# Multipath Mitigation for BOC Signals Based on Prompt-Assisted-Offset Correlator
## 2 Problem Formulation
### Double-Sideband Reception Model for BOC Signals  
In the time domain, the BOC signal can be viewed as the product of a BPSK signal and a subcarrier in the form of a square wave, which is written as follows:  
\[s(t)=c(t)\text{sign}\big{(}\sin(2\pi f_{\text{sc0}}t+\phi_{0})\big{)}\cos(2\pi f _{\text{c0}}t+\theta_{0}) \tag{1}\]  
where \(c(t)\) represents the spreading code, \(f_{\text{sc0}}\) is the nominal subcarrier frequency, \(\phi_{0}\) is the initial subcarrier phase, \(f_{\text{c0}}\) is the nominal carrier frequency, and \(\theta_{0}\) is the initial carrier phase. A BOC-modulated GNSS signal is usually abbreviated as BOC(m, n), indicating that its subcarrier frequency is \(m\times 1.023\) MHz and its code rate is \(n\times 1.023\) MHz. The order of the BOC signal is defined as \(k=2m/n\), which is generally an integer. Figure 1 shows the normalized ACF of BPSK(n), BOC(2n, n), and BOC(6n, n) signals. It can be seen from Figure 1 that the ACF of the BOC signal with a higher order has more peaks. In practice, the signal transmission bandwidth of GNSS satellites is limited. For BOC signals with a high subcarrier frequency, the actual subcarrier emitted by GNSS satellites is not a square wave but approximates a sine wave. For example, the transmission bandwidth for Galileo E1A signal with the BOC(15, 2.5) modulation is 40.96 MHz [26], so the subcarrier harmonics are filtered out at the transmitting end. Moreover, the front-end bandwidth of GNSS receivers is also limited and usually smaller than the GNSS signal transmission bandwidth. The subcarrier of the received BOC signal can be reasonably considered as a sine wave. Therefore, the received BOC signal can be written as:  
\[r_{d}(t)=Ac(t-\tau)\cos(2\pi f_{sc}t+\phi)\cos(2\pi f_{c}t+\theta) \tag{2}\]  
where \(A\) is the amplitude of the received BOC signal, \(\tau\) is the propagation delay, \(f_{sc}\) is the subcarrier frequency with Doppler, \(\phi\) is the subcarrier phase, \(f_{c}\) is the carrier frequency with Doppler, and \(\theta\) is the carrier phase. For the sake of simplicity, the noise is omitted in Equation (2), which does not affect the following discussions. Using the trigonometric transformation, Equation (2) can be rewritten as follows:  
\[ \begin{cases}r_{d}(t)=&r_{u}(t)+r_{l}(t)\\ r_{u}(t)=&\frac{A}{2}c(t-\tau)\cos\left(2\pi(f_{c}+f_{sc})t+( \theta+\phi)\right)\\ r_{l}(t)=&\frac{A}{2}c(t-\tau)\cos\left(2\pi(f_{c}-f_{sc})t+( \theta-\phi)\right)\end{cases}  \tag{3}\]  
where \(r_{u}(t)\) is the upper sideband (USB) of the BOC signal, and \(r_{l}(t)\) is the lower sideband (LSB) of the BOC signal. It can be seen from Equation (3) that both \(r_{u}(t)\) and \(r_{l}(t)\) are BPSK-modulated signals. Equation (3) shows a double-sideband reception model for BOC signals, where the received BOC signal is regarded as the sum of two BPSK signals. Based on this double-sideband signal model, the DBT method jointly processes the two BPSK signals in the USB and LSB to track the BOC signal. Since the ACFs of the two BPSK signals have only one peak, the DBT method solves the tracking ambiguity problem. The DBT loop employs two sets of correlators for the USB and LSB of the BOC signal, so it is compatible with classical GNSS receiver architectures designed for BPSK signals. Using the DBT loop to track BOC signals can yield three ranging measurements: the code delay, subcarrier phase, and carrier phase. The subcarrier ranging measurement is more accurate than the code ranging measurement, but it has integer ambiguity. In the  
Figure 1: Normalized auto-correlation functions of BPSK(n), BOC(2n,n), and BOC(6n,n) signals. absence of multipath, code ranging errors are usually less than half of the subcarrier wavelength.