# Robust Low-Sidelobe Transmit Beamforming under Peak-to-Average-Power Ratio Constraint
## 3 The Proposed Algorithm  
In this section, a robust low-sidelobe TBF method under a low-PAPR constraint and a robust constraint is proposed. Note that the PAPR constraint \(r_{\text{PAPR}}\leq\zeta\) in (10) is difficult to handle since it is a fractional constraint related to \(\boldsymbol{\omega}\). Therefore, we first convert this constraint into a substitutive expression that is tractable. Since \(\mathbf{a}\left(\theta_{0}\right)^{H}\boldsymbol{\omega}=1\), we have the following inequality:  
\[1=\abs{\mathbf{a}^{H}(\theta_{0})\boldsymbol{\omega}}^{2}\leq\lVert\mathbf{a} (\theta_{0})\rVert^{2}\lVert\boldsymbol{\omega}\rVert^{2}=N\lVert\boldsymbol{ \omega}\rVert^{2} \tag{11}\]  
Therefore, we can obtain  
\[\lVert\omega\rVert^{2}\geq 1/N \tag{12}\]  
By using (12), the PAPR constraint can be rewritten as  
\[r_{\text{PAPR}}=N\frac{\max_{i=1,2,\cdots,N}\abs{\omega_{i}}^{2}}{\lVert \boldsymbol{\omega}\rVert^{2}}\leq N\frac{\max_{i=1,2,\cdots,N}\abs{\omega_{i }}^{2}}{1/N}=N^{2}\underset{i=1,2,\cdots,N}{\max}\abs{\omega_{i}}^{2}\leq\zeta \tag{13}\]  
In other words,  
\[N^{2}\underset{i=1,2,\cdots,N}{\max}\abs{\omega_{i}}^{2}\leq\zeta \tag{14}\]  
which is equivalent to  
\[\underset{i=1,2,\cdots,N}{\max}\abs{\omega_{i}}^{2}\leq\sqrt{\zeta}/N \tag{15}\]Obviously, the PAPR constraint is substituted by many convex constraints. Then, problem (10) is converted into  
\[ \begin{split}\min_{\mathbf{\omega}}\max_{\theta_{\text{q}}\in\Theta_{ \text{G}}}&\left|\mathbf{\omega}^{H}\mathbf{a}\big{(}\theta_{\text{q} }\big{)}\right|\\ \text{s.t.}&\left|\mathbf{\omega}^{H}\mathbf{a}\big{(} \theta_{0}\big{)}\right|=1\\ &\left|\omega_{i}\right|\leq\sqrt{\xi}/N,\quad\forall i\\ &\mathbf{\omega}^{H}\mathbf{\omega}\leq b\end{split}  \tag{16}\]  
In practical beamforming applications, the assumed signal steering vector often suffers from a certain error; i.e., there is a certain mismatch between the assumed signal steering vector and its actual value. In order to improve the robustness of beamforming against an arbitrary steering vector mismatch, the actual steering vector is defined as  
\[\hat{\mathbf{a}}=\mathbf{a}+\mathbf{a}_{\text{e}} \tag{17}\]  
where \(\mathbf{a}_{\text{e}}\) is the steering vector error. The uncertainty region of \(\hat{\mathbf{a}}\) can be modeled as a ball set \(\varepsilon\), namely,  
\[\varepsilon=\{\mathbf{a}_{\text{e}}\ |\ \|\mathbf{a}_{\text{e}}\|\leq \varepsilon_{\text{e}}\} \tag{18}\]  
where \(\varepsilon_{\text{e}}\) is the radius of the ball. Then, problem (16) is translated into  
\[ \begin{split}\min_{\mathbf{\omega}}\max_{\theta_{\text{q}}\in\Theta_ {\text{G}},\hat{\mathbf{\omega}}(\theta)\in\varepsilon}&\left|\mathbf{ \omega}^{H}\hat{\mathbf{a}}\big{(}\theta_{\text{q}}\big{)}\right|\\ \text{s.t.}&\left|\mathbf{\omega}^{H}\mathbf{a}\big{(} \theta_{0}\big{)}\right|=1\\ &\left|\alpha_{i}\right|\leq\sqrt{\xi}/N,\quad\forall i\\ &\mathbf{\omega}^{H}\mathbf{\omega}\leq b\end{split}  \tag{19}\]  
Furthermore, problem (19) can be equivalent to  
\[ \begin{split}\min_{\mathbf{\omega}}\max_{\theta_{\text{q}}\in\Theta_ {\text{G}}}&\max_{\mathbf{\omega}\in\varepsilon}\left|\mathbf{\omega}^{ H}\hat{\mathbf{a}}\big{(}\theta_{\text{q}}\big{)}\right|\\ \text{s.t.}&\left|\mathbf{\omega}^{H}\mathbf{a}\big{(} \theta_{0}\big{)}\right|=1\\ &\left|\alpha_{i}\right|\leq\sqrt{\xi}/N,\quad\forall i\\ &\mathbf{\omega}^{H}\mathbf{\omega}\leq b\end{split}  \tag{20}\]  
Note that (20) is a nonconvex constrained optimization problem due to the nonconvex objective function, and its objective function has an infinite dimension. In the following, we will translate (20) into a tractable convex problem by using the special structure of the objective function and constraints.