Shannon wavelet

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1 Shannon wavelet or sinc wavelet
2 Real Shannon wavelet
3 Complex Shannon wavelet
4 References

[edit] Shannon wavelet or sinc wavelet

Two kinds of Shannon wavelets can be implemented:

Real Shannon wavelet
Complex Shannon wavelet

The signal analysis by ideal pass-band filters define a decomposition known as Shannon wavelets (or sinc wavelets). The Haar and sinc systems are Fourier duals of each other.

[edit] Real Shannon wavelet

The spectrum 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

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

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

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

or alternatively as

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

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 $C^\infty$ -class, but it decreases slowly at infinity and has no bounded support (see Function of compact support), since band-limited signals cannot be time-limited.