# Beamforming Techniques for Resilient Navigation with Small Antenna Arrays +
## 4 Performance Analysis
### Urban Scenario  
Urban scenarios are characterized by the abounding presence of obstacles and tall buildings, thus having sky visibility reduced to a small region in the vertical direction. Hence, the signals are prone to be received from high elevation angles. In that sense, the LOSS and the replica are considered herein to impinge from \(80^{\circ}\) and \(60^{\circ}\) elevation, respectively. An azimuth angle of \(30^{\circ}\) is adopted for both.  
Figure 1 illustrates the cancellation of the replica as a function of the delay spacing between the latter and the LOSS, normalized to \(T_{\text{c}}\). Any spacing below one is considered as correlated. The top plot depicts the ULA2 case, and the bottom plot depicts the URA4 case. When it comes to conventional beamformers, the PHA presents very poor mitigation performance, which remains the same irrespective of the replica delay. The CAP is even observed to amplify the undesired replica when highly correlated with the LOSS, in an attempt to level the power of the replica to that of the LOSS. It is not until signals become uncorrelated that the CAP provides acceptable performance, thus making it unsuitable for coherent multipath and spoofing mitigation. The LMMSE seems to provide better performance compared to the PHA and CAP in terms of replica cancellation, but, in contrast, it is found to incur dramatic LOSS degradation, reaching power drops of up to 30 dB. These results apply to both ULA2 and URA4.  
Interestingly, the poor performances described above are overcome by the use of more sophisticated beamformers. This is the case for the HYB and PBC. They show the best replica cancellation performance even for highly correlated signals, and it improves rapidly when increasing the replica delay. The latter occurs particularly for the URA4 case, where the cancellation shows a steeper behavior than in ULA2. More specifically, a cancellation of more than 10 dB is achieved in URA4 for as few as 0.3 replica delay, whereas the same performance requires 0.5 replica delay in ULA2, a point where the cancellation in URA4 already exceeds 15 dB. In that sense, it is observed that the gain provided by the use of URA4 with respect to the smaller ULA2 lies in the range of 3-5 dB.  
The above observations can also be inferred from the MEE results depicted in Figure 2, which ultimately unveil the impact of mitigation on the receiver performance. The CAP presents a detrimental behavior particularly when signals are highly correlated, with MEE values exceeding 60 m. The PHA provides a slight gain compared to when no beamforming (labeled as \"No BF\" in the figure) is performed, by decreasing the MEE from 25 m down to 20 m. However, acceptable MEE is not achieved until the signals become uncorrelated. In that sense, conventional beamformers are outperformed by the HYB and PBC, where the MEE barely exceeds 10 m and 5 m for ULA2 and URA4, respectively, and as few as 0.5 replica delay are required for the MEE to drop down below 5 m in both.  
Figure 1: Cancellation of replica with respect to its delay in an urban scenario with ULA2 (**top**) and URA4 (**bottom**) array distributions.