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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 right of the object one may observe images that are the Fourier transform of the object image. Somewhat further it is the inverted object image, still further it becomes the inverted Fourier transform and still further it is the upright image etc. These images are obtained by the Fourier operator applied to the object image, its second power (the inverted image), the 3rd power (the 1

The work of the ﬁrst author is partially supported by the Fund for Scientiﬁc Research (FWO), projects “CORFU: Constructive study of orthogonal functions”, grant #G.0184.02 and, “SMA: Structured matrices and their applications”, grant G#0078.01, “ANCILA: Asymptotic analysis of the convergence behavior of iterative methods in numerical linear algebra”, grant #G.0176.02, the K.U.Leuven research project “SLAP: Structured linear algebra package”, grant OT-00-16, the Belgian Programme on Interuniversity Poles of Attraction, initiated by the Belgian State, Prime Minister’s Oﬃce for Science, Technology and Culture. The scientiﬁc responsibility rests with the author.

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inverted Fourier image) and the 4th power (the original image), etc. The images in between are the result of intermediate (fractional) powers of the Fourier operator applied to the image object. Like for the Fourier transform, there exists a discrete version of the fractional Fourier transform. It is based on an eigenvalue decomposition of the discrete Fourier transform matrix. If F = EΛE −1 is this decomposition then F a = EΛa E −1 is the corresponding discrete fractional Fourier transform. As far as we know, there are not many public domain software routines available for the computation of the (discrete) fractional Fourier transform. We are aware of only two routines. There is the one that can be found on the web site [12] of the book [17] previously described in Ozaktas [16] and another one which is part of a package compiled by O’Neill. In this note we shall analyse the advantages and disadvantages of the two...