## 2 Gabor Filters and Their Properties  
A characteristic feature of Gabor filters is optimal localisation in the time and frequency domains. Individual filters can be adjusted to the characteristics of the processed signal. Gabor filter banks are used in time-frequency analysis. The features determined for the set of filters can then be combined using data fusion [9]. Multiresolution analysis may also be useful in the process of algorithm development and optimisation [10]. The disadvantage of Gabor filters is that their bandwidth is limited to one octave of frequencies. Log-Gabor filters do not have this limitation, as they also better attenuate lower frequencies. Gabor filters are widely used in various image analysis problems. This applies to both individual filters tailored to a specific application and a set of filters. Most applications of Gabor filters concern various classification problems. Very often, they are used to detect various types of defects [11] or identify features [12]. Gabor filters are also used to determine a shift based on recorded images [13]. Improving the quality of images, as well as achieving resistance to interference, can be achieved by using a log-Gabor filter [14]. Initial quality improvement in the case of images is most often understood as increasing the contrast between the details of interest or between the object of analysis and the background. The Gabor transform is also used in the analysis of one-dimensional signals, such as in nuclear magnetic resonance spectroscopy [15]. The transform is also useful for investigating various signals in mechanics [16], including vibration analysis [17]. The Gabor function in the one-dimensional case is a continuous complex function with two parameters related to the centre frequency \(\omega_{t}\) and the bandwidth \(\sigma_{t}\). The Gabor function can be written as follows:  
\[g(t)=\frac{1}{\sigma_{t}\cdot\sqrt{2\pi}}e^{-\frac{t^{2}}{2\sigma_{t}}}.e^{i \omega_{t}t}. \tag{1}\]  
This function is therefore the product of the Gaussian envelope with a complex sine wave. The Gabor function consists of a real part and an imaginary part, which can be written as:  
\[g(t)=\frac{1}{\sigma_{t}\cdot\sqrt{2\pi}}e^{-\frac{t^{2}}{2\sigma_{t}^{2}}}.( cos(\omega_{t}t)+i\cdot sin(\omega_{t}t)), \tag{2}\]  
where \(\omega_{t}=2\pi f_{t}\) is the central pulsation of the filter, and the parameter \(\sigma_{t}\) is responsible for the location of the filter in the time domain. The real part of the filter has even symmetry, and the imaginary part has odd symmetry. There is a phase shift of pi/2 between the real and imaginary parts. The optimal representation in the fundamental domain and in the frequency domain results from the shape of the Gabor function itself. A sine wave has the best localization in the frequency domain without the possibility of localisation in time. In turn, a single signal value (a sample) represents the exact position, but extends over the entire frequency range. Superimposing a Gaussian shape on sine waves introduces localisation in the time domain while limiting it in the frequency domain. To process and analyse digital, i.e., sampled, signals, we use the sampled version of the Gabor function. This sampled function is treated as the impulse response of a finite impulse response filter. Signal processing therefore involves convolving signal \(x(t)\) with such an impulse response:  
\[y(t)=x(t)*g(t). \tag{3}\]  
The output signal consists of a real and imaginary part:  
\[y(t)=y_{r}(t)+y_{i}(t). \tag{4}\]  
A filter that uses only the real part of the Gabor function is called group detection. The imaginary part is called edge detection [18]. Determining the modulus from the resulting Gabor filter response shows the amplitude (envelope) and phase of the signal at the frequency of this filter:  
\[y_{a}(t)=\sqrt{y_{r}^{2}(t)+y_{i}^{2}(t)}, \tag{5}\]  
\[\phi(t)=arctan\bigg{(}\frac{y_{i}(t)}{y_{r}(t)}\bigg{)}. \tag{6}\]  
For theoretical signals, such an envelope waveform should be smooth.