## 4 Determination of Rasterization Parameter Values
### Chirp-Z Transform (Chirp Transform)  
Fourier transform of the \(N\)-element sequence of signal samples \(x(n)\), where \(0\leq n\leq N-1\), determined for \((M-1)\) normed frequency values \(f_{k}=f_{o}+k\cdot\Delta f\), where \(0\leq k\leq M-1\), is determined by the relationship [26]:  
\[X(k)=\sum_{n=0}^{N-1}x(n)\cdot e^{-j2\pi f_{k}n}. \tag{3}\]  
Replacing:  
\[A=e^{-j2\pi f_{0}} \tag{4}\]  
and  
\[W=e^{-j2\pi\Delta f/2} \tag{5}\]  
there is obtained:  
\[X(k)=\sum_{n=0}^{N-1}x(n)A^{n}W^{2 kn}. \tag{6}\]  
Taking into account:  
\[2 kn=n^{2}+k^{2}-(k-n)^{2}, \tag{7}\]  
Fourier transform can be written as:  
\[X(k)=W^{k^{2}}\sum_{n=0}^{N-1}\left[x(n)A^{n}W^{n^{2}}\right]W^{-(k-n)^{2}}. \tag{8}\]  
The above expression corresponds to the convolution of two signals: \(y_{1}(n)=x(n)A^{n}W^{n^{2}}\) and \(y_{2}(n)=W^{-n^{2}}\), which in the spectral domain is realized as the product of the spectra of \(Y_{1}(k)\) and \(Y_{2}(k)\). The calculation of the CZT transform therefore requires three Fourier transforms to be determined: two simple and one inverse (Figure 6) [26; 38; 39; 40]. 2.1 Synthetic Signals Obtained by Decoding of the BMP File Subjected to the Differentiation Operation  
The original Raster Generator application version1.0.2 was used to recreate the data and present them in the form of graphic images (Figure 7). The software has three basic modules:  
* Graphic display and correction of rastering parameters;
* Low-pass, high-pass, median and edge detection;
* Signal analysis in time domain. Figure 6: The block diagram of the determining the Chirp-Z transform. These allow the analysis of the emission signals, which are revealed for all sources of these emissions in the form of graphic lines (e.g., laser printers, computer monitors, graphic displays, graphic interfaces, etc.). The current version of the software enables manual estimation of rastering parameters in the absence of knowledge of the operating mode of the eavesdropped source of unwanted emissions. The estimation accuracy allows for efficient summing up to 60 replicas of the same image obtained from a single realization of the recorded revealing signal. The signals described in point 3 (Figure 2) were used in the simulations. They show the test image often used in analyses of revealing emissions (Figure 2a) and the image of the MS Word application window used for text data processing. A differentiation of signals reflects the influence of an ideal information infiltration channel (Figure 8). For such images, an analysis of the accuracy of the line length estimation depending on the length of the transform (FFT and CZT) was conducted. The simulation results presented in Table 6 show that the precise estimation was obtained only for the test pattern in the form of vertical stripes (strict periodicity). The estimation for the MS Word test image is burdened with an error that has a significant impact on the image clarity after summation (Figure 9). Figure 9a,b present images after applying of 30-fold coherent summation. The line lengths of these images were estimated on the basis of analyzing the FFT algorithm (Table 6). The obtained results of line lengths are not the accurate values. Therefore, the summation of images does not improve the image quality. This phenomenon is shown on Figure 9, where reconstructed images  
Figure 8: Rasterized images on base on test signals obtained from the BMP file after the differentiation operation: (**a**) strips \(8\times 8\) pixels-fragment, (**b**) window of MS Word editor. Figure 7: A window of Raster Generator with an exemplary reconstructed image based on a recorded revealing signal. are blurred and illegible. However, this is typical of incorrect determination of raster parameters. Simultaneously, this points to a necessity to correct such parameters. Increasing the accuracy of the estimate requires the extension of the FFT transform. It can be concluded that satisfactory accuracy would be obtained with \(N=2^{27}=134,217,728\).