Novel convolution and correlation theorems for the fractional Fourier transform

The fractional Fourier transform (FRFT) plays an important role in many fields of optics and signal processing. Many results in Fourier analysis have been extended to the FRFT, including the convolution and correlation theorems. However, the convolution and correlation theorems don't have the elegance and simplicity comparable to that of the Fourier transform (FT). In this paper, we will propose a new convolution as well as correlation structure for FRFT which have similar time domain to frequency domain mapping results as the classical FT. First, we introduce a new convolution structure for the FRFT, which is expressed by a one dimensional integral and easy to implement in filter design. The conventional convolution theorem in FT domain is shown to be special cases of our achieved results. Then, we derive a convolution theorem for discrete time signal in discrete time FRFT (DTFRFT) domain. Last, based on the new convolution structure, we present a new kind of correlation operation for the FRFT that also generalizes very nicely the classical result for FT.