## 2 Preliminary
### The Challenges in MF Integration with a Long CPI
#### 2.2.2 Mf-Dfm  
In addition to RCM, MF-DFM is also an important phenomenon to be considered in MF-PBR. According to (5), the Doppler frequency of the target is proportional to the radial velocity \(V_{b}\), and inversely proportional to the signal wavelength \(\lambda_{m}\), i.e., proportional to the carrier frequency \(f_{m}\). Meanwhile, the Doppler resolution of the radar system is inversely proportional to \(T_{int}\), which indicates that the longer CPI is, the higher the Doppler resolution capability of the system, or the smaller the resolution unit \(\Delta f_{D}\) will be. If subcarrier segmentation is employed, the Doppler frequency of the target echo still changes as the carrier frequency of the RF signal, and the energy of the target will span multiple Doppler units when performing MF integration, i.e., MF-DFM, as shown in Figure 2. To better quantify this phenomenon, the factor of MF-DFM is defined as follows:  
\[O_{D}=\frac{2\Delta f_{c}V_{b}T_{int}cos(\beta/2)}{c}. \tag{10}\]The derivation of (10) is available in Appendix A, where \(\Delta f_{c}=f_{M}-f_{1}\) is the maximum separation of the carrier frequency. We can find that \(O_{D}\) is proportional to \(\Delta f_{c}\), \(V_{b}\), and \(T_{int}\). By fixing \(V_{b}\) and \(\Delta f_{c}\), \(O_{D}\) increases as \(T_{int}\) increases. If \(O_{D}\geq 1\), then MF-DFM occurs as,  
\[T_{int}\geq\frac{c}{2\Delta f_{c}V_{b}cos(\beta/2)}. \tag{11}\]  
From (11), \(T_{MF-DFM}\) is defined as follows:  
\[T_{MF-DFM}=\underset{T_{MF-DFM}}{\text{argmax}}\bigg{[}V_{b}T_{MF-DFM}\leq\frac {c}{2\Delta f_{c}cos(\beta/2)}\bigg{]}. \tag{12}\]  
For the case of compact arrangement of MF, let \(\Delta f_{c}=(M-1)B\). If \(M\geq 2\), we have \(\Delta f_{c}\geq B\). Compared with (9), we have,  
\[T_{MF-DFM}\leq T_{RCM}. \tag{13}\]  
From the analysis above, we have proved that MF-DFM will appear prior to RCM as the integration time (\(T_{int}\)) increases. To simplify the problem and avoid the occurrence of both MF-DFM and RCM in the detection problem, we constrain \(T_{int}\) between \(T_{MF-DFM}\) and \(T_{RCM}\), i.e., \(T_{int}\in(T_{MF-DFM},T_{RCM}]\). This means the RCM can be avoided and the proposed algorithm aims to overcome the MF-DFM in MF integration. For quantifying the \(O_{D}\), Figure 3 shows the contour plots of \(O_{D}\) with respect to the coherent integration time \(T_{int}\), and the maximum separation of the carrier frequency \(\Delta f_{c}\). The results are obtained by simulation according to Equation (10). The simulation settings are as follows: the range of CPI \(T_{int}\) from 0.05 s to 0.36 s and \(\Delta f_{c}\) from 0.05 GHz to 0.5 GHz. In addition, we temporarily ignore the geographical factor by setting \(cos(\beta/2)=1/2\) (\(\beta=120^{\circ}\)). Figure 3a,b give the contour plots when \(V_{b}\) equals 60 km/h and 120 km/h, respectively. We can see that \(O_{D}\) increases as \(T_{int}\), \(V_{b}\), and \(\Delta f_{c}\) increase. The value of \(O_{D}\) indicates the extent of MF-DFM. For example, when \(\Delta f_{c}=0.3\) GHz, \(T_{int}=0.2\) s, and \(V_{b}=60\) km/h, \(O_{D}\) is equal to 3.1, which means that the Doppler shift of the target will spread to more than 3 consecutive Doppler units. For a better understanding of the meaning of CPI, the time limit of RCM, i.e., \(T_{RCM}\), is also plotted in Figure 3. The time limit of MF-DFM (\(O_{D}=1\)) is determined according to the value of the \(\Delta f_{c}\) and \(V_{b}\). Figure 3 also gives the time limit of MF-DFM when \(\Delta f_{c}=0.28\) GHz (12.675 GHz \(-\) 12.395 GHz). It is noted that only MF-DFM in the MFI is discussed above. However, a more comprehensive case will occur with both RCM and MF-DFM when the CPI continues to increase. Thanks to the existing techniques of KT, the RCM can be effectively solved. In addition, the detection problem with MFI will become extremely complicated if a target with high maneuverability and high speed is considered. More efforts should be performed for the cases where the target is maneuverable, of high speed, and even extended.