标题： 有限维中Gabor框架的噪声和擦除鲁棒性 显示英文标题

标题： Robustness to noise and erasures of Gabor frames in finite dimensions

摘要： 在本文中，我们研究了结构框架对测量噪声和丢失的鲁棒性，重点研究任意时间-频率位移集 $\Lambda$ 的 Gabor 框架 $(g, \Lambda)$。这种框架的性质在许多信号处理应用中很重要，从无线通信到相位恢复和量化。我们旨在分析此类框架的框架界限与其结构和基数的依赖关系，并提供几乎紧致的 Gabor 框架的构造，其 $\vert \Lambda \vert$ 很小。我们表明，具有随机窗口 $g$ 和框架集 $\Lambda$ 的 Gabor 框架的框架界限表现出与具有独立条目的随机框架的框架界限相似的行为。此外，我们在测量丢失的情况下研究框架界限的统一估计。我们证明了相互正交基框架在丢失率高达 50% 时具有数值鲁棒性，并表明由 Alltop 窗口生成的 Gabor 框架可以提供类似于之前已知的最佳确定性构造的结果。 显示英文摘要

摘要： In this paper, we investigate the robustness of structured frames to measurement noise and erasures, with the focus on Gabor frames $(g, \Lambda)$ with arbitrary sets of time-frequency shifts $\Lambda$. This property of frames is important in many signal processing applications, from wireless communication to phase retrieval and quantization. We aim to analyze the dependence of the frame bounds of such frames on their structure and cardinality and provide constructions of nearly tight Gabor frames with small $\vert \Lambda \vert$. We show that the frame bounds of Gabor frames with a random window $g$ and frame set $\Lambda$ show similar behavior to the frame bounds of random frames with independent entries. Moreover, we study uniform estimates for the frame bounds in the case of measurement erasures. We prove numerical robustness to erasures for mutually unbiased bases frames with erasure rate up to 50\% and show that the Gabor frame generated by the Alltop window can provide results similar to the best previously known deterministic constructions.