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

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.

The interaction of fundamental torsional guided waves from axial and oblique defects in pipes

A quantitative study of the interaction of the T(0,1) torsional mode with axial and oblique defects in a pipe is presented in this paper. A mode decomposition technique employing the chirplet transform is used to separate the multimodal signals reflected from the defects. Reflection signals are obtained from experiments on a carbon steel pipe. The influence of the crack length and inclination angle on the reflection is investigated. The reflection from an axial defect is found to consist of a series of wave pulses with gradually decaying amplitude. The results show that the reflection coefficient of an axial crack initially increases with the crack length but finally reaches an oscillating regime. Furthermore, for an oblique crack, it is revealed that the reflection coefficient is linearly dependent on the equivalent circumferential extent of the defect and is independent of the axial length.