Operator theory-based discrete fractional Fourier transform

Abstract

The fractional Fourier transform is of importance in several areas of signal processing with many applications including optical signal processing. Deploying it in practical applications requires discrete implementations, and therefore defining a discrete fractional Fourier transform (DFRT) is of considerable interest. We propose an operator theory-based approach to defining the DFRT. By deploying hyperdifferential operators, a DFRT matrix can be defined compatible with the theory of the discrete Fourier transform. The proposed DFRT only uses the ordinary Fourier transform and the coordinate multiplication and differentiation operations. We also propose and compare several alternative discrete definitions of coordinate multiplication and differentiation operations, each of which leads to an alternative DFRT definition. Unitarity and approximation to the continuous transform properties are also investigated in detail. The proposed DFRT is highly accurate in approximating the continuous transform.

Optimal Fractional Fourier Filtering for Graph Signals

“…The diagonal matrix multiplication in the center can be performed in O(N ), where N is the number of samples in the signal. The discrete definition of the FRT and its fast computation are well-studied in the literature and the following references can be listed to name a few, [35], [67]- [74]. A detailed recent review on this issue can also be found in [75].…”