Continuous wavelet transform: Difference between revisions

Revision as of 15:07, 23 June 2022

Revision as of 15:08, 23 June 2022

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:<math>0<C_\psi <\infty</math>

is called an admissible wavelet. An admissible wavelet implies that <math>\hat{\psi}(0) = 0</math>, so that an admissible wavelet must integrate to zero. To recover the original signal <math>x(t)</math>, the second inverse continuous wavelet transform can be exploited.

:<math>x(t)=\frac{1}{2\pi\overline\hat{\psi}(1)}\int_{-\infty0}^{\infty}\int_{-\infty}^{\infty} \frac{1}{a^2}X_w(a,b)\exp\left(i\frac{t-b}{a}\right)\, db\ da</math>

This inverse transform suggests that a wavelet should be defined as

:<math>\psi(t)=w(t)\exp(it) </math>