Continuous wavelet transform: Difference between revisions

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{{Use dmy dates|date=July 2013}}

[[File:Continuous wavelet transform.svg|thumb|320px|right|Continuous [[wavelet]] transform of frequency breakdown signal. Used [[symlet]] with 5 vanishing moments.]]

In [[mathematics]], the '''continuous wavelet transform''' (CWT)''' is a formal (i.e., non-numerical) tool that provides an overcomplete representation of a signal by letting the translation and scale parameter of the [[Wavelet|wavelets]] vary continuously.

The continuous wavelet transform of a function <math>x(t)</math> at a scale (a>0) <math>a\in\mathbb{R^{+*}}</math> and translational value <math>b\in\mathbb{R}</math> is expressed by the following integral

:<math display="block">X_w(a,b)=\frac{1}{|a|^{1/2}} \int_{-\infty}^{\infty} x(t)\overline\psi\left(\frac{t-b}{a}\right)\, dt</math>

where <math>\psi(t)</math> is a continuous function in both the time ___domain and the frequency ___domain called the mother wavelet and the overline represents operation of [[complex conjugate]]. The main purpose of the mother wavelet is to provide a source function to generate the daughter wavelets which are simply the translated and scaled versions of the mother wavelet. To recover the original signal <math>x(t)</math>, the first inverse continuous wavelet transform can be exploited.

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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 [[Fastfast Fourier Transformtransform]] (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>|X_w(a,b)|^2</math> .

==Applications of the wavelet transform==