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
Copyright Status
Unknown
Socpus ID
0035981823 (Scopus)
Source API URL
https://api.elsevier.com/content/abstract/scopus_id/0035981823
STARS Citation
Gu, Qing and Han, Deguang, "Phases For Dyadic Orthonormal Wavelets" (2002). Scopus Export 2000s. 2580.
https://stars.library.ucf.edu/scopus2000/2580