Title

Phases For Dyadic Orthonormal Wavelets

Abstract

We consider real-valued functions α(s) and wavelets ψ ∈ L2(ℝ) such that eiα(s)|ψ̂(s)| is the Fourier transform of a wavelet. Such a function α(s) is called an attainable phase for the wavelet ψ. It is known that for all multiresolution analysis (MRA) wavelets, the phase function α(s) = 1/2s is attainable, and any real function α(s) is attainable by any minimally-supported-frequency (MSF) wavelet. Besides this, very little is known in the literature about attainable phases for wavelets. We study the problem of determining functions which are attainable phases for some (non-MSF) wavelets. We prove that there exists a non-MSF wavelet for which there is no attainable "set-wise" linear phase. This answers a basic question about wavelet phases. Although we do not know whether for any irrational number a, as is attainable by some non-MSF wavelets, we show that there exist certain rational numbers a such that as is not attainable by any non-MSF wavelet. We also prove that there exists a large class of rational numbers a such that as is attainable by some non-MSF wavelets. We examine the relationship between different classes of wavelets admitting linear phases. In particular we present an example of a non-MSF wavelet which is not an MRA wavelet but admits linear phase 1/2s. © 2002 American Institute of Physics.

Publication Date

5-1-2002

Publication Title

Journal of Mathematical Physics

Volume

43

Issue

5

Number of Pages

2690-2706

Document Type

Article

Personal Identifier

scopus

DOI Link

https://doi.org/10.1063/1.1462416

Socpus ID

0035981823 (Scopus)

Source API URL

https://api.elsevier.com/content/abstract/scopus_id/0035981823

This document is currently not available here.

Share

COinS