IEEE Transactions on Signal Processing vol. 61 no. 1, pp. 28–37 (2013)

Fast FIR algorithms for the continuous wavelet transform from constrained least squares

George M. Leigh, Member, IEEE

Copyright © 2012 IEEE. Personal use of this material is permitted. However, permission to use this material for any other purposes must be obtained from the IEEE by sending a request to This email address is being protected from spambots. You need JavaScript enabled to view it. .

This work is part of AMIRA International Project P843, supported by the Australian Research Council and mining industry participants Anglo Platinum, AngloGold Ashanti, Barrick Gold, BHP Billiton, Codelco, Newcrest Mining, Newmont Mining, OZ Minerals, Peñoles, Rio Tinto, Teck Cominco, Vale, Vale Inco, Xstrata Copper, Datamine Group, Geotek, Golder Associates, ioGlobal, and Metso Minerals.

G. M. Leigh conducted this work at the University of Queensland, Sustainable Minerals Institute, Julius Kruttschnitt Mineral Research Centre, Isles Rd, Indooroopilly, Qld 4068, Australia. He is currently with the Department of Agriculture, Fisheries and Forestry, PO Box 6097, St Lucia, Qld 4067, Australia (phone: +61-7-3255-4532; fax: +61-7-3346-2167; e-mail: This email address is being protected from spambots. You need JavaScript enabled to view it. ).

Abstract—New algorithms for the continuous wavelet transform are developed that are easy to apply, each consisting of a single-pass finite impulse response (FIR) filter, and several times faster than the fastest existing algorithms. The single-pass filter algorithm, named WT-FIR-1, is made possible by applying constraint equations to least-squares estimation of filter coefficients, which removes the need for separate low-pass and high-pass filters. Non-dyadic two-scale relations are developed and it is shown that filters based on them can work more efficiently than dyadic ones.Example applications to the Mexican hat wavelet are presented.

Index Terms—Algorithm design and analysis, Continuous wavelet transforms, Finite impulse response filter, Signal processing algorithms

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