep-features output by the classifier to enhance the domain adaptation ability of the model. By optimizing the feature-matching loss, the source domain and target domain are forced to align in the deep-feature space. High-order statistics (such as high-order moments and high-order cumulants) are typically used to describe the intrinsic distribution of signals and are also commonly used as recognition features of different signals. When a random signal follows a Gaussian distribution, its statistical characteristics can be understood through a mathematical expectation (first-order statistic) and an autocorrelation function (second-order statistic). However, the expression of statistical characteristics of non-Gaussian distribution signals by low-order statistics is limited, which may impact the effectiveness of domain matching. We propose to reduce the distance between the high-order moments of the source domain sample features and target domain sample features by optimizing the high-order moment matching loss. However, the calculation of higher-order tensors introduces higher temporal and spatial complexities. When the number of neurons in the bottleneck layer is \(m\), the dimension of the \(p\) order statistics reaches \(O(m^{p})\). In a bid to reduce complexity and achieve fine-grained domain alignment of features, we selected the last fully connected layer of the classifier as the bottleneck layer and used this layer to calculate third-order statistics of the features.