## 3 Theoretical Performance of DPMM Using MLE under One LOS and One Reflected Path Environment
### The Root Mean Square Error of the ML Estimators  
In this subsection we evaluate the ML estimators with MC simulations in each received signal case as listed in Table 2. The baseband I/Q signal is the GPS L1 coarse/acquisition (C/A) signal, which is generated according to Equation (5) with a CNR of 42 dB-Hz. The double-side bandwidth of the pre-correlation filter is 16 MHz. The I/Q data are sampled in 32 MHz for 100 ms each time to generate the ACF and CCFs for ML estimators. Here we define the short delay range as when the delay of multipath, relative to the LOS signal, is from 0 to the inverse of the bandwidth of the signal, i.e., from 0 to 62.5 ns or 0 to 18.75 m.  
Figure 3 shows the root mean square error (RMSE) of the TOA of the LOS signal \(\hat{t_{0}}\) of each ML estimator as a function of the relative delay of the MP signal to the LOS signal in the two-path environments defined by Weak LHCP Signal Cases 1 and 2 in Table 2. The performance of DP-R2L2-MLE (in magenta diamonds lines) under these environments is almost the same as that of RHCP-R2-MLE (in blue circles line), which is based on a single-polarization antenna. Figure 4 illustrates the RMSEs of \(\hat{t_{0}}\) estimated by ML estimators in Strong LHCP Signal Case 1 defined in Table 2 as _DPG_ increases. When _DPG_ = 1.25, the DP-R2L2-MLE performs almost the same as RHCP-R2-MLE. When the relative delay of the MP signal to the LOS signal is within 7 m to 25 m, the DP-R2L2-MLE with _DPG_ higher than 1.5 (magenta lines) surpasses the RHCP-R2-MLE (blue line). As _DPG_ increases, the improvement becomes larger (the results of DP-R2L2-MLE in Strong LHCP Signal Case 2 are similar to that in Strong LHCP Signal Case 1; hence, the corresponding RMSEs are omitted). From the results of the two simulations illustrated by Figures 3 and 4, we can conclude that: when there is not enough power in the LHCP channel, i.e., when the power of the MP signal received by the LHCP antenna is lower than \(-\)12 dB, dual-polarization antennas hardly help in multipath mitigation based on parameter estimation; when it is higher than \(-\)12 dB, dual-polarization antennas contribute to mitigating the short delay multipath. Figure 3: The RMSE of the TOA estimation of the LOS signal using MLE with different models in the two-path environments defined by Weak LHCP Signal Cases 1 and 2 in Table 2.  
In Figure 5, we compare the ML estimators in Strong LHCP Signal Case 2 when \(DPG=2\). The DP-R2L1-MLE (red line) outperforms the DP-R2L2-MLE (magenta diamonds line) in the short delay range (except for the range under 5 m). The uncertainty of the TOA of the first MP \(\hat{t_{1}}\) in the R2L1 model is greatly reduced compared with that in the R2L2 model, for we can get an almost unbiased estimation of the MP signal from the LHCP channel. The accurate estimation of the MP, in turn, enhances the TOA estimation of the LOS signal because they affect each other mutually [38]. However, when the MP signal is extremely close to the LOS signal (i.e., less than 5 m), DP-R2L1-MLE performs slightly worse than other algorithms. There are two possible reasons. Firstly, the one-path ML estimator actually performs better than the two-path ML estimator for the ill-conditioned parameter estimation problem in the extreme close-in range as concluded in [8; 9; 24; 36]. Secondly, the DP-R2L2-MLE has a high probability of degrading to a one-path ML estimator while DP-R2L1-MLE always works under the two-path model even when the LOS and MP signals are in the extremely close range. To be more specific, the estimated TOAs of MP (\(\hat{t_{1}}\)) in DP-R2L2-MLE are random within the searching space and the estimated amplitudes of MP (\(\hat{A_{1}}\)) are often close to zero, whereas the same estimated parameters in DP-R2L1-MLE are neither random nor close to zero, but are close to the true value.