Continuous wavelets of compact support can be built , 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 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 by Gnedenko and Kolmogorov applied for compactly supported signals .
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 .
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) .
Let be distributions such that .
Let , and .
Without loss of generality assume that and .
The random variable holds, as ,
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,
Figure. Unicyclic beta scale function and wavelet for different parameters: a) , b) , c) , .
Beta wavelet spectrum
The beta wavelet spectrum can be derived in terms of the Kummer hypergeometric function .
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
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:
Figure. Magnitude of the spectrum of beta wavelets, for Symmetric beta wavelet , ,
Figure. Magnitude of the spectrum of beta wavelets, for: Asymmetric beta wavelet , , , .
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  and its derivative are utilized in several real-time engineering applications such as image compression,bio-medical signal compression, image recognition  etc.
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