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Spectrum of the Meyer wavelet (numerically computed). The Meyer wavelet is an orthogonal wavelet proposed by Yves Meyer .[ 1] continuous wavelet , it has been applied in a number of cases, such as in adaptive filters ,[ 2] random fields ,[ 3] [ 4]
The Meyer wavelet is infinitely differentiable with infinite support and defined in frequency domain in terms of function
ν
{\displaystyle \nu }
Ψ
(
ω
)
:=
{
1
2
π
sin
(
π
2
ν
(
3
|
ω
|
2
π
−
1
)
)
e
j
ω
/
2
if
2
π
/
3
<
|
ω
|
<
4
π
/
3
,
1
2
π
cos
(
π
2
ν
(
3
|
ω
|
4
π
−
1
)
)
e
j
ω
/
2
if
4
π
/
3
<
|
ω
|
<
8
π
/
3
,
0
otherwise
,
{\displaystyle \Psi (\omega ):={\begin{cases}{\frac {1}{\sqrt {2\pi }}}\sin \left({\frac {\pi }{2}}\nu \left({\frac {3|\omega |}{2\pi }}-1\right)\right)e^{j\omega /2}&{\text{if }}2\pi /3<|\omega |<4\pi /3,\\{\frac {1}{\sqrt {2\pi }}}\cos \left({\frac {\pi }{2}}\nu \left({\frac {3|\omega |}{4\pi }}-1\right)\right)e^{j\omega /2}&{\text{if }}4\pi /3<|\omega |<8\pi /3,\\0&{\text{otherwise}},\end{cases}}}
where
ν
(
x
)
:=
{
0
if
x
<
0
,
x
if
0
<
x
<
1
,
1
if
x
>
1.
{\displaystyle \nu (x):={\begin{cases}0&{\text{if }}x<0,\\x&{\text{if }}0<x<1,\\1&{\text{if }}x>1.\end{cases}}}
There are many different ways for defining this auxiliary function, which yields variants of the Meyer wavelet.
For instance, another standard implementation adopts
ν
(
x
)
:=
{
x
4
(
35
−
84
x
+
70
x
2
−
20
x
3
)
if
0
<
x
<
1
,
0
otherwise
.
{\displaystyle \nu (x):={\begin{cases}x^{4}(35-84x+70x^{2}-20x^{3})&{\text{if }}0<x<1,\\0&{\text{otherwise}}.\end{cases}}}
Meyer scale function (numerically computed) The Meyer scaling function is given by
Φ
(
ω
)
:=
{
1
2
π
if
|
ω
|
<
2
π
/
3
,
1
2
π
cos
(
π
2
ν
(
3
|
ω
|
2
π
−
1
)
)
if
2
π
/
3
<
|
ω
|
<
4
π
/
3
,
0
otherwise
.
{\displaystyle \Phi (\omega ):={\begin{cases}{\frac {1}{\sqrt {2\pi }}}&{\text{if }}|\omega |<2\pi /3,\\{\frac {1}{\sqrt {2\pi }}}\cos \left({\frac {\pi }{2}}\nu \left({\frac {3|\omega |}{2\pi }}-1\right)\right)&{\text{if }}2\pi /3<|\omega |<4\pi /3,\\0&{\text{otherwise}}.\end{cases}}}
In the time domain , the waveform of the Meyer mother-wavelet has the shape as shown in the following figure:
waveform of the Meyer wavelet (numerically computed) Valenzuela and de Oliveira [ 5]
ϕ
(
t
)
=
{
2
3
+
4
3
π
t
=
0
,
sin
(
2
π
3
t
)
+
4
3
t
cos
(
4
π
3
t
)
π
t
−
16
π
9
t
3
otherwise
,
{\displaystyle \phi (t)={\begin{cases}{\frac {2}{3}}+{\frac {4}{3\pi }}&t=0,\\{\frac {\sin({\frac {2\pi }{3}}t)+{\frac {4}{3}}t\cos({\frac {4\pi }{3}}t)}{\pi t-{\frac {16\pi }{9}}t^{3}}}&{\text{otherwise}},\end{cases}}}
and
ψ
(
t
)
=
ψ
1
(
t
)
+
ψ
2
(
t
)
,
{\displaystyle \psi (t)=\psi _{1}(t)+\psi _{2}(t),}
where
ψ
1
(
t
)
=
4
3
π
(
t
−
1
2
)
cos
[
2
π
3
(
t
−
1
2
)
]
−
1
π
sin
[
4
π
3
(
t
−
1
2
)
]
(
t
−
1
2
)
−
16
9
(
t
−
1
2
)
3
,
{\displaystyle \psi _{1}(t)={\frac {{\frac {4}{3\pi }}(t-{\frac {1}{2}})\cos[{\frac {2\pi }{3}}(t-{\frac {1}{2}})]-{\frac {1}{\pi }}\sin[{\frac {4\pi }{3}}(t-{\frac {1}{2}})]}{(t-{\frac {1}{2}})-{\frac {16}{9}}(t-{\frac {1}{2}})^{3}}},}
ψ
2
(
t
)
=
8
3
π
(
t
−
1
2
)
cos
[
8
π
3
(
t
−
1
2
)
]
+
1
π
sin
[
4
π
3
(
t
−
1
2
)
]
(
t
−
1
2
)
−
64
9
(
t
−
1
2
)
3
.
