Continuous Wavelet Transform

A continuous wavelet transform (CWT) is used to divide a continuous-time function into wavelets. Unlike Fourier transform, the continuous wavelet transform possesses the ability to construct a time-frequency representation of a signal that offers very good time and frequency localization. In mathematics, the continuous wavelet transform of a continuous, square-integrable function at a scale and translational value is expressed by the following integral

where is a continuous function in both the time domain and the frequency domain called the mother wavelet and 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, inverse continuous wavelet transform can be exploited.

is the dual function of . And the dual function should satisfy

Sometimes, where

C_\psi = \frac{1}{2}\int_{-\infty}^{+\infty} \frac{\left| \hat \psi(\zeta) \right|^2}{\left| \zeta \right|} d\zeta

is called the admissibility constant and is the Fourier transform of . For a successful inverse transform, the admissibility constant has to satisfy the admissibility condition:

It is possible to show that the admissibility condition implies that, so that a wavelet must integrate to zero.

Read more about Continuous Wavelet Transform:  Mother Wavelet, Scaling Function, Scale Factor, Continuous Wavelet Transform Properties, Applications of The Wavelet Transform

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