# Motion Compensation for Long Integration Times and DoA Processing in Passive Radars
## 6 Multi-Stage Range-Doppler Migration Compensation  
In expression (7), the spreading effects are detailed due to the target bistatic delay \(m_{lg0}=(\tau_{lg0}f_{s})\) and its variation during the \(T_{int}\). Then, the main objective is the estimation of target dynamics (range, Doppler, and Doppler rate parameters) in each CPI to perform coherent integration reducing integration losses. This technique is applied to every detected target, so targets must be detected before starting the compensation process. To implement target motion compensation techniques based on inverse discrete Fourier transform (IDFT) and discrete Fourier transform (DFT), targets dynamic characterized by constant bistatic acceleration and rectilinear trajectory in the bistatic range and Doppler domain during the CPI are assumed. The proposed strategy is based on the following steps:  
* **First Stage:** Although a first detection at single element level would simplify the process, bistatic tracker results after beamforming technique are used (Figure 6) due to the high propagation losses and the reduced integration gain. As target contributions are spread throughout the 3D matrix \(\mathbf{S}_{CAF}\), the probability of detecting such a target can be improved by increasing the probability of false alarm and using the bistatic tracker to control the excess of false alarms. * **Doppler migration compensation filter bank:** Once possible target trajectories are declared in the area of interest, the associated bistatic velocities and accelerations can be used for the definition of \(N_{fDr}\) discrete \(a_{bis}\) values, associated with Doppler rates from \(f_{Drmin}\) to \(f_{Drmax}\), which will be considered to perform Doppler migration compensation and to improve target detection capabilities. \(N_{fDr}\) should be selected as a compromise between accuracy and computational cost. For one specific value \(f_{Drm}^{i}\), the compensation technique is based on applying inverse discrete Fourier transform (IDFT) expressed in Equation (8). In other words, IDFT performing across the \(p\)-axis (\(IDFT_{p}\)) or Doppler axis of the RDMs. If the expression  
Figure 6: IDEPAR array signal processing processing scheme [8]. of \(S_{surv}[n]\) is substituted, the equality (9) is obtained where a \(f_{Dr}\)-dependent term responsible for the Doppler walk appears. \[ \begin{split}\text{IDFT}_{p}(S^{CAF}[m,p])\\ =\sum\limits_{p=-P/2}^{(P/2)-1}\sum\limits_{n=0}^{N-1}S_{ref}^{*}[n -m]\cdot S_{surv}[n]\cdot\exp\left(-j2\pi\frac{f_{D}n}{f_{s}}n\right)\cdot\exp \left(j2\pi\frac{f_{D}n}{f_{s}}n\right)\\ =\sum\limits_{n=0}^{N-1}S_{ref}^{*}[n-m]\cdot S_{surv}[n]\end{split}  \tag{8}\]  
\[ \begin{split}\text{IDFT}_{p}(S^{CAF}[m,p])\\ =\sum\limits_{n=0}^{N-1}S_{ref}^{*}[n-m]\cdot\xi_{surv}\cdot\xi \big{[}n-m_{target}[n]\big{]}\exp\left(j2\pi\frac{f_{D}n}{f_{s}}+\phi_{t0} \right)\exp\left(j\pi\frac{f_{D}n^{2}}{f_{s}}\right)\end{split}  \tag{9}\]  
Then, the filter \(H_{D}\), defined in (10), will be implemented to obtain a corrected version of the IDFT of the CAF in (11), resulting in the convolution of the reference signal with the corrected version of the surveillance one \(S_{surv}^{\prime}[n]\). \[H_{D}[n,f_{Dr}^{i}]=\exp\left(-j\pi\frac{f_{Dr}^{i}n^{2}}{f_{s}}\right) \tag{10}\]  
\[ \begin{split}\text{IDFT}^{\prime}_{p}(S^{CAF}[m,p])\\ =\sum\limits_{n=0}^{N-1}S_{ref}^{*}[n-m]\cdot\xi_{surv}\cdot\xi \big{[}n-m_{target}[n]\big{]}\cdot\exp\left(j2\pi\frac{f_{D}n}{f_{s}}+\phi_{t0 }\right)\\ =\sum\limits_{n=0}^{N-1}S_{ref}^{*}[n-m]\cdot S_{surv}^{\prime}[n ]\end{split}  \tag{11}\]
* **Second Stage:** The inputs of each filter \(H_{D}[n,f_{Dr}^{i}]\) are the CAFs generated by each single antenna of the surveillance array (Figure 7). The detection and tracking stages are performed with higher \(P_{FA}\) requirements to the outputs of each filter \(H_{D}[n,f_{Dr}^{i}]\), where an SIR improvement is expected to be associated with Doppler migration compensation.