Fractional convolution, fractional correlation and their translation invariance properties

Authors:
Rafael Torres;Pierre Pellat-Finet;Yezid Torres
Affiliations:
GOTS, Universidad Industrial de Santander, AA 678 Bucaramanga, Colombia;GOTA, Laboratoire de Mathématiques et Applications des Mathématiques, Université de Bretagne Sud, BP 92116, 56321 Lorient cedex, France and Département d'optique, UMR CNRS 6082 ...;GOTS, Universidad Industrial de Santander, AA 678 Bucaramanga, Colombia
Venue:
Signal Processing
Year:
2010

Citing 2
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Abstract

The fractional convolution and correlation of two functions, as usually defined, exhibit only partial invariance properties which prevent their actual use in many applications of signal processing. Indeed, an effective translation invariance is obtained by slightly modifying the former definitions and by introducing fractional translation operators. The translation invariances of the so-defined fractional convolution and correlation are expressed by integrals which are shown to be the exact counterparts, in the fractional domain, of the integrals that are used to express the well-known translation invariances of the usual convolution and correlation. The former theoretical results are applied to proving a fractional sampling theorem and are illustrated by numerical simulations. They are also compared to those obtained by some authors on the basis of alternative definitions.