## 2 Generation of F-F Image  
The real part of the discrete frequency spectrum (DFT) of the diagnostic signal after the removal of the element representing the constant component is treated as the vector \(\mathbf{S}=[s_{k}]\); \(k=1,\,2,\,3,\,\ldots\,\,k_{s};\,k_{s}=1/2(T/\Delta t)\) is the vector \(\mathbf{S}\) length; \(T\), measurement duration; \(\Delta t\), vibration signal sampling period. The parameter \(s_{k}\) represents the amplitude of the spectral band with the frequency \(k\Delta f\); \(\Delta f=1/T\) is the spectrum frequency resolution. The vector \(\mathbf{S}\) is converted into the array \(\mathbf{P}\):  
\[\mathbf{S}=[s_{k}]\rightarrow\mathbf{P}=[p_{n,m}] \tag{1}\]  
where:  
\(n=1,\,2,\,3,\,\ldots\,\,n_{p},\,n_{p}\) is line length;  
\(m=1,\,2,\,3,\,\ldots\,\,m_{p},\,m_{p}\) is column length;  
in such a way that successive parts of the vector \(\mathbf{S}\) of length \(n_{p}\) form successive rows of the \(\mathbf{P}\) table, where \(k_{s}\leq n_{p}m_{p}\). The quantity \(f_{\mathbf{P}}=n_{p}\Delta f\) will be called the characteristic frequency of the table \(\mathbf{P}\). In the F-F image created on the basis of the \(\mathbf{P}\) table, the amplitudes of individual frequency components are represented by the color or degree of intensity of a given color, uniquely related to the adopted system of contour lines (similarly to cartographic maps) [6,7]. The F-F image is organized such that the \(p_{1,1}\) element is in the lower left corner. The frequency of the spectral lines associated with the cells of the \(\mathbf{P}\) table is uniquely described by the indices \(n\) and \(m\):  
\[f_{p_{n,m}}=(m-1)n_{p}\Delta f+n\Delta f \tag{2}\]  
Thus, successive elements in the row are distant by the frequency \(\Delta f\), while successive elements of the column are distant by the characteristic frequency \(f_{\mathbf{P}}\). The shift of the successive parts of the vector \(\mathbf{S}\), such that their first element occupies appropriate cells of the first column of table \(\mathbf{P}\), can also be considered as a shift of the given part of the vector \(\mathbf{S}\) by the frequency \(f_{\mathbf{P}}\) in the frequency domain. This means that the frequency component \(f_{\mathbf{P}}\) and its harmonics and their sidebands distant by the frequency \(f_{b}\) or its higher harmonics, resulting from the modulation process, appear in the appropriate columns described by the indices \(f_{\mathbf{P}}\) and \(f_{\mathbf{P}}\pm\eta f_{b}\) (\(n=1,\,2,\,3,\,\ldots\,\,;\) harmonic order) and create vertical patterns in the F-F image. The measurement signal, discrete in time and amplitude domain, significantly affects the result of the approximation using the Fourier series. Determining the DFT is equivalent to finding the transform of the product of the signal and the rectangular window with the same time duration. Consequently, only the frequency components of the signal whose period is commensurate with the duration of the rectangular window are represented by a single spectral line; the remaining incommensurable frequency components are represented by the diffuse spectrum. The diffuse spectrum decreases with the increase in frequency resolution of the spectrum and thus with the duration of the analyzed signal. As mentioned in the introduction, in the signal, one should expect frequency components with frequencies both commensurate and incommensurate with the signal duration (also with frequencies closely related to each other) and frequency components resulting from the modulation processes taking place in the tested electro-mechanical system.