Shannon wavelet

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In functional analysis, the Shannon wavelet (or sinc wavelets) is a decomposition that is defined by signal analysis by ideal bandpass filters. Shannon wavelet may be either of real or complex type.

Shannon wavelet is not well-localized (noncompact) in the time domain, but its Fourier transform is band-limited (compact support). Hence Shannon wavelet has poor time localization but has good frequency localization. These characteristics are in stark contrast to those of the Haar wavelet. The Haar and sinc systems are Fourier duals of each other.

## Contents

* 1 Definition
* 1.1 Scaling function
* 1.2 Mother wavelet
* 1.3 Property of mother wavelet and scaling function
* 2 Reconstruction of a Function by Shannon Wavelets
* 3 Real Shannon wavelet
* 4 Complex Shannon wavelet
* 5 References

## Definition

Sinc function is the starting point for the definition of the Shannon wavelet.

### Scaling function

First, we define the scaling function to be the sinc function.

ϕ(Sha)(t):=sin⁡πtπt=sinc⁡(t).

And define the dilated and translated instances to be

ϕkn(t):=2n/2ϕ(Sha)(2nt−k)

where the parameter n,k means the dilation and the translation for the wavelet respectively.

Then we can derive the Fourier transform of the scaling function:

Φ(Sha)(ω)=12πΠ(ω2π)={12π,if |ω|≤π,0if otherwise. where the (normalised) gate function is defined by

Π(x):={1,if |x|≤1/2,0if otherwise. Also for the dilated and translated instances of scaling function: Φkn(ω)=2−n/22πe−iω(k+1)/2nΠ(ω2n+1π)

### Mother wavelet

Use Φ(Sha) and multiresolution approximation we can derive the Fourier transform of the Mother wavelet:

Ψ(Sha)(ω)=12πe−iω(Π(ωπ−32)+Π(ωπ+32))

And the dilated and translated instances:

Ψkn(ω)=2−n/22πe−iω(k+1)/2n(Π(ω2nπ−32)+Π(ω2nπ+32))

Then the shannon mother wavelet function and the family of dilated and translated instances can be obtained by the inverse Fourier transform:

ψ(Sha)(t)=sin⁡π(t−(1/2))−sin⁡2π(t−(1/2))π(t−1/2)=sinc⁡(t−12)−2sinc⁡(2(t−12))

ψkn(t)=2n/2ψ(Sha)(2nt−k)

### Property of mother wavelet and scaling function

* Mother wavelets are orthonormal, namely,

<ψkn(t),ψhm(t)>=δnmδhk={1,if h=k and n=m0,otherwise

* The translated instances of scaling function at level n=0 are orthogonal

<ϕk0(t),ϕh0(t)>=δkh

* The translated instances of scaling function at level n=0 are orthogonal to the mother wavelets

<ϕk0(t),ψhm(t)>=0

* Shannon wavelets has an infinite number of vanishing moments.

## Reconstruction of a Function by Shannon Wavelets

Suppose f(x)∈L2(ℝ) such that supp⁡FT⁡{f}⊂[−π,π] and for any dilation and the translation parameter n,k,

|∫−∞∞f(t)ϕk0(t)dt|<∞, |∫−∞∞f(t)ψkn(t)dt|<∞

Then

f(t)=∑k=∞∞αkϕk0(t) is uniformly convergent, where αk=f(k)

## Real Shannon wavelet

Real Shannon wavelet

The Fourier transform of the Shannon mother wavelet is given by:

Ψ(Sha⁡)(w)=∏(w−3π/2π)+∏(w+3π/2π).

where the (normalised) gate function is defined by

∏(x):={1,if |x|≤1/2,0if otherwise.

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

ψ(Sha⁡)(t)=sinc⁡(t2)⋅cos⁡(3πt2)

or alternatively as

ψ(Sha⁡)(t)=2⋅sinc⁡(2t)−sinc⁡(t),

where

sinc⁡(t):=sin⁡πtπt

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

This wavelet belongs to the C∞-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:

ϕ(Sha)(t)=sin⁡πtπt=sinc⁡(t).

## Complex Shannon wavelet

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

ψ(CSha)(t)=sinc⁡(t)⋅e−2πit,

## 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.

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