## 3 Theoretical Performance of DPMM Using MLE under One LOS and One Reflected Path Environment
### The Root Mean Square Error of the ML Estimators
This phenomenon indicates that DP-R2L2-MLE often mistakes the noise as  
Figure 4: The RMSE of the TOA estimation of LOS signal using MLE with different signal models in the two-path environments defined by Strong LHCP Signal Case 1 in Table 2.  
Figure 5: The RMSE of the TOA estimation of LOS signal using MLE with different signal models in the two-path environment defined by Strong LHCP Signal Case 2 when \(DPG=2\). the second path and thus, its estimation of the first path is still the LOS and MP composite signal while DP-R2L1-MLE always gives the parameters of two paths that are close to their true values. Consequently, the performance of DP-R2L1-MLE is slightly worse than that of DP-R2L2-MLE when the MP signal is extremely close to the LOS signal. From the simulations above, the answer to the fundamental question on the effective conditions of multipath mitigation using a dual-polarization antenna is summarized as follows:  
1. Dual-polarization antenna technology contributes to multipath mitigation when the LHCP antenna receives enough of MP signal power (higher than \(-12\) dB). 2. If condition (1) is satisfied, dual-polarization antenna technology contributes to mitigating the short delay multipath. Based on the effective condition, when we design a MLE algorithm for DPMM, we may need a CNR indicator to detect whether the power of the received LHCP signal is high enough to ensure the additional computation load for DPMM is worthwhile. Based on this motivation, we propose an indicator-integrated DP-R2L1-MLE (iiDP-R2L1-MLE) algorithm following the steps below:  
1. Estimate the CNR of RHCP (\(CNR_{\text{RHCP}}\)) and LHCP (\(CNR_{\text{LHCP}}\)) channels. 2. If \(CNR_{\text{RHCP}}-CNR_{\text{LHCP}}>\textit{threshold}\), which means there is not enough power in the LHCP channel, we use the RHCP-R2-MLE to get the TOA estimation of the LOS signal; otherwise we use DP-R2L1-MLE (the _threshold_ we use here is 12 dB based on the effective condition concluded by the simulation results of Figures 3 and 4). Figure 6 is the performance of the iiDP-R2L1-MLE in different received signal cases defined in Table 2. In Weak LHCP Signal Cases 1 and 2 (\(DPG=1.25\)) when there is not enough power in the LHCP channel, the performance of DP-R2L1-MLE (in green diamonds and green circles) degrades to that of RHCP-R2-MLE (in blue circles). In Strong LHCP Signal Case 2 (\(DPG=2\)), it performs better than DP-R2L2-MLE and RHCP-R2-MLE within the short delay range from 6 m to 20 m. Comparing the DP-R2L1-MLE in Strong LHCP Signal Case 1 with that in Case 2 where the power of the LOS signal in LHCP channel is different, it deteriorates when the MP signal is about 20 m away from the LOS signal. This result indicates that the increasing of \(B_{0}\) does affect the performance of DP-R2L1-MLE for the growing bias of ignoring \(B_{0}\) in the R2L1 model. Nevertheless, usually the power of the LOS signal received by the LHCP antenna cannot be too high, and as long as the MP signal dominates in the LHCP channel, ignoring the modeling of the LOS component in the LHCP channel is valid. Figure 6: The RMSE of the TOA estimation of the LOS signals of the proposed iiDP-R2L1-MLE in different received signal cases defined in Table 2.  
To sum up, within the scope of MLE, multipath mitigation can benefit from the dual-polarization antenna technology when the LHCP antenna receives enough power of the MP signal (higher than \(-12\) dB as the power of the received LOS signal in the RHCP channel is normalized to \(0\) dB) and when short delay MP exists whose relative delay to the LOS signal is less than about \(75\) ns or \(25\) m. Under this circumstance, the proposed iIDP-R2L1-MLE algorithm based on the R2L1 model performs better than the conventional RHCP-R2-MLE algorithm.