# Robust Low-Sidelobe Transmit Beamforming under Peak-to-Average-Power Ratio Constraint
## 4 Numerical Results and Analyses
### Experiment 2: PAPR Tolerance vs. Sidelobe Level  
In this experiment, we revealed the relationship between the variation in the PAPR tolerance and the beampattern formed by the proposed method with different norm upper bounds of the weight vector. Here, as shown in Table 5, the PAPR tolerance range is set as \([1.1,2]\), and the upper bound is varied to equal \(0.0325,0.035,0.045,\) and \(1\).  
Taking LSS target detection as an example, we need to avoid out-of-band radiation, which affects signals in adjacent frequency bands, and in-band distortion, which causes the rotation, attenuation, and displacement of signals. We also want the ability to handle large-scale and fast-moving interference on the ground. That is, we expect to obtain a  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Method**} & \multirow{2}{*}{**PAPR**} & \multicolumn{2}{c}{**PSL (dB)**} & \multicolumn{2}{c}{**ISL (dB)**} & \multirow{2}{*}{**Norm**} \\  & & **Ground** & **Air** & **Ground** & **Air** & \\ \hline Static & \(1\) & \(-13.24\) & \(-13.24\) & \(2.02\) & \(2.02\) & \(5.66\) \\ SMI & \(2.27,2.62\) & \(-13.05,-13.80\) & \(-3.28,-4.27\) & \(10.86,8.33\) & \(15.49,12.74\) & \(0.49,0.43\) \\ Norm constraint & \(1.53,1.53\) & \(-26.76,-28.59\) & \(-12.60,-12.86\) & \(-3.26,-4.70\) & \(3.89,3.37\) & \(0.19,0.19\) \\ Worst case & \(2.23,2.54\) & \(-13.30,-14.02\) & \(-3.41,-4.32\) & \(10.52,8.11\) & \(15.09,12.60\) & \(0.50,0.44\) \\ Proposed method & \(1.03\) & \(-34.25\) & \(-7.89\) & \(-7.67\) & \(7.83\) & \(0.18\) \\ \hline \hline \end{tabular}
\end{table}
Table 4: The detailed information of different TBF methods.  
Figure 3: The beampatterns of the proposed TBF method and four typical counterparts when the interference region is close to the mainlobe. (**a**) \(k=64\); (**b**) \(k=500\).  
\begin{table}
\begin{tabular}{l l} \hline \hline
**PAPR** & **[1.1, 2]** \\ \hline Upper limit of norm & \(0.0325,0.035,0.045,1\) \\ Snapshots & \(4000\) \\ \hline \hline \end{tabular}
\end{table}
Table 5: Settings of experiment 2.  
low PAPR, a low SLL, and high robustness at the same time. However, these goals are mutually contradictory. Therefore, seeking for a compromise proposal to select the proper \(b\) is vital to the ultimate performance of the proposed method. Figure 4 shows beampatterns with different constant upper limits of the norm and varying PAPR tolerances. We further present Figure 5 in order to display the more explicit relationship between the PAPR and SLL.  
As shown in Figure 5, when the upper limit value is infinitely close to the lower bound \(1/N\), the sidelobe level is high and the inhibition effect is relatively weak. In addition, with the relaxation of the PAPR constraint, the improvement effect on the sidelobe level is not very substantial. As the value of \(b\) gradually increases, the sidelobe inhibition effect gradually improves, and the PAPR constraint has far more influence on this improvement. This situation continues until \(b\approx 0.045\). After that, the increase in \(b\) will no longer reduce the sidelobe level, and the influence degree of PAPR variation will no longer increase. According to the beampattern, when the sidelobe facing the ground side decreases, the counterpart on the air side will rise to a certain extent. However, the maximum sidelobe level is at least 10 dB lower than the mainlobe level, so it will not significantly affect the transceiver performance of the mainlobe or the overall performance of radar.  
Figure 4: Beampatterns with different constant upper limits of the norm. (**a**) \(b=0.0325\); (**b**) \(b=0.035\); (**c**) \(b=0.045\); (**d**) \(b=1\).