## 3 TF Transforms and Results
### TF Analysis  
The Short-Time Fourier Transform (STFT) [18; 29] is relatively simple transformation, which, due to it is low computational complexity, can be successfully used on various Single Board Computers (SBCs). The STFT involves performing the Fast Fourier Transform (FFT) over small time intervals. The STFT transformation has several parameters that can be controlled to adjust the transformation to the studied phenomenon. Such parameters are, e.g., the size of the signal window (i.e., the time interval) for which the sequential FFT calculations are performed. The most frequently used windowing functions are: Hann, Hamming or Kaiser [21]. The Kaiser function allows one to carry out the parametrization of the windowing characteristic. The STFT transformation is characterized by a constant window size during computations. Simultaneously, the size affects the resolution in time and frequency. Thus, the window may be wide in the time domain, but then, narrow in the frequency domain or vice versa. However, the resolution of calculations over time can be additionally increased by overlapping the successive windows of signals [17]. In this paper, the overlapping of 75% was used, which ultimately resulted in obtaining a resolution in  
Figure 2: Exemplary \(U_{BN}\) burst and magnetizing current course \(I_{e}\). \(\Delta f=488\) Hz and \(\Delta t=500\)\(\upmu\)s. The STFT and spectrogram is given, respectively, by the following formulas [29]:  
\[X(t,f)=\int_{-\infty}^{\infty}x(\tau)\omega(\tau-t)e^{-j2\pi f\tau}d\tau \tag{1}\]  
\[SPEC(t,f)=\left|X(t,f)\right|\text{ }^{2} \tag{2}\]  
where \(\omega(t)\)--window function, \(x(t)\)--input signal, \(X(t,f)\)--result of the STFT (matrix of complex number), and \(SPEC(t,f)\)--spectrogram of STFT. The FSST is the result of a post-processing method applied to the STFT transformation distribution. This method allows one to improve the concentration of the signal consisting of many components [6]. A Kaiser window of a size of 512 samples was used during the calculation of the STFT and the FSST as well. The applied frequency resolution was the same as in case of the STFT, but the resolution over time domain reached \(\Delta t=0.4\)\(\upmu\)s.  
The Continuous Wavelet Transform (CWT) is another transform commonly used to analyze non-stationary signals [30]. As in the case of the STFT, the square of the transform module defines a spectrogram, in the case of the CWT, the result is in a form of scalogram. Computations are also performed for the sequence of windowed signal; however, compared to the STFT, a window size is not fixed in this case. There are many types of base wavelets to choose from, which make it possible to emphasize various properties of the transformed signal at a given moment in time. In this paper, a Morlet wavelet is applied to obtain a constant variance for time and frequency domains. Similar to the STFT, in the case of the CWT, it is also possible to perform the scalograms sharpening using the SST, resulting in WSST transform. For the WSST, the resolution in the frequency domain is greater than for the CWT; however, both increase logarithmically. The resolution in time domain for both transformations was the same and equal to \(\Delta t=0.4\)\(\upmu\)s. The CWT is given by formula [5]:  
\[CWT(a,b)=\frac{1}{\sqrt{a}}\int_{-\infty}^{\infty}x(t)h^{*}\bigg{(}\frac{t-b}{ a}\bigg{)}dt \tag{3}\]  
where \(a\)--parameter responsible for scaling the wavelet, \(b\)--parameter responsible for translation effect (shift in time axis), \(h^{*}\)--complex conjugate of the scaled and shifted wavelet function, and \(x(t)\)--input signal. Another considered transformation is derived from the WVD transformation [31]. The WVD itself is the transformation that offers a wide range of analysis in the case of many time-varying signals. It overcomes the limited resolution of the transforms based on Fourier and wavelet methods through an autocorrelation operation. However, simultaneously, a certain difficulty brought by WVD is a cross-terms effect.