A convolution and product theorem for the fractional Fourier transform

TLDR

The fractional Fourier transform (FRFT) generalizes the Fourier transform and is widely used in signal processing and optics, but existing convolution formulas for the FRFT do not recover the classical convolution theorem. This paper introduces a new convolution structure for the FRFT. The structure preserves the classical convolution theorem and is straightforward to implement in filter design. See Appl.

Abstract

The fractional Fourier transform (FRFT), which is a generalization of the Fourier transform, has many applications in several areas, including signal processing and optics. Almeida (see ibid., vol.4, p.15-17, 1997) and Mendlovic et al. (see Appl. Opt., vol.34, p.303-9, 1995) derived fractional Fourier transforms of a product and of a convolution of two functions. Unfortunately, their convolution formulas do not generalize very well the classical result for the Fourier transform, which states that the Fourier transform of the convolution of two functions is the product of their Fourier transforms. This paper introduces a new convolution structure for the FRFT that preserves the convolution theorem for the Fourier transform and is also easy to implement in the designing of filters.