# LSOS: An FG Position Method Based on Group Phase Ranging Ambiguity Estimation of BeiDou Pseudolite
## 3 Algorithmic Principle
### Phase Quantization Process  
Generally, frequency and phase belong to the inherent properties of periodic signals. The change of frequency will inevitably cause the change of phase. One of the most remarkable characteristics of periodic signals is the periodic change of phase difference introduced by frequency difference. Usually, the change of frequency and phase can be converted into each other. The high-precision measurement of frequency can be converted into the measurement of the phase comparison of two periodic signals. The phase change can reflect the regular change between periodic signals more precisely.  
Phase information is a physical quantity, which reflect the state of periodic signals at any moment. The process of phase change among several periodic signals directly reflects the process of quantization between periodic signals in time. The following takes any two frequency points as an example to introduce the phase change relationship between periodic signals, as shown in Figure 2.  
Suppose that the selected signal frequencies are f\({}_{1}\) and f\({}_{2}\), f\({}_{1}\) < f\({}_{2}\). The corresponding signal period is \(T_{1}\) and \(T_{2}\). If the phase coincidence point between the two signals is identified as phase point 0, it can be seen from Figure 2 that if the phase of the two signals is to be the same, the time-domain relationship between the two frequency signals should be satisfied  
\[t=\frac{M}{f_{2}-f_{1}}\quad\ f_{0}=(f_{1},f_{2})\quad\ T_{\rm min}=\frac{1}{( f_{1},f_{2})} \tag{3}\]  
Figure 2: Phase quantization relationship between periodic signals.  
M is a positive integer, when M = 1, \((f_{1},f_{2})\) is the maximum common factor frequency of \(f_{1}\) and \(f_{2}\), and \(T_{\rm min}\) is a least common multiple period between two signals, which reflects the variation of the phase difference of the periodic signal in the least common multiple period and the variation of the temporal and spatial signal. The quantization is generally a quantitative change relation with a unit interval of \(f_{2}-f_{1}\) as the basis.  
In this case, the phase difference between the two phases can be expressed as  
\[\Delta t=n\frac{f_{0}}{f_{1}f_{2}} \tag{4}\]  
n is a positive integer, \(n=0,1,2,3,\ldots\ldots\ldots\) /\(f_{0}\), \(\frac{f_{0}}{f_{1}f_{2}}\) is the quantized phase shift discrimination rate between the two signals, namely the unit quantum. The change of phase difference is always an integer multiple of the phase quantum, which can be from 0 to /\(f_{0}\).  
From the above analysis, there is always a minimum quantization interval in the whole quantization process. The interval is the quantization resolution between two signals, which shows a quantization relation between the mutual phases. This change in phase difference is a step change in the smallest unit, which is called the quantized phase-shift resolution. The least common multiple of a periodic signal is shown in time as the least common multiple period between any two periodic signals. There is no such thing as the same phase difference in any given least common multiple cycle. The variation of phase difference is an integral multiple of phase shift resolution, which reflects the whole-cycle phase estimation during periodic phase coincidence.