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

## TL;DR

This paper introduces a new family of hyperbolic chirplet wavelets inspired by Altes' log-periodic waveforms, enabling efficient detection of accelerating oscillations in complex systems, with applications in finance, engineering, and beyond.

## Contribution

It formalizes Altes waveforms as hyperbolic chirplets, enhancing multiresolution analysis and detection of critical phenomena in various fields.

## Key findings

- Development of flexible, admissible wavelets with good time-frequency localization
- Efficient implementation of the hyperbolic chirplet transform (HCT)
- Demonstration of applications in financial, mechanical, and biological systems

## Abstract

This work revisits a class of biomimetically inspired log-periodic waveforms first introduced by R.A. Altes in the 1970s for generalized target description. It was later observed that there is a close connection between such sonar techniques and wavelet decomposition for multiresolution analysis. Motivated by this, we formalize the original Altes waveforms as a family of hyperbolic chirplets suitable for the detection of accelerating time-series oscillations. The formalism results in a remarkably flexible set of wavelets with desirable properties of admissibility, regularity, vanishing moments, and time-frequency localization. These "Altes wavelets" also facilitate efficient implementation of the scale invariant hyperbolic chirplet transform (HCT). From a practical perspective, log-periodic oscillations with an acceleration towards criticality can serve as indicators of an incipient bifurcation. Such signals abound in nature, often as precursors to phase transitions in the non-linear dynamics of complex systems. For example, the authors' interest lies in automatic detection of the well documented phenomenon of log-periodic price dynamics during financial bubbles and preceding market crashes. However, the methodology presented here is more widely applicable in such diverse domains as prediction of critical failures in mechanical systems, and fault detection in electrical networks. Examples beyond failure diagnostics include animal species identification via call recordings, commercial \& military radar, and there are many more. A synthetic application is presented in this report for illustrative purposes.