Beta wavelet

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 $alpha$ and . 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] .

Beta distribution

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

.

The normalising factor is ,

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

Gnedenko-Kolmogorov central limit theorem revisited

Let $p\_\{i\}(t)$ be a probability density of the random variable $t\_\{i\}$, $i=1,2,3..N$ 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 $t\_\{i\}$ are, respectively

$m\_\{i\}=int\_\{-infty\; \}^\{+infty\; \}\; au\; cdot\; p\_\{i\}(\; au\; )d\; au\; ,$ $sigma\; \_\{i\}^\{2\}=int\_\{-infty\; \}^\{+infty\; \}(\; au\; -m\_\{i\})^\{2\}cdot\; p\_\{i\}(\; au\; )d\; au$.

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 $\{p\_\{i\}(t)\}$ be distributions such that $Supp\{(p\_\{i\}(t))\}=(a\_\{i\},b\_\{i\})(forall\; i)$.

Let , and .

Without loss of generality assume that $a=0$ and $b=1$. The random variable $t$ holds, as $N\; ightarrow\; infty$,$p(t)approx$

where $alpha\; =frac\{m(m-m^\{2\}-sigma\; ^\{2\})\}\{sigma\; ^\{2,$ and

Beta wavelets

Since is unimodal, the wavelet generated by

has only one-cycle (a negative half-cycle and a positive half-cycle).

The main features of beta wavelets of parameters $alpha$ and are:

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

The parameter 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 $t\_\{zerocross\}$ from the first to the second half cycle is given by

The (unimodal) scale function associated with the wavelets is given by

$aleq\; tleq\; b$.

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

Beta wavelet spectrum

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

Let denote the Fourier transform pair associated with the wavelet.

This spectrum is also denoted by $Psi\; \_\{BETA\}(omega)$ for short. It can be proved by applying properties of the Fourier transform that

where .

Only symmetrical cases have zeroes in the spectrum. A few asymmetric beta wavelets are shown in Fig. Inquisitively, they are parameter-symmetrical in the sense that they hold

Higher derivatives may also generate further beta wavelets. Higher order beta wavelets are defined by

This is henceforth referred to as an $N$-order beta wavelet. They exist for order . After some algebraic handling, their close expression can be found:

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", vol.20, n.3, pp.27-33, 2005.

* http://www.iecom.org.br/
* http://www2.ee.ufpe.br/codec/WEBLET.html
* http://www2.ee.ufpe.br/codec/beta.html

* [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 Function and Related Functions, 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.