# Dual Cancelled Channel STAP for Target Detection and DOA Estimation in Passive Radar
## 4 Performance Analysis in Simulated Scenarios
### DOA Estimation Performance  
As discussed in Section 3, once a target is detected, AB-STAP is able to perform the target DOA estimation via the closed form expression in (7) and (8) by exploiting the outputs of the A and B channels. The DOA estimation accuracy provided by this approach was evaluated and compared with the full array STAP estimator in (2) by a Monte Carlo analysis applied against the same simulated scenario. Specifically, the full array MLE was operated with a bank of filters equally spaced in angle by \(\phi\phi=0.1^{\circ}\) within the array nominal beamwidth (BW \(\cong\) 38\({}^{\circ}\)), corresponding to a step smaller than BW/100. The AB-STAP scheme employed N\({}_{0}=2\) elements for the A and B channels. Both the estimators were applied after the post-Doppler transformation with \(L=3\). The bistatic velocity and DOA of the simulated target were respectively \(v_{b}=7\) m/s and \(\phi_{t}=\phi_{t}=90^{\circ}\).  
In Figure 5, the resulting estimation accuracy is shown as a function of target input SNR. The results are compared in terms of standard deviation in Figure 5a and bias error in Figure 5b, both normalised to the nominal BW.  
We notice that, for \(K=24\) training data (dashed curves), the accuracy of AB-STAP and full array STAP estimators were comparable, showing an almost identical standard deviation. When the training data was lowered to \(K=16\) (solid curves), the standard deviation and bias error increased in the full array case. Instead, the AB-STAP showed negligible adaptivity losses even for smaller sample support. In this case, AB-STAP allowed for an improvement of the standard deviation by approximately 20% with respect to full array STAP.  
The above considerations are also supported by the results in Figure 6, where the DOA estimation performance is represented as a function of K, for a fixed SNR of \(-45\) dB.  
In Figures 5a and 6, the Cramer-Rao bound (CRB) is also reported for both the considered estimators (dash-dotted curves). This provides a useful reference for the DOA estimation accuracy and allows for better assessment of the effect of the selected strategies on the ideal estimation performance.  
From [34; 35], the CRB for the estimator in (2) can be expressed as  
\[\sigma_{\phi}^{2}=\left\{2|A|^{2}\left[\left(\dot{\mathbf{s}}^{H}(\phi)\mathbf{Q}^{-1} \dot{\mathbf{s}}(\phi)\right)-\frac{\left|\dot{\mathbf{s}}^{H}(\phi)\mathbf{Q}^{-1}\mathbf{s} (\phi)\right|^{2}}{\mathbf{s}^{H}(\phi)\mathbf{Q}^{-1}\mathbf{s}(\phi)}\right]\right\}^{-1} \tag{9}\]  
where \(\sigma_{\phi}\) is the standard deviation of the estimation error and \(\dot{\mathbf{s}}(\phi)=\partial\mathbf{s}(\phi)/\partial\phi\). The corresponding CRB for the AB-STAP scheme can be obtained by replacing \(\mathbf{s}\), \(\dot{\mathbf{s}}\), and \(\mathbf{Q}\) with their corresponding two-dimensional quantities, \(\mathbf{s}_{AB}\), \(\dot{\mathbf{s}}_{AB}\), and \(\mathbf{Q}_{AB}\).  
Figure 5: Comparison of DOA estimation accuracy as a function of the target input SNR: (a) standard deviation in BW units; (b) bias error in BW units.  
Both the estimators appeared as asymptotically unbiased and efficient due to their ML nature. The amount of loss of the simulated results with respect to the CRB is a measure of the adaptivity loss, due to the available training data. Again, this confirms the robustness of the AB-STAP approach. In fact, despite its slightly worse asymptotic performance (see the difference in the CRB curves), it performed better than the full array scheme, as the availability of training data reduced.  
Finally, it is worth recalling that the full array estimator requires finding the maximum among a discrete set of angles (bank of filters) for DOA estimation. Therefore, it can be subject to performance saturation effects and involves a non-negligible computational load, which increases with the desired accuracy. Conversely, the closed form estimator of the AB-STAP scheme is not affected by saturation and offers a reduced complexity and computationally attractive solution.