# High-Precision Pseudo-Noise Ranging Based on BOC Signal: Zero-Bias Mitigation Methods
## 1 Introduction  
Binary offset subcarrier (BOC) modulation is a modern navigation signal modulation method that is widely used in global navigation satellite systems (GNSS) including Global Positioning System (GPS), Galileo, and BeiDou navigation satellite system (BDS) [1; 2; 3]. Compared with traditional binary phase shift keying (BPSK) modulation, BOC-modulated signal has better spectrum utilization, anti-interference capability, and theoretically higher pseudo-noise ranging performance [4]. In general, thermal noise and channel interference are the main factors limiting the tracking accuracy of the BOC signal. However, in high-precision applications where the channel environment is moderate, the pseudo-noise ranging deviation caused by digital signal sampling distortion can be significant [5; 6; 7]. There are two types of sampling distortion for a digital phase discriminator: Resolution error and zero-bias. The resolution error of the discriminator is caused by insufficient sampling of the digital signal [8]. Under the condition of commensurate sampling where the sampling frequency and the spreading symbol rate satisfy an integer-multiple relationship, the phase resolution error of the digital discriminator will increase significantly so that the phase resolution of the digital discriminator will be equal to the sampling interval of the digital signal [9; 10]. In addition, the zero-bias of the discriminator arises from the asymmetry of the digital correlation curve of the pseudo-random spread spectrum signal [11; 12]. Randomly flipping the chips of the pseudo-random spreading sequence destroys the symmetry of the digital correlation curve. The BOC signal modulates the sinusoidal or cosine subcarrier on the BPSK signal [1]. Tracking ambiguity may occur in the full match reception of the BOC signal [13; 14]. The dual estimation loop (DEL) reception method separately tracks the code phase and subcarrier phase and can eliminate this ambiguity [15; 16; 17]. In digital implementation, both the code phase discriminator and the subcarrierphase discriminator in the DEL are affected by sampling distortion. Many studies have focused on the impact of sampling distortion on the code phase discriminator [5; 6; 7; 8; 9; 10; 11; 12; 18; 19; 20; 21], while only a few of them study of the digital subcarrier phase discriminator. The effects of sampling and quantization of all-digital receivers are listed in this paragraph. [PERSON] first studied the relationship between sampling frequency and spreading code rate, and suggested that in the case of commensurate sampling, the digital correlation curve will produce significant distortion [6]. To eliminate this distortion, [PERSON] proposed a noncommensurate sampling method that can efficiently improve the resolution of the digital correlation curve [8]. In addition, [PERSON] and [PERSON] proposed a method based on phase jitter, which can further improve the phase resolution [20; 21]. When phase resolution is enhanced, the zero-bias effect is highlighted [11]. [PERSON] proposed two ways to eliminate zero-bias based phase errors, one is zero-bias compensation [11] and the other is zero-bias mitigation [11; 22]. [PERSON] developed an implementation of zero-bias mitigation by designing the optimal sampling frequency. However, in a typical receiver, the optimal sampling frequency determined by [PERSON]'s method is usually not easy to be implemented. In addition, [PERSON] analyzed the zero-bias characteristics of a digital code-based phase discriminator and finds a close correlation between zero-bias, sampling frequency and correlation interval. [PERSON] analyzed the comprehensive error of the discriminator and obtained some useful information but did not give a quantitative description of the zero-bias. In this paper, sampling distortion of BOC signal reception is studied. The influence of sampling distortion on the code phase discriminator and subcarrier phase discriminator in the double loop is analyzed. The zero-bias of the BOC signal is modeled using noncommensurate sampling.