Computation of the Fractional Fourier Transform

Preprint, February 1, 2004

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Computation of the Fractional Fourier Transform
Adhemar Bultheel and H´ctor E. Mart´ e ınez Sulbaran 1
Dept. of Computer Science, Celestijnenlaan 200A, B-3001 Leuven

Abstract In this note we make a critical comparison of some matlab programs for the digital computation of the fractional Fourier transform that are freely available and we describe our own implementation that filters the best out of the existing ones. Two types of transforms are considered: First the fast approximate fractional Fourier transform algorithm for which two algorithms are available. The method is described in H.M. Ozaktas, M.A. Kutay, and G. Bozda˘i. Digital computation of the fractional Fourier transform. g IEEE Trans. Signal Process., 44:2141–2150, 1996. There are two implementations: one is written by A.M. Kutay the other is part of package written by J. O’Neill. Secondly the discrete fractional Fourier transform algorithm described in the master thesis C. Candan. The discrete fractional Fourier transform, ¸ Bilkent Univ., 1998 and an algorithm described by S.C. Pei, M.H. Yeh, and C.C Tseng: Digital fractional Fourier transform based on orthogonal projections IEEE Trans. Signal Process., 47:1335–1348, 1999. Key words: Fractional Fourier transform

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Introduction

The idea of fractional powers of the Fourier transform operator appears in the mathematical literature as early as 1929 [23,9,11]. Later on it was used in quantum mechanics [14,13] and signal processing [1], but it was mainly the optical interpretation and the applications in optics that gave a burst of publications since the 1990’s that culminated in the book of Ozaktas et al [17]. The reason for its success in optical applications can be explained as follows. Consider a system which consists of a point light source on the left. The light illuminates an object after traversing a set of optical components like e.g. thin lenses. It is then well known that at certain points to the

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