## 3 Proposed Algorithm
### Me-Based PSO Algorithm for Phase Estimation  
The processing gain of NCI-based processing is limited as compared with coherent processing. In the following, an ME-based phase estimation method is proposed to eliminate the effect of the residual phase error. In order to ensure the availability of coherence, it is intuitive to estimate and compensate for the residual phase for further improving the processing gain of MF-RFT processing. Given a certain range and velocity, i.e., \(r\) and \(V_{b}\), the output of MF-DFB with phase compensation can be expressed as,  
\[S_{m}(f_{d},\phi_{m};r,V_{b})=DFT_{\eta}[R_{m}(r,\eta_{k})H_{m}(V_{b})\exp(- j\phi_{m})],\quad\phi_{m}\in[0,2\pi], \tag{23}\]where \(\phi_{m}\) is utilized to cancel the phase error. To determine \(\phi_{m}\), we formulate it as an optimization problem based on the global minimum-entropy criterion [39], as follows:  
\[E(\phi_{m};r,V_{b})=-\frac{1}{Dim(f_{d})}\sum_{f_{d}}\Biggl{[}\frac{|\sum_{m=1}^ {M}S_{m}(f_{d},\phi_{m})|^{2}}{C}\ln\frac{|\sum_{m=1}^{M}S_{m}(f_{d},\phi_{m})|^ {2}}{C}\Biggr{]} \tag{24}\]  
where \(Dim(f_{d})\) is the dimension of the Doppler domain, and \(C\) is a normalized term, which is calculated as,  
\[C=\sum_{f_{d}}\[PERSON]{|}\sum_{m=1}^{M}S_{m}(f_{d},\phi_{m};r,V_{b})\Biggr{|}^{2}. \tag{25}\]  
Consequently, finding \(\phi_{m}\) can be achieved by minimizing the optimization problem as,  
\[ \begin{split}&\phi_{m}=\underset{\phi_{m}}{argmin}[E(\phi_{m};r,V_{ b})],\\ & 0<\phi_{m}\leq 2\pi,m=1,\ldots,M.\end{split}  \tag{26}\]  
To ensure the uniqueness of the solution, we usually constrain \(\phi_{1}\)=0. The above optimization problem is high-dimensional and nonlinear, so it is non-trivial and NP-hard to prove its convexity. In the case when M is small, exhausted searching (ES) can be used to obtain the optimal minimum of (26). However, in practical conditions, ES is not a desirable choice since its computational complexity increases with the exponential of M, i.e., \(O(N^{M})\). As a greedy algorithm, the dynamic program (DP) can solve the above optimization problem with a lower computational complexity but DP cannot guarantee the global optimal when the SNR is extremely low. To reduce the complexity of the algorithm as well as guaranteeing the optimality of the solution, we choose the PSO algorithm in [40] to solve (26). Compared with ES, PSO is an efficient algorithm that can converge within a certain number of iterations and achieves satisfactory accuracy, although the result is not necessarily the optimum. The parameter notation in the ME-based PSO (ME-PSO) algorithm is summarized as follows:  
1. \(N_{p}\): The population of particles which is initialized, and \(i\) is its subscript index;
2. \(M\): The dimension of the particles' state, which is the same as the number of the used fusion channel;
3. \(k_{max}\): The maximum number of iterations; \(k\) is the iteration index;
4. \(\Phi_{i}^{k}\in R^{M\times 1}\): The phase vectors of the \(i\)-th particle at the \(k\)-th iteration, where \(R\) is denoted as fields of a real number;
5. \(\textbf{V}_{i}^{k}\in R^{M\times 1}\): The velocity vectors of the \(i\)-th particle at the \(k\)-th iteration;
6. \(w\): The inertia weights, which are used to control global exploration and local exploration of the particles;
7. \(c_{1},c_{2}\): The two independent and uniformly distributed random numbers accounting for random movements of the particle. In experiments, we generally set \(c_{1}\) = 0.5 and \(c_{2}\) = 0.5. \(r_{1}\) and \(r_{2}\) are random numbers with intervals of [0,1], denoted by rand ([0,1]);
8. \(\textbf{pbest}_{i}^{k}\in R^{M\times 1}\): The personal best of the \(i\)-th particle at the \(k\)-th iteration;
9. \(\textbf{gbest}^{k}\in R^{M\times 1}\): The global best at the \(k\)-th iteration. The procedures of the proposed ME-based PSO algorithm for phase estimation is shown in Algorithm 1. **Step 1: Carry out the demodulation and batch-based range compression to accumulate energy within each frequency channel.