# Dual Cancelled Channel STAP for Target Detection and DOA Estimation in Passive Radar
## 3 Dual Cancelled Channel STAP for Passive Radar  
The dual cancelled channel STAP approach adaptively forms two clutter cancelled channels that are then directly exploited for both target detection and DOA estimation. Specifically, we consider the AB-STAP technique, originally presented in [26]. This is particularly suitable for the passive radar case, thanks to its simple architecture, reduced computational load, and robustness against highly non-homogeneous clutter scenarios. The adopted scheme is sketched in Figure 2. It is obtained by modifying the scheme proposed in [18], so as to integrate the considered dual cancelled channel approach. Specifically, the processing steps that characterise the AB-STAP scheme, and that differentiate it from the original one, develop downstream of the post-Doppler transformation stage. These processing steps are described in the following:  
(i) Formation of two spatially displaced channels: dividing the antenna array into two (possibly overlapped) sub-arrays, the data vector \(\mathbf{x}\) is split into two \((N_{0}L\times 1)\) vectors, \(\mathbf{x}_{A}\) and \(\mathbf{x}_{B}\), corresponding to channels A and B, respectively. One includes the samples from the first \(N_{0}\) antennas, the other the samples from the last \(N_{0}\) antennas \((N/2\leq N_{0}<N)\). (ii) Cancellation of clutter on each channel: the same post-Doppler STAP is applied on both channels by filtering \(\mathbf{x}_{A}\) and \(\mathbf{x}_{B}\) with a sample matrix inverse scheme and obtaining  
\[y_{A(B)}=\mathbf{w}_{A(B)}^{H}\mathbf{x}_{A(B)}=\mathbf{s}_{A(B)}^{H}(\phi_{L})\hat{\mathbf{Q}} _{A(B)}^{-1}\mathbf{x}_{A(B)} \tag{3}\]  
where \(\mathbf{s}_{A(B)}(\phi_{L})\) and \(\hat{\mathbf{Q}}_{A(B)}\) are the steering vector and the estimated covariance matrix of channel A (B), respectively. Notice that, while \(\phi\) represents the generic unknown target DOA, we denote by \(\phi_{L}\) the array look direction. This contributes to the reduction in the computational complexity, since it involves a smaller size of the inverse matrix operation. Moreover, in a uniform linear array, channels A and B exhibit disturbance with the same structure, being \(\mathbf{s}_{B}(\phi_{L})=\mathbf{s}_{A}(\phi_{L})\exp\{j2\pi(N-N_{0})\)\(d/\lambda\cos\phi_{L}\}\). As a result, a better estimate can be obtained by averaging the estimates made on the two channels, that is, \(\hat{\mathbf{Q}}_{A}=\hat{\mathbf{Q}}_{B}=\frac{1}{2}\big{(}\mathbf{X}_{kA}\mathbf{X}_{kA}^{H} +\mathbf{X}_{kB}\mathbf{X}_{kB}^{H}\big{)}\), with \(\mathbf{X}_{kA}\) and \(\mathbf{X}_{kB}\) being the \((N_{0}L\times K)\) sets of training data for channels A and B, respectively. Although not totally independent (if \(N_{0}>N/2\)), averaging the two estimates yields a more stable estimate of the true covariance matrix, as well as a single matrix inversion. (iii) Recombination of channels for target detection: using only one of the two channels to detect targets may result in significant performance degradation in terms of cancellation capability, with the equivalent antenna length and employed spatial DOF being reduced. To recover this degradation and avoid detection losses, an optimal coherent recombination of the two channels is performed by solving a small \((2\times 2)\) adaptive problem. Arranging the two outputs in the \((2\times 1)\) vector \(\mathbf{x}_{AB}=\left[y_{A}\,y_{B}\right]^{T}\) and applying the AMF detection scheme, we obtain  
\[\frac{\left|\mathbf{s}_{AB}^{H}\hat{\mathbf{Q}}_{AB}^{-1}\mathbf{x}_{AB}\right|^{2}}\mathbf{{s}_{AB}^{H}\hat{\mathbf{Q}}_{AB}^{-1}\mathbf{s}_{AB}}\gtrless\eta_{AB} \tag{4}\]  
Figure 2: Processing scheme for post-Doppler AB-STAP in passive radar. where \(\eta_{AB}\) is the detection threshold, and the steering vector \(\mathbf{s}_{AB}\) is given by  
\[\mathbf{s}_{AB}(\phi_{L})=\left[ \begin{array}{c}\mathbf{w}_{A}^{H}\mathbf{s}_{A}(\phi_{L}) \\ \mathbf{w}_{B}^{H}\mathbf{s}_{B}(\phi_{L})\end{array} \right]=\mathbf{w}_{A}^{H}\mathbf{s}_{A}( \phi_{L})\left[ \begin{array}{c}1\\ 1\end{array} \right].