Enhanced Fractional Fourier Transform (FRFT) scheme based on closed Newton-Cotes rules

TL;DR

This paper introduces a novel approach to enhance the accuracy of the fractional Fourier transform by integrating closed Newton-Cotes quadrature rules, enabling more precise Fourier and Laplace transform inversions.

Contribution

It proposes a composite FRFT method based on Newton-Cotes weights, improving accuracy and demonstrating algebraic and numerical consistency over existing methods.

Findings

Composite FRFTs are commutative and consistent.

The method outperforms standard FRFT in Fourier and Laplace inversions.

Improvement over Newton-Cotes integration is notable but less pronounced.

Abstract

The paper improves the accuracy of the one-dimensional fractional Fourier transform (FRFT) by leveraging closed Newton-Cotes quadrature rules. Using the weights derived from the Composite Newton-Cotes rules of order QN, we demonstrate that the FRFT of a QN-long weighted sequence can be expressed as two composites of FRFTs. The first composite consists of an FRFT of a Q-long weighted sequence and an FRFT of an N-long sequence. Similarly, the second composite comprises an FRFT of an N-long weighted sequence and an FRFT of a Q-long sequence. Empirical results suggest that the composite FRFTs exhibit the commutative property and maintain consistency both algebraically and numerically. The proposed composite FRFT approach is applied to the inversion of Fourier and Laplace transforms, where it outperforms both the standard non-weighted FRFT and the Newton-Cotes integration method, though the improvement over the latter is less pronounced.