## 3 Theoretical Performance of DPMM Using MLE under One LOS and One Reflected Path Environment
### Maximum Likelihood Estimators
\(R_{\text{ssI}}(t)=\sum\limits_{n=1}^{N}s(nT-t)\times x_{\text{I}}(nT)\) and \(R_{\text{ssQ}}(t)=\sum\limits_{n=1}^{N}s(nT-t)\times x_{\text{Q}}(nT)\) are the in-phase and quadrature (I/Q) cross-correlation functions (CCFs) of the RHCP channel, respectively while \(R_{\text{syI}}(t)\) and \(R_{\text{syQ}}(t)\) are the corresponding LHCP terms. \(\ln C=\frac{1}{2\sigma^{2}}\sum\limits_{n=1}^{N}\left[\left|x_{\text{RHCP}}( nT)\right|^{2}+\left|x_{\text{LHCP}}(nT)\right|^{2}\right]+N\ln\left(2\pi \sigma^{2}\right)\) is a constant term irrelevant to the minimization. Both the ACF and CCFs have high resolution within the range from \(-1\) to \(+1\) code chip. The \(R_{\text{ss}}(t)\) can be calculated offline and \(R_{\text{ssI}}(t)\), \(R_{\text{ssQ}}(t)\), \(R_{\text{syI}}(t)\), \(R_{\text{syQ}}(t)\) are generated by \(N\) points of data using signal compression (SC) technology [36]. Therefore, for each pair of \([t_{0},\ t_{1}]\), the values of ACF and CCFs are known, and the minimization of \((\boldsymbol{\eta})\) with respect to the rest parameters \([a,b,c,d,e,f,p,q]\) becomes a quadratic optimization problem that can be easily solved. By linearly searching the space of \([t_{0},t_{1}]\), we obtain the minimum value of \((\boldsymbol{\eta})\) and the optimal TOA estimation \(\hat{t_{0}}\) of the LOS signal. The optimization method are similar with that in MMT and we add some constraints [37] to accelerate and improve the algorithm. For convenience, we name the signal model denoted by Equation (7) as \"R2L2\" and the ML estimator we have derived in Equations (10)-(12) as \"DP-R2L2-MLE\" hereinafter. For the purpose of finding the multipath mitigation improvement of the dual-polarization antenna technology over the single-polarization one, we compare the DP-R2L2-MLE with other ML estimators that use other signal models:  
* RHCP-R1-MLE: the widely used one-path ML estimator that assumes only the LOS signal in the RHCP antenna. It models the LOS signal in the RHCP channel in Equation (7) and the log-likelihood function is a special case in DP-R2L2-MLE, thus, the terms \({}_{\text{LHCP}}(\boldsymbol{\eta})\) in Equation (10) and \([c,d]\) in Equation (11) are zeros. We name this signal model as \"R1\" and the ML estimator as \"RHCP-R1-MLE\". * RHCP-R2-MLE: the two-path ML estimator or the two-path MMT estimator that models the LOS and MP signals in the RHCP antenna. Its model has only the \(x_{\text{RHCP}}(nT)\) term in Equation (7), hence, the term \({}_{\text{LHCP}}(\boldsymbol{\eta})\) in Equation (10) is zero. We name this signal model as \"R2\" and the corresponding ML estimator as \"RHCP-R2-MLE\". * DP-R2L1-MLE: in Strong LHCP Signal Case 2, \(B_{0}\) is much smaller than \(B_{1}\), hence, \(B_{0}\) can be ignored to yield a modified signal model as Equation (13) shows. In this model, the terms\([e,f]\) in Equation (12) are zeros and we denote the model in Equation (13) as \"R2L1\" and the corresponding ML estimator as \"DP-R2L1-MLE\". \[ \begin{array}{l}x_{\rm{RHCP}}(nT)=A_{0}{\rm{e}}^{{\rm{i}}\theta_{0}}s(nT-t_{0} )+A_{1}{\rm{e}}^{{\rm{i}}\theta_{1}}s(nT-t_{1})+w_{\rm{R}}(nT)\\ x_{\rm{LHCP}}(nT)=\hskip 56.905512 ptB_{1}{\rm{e}}^{{\rm{i}}\theta_{1}}s(nT-t_{1})+w_{ \rm{L}}(nT)\end{array}  \tag{13}\]