## 3 Analysis and Discussion
### Analysis of AFEM Setting
#### 3.3.4 Others  
In order to further improve the richness and adaptiveness of the features extracted by AFEM, we use a power function instead of a logarithmic function for non-linear activation in the activation layer after the linear transformation layer, and the transformation relationship between both is as follows:  
\[S_{1}(Z)=log(S(Z)) \tag{13}\]  
\[S_{2}(Z)=(S(Z))^{\gamma} \tag{14}\]  
\[S_{2}(Z)=e^{\gamma S_{1}(Z)}=\sum_{n=0}^{-\infty}\frac{(\gamma S_{1}(Z))^{n}}{n!} \tag{15}\]  
\[s_{2}(n)=\sigma(n)+\gamma s_{1}(n)+\frac{\gamma^{2}}{2}s_{1}(n)*s_{1}(n)+ \frac{\gamma^{3}}{3}s_{1}(n)*s_{1}(n)*s_{1}(n)+... \tag{16}\]  
where \(*\) denotes the convolution, \(\gamma\) is the power exponent, \(s(n)\) is the original signal, and \(S(Z)\) is a kind of \(Z\)-transform for \(s(n)\), which corresponds to the output of the linear transformation layer in the \(M_{1}\) model (Figure 5-\(M_{1}\)). \(S_{1}(Z)\) and \(S_{2}(Z)\) represent the results after the log-transform and power exponential transform of \(S(Z)\), respectively, and \(S_{2}(Z)\) is related to \(S_{1}(Z)\) by Equation (15), which is expanded to give Equation (16). It demonstrates that the features' power-transform is a linear combination of log-transformed convolution. Moreover, the overlapping combined components of the log-transform are compressed into the power-transform by convolution; the power-transform gives richer feature information so that a good result is obtained in the spoofing speech detection (SPD) task. The power exponent, in particular, is configured as a learnable weight that self-adapts to changes in the task. After a previous test, it is appropriate to set the initial value to \(-1/7\), which can reduce the calculation time by about 3% (as shown in Table 7) while keeping the verification accuracy largely unchanged (Figure 12). \begin{table}
\begin{tabular}{l l l l l l l} \hline \hline
**Model** & **Method** & **Features** & **Non-Linearity** & **Accuracy \%** & **Increase \%** & **Training Times (h)** \\ \hline \(M_{0}\) & Hand-Crafted & Amplitude & Log & 87.6 & - & 3.5 \\  & & Amplitude & Log & 89.2 & 1.8 & 3.3 \\ \(M_{1}\) & & Amplitude & Power & 89.2 & 1.8 & 3.2 \\  & & & Log & 93.6 & 6.8 & 3.4 \\  & & & Power & 94.1 & 7.4 & 2.9 \\ \(M_{2}\) & & Amplitude and Phases & Log & 93.8 & 7.1 & 3.5 \\  & & & Power & 94.4 & 7.6 & 3.2 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: The effect of different activation functions (\(\gamma=-1/7\), Epoch = 200). On the other hand, the weights learned in the linear transformation layer are shown in Figure 13. The normalization can help reduce the noise of the weights. It can also be seen that there is a significant difference between the learned weights and the initial weights (Figure 13a,b), especially in the low-frequency region where AFEM retains additional features. In addition, during the range of 0-30 Hz, AFEM learns several influential frequency bands from the input data. The learned weight matrix shows apparent fluctuations, and the weight features are clear in the range of 0-30 Hz. However, after 30 Hz, the weight matrix is approximately high-pass filtered. There is a boundary effect similar to that in the speech spectrum with obvious noise interference in the weight matrix near 50 Hz. Furthermore, the validation accuracy of the network does not improve by increasing the weight matrix to 100 dimensions, as shown in Figure 13d-right. The learned 100-dimensional weight matrix is very similar to that in 40 dimensions (Figure 13c-right), which seems to be some linear transformation of 40 dimensions. One possible explanation for an AFEM with the same size STFT is that the increased dimensionality makes it possible for the linear transform layer to have one or no FFTs passing through each row. It leads to the feature recognition effect needing to improve. Therefore, the size of the weight matrix is not a controlling factor in improving the performance of the AFEM. Furthermore, 40 dimensions are suitable here, considering the calculation cost and additional parameter tuning.