Mellin Convolution for Signal Filtering and Its Application to the Gaussianization of Lévy Noise

Noises are usually assumed to be Gaussian so that many existing signal processing techniques can be applied with no worry. However, in many real world natural or man-made systems, noises are usually heavy-tailed. It is increasingly desirable to address the problem of finding an opportune filter function for a given input noise in order to generate a desired output noise. By filtering theory, the probability density function of the output noise can be expressed by the integral of the product of the density of the input noise and the filter function. Adopting Mellin transformation rules, the Mellin transform of the unknown filter is determined by the Mellin transforms of the known density of the input noise and the desired density for the output noise. Finally, after the inversion, the Mellin–Barnes integral representation of the filter function is derived. The method is applied to compute the filter function to convert a Lévy noise into a Gaussian noise. INTRODUCTION In many real world natural or man-made systems, noises are usually heavy-tailed [1] with lots of spikes and for this they are difficult to be managed and should be further processed [2]. For example, the networked induced delays in networked control systems (NCS) are of spiky nature which hints that the generating ∗Address all correspondence to this author. dynamics should be characterized by fractional order differential models as pointed out in [3] based on [4]. Therefore, in fractional order control [5] and fractional order signal processing [1], it is increasingly desirable to address the problem of finding an opportune filter function for an input noise of a given distribution in order to generate an output noise of the desired distribution. In this contribution, noises with continuos probability density function (PDF) are taken into account. The Mellin transform tool is considered and in particular its convolution-type integral, where as convolution integral is intended that integral including the product of two functions whose transform is the product of the corresponding transformed functions. By filtering theory, the PDF of the output noise can be expressed by the integral of the product of the density of the input noise and the filter function. Adopting Mellin transformation rules, the Mellin transform of the unknown filter can be determined by the Mellin transforms of the known density of the input noise and the desired density for the output noise. Finally, after the inversion, the Mellin– Barnes integral representation of the filter function is derived. The method is applied to compute the filter function to convert a Lévy noise into a Gaussian noise. MELLIN CONVOLUTION FOR SIGNAL FILTERING Mellin transform is a powerful mathematical tool that better than other methods permits to successfully evaluate integrals [6– 1 Copyright c © 2011 by ASME DETC2011-47 Proceedings of the ASME 2011 International Design Engineering Technical Conferences & Computers and Information in Engineering Conference IDETC/CIE 2011 August 28-31, 2011, Washington, DC, USA