The Altes Family of Log-Periodic Chirplets and the Hyperbolic Chirplet Transform

Donnacha Daly,Didier Sornette

doi:10.3390/sym13101922

Donnacha Daly, Didier Sornette

Open Access

https://doi.org/10.3390/sym13101922

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Abstract

This work revisits a class of biomimetically inspired waveforms introduced by R.A. Altes in the 1970s for use in sonar detection. Similar to the chirps used for echolocation by bats and dolphins, these waveforms are log-periodic oscillations, windowed by a smooth decaying envelope. Log-periodicity is associated with the deep symmetry of discrete scale invariance in physical systems. Furthermore, there is a close connection between such chirping techniques, and other useful applications such as wavelet decomposition for multi-resolution analysis. Motivated to uncover additional properties, we propose an alternative, simpler parameterisation of the original Altes waveforms. From this, it becomes apparent that we have a flexible family of hyperbolic chirps suitable for the detection of accelerating time-series oscillations. The proposed formalism reveals the original chirps to be a set of admissible wavelets with desirable properties of regularity, infinite vanishing moments and time-frequency localisation. As they are self-similar, these “Altes chirplets” allow efficient implementation of the scale-invariant hyperbolic chirplet transform (HCT), whose basis functions form hyperbolic curves in the time-frequency plane. Compared with the rectangular time-frequency tilings of both the conventional wavelet transform and the short-time Fourier transform, the HCT can better facilitate the detection of chirping signals, which are often the signature of critical failure in complex systems. A synthetic example is presented to illustrate this useful application of the HCT.

Highlights

It has been noted that localisationefficiency should be sacrificed for chirping flexibility, and instead we seek parameters that allow wide-band chirping for a selection of chirp-rates with minimised sampling rate and Fourier transform size
The short-time Fourier transform (STFT) picks out the sinusoid, which has been injected at ω = π3
Assuming we are only interested in detecting log-periodicities, it is clear that the STFT is relatively poor, as is to be expected from its constant time-resolution at all frequencies

Summary

Introduction

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. Over the course of the 1970s, Richard A. Altes developed the theory behind a new family of waveforms with optimal Doppler tolerance for sonar applications. Inspired by mammalian acoustic echo-location calls such as those of bats and dolphins, their design evolved to a set of carefully parameterised hyperbolic chirps with useful time-frequency (TF) localisation properties [1,2,3,4,5]. Patrick Flandrin and their colleagues at the French national center for scientific research (CNRS) exposed the close mathematical parallels between sonar-based target description using these chirps, and multi-resolution wavelet analysis [6]. The current work extends these previous, and somewhat neglected ideas, with three key contributions

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