# Robust Low-Sidelobe Transmit Beamforming under Peak-to-Average-Power Ratio Constraint
## 1 Introduction  
With the persistently increasing complexity of the radar working environment, traditional analog beamforming technology has been unable to meet increasingly stringent requirements. The development and application of digital beamforming (DBF) technology has improved the performance of radar in harsh environments. Due to its high level of flexibility and interference suppression ability, DBF is now widely used in both radar transmitters and receivers. Many scientific achievements have been made in various fields, such as radar [1], wireless communication [2], medical imaging [3], radio astronomy [4], etc. With the gradual progress of digital technology, DBF-based transmitting technology has recently become one of the hot topics in array signal processing and is increasingly exploited in radar and communication systems. In order to alleviate the adverse effects of uncertain system factors and harsh working environments on beamformers, improving the robustness of beamformers has become an important issue in recent years. In its infancy, the research on robustness mainly focused on receive beamforming. Several representative robust adaptive beamforming (RAB) methods have been widely studied, including sample matrix inversion (SMI) beamforming [5], diagonal loading (DL) [6], and eigenspace-based beamforming [7]. For example, several algorithms were developed based on the SMI algorithm [8] and are widely usedbecause of their good real-time performance, fast convergence, and low computational complexity [9; 10]. Although the SMI algorithm performs well for small-scale arrays since the number of observations usually greatly exceeds that of sensors and the RMB rule is always satisfied [11], the estimation error of the sample covariance matrix can significantly impair the performance of modular or massive-scale arrays due to finite training data, leading to a rise in the sidelobe level, the shift of the mainlobe, the degradation of the output signal-to-interference-plus-noise ratio (SINR), etc. In addition, some problems exist in the other two RAB methods mentioned above, such as the difficulty determining the loading factor in DL beamforming and the high requirement for the SINR in eigenspace-based beamforming. On the other hand, the robustness of RAB is enhanced by introducing additional objective functions or constraints, such as the positive semi-definiteness (PSD) of the signal-plus-noise covariance matrix [12], the norm of steering vectors [13; 14; 15; 16], and the stochastic distortionless response [17; 18]. Several extensions of the standard Capon beamformer (SCB) have been provided that improve the robustness to uncertain steering vectors. To deal with the steering vector mismatch, worst-case performance optimization (WCPO) [19], which limits the desired signal steering vector to various uncertainty sets, was performed by modeling the uncertainty set as an ellipsoid or sphere with a specific look direction. However, it is worth noting that more sophisticated prior information has to be available when the uncertainty set is no longer simply spherical. The work in [20] introduced a norm constraint into SCB and thus provided norm-constrained Capon beamforming (NCCB) [21] to develop a generalized doubly constrained robust (GDCR) beamformer. Ref. [22] proved that the ripple control over the steering vector uncertainty set can be transformed into a norm constraint for the weight vector, and a doubly constrained robust Capon beamformer (DCRCB) [23] proved to be the preferred choice for applications requiring high SINRs by combining a constant-norm constraint with a spherical uncertainty set constraint. Inspired by these works, [PERSON] described the uncertainty set as a rhombus to represent the \(l_{\infty}\)-norm constraint and devise a computationally acceptable strategy for efficiently solving large-scale \(l_{\infty}\)-beamforming problems via ADMM [24]. However, these RAB methods rarely consider the influence of signal distortion and often consume large computational overhead.