## 4 Performance Evaluation
### Numerical Results
#### 4.1.1 Case Study  
For verifying the effectiveness of the proposed method, we provide a case study, where the simulated parameters are set as follows. First, the channel occupation is based on the investigation of the IOs that emit the DVB-S signal. Among the investigated IOs, the Apstar-6C is chosen since it can provide multiple adjacent frequency channels with the same bandwidth and polarization [46]. Actually, the DVB-S standard transmits multi-frequency signals via a single carrier per transponder with time division multiplex (TDM) or multi-carrier frequency division multiplex (FDM) [47]. Apstar-6C employs the latter, and as a result, the concern about illuminating the target asynchronously can be ignored in this case. In the radar system using the Apstar-6C as the IO, the channel number for fusion \(M=5\), the carrier frequencies range from 12.395 GHz to 12.675 GHz, the sample rate \(f_{s}=100\) MHz, and the signal bandwidth \(B=36\) MHz. The noise in different frequency bands is assumed to be IID. We consider a case, where a target is moving with a constant speed and distance of 16 m/s and 40 m, respectively. For avoiding the RCM, the CPI \(T_{int}\) for each channel is selected as 0.1678 s. As a result, the processing gain \(G_{p}=10\times\log_{10}(\text{B}\times T_{int})=71.23\) dB. The simulation settings are summarized in Table 1.  
According to the simulation settings, the largest separation of the carrier frequency is 280 MHz (i.e., \(f_{c,5}-f_{c,1}\)), the Doppler frequency resolution and range resolution of the system are 5.96 Hz and 4.04 m, respectively. During the observation time of 0.1678 s, the displacement of the target is 2.6848 m, which does not exceed the range resolution. Consequently, RCM does not occur. However, according to (10), the variation in the target's Doppler frequency ranges from \(-660\) Hz to \(-680\) Hz at different frequency bands. Consequently, the target's energy will disperse in up to three sequential Doppler cells. This Doppler spread will be further intensified as the CPI and channel number continue to increase. For confirming the analysis above, the MF-DFM of the moving target is shown in Figure 6, where it is noise free. The performance comparison of different methods is given in Figure 7, where a three-dimensional (3D) RDM is provided. For the sake of comparison, this case only focuses on the migration of MF-DFM, so the residual phase error is temporarily not considered. For a comprehensive comparison, the MF-NCI with equal-length CPI (MF-NCI-EL) [48], MF-NCI with unequal-length CPI (MF-NCI-UEL) in [11], and MF-RFT-NCI in (22) are selected. For a fair comparison, the CPIs at each frequency channel in MF-NCI-UEL are specially selected and their total CPI is the same as that in the other detectors (i.e., \(M\times T_{int}\)). The input \(SNR_{SC,m}\), are all set as \(-60\) dB. From Figure 7a-d, one can observe that the output SNR from MF-NCI-EL is the worst due to its lack of Doppler spread correction. Therefore, the target's energy cannot be concentrated in the same Doppler frequency bin, and the weak target is submerged below the noise floor, as shown in Figure 7a. From Figure 7b, \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameter** & **Value** & **Parameter** & **Value** \\ \hline Freq. of Ch. 1 & 12.395 GHz & Sample rate & 100 MHz \\ Freq. of Ch. 2 & 12.435 GHz & Bandwidth & 36 MHz \\ Freq. of Ch. 3 & 12.515 GHz & Modulation & QPSK \\ Freq. of Ch. 4 & 12.595 GHz & Target’s velocity & \(-16\) m/s \\ Freq. of Ch. 5 & 12.675 GHz & Target’s range & 40 m \\ CPI & 0.1678 s & Input \(\text{SNR}_{SC}/\text{SNR}_{RC}\) & \(-63/10\) dB \\ Range resolution & 4.04 m & Doppler frequency resolution & 5.96 Hz \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Simulation settings for case study. Figure 6: The MF-DFM when it is noise free. the MF-DFM can be alleviated via MF-NCI-UEL but the output SNR is minor due to the non-ideal range misalignment. Consequently, the faint target also cannot be effectively detected by the subsequent CFAR detector.