Wavelets for Elliptical Waveguide Problems

TL;DR

This paper introduces new elliptic cylindrical wavelets based on Mathieu differential equations, enabling customizable filter notches with potential applications in optics, microwaves, and electromagnetism.

Contribution

It presents a novel wavelet construction method using Mathieu equations, linking analyzing filters to Floquet's solutions for improved design flexibility.

Findings

Transfer functions relate to Mathieu equations with odd characteristic exponents.

Number of filter notches can be easily designed.

Wavelets have potential applications in optics, microwaves, and electromagnetism.

Abstract

New elliptic cylindrical wavelets are introduced, which exploit the relationship between analysing filters and Floquet's solution of Mathieu differential equations. It is shown that the transfer function of both multiresolution filters is related to the solution of a Mathieu equation of odd characteristic exponent. The number of notches of these analysing filters can be easily designed. Wavelets derived by this method have potential application in the fields of optics, microwaves and electromagnetism.