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Algorithms 2018, 11(3), 24; https://doi.org/10.3390/a11030024

A Class of Algorithms for Continuous Wavelet Transform Based on the Circulant Matrix

1
Department of Mathematics, Jinggangshan University, Ji’an 343009, Jiangxi Province, China
2
Department of Mathematics, Tongji University, Shanghai 200092, China
*
Author to whom correspondence should be addressed.
Received: 20 November 2017 / Revised: 6 February 2018 / Accepted: 22 February 2018 / Published: 27 February 2018
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Abstract

The Continuous Wavelet Transform (CWT) is an important mathematical tool in signal processing, which is a linear time-invariant operator with causality and stability for a fixed scale and real-life application. A novel and simple proof of the FFT-based fast method of linear convolution is presented by exploiting the structures of circulant matrix. After introducing Equivalent Condition of Time-domain and Frequency-domain Algorithms of CWT, a class of algorithms for continuous wavelet transform are proposed and analyzed in this paper, which can cover the algorithms in JLAB and WaveLab, as well as the other existing methods such as the c w t function in the toolbox of MATLAB. In this framework, two theoretical issues for the computation of CWT are analyzed. Firstly, edge effect is easily handled by using Equivalent Condition of Time-domain and Frequency-domain Algorithms of CWT and higher precision is expected. Secondly, due to the fact that linear convolution expands the support of the signal, which parts of the linear convolution are just the coefficients of CWT is analyzed by exploring the relationship of the filters of Frequency-domain and Time-domain algorithms, and some generalizations are given. Numerical experiments are presented to further demonstrate our analyses. View Full-Text
Keywords: circulant matrix; continuous wavelet transform; linear convolution; circular convolution; FFT circulant matrix; continuous wavelet transform; linear convolution; circular convolution; FFT
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Yi, H.; Xin, S.-Y.; Yin, J.-F. A Class of Algorithms for Continuous Wavelet Transform Based on the Circulant Matrix. Algorithms 2018, 11, 24.

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