Compactly Supported One-cyclic Wavelets Derived from Beta Distributions

TL;DR

This paper introduces a new class of one-cyclic, compactly supported wavelets derived from beta distributions, offering a soft, parameter-tunable alternative to Haar wavelets with potential applications in signal analysis.

Contribution

It presents a novel construction of beta-based wavelets with one cycle, including explicit formulas and spectral analysis, expanding the toolkit for wavelet-based signal processing.

Findings

Beta wavelets have explicit expressions and spectra.

They are soft, parameter-tunable variants of Haar wavelets.

The wavelets are supported by the Central Limit Theorem for compact signals.

Abstract

New continuous wavelets of compact support are introduced, which are related to the beta distribution. They can be built from probability distributions using 'blur'derivatives. These new wavelets have just one cycle, so they are termed unicycle wavelets. They can be viewed as a soft variety of Haar wavelets whose shape is fine-tuned by two parameters a and b. Close expressions for beta wavelets and scale functions as well as their spectra are derived. Their importance is due to the Central Limit Theorem applied for compactly supported signals.