Reduced Order Fractional Fourier Transform A New Variant to Fractional Signal Processing Definition and Properties

TL;DR

This paper introduces the Reduced Order Fractional Fourier Transform, a simplified variant with derived properties, that reduces to the conventional Fourier transform at 90 degrees and simplifies convolution operations.

Contribution

A new Reduced Order Fractional Fourier Transform is proposed, offering easier analytical handling and a simplified convolution theorem compared to traditional fractional Fourier transforms.

Findings

The transform reduces to the conventional Fourier transform at 90 degrees.

Analytical expressions for various signals are derived in the reduced form.

The convolution theorem simplifies to multiplication in the fractional frequency domain.

Abstract

In this paper, a new variant to fractional signal processing is proposed known as the Reduced Order Fractional Fourier Transform. Various properties satisfied by its transformation kernel is derived. The properties associated with the proposed Reduced Order Fractional Fourier Transform like shift, modulation, time-frequency shift property are also derived and it is shown mathematically that when the rotation angle of Reduced Order Fractional Fourier Transform approaches 90 degrees, the proposed Reduced Order Fractional Fourier Transform reduces to the conventional Fourier transform. Also, the Reduced Order Fractional Fourier Transform of various kinds of signals is also derived and it is shown that the obtained analytical expressions of different Reduced Order Fractional Fourier Transform are a reduced form of the conventional fractional Fourier transform. It is also shown that proposed definition of Fractional Fourier Transform is easier to be handled analytically. Finally, the convolution theorem associated with the proposed Reduced Order Fractional Fourier Transform is derived with its various properties like shift convolution, modulation convolution, and time-frequency shift modulation properties. It has been shown that with this proposed new definition of fractional Fourier transform, the convolution theorem has been reduced to multiplication in the fractional frequency domain.