## 5 Simulations
### Baseline Convolution Networks  
To evaluate the impact of CFO on classification performance, we use CNN as a baseline. The baseline network uses training and optimizing methods described in [16], which can representatively evaluate classification performance under different degrees of offset. We train the baseline networks on the CFO1 and CFO2 datasets. For test scenarios, the CFO is shifted from \(-6\%\) to \(6\%\) of the sample rate. Figure 4 shows the classification accuracy in the CFO1 and CFO2 scenarios. For frequency offset outside the training range, the classification performance of CNN classifiers declines rapidly. The consistent decline in performance occurred in both training datasets. This shows \begin{table}
\begin{tabular}{c c} \hline \hline
**Hyperparameters** & **Value** \\ \hline Loss Function & Categorical Crossentropy \\ Optimizer & ADAM \\ Dropout Rate & 0.4 \\ Initialize & Lecun Normal [18] \\ Learning Rate & 0.0024 (baseline), 0.00081(SSTM) \\ Batch Size & 256 \\ Epochs & 500 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Hyperparameters in signal spatial transformer module (SSTM)-based method training. Figure 4: Overall classification accuracy of baseline classifier for various training scenarios vs. frequency offset and SNR. (**a**) CFO1; (**b**) CFO2. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Layer Type** & **Input Size** & **Parameters** & **Activation Function** \\ \hline Convolution Layer & \(128\times 128\) & \(6\times 6\) filter kern & Leaky ReLU \\ Pooling Layer & \(128\times 128\times 64\) & \(2\times 2\) max pooling & None \\ Convolution Layer & \(64\times 64\times 64\) & \(3\times 3\) filter kernel & Leaky ReLU \\ Pooling Layer & \(64\times 64\times 32\) & \(2\times 2\) max pooling & None \\ Flatten & \(32\times 32\times 32\) & Flatten feature map & None \\ Dense & 32,768 & 64 neurons & ReLU \\ Dense & 64 & 8 neurons & Softmax \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Structure of baseline classifier. that the baseline classifier is sensitive to frequency offsets. Besides, at all frequency offsets, as the SNR is shifted from \(-10\) to \(10\) dB, the classification accuracy continues to increase significantly. However, when SNR exceeds \(10\) dB, the classification accuracy cannot be further improved. We believe this is because as the SNR increases, higher SNR does not provide more useful information to help optimize network parameters. We test on signal captures with sample rate range from \(1\) to \(8\) multiples of the bandwidth to verify the effect of sample rate offset on modulation recognition performance. We train the baseline networks on the SRO1 and SRO2 datasets. As presented in Figure 5, for sample rate offsets outside the training range, the classification accuracy decreases rapidly, and this reduction occurs in both test set scenarios. Besides, this reduction in classification performance shows consistency across all SNR scenarios. At the same sample rate offset, the classification accuracy versus SNR is similar to frequency offset scenario. In Figure 6, we present a sectional view of overall accuracy versus the frequency offset and sample rate when SNR is fixed at \(20\) dB. These simulations help to show the trends observed in Figures 4 and 5. Numerical analysis for all training scenarios in typical SNRs and offsets is shown in Table 4. In all offsets and training set scenarios, as the SNR increase, the classification performance can be significantly improved. For low frequency and sample rate offsets, the classifiers on all training sets have similar performance. For medium frequency and sample rate offsets, the classifiers obtained on the SRO2 and CFO2 training sets perform significantly better than the classifiers obtained on the SRO1 and CFO1 training sets. As frequency and sample rate offsets increase, the classification performance decreases consistently across all SNR and training set scenarios. Figure 5: Overall classification accuracy of baseline classifier for various training scenarios vs. sample rate offsets and SNR.