The design of approximate Hilbert transform pairs of wavelet bases

Ivan W. Selesnick

doi:10.1109/78.995070

The design of approximate Hilbert transform pairs of wavelet bases

Ivan W. Selesnick

Electrical and Computer Engineering

Research output: Contribution to journal › Article › peer-review

Abstract

Several authors have demonstrated that significant improvements can be obtained in wavelet-based signal processing by utilizing a pair of wavelet transforms where the wavelets form a Hilbert transform pair. This paper describes design procedures, based on spectral factorization, for the design of pairs of dyadic wavelet bases where the two wavelets form an approximate Hilbert transform pair. Both orthogonal and biorthogonal FIR solutions are presented, as well as IIR solutions. In each case, the solution depends on an allpass filter having a flat delay response. The design procedure allows for an arbitrary number of vanishing wavelet moments to be specified. A Matlab program for the procedure is given, and examples are also given to illustrate the results.

Original language	English (US)
Pages (from-to)	1144-1152
Number of pages	9
Journal	IEEE Transactions on Signal Processing
Volume	50
Issue number	5
DOIs	https://doi.org/10.1109/78.995070
State	Published - May 2002

Keywords

Dual-tree complex wavelet transform
Hilbert transform
Wavelet transforms

ASJC Scopus subject areas

Signal Processing
Electrical and Electronic Engineering

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10.1109/78.995070

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title = "The design of approximate Hilbert transform pairs of wavelet bases",

abstract = "Several authors have demonstrated that significant improvements can be obtained in wavelet-based signal processing by utilizing a pair of wavelet transforms where the wavelets form a Hilbert transform pair. This paper describes design procedures, based on spectral factorization, for the design of pairs of dyadic wavelet bases where the two wavelets form an approximate Hilbert transform pair. Both orthogonal and biorthogonal FIR solutions are presented, as well as IIR solutions. In each case, the solution depends on an allpass filter having a flat delay response. The design procedure allows for an arbitrary number of vanishing wavelet moments to be specified. A Matlab program for the procedure is given, and examples are also given to illustrate the results.",

keywords = "Dual-tree complex wavelet transform, Hilbert transform, Wavelet transforms",

author = "Selesnick, {Ivan W.}",

note = "Funding Information: Manuscript received December 6, 2000; revised January 7, 2002. This work was supported in part by the National Science Foundation under CAREER Grant CCR-987452. The associate editor coordinating the review of this paper and approving it for publication was Dr. Ta-Hsin Li. The author is with the Electrical and Computer Engineering, Polytechnic University, Brooklyn, NY 11201 USA (e-mail: selesi@taco.poly.edu). Publisher Item Identifier S 1053-587X(02)03148-3.",

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AB - Several authors have demonstrated that significant improvements can be obtained in wavelet-based signal processing by utilizing a pair of wavelet transforms where the wavelets form a Hilbert transform pair. This paper describes design procedures, based on spectral factorization, for the design of pairs of dyadic wavelet bases where the two wavelets form an approximate Hilbert transform pair. Both orthogonal and biorthogonal FIR solutions are presented, as well as IIR solutions. In each case, the solution depends on an allpass filter having a flat delay response. The design procedure allows for an arbitrary number of vanishing wavelet moments to be specified. A Matlab program for the procedure is given, and examples are also given to illustrate the results.

KW - Dual-tree complex wavelet transform

KW - Hilbert transform

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The design of approximate Hilbert transform pairs of wavelet bases

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Keywords

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