## 4 Multiple Hypothesis Testing Technique
### Multiple Hypothesis Testing Technique
 \tag{40}\]  
\(\Lambda_{0,1(\bullet)}\), \(\Lambda_{0,2(\bullet)}\), \(\ldots\), \(\Lambda_{0,M(\bullet)}\) may not be mutually independent, which makes it very difficult to calculate the joint probability density function and the probability in Equation (40). Performance is investigated by means of numerical analysis and Monte Carlo simulation in terms of \(P_{d}\). Firstly, we set \(P_{fa}=10^{-5}\) and assume that the \"local\" \(P_{fa}\) pertaining to the different roads are all the same. For given probabilities of false alarm, the \"total\" \(P_{fa}\) can be divided equally to the \"local\" \(P_{fa}\). We calculated the \"local\" thresholds \(\eta_{0,1(\bullet)},\eta_{0,2(\bullet)},\ldots\eta_{0,M(\bullet)}\) by using Equations (18)-(26) and (33). Then, we employed the Monte Carlo technique to approximate the integration in Equation (40) and calculate \"total\" and \"local\" \(P_{d}\) by running 100,000 Monte Carlo runs. The probability of correct classification \(P_{c}\) is defined as the probability of deciding for the correct hypothesis when a target is present and belongs to one of the \(M\) roads, that is,  
\[ \begin{split} P_{c}&=\frac{P\big{(}\hat{H}_{1}|H_{1} \big{)}P(H_{1})+P\big{(}\hat{H}_{2}|H_{2}\big{)}P(H_{2})\ldots+P\big{(}\hat{H}_{ M}|H_{M}\big{)}P(H_{M})}{P(H_{1})+P(H_{2})\ldots+P(H_{M})}\\ &=\frac{P_{c|H_{1}}P(H_{1})+P_{c|H_{2}}(H_{2})+\ldots+P_{c|H_{M}} \big{)}P(H_{M})}{P(H_{1})+P(H_{2})\ldots+P(H_{M})}\end{split}  \tag{41}\]Under the assumption \(P(H_{1})=P(H_{2})=\ldots\ldots=P(H_{M})\), Equation (41) can be expressed as,  
\[P_{c}=\frac{1}{M}\sum_{l=1}^{M}P_{c|H_{l}} \tag{42}\]  
As shown in the front part, for given probabilities of false alarm, the \"total\" \(P_{fa}\) can be divided equally to the \"local\" \(P_{fa}\). The thresholds \(\eta_{0,1(\bullet)}=\eta_{0,2(\bullet)}=\ldots\eta_{0,M(\bullet)}\). So \(P_{c|H_{l}}\) can be expressed using the inclusion-exclusion principle [35; 36],  
\[ \begin{array}{l}P_{c|H_{l}}=P\Big{\{}\Big{(}\Lambda_{0,|(\bullet)}>\eta_{0,|( \bullet)}|H_{l}\,\Big{)}_{i=1,i}^{M}\Big{(}\Lambda_{i|(\bullet)}>1|H_{l}\, \Big{)}\Big{\}}\\ \text{or}\ P_{c|H_{l}}=P\Big{(}\Lambda_{0,|(\bullet)}>\eta_{0,|(\bullet)}\cdot \Lambda_{i|(\bullet)}>1,\ldots,\Lambda_{M,|(\bullet)}>1|H_{l}\,\Big{)}i=1, \ldots,M,l=1,\ldots,M,i\
eq l\end{array}  \tag{43}\]  
As shown in the front part, it is very difficult to calculate the joint probability density function and the probability in the Equations (42) and (43). For performance analysis, we resorted to numerical integration to calculate them, which is a show of good performance under the Monte Carlo simulation. Figure 4 shows plots of the Pd and Pc versus input \(SCNR\) (\(SCNR_{in}\)) of the proposed MHT technique based on the AMF and the ABORT with parameters \(L=49\), \(K=6\), \(M=4\), and \(P_{fa}=10^{-5}\). The \"total\" probabilities of false alarm are divided equally to the \"local\" Pfa. The Monte Carlo runs are 100,000. Moving targets are modeled as point targets of the Swerling 0 case. In Figure 4a, Pd versus \(SCNR_{in}\) of the two detection techniques for \(\psi_{H_{1}}=1.0 rad\), \(\psi_{H_{2}}=1.5 rad\), \(\psi_{H_{3}}=1.1 rad\), \(-1.2 rad\), \(\psi_{H_{4}}=-1.5 rad\) are shown. In Figure 4b, Pc versus \(SCNR_{in}\) for \(\psi_{H_{1}}=1.0 rad\), \(\psi_{H_{2}}=1.5 rad\), \(\psi_{H_{3}}=1.1 rad\), \(-1.2 rad\), \(\psi_{H_{4}}=-1.5 rad\) is shown. In Figure 4c, Pd versus \(SCNR_{in}\) for \(\psi_{H_{1}}=1.0 rad\), \(\psi_{H_{2}}=1.5 rad\), \(\psi_{H_{3}}=1.1 rad\), \(-1.2 rad\), \(\psi_{H_{4}}=0.9 rad\) is shown. In Figure 4d, Pc versus \(SCNR_{in}\) for \(\psi_{H_{1}}=1.0 rad\), \(\psi_{H_{2}}=1.5 rad\), \(\psi_{H_{3}}=1.1 rad\), \(-1.2 rad\), \(\psi_{H_{4}}=0.9 rad\) is shown. The figures show that Pd and Pcincrease with \(SCNR_{in}\) and it is easier to detect the presence of a target rather than classify them to the corresponding road. Pd and Pc increase with the phase difference for different corresponding roads and it is easier to detect and classify the targets for larger phase differences.