{\displaystyle \psi _{2}(t)={\frac {{\frac {8}{3\pi }}(t-{\frac {1}{2}})\cos[{\frac {8\pi }{3}}(t-{\frac {1}{2}})]+{\frac {1}{\pi }}\sin[{\frac {4\pi }{3}}(t-{\frac {1}{2}})]}{(t-{\frac {1}{2}})-{\frac {64}{9}}(t-{\frac {1}{2}})^{3}}}.}
^ Meyer, Yves (1990). Ondelettes et opérateurs: Ondelettes . Hermann. ISBN 9782705661250 ^ Xu, L.; Zhang, D.; Wang, K. (2005). "Wavelet-based cascaded adaptive filter for removing baseline drift in pulse waveforms". IEEE Transactions on Biomedical Engineering . 52 (11): 1973–1975. doi :10.1109/tbme.2005.856296 . hdl :10397/193 PMID 16285403 . S2CID 6897442 . ^ Elliott, Jr., F. W.; Horntrop, D. J.; Majda, A. J. (1997). "A Fourier-Wavelet Monte Carlo method for fractal random fields" . Journal of Computational Physics . 132 (2): 384–408. Bibcode :1997JCoPh.132..384E . doi :10.1006/jcph.1996.5647 ^ Abbasion, S.; et al. (2007). "Rolling element bearings multi-fault classification based on the wavelet denoising and support vector machine". Mechanical Systems and Signal Processing . 21 (7): 2933–2945. Bibcode :2007MSSP...21.2933A . doi :10.1016/j.ymssp.2007.02.003 . ^ Valenzuela, Victor Vermehren; de Oliveira, H. M. (2015). "Close expressions for Meyer Wavelet and Scale Function". Anais de XXXIII Simpósio Brasileiro de Telecomunicações . p. 4. arXiv :1502.00161 doi :10.14209/SBRT.2015.2 . S2CID 88513986 .
Look up
wavelet in Wiktionary, the free dictionary.
Wikimedia Commons has media related to
Wavelet .
![{\displaystyle \psi _{1}(t)={\frac {{\frac {4}{3\pi }}(t-{\frac {1}{2}})\cos[{\frac {2\pi }{3}}(t-{\frac {1}{2}})]-{\frac {1}{\pi }}\sin[{\frac {4\pi }{3}}(t-{\frac {1}{2}})]}{(t-{\frac {1}{2}})-{\frac {16}{9}}(t-{\frac {1}{2}})^{3}}},}](https://wikimedia.riteme.site/api/rest_v1/media/math/render/svg/45417c0872e508bc48dad9a6b0c84370ffaf16f8)
![{\displaystyle \psi _{2}(t)={\frac {{\frac {8}{3\pi }}(t-{\frac {1}{2}})\cos[{\frac {8\pi }{3}}(t-{\frac {1}{2}})]+{\frac {1}{\pi }}\sin[{\frac {4\pi }{3}}(t-{\frac {1}{2}})]}{(t-{\frac {1}{2}})-{\frac {64}{9}}(t-{\frac {1}{2}})^{3}}}.}](https://wikimedia.riteme.site/api/rest_v1/media/math/render/svg/61771078aee24f360757f7db5cdc33832e15fe98)
