New product and correlation theorems for the offset linear canonical transform and its applications

The offset linear canonical transform (OLCT) plays an important role in optic signal processing. Many properties for this transform are already known, however, the product and correlation theorems doesn't have the simplicity and elegance comparable to that of the Fourier transform (FT). In this paper, we will introduce new product and correlation theorems for the OLCT, which have similar results for FT. First, the convolution and product theorem is introduced which states that a generalized convolution in the time domain is equivalent to simple multiplication operations for OLCT with a scaling factor. The new product structure does exactly parallel the product theorem for the FT. Moreover, the classical convolution and product theorem in the FT domain is shown to be the special case of our derived results. Furthermore, we propose a new correlation operation based on the convolution structure.Then, using the introduced convolution structure, we investigate the sampling theorem for the band-limited signal in the OLCT domain. We also discuss the applications of the new convolution for designing of multiplicative filter in the OLCT domain.