310 IEEE SIGNAL PROCESSING LETTERS, VOL. 3, NO. 12, DECEMBER 1996 elationship Between the Fourier Fractional Fourier Transforms Ahmed I. Zayed Abstract- In recent years, the fractional Fourier transform where we obtain the expression shown on the top of the next has been the focus of many research papers. In this letter, we page, with show that the fractional Fourier transform is nothing more than a variation of the standard Fourier transform and, as such, many a(.) = (cot a)/2, b(a) = seca of its properties, such as its inversion formula and sampling theorems, can be deduced from those of the Fourier transform and by a simple change of variable. e(.) = di-qEG. Throughout this paper the constants a(ol), b(a), and c(.) will denote these values, and for simplicity we shall supress their HE fractional Fourier transform (FRFT) has been inves- dependency on a! and write them as a, b and c. Special cases, where a = 0, n/2and n, yield the following tigated in a number of papers [l], [2], [5] and has proved to be very useful in solving some problems in quantum physics FRFT of f: [7], optics, and signal processing [3], [4], [6], [lo], [ll]. Its connection with the wavelet transform has also been recently reported [9]. In a very recent paper [8], Xia derived sampling where f denotes the ordinary Fourier transform of f. There- expansions for the fractional Fourier transform of band-limited fore, from now on we shall confine our attention to Fa for and time-limited signals. a! # O,n/2,n. The purpose of this letter is to show that the so-called FRFT is nothing more than a variation of the standard Fourier 111. THE MAIN RESULT transform. As such, many of its properties, such as its inversion formula and sampling theorems for band-limited and time- The main result of this paper will be stated as a theorem. limited signals, can be derived very easily from those of Theorem 1: The FRFT with angle a of a signal f for the Fourier transform by a simple change of variable. To # O,n/2,7r demonstrate the idea, we shall derive the inversion formula of the FRFT and a sampling theorem recently obtained by Xia from the corresponding results in the Fourier analysis. can be reduced to the standard Fourier transform I. INTRODUCTION 11. PRELIMINARIES i roo We define the Fourier transform of a function f as where so that its inverse is with The fractional Fourier transform with angle a! of a signal f(t) is defined as 00 and Manuscript received June 12, 1996. The associate editor coordinating the review of this manuscript and approving it for publication was Dr. M. Unser. The author is with the Department of Mathematics, University of Central Florida, Orlando, FL 32816 USA (e-mail: fdzayed at ucf1vm.cc.ucf.edu). Publisher Item Identifier S 1070-9908(96)09034-7. Proofi The proof is straightforward and left to the reader to verify. 0 Many properties of the FRFT, such as its inversion formula and sampling theorems, can now be deduced from those of the Fourier transform by a simple change of variable. To illustrate the idea, we shall derive the inversion formula and the sampling expansion of the FRFT from that of (1) as corollaries. 1070-9908/96$05.00 0 1996 IEEE ZAYED: FOURIER AND FRACTIONAL FOURIER TRANSFORMS 311 CoroZZary2: The inversion formula of the FRFT can be ul(2ab) = usina, we obtain derived from that of the Fourier transform (1). Proof From (1) and (2), we have If we replace U sin a by t, we obtain 00 F,[f](t) = ejt2cota/2 e-jau2, sin’ cy Fa [fI(% sin a) which, in view of the relations n=--03 t,)] U(-.) = -a(a), b(-a) = b(a) X sin[acsccy(t -acsca(t - t,) and c(-a)= Jl+jcota where t, = nrsina/a. This last formula is the same as Formula (9) in [8], except for the constant c, which has been leads to [8, eq. (5)] since absorbed in the function ga. 0 REFERENCES L. B. Almeida, “An introduction to the angular Fourier transform,” in Proc. Con$ IEEE Acoust., Speech, Signal Processing, Minneapolis, MN, Apr. 1993. L. B. Almeida, “The fractional Fourier transform and time-frequency representations,” IEEE Trans. Signal Processing, vol. 42, pp. 3084-3091, 1994. T. Alieva, V. Lopez, F. Aguillo-Lopez, and L. B. Almeida, “The angular Likewise, the sampling theorems for the FRFT of band- Fourier transform in optical propagation problems,” J. 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