Point Processes and spatial statistics in time-frequency analysis
Barbara Pascal, Rémi Bardenet
TL;DR
This work reframes time-frequency analysis around the zeros of spectrograms, treating them as a spatial Point Process linked to Gaussian Analytic Functions (GAF). By mapping time-frequency representations to analytic functions via transforms like the Bargmann transform, the zeros acquire tractable probabilistic structure (stationarity, isotropy, hyperuniformity) that supports zero-based detection and denoising through spatial statistics. The authors develop explicit tools—from Edelman-Kostlan intensities to Ripley-type statistics and hole probabilities—and propose practical zero-based Monte Carlo tests and algorithms for signal detection. This framework offers principled, robust procedures for extracting signals from noisy measurements and opens multiple avenues for extending analytic-function-based time-frequency methods to broader classes of transforms and windows.
Abstract
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 $ω$. The squared modulus $(t, ω) \mapsto \vert\mathcal{V}s(t,ω)\vert^2$ of the time-frequency representation is known as the spectrogram of $s$; in the musical score analogy, a peaked spectrogram at $(t_0,ω_0)$ corresponds to a musical note at angular frequency $ω_0$ 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 $z = ω+ \mathrm{i}t$, 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.