Continuous wavelets of compact support alpha can be built,[ 1] beta distribution . The process is derived from probability distributions using blur derivative. These new wavelets have just one cycle, so they are termed unicycle wavelets. They can be viewed as a soft variety of Haar wavelets whose shape is fine-tuned by two parameters
α
{\displaystyle \alpha }
β
{\displaystyle \beta }
Central Limit Theorem by Gnedenko and Kolmogorov applied for compactly supported signals.[ 2]
The beta distribution is a continuous probability distribution defined over the interval
0
≤
t
≤
1
{\displaystyle 0\leq t\leq 1}
α
{\displaystyle \alpha }
β
{\displaystyle \beta }
P
(
t
)
=
1
B
(
α
,
β
)
t
α
−
1
⋅
(
1
−
t
)
β
−
1
,
1
≤
α
,
β
≤
+
∞
{\displaystyle P(t)={\frac {1}{B(\alpha ,\beta )}}t^{\alpha -1}\cdot (1-t)^{\beta -1},\quad 1\leq \alpha ,\beta \leq +\infty }
The normalising factor is
B
(
α
,
β
)
=
Γ
(
α
)
⋅
Γ
(
β
)
Γ
(
α
+
β
)
{\displaystyle B(\alpha ,\beta )={\frac {\Gamma (\alpha )\cdot \Gamma (\beta )}{\Gamma (\alpha +\beta )}}}
where
Γ
(
⋅
)
{\displaystyle \Gamma (\cdot )}
generalised factorial function of Euler and
B
(
⋅
,
⋅
)
{\displaystyle B(\cdot ,\cdot )}
Beta function .[ 3]
Gnedenko-Kolmogorov central limit theorem revisited
edit
Let
p
i
(
t
)
{\displaystyle p_{i}(t)}
t
i
{\displaystyle t_{i}}
i
=
1
,
2
,
3..
N
{\displaystyle i=1,2,3..N}
p
i
(
t
)
≥
0
{\displaystyle p_{i}(t)\geq 0}
(
∀
t
)
{\displaystyle (\forall t)}
∫
−
∞
+
∞
p
i
(
t
)
d
t
=
1
{\displaystyle \int _{-\infty }^{+\infty }p_{i}(t)dt=1}
Suppose that all variables are independent.
The mean and the variance of a given random variable
t
i
{\displaystyle t_{i}}
m
i
=
∫
−
∞
+
∞
τ
⋅
p
i
(
τ
)
d
τ
,
{\displaystyle m_{i}=\int _{-\infty }^{+\infty }\tau \cdot p_{i}(\tau )d\tau ,}
σ
i
2
=
∫
−
∞
+
∞
(
τ
−
m
i
)
2
⋅
p
i
(
τ
)
d
τ
{\displaystyle \sigma _{i}^{2}=\int _{-\infty }^{+\infty }(\tau -m_{i})^{2}\cdot p_{i}(\tau )d\tau }
The mean and variance of
t
{\displaystyle t}
m
=
∑
i
=
1
N
m
i
{\displaystyle m=\sum _{i=1}^{N}m_{i}}
σ
2
=
∑
i
=
1
N
σ
i
2
{\displaystyle \sigma ^{2}=\sum _{i=1}^{N}\sigma _{i}^{2}}
The density
p
(
t
)
{\displaystyle p(t)}
t
=
∑
i
=
1
N
t
i
{\displaystyle t=\sum _{i=1}^{N}t_{i}}
Central Limit Theorem for distributions of compact support (Gnedenko and Kolmogorov). [ 2]
Let
{
p
i
(
t
)
}
{\displaystyle \{p_{i}(t)\}}
S
u
p
p
{
(
p
i
(
t
)
)
}
=
(
a
i
,
b
i
)
(
∀
i
)
{\displaystyle Supp\{(p_{i}(t))\}=(a_{i},b_{i})(\forall i)}
Let
a
=
∑
i
=
1
N
a
i
<
+
∞
{\displaystyle a=\sum _{i=1}^{N}a_{i}<+\infty }
b
=
∑
i
=
1
N
b
i
<
+
∞
{\displaystyle b=\sum _{i=1}^{N}b_{i}<+\infty }
Without loss of generality assume that
a
=
0
{\displaystyle a=0}
b
=
1
{\displaystyle b=1}
The random variable
t
{\displaystyle t}
N
→
∞
{\displaystyle N\rightarrow \infty }
p
(
t
)
≈
{\displaystyle p(t)\approx }
{
k
⋅
t
α
(
1
−
t
)
β
,
o
t
h
e
r
w
i
s
e
{\displaystyle {\begin{cases}{k\cdot t^{\alpha }(1-t)^{\beta }},\\otherwise\end{cases}}}
where
α
=
m
(
m
−
m
2
−
σ
2
)
σ
2
,
{\displaystyle \alpha ={\frac {m(m-m^{2}-\sigma ^{2})}{\sigma ^{2}}},}
β
=
(
1
−
m
)
(
α
+
1
)
m
.
