# Dual Cancelled Channel STAP for Target Detection and DOA Estimation in Passive Radar
## 2 Passive Radar STAP Scheme  
In this section, we briefly recall the processing scheme proposed in [18] for passive radar STAP, here referred to as fullarray STAP since it jointly exploits all the available spatial DOF. The interested reader is referred to [18] for a more detailed description. We assumed the same signal and clutter model adopted in [10], for a dual channel system, and in [18], for the corresponding multichannel case. We considered a radar receiver installed on a platform moving at velocity \(v_{p}\) and exploiting a stationary illuminator of opportunity (Figure 1). The \(N\) receiving channels, displaced by \(d\) in the along-track direction, form a side-looking uniform linear array (ULA). We denote by \(\phi\) the angle between antenna end-fire and receiver to scatterer line of sight. After a preliminary stage including synchronisation and reconstruction of the reference signal, a batch processing architecture is separately applied at each receiving channel, recreating the conventional fast-time/slow-time framework of a pulsed radar operating at an equivalent pulse repetition time (PRT) given by the batch duration \(T\). For each batch, the range compression is performed by resorting to a reciprocal filtering strategy [28; 29; 30; 31]. This has the role of both improving the signal ambiguity function and providing a time-invariant impulse response by eliminating the temporal variability associated to the employed waveform of opportunity [10]. The range compressed data from the cell under test are arranged into a \((NM\times 1)\) space-time vector \(\widetilde{\mathbf{x}}\), with \(M\) being the number of batches in the coherent processing interval (CPI). In the target presence, this is expressed as \(\widetilde{\mathbf{x}}~{}=~{}A~{}\widetilde{\mathbf{s}}+\widetilde{\mathbf{d}}\), where \(A\) denotes the target complex amplitude; \(\widetilde{\mathbf{s}}\) the space-time steering vector; and \(\widetilde{\mathbf{d}}\) a disturbance vector, assumed Gaussian with covariance matrix \(\widetilde{\mathbf{Q}}\). For a side-looking ULA, vector \(\widetilde{\mathbf{s}}(f_{D},\phi)~{}=~{}\mathbf{s}_{t}(f_{D})~{}\bigotimes~{}\mathbf{s}_ {s}(\phi)\) has time and space components \(\mathbf{s}_{t}(f_{D})~{}=~{}[1,e^{-j2\pi f_{D}T},\ldots,\)  
Figure 1: Considered system geometry for a multichannel mobile passive radar exploiting a stationary transmitter as illuminator of opportunity. \(e^{-j2\pi f_{D}MT})^{H}\) and \(\mathbf{s}_{s}(\phi)=[1,e^{j2\pi d/\lambda\cos\phi},\ldots,e^{j2\pi Nd/\lambda \cos\phi}]^{H}\), \(f_{D}\) denoting the target Doppler frequency, \(\lambda\) the carrier wavelength, \(H\) the Hermitian transpose, and \(\bigotimes\) the Kronecker product. In [18], a post-Doppler STAP approach [21; 22; 23] is suggested as appropriate for mobile passive radar, where the typical long integration times provide a good decoupling of the clutter Doppler components. The same adjacent-bin post-Doppler (ABPD) strategy is adopted here by operating adaptively on the \(N\) spatial channels and a subset of \(L\) adjacent Doppler processed data around the cell under test. We denote by \(\mathbf{x}=A\ \mathbf{s}+\mathbf{d}\) the \((NL\times 1)\) data vector in the post-Doppler domain, where \(\mathbf{s}(\phi)=\mathbf{s}_{d}\bigotimes\mathbf{s}_{s}(\phi)\) is the steering vector and \(\mathbf{Q}\) the disturbance covariance matrix. The well-known adaptive matched filter (AMF) detector [32] is given by  
\[\frac{\left|\mathbf{s}^{H}\hat{\mathbf{Q}}^{-1}\mathbf{x}\right|^{2}}{\mathbf{s}^{H}\hat{\mathbf{Q }}^{-1}\mathbf{s}}\gtrgtr\eta \tag{1}\]  
where \(\eta\) is the detection threshold, selected according to a desired false alarm probability (PFA). Matrix \(\mathbf{Q}\) is replaced with its ML estimate \(\hat{\mathbf{Q}}=\mathbf{X}_{k}\mathbf{X}_{k}^{H}\), obtained by a set of training data \(\mathbf{X}_{k}=[\mathbf{x}_{1},\ldots,\mathbf{x}_{K}]\) from \(K\) adjacent range cells, assumed independent, identically distributed and target-free.