A closed form expression for the Gaussian-based Caputo-Fabrizio fractional derivative for signal processing applications

The Gaussian function has been employed in a vast number of practical and theoretical applications since it was proposed. Likewise, Gaussian function and its ordinary derivatives are considered as powerful tools for signal processing and control applications, e.g., smoothing, sampling, change detection, blob detection, and transforms based on the Hermite polynomials. Nonetheless, it has impressive characteristics hidden amongst its fractional derivatives eager to be explored and studied in-depth. This work proposes a closed formula for the (n+ν)-order fractional derivative of the Gaussian function, based on the Caputo-Fabrizio definition, as an approach for analysing those attributes. The obtained expression was numerically tested with several fractional orders, and their resulting behaviours were eventually analysed. Finally, three practical applications on signal processing via this closed formula were discussed, i.e., customisable wavelets, image processing filters, and Rayleigh distributions.