# Robust Low-Sidelobe Transmit Beamforming under Peak-to-Average-Power Ratio Constraint
## 2 Signal Model and Problem Formulation  
Consider a narrowband signal transmitted by a linear array with \(N\) elements located at known locations \(d_{n}\in\mathbb{R},n=1,2,\ldots,N\), representing the distance from the \(n\)-th element to the reference element, as shown in Figure 1. The transmit beampattern can be expressed as  
\[ \begin{split}& y(t)=h_{1}\omega_{1}^{*}s(t)+h_{2}\omega_{2}^{*}s(t)e^{ 2\pi d_{2}\sin\theta/\lambda}+\cdots\\ &+h_{N}\omega_{N}^{*}s(t)e^{2\pi d_{N}\sin\theta/\lambda}\\ &=\mathbf{\omega}^{H}\mathbf{a}(\theta)s(t)\end{split}  \tag{1}\]  
where \(\mathbf{\omega}=[\omega_{1},\omega_{2},\ldots,\omega_{N}]^{T}\) is the transmit weight vector, and \(\mathbf{a}(\theta)\) is the array steering vector in the signal of interest (SOI) direction \(\theta\), which is defined as  
\[\mathbf{a}(\theta)=\left[h_{1}(\theta),h_{2}(\theta)e^{j2\pi d_{2}\sin\theta/ \lambda},\cdots,h_{N}(\theta)e^{j2\pi d_{N}\sin\theta/\lambda}\right]^{T} \tag{2}\]  
where \(\lambda\) is the wavelength, \(h_{n}(\theta)\), \(n=1,2,\ldots,N\) is the pattern radiated by the \(n\)-th element, and \(h_{n}(\theta)=1\) when the element is isotropic. Then, the transmit beampattern in (1) can be expressed as  
\[F(\theta)=\mathbf{\omega}^{H}\mathbf{a}(\theta) \tag{3}\]  
In the practical situation of LSS target detection, there are large amounts of clutter and interference existing on the ground, which increases the difficulty of LSS target detection. Therefore, we need to design an array weight vector to markedly suppress the sidelobe levels of the transmit beampattern on the near-ground side so that the reflected ground clutter can be significantly reduced. Based on this consideration, the method in this paper aims to minimize the PSL on the ground side while preserving the array response in the  
Figure 1: Transmit array layout of a linear array. SOI direction. Then, the corresponding TBF problem can be formulated as the following min-max optimization problem:  
\[ \begin{split}\min_{\mathbf{\omega}}&\max_{\theta_{\text{q}} \in\Theta_{\text{G}}}\left|\mathbf{\omega}^{H}\mathbf{a}\left(\theta_{\text{q}} \right)\right|\\ &\text{s.t.}&\left|\mathbf{\omega}^{H}\mathbf{a}\left( \theta_{0}\right)\right|=1\end{split}  \tag{4}\]  
where \(\Theta_{\text{G}}\) indicates the direction set corresponding to the ground-side clutter region, \(\mathbf{a}\left(\theta_{\text{q}}\right)\) is the array steering vector in the direction \(\theta_{\text{q}}\), and \(\theta_{\text{q}}\in\Theta_{\text{G}}\). In practice, the element spacing of the linear array may not be exactly identical because of the measurement error. Therefore, the actual gain and phase of the signals of each element are also in error with the ideal situation, which leads to the mismatch of the steering vector. The mismatched array steering vector inevitably affects the performance of the beampattern and leads to the degradation of the spatial filtering performance of the array. In fact, if the gain and phase errors resulting from inaccurate element spacing and nonideal RF chains are incorporated into the array element weight vector, the actual beam response is  
\[\hat{F}(\theta)=\hat{\mathbf{\omega}}^{H}\hat{\mathbf{a}}(\theta)=\sum_{n=1}^{N} \hat{g}_{\text{n}}e^{i\left(\hat{\theta}_{\text{n}}-2\pi\hat{p}_{\text{n}} \sin(\theta)/\lambda\right)} \tag{5}\]  
where \(\hat{\mathbf{\omega}}\) and \(\hat{\mathbf{a}}(\theta)\) are the actual array weighting vector and the actual array steering vector, \(\hat{g}_{\text{n}}\) is the actual amplitude weighting factor, \(\hat{\varphi}_{\text{n}}\) is the actual phase weighting factor, and \(\hat{p}_{\text{n}}\) is the actual position of the \(n\)-th array element. When the above error is small, the actual amplitude-squared expectation of the beam response is  
\[