{\displaystyle \beta ={\frac {(1-m)(\alpha +1)}{m}}.}
Since
P
(
⋅
|
α
,
β
)
{\displaystyle P(\cdot |\alpha ,\beta )}
unimodal , the wavelet generated by
ψ
b
e
t
a
(
t
|
α
,
β
)
=
(
−
1
)
d
P
(
t
|
α
,
β
)
d
t
{\displaystyle \psi _{beta}(t|\alpha ,\beta )=(-1){\frac {dP(t|\alpha ,\beta )}{dt}}}
The main features of beta wavelets of parameters
α
{\displaystyle \alpha }
β
{\displaystyle \beta }
S
u
p
p
(
ψ
)
=
[
−
α
β
α
+
β
+
1
,
β
α
α
+
β
+
1
]
=
[
a
,
b
]
.
{\displaystyle Supp(\psi )=[-{\sqrt {\frac {\alpha }{\beta }}}{\sqrt {\alpha +\beta +1}},{\sqrt {\frac {\beta }{\alpha }}}{\sqrt {\alpha +\beta +1}}]=[a,b].}
l
e
n
g
t
h
S
u
p
p
(
ψ
)
=
T
(
α
,
β
)
=
(
α
+
β
)
α
+
β
+
1
α
β
.
{\displaystyle lengthSupp(\psi )=T(\alpha ,\beta )=(\alpha +\beta ){\sqrt {\frac {\alpha +\beta +1}{\alpha \beta }}}.}
The parameter
R
=
b
/
|
a
|
=
β
/
α
{\displaystyle R=b/|a|=\beta /\alpha }
t
z
e
r
o
c
r
o
s
s
{\displaystyle t_{zerocross}}
t
z
e
r
o
c
r
o
s
s
=
(
α
−
β
)
(
α
+
β
−
2
)
α
+
β
+
1
α
β
.
{\displaystyle t_{zerocross}={\frac {(\alpha -\beta )}{(\alpha +\beta -2)}}{\sqrt {\frac {\alpha +\beta +1}{\alpha \beta }}}.}
The (unimodal) scale function associated with the wavelets is given by
ϕ
b
e
t
a
(
t
|
α
,
β
)
=
1
B
(
α
,
β
)
T
α
+
β
−
1
⋅
(
t
−
a
)
α
−
1
⋅
(
b
−
t
)
β
−
1
,
{\displaystyle \phi _{beta}(t|\alpha ,\beta )={\frac {1}{B(\alpha ,\beta )T^{\alpha +\beta -1}}}\cdot (t-a)^{\alpha -1}\cdot (b-t)^{\beta -1},}
a
≤
t
≤
b
{\displaystyle a\leq t\leq b}
A closed-form expression for first-order beta wavelets can easily be derived. Within their support,
ψ
b
e
t
a
(
t
|
α
,
β
)
=
−
1
B
(
α
,
β
)
T
α
+
β
−
1
⋅
[
α
−
1
t
−
a
−
β
−
1
b
−
t
]
⋅
(
t
−
a
)
α
−
1
⋅
(
b
−
t
)
β
−
1
{\displaystyle \psi _{beta}(t|\alpha ,\beta )={\frac {-1}{B(\alpha ,\beta )T^{\alpha +\beta -1}}}\cdot [{\frac {\alpha -1}{t-a}}-{\frac {\beta -1}{b-t}}]\cdot (t-a)^{\alpha -1}\cdot (b-t)^{\beta -1}}
Figure. Unicyclic beta scale function and wavelet for different parameters: a)
α
=
4
{\displaystyle \alpha =4}
