Wavelets and their Applications

Chapter 6

A Short 1D Illustrated Handbook

6.1. Introduction

Interesting signals generally exhibit numerous non-stationary characteristics that constitute a considerable part of the information contained in a series: drifts, trends, breakdowns, beginnings and ends of events, transitory phenomena. Most classical mathematical approaches widely used are well adapted to the study of stationary processes. Among others we may cite the spectral approach linked to the Fourier transform and that resulting from the ARMA processes [AZE 84].

This chapter takes you through the wavelet decomposition of signals and compiles “illustrated handbooks” associated with classical situations frequently encountered in statistics or signal processing. Its application is pedagogical as well as practical, and is based on the analysis of several elementary examples to learn how to recognize forms. The signals are presented followed by tests and trials, and we comment on the results. This chapter consists of three parts.

The first part¹, which is also the longest, concerns discrete signal analysis. It is intentionally over-detailed and exploits the capacity of wavelets to decompose a signal into a sum of approximation and detail signals. The topics dealt with relate to the identification of trends, periodic signals, noises, breakdowns and discontinuities. The commented analysis of 10 simulated signals constitutes the first “illustrated handbook”. It is supplemented by an examination of two real signals: an ...