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m →Continuous wavelet transform properties: Replaced time-like \sigma convention with frequency-like parameter, to better match convention used in Morlet wavelet. |
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==Continuous wavelet transform properties==
In definition, the continuous wavelet transform is a [[convolution]] of the input data sequence with a set of functions generated by the mother wavelet. The convolution can be computed by using a [[fast Fourier transform]] (FFT) algorithm. Normally, the output <math>X_w(a,b)</math> is a real valued function except when the mother wavelet is complex. A complex mother wavelet will convert the continuous wavelet transform to a complex valued function. The power spectrum of the continuous wavelet transform can be represented by <math>\frac{1}{a}|X_w(a,b)|^2</math> <ref>{{cite journal |last1=Torrence |first1=Christopher |last2=Compo |first2=Gilbert |title=A Practical Guide to Wavelet Analysis |journal=Bulletin of the American Meteorological Society |date=1998 |volume=79 |issue=1 |pages=61-78 |access-date=29 April 2022}}</ref><ref>{{cite journal |last1=Liu |first1=Yonggang |title=Rectification of the Bias in the Wavelet Power Spectrum |journal=JOURNAL OF ATMOSPHERIC AND OCEANIC TECHNOLOGY |date=December 2007 |volume=24 |pages=2093-2102 |access-date=29 April 2022}}</ref>.
[[File:Wavelet scale sweep for FM signal.gif|thumb|300px|Visualizing the effect of changing a [[Morlet wavelet|Morlet wavelet's]] <math>\sigma</math> parameter, which interpolates between the original time-series and a [[Fourier transform]]. Here, a [[Frequency modulation|frequency-modulated]] tone (plus noise) is analyzed; <math>1/\sigma</math> is adjusted from 1 to 200, in steps of unity.]]
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