# Multipath Mitigation for BOC Signals Based on Prompt-Assisted-Offset Correlator
## 3 Materials and Methods
### PAOC Technique for Subcarrier Multipath Mitigation  
There are two specially designed subcarrier discriminators in the PAOC technique. One subcarrier discriminator uses the offset correlator and is called the S-OC discriminator. The other subcarrier discriminator uses the prompt correlator and is called the S-P discriminator. The forward offset of the offset correlator relative to the prompt correlator is assumed to be \(\tau_{oc}\) chips. The S-OC discriminator is designed as:  
\[\text{Disc}(\Delta\phi_{oc})=\arctan\left(\frac{QR_{oc}^{u}-QR_{oc}^{l}}{IR_{oc }^{u}+IR_{oc}^{l}}\right) \tag{14}\]  
where \(IR\) denotes the in-phase correlator output, \(QR\) denotes the quadrature-phase correlator output, the subscript \(oc\) denotes the offset correlator, and the superscripts \(u\) and \(l\) denote the USB and LSB, respectively. The S-P discriminator is designed as:  
\[\text{Disc}(\Delta\phi_{p})=\arctan\left(\frac{QR_{p}^{u}-QR_{p}^{l}}{IR_{p}^ {u}+IR_{p}^{l}}\right) \tag{15}\]  
where the subscript \(p\) denotes the prompt correlator. When the multipath relative delay is greater than \(1-\tau_{oc}\) chips, the S-OC discriminator output contains no multipath error and can be modeled as:  
\[\Delta\phi_{oc}=\Delta\phi_{d}+\epsilon_{oc} \tag{16}\]  
where \(\Delta\phi_{d}\) is the subcarrier phase difference between the local signal and the direct signal, and \(\epsilon_{oc}\) is the S-OC noise error. The S-P discriminator output contains multipath errors and can be modeled as:  
\[\Delta\phi_{p}=\Delta\phi_{d}+\Delta\phi_{ME}+\epsilon_{p} \tag{17}\]  
where \(\Delta\phi_{ME}\) is the subcarrier multipath error, and \(\epsilon_{p}\) is the S-P noise error. Because the prompt correlator output has a higher SNR than the offset correlator, the variance of \(\epsilon_{p}\) is smaller than that of \(\epsilon_{oc}\). Combining the discriminator outputs shown in Equations (16) and (17), a coarse estimate of the subcarrier multipath error can be obtained as:  
\[\Delta\hat{\phi}_{ME}=\Delta\phi_{p}-\Delta\phi_{oc}=\Delta\phi_{ME}+e_{p}- \epsilon_{oc} \tag{18}\]  
The subcarrier multipath error estimate \(\Delta\hat{\phi}_{ME}\) shown in Equation (18) is noisy. To obtain a fine estimate of the subcarrier multipath error, a first-order low-pass filter is used to reduce the noise error of \(\Delta\hat{\phi}_{ME}\). The low-pass filtering process can be expressed as:  
\[\Delta\hat{\phi}_{MEF}[k]=\frac{N-1}{N}\Delta\hat{\phi}_{MEF}[k-1]+\frac{1}{N }\Delta\hat{\phi}_{ME}[k] \tag{19}\]  
where \(N\) is the smoothing constant of the low-pass filter, and \(k\) is the current epoch. The initial value \(\Delta\hat{\phi}_{MEF}[0]\) can be set as 0. There is an implicit prerequisite for using the low-pass filter to reduce noise errors, which is that the subcarrier multipath error changes slowly. For a low-dynamic GNSS receiver, the fading frequencies of received multipath signals are usually very small. Therefore, the multipath errors of a static or low-dynamic GNSS receiver change slowly, which allows the low-pass filter to be used. The smoothing constant \(N\) determines the noise error of the filtered estimate \(\Delta\hat{\phi}_{MEF}\). As the smoothing constant increases, the noise error of \(\Delta\hat{\phi}_{MEF}\) reduces, while the dynamic performance of the GNSS receiver degrades. In practice, the smoothing constant \(N\) can be set flexibly as needed. Considering that the fading frequency of multipath signals received by a static or low-dynamic receiver is generally less than 1 Hz, the smoothing time of the low-pass filter is preferably set to be no longer than 1 s. More discussions about the low-pass filter will be given in Section 4. After low-pass filtering, an accurate estimate of the subcarrier multipath error can be obtained as:  
\[\Delta\hat{\phi}_{MEF}=\Delta\phi_{ME}+\epsilon_{res} \tag{20}\]  
where \(\epsilon_{res}\) is the residual noise error with a small variance.