# Multipath Mitigation for BOC Signals Based on Prompt-Assisted-Offset Correlator
## 4 Performance Evaluation and Results
### Thermal Noise Performance Analysis  
The PAOC technique employs two subcarrier discriminators in the PAOC-SPLL, i.e., the S-OC and S-P discriminators. The S-P discriminator is the same as the subcarrier discriminator used in the original DBT loop. Thus, the thermal noise error of the S-P discriminator output can be expressed as [10]:  
\[\sigma_{s-p}^{2}=\frac{B_{SPLL}(1-0.5B_{SPLL}T)}{C/N_{0}\int_{-\beta_{r}/2}^{+ \beta_{r}/2}G_{s}(f)df} \tag{30}\]  
where \(B_{SPLL}\) is the noise bandwidth of the PAOC-SPLL, \(T\) is the integration time, \(C/N_{0}\) is the carrier-to-noise ratio, \(G_{s}(f)\) is the normalized power spectral density (PSD) of each single sideband of the BOC signal, and \(\beta_{r}\) is the received bandwidth. The S-OC discriminator uses the offset correlator instead of the prompt correlator. According to the relationship between the offset and prompt correlators, the thermal noise error of the S-OC discriminator output can be calculated as:  
\[\sigma_{s-oc}^{2}=\frac{B_{SPLL}(1-0.5B_{SPLL}T)\int_{-\beta_{r}/2}^{+\beta_{r}/ 2}G_{s}(f)df}{C/N_{0}\left(\int_{-\beta_{r}/2}^{+\beta_{r}/2}G_{s}(f)\cos(2 \pi f\tau_{oc})df\right)^{2}} \tag{31}\]  
The derivation of Equation (31) is given in Appendix A. The outputs of the S-P and S-OC discriminators are combined in the subcarrier PAOC complementary filter to obtain an accurate subcarrier estimate \(\Delta\phi_{PAOC}\). Therefore, the noise variance \(\sigma_{s-PAOC}^{2}\) of the PAOC subcarrier estimate \(\Delta\phi_{PAOC}\) is between \(\sigma_{s-p}^{2}\) and \(\sigma_{s-oc}^{2}\). When the smoothing constant of the subcarrier PAOC complementary filter is one, the PAOC technique degenerates to the OC technique, and \(\sigma_{s-PAOC}^{2}\) is equal to \(\sigma_{s-oc}^{2}\). As the smoothing constant increases, \(\sigma_{s-PAOC}^{2}\) gradually approaches \(\sigma_{s-p}^{2}\). The PAOC technique employs two carrier discriminators in the PAOC-PLL, i.e., the C-OC and C-P discriminators. The C-P discriminator is the same as the carrier discriminator used in the original DBT loop. Thus, the thermal noise error of the C-P discriminator output can be expressed as [10]:  
\[\sigma_{c-p}^{2}=\frac{B_{PLL}(1-0.5B_{PLL}T)}{C/N_{0}\int_{-\beta_{r}/2}^{+ \beta_{r}/2}G_{s}(f)df} \tag{32}\]  
where \(B_{PLL}\) is the noise bandwidth of the PAOC-PLL. Figure 7: MEEs of the DBT, OC, and PAOC techniques for the BOC(15, 2.5) signal with a bandwidth of 40.96 MHz: (**a**) subcarrier MEE; (**b**) carrier MEE. The C-OC discriminator uses the offset correlator instead of the prompt correlator. According to the relationship between the offset and prompt correlators, the thermal noise error of the C-OC discriminator output can be calculated as:  
\[\sigma_{c-oc}^{2}=\frac{B_{PLL}(1-0.5B_{PLL}T)\int_{-\beta_{r}/2}^{+\beta_{r}/2 }G_{s}(f)df}{C/N_{0}\big{(}\int_{-\beta_{r}/2}^{+\beta_{r}/2}G_{s}(f)\cos(2\pi f \tau_{oc}T_{c})df\big{)}^{2}} \tag{33}\]  
The derivation of Equation (33) is given in Appendix A. The outputs of the C-P and C-OC discriminators are combined in the carrier PAOC complementary filter to obtain an accurate carrier estimate \(\Delta\theta_{PAOC}\). Therefore, the noise variance \(\sigma_{c-PAOC}^{2}\) of the PAOC carrier estimate \(\Delta\theta_{PAOC}\) is between \(\sigma_{c-p}^{2}\) and \(\sigma_{c-oc}^{2}\). When the smoothing constant of the carrier PAOC complementary filter is one, \(\sigma_{c-PAOC}^{2}\) is equal to \(\sigma_{c-oc}^{2}\). As the smoothing constant increases, \(\sigma_{c-PAOC}^{2}\) gradually approaches \(\sigma_{c-p}^{2}\). Figure 8 shows the subcarrier and carrier noise errors of three techniques for the BOC(15, 2.5) signal, where the noise bandwidths of the SPLL and the PLL are 1 Hz and 10 Hz, respectively. With the identical forward offset, the PAOC technique has better noise performance than the OC technique. As the smoothing constant \(N\) increases, thermal noise errors of the PAOC technique gradually decrease and approach those of the DBT method.