# Multipath Mitigation for BOC Signals Based on Prompt-Assisted-Offset Correlator
## 2 Problem Formulation
### Multipath Effect Analysis for BOC Signals  
In the DBT loop, the received BOC signal is first mixed with the in-phase carrier and quadrature-phase carrier generated by the carrier numerically controlled oscillator (NCO), respectively. This process is called the carrier wipe-off. Two zero intermediate-frequency (IF) signals are obtained after wiping off the carrier, which are named the in-phase signal and the quadrature-phase signal, respectively. In correlator channels, the in-phase signal is correlated with local codes generated by the code NCO to calculate in-phase correlation values, and the quadrature-phase signal is correlated with local codes to calculate quadrature-phase correlation values. The complex correlation functions of the USB and LSB signals can be written as:  
\[R_{u}= \frac{A}{2}\bigg{(}R(\Delta\tau_{d})+\sum_{m=1}^{M}\alpha_{m}R( \Delta\tau_{d}-\tau_{m})\exp\big{(}j(\theta_{m}+\phi_{m})\big{)}\bigg{)}\exp \big{(}j(\Delta\theta_{d}+\Delta\phi_{d})\big{)} \tag{8}\] \[R_{l}= \frac{A}{2}\bigg{(}R(\Delta\tau_{d})+\sum_{m=1}^{M}\alpha_{m}R( \Delta\tau_{d}-\tau_{m})\exp\big{(}j(\theta_{m}-\phi_{m})\big{)}\bigg{)}\exp \big{(}j(\Delta\theta_{d}-\Delta\phi_{d})\big{)} \tag{9}\]  
where \(\Delta\tau_{d}\), \(\Delta\theta_{d}\), and \(\Delta\phi_{d}\) are the code delay difference, carrier phase difference, and subcarrier phase difference between the local signal and the direct signal, respectively. \(R(\cdot)\) denotes the normalized ACF of the spreading code. When the code delay difference between the local signal and the received signal is greater than one chip, the correlation value is very close to zero. The real part of the complex correlation function is the in-phase correlation function, and the imaginary part of the complex correlation function is the quadrature-phase correlation function. In the absence of multipath, the correlation functions of USB and LSB signals have the same shape. The in-phase correlation function of each single sideband (SSB) signal is an isosceles triangle, as shown by the blue dashed lines in Figure 2. In the DBT loop, each correlator channel for the USB and LSB signals has three correlators, which are denoted as early (E), prompt (P), and late (L), respectively. The spacing between every two adjacent correlators is identical, and the spacing between E and L correlators is called the early-minus-late (EML) spacing. The EML code discriminator is designed based on the symmetry of the correlation function. However, the symmetry of the correlation function is broken by the multipath as shown in Figure 2, where the multipath relative amplitude is 0.5, the multipath relative carrier phase is \(\pi/6\), and the multipath relative delay is 0.4 chips. Thus, the multipath leads to bias in the EML discrimination output, and the code multipath error is related to the EML spacing. Using narrow EML spacing can effectively reduce the code multipath error, which is the principle of the NC technique. The P correlation values are used for subcarrier and carrier discrimination in the DBT loop. In the absence of multipath, the quadrature-phase P correlation values of the USB and LSB signals are 0. However, in the presence of multipath, the quadrature-phase P correlation values of the USB and LSB signals are not 0. As a result, the outputs of the carrier and subcarrier discriminators are affected by the multipath. As shown in Equations (8) and (9), the carrier phase and subcarrier phase are coupled to each other in the USB and LSB correlation values, which implies that the carrier and subcarrier multipath errors interact with each other. Thus, the DBT method subtly combines the USB and LSB correlation values to separate the carrier phase from the subcarrier phase.