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The wavelet transform is a transformation to basis functions that are localized in frequency (similar in that sense to Fourier-related transforms). As basis functions one uses wavelets. The big advantage over the Fourier transform is the temporal (or spatial) locality of the base functions (see also short time Fourier transform) and the smaller complexity (O(N) instead of O(N log N) for the fast Fourier transform (where N is the data size).
Important applications are:
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The continuous wavelet transform is defined as
The original function can be reconstructed with the inverse transform
Continuous wavelet transform (CWT)
where represents translation, represents scale and is the transforming function or mother wavelet.
where
is called the admissibility constant. For a successful inverse transform, the admissibility constant has to satisfy the admissibility condition:
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