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.

An Introduction to Frames

“…In addition, the filters have to be FIR and satisfy the so-called half sample delay condition, which implies that all of the filters have to be designed simultaneously. From this condition it also follows that the two highpass filters form an approximate Hilbert transform pair, and it thus makes sense to regard the outputs of the two trees as the real and imaginary parts of complex functions [145]. Different design solutions exist, amongst them the linear phase biorthogonal one and the quarter-shift one [115,147].…”

Steerable Filters Generated with the Hypercomplex Dual-Tree Wavelet Transform

“…An outline of Selesnick's filter design technique for designing biorthogonal wavelets [4] is hereby given. For a more detailed This project was supported by the Nanorobotics EPSRC Basic Technology grant GR/S85696/01 introduction to the dual-tree wavelet transform see Selesnick's joint publication with Kingsbury [1].…”

Changes in Phytoplankton Biomass in the Western Scheldt Estuary During the Period 1978–2006

“…argument of a complex number), similar to the amplitude and phase of, for example, a sine wave. We refer the interested reader to the specialized literature for more technical details (e.g., Percival and Walden 2000;Selesnick 2001Selesnick , 2002Whitcher and Craigmile 2005).…”