Exponential decay of scattering coefficients

TL;DR

This paper establishes a link between the decay of a signal's scattering coefficients and its Fourier transform, showing that high-order scattering coefficients are bounded by high-frequency energy, thus enhancing understanding of signal properties.

Contribution

It demonstrates that the energy in high-order scattering coefficients is bounded by high-frequency energy, relaxing previous wavelet admissibility conditions and extending prior results.

Findings

High-order scattering coefficients decay with high-frequency energy

The decay of scattering coefficients characterizes signal frequency content

Generalizes Mallat's results by relaxing wavelet conditions

Abstract

We study an aspect of the following general question: which properties of a signal can be characterized by its scattering transform? We show that the energy contained in high order scattering coefficients is upper bounded by the energy contained in the high frequencies of the signal. This result links the decay of the scattering coefficients of a signal with the decay of its Fourier transform. Additionally, it allows to generalize some results of Mallat (2012), by relaxing the admissibility condition on the wavelet family.