# Robust Low-Sidelobe Transmit Beamforming under Peak-to-Average-Power Ratio Constraint
## 4 Numerical Results and Analyses
### Experiment 1: The Proposed TBF versus Several Typical Beamformers  
In this experiment, we compared the transmit beampatterns provided by the proposed method and several typical counterparts. As is known, traditional TBF methods are often sensitive to inaccurate covariance matrix estimation. With few snapshots, the sampling covariance matrix can only provide a coarse covariance matrix estimate. In addition, the direction of arrival of interferences, which are close to the mainlobe, can hinder the formation of notches and reduce the gain of the mainlobe, thus causing the significant deterioration of the TBF performance. In this experiment, we assumed that 80 interferences exist in a continuous region on the ground side. The specific parameters are summarized in Table 2. According to the RMB rule [11], fully adaptive processing requires the number of snapshots to be not less than twice the system's degrees of freedom, i.e., \(2N\). Based on this principle, Figure 2 displays the transmit beampatterns generated by the proposed TBF method and four typical counterparts with different numbers of snapshots. The detailed PSL and ISL of transmit beampatterns of these TBF methods are shown in Table 3, where the data before and after ', correspond to 64 and 500 snapshots, respectively. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**PAR** & **1.2** & **Norm Upper Bound** & **0.8** \\ \hline Interference direction & \(-59.5^{\circ}:0.5^{\circ}:-20^{\circ}\) & Snapshots & 32, 64, 500 \\ Interference number & 80 & INRs & 10 dB \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Parameter settings of comparison experiment. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Method** & **PAPR** & \multicolumn{3}{c}{**PSL (dB)**} & \multicolumn{3}{c}{**ISL (dB)**} & \multirow{3}{*}{**Norm**} \\  & & **Ground** & **Air** & **Ground** & **Air** & \\ \hline Static & \(1\) & \(-13.24\) & \(-13.24\) & \(2.02\) & \(2.02\) & \(5.66\) \\ SMI & \(3.43,1.88\) & \(-12.84,-12.90\) & \(-5.929,-12.89\) & \(6.83,3.02\) & \(14.18,4.09\) & \(0.22,0.19\) \\ Norm constraint & \(1.73,1.58\) & \(-13.16,-13.62\) & \(-11.50,-12.84\) & \(2.78,2.31\) & \(5.48,3.72\) & \(0.19,0.19\) \\ Worst case & \(3.42,1.87\) & \(-12.81,-12.92\) & \(-5.99,-12.89\) & \(6.79,3.01\) & \(14.06,4.07\) & \(0.22,0.19\) \\ Proposed method & \(1.03\) & \(-34.25\) & \(-7.89\) & \(-7.67\) & \(7.83\) & \(0.18\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Detailed PSL and ISL of five TBF methods with different snapshots. As shown in Figure 2b, when the number of snapshots is large enough, traditional adaptive beamforming has excellent performance. However, in practice, the sampling frequency is often limited, as the larger the sampling rate is, the higher the data rate is, which will impose a great burden on the storage and processing of the system. As shown in Figure 2a,c, when interferences change rapidly, such as in the vehicle-mounted environment, the performance of the adaptive methods will drastically decrease due to insufficient snapshots. The smaller the snapshots, the more the performance decreases. The results of this experiment are shown in Table 3.  
According to Table 3, although deep notches against interferences can be formed, the PSLs and ISLs on both sides are much higher than those of the proposed method in most cases after a further considerable rise when the snapshots decrease to 64. In addition, the actual PAPRs of the three adaptive methods are quite large compared with that of the proposed method, which increases the power consumption. Static beamforming and the proposed method are not adaptive, which means they cannot be affected by snapshots, which is why the data of these two remain consistent. However, the former can neither effectively suppress the interference nor suppress the sidelobe level. In addition to the snapshots, the DOAs of interferences also affect the TBF performance, such as the gain, notch depth, and so on.