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Estimation and inference of signals via the stochastic geometry of spectrogram level sets
arXiv - CS - Sound Pub Date : 2021-05-06 , DOI: arxiv-2105.02471
Subhroshekhar Ghosh, Meixia Lin, Dongfang Sun

Spectrograms are fundamental tools in the detection, estimation and analysis of signals in the time-frequency analysis paradigm. Signal analysis via spectrograms have traditionally explored their peaks, i.e. their maxima, complemented by a recent interest in their zeros or minima. In particular, recent investigations have demonstrated connections between Gabor spectrograms of Gaussian white noise and Gaussian analytic functions (abbrv. GAFs) in different geometries. However, the zero sets (or the maxima or minima) of GAFs have a complicated stochastic structure, which makes a direct theoretical analysis of usual spectrogram based techniques via GAFs a difficult proposition. These techniques, in turn, largely rely on statistical observables from the analysis of spatial data, whose distributional properties for spectrogram extrema are mostly understood empirically. In this work, we investigate spectrogram analysis via an examination of the stochastic, geometric and analytical properties of their level sets. This includes a comparative analysis of relevant spectrogram structures, with vs without the presence of signals coupled with Gaussian white noise. We obtain theorems demonstrating the efficacy of a spectrogram level sets based approach to the detection and estimation of signals, framed in a concrete inferential set-up. Exploiting these ideas as theoretical underpinnings, we propose a level sets based algorithm for signal analysis that is intrinsic to given spectrogram data. We substantiate the effectiveness of the algorithm by extensive empirical studies, and provide additional theoretical analysis to elucidate some of its key features. Our results also have theoretical implications for spectrogram zero based approaches to signal analysis.

中文翻译:

通过频谱图水平集的随机几何来估计和推断信号

频谱图是时频分析范式中信号的检测,估计和分析的基本工具。传统上,通过频谱图进行信号分析的方法是探究其峰值(即最大值),并辅之以对零或最小值的最新关注。特别是,最近的研究表明,在不同的几何形状中,高斯白噪声的Gabor谱图与高斯分析函数(abbrv。GAF)之间存在联系。然而,GAF的零集(或最大值或最小值)具有复杂的随机结构,这使得通过GAF对基于常规频谱图的技术进行直接理论分析变得困难。反过来,这些技术很大程度上依赖于空间数据分析中的统计可观察性,凭经验了解其频谱图极值的分布特性。在这项工作中,我们通过检查其水平集的随机,几何和分析特性来研究频谱图分析。这包括对相关频谱图结构的比较分析,有无信号与高斯白噪声耦合存在与否。我们获得了定理,该定理证明了以频谱图水平集为基础的方法在检测和估计信号方面的功效,并以具体的推论设置为框架。利用这些思想作为理论基础,我们提出了一种基于水平集的信号分析算法,该算法对于给定的频谱图数据而言是固有的。我们通过大量的经验研究证实了该算法的有效性,并提供其他理论分析,以阐明其一些主要功能。我们的结果对于基于频谱图零信号分析方法也具有理论意义。
更新日期:2021-05-07
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