Comment: Fails WP:GNG , requires significant coverage in multiple independent secondary sources. Dan arndt (talk ) 06:50, 5 January 2023 (UTC)
Causal wavelet
In mathematics, a causal wavelet is a continuous wavelet used in the real time continuous wavelet transform . The causal mother wavelet is defined as:[1]
h
(
t
)
=
ψ
(
t
)
I
t
≥
0
=
e
−
t
2
2
e
j
2
π
f
0
t
I
t
≥
0
{\displaystyle h(t)=\psi (t)I_{t\geq 0}=e^{-{\frac {t^{2}}{2}}}e^{j2\pi f_{0}t}I_{t\geq 0}}
where
ψ
(
t
)
{\displaystyle \psi (t)}
is a Morlet wavelet .
Hence, the Fourier transform of causal mother wavelet[1]
|
H
(
ω
)
|
2
=
4
4
(
ω
−
2
π
f
0
)
+
1
{\displaystyle |H(\omega )|^{2}={\frac {4}{4(\omega -2\pi f_{0})+1}}}
and satifies the Admissibility Criterion
∫
0
∞
|
H
(
ω
)
|
2
ω
d
ω
<
∞
{\displaystyle \int _{0}^{\infty }{\frac {|H(\omega )|^{2}}{\omega }}\,d\omega <\infty }
, then the causal wavelet transform is reversible. Furthermore, we can observe that
|
H
(
ω
)
|
{\displaystyle |H(\omega )|}
reach the maximum value at
ω
=
2
π
f
0
{\displaystyle \omega =2\pi f_{0}}
. Therefore, when the
f
0
{\displaystyle f_{0}}
is high, the convolution of causal wavelet is a high pass filter and vice versa. While we usually chose
f
0
=
5
2
π
{\displaystyle f_{0}={\frac {5}{2\pi }}}
for a Morlet wavelet, hence we have
h
(
t
)
=
e
−
t
2
2
e
j
5
t
I
t
≥
0
{\displaystyle h(t)=e^{-{\frac {t^{2}}{2}}}e^{j5t}I_{t\geq 0}}
and the real form
h
(
t
)
=
e
−
t
2
2
cos
5
t
I
t
≥
0
{\displaystyle h(t)=e^{-{\frac {t^{2}}{2}}}\cos {5t}I_{t\geq 0}}
.
Moreover, we define the causal wavelet transform as[1]
W
h
[
x
(
t
)
]
(
a
,
b
)
=
1
b
∫
−
∞
∞
x
(
t
)
h
(
t
−
a
b
)
¯
d
t
=
∫
−
∞
∞
x
(
t
)
h
(
a
,
b
)
(
t
)
¯
d
t
{\displaystyle W_{h}[x(t)](a,b)={\frac {1}{\sqrt {b}}}\int _{-\infty }^{\infty }x(t){\overline {h({\frac {t-a}{b}})}}\,dt=\int _{-\infty }^{\infty }x(t){\overline {h_{(a,b)}(t)}}\,dt}
where
h
(
a
,
b
)
(
t
)
=
1
b
h
(
t
−
a
b
)
{\displaystyle h_{(a,b)}(t)={\frac {1}{\sqrt {b}}}h({\frac {t-a}{b}})}
is called the daughter wavelet of the causal wavelet.[1]
Simulation
Simulation on causal signal
Consider a causal signal
x
(
t
)
{\displaystyle x(t)}
, which there is a value
t
0
{\displaystyle t_{0}}
such that
x
(
t
)
=
0
,
∀
t
<
t
0
{\displaystyle x(t)=0,\forall t<t_{0}}
, for example [1]
x
(
t
)
=
sin
5
(
t
−
512
)
16
cos
−
(
t
−
512
)
32
I
t
≥
512
{\displaystyle x(t)=\sin {\frac {5(t-512)}{16}}\cos {\frac {-(t-512)}{32}}I_{t\geq 512}}
and we define the causal mother wavelet as
h
(
t
)
=
e
−
t
2
2
cos
5
t
I
t
≥
0
{\displaystyle h(t)=e^{-{\frac {t^{2}}{2}}}\cos {5t}I_{t\geq 0}}
.
