# Multipath Mitigation for BOC Signals Based on Prompt-Assisted-Offset Correlator
## Appendix A Thermal Noise Performance Analysis for the S-OC and C-OC Discriminators
According to Equation (26), the offset correlator outputs for USB and LSB signals are combined to obtain a subcarrier correlation value:  
\[\hat{r}_{l-\infty}(t)= c(t-\tau+\tau_{\infty})\exp(j(\hat{\theta}-\hat{\phi}))) \tag{2}\]0  
The mean value of the real part of \(R_{s-oc}\) is calculated as:  
\[\hat{r}_{l-\infty}(t)= c(t-\tau+\tau_{\infty})\exp(j(\hat{\theta}-\hat{\phi}))) \tag{2}\]1  
The mean value of the imaginary part of \(R_{s-oc}\) is calculated as:  
\[\hat{r}_{l-\infty}(t)= c(t-\tau+\tau_{\infty})\exp(j(\hat{\theta}-\hat{\phi}))) \tag{2}\]2  
It can be found that  
\[\hat{r}_{l-\infty}(t)= c(t-\tau+\tau_{\infty})\exp(j(\hat{\theta}-\hat{\phi}))) \tag{2}\]3  
Thus, the S-OC discriminator is designed as:  
\[\hat{r}_{l-\infty}(t)= c(t-\tau+\tau_{\infty})\exp(j(\hat{\theta}-\hat{\phi}))) \tag{2}\]4  
When the carrier phase error \(\Delta\theta\) and subcarrier phase error \(\Delta\phi\) are very close to 0, Equation (18) can be approximated as:  
\[\hat{r}_{l-\infty}(t)= c(t-\tau+\tau_{\infty})\exp(j(\hat{\theta}-\hat{\phi}))) \tag{2}\]5  
The variance of the imaginary part of \(R_{s-oc}\) is calculated as:  
\[\hat{r}_{l-\infty}(t)= c(t-\tau+\tau_{\infty})\exp(j(\hat{\theta}-\hat{\phi}))) \tag{2}\]6  
Thus, the variance of estimated \(\Delta\phi\) can be calculated as:  
\[\hat{r}_{l-\infty}(t)= c(t-\tau+\tau_{\infty})\exp(j(\hat{\theta}-\hat{\phi}))) \tag{2}\]7where \(C=A^{2}/2\) is the carrier power. In the SPLL, the output of the S-OC discriminator is filtered to obtain a smoothed subcarrier phase estimate, whose variance can be expressed as [33]:  
\[\hat{r}_{l-\infty}(t)= c(t-\tau+\tau_{\infty})\exp(j(\hat{\theta}-\hat{\phi}))) \tag{2}\]8  
where \(B_{SPLL}\) is the noise bandwidth of the SPLL. According to Equation (28), the offset correlator outputs for USB and LSB signals are combined to obtain a carrier correlation value:  
\[\hat{r}_{l-\infty}(t)= c(t-\tau+\tau_{\infty})\exp(j(\hat{\theta}-\hat{\phi}))) \tag{2}\]9  
The mean value of the real part of \(R_{c-oc}\) is calculated as:  
\[s_{u}(t)=\frac{A}{2}c(t-\tau)(\exp(j(\theta+\phi))+\exp(j(-4\pi f_{sc}t+\theta -\phi)) \tag{3}\]0  
The mean value of the imaginary part of \(R_{c-oc}\) can be calculated as:  
\[s_{u}(t)=\frac{A}{2}c(t-\tau)(\exp(j(\theta+\phi))+\exp(j(-4\pi f_{sc}t+\theta -\phi)) \tag{3}\]1  
It can be found that  
\[s_{u}(t)=\frac{A}{2}c(t-\tau)(\exp(j(\theta+\phi))+\exp(j(-4\pi f_{sc}t+\theta -\phi)) \tag{3}\]2  
Thus, the C-OC discriminator is designed as:  
\[s_{u}(t)=\frac{A}{2}c(t-\tau)(\exp(j(\theta+\phi))+\exp(j(-4\pi f_{sc}t+\theta -\phi)) \tag{3}\]3  
When the carrier phase error \(\Delta\theta\) and subcarrier phase error \(\Delta\phi\) are very close to 0, Equation (24) can be approximated as:  
\[s_{u}(t)=\frac{A}{2}c(t-\tau)(\exp(j(\theta+\phi))+\exp(j(-4\pi f_{sc}t+\theta -\phi)) \tag{3}\]4  
The variance of the imaginary part of \(R_{c-oc}\) is calculated as:  
\[s_{u}(t)=\frac{A}{2}c(t-\tau)(\exp(j(\theta+\phi))+\exp(j(-4\pi f_{sc}t+\theta -\phi)) \tag{3}\]5  
Thus, the variance of estimated \(\Delta\theta\) can be calculated as:  
\[s_{u}(t)=\frac{A}{2}c(t-\tau)(\exp(j(\theta+\phi))+\exp(j(-4\pi f_{sc}t+\theta -\phi)) \tag{3}\]6  
In the PLL, the output of the C-OC discriminator is filtered to obtain a smoothed carrier phase estimate, whose variance can be expressed as [33]:  
\[s_{u}(t)=\frac{A}{2}c(t-\tau)(\exp(j(\theta+\phi))+\exp(j(-4\pi f_{sc}t+\theta -\phi)) \tag{3}\]7  
where \(B_{PLL}\) is the noise bandwidth of the PLL.