Beta wavelet

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Continuous wavelets of compact support can be built [1], which are related to the 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 and β. Close expressions for beta wavelets and scale functions as well as their spectra are derived. Their importance is due to the Central Limit Theorem by Gnedenko&Kolmogorov applied for compactly supported signals [2].

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[edit] Beta distribution

The beta distribution is a continuous probability distribution defined over the interval 0\leq t\leq 1. It is characterised by a couple of parameters, namely α and β according to:

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(\alpha ,\beta )=\frac{\Gamma (\alpha )\cdot \Gamma (\beta )}{\Gamma (\alpha +\beta )},

where \Gamma (\cdot ) is the generalised factorial function of Euler and B(\cdot ,\cdot ) is the Beta function [4].

[edit] Gnedenko-Kolmogorov Central limit theorem revisited

Let pi(t) be a probability density of the random variable ti, i = 1,2,3..N i.e.

p_{i}(t)\ge 0, (\forall t) and \int_{-\infty }^{+\infty }p_{i}(t)dt=1.

Suppose that all variables are independent.

The mean and the variance of a given random variable ti are, respectively

m_{i}=\int_{-\infty }^{+\infty }\tau \cdot p_{i}(\tau )d\tau , \sigma _{i}^{2}=\int_{-\infty }^{+\infty }(\tau -m_{i})^{2}\cdot p_{i}(\tau )d\tau.

The mean and variance of t are therefore m=\sum_{i=1}^{N}m_{i} and \sigma^2 =\sum_{i=1}^{N}\sigma _{i}^{2}.

The density p(t) of the random variable corresponding to the sum t=\sum_{i=1}^{N}t_{i} is given by the

Central Limit Theorem for distributions of compact support (Gnedenko and Kolmogorov) [2].

Let {pi(t)} be distributions such that Supp\{(p_{i}(t))\}=(a_{i},b_{i})(\forall i).

Let a=\sum_{i=1}^{N}a_{i}<+\infty, and b=\sum_{i=1}^{N}b_{i}<+\infty.

Without loss of generality assume that a = 0 and b = 1.

The random variable t holds, as N\rightarrow \infty, p(t)\approx \begin{cases} {k \cdot t^{\alpha }(1-t)^{\beta}}, \\otherwise \end{cases}

where \alpha =\frac{m(m-m^{2}-\sigma ^{2})}{\sigma ^{2}}, and \beta =\frac{(1-m)(\alpha +1)}{m}.

[edit] Beta wavelets

Since P(\cdot |\alpha ,\beta ) is unimodal, the wavelet generated by

\psi _{beta}(t|\alpha ,\beta )=(-1)\frac{dP(t|\alpha ,\beta )}{dt} has only one-cycle (a negative half-cycle and a positive half-cycle).

The main features of beta wavelets of parameters α and β are:

Supp(\psi )=[ \frac{-1}{\sqrt{{\beta }/ \alpha }}\sqrt{\alpha + \beta +1},\sqrt{ \frac{\beta }{\alpha }} \sqrt{\alpha +\beta +1}]=[a,b].

lengthSupp(\psi )=T(\alpha ,\beta )=(\alpha +\beta )\sqrt{\frac{\alpha +\beta +1}{\alpha \beta }}.

The parameter R = b / | a | = β / α is referred to as “cyclic balance”, and is defined as the ratio between the lengths of the causal and non-causal piece of the wavelet. The instant of transition tzerocross from the first to the second half cycle is given by

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

\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\leq t\leq b.

A close expression for first-order beta wavelets can easily be derived. Within their support,

\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, β = 3 b) α = 3, β = 7 c) α = 5, β = 17.
Figure. Unicyclic beta scale function and wavelet for different parameters: a) α = 4, β = 3 b) α = 3, β = 7 c) α = 5, β = 17.

[edit] Beta wavelet spectrum

The beta wavelet spectrum can be derived in terms of the Kummer hypergeometric function [5].

Let \psi _{beta}(t|\alpha ,\beta )\leftrightarrow \Psi _{BETA}(\omega |\alpha ,\beta ) denote the Fourier transform pair associated with the wavelet.

This spectrum is also denoted by ΨBETA(ω) for short. It can be proved by applying properties of the Fourier transform that

\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(\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) cases have zeroes in the spectrum. A few asymmetric (\alpha \neq \beta ) beta wavelets are shown in Fig. Inquisitively, they are parameter-symmetrical in the sense that they hold | ΨBETA(ω | α,β) | = | ΨBETA(ω | β,α) | .

Higher derivatives may also generate further beta wavelets. Higher order beta wavelets are defined by \psi _{beta}(t|\alpha ,\beta )=(-1)^{N}\frac{d^{N}P(t|\alpha ,\beta )}{dt^{N}}.

This is henceforth referred to as an N-order beta wavelet. They exist for order N\leq Min(\alpha ,\beta )-1. After some algebraic handling, their close expression can be found:

\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 ΨBETA(ω) of beta wavelets, | ΨBETA(ωα,β) | for Symmetric beta wavelet α = β = 3, α = β = 4, α = β = 5
Figure. Magnitude of the spectrum ΨBETA(ω) of beta wavelets, | ΨBETA(ωα,β) | \times \omega for Symmetric beta wavelet α = β = 3, α = β = 4, α = β = 5
Figure. Magnitude of the spectrum ΨBETA(ω) of beta wavelets, | ΨBETA(ωα,β) | for: Asymmetric beta wavelet α = 3, β = 4, α = 3, β = 5.
Figure. Magnitude of the spectrum ΨBETA(ω) of beta wavelets, | ΨBETA(ωα,β) | \times \omega for: Asymmetric beta wavelet α = 3, β = 4, α = 3, β = 5.

[edit] References

  • [1] H.M. de Oliveira, G.A.A. Araújo, Compactly Supported One-cyclic Wavelets Derived from Beta Distributions, Journal of Communication and Information Systems, (former Journal of the Brazilian Telecommunications Society), vol.20, n.3, pp.27-33, 2005.
  • [2] B.V. Gnedenko and A.N. Kolmogorov, Limit Distributions for Sums of Independent Random Variables, Reading, Ma: Addison-Wesley, 1954.
  • [3] W.B. Davenport, Probability and Random Processes, McGraw-Hill /Kogakusha, Tokyo, 1970.
  • [4] P.J. Davies, Gamma Fuction and Related Fuctions, in: M. Abramowitz; I. Segun (Eds.), Handbook of Mathematical Functions, New York: Dover, 1968.
  • [5] L.J. Slater, Confluent Hypergeometric Function, in: M. Abramowitz; I. Segun (Eds.), Handbook of Mathematical Functions, New York: Dover, 1968.
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