Moreover, we define the inner product in
L
2
(
R
)
{\displaystyle L^{2}(\mathbb {R} )}
⟨
f
(
t
)
,
g
(
t
)
⟩
=
∫
−
∞
∞
f
(
t
)
g
(
t
)
d
t
{\displaystyle \langle f(t),g(t)\rangle =\int _{-\infty }^{\infty }f(t)g(t)\,dt}
, for
f
(
t
)
,
g
(
t
)
∈
L
2
(
R
)
{\displaystyle f(t),g(t)\in L^{2}(\mathbb {R} )}
.
Then, we can define the Signal-to-noise ratio (SNR) w.r.t. the wavalet
ψ
(
t
)
{\displaystyle \psi (t)}
as
S
N
R
ψ
(
x
(
t
)
)
=
max
a
,
b
∫
−
∞
∞
|
x
(
t
)
|
2
d
t
∫
−
∞
∞
|
x
(
t
)
−
ψ
(
a
,
b
)
(
t
)
|
2
d
t
=
max
a
,
b
⟨
x
(
t
)
,
x
(
t
)
⟩
⟨
x
(
t
)
,
x
(
t
)
⟩
+
⟨
ψ
(
a
,
b
)
(
t
)
,
ψ
(
a
,
b
)
(
t
)
⟩
−
2
⟨
x
(
t
)
,
ψ
(
a
,
b
)
(
t
)
⟩
{\displaystyle SNR_{\psi }(x(t))=\mathop {\max } _{a,b}{\frac {\int _{-\infty }^{\infty }|x(t)|^{2}\,dt}{\int _{-\infty }^{\infty }|x(t)-\psi _{(a,b)}(t)|^{2}\,dt}}=\mathop {\max } _{a,b}{\frac {\langle x(t),x(t)\rangle }{\langle x(t),x(t)\rangle +\langle \psi _{(a,b)}(t),\psi _{(a,b)}(t)\rangle -2\langle x(t),\psi _{(a,b)}(t)\rangle }}}
.
We can see that the causal wavelet
h
(
t
)
{\displaystyle h(t)}
is always better than the Morlet wavelet
g
(
t
)
{\displaystyle g(t)}
for the SNR of the causal signal
x
(
t
)
{\displaystyle x(t)}
.
Since
arg
max
a
,
b
⟨
x
(
t
)
,
x
(
t
)
⟩
⟨
x
(
t
)
,
x
(
t
)
⟩
+
⟨
ψ
(
a
,
b
)
(
t
)
,
ψ
(
a
,
b
)
(
t
)
⟩
−
2
⟨
x
(
t
)
,
ψ
(
a
,
b
)
(
t
)
⟩
=
arg
max
a
,
b
{
2
⟨
x
(
t
)
,
ψ
(
a
,
b
)
(
t
)
⟩
−
⟨
ψ
(
a
,
b
)
(
t
)
,
ψ
(
a
,
b
)
(
t
)
⟩
}
{\displaystyle \mathop {\arg \max } _{a,b}{\frac {\langle x(t),x(t)\rangle }{\langle x(t),x(t)\rangle +\langle \psi _{(a,b)}(t),\psi _{(a,b)}(t)\rangle -2\langle x(t),\psi _{(a,b)}(t)\rangle }}=\mathop {\arg \max } _{a,b}\{2\langle x(t),\psi _{(a,b)}(t)\rangle -\langle \psi _{(a,b)}(t),\psi _{(a,b)}(t)\rangle \}}
and
⟨
h
(
a
,
b
)
(
t
)
,
h
(
a
,
b
)
(
t
)
⟩
<
⟨
g
(
a
,
b
)
(
t
)
,
g
(
a
,
b
)
(
t
)
⟩
{\displaystyle \langle h_{(a,b)}(t),h_{(a,b)}(t)\rangle <\langle g_{(a,b)}(t),g_{(a,b)}(t)\rangle }
for the same
a
,
b
{\displaystyle a,b}
, so as
⟨
x
(
t
)
,
h
(
a
,
b
)
(
t
)
⟩
=
⟨
x
(
t
)
,
g
(
a
,
b
)
(
t
)
⟩
{\displaystyle \langle x(t),h_{(a,b)}(t)\rangle =\langle x(t),g_{(a,b)}(t)\rangle }
for the same
a
,
b
{\displaystyle a,b}
,
we get
S
N
R
h
(
x
(
t
)
)
>
S
N
R
g
(
x
(
t
)
)
{\displaystyle SNR_{h}(x(t))>SNR_{g}(x(t))}
.
References