# Low-Cost Dual-Frequency GNSS Receivers and Antennas for Surveying in Urban Areas
## 3 Results and Discussion
### Relative Positioning Performance
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Sessions (min)** & \(\mathcal{\lambda}\)**e (mm)** & \(\mathcal{\lambda}\)**n (mm)** & \(\mathcal{\lambda}\)**h (mm)** & \(\mathbf{d_{2D}}\) (mm)** & \(\mathbf{d_{3D}}\) (mm)** \\ \hline
0–60 & 0.7 & \(-2.8\) & \(-1.3\) & 2.9 & 3.2 \\
0–30 & 0.7 & \(-2.1\) & \(-1.2\) & 2.2 & 2.5 \\
30–60 & 0.7 & \(-2.9\) & \(-0.9\) & 3.0 & 3.1 \\
0–20 & 0.1 & \(-1.1\) & \(-0.4\) & 1.1 & 1.2 \\
20–40 & 1.0 & \(-3.0\) & \(-1.9\) & 3.2 & 3.7 \\
40–60 & 0.6 & \(-2.7\) & \(-0.6\) & 2.8 & 2.8 \\
0–15 & 0.2 & \(-0.5\) & \(-0.9\) & 0.5 & 1.0 \\
15–30 & 0.6 & \(-2.2\) & \(-0.7\) & 2.3 & 2.4 \\
30–45 & 1.8 & \(-2.9\) & \(-2.9\) & 3.4 & 4.5 \\
45–60 & 0.6 & \(-2.6\) & \(-0.7\) & 2.7 & 2.8 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Positioning performance results at ST 1. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & \(\mathcal{\lambda}\)**e (mm)** & \(\mathcal{\lambda}\)**n (mm)** & \(\mathcal{\lambda}\)**h (mm)** & \(\mathbf{d_{2D}}\) (mm)** & \(\mathbf{d_{3D}}\) (mm)** \\ \hline
0–60 & \(-0.5\) & 0.0 & 8.5 & 0.5 & 8.5 \\
0–30 & \(-4.3\) & \(-0.9\) & 3.0 & 4.4 & 5.3 \\
3 The results of the horizontal (d\({}_{\text{2D}}\)), vertical (\(\lambda\)h), and spatial absolute (d\({}_{\text{3D}}\)) position differences shown in the last four tables are also presented graphically in Figure 10. In open-sky conditions, the differences are less than 5.0 mm for horizontal, vertical, and spatial positions in all sessions. Under urban conditions, buildings and trees near the stations degrade the signal quality, and low-cost instruments prove to have poorer positioning results. The differences for the horizontal and vertical positions are smaller than 10 and 15 mm in 85% and 82.5% of the sessions, respectively. For the spatial positions, the differences are smaller than 15 mm in 77.5% of the considered sessions in urban areas. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Sessions (min)** & \(\mathcal{J}\)**e (mm)** & \(\mathcal{J}\)**n (mm)** & \(\mathcal{J}\)**h (mm)** & \(\mathbf{d}_{\text{2D}}\) (mm)** & \(\mathbf{d}_{\text{3D}}\) (mm) \\ \hline
0–60 & \(-4.8\) & \(0.4\) & \(10.3\) & \(4.8\) & \(11.4\) \\
0–30 & \(-6.6\) & \(-0.9\) & \(13.3\) & \(6.7\) & \(14.9\) \\
30–60 & \(-4.8\) & \(3.5\) & \(5.2\) & \(5.9\) & \(7.9\) \\
0–20 & \(-6.0\) & \(0.6\) & \(6.8\) & \(6.0\) & \(9.1\) \\
20–40 & \(-3.8\) & \(5.2\) & \(-11.6\) & \(6.4\) & \(13.3\) \\
40–60 & \(-6.4\) & \(-1.3\) & \(13.6\) & \(6.5\) & \(15.1\) \\
0–15 & \(-6.9\) & \(-4.9\) & \(-9.8\) & \(8.5\) & \(12.9\) \\
15–30 & \(-5.3\) & \(3.9\) & \(4.6\) & \(6.6\) & \(8.0\) \\
30–45 & \(-6.3\) & \(0.7\) & \(6.4\) & \(6.3\) & \(9.0\) \\
45–60 & \(-4.6\) & \(1.2\) & \(13.6\) & \(4.8\) & \(14.4\) \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Positioning performance results at ST 3. Figure 10: _Cont._  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Sessions (min)** & \(\mathcal{J}\)**e (mm)** & \(\mathcal{J}\)**n (mm)** & \(\mathcal{J}\)**h (mm)** & \(\mathbf{d}_{\text{2D}}\) (mm)** & \(\mathbf{d}_{\text{3D}}\) (mm) \\ \hline
0–60 & \(-8.8\) & \(8.4\) & \(-1.5\) & \(12.2\) & \(12.3\) \\
0–30 & \(-5.2\) & \(2.4\) & \(4.3\) & \(5.7\) & \(7.2\) \\
30–60 & \(-3.3\) & \(4.4\) & \(6.9\) & \(5.5\) & \(8.8\) \\
0–20 & \(-8.3\) & \(0.3\) & \(-6.5\) & \(8.3\) & \(10.5\) \\
20–40 & \(-7.4\) & \(5.0\) & \(-8