Shannon wavelet

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In functional analysis, a Shannon wavelet may be either of real or complex type. Signal analysis by ideal bandpass filters defines a decomposition known as Shannon wavelets (or sinc wavelets). The Haar and sinc systems are Fourier duals of each other.

Real Shannon wavelet[edit]

Real Shannon wavelet

The Fourier transform of the Shannon mother wavelet is given by:

where the (normalised) gate function is defined by

The analytical expression of the real Shannon wavelet can be found by taking the inverse Fourier transform:

or alternatively as

where

is the usual sinc function that appears in Shannon sampling theorem.

This wavelet belongs to the -class of differentiability, but it decreases slowly at infinity and has no bounded support, since band-limited signals cannot be time-limited.

The scaling function for the Shannon MRA (or Sinc-MRA) is given by the sample function:

Complex Shannon wavelet[edit]

In the case of complex continuous wavelet, the Shannon wavelet is defined by

,

References[edit]