白噪声与高斯解析函数的时频变换
Time-frequency transforms of white noises and Gaussian analytic functions
摘要：
最近已将高斯解析函数（GAF）族与高斯白噪声的伽柏变换相关联[Bardenet等人，2017年]。这回答了Flandrin [2015]的开创性工作，Flandrin观察到白噪声的Gabor变换的零点具有非常规则的分布，并提出了基于频谱图零点的滤波算法。与GAF的数学联系提供了丰富的概率结果，以告知这样的信号处理过程的设计。本文系统地研究了信号的希尔伯特空间上的高斯白噪声的时频变换与广义自相关函数之间的联系。我们的主要观察是对（变换，GAF）和生成函数的经典正交多项式之间的概念对应。这种对应关系涵盖了一些经典的时频变换，如Gabor变换和Daubechies-Paul解析小波变换。它还推出了新的窗口离散傅里叶变换，映射白噪声的基本GAF。因此，所有这些变换可能对研究程序“用零滤波”感兴趣。我们还确定了GAF的零点是白噪声的Gabor变换的极值，并推导出它们的第一强度。此外，我们还讨论了定义白噪声及其在无限维希尔伯特空间上的变换的重要细节。最后，我们提供定量估计这些白噪声，这是实际的兴趣，当涉及到实现基于GAF的信号处理算法的有限维近似。
英文摘要：
A family of Gaussian analytic functions (GAFs) has recently been linked to the Gabor transform of white Gaussian noise [Bardenet et al., 2017]. This answered pioneering work by Flandrin [2015], who observed that the zeros of the Gabor transform of white noise had a very regular distribution and proposed filtering algorithms based on the zeros of a spectrogram. The mathematical link with GAFs provides a wealth of probabilistic results to inform the design of such signal processing procedures. In this paper, we study in a systematic way the link between GAFs and a class of time-frequency transforms of Gaussian white noises on Hilbert spaces of signals. Our main observation is a conceptual correspondence between pairs (transform, GAF) and generating functions for classical orthogonal polynomials. This correspondence covers some classical time-frequency transforms, such as the Gabor transform and the Daubechies-Paul analytic wavelet transform. It also unveils new windowed discrete Fourier transforms, which map white noises to fundamental GAFs. All these transforms may thus be of interest to the research program `filtering with zeros'. We also identify the GAF whose zeros are the extrema of the Gabor transform of the white noise and derive their first intensity. Moreover, we discuss important subtleties in defining a white noise and its transform on infinite dimensional Hilbert spaces. Finally, we provide quantitative estimates concerning the finite-dimensional approximations of these white noises, which is of practical interest when it comes to implementing signal processing algorithms based on GAFs.
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