# Two-Stage Beamforming Technique for GNSS Applications +
## 3 Experimental Results and Discussion
### Field Results  
In this section, we present the experimental results of the proposed solution. We compared the proposed solution against the traditional single-antenna system; the first antenna of the four antennas was selected as a reference. All the signal processing was conducted offline using MATLAB implementation. In this work, we modified the Multi-GNSS SDR Receiver (FGI-GSRx), based on the guidance provided by [PERSON] et al. [15], to include the beamforming approach. The FGI-GSRx MATLAB software is capable of offering a positioning solution with multiple GNSS signals; in our implementation, we have considered GPS L1, Galileo E1b, and BeiDou B1I signals.  
Figure 3 shows the skylpot of the acquired satellites from a single-antenna system and the proposed solution, respectively. It is clear, as expected, that the array antenna system acquired more satellites than the single-antenna system. The proposed solution has acquired 28 satellites, while the single-antenna system has only acquired 21 satellites.  
Figure 4 shows that the proposed solution provides higher CN0 estimates for all the acquired satellites, as compared to the single-antenna system. It is worth mentioning that the traditional MVDR beamformer fails to provide any valuable weight vectors. Accordingly, the traditional MVDR results were excluded due to loss of the tracking signals. This can be attributed to the presence of the multipath correlated signals and the sensitivity of the traditional MVDR to the sample size used to construct the array antenna covariance matrix; in this test, only four acquired complex samples have been exploited. Consequently, the MVDR algorithm fails to provide the correct weight vectors.  
Figure 2: The deployed 4-channel phase aligned record and playback system in the experiment. (**a**) hardware configuration of the setup showing the 4 channel VSTs in the upper PXI chassis and the associated FPGA cards and RAID disks in the lower chassis; (**b**) live test experiment using the setup connected to a rectangular antenna array, where each element is wired to a low noise amplifier (LNA) into the input of the corresponding VST.  
Figure 5 shows the positioning estimation error in meters for the single-antenna system and the proposed approach, respectively. The obtained positioning estimation error is  
Figure 4: CN0 values for acquired satellites. Proposed solution has acquired more satellites than single-antenna system. Furthermore, proposed technique has estimated high CN0 values for all acquired signals.  
Figure 3: Skyplot representations of the acquired satellites from both the single-antenna system and the proposed solution, respectively.  
presented as the deviation from the true 3D position in East (E), North (N), and Up (U) coordinates. The true reference position has been estimated using Real-time kinematic (RTK) technique. It is clear that the proposed approach has provided a more accurate estimation than the single-antenna system. The 95% 3D position error for the proposed solution and for the single-antenna system is 5.2 and 7.7 m, respectively. Furthermore, the position dilution of precision (pdop) is 0.84 for the proposed approach and 1.07 for the single-antenna system. Finally, it can be observed, from Figure 5, that the proposed solution provides a more stable estimation over time than the single-antenna system, which can be attributed to the proposed solution capability of estimating and mitigating the multipath effect.