# Sampling Rate Impact on Precise Point Positioning with a Low-Cost GNSS Receiver
## 1 Introduction
### The Precise Point Positioning Method  
The PPP method is commonly used for processing measurements or GNSS observations of a single GNSS receiver due to its capability to compute position with high accuracy. According to [17], the model for dual-frequency GNSS receivers with pseudo-range and carrier-phase observables on \(\mathrm{L}_{1}\) and \(\mathrm{L}_{2}\) between a receiver \(\mathrm{j}\) and satellite \(\mathrm{k}\) is:  
\[\mathrm{P}_{\mathrm{L}_{i}}^{\mathrm{k}} =\mathrm{\rho}_{\mathrm{j}}^{\mathrm{k}}+\mathrm{c}\left(\mathrm{ d}\mathrm{T}_{\mathrm{j}}-\mathrm{d}t^{\mathrm{k}}\right)+\mathrm{d}_{ \mathrm{orb}}^{\mathrm{k}}+\mathrm{d}_{\mathrm{trop}}^{\mathrm{k}}+\mathrm{d}_ {\mathrm{ion},\mathrm{L}_{i}}^{\mathrm{k}}+\mathrm{d}_{\mathrm{hd},P_{\mathrm{ L}_{i}}}^{\mathrm{j}}-\mathrm{d}_{\mathrm{hd},P_{\mathrm{L}_{i}}}^{\mathrm{k}} \tag{1}\] \[+\mathrm{d}_{\mathrm{mult},P_{\mathrm{L}_{i}}}+\varepsilon_{P_{ \mathrm{L}_{i}}}\] \[\mathrm{\Phi}_{\mathrm{L}_{i}}^{\mathrm{k}} =\mathrm{\rho}_{\mathrm{j}}^{\mathrm{k}}+\mathrm{c}\left(\mathrm{ d}\mathrm{T}_{\mathrm{j}}-\mathrm{d}t^{\mathrm{k}}\right)+\mathrm{d}_{ \mathrm{orb}}^{\mathrm{k}}+\mathrm{d}_{\mathrm{trop}}^{\mathrm{k}}-\mathrm{d}_ {\mathrm{ion},\mathrm{L}_{i}}^{\mathrm{k}}+\lambda_{\mathrm{L}_{i}}\mathrm{N} _{\mathrm{L}_{i}}+\mathrm{d}_{\mathrm{hd},\mathrm{\Phi}_{\mathrm{L}_{i}}}^{ \mathrm{j}}\] (2) \[-\mathrm{d}_{\mathrm{hd},\mathrm{\Phi}_{\mathrm{L}_{i}}}^{\mathrm{ k}}+\mathrm{d}_{\mathrm{mult},\mathrm{\Phi}_{\mathrm{L}_{i}}}^{\mathrm{k}}+\varepsilon_{\mathrm{ \Phi}_{\mathrm{L}_{i}}}\]  
where: \(\mathrm{\Phi}_{\mathrm{L}_{i}}^{\mathrm{k}}\) and \(\mathrm{P}_{\mathrm{L}_{i}}^{\mathrm{k}}\) are the carrier-phase and pseudo-range measurements on the frequency \(\mathrm{L}_{i}\), respectively; \(\mathrm{\rho}_{\mathrm{j}}^{\mathrm{k}}\) is the geometric distance between receiver \(\mathrm{j}\) and satellite \(\mathrm{k}\), respectively; \(\mathrm{c}\) is the theoretical speed of light; \(\mathrm{d}\mathrm{T}_{\mathrm{j}}\) and \(\mathrm{d}t^{\mathrm{k}}\) are the clock errors of the receiver and satellite, respectively; \(\mathrm{d}_{\mathrm{orb}}^{\mathrm{k}}\) is the satellite orbit error in \(\mathrm{m}\); \(\mathrm{d}_{\mathrm{trop}}^{\mathrm{k}}\) is the troposphere delay in \(\mathrm{m}\); \(\mathrm{d}_{\mathrm{ion},\mathrm{L}_{i}}^{\mathrm{k}}\) is the first order ionosphere effect on the frequency \(\mathrm{L}_{i}\) in \(\mathrm{m}\); \(\mathrm{d}_{\mathrm{hd},P_{\mathrm{L}_{i}}}^{\mathrm{j}}\) and \(\mathrm{d}_{\mathrm{hd},P_{\mathrm{L}_{i}}}^{\mathrm{j}}\) are the satellite and receiver hardware delays, respectively, for the \(\mathrm{L}_{i}\) pseudo range in \(\mathrm{m}\);\(\mathrm{d}_{\mathrm{hd},\phi_{\mathrm{L_{i}}}}^{\mathrm{j}}\) and \(\mathrm{d}_{\mathrm{hd},\phi_{\mathrm{L_{i}}}}^{\mathrm{k}}\) are the satellite and receiver hardware delays, respectively, for the \(\mathrm{L_{i}}\) carrier phase in m; \(\mathrm{d}_{\mathrm{mult},\mathrm{P_{L_{i}}}}\) and \(\mathrm{d}_{\mathrm{mult},\phi_{\mathrm{L_{i}}}}\) are the multipath errors of the pseudo range and carrier phase on the frequency \(\mathrm{L_{i}}\) in m, respectively; \(\varepsilon_{\mathrm{P_{L_{i}}}}\) and \(\varepsilon_{\mathrm{\phi_{L_{i}}}}\) are the noise errors on the pseudo range and carrier phase on the frequency \(\mathrm{L_{i}}\) in m, respectively; \(\lambda_{\mathrm{L_{i}}}\) is the wavelength of the frequency \(\mathrm{L_{i}}\) in m; and \(\mathrm{N_{L_{i}}}\) is the ambiguity on the frequency \(\mathrm{L_{i}}\) in cycles. On the other hand, to reduce errors in the positioning, the use of precise orbit and clock corrections is fundamental [17; 31; 32].