The chirplet transform: physical considerations

TLDR

The study examines a multidimensional parameter space derived from inner products of parameterizable chirp functions with a signal. The authors propose using quadratic chirp functions (q‑chirps) to create a parameter space that unifies the time‑frequency and time‑scale planes. They construct a time‑frequency‑scale volume that incorporates STFT and wavelet slices, adds shear axes, and defines a new q‑chirplet transform to generate signals in this multidimensional space. The resulting chirplets generalize wavelets by allowing 2‑D affine transformations—translations, dilations, rotations, and shears—in the time‑frequency plane, unlike traditional wavelets limited to 1‑D translations and dilations.

Abstract

We consider a multidimensional parameter space formed by inner products of a parameterizable family of chirp functions with a signal under analysis. We propose the use of quadratic chirp functions (which we will call q-chirps for short), giving rise to a parameter space that includes both the time-frequency plane and the time-scale plane as 2-D subspaces. The parameter space contains a "time-frequency-scale volume" and thus encompasses both the short-time Fourier transform (as a slice along the time and frequency axes) and the wavelet transform (as a slice along the time and scale axes). In addition to time, frequency, and scale, there are two other coordinate axes within this transform space: shear in time (obtained through convolution with a q-chirp) and shear in frequency (obtained through multiplication by a q-chirp). Signals in this multidimensional space can be obtained by a new transform, which we call the "q-chirplet transform" or simply the "chirplet transform". The proposed chirplets are generalizations of wavelets related to each other by 2-D affine coordinate transformations (translations, dilations, rotations, and shears) in the time-frequency plane, as opposed to wavelets, which are related to each other by 1-D affine coordinate transformations (translations and dilations) in the time domain only.