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.
Definition [ edit ]
Sinc function is the starting point for the definition of the Shannon wavelet.
Scaling function [ edit ]
First, we define the scaling function to be the sinc function.
ϕ
(Sha)
(
t
)
:=
sin
π
t
π
t
=
sinc
(
t
)
.
{\displaystyle \phi ^{\text{(Sha)}}(t):={\frac {\sin \pi t}{\pi t}}=\operatorname {sinc} (t).}
And define the dilated and translated instances to be
ϕ
k
n
(
t
)
:=
2
n
/
2
ϕ
(Sha)
(
2
n
t
−
k
)
{\displaystyle \phi _{k}^{n}(t):=2^{n/2}\phi ^{\text{(Sha)}}(2^{n}t-k)}
where the parameter
n
,
k
{\displaystyle 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
)
=
1
2
π
Π
(
ω
2
π
)
=
{
1
2
π
,
if
|
ω
|
≤
π
,
0
if
otherwise
.
{\displaystyle \Phi ^{\text{(Sha)}}(\omega )={\frac {1}{2\pi }}\Pi ({\frac {\omega }{2\pi }})={\begin{cases}{\frac {1}{2\pi }},&{\mbox{if }}{|\omega |\leq \pi },\\0&{\mbox{if }}{\mbox{otherwise}}.\\\end{cases}}}
where the (normalised) gate function is defined by
Π
(
x
)
:=
{
1
,
if
|
x
|
≤
1
/
2
,
0
if
otherwise
.
{\displaystyle \Pi (x):={\begin{cases}1,&{\mbox{if }}{|x|\leq 1/2},\\0&{\mbox{if }}{\mbox{otherwise}}.\\\end{cases}}}
Also for the dilated and translated instances of scaling function:
Φ
k
n
(
ω
)
=
2
−
n
/
2
2
π
e
−
i
ω
(
k
+
1
)
/
2
n
Π
(
ω
2
n
+
1
π
)
{\displaystyle \Phi _{k}^{n}(\omega )={\frac {2^{-n/2}}{2\pi }}e^{-i\omega (k+1)/2^{n}}\Pi ({\frac {\omega }{2^{n+1}\pi }})}
Mother wavelet [ edit ]
Use
Φ
(Sha)
{\displaystyle \Phi ^{\text{(Sha)}}}
and multiresolution approximation we can derive the Fourier transform of the Mother wavelet:
Ψ
(Sha)
(
ω
)
=
1
2
π
e
−
i
ω
(
Π
(
ω
π
−
3
2
)
+
Π
(
ω
π
+
3
2
)
)
{\displaystyle \Psi ^{\text{(Sha)}}(\omega )={\frac {1}{2\pi }}e^{-i\omega }{\bigg (}\Pi ({\frac {\omega }{\pi }}-{\frac {3}{2}})+\Pi ({\frac {\omega }{\pi }}+{\frac {3}{2}}){\bigg )}}
And the dilated and translated instances:
Ψ
k
n
(
ω
)
=
2
−
n
/
2
2
π
e
−
i
ω
(
k
+
1
)
/
2
n
(
Π
(
ω
2
n
π
−
3
2
)
+
Π
(
ω
2
n
π
+
3
2
)
)
{\displaystyle \Psi _{k}^{n}(\omega )={\frac {2^{-n/2}}{2\pi }}e^{-i\omega (k+1)/2^{n}}{\bigg (}\Pi ({\frac {\omega }{2^{n}\pi }}-{\frac {3}{2}})+\Pi ({\frac {\omega }{2^{n}\pi }}+{\frac {3}{2}}){\bigg )}}
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
2
π
(
t
−
(
1
/
2
)
)
π
(
t
−
1
/
2
)
=
sinc
(
t
−
1
2
)
−
2
sinc
(
2
(
t
−
1
2
)
)
{\displaystyle \psi ^{\text{(Sha)}}(t)={\frac {\sin \pi (t-(1/2))-\sin 2\pi (t-(1/2))}{\pi (t-1/2)}}=\operatorname {sinc} {\bigg (}t-{\frac {1}{2}}{\bigg )}-2\operatorname {sinc} {\bigg (}2(t-{\frac {1}{2}}){\bigg )}}
ψ
k
n
(
t
)
=
2
n
/
2
ψ
(Sha)
(
2
n
t
−
k
)
{\displaystyle \psi _{k}^{n}(t)=2^{n/2}\psi ^{\text{(Sha)}}(2^{n}t-k)}
Property of mother wavelet and scaling function [ edit ]
Mother wavelets are orthonormal, namely,
<
ψ
k
n
(
t
)
,
ψ
h
m
(
t
)
>=
δ
n
m
δ
h
k
=
{
1
,
if
h
=
k
and
n
=
m
0
,
otherwise
{\displaystyle <\psi _{k}^{n}(t),\psi _{h}^{m}(t)>=\delta ^{nm}\delta _{hk}={\begin{cases}1,&{\text{if }}h=k{\text{ and }}n=m\\0,&{\text{otherwise}}\end{cases}}}
The translated instances of scaling function at level
n
=
0
{\displaystyle n=0}
are orthogonal
<
ϕ
k
0
(
t
)
,
ϕ
h
0
(
t
)
>=
δ
k
h
{\displaystyle <\phi _{k}^{0}(t),\phi _{h}^{0}(t)>=\delta ^{kh}}
The translated instances of scaling function at level
n
=
0
{\displaystyle n=0}
are orthogonal to the mother wavelets
<
ϕ
k
0
(
t
)
,
ψ
h
m
(
t
)
>=
0
{\displaystyle <\phi _{k}^{0}(t),\psi _{h}^{m}(t)>=0}
Shannon wavelets has an infinite number of vanishing moments.
