## 4 Multiple Hypothesis Testing Technique  
### Adaptive Detection Techniques  
Research in the area of moving targets detection in the presence of clutter has resulted in a number of adaptive detection algorithms. AMF [31; 32] and ABORT [29; 30] are the two well-known techniques. The AMF exhibits robust behavior with respect to steering vector mismatches and more computational efficiency than many other detectors as the generalized likelihood ratio test (GLRT) and adaptive coherence estimator (ACE) [15; 22; 23]. The ABORT can reject interference signals orthogonal to the nominal steering direction, which modifies the null hypothesis to the unwanted signal assumed to be orthogonal to the nominal steering vector [29; 30]. Traditional _AMF_ can be expressed in the formula as [28; 29; 30; 31; 28],  
\[\Lambda_{AMF}=\frac{\left|\overset{\  
beta distribution [23; 29; 30]. When the moving target steering vector mismatch did not exit, \(\tau\) follows the complex central beta distribution. The conditional \(P_{fl}\)\(P_{fl(\bullet)|\tau}\) and \(P_{d}\)\(P_{d(\bullet)|\tau}\) can also be expressed as,  
\[P_{fl(\bullet)|\tau}=\int_{0}^{\widetilde{\eta}_{(\bullet)}}\Pr\Big{(}\Lambda_ {(\bullet)}|\tau\Big{)}d\Lambda_{(\bullet)},\ P_{d(\bullet)|\tau}=\int_{ \widetilde{\eta}_{(\bullet)}}^{+\infty}\Pr\Big{(}\Lambda_{(\bullet)}|\tau \Big{)}d\Lambda_{(\bullet)} \tag{23}\]  
The alternative threshold \(\widetilde{\eta}_{(\bullet)}\) for \(AMF\) and \(ABORT\) are defined as,  
\[\widetilde{\eta}_{AMF}=\tau\eta^{\prime}_{AMF},\ \widetilde{\eta}_{ABORT}=\eta_{ABORT}-\tau \tag{24}\]  
As shown in [23; 29; 30], the \(P_{fl(\bullet)|\tau}\) can also be expressed in the simple formulation,  
\[P_{fl(\bullet)|\tau}=1/\Big{(}1+\widetilde{\eta}_{(\bullet)}\Big{)}^{L} \tag{25}\]  
\(P_{d(\bullet)|\tau}\) is [23; 29; 30],  
\[P_{d(\bullet)|\tau}=1-\frac{1}{\Big{(}1+\widetilde{\eta}_{(\bullet)}\Big{)}^{ L}}\sum_{l=1}^{L}\binom{L}{l}\cdot\widetilde{\eta}_{(\bullet)}^{l}G_{l} \Bigg{(}\frac{SCNR_{out}\tau}{1+\widetilde{\eta}_{(\bullet)}}\Bigg{)} \tag{26}\]  
where \(G_{l}(*)\) is the incomplete Gamma function \(L=K_{nm}-K+1\). According to the Equations (19)-(26), we can analyze the detection performance loss for AMF and ABORT under different mismatch angles \(\theta\).  
Figure 1 is a plot of the Pd versus \(\cos^{2}(\theta)\) of the above two detection algorithms with parameters \(L=49\), \(K=6\), \(P_{fl}=10^{-5}\), \(SCNR_{out}=20 dB\). As shown in Figure 1, with the increase of the mismatch angle \(\theta\), the Pd of the two detection algorithms decrease quickly. AMF is much more robust for target steering vector mismatches and has similar detection performance to ABORT for the large \(\cos^{2}(\theta)\) case. For ABORT, if the mismatch angle \(\theta\) is so large, then the Pd may approximate to zero. Figure 2 shows the ROC (Receiver Operating Characteristic) curves, which are Pd versus Pfa in log scale with parameters \(L=49\), \(K=6\), \(SCNR_{out}=10 dB\), \(\theta=0\), \(pi/5\). Figure 2 shows that AMF and ABORT have approximate detection performance for the \(\theta=0\) case, especially under low Pfa. For the \(\theta=pi/5\) case, AMF has a much better detection performance than the ABORT. With the increase of mismatch angle \(\theta\), the performance of the ROC for the two detection algorithms decrease very obviously.  
Figure 1: Pd versus \(\cos^{2}(\theta)\) for AMF and ABORT.  
As shown in Figures 1 and 2, when the steering vector mismatch increases, the detection performance for the two adaptive detection techniques decreases largely. AMF has much better mismatch discrimination capabilities than ABORT. In this paper, based on the a priori knowledge of the road network database, the expected steering vectors are derived directly from the interferometric phase map and azimuth displacement map. In Equations (19) and (20), the searched moving target steering vectors are replaced by steering vector estimation results from the interferometric phase map and displacement map.