## 3 Results and Discussion
### Experiments and Discussion
GFSK and QPSK are also very close. These results are identical to the facts we observed on the confusion matrix. From \(-4\) dB to \(6\) dB, the distance between 16 QAM and 64 QAM has expanded and the accuracy has increased, AM-DSB and WBFM have also partly separated from each other. We can also find that the separated features of different modulation categories are far away from each other even for an SNR of \(-4\) dB. Then we analyze the influence of the hyper parameters of AM-softmax. AM-softmax has two parameters: \(m\) means margin and \(s\) means scale. \(m\) controls the cosine margin which can be written as the difference between the cosine values of both sides of the dividing interval. Table 2 is the performance of hybrid parallel network with different margin \(m\). We stacked 4 hybrid parallel modules with \(s=10\), and other parameters were all fixed. We can find that the difference between the maximum accuracy and the minimum of the accuracy is less than 2%, which means the parameter \(m\) has little influence on our proposed method. This can be explained according to the results above. In Figure 7, the accuracy of AM-DSB is always non-ideal while the others have already got high values. In Figure 8 we can also find that, except AM-DSB, almost all the modulation categories are well classified at \(6\) dB; even at \(-4\) dB, the classification accuracy of most modulation types has been over 80%. In Figure 9, almost all features at hidden layers are discriminate between each other since \(0\) dB and the distance between each category of feature is very large except confused types. These results all point to the facts that most modulation types have been classified very well and the cosine angular distance within each category of features is very small except BWFM and AM-DSB. As a result, no matter whether the cosine angular margin \(m\) is small or large, it has limited influence on accuracy. The performance of AM-softmax with different scales \(s\) is illustrated from Table 3. We fix the structure of network with 4 hybrid parallel modules and \(m\) is set to \(0.1\). With the increase of the scale \(s\), the performance of the hybrid parallel network also gets better until \(s=10\). The parameter \(s\) controls the rate of convergence. If \(s\) is too small, the gradient will grow slow and is easy to fall into local optima. Therefore, \(s\) cannot be too small. As we can see in Table 3, when \(s\) is larger than \(10\), the accuracy result will become stable. As a \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline
**m** & 0.0 & 0.1 & 0.2 & 0.3 & 0.4 & 0.5 & 0.6 & 0.7 & 0.8 \\ \hline
**accuracy (\%)** & 89.76 & 90.51 & 90.36 & 89.87 & 90.62 & 90.66 & 90.06 & 90.13 & 88.71 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The performance of AM-softmax with different margin. Figure 9: The distribution of features extracted at the penultimate layer for different SNRs. **(a)** The distribution of features extracted at the penultimate layer for an SNR of \(-4\) dB. **(b)** The distribution of features extracted at the penultimate layer for an SNR of \(0\) dB. **(c)** The distribution of features extracted at the penultimate layer for an SNR of \(6\) dB. consequence, we can conclude that as long as \(s\) reaches a threshold, the growth of \(s\) will have little effect on the outcome. From Table 3, we know the threshold of \(s\) in this study is 10. In Figure 10, we compare the softmax function with AM-softmax function. We cascade 3 hybrid parallel modules with different output function. We choose \(m=0.1\) and \(s=10\). We applied AM-softmax and softmax respectively for the output of the same model, and each ran 100 times and calculated the average for each SNR. The average accuracy of AM-softmax and softmax are respectively 87.33% and 86.36%. The accuracy of AM-softmax is 1% greater than Softmax. However, the improvement of 1% is within the margin of error; thus, we believe that AM-softmax can achieve similar performance to softmax.