## 3 Proposed Algorithm
### Me-Based PSO Algorithm for Phase Estimation
**  
Based on the MF coherent PBR receiver, the target returns are sampled in fast-time in the intermediate frequency (IF). Then, the sample data are digitally down-converted (DDC) into the signal of the baseband, which is also known as demodulation. With the use of corresponding signals of RC, the batch-based range compression [43] is used for the first stage of coherent integration in the fast-time domain. The data, after batch-based range compression, are stored as a 2-D matrix in the range and slow-time domains, i.e., \(R_{m}(r,\eta_{k})\in C^{N/K\times K}\), for \(m=1,\ldots,M\), \(k=1,2,\ldots,K\). The signals at different frequency bands are also processed in parallel. **Step 2: Parameter initialization for MF-RFT and ME-PSO algorithm. **  
First, parameter initialization for the MF-RFT algorithm is given as follows: the coherent integration time \(T_{int}\), the number of batches in slow-time \(K\), the range search scope \([r_{min},r_{max}]\). Based on the relative prior information such as moving statues of the target to be detected, the expected search scope of the velocity \([V_{min},V_{max}]\), and the velocity interval \(\Delta V_{b}\) is defined as,  
\[\Delta V_{b}=\frac{c}{2f_{c,1}T_{int}} \tag{28}\]  
Figure 5: Flowchart of the proposed algorithm. Therefore, the values of the search range are \(N_{r}=\text{round}[(r_{max}-r_{min})/\Delta r]\), where \(\Delta r\) denotes the range resolution as (6), and round[] denotes the operation of rounding up to an integer. Similarly, the search value of the velocity is \(N_{v}=\text{round}[(V_{b,max}-V_{b,min})/\Delta V_{b}]\). In particular, the coherent integration time \(T_{int}\) should be specially selected, i.e., \(T_{int}\in(T_{MF-DFM},T_{RCM}]\) for avoiding the RCM. Second, the parameter initialization of the ME-PSO algorithm is given as follows: According to [44], the population of the particles \(N_{p}\) is set as 80. Most importantly, the particles \(\Phi_{m}\) are initialized randomly and as uniformly distributed in the searching space as possible. The maximum iteration \(k_{max}=50\). **Step 3: Apply the MF-RFT to eliminate the MF-DFM using a velocity search. **  
Given a search range \(r\), this step is performing the MF-DFB to estimate the target's velocity \(V_{b}\). Given the prior knowledge of the carrier frequency, the expected search scope of the velocity \([V_{min},V_{max}]\), and velocity interval \(\Delta V_{b}\), and the coefficients of MF-DFB, i.e., \(H_{m}(V_{b})\) can be determined using (15). **Step 4: Perform PSO algorithm for phase estimation. **  
Given a searching velocity \(V_{b}\), this step is to find the estimate of the residual phase error by solving the optimization problem in (26). It is noted that the accuracy of phase estimation is highly related to the dimension of the searching space of the range and Doppler frequency. To the best knowledge of the authors, the interesting range is as limited as possible. **Step 5: Go through all the searching parameters and obtain the estimates of the unknown parameters. **  
Go through all the searching scope of range, velocity, and phase. Repeat steps 3 and 4 to obtain a peak value in the range-Doppler domain, as  
\[(r^{*},V_{b}^{*},\phi_{m}^{*})=\underset{r,V_{b},\phi_{m}}{argmax}\ T_{MF- RFT-QCI} \tag{29}\]  
where \(T_{MF-RFT-QCI}\) is calculated as (27). **Step 6: Perform quasi-coherent integration and construct the integrated RDM. **  
With the estimation of range, velocity, and phase, the integrated RDM can be obtained using (27). It is noted that the above operations will lead to power deviation, so a normalization in MF is required before quasi-coherent MF integration. **Step 7: Carry out the CFAR detection to confirm a target. **  
Take the squared magnitude of the integrated RDM in step 6 as the test statistic, and compare it with the adaptive threshold for a given expected false alarm rate as,  
\[T_{MF-RFT-QCI}(r,f_{d}) \begin{array}{l}H_{1}\\ \gtrless\\ H_{0}\end{array} \gamma, \tag{30}\]  
where \(\gamma\) is the adaptive threshold obtained by the CFAR detector.