# Precise Point Positioning Using Dual-Frequency GNSS Observations on Smartphone
## 3 Experiments and Results
### Static Data Collection
Both kinds of receiver indicated a relatively higher positioning accuracy in dual-frequency. And the RMS of position errors are calculated from [20]:  
\[\text{RMS}_{E}=\sqrt{\frac{1}{n}\sum_{i=1}^{n}\Delta E_{i}^{2}} \tag{39}\]  
\[\text{RMS}_{N}=\sqrt{\frac{1}{n}\sum_{i=1}^{n}\Delta N_{i}^{2}} \tag{40}\]  
\[\text{RMS}_{U}=\sqrt{\frac{1}{n}\sum_{i=1}^{n}\Delta U_{i}^{2}} \tag{41}\]  
\[\text{RMS}_{3D}=\sqrt{\frac{1}{n}\sum_{i=1}^{n}\bigl{(}\Delta E_{i}^{2}+\Delta N _{i}^{2}+\Delta U_{i}^{2}\bigr{)}^{2}} \tag{42}\]  
where \(RMS_{E}\), \(RMS_{N}\), \(RMS_{U}\) are RMS values of position errors in east, north and up directions, respectively, \(RMS_{3D}\) is RMS values of 3D position errors, \(n\) is the number of positioning results, \(\Delta E_{i}\), \(\Delta N_{i}\) and \(\Delta U_{i}\) are the position errors in the east, north and vertical components of the \(i\)th epoch. The positioning accuracy of \"smartphone single-frequency\" could not be converged to 1 m. The RMS (Root Mean Square) positioning errors of \"smartphone double-frequency\" after convergence in the E, N, U directions were 21.8 cm, 4.1 cm, 11.0 cm. The positioning accuracy of \"smartphone double-frequency\" was at the same level of \"receiver single-frequency\", and had an evident gap with \"receiver dual-frequency\" (Table 4). \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Data Sets** & **East (cm)** & **North (cm)** & **Up (cm)** \\ \hline smartphone single-frequency & N/A & N/A & N/A \\ smartphone dual-frequency & 21.8 & 4.1 & 11.0 \\ receiver single-frequency & 14.6 & 25.8 & 27.1 \\ receiver dual-frequency & 0.2 & 0.1 & 0.5 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: RMS positioning errors of four data sets. \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Data Sets**} & \multicolumn{3}{c}{**Positioning Accuracy (m)**} \\ \cline{2-4}  & **1** & **0.5** & **0.2** & **0.1** \\ \hline smartphone single-frequency(min) & N/A & N/A & N/A & N/A \\ smartphone dual-frequency(min) & 102 & 107 & 116 & N/A \\ receiver single-frequency(min) & 35 & 158 & N/A & N/A \\ receiver dual-frequency(min) & 66 & 107 & 272 & 301 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: The convergence time to different accuracies. The positioning accuracy of dual-frequency observations was better than that of single-frequency observations due to the distinct processing methods for ionospheric delay. The PPP of single-frequency observations implemented the ionospheric broadcast model, and the ionospheric errors were large. However, the process of dual-frequency observations applied the ionospheric-free combination, eliminating the first-order ionospheric delay. The positioning accuracy of receiver observations was higher than that of the smartphone observations because of the different antennas. The smartphone is equipped with a linearly polarized antenna, which has poor multipath suppression and irregular gain pattern. The receiver carries a geodetic-grade right-hand circularly polarized antenna, and the data sets can be acquired with very high quality. The dual-frequency data sets had no results under several epochs due to the exclusion of some satellites with relatively larger carrier phase residuals (Figure 6 and Table 5). The carrier phase residuals of smartphone were higher than those of the receiver because larger measurement noises existed in the smartphone observations.