Shannon wavelet


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. The Haar and sinc systems are Fourier duals of each other.

Real Shannon wavelet

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

Complex Shannon wavelet

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