Shannon wavelet

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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.

Real Shannon wavelet[edit]

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| \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{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 - 1)-\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[edit]

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

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

References[edit]

  • S.G. Mallat, A Wavelet Tour of Signal Processing, Academic Press, 1999, ISBN 0-12-466606-X