## 4 Performance Evaluation
### Numerical Results
#### 4.1.2 Detection Performance  
Detection Performance in the Presence of MF-DFM  
In this section, the detection performance of the proposed algorithm is evaluated using Monte Carlo simulations. Assuming that the input SNR of the target echo varies from \(-70\) dB to \(-60\) dB, the output SNR of the target echo is obtained using different fusion methods, including the conventional MF-NCI-EL, MF-NCI-UEL, MF-RFT-NCI, and the proposed MF-RFT. Meanwhile, the results are obtained via 100 Monte Carlo simulations. For controlling the false alarms, the expected false alarm probability \(P_{fa}\) in the CFAR detector is set to \(10^{-5}\). In the CFAR detector, the total number of the training cells in range and Doppler axis Q = 60, and the shape of the guard cell is set to a rectangular window with a size of \(5\times 5\). In this setting, the CFAR loss is limited below 0.2 dB. Figure 9a shows the detection probability as a function of the input \(SNR_{SC}\) when only the MF-DFM is considered. From Figure 9a, the proposed method can obtain, respectively, 1.2 dB, 1.5 dB, and 5 dB of SNR improvement for the detection probability of 0.9, with the MF-RFT-NCI, MF-NCI-UEL, and MF-NCI-EL. These results further confirm the effectiveness of the MF-DFB for migrating MF-DFM. Figure 8: Output SNR of the proposed method with and without ME-PSO in the presence of both MF-DFM and residual phase errors. Figure 9: The detection probability vs. input SNR only in the presence of MF-DFM: **(a)** PD vs. SNR; **(b)** PFA vs. SNR. Reaching a constant false alarm and maximum detection probability condition in a PBR system is quite important. To verify the CFAR property of the considered methods, Figure 9b shows the measured \(P_{fa}\) as a function of different input \(SNR_{SC}\). The measured \(P_{fa}\) is defined as,  
\[\text{Measured }P_{fa}=\frac{FAR}{N_{R}N_{D}}, \tag{33}\]  
where \(FAR\) is the number of false alarms generated by the CFAR detector in the observation region, and \(N_{R}N_{D}\) is the total number of range-Doppler resolution units. This result indicates that the measured \(P_{fa}\) is consistent with the expected one, showing the capability of controlling \(P_{fa}\). Moreover, this result also verifies the fairness of the comparison of different fusion detectors in terms of detection probability in Figure 9a. Robust Detection Performance against Residual Phase Error  
Next, we further evaluate the detection performance of the proposed method in the presence of the residual phase error. Note that, in order to limit the computational burden, in this subsection, we set M = 3, and the selected carrier frequencies are 12.395, 12.515, and 12.675 GHz, respectively. The residual phase error is modeled as the uniform distribution [35], i.e., \(\phi_{m}\sim U[0,p]\), where \(0<p\leq 2\pi\) denotes the fluctuation in phase error. In addition, \(p\) and \(M\), the other simulation settings, are the same as in the previous simulation. For a better understanding of the effectiveness of the PSO-based algorithm, we select the MF-RFT without PSO-based phase estimation for comparison. The detection performance of the consider detector under different \(p\) is given in Figure 10, where \(p=\pi/2,\pi\), and \(2\pi\) are shown in Figure 10, respectively. From the results, we can conclude that the performance of the proposed MF-RFT-QCI is robust to the residual phase error, while the performance of the MF-RFT without ME-PSO drops significantly as the fluctuations in the residual phase error \(p\) increase. For achieving a PD of 90%, the proposed MF-RFT with ME-PSO can, respectively, achieve approx. 3.9 dB and 9.4 dB of processing gain over the MF-RFT without ME-PSO when \(p=\pi\) and \(p=2\pi\). These results confirm the effectiveness of the ME-PSO algorithm for phase estimation and conjugated phase compensation. Figure 10: The impact of the residual phase errors on the proposed algorithm with and without ME-PSO under different \(p\).