# Multipath Mitigation for BOC Signals Based on Prompt-Assisted-Offset Correlator
## 4 Performance Evaluation and Results
### Multipath Performance Analysis  
The outputs of offset correlators for the USB and LSB signals can be expressed as:  
\[R_{u-\infty c}= \frac{A}{2}\left(R(\Delta\tau_{d}-\tau_{\infty c})+\sum_{m=1}^{M} \alpha_{m}R(\Delta\tau_{d}-\tau_{\infty c}-\tau_{m})\exp\left(j(\theta_{m}+ \phi_{m})\right)\right)\exp\left(j(\Delta\theta_{d}+\Delta\phi_{d})\right) \tag{24}\] \[R_{l-\infty c}= \frac{A}{2}\left(R(\Delta\tau_{d}-\tau_{\infty c})+\sum_{m=1}^{M} \alpha_{m}R(\Delta\tau_{d}-\tau_{\infty c}-\tau_{m})\exp\left(j(\theta_{m}- \phi_{m})\right)\right)\exp\left(j(\Delta\theta_{d}-\Delta\phi_{d})\right) \tag{25}\]  
The synthesized correlation function used by the S-OC discriminator is calculated as:  
\[R_{s-\infty c}= R_{u-\infty c}+R_{l-\infty}^{*} \tag{26}\] \[= A\left(R(\Delta\tau_{d}-\tau_{\infty c})\cos(\Delta\theta_{d})+ \sum_{m=1}^{M}\alpha_{m}R(\Delta\tau_{d}-\tau_{\infty c}-\tau_{m})\cos(\Delta \theta_{d}+\theta_{m})\exp(j\phi_{m})\right)\exp(j\Delta\phi_{d})\]  
Figure 6: Implementation architecture of the PAOC technique in the DBT loop for BOC signals. Multipath signals cause a phase rotation of \(R_{s-oc}\), which is the subcarrier multipath error. Assuming that there is only one multipath signal, the subcarrier multipath error of the S-OC discriminator can be calculated as:  
\[\Delta\phi_{ME}=\arctan\left(\frac{\alpha_{m}R(\Delta\tau_{d}-\tau_{oc}-\tau_{ m})\cos(\Delta\theta_{d}+\theta_{m})\sin(\phi_{m})}{R(\Delta\tau_{d}-\tau_{oc})\cos( \Delta\theta_{d})+\alpha_{m}R(\Delta\tau_{d}-\tau_{oc}-\tau_{m})\cos(\Delta \theta_{d}+\theta_{m})\cos(\phi_{m})}\right) \tag{27}\]  
When \(|\Delta\tau_{d}-\tau_{oc}-\tau_{m}|\geq 1\), \(R(\Delta\tau_{d}-\tau_{oc}-\tau_{m})=0\). This demonstrates that the S-OC discriminator can eliminate subcarrier multipath errors caused by the multipath signal with a relative delay greater than \(1-\tau_{oc}\) chips. The subcarrier multipath error of the S-P discriminator is equal to that of the DBT method, which is given in Equation (11). The PAOC technique subtly combines the outputs of the S-OC and S-P discriminators. Thus, the subcarrier multipath performance of the PAOC technique is the same as that of the S-OC discriminator shown in Equation (27). The synthesized correlation function used by the C-OC discriminator is calculated as:  
\[R_{c-oc}= R_{u-oc}+R_{l-oc} \tag{28}\] \[= A\left(R(\Delta\tau_{d}-\tau_{oc})\cos(\Delta\phi_{d})+\sum_{m=1 }^{M}\alpha_{m}R(\Delta\tau_{d}-\tau_{oc}-\tau_{m})\cos(\Delta\phi_{d}+\phi_{m })\exp(j\theta_{m})\right)\exp(j\Delta\theta_{d})\]  
Assuming that there is only one multipath signal, the carrier multipath error of the C-OC discriminator can be calculated as:  
\[\Delta\theta_{ME}=\arctan\left(\frac{\alpha_{m}R(\Delta\tau_{d}-\tau_{oc}- \tau_{m})\cos(\Delta\phi_{d}+\phi_{m})\sin(\theta_{m})}{R(\Delta\tau_{d}- \tau_{oc})\cos(\Delta\phi_{d})+\alpha_{m}R(\Delta\tau_{d}-\tau_{oc}-\tau_{m}) \cos(\Delta\phi_{d}+\phi_{m})\cos(\theta_{m})}\right) \tag{29}\]  
When \(|\Delta\tau_{d}-\tau_{oc}-\tau_{m}|\geq 1\), \(R(\Delta\tau_{d}-\tau_{oc}-\tau_{m})=0\). This demonstrates that the C-OC discriminator can eliminate carrier multipath errors caused by the multipath signal with a relative delay greater than \(1-\tau_{oc}\) chips. The carrier multipath error of the C-P discriminator is equal to that of the DBT method, which is given in Equation (13). The PAOC technique subtly combines the outputs of the C-OC and C-P discriminators. Thus, the carrier multipath performance of the PAOC technique is the same as that of the C-OC discriminator shown in Equation (29). The multipath error envelope (MEE) is a commonly used criterion to evaluate the multipath performance, which shows the maximum error under different multipath relative delays [32]. Equation (27) indicates that the subcarrier multipath error reaches a maximum value when \(\theta_{m}=0\) and \(\pi\). Equation (29) indicates that the carrier multipath error reaches a maximum value when \(\theta_{m}=\pm\pi/2\).