# Dual Cancelled Channel STAP for Target Detection and DOA Estimation in Passive Radar
## 3 Dual Cancelled Channel STAP for Passive Radar
\tag{5}\]  
\(\hat{\mathbf{Q}}_{AB}\) is the \((2\times 2)\) estimated covariance matrix at the output of channels A and B. Notice that \(\mathbf{Q}_{AB}\) is a Toeplitz Hermitian matrix with terms \(\alpha\) and \(\rho\) respectively being the average disturbance power at the output of the two channels and their cross-correlation. This aspect can also be exploited to obtain a more stable estimate from the data. The optimal recombination of the two channels allows for recovery of most of the detection loss with respect to the full array case [26]. Moreover, due to the smaller number of adaptive DOF used by each step of the AB-STAP approach, a lower computational cost as well as reduced adaptivity losses are expected in comparison to full array STAP. This latter aspect in particular plays a fundamental role in a real passive radar scenario, where the potential non-homogeneity of bistatic clutter may limit the number of relevant training data usefully exploitable for filter adaptivity. (iv) Estimation of target DOA from the two cancelled channels: for range cells where a detection is declared, target DOA can be estimated from the two cancelled channels exploiting their different phase centres. Applying the MLE to the two outputs results in  
\[\hat{\phi}_{t,AB}=\arg\max_{\phi}\left\{\frac{\left|\mathbf{s}_{AB}^{H}(\phi) \hat{\mathbf{Q}}_{AB}^{-1}\mathbf{x}_{AB}\right|^{2}}{\mathbf{s}_{AB}^{H}(\phi)\hat{\mathbf{Q }}_{AB}^{-1}\mathbf{s}_{AB}(\phi)}\right\}. \tag{6}\]  
The maximisation is performed only with respect to the last stage of the processing chain, which only involves two-dimensional quantities. This allows us to find a closed form expression for the ML estimate of target DOA. By defining the quantities \(\mathbf{z}=\left[ \begin{array}{cc}z_{1}&z_{2}\end{array} \right]^{\mathrm{T}}= \mathbf{Q}_{AB}^{-1}\mathbf{x}_{AB}\), \(v=z_{1}z_{2}^{H}/|\mathbf{z}|^{2}\), and \(u=\rho/\alpha\), we obtain the ML estimate of target DOA as (see [27] for more details)  
\[\hat{\psi}=\arcsin\left(\frac{2\mathrm{Im}\{vu^{H}\}}{|2v+u|}\right)-\angle(2 v+u) \tag{7}\]  
\[\hat{\phi}_{t,AB}=\arccos\left\{\frac{\lambda}{2\pi d(N-N_{0})}\hat{\psi}+ \cos\phi_{L}\right\}. \tag{8}\]  
where \(\mathrm{Im}\{\zeta\}\) and \(\angle(\zeta)\) are imaginary part and phase of complex scalar \(\zeta\), respectively. The obtained DOA estimator does not involve a functional maximisation and it is implemented by a simple closed form expression, thus limiting the computational cost. It is worth noting that the value of \(N_{0}\) defines the number of spatial DOF used for clutter rejection on channels A and B, but it also affects the distance of their phase centres. Therefore, a trade-off exists between the theoretical sensitivity of DOA estimation and the signal-to-clutter plus noise ratio (SCNR) achievable at the output of the channels. Appropriate choices should be made according to the specific scenario. Notice that the SCNR at the output of the A and B channels, responsible for DOA estimation accuracy, is always lower than the SCNR available for target detection after channel recombination.