# Radio Frequency Fingerprinting for Frequency Hopping Emitter Identification
## 3 Proposed RF Fingerprinting-Based Emitter Identification Method
### User Emitter Classification
#### 3.3.1 Base Classifier: Deep Inception Network Classifier  
There are two main blocks to construct the custom deep learning-based classifier: a residual block [22] and an inception block [23]. The residual block is designed to enable flexible training as the depth of the network increases. In the case of the inception block, the main purpose is to filter out input features with different receptive field sizes. Details of the architecture and design strategies of the main blocks are described in Appendix A. The spectrogram consists of physical measurements calculated from the SF signals. It represents the power densities of the SFs along the time-frequency axes. Thus, the subtle differences exhibited by the SFs can be anywhere on the time-frequency axes of the spectrogram, and the size of the features can be varied. To train these SFs, we aimed to filter the spectrogram on multiple scales in the temporal and spatial domains by applying inception blocks to construct a custom deep learning classifier. We utilized the inception-A and reduction-A blocks to construct the base classifier: the DIN classifier. The inception-A and reduction-A blocks are the basic blocks for constructing the Inception-v4 models [24]. The role of the inception-A block is to filter the input features with multiple receptive field sizes and concatenate them as the filter axis, thereby expanding its dimensions. The role of the reduction-A block is to downsize the feature map on the grid side, that is, the time-frequency axes of the spectrogram. It can effectively manage the number of weights inside the classifier, similar to the pooling layer. We adopted the inception-A and reduction-A blocks, as shown in Figure 6. The structures of the blocks are the same as defined in [24]. However, the filter sizes \(N_{F}\) of the sublayers were set to 32 and 64, adjusted by the experiments. Batch normalization [25] and rectified linear unit activation units were applied immediately after every convolutional layer. The inception-A block was applied twice to expand the filter axis, and the reduction-A block was applied once to re-size the feature map on the grid axis. We applied these block sequences twice, adjusted by heuristic experiments. The total structure of the DIN classifier is provided in Table 1.  
Figure 6: Basic block units used to construct the DIN: (**a**) the Inception-A block in [24] and (**b**) the Reduction-A block in [24]. Finally, we obtained the deep learning classification framework, as in Equation (15). From the \(M\) training samples in \(\mathbf{S}=[\mathbf{s}_{1},\mathbf{s}_{2},...,\mathbf{s}_{M}]\) and output samples \(\mathbf{Y}=[\mathbf{y}_{1},\mathbf{y}_{2},...,\mathbf{y}_{M}]\), the cross-entropy loss, was applied such that loss function can be expressed as follows  
\[\text{loss}=-(1/M){\sum_{i=1}^{M}}\log\Big{(}e^{\mathbf{y}_{i}[c_{k}]}/{\sum_ {j=1}^{C}}\mathbf{y}^{\mathbf{y}_{i}[c_{j}]}\Big{)} \tag{16}\]  
where \(c_{k}\) is the true label of sample \(\mathbf{y}_{i}\) with the \(k\)th emitter ID, \(\mathbf{y}_{i}[c_{j}]\) is the \(j\)th element of output sample \(\mathbf{y}_{i}\). Based on the cross-entropy losses, the Adam optimizer [26] is utilized to update the weights of our DIN classifier. After finishing the training of the DIN classifier, the emitter ID of input sample \(\mathbf{y}_{i}\) can be estimated as follows  
\[p(c_{i};\mathbf{s}_{\text{SF}}) =\text{softmax}(\mathbf{y})_{c_{l}} \tag{17}\] \[=\frac{e^{\mathbf{y}[c_{i}]}}{{\sum_{j=1}^{C}}e^{\mathbf{y}[c_{j}]}}\] \[\widetilde{k} =\underset{c_{j}\in\mathbb{C}}{\operatorname{argmax}}\big{(}p \big{(}c_{j};\mathbf{s}_{\text{SF}}\big{)}\big{)}\] (18) \[=\underset{c_{j}\in\mathbb{C}}{\operatorname{argmax}}\Big{(} \text{softmax}(\mathbf{y})_{c_{j}}\Big{)}\]  
where \(p(c_{l};\mathbf{s}_{i,\text{SF}})\) is the probability that the emitter ID of the input sample is \(c_{l}\), which can be defined as the softmax output of sample \(\mathbf{y}_{i}\).