# Precise Point Positioning Using Dual-Frequency GNSS Observations on Smartphone
## 3 Experiments and Results
### Static Data Collection
Four data sets were processed according to the strategies in Table 1, which included the data sets of smartphone GPS L1/L5, Galileo E1/E5a dual-frequency observations (hereafter \"smartphone dual-frequency\"), GPS L1, GLONASS L1, Galileo E1 single-frequency observations (hereafter \"smartphone single-frequency\"), and the data sets of geodetic receiver GPS L1/L5, Galileo E1/E5a dual-frequency observations (hereafter \"receiver dual-frequency\"), GPS L1, GLONASS L1, Galileo E1 single-frequency observations (hereafter \"receiver single-frequency\"). The reference coordinates were calculated by Bernese software using 12-h static data of the geodetic receiver and the data from IGS stations (e.g., BADG, BJFS, DAEJ, YSSK). After processing, we can get the positioning result coordinates \(r_{ECEF}\) and the reference coordinates \(r_{r}\) in ECEF (Earth-Center Earth-Fixed). The position errors are often expressed in ENU coordinates, which could be derived from the following formula:  
\[r_{ENU}=E_{r}(r_{ECEF}-r_{r}) \tag{37}\] \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Satellite**} & **STD of MP\({}_{1}\) under** & **STD of MP\({}_{5}\) under** & **STD of MP\({}_{1}\)** & **STD of MP\({}_{5}\) on** \\  & **the Antenna** & **the Antenna** & **on the Edge** & **the Edge** \\ \hline G24 & 0.68 & 0.70 & 0.78 & 0.71 \\ G30 & 0.64 & 0.74 & 0.75 & 0.83 \\ E02 & 0.71 & 0.72 & N/A & N/A \\ E25 & 0.78 & 0.74 & N/A & N/A \\ E04 & N/A & N/A & 0.67 & 0.76 \\ E09 & N/A & N/A & 0.82 & 0.67 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: STD of _MP\({}_{1}\)_ and _MP\({}_{5}\)_ in two cases. Figure 4: The number of satellites with simultaneous L1 and L5 frequency data record and PDOP. where \(E_{r}\) is the rotation matrix of the ECEF coordinates to the ENU coordinates and can be expressed as:  
\[E_{r}=\left[ \begin{array}{ccc}-\sin\lambda&\cos\lambda&0\\ -\sin\phi\cos\lambda&-\sin\phi\sin\lambda&\cos\phi\\ \cos\phi\cos\lambda&\cos\phi\sin\lambda&\sin\phi\end{array} \right] \tag{38}\]  
where \(\phi\) and \(\lambda\) are the geodetic latitude and longitude of reference coordinates, respectively. Figure 5 showed the positioning errors of PPP solutions of smartphone and receiver. In the E direction, the positioning errors of the four data sets converged to 0.5 m at 9,000 epochs approximately, and converged to 0.2 m at 20,000 epochs (Figure 5a). In the N direction, \"receiver single-frequency\" showed the longest convergence time with the worst positioning accuracy (Figure 5b). Horizontal components of PPP mode from the different data sets achieved an accuracy of decimeter-level. In the U direction, obvious inconsistence existed in the four data sets (Figure 5c). The position errors of dual-frequency data sets (\"receiver dual-frequency\" and \"smartphone dual-frequency\") showed similar trends and obtained the best results, the accuracy of \"receiver single-frequency\" was close to dual-frequency, and the \"smartphone single-frequency\" achieved the worst results. Figure 5: Position errors of smartphone and receiver ((a): east position errors; (b): north position errors; (c): up position errors). The convergence time is defined as the time when a 3-dimensional positioning accuracy of 1 dm is reached and maintained for 20 epochs at least [30]. For smartphones, the accuracy is difficult to obtain. In this paper, we chose 1 m as the threshold of convergence because the accuracy of 1 m can meet the needs of most non-professional fields and some professional fields with low-precision requirements. Table 3 shows the convergence time and positioning accuracy of four data sets. The solutions of \"smartphone single-frequency\" were hard to converge to 1 m. The convergence time of \"smartphone dual-frequency\" was 102 min to 1 m, 107 min to 0.5 m, and 116 min to 0.2 m, respectively. Meanwhile, for the results of \"receiver single-frequency\", the convergence time was 35 min to 1 m, and 158 min to 0.5 m. The results of \"receiver dual-frequency\" could reach 0.1 m accuracy in 301 min.