Signal Analysis based on Complex Wavelet Signs

TL;DR

This paper introduces a novel signal analysis method using the sign of complex wavelet coefficients, called the signature, which detects salient features like jumps and cusps by analyzing the behavior of wavelet signs.

Contribution

The paper defines the signature based on complex wavelet signs, characterizes its behavior at regular and salient points, and demonstrates its invariance properties and applicability to various signals.

Findings

Signature is zero at regular points

Non-zero at jumps or cusps

Vanishing for white noise and Brownian motion

Abstract

We propose a signal analysis tool based on the sign (or the phase) of complex wavelet coefficients, which we call a signature. The signature is defined as the fine-scale limit of the signs of a signal's complex wavelet coefficients. We show that the signature equals zero at sufficiently regular points of a signal whereas at salient features, such as jumps or cusps, it is non-zero. At such feature points, the orientation of the signature in the complex plane can be interpreted as an indicator of local symmetry and antisymmetry. We establish that the signature rotates in the complex plane under fractional Hilbert transforms. We show that certain random signals, such as white Gaussian noise and Brownian motions, have a vanishing signature. We derive an appropriate discretization and show the applicability to signal analysis.