# Motion Compensation for Long Integration Times and DoA Processing in Passive Radars
## 6 Multi-Stage Range-Doppler Migration Compensation
The \(N_{/Dr}\) tracking results are combined using a weighted centroid of the target detected pixels and their SIRs in each CPI to obtain target trajectories in \(T_{acq}\) with more accuracy. At the output of the considered detector, the detection matrix is generated, and a target can be observed in a set of cells depending on its size and kinematics. The target position (range-Doppler in the RDM) could be estimated by the data extractor. This extractor applies algorithms such as the flood-fill algorithm, which connects all of the hints associated with a target and estimates the target position in the system coordinates. These estimated values define a point in the input 2D space, which is denoted as the plot associated with the target. Once the connected hints are detected, the centroid related to each blob is calculated as its center of mass considering the corresponding pixels intensity or outputs of the envelope detector. For each declared target trajectory, a mean value of Doppler Shift rate during \(T_{acq}\), \(\overline{f_{Dr}}\), is estimated to define a constrained version of the filter bank with \(N_{/Dr}^{\prime}<N_{/Dr}\) filters designed for \(f_{Drr}^{c,1}...f_{Drr}^{c,N_{/Dr}^{\prime}}\) values close to \(\overline{f_{Dr}}=\overline{u_{bis}}/\lambda\) (Figure 8). For the next stage and for each CPI, only the corrected CAFs associated with the filter designed the \(f_{Dr}^{c,1}\) that provides the highest SIR are selected. This strategy is based on refining the Doppler migration compensation in case the target bistatic acceleration is not constant during \(T_{acq}\). * **Third Stage:** Before range migration correction is performed, the discrete Fourier transform (DFT) of expression (11) is used and detailed in (12). In this case, DFT is performed across the \(m\)-axis (\(DFT_{m}\)) or bistatic range axis of the RDMs. With a change of variable \(y=n-m\), the result (13) can be expressed in the function of the DFT of the surveillance signal corrected version. \[\mathrm{DFT}_{m}(\mathrm{IDFT}^{\prime}_{p}(S^{CAF}[m,p])^{\prime})=\sum_{m=0} ^{N-1}\sum_{n=0}^{N-1}S^{*}_{ref}[n-m]\cdot S^{\prime}_{suro}[n]\exp\left(-j2 \pi\frac{k}{N}m\right)\] (12) \[\mathrm{DFT}_{m}(\mathrm{IDFT}^{\prime}_{p}(S^{CAF}[m,p])^{\prime})\] \[=\sum_{y=0}^{N-1}S^{*}_{ref}[y]\exp\left(j2\pi\frac{k}{N}y\right) \sum_{n=0}^{N-1}S^{\prime}_{suro}[n]\exp\left(-j2\pi\frac{k}{N}n\right)\] (13) \[=\mathrm{IDFT}_{y}(S^{*}_{ref}[y])\cdot\mathrm{DFT}_{n}(S^{ \prime}_{suro}[n])\]  
The use of the expression of \(S^{\prime}_{suro}[n]\) and the properties of the DFT leads to a \(f_{\mathrm{D}}\)-dependent exponential, responsible for the range migration, defined in (14). Then, the filter \(H_{R}\), (15) is implemented to obtain an updated version of the surveillance signal, \(S^{\prime\prime}_{suro}[n]\), and to implement the range target motion compensation expressed in (16). \[\mathrm{DFT}_{m}(\mathrm{IDFT}^{\prime}_{p}(S^{CAF}[m,p])^{\prime})= \mathrm{IDFT}_{y}(S^{*}_{ref}[y])\xi_{suro}\mathrm{DFT}_{n}\Big{(}\tilde{s}[ n],k-\frac{f_{\mathrm{D}}N}{f_{s}}\Big{)} \tag{14}\] \[\times\exp\left(-j2\pi\frac{k}{N}m_{target0}+\phi_{t0}\right)\exp \left(j2\pi\frac{k}{N}\frac{f_{\mathrm{D}}\mathrm{net}}{f_{s}f_{s}}\right)\]  
\[H_{R}[k,n]=\exp\left(-j2\pi\frac{k}{N}\frac{f_{\mathrm{D}}n}{f_{c}f_{s}} \right)\qquad k=0,\ldots,N-1 \tag{15}\]  
Figure 8: Doppler migration filter banks for a specific target. Figure 7: Doppler migration compensation scheme for \(f^{i}_{Dr}\). \[ \begin{split}\text{DFT}_{m}^{\prime}(\text{IDFT}_{p}^{\prime}(S^{ CAF}[m,p])^{\prime})=\text{IDFT}_{y}(S_{ref}^{*}[y])\xi_{surv}\text{DFT}_{n} \Big{(}\Big{(}\xi[n],k-\frac{f_{D}N}{f_{s}}\Big{)}\\ \times\exp\left(-j2\pi\frac{k}{N}m_{target0}+\phi_{i0}\right)= \text{IDFT}_{y}(S_{ref}^{*}[y])\cdot\text{DFT}_{n}(S_{surv}^{\prime\prime}[n] )\end{split}