β
=
3
{\displaystyle \beta =3}
α
=
3
{\displaystyle \alpha =3}
β
=
7
{\displaystyle \beta =7}
α
=
5
{\displaystyle \alpha =5}
β
=
17
{\displaystyle \beta =17}
Beta wavelet spectrum
edit
The beta wavelet spectrum can be derived in terms of the Kummer hypergeometric function.[ 4]
Let
ψ
b
e
t
a
(
t
|
α
,
β
)
↔
Ψ
B
E
T
A
(
ω
|
α
,
β
)
{\displaystyle \psi _{beta}(t|\alpha ,\beta )\leftrightarrow \Psi _{BETA}(\omega |\alpha ,\beta )}
This spectrum is also denoted by
Ψ
B
E
T
A
(
ω
)
{\displaystyle \Psi _{BETA}(\omega )}
Ψ
B
E
T
A
(
ω
)
=
−
j
ω
⋅
M
(
α
,
α
+
β
,
−
j
ω
(
α
+
β
)
α
+
β
+
1
α
β
)
⋅
e
x
p
{
(
j
ω
α
(
α
+
β
+
1
)
β
)
}
{\displaystyle \Psi _{BETA}(\omega )=-j\omega \cdot M(\alpha ,\alpha +\beta ,-j\omega (\alpha +\beta ){\sqrt {\frac {\alpha +\beta +1}{\alpha \beta }}})\cdot exp\{(j\omega {\sqrt {\frac {\alpha (\alpha +\beta +1)}{\beta }}})\}}
where
M
(
α
,
α
+
β
,
j
ν
)
=
Γ
(
α
+
β
)
Γ
(
α
)
⋅
Γ
(
β
)
⋅
∫
0
1
e
j
ν
t
t
α
−
1
(
1
−
t
)
β
−
1
d
t
{\displaystyle M(\alpha ,\alpha +\beta ,j\nu )={\frac {\Gamma (\alpha +\beta )}{\Gamma (\alpha )\cdot \Gamma (\beta )}}\cdot \int _{0}^{1}e^{j\nu t}t^{\alpha -1}(1-t)^{\beta -1}dt}
Only symmetrical
(
α
=
β
)
{\displaystyle (\alpha =\beta )}
(
α
≠
β
)
{\displaystyle (\alpha \neq \beta )}
|
Ψ
B
E
T
A
(
ω
|
α
,
β
)
|
=
|
Ψ
B
E
T
A
(
ω
|
β
,
α
)
|
.
{\displaystyle |\Psi _{BETA}(\omega |\alpha ,\beta )|=|\Psi _{BETA}(\omega |\beta ,\alpha )|.}
Higher derivatives may also generate further beta wavelets. Higher order beta wavelets are defined by
ψ
b
e
t
a
(
t
|
α
,
β
)
=
(
−
1
)
N
d
N
P
(
t
|
α
,
β
)
d
t
N
.
{\displaystyle \psi _{beta}(t|\alpha ,\beta )=(-1)^{N}{\frac {d^{N}P(t|\alpha ,\beta )}{dt^{N}}}.}
This is henceforth referred to as an
N
{\displaystyle N}
N
≤
M
i
n
(
α
,
β
)
−
1
{\displaystyle N\leq Min(\alpha ,\beta )-1}
Ψ
b
e
t
a
(
t
|
α
,
β
)
=
(
−
1
)
N
B
(
α
,
β
)
⋅
T
α
+
β
−
1
∑
n
=
0
N
s
g
n
(
2
n
−
N
)
⋅
Γ
(
α
)
Γ
(
α
−
(
N
−
n
)
)
(
t
−
a
)
α
−
1
−
(
N
−
n
)
⋅
Γ
(
β
)
Γ
(
β
−
n
)
(
b
−
t
)
β
−
1
−
n
.
