# Robust Low-Sidelobe Transmit Beamforming under Peak-to-Average-Power Ratio Constraint
## 2 Signal Model and Problem Formulation
\begin{split}\mathbb{E}\Big{\{}|\hat{F}(\theta)|^{2}\Big{\}}& =|F(\theta)|^{2}e^{-\left(\varphi_{\text{p}}^{2}+\sigma_{\lambda }^{2}\right)}+\left(\sigma_{\text{p}}^{2}+\sigma_{\lambda}^{2}+\sigma_{\text{ g}}^{2}\right)\sum_{n=1}^{N}g_{\text{n}}^{2}\\ &=|F(\theta)|^{2}e^{-\left(\sigma_{\text{p}}^{2}+\sigma_{\lambda }^{2}\right)}+\left(\sigma_{\text{p}}^{2}+\sigma_{\lambda}^{2}+\sigma_{\text{ g}}^{2}\right)\|\omega\|^{2}\end{split}  \tag{6}\]  
where \(\hat{F}(\theta)\) is the actual beampattern, and \(\sigma_{\text{g}}\) and \(\sigma_{\text{g}}\) are, respectively, the variance of the amplitude and the phase error of the weights. \(\sigma_{\lambda}=2\pi\sigma_{\text{P}}/\lambda\), where \(\sigma_{\text{P}}\) is the variance of the antenna position error. Equation (6) consists of two parts. The first part is the ideal beampattern multiplied by an attenuation coefficient, which will not affect the gain of beam formation. The second part is the product of the sum of the errors and the norm of the array weighting vector, which mainly affects the sidelobe level of the beampattern. The sensitivity function is defined as  
\[T_{\text{se}}=\sum_{n=1}^{N}g_{\text{n}}^{2}=\|\mathbf{\omega}\|^{2} \tag{7}\]  
Therefore, for a certain error, the smaller the \(\|\omega\|^{2}\), the higher the robustness of beamforming, and \(\|\mathbf{\omega}\|^{2}\) can be used as an important indicator for the robustness of beamforming. In order to design a robust beamformer, \(\|\omega\|^{2}\) should be small. Therefore, we set the following robust constraint:  
\[\|\omega\|^{2}\leq b \tag{8}\]  
where \(b\) is the upper bound of the prescribed norm, and different values can be specified according to the actual requirements. In addition to improving robustness, maximizing the transmit power is important to improve the reliability of the sensing system. It is well known that receive and transmit beam designs are reciprocal because of the transmit-receive reciprocity of the antenna. In practice, however, the transmit beam design becomes more difficult due to the energy and PAPR constraints. Ideally, to obtain a stronger signal-to-noise ratio (SNR) with better performance, a radar transmitter for detecting LSS targets should transmit all the energy to the area where the target is located. For TBF, to maximize the radar transmit power, the transmit signal needs to meet the CM constraint or have a low PAPR. To obtain a better performance of the transmit beam and to avoid excessive energy fading, we introduce a PAPR constraint on the transmit weights. The low-PAPR constraint ensures that the amplitude factor fluctuates within a certain range while improving the utilization of the transmit signal power. According to the definition of the PAPR, the constraint on transmit weights can be expressed as  
\[r_{\text{PAPR}}=\frac{\max_{i=1,2,\cdots,N}\abs{\omega_{i}}^{2}}{\lVert \boldsymbol{\omega}\rVert^{2}/N}=N\frac{\max_{i=1,2,\cdots,N}\abs{\omega_{i}} ^{2}}{\lVert\boldsymbol{\omega}\rVert^{2}} \tag{9}\]  
Considering the low-PAPR constraint, the preservation of the mainlobe response, and the robustness norm constraint, problem (4) can be further expressed as  
\[ \begin{split}\underset{\boldsymbol{\omega}}{\text{min}}& \underset{\theta_{\text{q}}\in\Theta_{\text{G}}}{\max}\abs{ \boldsymbol{\omega}^{H}\mathbf{a}\left(\theta_{\text{q}}\right)}\\ \text{s.t.}&\abs{\boldsymbol{\omega}^{H}\mathbf{a} \left(\theta_{0}\right)}=1\\ & r_{\text{PAPR}}\leq\zeta\\ &\boldsymbol{\omega}^{H}\boldsymbol{\omega}\leq b\end{split}  \tag{10}\]  
where \(\zeta\) is the preset upper bound of PAPR tolerance.