# Rao and Wald Tests for Moving Target Detection in Forward Scatter Radar
## 5 Numerical Evaluation
### Performance Analysis of the Proposed Detectors with Secondary Data  
In this part, the detection performance of the proposed Rao-SD-FSR and Wald-SD-FSR was analyzed when the secondary data are available. In Figure 2, the probability of detection \(P_{d}\) is plotted as a function of the DNSR. The probabilities of detection of the proposed Rao-SD-FSR and Wald-SD-FSR are obtained from the test statistics (36) and (43). For comparison, the simulated and theoretical probabilities of detection of the GLRT-SD-FSR and the ideal optimum detector are also given.  
We can see that the proposed Rao-SD-FSR and Wald-SD-FSR detectors and the GLRT-SD-FSR coincide with each other. Moreover, simulation results of the three detectors are all in agreement with the theoretical result of the GLRT-SD-FSR. Thus, both simulation results and the theoretical analysis demonstrate the equivalence of the complex parameter Wald detector, complex Rao detector, and the GLRT detector for the detection problem (1).  
From Figure 2, we can also see that the probabilities of detection of all the detectors improve as the DNSR increases. Meanwhile, the proposed Rao-SD-FSR and Wald-SD-FSR outperform the CVD. The detection performance gains of the Rao-SD-FSR and Wald-SD-FSR with respect to the CVD are about 3 dB and 10 dB for \(P_{d}=0.9,R=4000\) and \(P_{d}=0.9,R=600\). Moreover, the proposed Rao-SD-FSR and Wald-SD-FSR detectors achieve similar detection performance to the ideal optimum detector. The performance losses of the Rao-SD-FSR and Wald-SD-FSR detectors compared with the ideal optimum GLRT are about 0.5 dB and 1 dB for \(P_{d}=0.9\), \(R=4000\) and \(P_{d}=0.9\), \(R=600\). When comparing Figure 2a with Figure 2b, it can be seen that the detection performance degradation of the proposed Rao-SD-FSR and Wald-SD-FSR with respect to the ideal optimum detector when the target approaches the near field is higher than those when the target is in the far field.  
The receiver operating characteristic (ROC) curve, which is plotted with the probability of false alarm \(P_{fa}\) as the horizontal coordinate and the probability of detection \(P_{d}\) as the  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameters** & **Symbol** & **Value** \\ \hline Carrier Frequency & \(f_{c}\) & 2.4 GHz \\ Horizontal Dimension & \(L_{h}\) & 4 m \\ Vertical Dimension & \(L_{o}\) & 1 m \\ Target Velocity & \(v\) & 25 m/s \\ Base Line & \(R\) & 4000 m, 600 m \\ Distance & \(R_{0}\) & \(R/2\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: The target parameters and radar system parameters.  
Figure 2: Probability of detection versus DNSR (**a**) \(R=4000\), \(K=N\) (**b**) \(R=600,K=N\).  
vertical coordinate, is a widely used evaluation tool for signal detection. To see the impact of the observation time \(T\) on the performance of the Rao-SD-FSR and Wald-SD-FSR, the receiver operating characteristic (ROC) of the Rao-SD-FSR and Wald-SD-FSR is plotted for different observation times when \(R=4000\), \(DNSR=35\) dB and \(R=600\), \(DNSR=30\) dB in Figure 3a,b. Figure 3 indicates that the detection performance of the proposed detectors improves with an increase in the observation time and approaches that of the ideal optimum detector when the long observation time is used. Thus, the increase in the observation time can achieve a detection performance gain.  
In Figure 4, probabilities of detection of the Rao-SD-FSR and Wald-SD-FSR are plotted as a function of the number of the secondary data \(K\). We can see that the probabilities of detection of the Rao-SD-FSR and Wald-SD-FSR become higher when the number of the secondary data increases. This is due to the fact that the estimations of the unknown parameters become more accurate as the secondary data increase.