{\displaystyle \Psi _{beta}(t|\alpha ,\beta )={\frac {(-1)^{N}}{B(\alpha ,\beta )\cdot T^{\alpha +\beta -1}}}\sum _{n=0}^{N}sgn(2n-N)\cdot {\frac {\Gamma (\alpha )}{\Gamma (\alpha -(N-n))}}(t-a)^{\alpha -1-(N-n)}\cdot {\frac {\Gamma (\beta )}{\Gamma (\beta -n)}}(b-t)^{\beta -1-n}.}
Figure. Magnitude of the spectrum
Ψ
B
E
T
A
(
ω
)
{\displaystyle \Psi _{BETA}(\omega )}
|
Ψ
B
E
T
A
(
ω
α
,
β
)
|
{\displaystyle |\Psi _{BETA}(\omega \alpha ,\beta )|}
×
ω
{\displaystyle \times \omega }
α
=
β
=
3
{\displaystyle \alpha =\beta =3}
α
=
β
=
4
{\displaystyle \alpha =\beta =4}
α
=
β
=
5
{\displaystyle \alpha =\beta =5}
Figure. Magnitude of the spectrum
Ψ
B
E
T
A
(
ω
)
{\displaystyle \Psi _{BETA}(\omega )}
|
Ψ
B
E
T
A
(
ω
α
,
β
)
|
{\displaystyle |\Psi _{BETA}(\omega \alpha ,\beta )|}
×
ω
{\displaystyle \times \omega }
α
=
3
{\displaystyle \alpha =3}
β
=
4
{\displaystyle \beta =4}
α
=
3
{\displaystyle \alpha =3}
β
=
5
{\displaystyle \beta =5}
Wavelet theory is applicable to several subjects. All wavelet transforms may be considered forms of time-frequency representation for continuous-time (analog) signals and so are related to harmonic analysis. Almost all practically useful discrete wavelet transforms use discrete-time filter banks. Similarly, Beta wavelet[ 1] [ 5] [ 5] [ 6] [ 7] [ 8]
^ a b de Oliveira, Hélio Magalhães; Schmidt, Giovanna Angelis (2005). "Compactly Supported One-cyclic Wavelets Derived from Beta Distributions" . Journal of Communication and Information Systems . 20 (3): 27–33. arXiv :1502.02166 doi :10.14209/jcis.2005.17 ^ a b Gnedenko, Boris Vladimirovich; Kolmogorov, Andrey (1954). Limit Distributions for Sums of Independent Random Variables . Reading, Ma: Addison-Wesley. ^ Davis, Philip J. (1968). "Gamma Function and Related Functions" . In Abramowitz, Milton ; Stegun, Irene (eds.). Handbook of Mathematical Functions . New York : Dover . pp. 253–294. ISBN 0-486-61272-4 ^ Slater, Lucy Joan (1968). "Confluent Hypergeometric Function" . In Abramowitz, Milton ; Stegun, Irene (eds.). Handbook of Mathematical Functions . New York : Dover . pp. 503–536. ISBN 0-486-61272-4 ^ a b Ben Amar, Chokri; Zaied, Mourad; Alimi, Adel M. (2005). "Beta wavelets. Synthesis and application to lossy image compression" . Advances in Engineering Software . 36 (7). Elsevier: 459–474. doi :10.1016/j.advengsoft.2005.01.013 . ^ Kumar, Ranjeet; Kumar, Anil; Pandey, Rajesh K. (2012). "Electrocardiogram Signal compression Using Beta Wavelets" . Journal of Mathematical Modelling and Algorithms . 11 (3). Springer Verlag : 235–248. doi :10.1007/s10852-012-9181-9 . S2CID 4667379 . ^ Kumar, Ranjeet; Kumar, Anil; Pandey, Rajesh K. (2013). "Beta wavelet based ECG signal compression using lossless encoding with modified thresholding" . Computers & Electrical Engineering . 39 (1). Elsevier : 130–140. doi :10.1016/j.compeleceng.2012.04.008 . ^ Zaied, Mourad; Jemai, Olfa; Ben Amar, Chokri (2008). "Training of the Beta wavelet networks by the frames theory: Application to face recognition" . 2008 First Workshops on Image Processing Theory, Tools and Applications . IEEE . pp. 1–6. doi :10.1109/IPTA.2008.4743756 . eISSN 2154-512X . ISBN 978-1-4244-3321-6 ISSN 2154-5111 . S2CID 12230926 .
W.B. Davenport , Probability and Random Processes, McGraw-Hill, Kogakusha, Tokyo, 1970. 
![{\displaystyle Supp(\psi )=[-{\sqrt {\frac {\alpha }{\beta }}}{\sqrt {\alpha +\beta +1}},{\sqrt {\frac {\beta }{\alpha }}}{\sqrt {\alpha +\beta +1}}]=[a,b].}](https://wikimedia.org/api/rest_v1/media/math/render/svg/faa06c49ad84eb65113d1f1596540aba1e016d85)
![{\displaystyle \psi _{beta}(t|\alpha ,\beta )={\frac {-1}{B(\alpha ,\beta )T^{\alpha +\beta -1}}}\cdot [{\frac {\alpha -1}{t-a}}-{\frac {\beta -1}{b-t}}]\cdot (t-a)^{\alpha -1}\cdot (b-t)^{\beta -1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/13fd943e6f25952bc077b5a48c60d5462379a0ad)
