## 3 Proposed Algorithm
### Definition of MF-RFT  
According to [30], RFT achieves coherent integration of the target's energy via DFB obtained by searching motion parameters. As an extension of RFT, MF-RFT enables the elimination of the MF-DFM effect by designing a sequence of MF-DFB. The derivation of MF-DFB is given in the following. From (2), we define the Doppler term of the target, \(K_{m}=\exp(j2\pi f_{d,m}t)\). By substituting (5) into \(K_{m}\), \(K_{m}\) can be rewritten as follows:  
\[ \begin{split} K_{m}=&\exp\left(\frac{j2\pi f_{m}V_{ b}t}{c}\right)\\ =&\exp\left[\frac{j2\pi(f_{1}+f_{m}-f_{1})V_{b}t}{c} \right]\\ =&\exp(j2\pi f_{d,1}t)H_{m},\end{split}  \tag{14}\]  
where \(f_{d,1}=f_{1}V_{b}/c\), is the Doppler centroid of the moving target, as shown in Figure 4. As consequently, the coefficients of MF-DFB are defined as follows:  
\[H_{m}(V_{b})=\exp\left[\frac{j2\pi(f_{m}-f_{1})V_{b}t}{c}\right] \tag{15}\]Obviously, given a set of selected frequency channels, i.e., \(\vec{f}=[f_{1},\ldots,f_{M}]\), and a velocity of interest \(V_{b}\), the coefficients of \(H_{m}\) are uniquely determined. In other words, leveraging the prior knowledge of the carrier frequency can effectively reduce the computational burden of the MF-DFB. In practice, the target's velocity is usually unknown. However, it is possible to employ a velocity search to estimate the target's velocity. This means the coherent synthesis of multiple RDMs, i.e., \(I_{MF}(r,f_{d})\), reaches its maximum only when the searching parameter matches the true target's velocity \(V_{b}^{*}\), i.e., \(V_{b}=V_{b}^{*}\). The coherent synthesis of multiple RDMs, i.e., \(I_{MF}(r,f_{d})\), is defined as follows:  
\[I_{MF}(r,f_{d};V_{b})=\left|\sum_{m=1}^{M}I_{MF-DFB,m}(r,f_{d};V_{b})\right|^{ 2}, \tag{16}\]  
where \(I_{MF-DFB,m}(r,f_{d};V_{b})\) is the \(m\)-th filtered RDM, which is calculated as,  
\[I_{MF-DFB,m}(r,f_{d};V_{b})=\int_{-\eta_{K}/2}^{\eta_{K}/2}R_{m}(r,\eta_{k})H_ {m}(V_{b})\exp(-j\frac{2\pi f_{d}k\eta_{k}}{K})d\eta_{k}, \tag{17}\]  
where \(R_{m}(r,\eta_{k})\) is the two-dimensional (2-D) matrix obtained by batch-based range compression and stacking along slow-time. \(\eta_{k}\) is the index of slow-time. As a result, the Doppler frequencies of the target in the \(m\)-th frequency band can be corrected to the DC of the target because of the MF-DFB. To implement the calculation of (17), discrete Fourier transform (DFT) can be employed. Accordingly, Equation (17) can be rewritten as follows:  
\[I_{MF-DFB,m}(r,f_{d};V_{b})=DFT_{\eta}[R_{m}(r,\eta_{k})H_{m}(V_{b})],\quad k= 1,2,\ldots,K \tag{18}\]  
Figure 4: Geometric schematics of the proposed MF-RFT. where \(DFT_{\eta}\) denotes the DFT along the slow-time dimension. As a result, the estimation of the unknown target's velocity is obtained by solving the following optimization problem as follows:  
\[\hat{V}_{b}=\underset{r,V_{b}}{argmax}\left[\left|\sum_{m=1}^{M} DFT_{\eta}[R_{m}(r,\eta)H_{m}(V_{b})]\right|^{2}\right], \tag{19}\] \[r\in[r_{min},r_{max}],\qquad V_{b}\in[V_{b,min},V_{b,max}].\]  
The above optimization problem can be solved by first fixing a small region of interest (ROI) \(r\) and then performing a parameter search of \(V_{b}\). As shown in Figure 4, the filter coefficients of MF-DFB are then be determined after the velocity search. The final output of MF-RFT is expressed as follows:  
\[T_{MF-RFT}(r,f_{d})=\left|\sum_{m=1}^{M}DFT_{\eta}[R_{m}(r,\eta_{k})H_{m}(\hat{ V}_{b})\right|^{2}. \tag{20}\]  
Using (20), the MF-DFM effect can be eliminated by the proposed MF-RFT so that the target envelope can be locked in the same Doppler cell. However, the derivation above ignores the residual phase error in the target unit. The residual phase error will lead to coherent integration loss. In [35], the authors have shown that the coherent integration loss depends on the distribution of the phase error.