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Continuous wavelet transform

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 x ( t ) {displaystyle x(t)} at a scale (a>0) a ∈ R + ∗ {displaystyle ain mathbb {R^{+*}} } and translational value b ∈ R {displaystyle bin mathbb {R} } is expressed by the following integral where ψ ( t ) {displaystyle psi (t)} 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 x ( t ) {displaystyle x(t)} , the first inverse continuous wavelet transform can be exploited. ψ ~ ( t ) {displaystyle { ilde {psi }}(t)} is the dual function of ψ ( t ) {displaystyle psi (t)} and is admissible constant, where hat means Fourier transform operator. Sometimes, ψ ~ ( t ) = ψ ( t ) {displaystyle { ilde {psi }}(t)=psi (t)} , then the admissible constant becomes

[ "Discrete wavelet transform", "Wavelet packet decomposition" ]
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