The Continuous Wavelet Transform (CWT) is an important mathematical tool in signal\nprocessing, which is a linear time-invariant operator with causality and stability for a fixed scale and\nreal-life application. A novel and simple proof of the FFT-based fast method of linear convolution is\npresented by exploiting the structures of circulant matrix. After introducing Equivalent Condition\nof Time-domain and Frequency-domain Algorithms of CWT, a class of algorithms for continuous\nwavelet transform are proposed and analyzed in this paper, which can cover the algorithms in\nJLAB andWaveLab, as well as the other existing methods such as the cwt function in the toolbox of\nMATLAB. In this framework, two theoretical issues for the computation of CWT are analyzed. Firstly,\nedge effect is easily handled by using Equivalent Condition of Time-domain and Frequency-domain\nAlgorithms of CWT and higher precision is expected. Secondly, due to the fact that linear convolution\nexpands the support of the signal, which parts of the linear convolution are just the coefficients of\nCWT is analyzed by exploring the relationship of the filters of Frequency-domain and Time-domain\nalgorithms, and some generalizations are given. Numerical experiments are presented to further\ndemonstrate our analyses.
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