## 4 Testing
### Calibration Results  
An open source MATLAB-based single antenna software receiver in [18] was further developed for multi-antenna receiver and the tracking, acquisition and position solution parts of the original software were modified. As shown in Figure 2 showing the receiver structure, only the satellite signals received  
Figure 4: Data collection set up. at the reference antenna are acquired, tracked and used for the positioning and the locally generated codes obtained from processing the signal of this antenna are used to measure the phases and amplitudes of the signals at the other antennas. Therefore, the estimated discriminator outputs at different antennas differ only on phases and amplitudes (changing only by changes in the signal DOA). For illustration, in Figure 5, the tracking results of the multi-antenna receiver for the signal of one satellite are shown. In this figure, the scatter plots and measured amplitudes and phases of antenna elements are shown. Outputs of phase lock loops (PLL) and delay lock loops (DLL), early, late and prompt correlation results and C/N\({}_{0}\) variations are also shown. Figure 6 shows the position errors in the ENU coordinate system for the first 200 s of the calibration stage that includes 80 s in the static mode followed by two circular motions (each 60 s). The reference trajectory is obtained using NovAtel's Inertial Explorer software. This figure also shows the satellites constellation at the time of data collection. The GPS satellite PRN number and their azimuths and elevation angles are presented in Table 1. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**PRN** & **20** & **16** & **30** & **32** & **31** & **23** \\ \hline Azimuth (Degree) & 204.1 & 146.6 & 78.3 & 163.7 & 58.8 & 292.2 \\ Elevation (Degree) & 52.1 & 37.5 & 45.2 & 29.1 & 28.9 & 68.6 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Azimuth and elevation angles of satellites at the time of the data collection employed for the array calibration. Figure 5: Tracking results of the multi-antenna receiver for one satellite signal. The measured amplitudes and phases for these satellite signals _versus_ time are shown in Figures 7 and 8, respectively. In each figure, the measured values for Antenna 2 and 3, from the discriminator outputs after filtering, are shown (as mentioned before, the phase of reference antenna is kept almost zero and its amplitude is kept to 1). An exponential filter is used for filtering with a smoothing factor of \(\mu=0.05\) as:  
\[ \begin{array}{l}\left|\mathbf{z}_{i}\right|=\mu\left|\tilde{\mathbf{y}}_{i} \right|+\left(1-\mu\right)\left|\mathbf{z}_{i-1}\right|\\ \angle\mathbf{z}_{i}=\mu\angle\tilde{\mathbf{y}}_{i}+\left(1-\mu\right)\angle \mathbf{z}_{i-1}\end{array}  \tag{18}\]  
in which, \(\left|\mathbf{z}_{i}\right|\) and \(\angle\mathbf{z}_{i}\) are the amplitude and phase of the filter output vector. Due to the normalization with respect to the reference antenna, Doppler frequency and its variations, which are precisely tracked by the receiver PLL, are removed from the measurements. These plots indicate that the measured amplitudes from each antenna not only are different but also vary by the changes in the signal DOA in the body frame coordinate. For example, the amplitude of the signal at Antenna 2 varies in the range of \(\pm 3\mathit{dB}\) with respect to the reference antenna. The plots in Figure 8 show that the measured relative phases also vary depending on the satellite signal DOA and the array orientation. For example, the signal received from the satellite with PRN number 23 which is the one with the highest elevation has the least variation in the circular motion and the signals of the satellites with low elevation angles show the highest variations. Figure 6: Estimated position solutions and the corresponding errors by the software receiver. Figure 9 shows the amplitudes and phases of the estimated calibration coefficients from the first stage of the calibration process.