Shannon wavelet

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Shannon wavelet or sinc wavelet

Two kinds of Shannon wavelets can be implemented:

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

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)%2B\prod \left( \frac {w%2B 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 - \frac{1}{2}),

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

References