## 4 Numerical Experiments
### Results
The Hough transform was applied on the curve fragment derived from GPS data (see the green line in Figure 9), allowing the precise Doppler rate to be assessed. Next, the error between the reference (GPS data) and two estimated lines was computed. However, due to the limited precision of the DVB-T transmitter localization, as well as the smoothing of the GPS-based trajectory, the actual crossing point is mismatched. The additional sources of mismatch error may be connected with the GPS logger. This device's coordinates estimation accuracy is limited, and as well as its own location inside the aircraft also mat  
Figure 8: The energy distributions of the measured signal obtained using the concentrated STFT. Figure 9: Results of the Hough transform obtained for the two methods investigated—the classical and the reassigned STFT. which had an impact on the accuracy is the information about the transmitter (T\({}_{\rm X}\)) position. For the analysis, the authors took the T\({}_{\rm x}\) position from an open database and validated the T\({}_{\rm x}\) coordinates using Google Maps. However, the accuracy of the T\({}_{\rm x}\) position is also often given with a precision of several meters, which might have had an impact on the presented results. Therefore, an additional simulation was performed in which the geometry was appropriately modified by changing the transmitter position that aims to reduce this fault. After these modifications the error was eliminated, and the results for both the initial and modified trajectories are depicted in Figure 10. The plots show the absolute value of the estimation error \(\delta\). The proposed method allowing signal concentration in the TF to be obtained improved the precision of the Doppler rate estimation. As can be observed, the error was reduced for both parameters: the Doppler rate, and for the time when the target crossed the baseline. This result indicates the effectiveness of the proposed approach. For the Hough transform, the computational complexity increases at a rate of \(\mathcal{O}\left(A^{m-2}\right)\) where \(A\) denotes the size of the image space and \(m\) corresponds to the number of parameters applied in the processing pipeline. In the proposed method, the Hough transform is the same as in the classical approach with STFT. Therefore, the only difference results from the signal processing associated with the reassignment operation. For this reason, only the STFT and the reassigned STFT computational complexity are compared in Table 2[40]. The increase in computational complexity in the concentrated spectrogram technique results from the fact that 3 distributions are necessary to be implemented in accordance to Equation (20). Namely, the first distribution is the classical STFT with the original window according to Equation (11). The second one corresponds to Equation (16), and the last one to Equation (17). Hence, the precision of the Doppler rate can be improved at the expense of computational complexity. In fact, the processing time may be reduced through the manipulation of the processing parameters. For the purpose of this paper, the number of frequency bins and the window shift were assumed with some redundancy for high resolution of distributions. In practice, these parameters can be reduced to ensure fast processing. For the parameters defined above, the computation time for the spectrogram (\(t_{S}\)) and the reassigned spectrogram (\(t_{R}\)) was respectively \(t_{S}=0.42\) s and \(t_{R}=1641.09\) s, but for 1024 points of the FFT this \begin{table}
\begin{tabular}{c c} \hline \hline  & **STFT** & **Reassigned STFT** \\ \hline Computational complexity & \(\mathcal{O}\left(KN\log_{2}(N)\right)\) & \(3\cdot\mathcal{O}\left(KN\log_{2}(N)\right)\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Computational complexity for the STFT and the reassigned STFT in the \(\mathcal{O}\) notation.