## 5 Conclusions  
A novel two-stage precise calibration method using satellite signals for GNSS applications was proposed. In this method, all of the main uncertainties in the calibration process were modeled and it  
Figure 14: Normalized array antenna beam patterns after beamforming for PRNs 20, 16, 30, 32, 31 and 23.  
Figure 15: Comparing C/N\({}_{0}\) for a single antenna receiver and the beamformer for PRNs 20, 16, 30, 32, 31 and 23.  
was shown that they could be separated into two parts; one is constant and the other one consists of DOA-dependent parameters. One solution for the associated optimization to this problem was introduced in the form of two eigenvalue decomposition problems. Although the estimated constant term could provide coarse estimates of satellite steering vectors, for more precise calibration the second stage provided compensation amplitude and phase for each incident signal corresponding to its DOA in a form of a set of look up tables for each antenna element. The proposed two-stage calibration process not only simplified the interpretation of uncertainties, but also was used to come up with an iterative method with a fast convergence to accurately estimate satellite signal steering vectors. The effectiveness of this method in both calibration and receiver operation modes was evaluated by using a set of collected GPS signals in two different GNSS multi-antenna applications. Although an IMU unit was employed in this case for calibration and its performance evaluation, heading, roll and pitch angles could also have been provided by the carrier phase measurements obtained by the antenna array using standard GNSS attitude determination methods. In this paper, the measurement noise and carrier phase multipath were not explicitly considered in the mathematical model of the calibration although the measurements might be affected by them. This may reduce the accuracy of the calibration results. For example, although carrier phase multipath might be compensated in the second stage of the calibration process for certain observables, the compensation values are not necessarily applicable to the other observations. Moreover, a weighting model could be used to reduce the effect of noisier observations (e.g., signals from satellites with lower elevation angles) instead of using the equal weight model employed in this paper. Herein the results of calibration were verified through attitude determination and beamforming applications and employing the IMU measurements as a reference. However, estimating the accuracy of the calibration procedure as an independent block has not yet been investigated. Studying an approach to express the accuracy of the calibration results as a function of azimuth and elevation angles is suggested as future work. Furthermore, repeating the performed data collection for a certain antenna array for a long period of time in different environments and checking the consistency between the results to observe the effect of carrier phase multipath is suggested for further research on this paper.  
Authors namely, [PERSON], [PERSON], [PERSON] and [PERSON], have cooperated in technical parts, data collections and analyses and writing of this paper. Fund for this research was provided by [PERSON]'s iCORE grant.  
The authors declare no conflict of interest.