# Multipath Mitigation for BOC Signals Based on Prompt-Assisted-Offset Correlator
## 2 Problem Formulation
### Multipath Model for BOC Signals  
In practice, the BOC signal is probably reflected or diffracted by obstacles around the GNSS receiver, resulting in multipath signals. Compared with the direct signal, the multipath signal has a longer propagation path. Moreover, the power and carrier Doppler of a multipath signal are usually different from those of the direct signal. The compound signal consisting of the direct signal and multipath signals can be expressed as follows:  
\[r_{c}(t)= r_{d}(t)+\sum_{m=1}^{M}r_{m}(t) \tag{5}\] \[r_{m}(t)= a_{m}Ac(t-\tau-\tau_{m})\cos(2\pi f_{sc}t+\phi+\phi_{m})\cos \left(2\pi(f_{c}+f_{m})t+\theta+\theta_{m}\right) \tag{6}\]  
where \(r_{d}(t)\) is the direct signal shown in Equation (2), \(r_{m}(t)\) is the m-th multipath signal, \(M\) is the number of multipath signals, \(\alpha_{m}\) is the multipath relative amplitude, \(\tau_{m}\) is the multipath relative delay, \(\phi_{m}=-2\pi f_{sc}\tau_{m}\) is the multipath relative subcarrier phase, \(f_{m}\) is the Doppler difference between the direct signal and m-th multipath signal, which is called the fading frequency, and \(\theta_{m}\) is the multipath relative phase.  
The multipath fading frequency is related to the geometric distribution of the satellite, reflecting platform, and GNSS receiver. Fast-varying multipath signals with large fading frequencies will be suppressed by correlators and loop filters of the GNSS receiver [27]. When the reciprocal of the fading frequency is less than the integration time, the multipath signal is completely averaged out and has no effect on the tracking loop. By contrast, slow-varying multipath signals with small fading frequencies severely affect the tracking loop and cause large ranging errors. In conclusion, the slow-varying multipath poses a greater threat to GNSS positioning accuracy than the fast-varying multipath. For a stationary GNSS receiver, the multipath fading frequency is very small, typically less than 1 Hz [28; 29]. Most multipath signals received by GNSS receivers used in surveying, monitoring stations, and other low-dynamic scenarios have small fading frequencies and change slowly [30; 31]. Therefore, this paper investigates the slow-varying multipath problem for low-dynamic GNSS receivers.  
According to the double-sideband reception model for BOC signals, the compound signal shown in Equation (5) can be transformed as follows:\[\left\{\begin{aligned} r_{c}(t)=& r_{uc}(t)+r_{lc}(t)\\ r_{uc}(t)=&\frac{A}{2}\bigg{(}c(t-\tau)\cos(2\pi f_{u}t +\theta_{u})+\sum_{m=1}^{M}\alpha_{m}c(t-\tau-\tau_{m})\cos\big{(}2\pi(f_{u}+f_ {m})t+\theta_{u}+\theta_{m}+\phi_{m}\big{)}\bigg{)}\\ r_{lc}(t)=&\frac{A}{2}\bigg{(}c(t-\tau)\cos(2\pi f_{ l}t+\theta_{l})+\sum_{m=1}^{M}\alpha_{m}c(t-\tau-\tau_{m})\cos\big{(}2\pi(f_{l}+f_ {m})t+\theta_{l}+\theta_{m}-\phi_{m}\big{)}\bigg{)}\\ \end{aligned}\right. \tag{7}\]  
where \(r_{uc}(t)\) is the USB of the compound signal, \(r_{lc}(t)\) is the LSB of the compound signal, \(f_{u}=f_{c}+f_{sc}\) and \(f_{l}=f_{c}-f_{sc}\) are the carrier frequencies of the direct USB and LSB signals, respectively, and \(\theta_{u}=\theta+\phi\) and \(\theta_{l}=\theta-\phi\) are the carrier phases of the USB and LSB direct signals, respectively.  
Equation (7) shows that the received BOC signal can be regarded as the sum of USB and LSB signals in the presence of multipath. The BOC multipath signal can be decomposed into two BPSK multipath signals located in the USB and LSB, respectively. Therefore, it is feasible to track the compound BOC signal with the DBT loop. However, multipath signals lead to tracking errors. It is necessary to analyze the effect of multipath on the DBT loop.