New convolution theorem for the linear canonical transform and its translation invariance property

The linear canonical transform (LCT), which is a generalization of the Fourier transform (FT), has many applications in several areas, including signal processing and optics. Many properties for this transform are already known, but an extension of convolution theorem of FT is still not having a widely accepted closed form expression. In the literature of recent past different authors have tried to formulate convolution theorem for LCT, but none have received acclamation because their definition do not generalize very nicely the classical result for the FT. Moreover, those definitions exhibit only partial invariance properties which prevent their actual use in many applications of signal processing. The purpose of this paper is to introduce a new convolution structure for the LCT that preserves the translation invariance property. Indeed, an effective translation invariance is obtained by slightly modifying the former definitions and by introducing linear canonical translation operators.