# LSOS: An FG Position Method Based on Group Phase Ranging Ambiguity Estimation of BeiDou Pseudolite
## 3 Algorithmic Principle
### Group Phase Estimation  
According to frequency signal phase quantization processing in the previous section, the unit quantization theory can be better applied to phase measurement. The phase difference between periodic signals varies periodically in least common multiple, and any periodic phenomenon can be represented by a group. After further study of this phenomenon, it is found that the variation features have some symmetry, which is called group phase theory in the following. The group phase character belongs to the finite periodic cycle class. According to the different frequency relations, the characteristics of multi-frequency point group phase estimation are analyzed.  
The set of phase differences in the least common multiple periods between any two periodic signals is called a cell group, and each cell group consists of several subsets. If we think of each least common multiple element in the actual least common multiple cycle as a subset, each subset can be used as \(GU_{1}\), \(GU_{2}\), \(\ldots\)\(GU_{n}\).  
\[G_{n}=\{t|\Delta t^{n}=1\} \tag{5}\]  
\[G_{n}=\{GU_{1}\quad GU_{2}\quad\ldots\quad GU_{n}\} \tag{6}\]  
Here t is the generator and n is the order. The phase difference group is a finite cyclic group, isomorphic to the addition group. This group can be discussed in terms of the theory of finite additive groups. When the nominal value between two periodic signals is the same, the generator in a minimum common multiple periods is not unique, but the minimum generator between signals is unique.  
Obviously, the minimum generator is the highest phase shift resolution between two signals. In the phase comparison and frequency measurement, the most concerned is the quantized phase shift resolution. A cyclic group of order n has \(\varphi\) (n) generators, and \(\varphi\) (n) is Euler function. When n is an integer greater than 1, and n = pk\({}_{n}\) is the standard factorization of n. n has T(n) = (k\({}_{1}\) + 1) (k\({}_{2}\) + 1) \(\ldots\) (k\({}_{m}\) + 1) positive factors, where T(n) of n is the number of positive factors of n. Then cyclic groups of order n have only T(n) cyclic group subgroups. If the least common multiple period between two periodic signals is T\({}_{\rm minc}\), if f\({}_{1}\) < f\({}_{2}\), then the order of the phase group is T\({}_{\rm minc}\)\(\times\) f\({}_{2}\). Obviously, such a cyclic phase group has a strong regularity, and it is this regularity that makes all kinds of comparisons work smoothly.  
In any \(GU_{i}\), the phase changes between signals are complex and changeable. In the same cyclic group, the changes are in line with the characteristics of cyclic group. The construction of cyclic groups is completely determined by the generator (that is, the resolution of the quantized phase shift between the two signals).  
\[GP_{n}=\left\{G_{1}\quad G_{2}\quad\ldots\quad G_{Num}\right\} \tag{7}\]  
Assuming that there are N frequency points of the signal, the order of the frequency is \(f_{1}\quad f_{2}\quad\ldots\quad f_{N}\). Therefore, the corresponding number of least common multiple periods can be obtained according to different frequency combinations, and the total number of phase quantization period combinations of \(Num=C_{N}^{2}+C_{N}^{3}+C_{N}^{4}+\ldots+C_{N}^{N}\) species can be obtained. The group phase quantization period can be expressed as \(T=\left[T_{1}\quad T_{2}\quad\ldots\quad T_{Num}\right]\). Thus, the element group can be extended to group-phase subgroup cells with different number of periods, and the characteristic period can be estimated from the minimum resolution element, the element group to the group-phase multi-level least common multiple.  
\[GP_{n}=\left\{G_{1}\quad G_{2}\quad\ldots\quad G_{Num}\right\} \tag{8}\]