# Radio Frequency Fingerprinting for Frequency Hopping Emitter Identification
## 3 Proposed RF Fingerprinting-Based Emitter Identification Method
### User Emitter Classification
#### 3.3.2 Ensemble Approach for Multimodal Signal Fingerprints  
The ensemble approach is a well-known method that ensures better generalization performance of classification models [27]. It combines the results of multiple base classifiers trained on the same training dataset and makes a final decision based on these results. Stacking is a combined method that uses the final model to combine the outputs of the base model [27]. It is useful when multimodal features are present in applications such as video signal processing where audio, video, and text segments exist simultaneously [28].  
It was reported that multiple SFs, that is, the RT, SS, and FT signals, can be considered as multimodal features for an accurate RF fingerprinting model [6]. To utilize the multimodality features of the SFs, we adapted the stacking ensemble approach to the DIN model as presented in Figure 7. The SFs \(\mathbf{s}_{\text{SF}}\) were extracted from hop signal \(\mathbf{s}\) in Equation (10). These SFs can act as independent features for emitter identification. Thus, each of the SFs,  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Type** & **Filter Size/Stride** & **Output Shape** \\  & **/Padding** & **(for the SS Input)** \\ \hline Input signal & - & 205 \(\times\) 340 \(\times\) 1 \\ Conv\_1 & 3 \(\times\) 3/2/0 & 102 \(\times\) 169 \(\times\) 32 \\ Conv\_2 & 3 \(\times\) 3/1/0 & 100 \(\times\) 167 \(\times\) 32 \\ Conv\_3 & 3 \(\times\) 3/1/1 & 100 \(\times\) 167 \(\times\) 32 \\ Max. pool & 3 \(\times\) 3/2/0 & 49 \(\times\) 83 \(\times\) 32 \\
2 \(\times\) inception & Inception-A [\(N_{F}=32\)] & 49 \(\times\) 83 \(\times\) 128 \\ \(1\times\) reduction & Reduction-A [\(N_{F}=32\)] & 24 \(\times\) 41 \(\times\) 192 \\
2 \(\times\) inception & Inception-A [\(N_{F}=64\)] & 24 \(\times\) 41 \(\times\) 256 \\
1 \(\times\) reduction & Reduction-A [\(N_{F}=64\)] & 20 \(\times\) 11 \(\times\) 384 \\ Avg. pool & Adaptive avg. pooling & 1 \(\times\) 1 \(\times\) 384 \\ Linear & Logits & 1 \(\times\) 1 \(\times\) 7 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Structure of the base classifier: the DIN classifier.  
i.e., RT, SS, and FT, is assumed to be independent of the others. For the ensemble approach, the probability that the emitter ID is \(c_{l}\) can be defined as follows  
\[p(c_{l};\mathbf{s})=\prod_{\mathbf{SF}\in\{\text{RT,SS,FT}\}}p(c_{j};\mathbf{s} _{\text{SF}}). \tag{19}\]  
According to the DIN classifier trained on the RT, FT, and SS signals presented in Section 3.3.1, the final decision was performed by a linear combination of each base classifier (i.e., DIN classifier) such that  
\[\begin{split}\widetilde{k}&=\underset{c_{j}\in \text{C}}{\text{argmax}}p\big{(}c_{j};\mathbf{s}\big{)}\\ &=\underset{c_{j}\in\text{C}}{\text{argmax}}\prod_{\text{SF}\in \{\text{RT,SS,FT}\}}p\big{(}c_{j};\mathbf{s}_{\text{SF}}\big{)}\\ &=\underset{c_{j}\in\text{C}}{\text{argmax}}\prod_{\text{SF}\in \{\text{RT,SS,FT}\}}\text{softmax}(\mathbf{y}_{\text{SF}})_{c_{j}}\end{split} \tag{20}\]