New physical wavelet 'Gaussian Wave Packet'

TL;DR

This paper introduces a new physical wavelet derived from an exact solution of the wave equation, characterized by explicit formulas, exponential localization, and directional properties, useful for continuous wavelet analysis.

Contribution

It presents a novel multidimensional mother wavelet based on a physical wave solution, expanding the tools available for wavelet analysis with explicit formulas and localization.

Findings

Wavelet is explicitly formulated and exponentially localized.

Wavelet behaves asymptotically like Morlet wavelet for large parameters.

Wavelet can be interpreted as a sum of advanced and retarded fields.

Abstract

An exact solution of the homogeneous wave equation, which was found previously, is treated from the point of view of continuous wavelet analysis (CWA). If time is a fixed parameter, the solution represents a new multidimensional mother wavelet for the CWA. Both the wavelet and its Fourier transform are given by explicit formulas and are exponentially localized. The wavelet is directional. The widths of the wavelet and the uncertainty relation are investigated numerically. If a certain parameter is large, the wavelet behaves asymptotically as the Morlet wavelet. The solution is a new physical wavelet in the definition of Kaiser, it may be interpreted as a sum of two parts: an advanced and a retarded part, both being fields of a pulsed point source moving at a speed of wave propagation along a straight line in complex space-time.