On two fundamental properties of the zeros of spectrograms of noisy signals

TL;DR

This paper investigates how the zeros of spectrograms of noisy signals are affected by noise and interference, revealing that zeros tend to outline the signal support and form detectable structures.

Contribution

It provides a formal mathematical analysis of zero patterns in noisy spectrograms, using intensity and Rouché's theorem to explain delineation and trapping effects.

Findings

Zeros delineate the support of signals in noisy spectrograms.

Interfering chirps create detectable zero structures even when nearly superimposed.

Mathematical arguments clarify the influence of SNR and interference on zero patterns.

Abstract

The spatial distribution of the zeros of the spectrogram is significantly altered when a signal is added to white Gaussian noise. The zeros tend to delineate the support of the signal, and deterministic structures form in the presence of interference, as if the zeros were trapped. While sophisticated methods have been proposed to detect signals as holes in the pattern of spectrogram zeros, few formal arguments have been made to support the delineation and trapping effects. Through detailed computations for simple toy signals, we show that two basic mathematical arguments, the intensity of zeros and Rouch\'e's theorem, allow discussing delineation and trapping, and the influence of parameters like the signal-to-noise ratio. In particular, interfering chirps, even nearly superimposed, yield an easy-to-detect deterministic structure among zeros.