# A Design Method of Code Correlation Reference Waveform in GNSS Based on Least-Squares Fitting
## 1 Introduction  
The binary offset carrier (BOC) signal, which is adopted by modernized GPS, GLONASS, Galileo and BeiDou satellite navigation systems [1; 2; 3], is designed for spectrum separation and ranging accuracy improvement. It can be considered as a pseudo-random noise (PRN) code multiplied by a'square sine' subcarrier, which is a sinusoidal carrier with one-bit quantization, as shown in Figure 1. For the BOC(m,n) signal, the code chipping rate is \(f_{c}=mf_{0}\), and the frequency of the sine-phased square-wave subcarrier is \(f_{sc}=nf_{0}\), \(f_{0}=1.023\) MHz. In particular, BOC(1,1) is commonly considered as the interoperability baseline for the modernized BeiDou BIC signal, the Galileo E1 signal and the GPS modernized L1C signal [4; 5]. High-order BOC signals, like BOC(10,5), BOC(15,2.5) and BOC(14,2), are designed for providing defense and security services, such as the Galileo Public Regulated Service (PRS), BeiDou military signals and GPS military signals [4; 6]. In modern Global Navigation Satellite Systems (GNSSs), the BOC signal is widely used for civil and military applications. These novel signals can improve the tracking accuracy and have better multipath mitigation performance with narrow correlators, when compared to the binary phase shift keying (BPSK) signal. However, two problems emerge in the application of BOC signals, which are the ambiguity in receiver tracking and the lack of multipath mitigation methods [7]. Better ranging accuracy can be achieved, since the BOC signal occupies a larger Gabor bandwidth than the binary phase shift keying (BPSK) signal with the same code frequency. However, the multi-peaked auto-correlation functions (ACF) of BOC signals lead to the possibility of receiver code tracking loop locking onto the side peaks [8], when applying traditional multipath mitigation techniques to the BOC signals. Many multipath mitigation techniques have been proposed and implemented for the BPSK signal in receiver processing. The narrow correlator technique is primarily proposed in [9] to narrow down the space between correlators in early and late branches. As the code noise variance and multipath error envelope are proportional to the chip spacing, it can enhance the performance of multipath mitigation and tracking precision. The double-delta correlator (\(\Delta\Delta\)) technique is a representative architecture for multipath mitigation techniques using a linear combination of four correlators [10]. By adding another set of early-minus-late correlators beyond the original narrow correlators, it further improves the multipath mitigation performance. The CCRW technique generates a unique local reference waveform instead of local code replicas [11; 12], which is distinct from the narrow correlator and double-delta correlator technique. Its resulting cross-correlation function (CCF) of the incoming signal and the local reference waveform is directly a discriminator. By carefully choosing the local code reference waveform, the code multipath mitigation performance can be better, at the expense of the carrier-to-noise ratio (CNR). The W2 waveform in [12] is chosen as the representative CCRW discriminator. The multipath mitigation techniques mentioned above are preferred in the receiver realizations, since they are implemented in the code delay lock loop (DLL) with acceptable hardware cost. However, they are less effective when dealing with BOC signals. Because the resulting DLL discriminator function may have more than one zero-crossing, i.e., they may be tracking on the wrong lock-points and have an intolerable bias in measurements. Furthermore, the filtering effect of the transmission channel is not considered in the CCRW design. Only the zero-crossing in the center of discriminator corresponds to the zero-code-phase error; the other ones would lead to biased pseudo-range measurements. Figure 2 shows the tracking error result for the CCRW W2 technique of BPSK(1) and BOC(1,1) signals.