## 2 Informative Frequency Band Selectors
### Kurtosis  
The kurtosis statistic is the most known impulsive measure in the probability and statistical theory, see [50]. It gives knowledge about the non-Gaussianity of the signal (for the Gaussian distribution the statistic is equal to 0) or in other words about the impulsiveness of the signal. The kurtosis for the vector \(x=(x_{1},x_{2},...,x_{N})\) is defined as follows [51]:  
\[\hat{K}(x)=\frac{\frac{1}{N}\sum_{i=1}^{N}\left(x_{i}-\overline{x}\right)^{4}} {\left(\frac{1}{N}\sum_{i=1}^{N}\left(x_{i}-\overline{x}\right)^{2}\right)^{2 }}-3, \tag{1}\]  
where \(\overline{x}\) is the sample mean and \(N\) is the sample length. It is the most frequently used sparsity index in the diagnosis of bearing faults [45; 52]. In our approach, we apply the kurtosis not to the raw signal, but to its time-frequency representation. The time-frequency representation of the signal is widely used because it gives the temporal and spectral content simultaneously and provides much more information about data than the one-dimensional representation in the time domain. The signal decomposition can be performed by the short-time Fourier transform (STFT) or many other techniques (Wavelets, Wigner, EMD, etc.), see, e.g., [53]. In our case, the STFT is applied. The interpretation of the STFT-based map is intuitive--it describes the energy flow in time for some narrow frequency band, i.e., sub-signal [18; 20; 46; 49; 54]. The spectrogram has some limitations. It is well known that selection of the window, its length, overlap size, etc., is a compromise between time and frequency domain resolution and strongly depends on the signal. By trials and error, we have found the \"optimal\" solution for the given signal. To calculate the STFT we assume 512 sampling points to calculate the discrete Fourier transform and Hamming windowing for window size 256. The number of samples of overlap between adjoining segments has been set on \(\left\lfloor 85\%\cdot\text{window}\right\rfloor\).  
The usage of the spectral kurtosis has been proposed in the rotating machinery diagnosis by [PERSON] [45]. After time-frequency signal decomposition the statistic \(\hat{K}(\cdot)\) applied to the individual frequency band \(f_{i}\): \(\hat{K}(f_{i})\) is called spectral kurtosis.  
One may find the usage of the filter bank for the signal decomposition, e.g., \(1/3\)-binary tree structure [19]. Then the statistic \(\hat{K}(\cdot)\) can be applied to the real part of the envelope of the signal at frequency \(f_{i}\). The representation of the spectral kurtosis in the \((f,\Delta f)\) plane is called a kurtogram [19].  
Currently, it is known that methods which are based on the kurtosis statistic have some limitations. The value of the kurtosis decreases when the repetition rate of the impulses increases. If the cyclic impulses become so frequent that they overlap over each other, then the kurtosis vanishes [55]. On the other hand, when the outliers (non-cyclic impulses) are observed in the recorded signal then the kurtosis takes maximal value. This issue was addressed in [3] where the authors proposed an interesting alternative, namely the protrugram. It is based on the kurtosis of the envelope spectrum rather than the signal. However, despite its advantages, the protrugram has some limitations as well. It considers only the periodicity without scoring the impulsive behaviour. What is more, in the case of outliers its envelope spectrum distorts, thus the protrugram's results can be misleading.