# 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 (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:

${\displaystyle \Psi ^{(\operatorname {Sha} )}(w)=\prod \left({\frac {w-3\pi /2}{\pi }}\right)+\prod \left({\frac {w+3\pi /2}{\pi }}\right).}$

where the (normalised) gate function is defined by

${\displaystyle \prod (x):={\begin{cases}1,&{\mbox{if }}{|x|\leq 1/2},\\0&{\mbox{if }}{\mbox{otherwise}}.\\\end{cases}}}$

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

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

or alternatively as

${\displaystyle \psi ^{(\operatorname {Sha} )}(t)=2\cdot \operatorname {sinc} (2t-1)-\operatorname {sinc} (t),}$

where

${\displaystyle \operatorname {sinc} (t):={\frac {\sin {\pi t}}{\pi t}}}$

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

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

${\displaystyle \phi ^{(Sha)}(t)={\frac {\sin \pi t}{\pi t}}=\operatorname {sinc} (t).}$

## Complex Shannon wavelet

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

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

## References

• 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.