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 .
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) , .
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.
 B.V. Gnedenko and A.N. Kolmogorov, Limit Distributions for Sums of Independent Random Variables, Reading, Ma: Addison-Wesley, 1954.
 W.B. Davenport, Probability and Random Processes, McGraw-Hill /Kogakusha, Tokyo, 1970.
 P.J. Davies, Gamma Function and Related Functions, in: M. Abramowitz; I. Segun (Eds.), Handbook of Mathematical Functions, New York: Dover, 1968.
 L.J. Slater, Confluent Hypergeometric Function, in: M. Abramowitz; I. Segun (Eds.), Handbook of Mathematical Functions, New York: Dover, 1968.
 B.C. Amar, M. Zaied, M.A. Alimi, "Beta wavelet synthesis and application to lossy image compression" Adv Eng Softw, 36 (2005), pp. 459–474
 Ranjeet Kumar, A. Kumar and Rajesh K. Pandey, “Electrocardiogram Signal compression Using Beta Wavelets” Journal of Mathematical Modeling and Algorithms, Vol. 11, pp. 235–248, 2012.
 Ranjeet Kumar, A. Kumar and Rajesh K Pandey “Beta Wavelet Based ECG Signal Compression using Loss-less Encoding with Modified Thresholding” Computers & Electrical Engineering, Vol. 39, Issue. 1, pp. 130– 140, 2013.
 Zaied, M., Jemai, O., Ben Amar, C., "Training of the Beta wavelet networks by the frames theory: Application to face recognition", Image Processing Theory, Tools and Applications, 2008. DOI: 10.1109/IPTA.2008.4743756