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 wavelet

Haar wavelet

In mathematics, the Haar wavelet is a certain sequence of rescaled "square-shaped" functions which together form a wavelet family or basis. Wavelet analysis is similar to Fourier analysis in that it allows a target function over an interval to be represented in terms of an orthonormal function basis...

s whose shape is fine-tuned by two parameters and . Closed-form expressions for beta wavelets and scale functions as well as their spectra are derived. Their importance is due to the Central Limit Theorem

Central limit theorem

In probability theory, the central limit theorem states conditions under which the mean of a sufficiently large number of independent random variables, each with finite mean and variance, will be approximately normally distributed. The central limit theorem has a number of variants. In its common...

 by Gnedenko&Kolmogorov applied for compactly supported signals [2].

Beta distribution

The beta distribution is a continuous probability distribution defined over the interval . It is characterised by a couple of parameters, namely and according to:

.

The normalising factor is ,

where is the generalised factorial function of Euler and is the Beta function [4].

Gnedenko-Kolmogorov central limit theorem revisited

Let be a probability density of the random variable , i.e.

, and .

Suppose that all variables are independent.

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

.

The mean and variance of are therefore and .

The density of the random variable corresponding to the sum is given by the

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

Let be distributions such that .

Let , and .

Without loss of generality assume that and .

The random variable holds, as ,


where 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 and are:





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 from the first to the second half cycle is given by



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

.

A closed-form 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 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 -order beta wavelet. They exist for order . After some algebraic handling, their closed-form expression can be found: