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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.

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:

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

- ,

- S.G. Mallat,
*A Wavelet Tour of Signal Processing*, Academic Press, 1999, ISBN 0-12-466606-X - C.S. Burrus, R.A. Gopinath, H. Guo,
*Introduction to Wavelets and Wavelet Transforms: A Primer*, Prentice-Hall, 1988, ISBN 0-13-489600-9.