Reconstruction of a Function by Shannon Wavelets [ edit ]
Suppose
f
(
x
)
∈
L
2
(
R
)
{\displaystyle f(x)\in L_{2}(\mathbb {R} )}
such that
supp
FT
{
f
}
⊂
[
−
π
,
π
]
{\displaystyle \operatorname {supp} \operatorname {FT} \{f\}\subset [-\pi ,\pi ]}
and for any dilation and the translation parameter
n
,
k
{\displaystyle n,k}
,
|
∫
−
∞
∞
f
(
t
)
ϕ
k
0
(
t
)
d
t
|
<
∞
{\displaystyle {\Bigg |}\int _{-\infty }^{\infty }f(t)\phi _{k}^{0}(t)dt{\Bigg |}<\infty }
,
|
∫
−
∞
∞
f
(
t
)
ψ
k
n
(
t
)
d
t
|
<
∞
{\displaystyle {\Bigg |}\int _{-\infty }^{\infty }f(t)\psi _{k}^{n}(t)dt{\Bigg |}<\infty }
Then
f
(
t
)
=
∑
k
=
∞
∞
α
k
ϕ
k
0
(
t
)
{\displaystyle f(t)=\sum _{k=\infty }^{\infty }\alpha _{k}\phi _{k}^{0}(t)}
is uniformly convergent, where
α
k
=
f
(
k
)
{\displaystyle \alpha _{k}=f(k)}
Real Shannon wavelet [ edit ]
Real Shannon wavelet
The Fourier transform of the Shannon mother wavelet is given by:
Ψ
(
Sha
)
(
w
)
=
∏
(
w
−
3
π
/
2
π
)
+
∏
(
w
+
3
π
/
2
π
)
.
{\displaystyle \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
∏
(
x
)
:=
{
1
,
if
|
x
|
≤
1
/
2
,
0
if
otherwise
.
{\displaystyle \prod (x):={\begin{cases}1,&{\mbox{if }}{|x|\leq 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 :
ψ
(
Sha
)
(
t
)
=
sinc
(
t
2
)
⋅
cos
(
3
π
t
2
)
{\displaystyle \psi ^{(\operatorname {Sha} )}(t)=\operatorname {sinc} \left({\frac {t}{2}}\right)\cdot \cos \left({\frac {3\pi t}{2}}\right)}
or alternatively as
ψ
(
Sha
)
(
t
)
=
2
⋅
sinc
(
2
t
)
−
sinc
(
t
)
,
{\displaystyle \psi ^{(\operatorname {Sha} )}(t)=2\cdot \operatorname {sinc} (2t)-\operatorname {sinc} (t),}
where
sinc
(
t
)
:=
sin
π
t
π
t
{\displaystyle \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
∞
{\displaystyle 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:
ϕ
(
S
h
a
)
(
t
)
=
sin
π
t
π
t
=
sinc
(
t
)
.
{\displaystyle \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
ψ
(
C
S
h
a
)
(
t
)
=
sinc
(
t
)
⋅
e
−
2
π
i
t
{\displaystyle \psi ^{(CSha)}(t)=\operatorname {sinc} (t)\cdot e^{-2\pi it}}
,
References [ edit ]
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 .