# Robust Low-Sidelobe Transmit Beamforming under Peak-to-Average-Power Ratio Constraint
## 1 Introduction
In recent years, transmit beamforming (TBF) has been extensively studied in the design of radar and communication systems [25]. In order to maximize the transmit power and reduce the dynamic range of DACs, constant modular (CM) constraints are generally required, but this severely limits the degrees of freedom (DoF) of weights. With the relaxation of CM, the peak-to-average-power ratio (PAPR) now raises concerns. In the incipient stage, PAPR was mainly used in the waveform design of OFDM and MIMO radar signals to control the dynamic range of transmit waveforms [26; 27; 28]. In massive-scale arrays, constant or approximately consistent amplitude constraints are imposed on the analog part of hybrid TBF. Therefore, a PAPR constraint is introduced into TBF as a trade-off between the power efficiency and the mainlobe ripple, peak sidelobe level (PSL), or other indicators by relaxing the unimodular constraints [29]. However, this TBF method is not yet suitable for RAB. In [30], [PERSON] imposed a PAPR constraint on transmit weights so as to avoid excessive energy attenuation and reduce the ripple of transmit weights. It has strong robustness to the steering vector mismatch. However, this method did not consider the sidelobe level of the beampattern, which considerably affects the performance of some radar systems suffering from strong interferences from a given region, especially for low-altitude and slow-speed small (LSS) radars. As is known, the detection performance and low interception of radar will be reduced if the transmit beampattern has a high sidelobe level. There now exist a large number of works on the control of the sidelobe level [31; 32; 33; 34; 35]. On the one hand, sidelobe control is realized by introducing the sidelobe-level constraint directly. In [31; 32], the sidelobe levels were precisely controlled for a ULA and an arbitrary geometric array, respectively. Furthermore, an additional weight function was defined to effectively control the sidelobe level in [33]. However, this method is computationally expensive and only applicable to a limited scope. On the other hand, received data reconstruction and PSL minimization were used to control the sidelobe level and to improve the robustness simultaneously. In [34], matrix reconstruction was exploited to improve the robustness and accurately control the sidelobe level. In [35], the alternating direction method of multipliers (ADMM) was used to solve sidelobe control combination constraints, which can effectively control the mainlobe, sidelobe, and nulling. In [36], the ratio between the minimum mainlobe level and the maximum sidelobe level was maximized by using a customized ADMM-based algorithm. Based on the above observations, robust TBF with sidelobe-level control (SLC) is lacking. However, it is imperative in certain application scenarios, such as LSS target detection. Generally, \"LSS target\" is a collective term for a variety of small craft and air drifters that fly at low or ultra-low altitudes and have slow speeds and other characteristics. Radar detection is now quite a popular means of LSS target detection compared with acoustic detection [37] and photoelectric detection [38] because of its advantages of all-time, all-weather operation and fast response time. However, it faces difficulties in the complex urban background where the target is located, such as fast-moving interference, strong ground clutter and noise, and a high-complexity environment, resulting in the low SNR of the radar echo signal and affecting the detection performance of the radar. To address the above problem, this paper proposes a robust TBF with sidelobe-level control under a PAPR constraint. This robust TBF approach can effectively deal with rapid changes in ground-side interference by minimizing the PSL in the preset region.