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Concept
Continuous wavelet transform
Summary
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 wavelets vary continuously.
The continuous wavelet transform of a function at a scale (a>0) 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 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 , the first inverse continuous wavelet transform can be exploited.
is the dual function of and
is admissible constant, where hat means Fourier transform operator. Sometimes, , then the admissible constant becomes
Traditionally, this constant is called wavelet admissible constant. A wavelet whose admissible constant satisfies
is called an admissible wavelet. An admissible wavelet implies that , so that an admissible wavelet must integrate to zero. To recover the original signal , the second inverse continuous wavelet transform can be exploited.
This inverse transform suggests that a wavelet should be defined as
where is a window. Such defined wavelet can be called as an analyzing wavelet, because it admits to time-frequency analysis. An analyzing wavelet is unnecessary to be admissible.
The scale factor either dilates or compresses a signal. When the scale factor is relatively low, the signal is more contracted which in turn results in a more detailed resulting graph. However, the drawback is that low scale factor does not last for the entire duration of the signal. On the other hand, when the scale factor is high, the signal is stretched out which means that the resulting graph will be presented in less detail. Nevertheless, it usually lasts the entire duration of the signal.
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