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## Summary

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 Real Shannon wavelet

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

$\Psi ^{(\operatorname {Sha} )}(w)=\prod \left({\frac {w-3\pi /2}{\pi }}\right)+\prod \left({\frac {w+3\pi /2}{\pi }}\right).$ where the (normalised) gate function is defined by

$\prod (x):={\begin{cases}1,&{\mbox{if }}{|x|\leq 1/2},\\0&{\mbox{if }}{\mbox{otherwise}}.\\\end{cases}}$ The analytical expression of the real Shannon wavelet can be found by taking the inverse Fourier transform:

$\psi ^{(\operatorname {Sha} )}(t)=\operatorname {sinc} \left({\frac {t}{2}}\right)\cdot \cos \left({\frac {3\pi t}{2}}\right)$ or alternatively as

$\psi ^{(\operatorname {Sha} )}(t)=2\cdot \operatorname {sinc} (2t)-\operatorname {sinc} (t),$ where

$\operatorname {sinc} (t):={\frac {\sin {\pi t}}{\pi t}}$ is the usual sinc function that appears in Shannon sampling theorem.

This wavelet belongs to the $C^{\infty }$ -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:

$\phi ^{(Sha)}(t)={\frac {\sin \pi t}{\pi t}}=\operatorname {sinc} (t).$ ## Complex Shannon wavelet

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

$\psi ^{(CSha)}(t)=\operatorname {sinc} (t)\cdot e^{-j2\pi t}$ ,