# Research on Improving Satellite Positioning Precision Based on Multi-Frequency Navigation Signals
## 2 Methodology and Equations
### Physical Augmentation Factor of Precision (PAFP)  
Combining the resource advantages of the communication satellites of the CAPS system, [PERSON] et al. proposed a method of using multiple frequencies to broadcast navigation signals separately, which can equivalently improve the spatial layout of the constellation [22]. The CAPS system selects communication band (including C-band and Ku-band) to transmit satellite navigation signals. There are more than 20 communication transponders on numerous communication satellite to transmit CAPS navigation signals. The working frequency bands and available frequency points are shown in Table 1.  
Assuming that the number of visible satellites is n, each satellite sends down one navigation signal, and the coefficient matrix obtained by linearizing the pseudo-range observation equation is:  
\[\mathrm{H}= \begin{bmatrix}\mathrm{h}\mathrm{x}_{1}&\mathrm{h}\mathrm{y}_{1}& \mathrm{h}\mathrm{z}_{1}&1\\ \mathrm{h}\mathrm{x}_{2}&\mathrm{h}\mathrm{y}_{2}&\mathrm{h}\mathrm{z}_{2}&1 \\ \mathrm{h}\mathrm{x}_{3}&\mathrm{h}\mathrm{y}_{3}&\mathrm{h}\mathrm{z}_{3}&1 \\ \vdots&\vdots&\vdots&\vdots&\vdots\\ \mathrm{h}\mathrm{x}_{i}&\mathrm{h}\mathrm{y}_{i}&\mathrm{h}\mathrm{z}_{i}&1 \\ \end{bmatrix}  \tag{3}\]  
where \([\mathrm{h}\mathrm{x}_{i},\mathrm{h}\mathrm{y}_{i},\mathrm{h}\mathrm{z}_{i}]\) (i = 1, 2, 3,..., n) is the cosine vector from the receiver position to each satellite, and n is the number of satellites involved in the calculation. Then, the weight coefficient matrix is: \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**The Name of** & **Up-Band (MHz)** & **Down-Band (MHz)** & **Available** & **The Number of** & **Available** \\  & **(MHz)** & **Bandwidth** & **Frequency** \\ \hline Standard C-band & 5925–6425 & 3700–4200 & 500 & 78 \\ Extended C-band & 6425–6725 & 3400–3700 & 500 & \\ Standard & 14,000–14,500 & 12,250–12,750 & 500 & \\ Extended Ku-band & 13,750–14,000 & 10,950–11,200 & 250 & 139 \\ Planning & 12,750–13,250 & 10,700–10,950, & \\ Ku-Band & 11,200–11,450 & 500 & \\ Broadcast Ku & 17,300–17,800 & 11,700–12,200 & 500 & \\ band & & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Frequency band and the number of available frequency points. \[\mathrm{D}=\left(\mathrm{H}^{\mathrm{T}}\mathrm{H}\right)^{-1}=\left[ \begin{array}{ c c c c c}\sum\limits_{i=1}^{n}\mathrm{h}\mathrm{x}_{i}{}^{2}&\sum\limits_{i=1}^{n}\mathrm{h} \mathrm{x}_{i}\mathrm{h}\mathrm{y}_{i}&\sum\limits_{i=1}^{n}\mathrm{h}\mathrm{x }_{i}\mathrm{h}\mathrm{z}_{i}&\sum\limits_{i=1}^{n}\mathrm{h}\mathrm{x}_{i}\\ \sum\limits_{i=1}^{n}\mathrm{h}\mathrm{x}_{i}\mathrm{h}\mathrm{y}_{i}&\sum \limits_{i=1}^{n}\mathrm{h}\mathrm{y}_{i}&\sum\limits_{i=1}^{n}\mathrm{h} \mathrm{y}_{i}\mathrm{h}\mathrm{z}_{i}&\sum\limits_{i=1}^{n}\mathrm{h}\mathrm{y }_{i}\\ \sum\limits_{i=1}^{n}\mathrm{h}\mathrm{x}_{i}\mathrm{h}\mathrm{z}_{i}&\sum \limits_{i=1}^{n}\mathrm{h}\mathrm{y}_{i}\mathrm{h}\mathrm{z}_{i}&\sum \limits_{i=1}^{n}\mathrm{h}\mathrm{z}_{i}&\sum\limits_{i=1}^{n}\mathrm{h} \mathrm{z}_{i}\\ \sum\limits_{i=1}^{n}\mathrm{h}\mathrm{x}_{i}&\sum\limits_{i=1}^{n}\mathrm{h} \mathrm{y}_{i}&\sum\limits_{i=1}^{n}\mathrm{h}\mathrm{z}_{i}&\sum\limits_{i= 1}^{n}\mathrm{h}\mathrm{z}_{i}\\ \sum\limits_{i=1}^{n}\mathrm{h}\mathrm{x}_{i}&\sum\limits_{i=1}^{n}\mathrm{h} \mathrm{y}_{i}&\sum\limits_{i=1}^{n}\mathrm{h}\mathrm{z}_{i}&\mathrm{n}\\ \end{array} \right]^{-1}=\left[ \begin{array}{cccc}\mathrm{P}_{11}&\mathrm{P}_{ 12}&\mathrm{P}_{13}&\mathrm{P}_{14}\\ \mathrm{P}_{21}&\mathrm{P}_{22}&\mathrm{P}_{23}&\mathrm{P}_{24}\\ \mathrm{P}_{31}&\mathrm{P}_{32}&\mathrm{P}_{33}&\mathrm{P}_{34}\\ \mathrm{P}_{41}&\mathrm{P}_{42}&\mathrm{P}_{43}&\mathrm{P}_{44}\\ \end{array} \right] \tag{4}\]  
Then, the PDOP value can be expressed as:  
\[\mathrm{PDOP}=\sqrt{\mathrm{P}_{11}+\mathrm{P}_{22}+\mathrm{P}_{33}} \tag{5}\]  
where \(\mathrm{P}_{ii}\) (i = 1, 2, 3, 4) is the diagonal element of the weight coefficient matrix of the pseudo-range observation equation.