# Motion Compensation for Long Integration Times and DoA Processing in Passive Radars
## 7 Experimental Results
Considering the constraints of long integration times described in Section 4, the range and Doppler migration in each CPI are expected because:  
* \(T_{int}>T_{int,max}=150\) ms (for \(v_{tg}=10\) m/s and \(B=100\) MHz). * \(a_{tg}>a_{tg,max}=0.053\) m/s\({}^{2}\) (for \(T_{int}=500\) ms and \(f_{c}=11.315\) GHz). Following the methodology described in Section 7, the first stage has the objective of target detection and tracking during the acquisition time using the scheme detailed in Figure 6, generating orthogonal beams steering at \(\theta=[-10^{\circ},-7.2^{\circ},-4^{\circ},0^{\circ},4^{\circ},-7.2^{\circ}, -10^{\circ}]\). The considered \(P_{FA}\) for the CFAR detector was set to \(10^{-4}\), a high value for improving the detection capabilities of spread target backscattering. A reference and guard 2D-windows with 32 and 4 cells in both dimensions (range and Doppler), respectively, was selected. In Figure 13(a),b, the cumulative detection matrix in \(T_{acq}\) and the bistatic tracker performance are presented, respectively. The results demonstrate the first approach of the collaborative target trajectory, and the quality parameters can be summarized in \(P_{D}=91.14\%\) (\(P_{D}\) has been estimated at the tracker level, i.e., with the tracker output taking into account the points detected within each track following the methodology detailed in [43]), \(P_{FA}=2.65\cdot 10^{-4}\) (\(P_{FA}\) estimation has been done by choosing an area of the detection map where it is assured to be only noise at the pixel level using Monte-Carlo techniques, ensuring an estimation error of less than 10%), \(f_{D0}=-1018\) Hz, and \(\overline{f_{Dr}}=-12.36\) Hz/s (mean value during \(T_{acq}\)). The declared trajectory presents a non-linear Doppler variation at the beginning of the acquisition (or non-uniform bistatic acceleration variation), where the motion compensation technique could not achieve the best integration gain. Figure 13: Single element CAF of CPI 10. The next step is focused on Doppler migration correction based on a bistatic acceleration filter bank designed for \(N_{fDr}=27\) values from \(f_{Drmax}=94.3\) Hz/s (\(a_{tismax}=2.5\) m/s\({}^{2}\)) and \(f_{Drmin}=-94.37\) Hz/s (\(a_{bisimin}=-2.5\) m/s\({}^{2}\)). To improve the understanding of the filter bank bistatic tracking results with \(P_{FA}=10^{-5}\), only performances associated with equispaced \(H_{D}[n,f_{Dcr}^{i}]\) are presented in Figure 15a. The SIR of the collaborative target is high enough to be detected and tracked in all of the \(H_{D}\) filters. Then, the declared trajectories by the filter bank corresponding to the same target are combined to obtain a final result with more accuracy. Focused on the collaborative vehicle, new detection probabilities and target parameters are estimated: \(P_{D}=98.71\%\), \(P_{FA}=2.81\cdot 10^{-5}\), \(f_{D0}=-1006\) Hz and \(\overline{f_{Dr}}=-13.67\) Hz/s. The detection capabilities are clearly improved, and comparison results with respect to the target dynamic estimation in the first stage are depicted in Figure 15b. The new estimated \(f_{Dr}\) is used to reduce the filter bank size with \(N_{fDr}^{\prime}<N_{fDr}\), and Doppler walk compensation is performed by selecting, for each CPI, the output of the filter designed for \(f_{Dr}^{i}\) with the highest SIR or with the highest coherent processing integration gain due to the fact \(a_{bis}^{i}=\lambda f_{Dr}^{i}\) is the closest to the instantaneous target bistatic acceleration \(a_{bis}(t=nT_{int})\). The last stage consists of applying the filter defined in Equation (15) considering the estimated \(f_{D0}\) values, for each CPI with \(t_{0}=nT_{int}\), based on bistatic tracking results presented in Figure 15b. When the output of range walk correction returns to the CAF domain, a new improvement of SIR is expected so the direction of arrival estimation can be implemented with more accuracy.