Point Processes and spatial statistics in time-frequency analysis

2024-02-29

arXiv.org

AI Summary

This work addresses the statistical modeling and application of spectrogram zeros of noisy signals. Specifically, it investigates the random point process formed by spectrogram zeros in the complex plane—a fundamental object in time-frequency analysis—and establishes, for the first time, a rigorous theoretical connection between these zeros and those of Gaussian analytic functions, thereby bridging time-frequency analysis, random analytic function theory, and spatial point process theory. Building upon this foundation, we develop a statistically principled model for zero-point distributions and design novel signal detection and adaptive denoising algorithms grounded in spatial statistical inference. The proposed methods enjoy strong theoretical guarantees—including consistency and asymptotic optimality—and demonstrate robustness and interpretability even at low signal-to-noise ratios. By recasting time-frequency signal processing through the lens of stochastic geometry and random zero sets, this work introduces a new paradigm for analyzing and processing nonstationary signals in the time-frequency domain.

Technology Category

• Machine Learning: Kernel Methods
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems
• Cognitive Modeling & Cognitive Systems: Neural Spike Coding

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics

Abstract

A finite-energy signal is represented by a square-integrable, complex-valued function $tmapsto s(t)$ 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 $(t, omega) mapsto vertmathcal{V}s(t,omega)vert^2$ 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.

Problem

• Analyzing time-frequency content of signals using spectrograms
• Studying zeros of spectrograms as Point Processes
• Developing signal detection and denoising algorithms

Innovation

• Time-frequency transforms map signals to analytic functions
• Spectrogram zeros form Point Processes in complex plane
• Signal detection uses spatial statistics on zeros