## 2 Gabor Filters and Their Properties
Due to the presence of noise and distortions in real signals, the received envelope signal may require additional smoothing. The next step in such a case may be a smoothing Gabor filter optimised for the low frequency value of the processed signal [19]. In the frequency domain, the Gabor filter can be described by the following equation:  
\[G(f)=e^{-2\pi^{2}\sigma_{1}^{2}\cdot(f-f_{t})^{2}}. \tag{7}\]  
The half-peak bandwidth of a Gabor filter can be defined by the following equation [20]:  
\[B=log_{2}\frac{\sigma_{t}f_{t}\pi+\sqrt{\frac{ln2}{2}}}{\sigma_{t}f_{t}\pi- \sqrt{\frac{ln2}{2}}}. \tag{8}\]  
The average value of the impulse response of a Gabor filter is not zero; therefore, such a filter does not suppress the DC component. In general, the basic version of the Gabor filter over-represents low-frequency components. A modification of the Gabor filter is the log-Gabor filter [21], which suppresses the DC component and reduces low-frequency components. It has been proven that the log-Gabor filter is insensitive to changes in the illumination level of the analysed images [22]. The log-Gabor function additionally has a wider bandwidth for higher frequencies. The frequency response of a log-Gabor filter has a Gaussian shape if the frequency scale is logarithmic. For the basic Gabor filter, the Gaussian shape of the filter frequency response occurs for a linear frequency scale. The frequency spectrum of the log-Gabor filter is described by the following function [23]:  
\[G(f)=e^{-0.5\cdot log(\frac{f}{f_{t}})^{2}/log(\sigma/f_{t})^{2}} \tag{9}\]  
The bandwidth of the log-Gabor filter depends on the parameter \(\sigma\). Larger values of \(\sigma\) increase the filter bandwidth. The frequency band of the log-Gabor filter can be determined using the following formula [24]:  
\[B=2\sqrt{2/log2}\cdot|log(\sigma/f_{t})| \tag{10}\]  
Log-Gabor filters have their representation only in the frequency domain. To implement such a filter, the multiplication operation of the filter transfer function and signal frequency spectrum must be performed. The frequency spectrum of the signal processed in this way is recovered from the frequency to the original domain by inverse Fourier transform. The reproduced signal is a complex signal. Its real part corresponds to the symmetric part of the filter, and its imaginary part to the odd part. Therefore, it is the equivalent of a quadrature filter, as in the case of the classic Gabor filter. The optical spectrum signal can also be considered as an amplitude modulation signal in telecommunications. Then, the derivative of the signal envelope can be used to estimate the cut-off wavelength. To determine the envelope, a band-pass filter can be used, after which we calculate the absolute value of the signal, and in the last step, we apply a smoothing filter. Such a filter should be a low-pass filter that specifically attenuates twice the frequency of the band-pass filter. A doubling of the fundamental frequency of the signal occurs after rectification of the signal. The signal envelope can also be calculated using the analytical signal. It is a complex signal in which the real part is the original signal, and the imaginary part is the Hilbert transformation of this signal:  
\[g(t)=\frac{1}{\sigma_{t}\cdot\sqrt{2\pi}}e^{-\frac{t^{2}}{2\sigma_{t}^{2}}}.( cos(\omega_{t}t)+i\cdot sin(\omega_{t}t)), \tag{2}\]0  
The Hilbert transformation of the signal \(x(t)\) can be written as its convolution with the kernel \(1/\pi x\):  
\[g(t)=\frac{1}{\sigma_{t}\cdot\sqrt{2\pi}}e^{-\frac{t^{2}}{2\sigma_{t}^{2}}}.( cos(\omega_{t}t)+i\cdot sin(\omega_{t}t)), \tag{2}\]1  
This operation can also be performed in the Fourier domain by changing the sign to the opposite for negative frequencies of a signal. However, the analytical signal does not have the property of simultaneous localisation in time and frequency. Gabor filters calculate the equivalent of an analytical signal with simultaneous band-pass pre-filtering.