## 3 Data Analysis
### Results for the Simulated Data  
In this section, the comparative results are presented for simulated data. For each of the previously defined signals \(s_{1},s_{2},s_{3}\) and \(s_{4}\) the selectors presented above have been applied. In Figure 2a the spectrogram of the signal \(s_{1}\) has been presented. The signal is the Gaussian White Noise, so it is not impulsive and not periodic. Figure 2: Results of the informative frequency band selection for 7 different methods for the signal \(s_{1}\). As one can see in Figure 2b-h, if the signal contains only Gaussian noise, all of the considered techniques (kurtosis, CVB, Alpha selectors, spectral Gini and spectral smoothness indexes), as well as the infogram, have relatively small amplitudes and do not indicate the informative frequency band, as expected. There is no frequency band within 256 tested, which significantly stands out from the others (in Figure 2e-h see the last 5 th level of the decomposition in the 1/3-binary tree structure). However, one can note that the spectral smoothness index takes maximum value within all frequency range up to the 3 rd level of the decomposition. It is related to the decomposition of the signal through the filter bank (for each decomposition level the signal is shorter). It changes the length of the vector, which is tested by the given statistic. As one can see in Equation (4), in the denominator of the statistic's definition there is the product of the vector's components. Hence, if the value of the vector components are relatively small in the absolute values, then the spectral smoothness index tends to infinite (marked in yellow in Figure 2g). One can notice that this happen especially in case of the long vectors with small values, here up to the 3 rd level of the decomposition. For the cyclic impulsive signal, described in Equation (9), we expect that the techniques point out the IFB between 2-3 kHz (the centre frequency is set on 2500 Hz). As one can see in Figure 3b-h, for signal \(s_{2}\) all techniques work well but the results for the CVB selector seem to be most unequivocal. The value of the CVB selector in the range of the IFB is significantly higher than for other frequency bins. If the cyclic impulses will have higher amplitudes then their variance will increase and the value of the CVB statistic will increase as well. For the tested signal, the \(ACI\) parameter determines the amplitude of the cyclic impulses and it is assumed that \(ACI\) is equal to 3. It means that the cyclic impulses are almost fully hidden in the Gaussian noise \(ACI\leq 3\sigma+\mu\), where \(\sigma=1\) and \(\mu=0\). Figure 3: Results of the informative frequency band selection for 7 different methods for the signal \(s_{2}\). Figure 4 presents the spectrogram and the selectors' results for the signal with the background of the mixture of Gaussian White Noise and non-cyclic impulses (signal \(s_{3}\)) described in Equation (10). For the non-cyclic impulsive signal \(s_{3}\), all techniques properly select the frequency band where the non-cyclic impulses appear. Note that, none of the methods takes into consideration the cyclic behaviour of the signal, only its impulsiveness. The only exception is the infogram, which takes into consideration both impulsiveness of the signal in the time domain and periodicity in the envelope spectrum; see [48]. However, one can expected it will not point out the frequency band of non-cyclic impulses as there is no periodicity of impulses. In fact, only the SE infogram is sensitive to the impulses. The SES infogram has dispersed values from range 0.45-0.7 kHz, whereas the SE infogram takes values from range 1-6.5 kHz and the maximum value it takes in the middle of this range (6 kHz). However, due to the use of the average of both of them in the infogram definition, see Equation (8), the result is in favor of non-cyclic impulses.