A finite-energy signal is represented by a square-integrable, complex-valued function $t\mapsto s(t)$ of a real variable $t$, interpreted as time. Similarly, a noisy signal is represented by a random process. Time-frequency analysis, a subfield of signal processing, amounts to describing the temporal evolution of the frequency content of a signal. Loosely speaking, if $s$ is the audio recording of a musical piece, time-frequency analysis somehow consists in writing the musical score of the piece. Mathematically, the operation is performed through a transform $\mathcal{V}$, mapping $s \in L^2(\mathbb{R})$ onto a complex-valued function $\mathcal{V}s \in L^2(\mathbb{R}^2)$ of time $t$ and angular frequency $\omega$. The squared modulus $(t, \omega) \mapsto \vert\mathcal{V}s(t,\omega)\vert^2$ of the time-frequency representation is known as the spectrogram of $s$; in the musical score analogy, a peaked spectrogram at $(t_0,\omega_0)$ corresponds to a musical note at angular frequency $\omega_0$ localized at time $t_0$. More generally, the intuition is that upper level sets of the spectrogram contain relevant information about in the original signal. Hence, many signal processing algorithms revolve around identifying maxima of the spectrogram. In contrast, zeros of the spectrogram indicate perfect silence, that is, a time at which a particular frequency is absent. Assimilating $\mathbb{R}^2$ to $\mathbb{C}$ through $z = \omega + \mathrm{i}t$, this chapter focuses on time-frequency transforms $\mathcal{V}$ that map signals to analytic functions. The zeros of the spectrogram of a noisy signal are then the zeros of a random analytic function, hence forming a Point Process in $\mathbb{C}$. This chapter is devoted to the study of these Point Processes, to their links with zeros of Gaussian Analytic Functions, and to designing signal detection and denoising algorithms using spatial statistics.

翻译：有限能量信号由平方可积的复值函数 $t\mapsto s(t)$ 表示，其中实变量 $t$ 解释为时间。类似地，含噪信号由随机过程表示。时频分析作为信号处理的子领域，旨在描述信号频率内容随时间演变的特征。通俗而言，若 $s$ 为某音乐片段的音频记录，时频分析大致相当于为该片段谱写乐谱。数学上，该操作通过变换 $\mathcal{V}$ 实现，将 $s \in L^2(\mathbb{R})$ 映射为时间 $t$ 与角频率 $\omega$ 的复值函数 $\mathcal{V}s \in L^2(\mathbb{R}^2)$。时频表示模的平方 $(t, \omega) \mapsto \vert\mathcal{V}s(t,\omega)\vert^2$ 称为信号 $s$ 的谱图；在乐谱类比中，$(t_0,\omega_0)$ 处的峰值谱图对应于定位在时刻 $t_0$ 的角频率 $\omega_0$ 音符。更一般地，直观上谱图的上水平集包含原始信号的相关信息。因此，许多信号处理算法围绕谱图极大值的识别展开。相反，谱图的零点则指示完全静默，即特定频率在某一时刻缺失。通过 $z = \omega + \mathrm{i}t$ 将 $\mathbb{R}^2$ 等同于 $\mathbb{C}$，本章聚焦于将信号映射为解析函数的时频变换 $\mathcal{V}$。此时，含噪信号谱图的零点即为随机解析函数的零点，因此形成 $\mathbb{C}$ 上的点过程。本章致力于研究这些点过程及其与高斯解析函数零点的关联，并利用空间统计原理设计信号检测与去噪算法。