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| In [[functional analysis]], a '''Shannon wavelet''' may be either of [[real number|real]] or [[complex number|complex]] type.
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| Signal analysis by ideal [[bandpass filter]]s defines a decomposition known as Shannon wavelets (or sinc wavelets). The Haar and sinc systems are Fourier duals of each other.
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| == Real Shannon wavelet ==
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| [[File:Wavelet Shan.svg|thumb|right|Real Shannon wavelet]]
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| The [[Fourier transform]] of the Shannon mother wavelet is given by:
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| : <math> \Psi^{(\operatorname{Sha}) }(w) = \prod \left( \frac {w- 3 \pi /2} {\pi}\right)+\prod \left( \frac {w+ 3 \pi /2} {\pi}\right). </math>
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| where the (normalised) [[gate function]] is defined by | |
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| : <math> \prod ( x):=
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| \begin{cases}
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| 1, & \mbox{if } {|x| \le 1/2}, \\
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| 0 & \mbox{if } \mbox{otherwise}. \\
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| \end{cases} </math>
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| The analytical expression of the real Shannon wavelet can be found by taking the inverse [[Fourier transform]]:
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| : <math> \psi^{(\operatorname{Sha}) }(t) = \operatorname{sinc} \left( \frac {t} {2}\right)\cdot \cos \left( \frac {3 \pi t} {2}\right)</math>
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| or alternatively as
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| : <math> \psi^{(\operatorname{Sha})}(t)=2 \cdot \operatorname{sinc}(2t - 1)-\operatorname{sinc}(t), </math>
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| where
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| : <math>\operatorname{sinc}(t):= \frac {\sin {\pi t}} {\pi t}</math>
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| is the usual [[sinc function]] that appears in [[Shannon sampling theorem]].
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| This wavelet belongs to the <math>C^\infty</math>-class of [[Smooth function#Differentiability classes|differentiability]], but it decreases slowly at infinity and has no [[Support_(mathematics)#Compact_support|bounded support]], since [[band-limited]] signals cannot be time-limited.
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| The [[Wavelet#Scaling_function|scaling function]] for the Shannon MRA (or ''Sinc''-MRA) is given by the sample function:
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| : <math>\phi^{(Sha)}(t)= \frac {\sin \pi t} {\pi t} = \operatorname{sinc}(t).</math>
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| == Complex Shannon wavelet ==
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| In the case of [[complex number|complex]] continuous wavelet, the Shannon wavelet is defined by
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| :<math> \psi^{(CSha) }(t)=\operatorname{sinc}(t).e^{-j2 \pi t}</math>,
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| == References ==
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| * S.G. Mallat, ''A Wavelet Tour of Signal Processing'', Academic Press, 1999, ISBN 0-12-466606-X
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| * [[C. Sidney Burrus|C.S. Burrus]], R.A. Gopinath, H. Guo, ''Introduction to Wavelets and Wavelet Transforms: A Primer'', Prentice-Hall, 1988, ISBN 0-13-489600-9.
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| [[Category:Continuous wavelets]]
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| [[Category:Functional analysis]]
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Andrew Berryhill is what his spouse enjoys to call him and he totally digs that name. The preferred hobby for him and his children is fashion and he'll be beginning some thing else along with it. He is an order clerk and it's some thing he really appreciate. Kentucky is exactly where I've usually been living.
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