A smaller number of transmit antennas can reduce the array system cost, while sparse antenna design can provide additional degrees of freedom to beampattern synthesis. It is also known that the min-max beampattern matching design criterion can synthesize beampattern with small mainlobe ripple, low peak sidelobe level, and specified null/notch. Based on this design metric, we formulate a unified sparse array design framework for beampattern synthesis. Our framework includes the colocated multiple-input multiple-output (MIMO) radar transmit beampattern synthesis, phase-only beampattern synthesis, and phased-array beampattern synthesis as special cases. In the design formulation, the antenna positions and phase-only waveforms (for MIMO radar beampattern) or array weight vector (for phase-only and traditional phased-array beampattern) are jointly determined. The resultant problem is challenging because the nonconvex nonsmooth min-max objective function, and phase-only/unimodular requirements make it become typically NP-hard, and the antenna position selection results in a difficult Boolean constraint. Nevertheless, we are able to find a local optimal solution efficiently by applying the Lawson approximation and majorization-minimization. Numerical examples show that the devised approach can achieve the desired transmit beampattern by jointly optimizing the phase-only waveforms/weight vector and antenna locations.

较少数量的发射天线可以降低阵列系统的成本，而稀疏的天线设计可以为波束图合成提供更多的自由度。还已知最小-最大波束图匹配设计准则可以合成具有小主瓣波纹，低峰值旁瓣电平和指定的空/陷波的波束图。基于此设计指标，我们制定了一个统一的稀疏阵列设计框架用于波束图合成。作为特殊情况，我们的框架包括并置的多输入多输出（MIMO）雷达发射波束图合成，仅相位波束图合成和相控阵波束图合成。在设计方案中，联合确定天线位置和仅相位波形（对于MIMO雷达波束图）或阵列加权矢量（对于仅相位和传统相控阵波束图）。由于非凸，非平滑的最小-最大目标函数以及仅相位/单模的要求使其变得典型地为NP困难，并且天线位置选择导致困难的布尔约束，因此产生的问题具有挑战性。但是，我们可以通过应用Lawson逼近和主化-最小化来有效地找到局部最优解。数值示例表明，通过共同优化仅相位波形/权重矢量和天线位置，该设计方法可以实现所需的发射波束方向图。且仅相位/单模的要求使其通常变得难于NP，并且天线位置选择会导致难以进行布尔约束。但是，我们可以通过应用Lawson逼近和主化-最小化来有效地找到局部最优解。数值示例表明，通过共同优化仅相位波形/权重矢量和天线位置，该设计方法可以实现所需的发射波束方向图。且仅相位/单模的要求使其通常变得难于NP，并且天线位置选择会导致难以进行布尔约束。但是，我们可以通过应用Lawson逼近和主化-最小化来有效地找到局部最优解。数值示例表明，通过共同优化仅相位波形/权重矢量和天线位置，该设计方法可以实现所需的发射波束方向图。