## 3 Theoretical Performance of DPMM Using MLE under One LOS and One Reflected Path Environment
### Maximum Likelihood Estimators  
Here we propose the ML estimator for the time of arrival (TOA) estimation of the LOS signal using a dual-polarization antenna whose signal model is given by Equation (5). The data in RHCP and LHCP channels are synchronously sampled on \([0,T_{\mathrm{s}}]\) at the Nyquist rate \(T^{-1}\) which yields the discrete time signal model in Equation (7):  
\[ \begin{split} x_{\mathrm{RHCP}}(nT)&=A_{0}e^{ \mathrm{i}\theta_{0}}s(nT-t_{0})+A_{1}e^{\mathrm{i}\theta_{1}}s(nT-t_{1})+w_{ \mathrm{R}}(nT)\\ x_{\mathrm{LHCP}}(nT)&=B_{0}e^{\mathrm{i}\theta_{0 }^{\prime}}s(nT-t_{0})+B_{1}e^{\mathrm{i}\theta_{1}^{\prime}}s(nT-t_{1})+w_{ \mathrm{L}}(nT)\end{split}  \tag{7}\]  
where \(n=1,2,3,\ldots,N\) is the sample point number. The noise components of distinct samples are independent; hence, the log-likelihood function of MLE to be maximized is easy to obtain in Equation (8):  
\[\ln p(\eta)=\ln\frac{1}{(2\pi\delta^{2})^{N}}-\frac{1}{2\delta^{2}}\left( \begin{array}{c}\sum\limits_{n=1}^{N}\left\{\begin{array}{c}\left[x_{ \mathrm{RHCP}}(nT)-A_{0}e^{\mathrm{i}\theta_{0}}s(nT-t_{0})-A_{1}e^{\mathrm{ i}\theta_{1}}s(nT-t_{1})\right]^{\mathrm{H}}\\ *\left[x_{\mathrm{RHCP}}(nT)-A_{0}e^{\mathrm{i}\theta_{0}}s(nT-t_{0})-A_{1}e^{ \mathrm{i}\theta_{1}}s(nT-t_{1})\right]\end{array}\right\}\\ +\sum\limits_{n=1}^{N}\left\{\begin{array}{c}\left[x_{\mathrm{ LHCP}}(nT)-B_{0}e^{\mathrm{i}\theta_{0}^{\prime}}s(nT-t_{0})-B_{1}e^{\mathrm{i} \theta_{1}^{\prime}}s(nT-t_{1})\right]^{\mathrm{H}}\\ *\left[x_{\mathrm{LHCP}}(nT)-B_{0}e^{\mathrm{i}\theta_{0}^{\prime}}s(nT-t_{0}) -B_{1}e^{\mathrm{i}\theta_{1}^{\prime}}s(nT-t_{1})\right]\end{array}\right\} \end{array} \right) \tag{8}\]  
where \([\ ]^{\mathrm{H}}\) denotes the Hermitian transpose and \(\delta^{2}\) is the variance of the white noise. By substituting the product of the amplitude and exponential term as is done in MMT [9] by Equation (9): \begin{table}
\begin{tabular}{c c c} \hline \hline
**Received Signal Cases** & \(\mathbf{B_{0}^{2}}\) & \(\mathbf{B_{1}^{2}}\) (DPG 1) \\ \hline Weak LHCP Signal & & \\ Case 1 (weak1) & \(-15\) dB & \(-15\) dB (\(1.125\)), \(-30\) dB (\(1\)) \\ \hline Weak LHCP Signal & & \\ Case 2 (weak2) & \(-30\) dB & \(-15\) dB (\(1.125\)) \\ \hline Strong LHCP Signal & & \\ Case 1 (strong1) & & \\ \hline Strong LHCP Signal & & \\ Case 2 (strong2) & & \\ \hline \hline \end{tabular} \({}^{1}\) The relative power of \(A_{1}^{2}\) is \(-6\) dB when calculating \(DPG\).  
\end{table} 
Table 2: The specific power level of the received LOS and MP signals in the LHCP antenna. \[ \begin{split} x_{\text{RHCP}}(nT)&=x_{1}(nT)+\text{j}x _{\text{Q}}(nT)\\ x_{\text{LHCP}}(nT)&=y_{1}(nT)+\text{j}y_{\text{Q}}(nT )\\ A_{0}\text{e}^{\text{j}\theta_{0}}&=a+\text{j}b\\ A_{1}\text{e}^{\text{j}\theta_{1}}&=c+\text{j}d \\ B_{0}\text{e}^{\text{j}\theta_{0}^{\prime}}&=e+\text{j}f \\ B_{1}\text{e}^{\text{j}\theta_{1}^{\prime}}&=p+\text{j}q \end{split}  \tag{9}\]  
the maximization of Equation (8) is equivalent to the minimization of the objective function in Equations (10)-(12) with respect to the parameters \(\boldsymbol{\eta}=[t_{0},t_{1},\ a,b,c,d,e,f,p,q]\) to be estimated:  
\[(\boldsymbol{\eta})=\ln C+\frac{1}{2\sigma^{2}}[\text{R}_{\text{HCP}}( \boldsymbol{\eta})+_{\text{LHCP}}(\boldsymbol{\eta})] \tag{10}\]  
\[ \begin{split} O_{\text{RHCP}}(\boldsymbol{\eta})&=( a^{2}+b^{2}+c^{2}+d^{2})R_{\text{ss}}(0)+2(ac+bd)R_{\text{ss}}(t_{1}-t_{0})\\ &-2 aR_{\text{ssI}}(t_{0})-2 bR_{\text{ssQ}}(t_{0})-2 cR_{\text{ssI} }(t_{1})-2 dR_{\text{ssQ}}(t_{1})\\ O_{\text{LHCP}}(\boldsymbol{\eta})&=(e^{2}+f^{2}+p ^{2}+q^{2})R_{\text{ss}}(0)+2(ep+fq)R_{\text{ss}}(t_{1}-t_{0})\\ &-2 eR_{\text{syI}}(t_{0})-2 fR_{\text{syQ}}(t_{0})-2 pR_{\text{syI} }(t_{1})-2 qR_{\text{syQ}}(t_{1})\end{split}  \tag{11}\]  
where \(R_{\text{ss}}(t_{1}-t_{0})=\sum\limits_{n=1}^{N}s(nT-t_{0})\times s(nT-t_{1})\) is the auto-correlation function (ACF) of the local code wave.