# Une ou deux composantes ? La r\'eponse de la diffusion en ondelettes

## TL;DR

This paper investigates how wavelet scattering networks can distinguish between one or multiple components in a signal, providing criteria for interference and methods to analyze complex Fourier series representations.

## Contribution

It introduces a simple numerical criterion for component interference and generalizes the framework to multiple sine waves using deep wavelet networks.

## Key findings

- Renormalizing second-order nodes helps identify psychoacoustic interference.
- A wavelet network of depth log2 N characterizes Fourier series components.
- The method achieves invariance to frequency shifts and phase changes.

## Abstract

With the aim of constructing a biologically plausible model of machine listening, we study the representation of a multicomponent stationary signal by a wavelet scattering network. First, we show that renormalizing second-order nodes by their first-order parents gives a simple numerical criterion to establish whether two neighboring components will interfere psychoacoustically. Secondly, we generalize the `one or two components' framework to three sine waves or more, and show that a network of depth $M = \log_2 N$ suffices to characterize the relative amplitudes of the first $N$ terms in a Fourier series, while enjoying properties of invariance to frequency transposition and component-wise phase shifts.