Beta wavelet

Continuous wavelets of compact support alpha can be built,[1] which are related to the beta distribution.

They can be viewed as a soft variety of Haar wavelets whose shape is fine-tuned by two parameters

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.

It is characterised by a couple of parameters, namely

is the generalised factorial function of Euler and

be a probability density of the random variable

The mean and the variance of a given random variable

is given by the Central Limit Theorem for distributions of compact support (Gnedenko and Kolmogorov).

Without loss of generality assume that

is unimodal, the wavelet generated by

{\displaystyle \psi _{beta}(t|\alpha ,\beta )=(-1){\frac {dP(t|\alpha ,\beta )}{dt}}}

The main features of beta wavelets of parameters

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

{\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 closed-form expression for first-order beta wavelets can easily be derived.

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

denote the Fourier transform pair associated with the wavelet.

It can be proved by applying properties of the Fourier transform that

cases have zeroes in the spectrum.

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

After some algebraic handling, their closed-form expression can be found:

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] and its derivative are utilized in several real-time engineering applications such as image compression,[5] bio-medical signal compression,[6][7] image recognition [9][8] etc.

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

Figure. Magnitude of the spectrum $\Psi _{BETA}(\omega )$ of beta wavelets, $|\Psi _{BETA}(\omega \alpha ,\beta )|$ $\times \omega$ for Symmetric beta wavelet $\alpha =\beta =3$ , $\alpha =\beta =4$ , $\alpha =\beta =5$

Figure. Magnitude of the spectrum $\Psi _{BETA}(\omega )$ of beta wavelets, $|\Psi _{BETA}(\omega \alpha ,\beta )|$ $\times \omega$ for: Asymmetric beta wavelet $\alpha =3$ , $\beta =4$ , $\alpha =3$ , $\beta =5$ .