## 4 Dual-Polarization Multipath Mitigation Algorithm  
The two-path ML estimator can be used to assess the theoretical performance of DPMM as analyzed above. However, the heavy computation limits its application in the multiple-path environment. Therefore, in this section, inspired by the iterative MEDLINE algorithms [38], we develop a sequential iterative maximum likelihood estimation (SIMLE) algorithm, which has reasonable computational load, to verify our theory. MEDLL is an approximate implementation of the MLE algorithm that uses the locally generated ACFs to match the CCF of the received signal. It is an iterative approximation of Equation (14):  
\[\min_{A_{i}\theta_{i},t_{i}}\left|\left|r_{\text{RHCP}}(t)-\sum\limits_{i=0}^{ M}A_{i}e^{i\theta_{i}}R_{\text{ss}}(t-t_{i})\right|\right| \tag{14}\]  
where \(r_{\text{RHCP}}(t)=R_{\text{ss}}(t)+\text{j}R_{\text{ss}}(t)\) is the original CCF of the RHCP channel, \(R_{\text{ss}}(t)\) is the ACF of the local code wave, \(\left[A_{i},\theta_{i},t_{i}\right]\) are the estimated path parameters and \(M\) is the multipath number. \(t\) is the relative delay to the time of the receiver. The MEDLL algorithm sequentially estimate the path number \(M\) based on the estimated SNR or the noise power and decomposes the M-dimensional searching problem to \(M\) one-dimensional searching problems to reduce the computational load. Consequently, the mutual relationships between correlated paths are not perfectly considered during iterations. In the one LOS and one reflected path environment, if the relative delay of MP to the LOS signal is in the short delay range, MEDLL often considers the composite signal as the LOS signal and the noise as the MP signal at the beginning of the iteration, which results in a wrong initial TOA estimation for the short delay MP signal. The algorithm with wrong initial status finally converges to a local minimum instead of the global one. Fortunately, when LHCP channel receives enough power of the MP signal, the LHCP channel gives a good initial TOA estimation of \(t_{1}\). We can use the CCF of the LHCP channel to assist the iterative MEDLL algorithm in the RHCP channel to improve the short delay multipath mitigation performance, which is the basic concept of our DP-SIMLE algorithm. The proposed DP-SIMLE algorithm minimizes Equation (15):  
\[\min_{A_{i},\theta_{i},B_{i}\theta_{i}^{\prime},t_{i}}\left\{ \begin{array}{l }\left|\left|r_{\text{RHCP}}(t)-\sum\limits_{i=0}^{M}A_{i}e^{i\theta_{i}}R_{ \text{ss}}(t-t_{i})\right|\right|+\\ \left|\left|r_{\text{LHCP}}(t)-\sum\limits_{i=1}^{M}B_{i}e^{i\theta_{i}^{ \prime}}R_{\text{ss}}(t-t_{i})\right|\right|\end{array} \right.\right\} \tag{15}\]  
where \(M\) is the multipath number. The iterative steps of DP-SIMLE algorithm that simultaneously estimates the path parameters and path number are given below:  
(1) Initialization:  
Initialize the estimated path number \(PathNum=0\). Calculate the noise power using the estimated CNR or SNR and the maximum value of the CCF of the RHCP channel, i.e., \(SignalPower=\underset{t}{\text{max}}\Big{(}\left|r_{\text{RHCP}}(t)\right|^{2} \Big{)}\), and \(NoisePower=SignalPower/10^{SNR/10}\). The \(t\) that maximizes \(\left|r_{\text{RHCP}}(t)\right|^{2}\) is denoted as \(\widehat{\text{f}_{\text{RHCP}}}\). Similarly, the \(t\) that maximizes \(\left|r_{\text{LHCP}}(t)\right|^{2}\) is denoted as \(\widehat{\text{f}_{\text{LHCP}}}\). If \(\widehat{\text{f}_{\text{RHCP}}}>\widehat{\text{f}_{\text{LHCP}}}\), we set \(r_{\text{LHCP}}(t)=0\), and the steps below degenerate to the traditional iterative MEDLL algorithm. This indicator will be triggered in some critical cases which we will talk about in Section 5.5. 2.