# Research on Improving Satellite Positioning Precision Based on Multi-Frequency Navigation Signals
## 2 Methodology and Equations
### Positioning Solution of Multi-Frequency Receiver  
The baseband system of the C/Ku dual-band navigation receiver in the user segment in the CAPS has 48 signal channels, which can simultaneously receive 48 downlink navigation signals, as shown in Figure 5. It uses virtual clock technology to correct satellite orbit errors, clock errors, and errors caused by atmospheric delay in the measurement pseudo-range and solves the receiver position by establishing pseudo-range equations to solve satellite positioning. Figure 5: CAPS navigation receiver: (**a**) is the physical diagram of the CAPS receiver, including multi-beam antenna, frequency converter, radio frequency device, signal channel, intelligent control terminal (including memory, CPU, display), power supply, etc. ; (**b**) is a variable frequency navigation radio frequency device with 48 channels. The process of satellite positioning using multi-frequency pseudo-range observations is as follows. The pseudo-range observation equation of the navigation satellite can be expressed as:  
\[\rho_{\mathrm{ij}}=\mathrm{c}\cdot(t_{\mathrm{c}}-t_{\mathrm{v}})=\sqrt{(X_{ \mathrm{i}}-X)^{2}+(Y_{\mathrm{i}}-Y)^{2}+(Z_{\mathrm{i}}-Z)^{2}}+\epsilon_{ \mathrm{ij}} \tag{15}\]  
where \(\rho_{\mathrm{ij}}\) is the distance from the receiver to the i-th satellite that transmits the j-th signal, that is, the pseudo-range from the satellite to the receiver; \(\mathrm{c}\) is the speed of light; \(t_{\mathrm{c}}\) is the total delay of the signal sent from the integrated baseband transmitting terminal of the ground station, through the radio frequency transmission channel, space uplink, satellite forwarding, and space downlink to the integrated baseband accept terminal of the subscriber receiver; \(t_{\mathrm{v}}\) is the time correction of the virtual clock; \((X_{\mathrm{i}},\,Y_{\mathrm{i}},\,Z_{\mathrm{i}})\) is the position coordinate of the i-th satellite; \((X,\,Y,\,Z)\) is the receiver position coordinates; and \(\epsilon_{\mathrm{ij}}\) is gaussian-distributed random errors due to multipath effects and receiver thermal noise, etc. The pseudo-range observation equation is expanded by the first-order Taylor series at the initial point \(\mathrm{O}\left(X_{0},Y_{0},Z_{0}\right)\). After omitting the higher-order terms, we can get:  
\[\rho_{\mathrm{ij}}-\rho_{\mathrm{i}}(0)=\mathrm{H}\cdot\mathrm{d}X+\epsilon_{ \mathrm{ij}} \tag{16}\]  
Its coefficient matrix is:  
\[\mathrm{H}= \begin{bmatrix}\mathrm{h}\mathrm{x}_{11}&\mathrm{h}\mathrm{y}_{11}& \mathrm{h}\mathrm{z}_{11}&1\\ \mathrm{h}\mathrm{x}_{21}&\mathrm{h}\mathrm{y}_{21}&\mathrm{h}\mathrm{z}_{21 }&1\\ \vdots&\vdots&\vdots&\vdots\\ \mathrm{h}\mathrm{x}_{i1}&\mathrm{h}\mathrm{y}_{j1}&\mathrm{h}\mathrm{z}_{i1} &1\\ \mathrm{h}\mathrm{x}_{12}&\mathrm{h}\mathrm{y}_{12}&\mathrm{h}\mathrm{z}_{12 }&1\\ \mathrm{h}\mathrm{x}_{22}&\mathrm{h}\mathrm{y}_{22}&\mathrm{h}\mathrm{z}_{22}&1 \\ \vdots&\vdots&\vdots&\vdots\\ \mathrm{h}\mathrm{x}_{22}&\mathrm{h}\mathrm{y}_{j2}&\mathrm{h}\mathrm{z}_{i2} &1\\ \vdots&\vdots&\vdots&\vdots\\ \mathrm{h}\mathrm{x}_{ij}&\mathrm{h}\mathrm{y}_{ij}&\mathrm{h}\mathrm{z}_{ij} &1\end{bmatrix}  \tag{17}\]  
where \(\left[\mathrm{h}\mathrm{x}_{\mathrm{ij}},\,\mathrm{h}\mathrm{y}_{\mathrm{ij}},\,\mathrm{h}\mathrm{z}_{\mathrm{ij}}\right]\) (i = 1, 2, \(\ldots\), \(\mathrm{n}\); j = 1, 2, \(\ldots\), \(\mathrm{k}\)) is the cosine vector of the direction from the receiver to the i-th satellite that transmits the j-th signal. The difference between the observed pseudo-range and the predicted pseudo-range value at the initial point can be expressed as:  
\[\mathrm{B}= \begin{bmatrix}\rho_{11}-\rho_{1}(0)\\ \rho_{21}-\rho_{2}(0)\\ \vdots\\ \rho_{11}-\rho_{\mathrm{i}}(0)\\ \vdots\\ \rho_{12}-\rho_{1}(0)\\ \rho_{22}-\rho_{2}(0)\\ \vdots\\ \rho_{\mathrm{i2}}-\rho_{\mathrm{i}}(0)\\ \vdots\\ \rho_{\mathrm{ij}}-\rho_{\mathrm{i}}(0)\end{bmatrix}