Atomic norm methods have recently been proposed for spectral super-resolution with flexibility in dealing with missing data and miscellaneous noises. A notorious drawback of these convex optimization methods however is their lower resolution in the high signal-to-noise (SNR) regime as compared to conventional methods such as ESPRIT. In this paper, we devise a simple weighting scheme in existing atomic norm methods and show that in theory the resolution of the resulting convex optimization method can be made arbitrarily high in the absence of noise, achieving the so-called separation-free super-resolution. This is proved by a novel, kernel-free construction of the dual certificate whose existence guarantees exact super-resolution using the proposed method. Numerical results corroborating our analysis are provided.

中文翻译：

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通过凸优化实现免分离光谱超分辨率

最近提出了原子范数方法用于光谱超分辨率，可以灵活地处理丢失数据和杂项噪声。然而，这些凸优化方法的一个众所周知的缺点是，与 ESPRIT 等传统方法相比，它们在高信噪比 (SNR) 范围内的分辨率较低。在本文中，我们在现有原子范数方法中设计了一种简单的加权方案，并表明理论上，在没有噪声的情况下，所得凸优化方法的分辨率可以任意高，实现所谓的无分离超分辨率。这是通过一种新颖的、无内核的双证书结构证明的，其存在保证了使用所提出的方法的精确超分辨率。提供了证实我们分析的数值结果。