# Motion Compensation for Long Integration Times and DoA Processing in Passive Radars
## 4 Long Integration Time Passive Radar Processing  
In a classical PR, the processing scheme is limited by the time a target remains in the same resolution cell. In this case, target bistatic time delay and Doppler shift can be assumed to be constant during the CPI, and the CAF output, expressed in (6), can be obtained from Equation (1) being \(n=(t-t_{0})f_{s}\), \(t_{0}=n_{CPI}\cdot PRI\), \(m_{tg0}=(\tau_{tg0}f_{s})\), \(p_{tg0}=(f_{D0}N/f_{s})\) and \(E_{T}\) the energy of \(\tilde{s}(t)\).  
Figure 3: Projections on the passive radar, illuminator of opportunity, and target plane.  
\[S^{CAF}\big{[}m_{tg0},p_{tg0}\big{]}=\sum_{n=0}^{N-1}S_{ref}^{*}\big{[}n-m_{tg0} \big{]}\cdot S_{surv}[n]\cdot\exp\big{(}-j2\pi\frac{p_{tg0}}{N}n\big{)} \tag{6}\] \[=\xi_{ref}^{*}\cdot\xi_{surv}\cdot E_{T}\cdot\exp\big{(}\phi_{t0} \big{)}\]  
Range walk (the movement of the target during \(T_{int}\) exceeds the range resolution) and Doppler walk (the acceleration of the target produces a Doppler cell migration during \(T_{int}\)) give rise to target contributions' spreading along range and Doppler and integration gain losses. The associated reduction of signal to interference level will deteriorate the probability of detection (\(P_{D}\)) for the selected probability of false alarm (\(P_{FA}\)). Additionally, the spreading effects complicate the estimation of the detected targets' bistatic range and Doppler and reduce the system's accuracy.  
To avoid range walk, the target movement during \(T_{int}\) should be smaller than the bistatic range resolution cell, \(\Delta_{R_{b}}=c/BW\). The main parameters are the radial components of the target velocity vector projected on to the bistatic plane, \(\vec{v}_{tg}=v_{tg}\phi_{tg}\), with respect to the IoO, \(v_{tg,T}\), and the PR receiver, \(v_{tg,FR}\). These components depend on the target trajectory described by the unitary vector \(\phi_{tg}\). Taking into consideration that \(v_{tg,T}\) and \(v_{tg,FR}\) are lower than or equal to \(v_{tg}\), a worst case \(T_{int}\) maximum value can be estimated as \(T_{int,max}=c/(2 BWv_{tg})\). This expression assumes a constant \(v_{tg}\) during \(T_{int}\).  
The bistatic velocity resolution depends on \(f_{c}\) and \(T_{int}\): \(\Delta_{v_{bin}}=c/(f_{c}T_{int})\). If target bistatic acceleration is assumed to be constant during the integration time, the maximum value that ensures the target remains in the same Doppler cell can be obtained as \(a_{bis,max}=c/(2f_{c}T_{int}^{2})\). The dependence on the signal frequency is an important factor, especially for DVB-S signals, due to the high carrier frequencies (in the Ku band).  
If \(T_{int}\) is higher than \(T_{int,max}\) and the target bistatic acceleration is higher than \(a_{bis,max}\), \(S^{CAF}\big{[}m_{tg0},p_{tg0}\big{]}\) has to be modified, as seen in Equation (7), where \(m_{tg}=(\tau_{tg}f_{s})\), to include the spreading effects.  
\[S^{CAF}\big{[}m_{tg0},p_{tg0}\big{]}=\sum_{n=0}^{N-1}S_{ref}^{*} \big{[}n-m_{tg0}\big{]}\cdot S_{surv}[n]\cdot\exp\big{(}-j2\pi\frac{p_{tg0}} {N}n\big{)} \tag{7}\] \[= \sum_{n=0}^{N-1}S_{ref}^{*}\big{[}n-m_{tg0}\big{]}\cdot S_{surv} ^{*}g^{*}\big{[}n-m_{tgg}\big{]}\exp\big{(}j2\pi f_{c}\frac{m_{tg}}{f_{s}} \big{)}\cdot\exp\big{(}-j2\pi\frac{p_{tg0}}{N}n\big{)}\] \[= \xi_{ref}^{*}\xi_{surv}\exp\big{(}\phi_{t0}\big{)}\sum_{n=0}^{N-1} S^{*}\big{[}n-m_{tg0}\big{]}\xi\Big{[}n-m_{tg0}+\frac{f_{D}n}{f_{c}f_{s}} \big{]}\exp\big{(}j\pi\frac{f_{D}n^{2}}{f_{s}}\big{)}\]