A convolution and product theorem for the fractional Fourier transform

IEEE SIGNAL PROCESSING LETTERS, VOL. 5, NO. 4, APRIL 1998

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A Convolution and Product Theorem for the Fractional Fourier Transform
Ahmed I. Zayed
Abstract― The fractional Fourier transform (FRFT), which is a generalization of the Fourier transform, has many applications in several areas, including signal processing and optics. In two recent papers, Almeida and Mendlovic et al. derived fractional Fourier transforms of a product and of a convolution of two functions. Unfortunately, their convolution formulas do not generalize very nicely the classical result for the Fourier transform, which states that the Fourier transform of the convolution of two functions is the product of their Fourier transforms. The purpose of this note is to introduce a new convolution structure for the FRFT that preserves the convolution theorem for the Fourier transform and is also easy to implement in the designing of ?lters. Index Terms― Convolution and product theorems, fractional Fourier transform.

Let be that subspace of the space of all integrable if and only if the functions with the property that Let and be in Fourier transform of is also in and denote their convolution by , i.e., (2) Then the fractional Fourier transform of , denoted by is given by ,

I. INTRODUCTION HE fractional Fourier transform (FRFT) has become the focus of many research papers in the last four years because of its recent applications in many ?elds, including optics and signal processing 쭯, , , , . is de?ned as The FRFT with angle of a signal (1) where we have the formulation shown on the bottom of the next page, with

(3) is the same See [1, Eq. (5)]. It should be noted that the space in Almeida’s notation, where is as the space the Wiener algebra consisting of functions that are Fourier transforms of functions in As for the FRFT of the product of two functions and i.e., we have (cf., [1, Eq. (2)])

T

(4) Unlike the convolution theorem for the Fourier transform, which states that the Fourier transform of the convolution of two functions is the product of their Fourier transforms, the one for the FRFT does not seem as nice or as practical. The reason, in our opinion, is that the convolution operation de?ned by (2) is not the right sort of convolution for the FRFT. In the general framework of convolution theory (see [13, Ch. 4]), it is known that to every integral transformation , one can, at least theoretically, associate with it a convolution operation, , such that (5) For example, the convolution operation associated with the Hankel transform is too complicated to be stated here, but the interested reader can ?nd the details in [13, Sect. 21.6]. In this letter, we propose a new convolution structure for the FRFT that is different from those introduced in  and . Unlike those introduced in  and , ours preserves property (5) and is easier to implement, in particular, in ?lter design.

and

Throughout this paper the constants, , and will denote these values, and for simplicity we may write them as , and . The special cases where , and yield the following FRFT of , where denotes the ordinary Fourier transform of . Therefore, from now on we shall con?ne our for . attention to Many properties of the FRFT are currently well known, including its product and convolution theorems, which have recently been derived by Almeida . His main result reads as follows.
Manuscript received November 30, 1997. The associate editor coordinating the review of this manuscript and approving it for publication was Prof. A. Tew?k. The author is with the Department of Mathematics, University of Central Florida, Orlando, FL 32816 USA (e-mail: zayed@pegasus.cc.ucf.edu). Publisher Item Identi?er S 1070-9908(98)02847-8.

1070쭯9908/98\$10.00 ? 1998 IEEE

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IEEE SIGNAL PROCESSING LETTERS, VOL. 5, NO. 4, APRIL 1998

By making the change of variable,

, we obtain

Fig. 1. Convolution for the FRFT.

II. CONVOLUTION THEOREM Let us introduce the following de?nition. , let us de?ne the De?nition 1: For any function functions and by and For any two functions and , we de?ne the convolution operation by

and this completes the proof of (6). As for (7), we have from De?nition 1

where is the convolution operation for the Fourier transform as de?ned by (2). Likewise, we de?ne the operation by But from the de?nition of the FRFT, we obtain

See Fig. 1 for a realization of the convolution operation . Now we state and prove our convolution theorem. Theorem 1: Let and denote the FRFT of and , respectively. Then (6) Moreover (7) Proof: From the de?nition of the FRFT and De?nition 1, we have which is the same as (7). Equation (6) is particularly useful in ?lter design. For example, if we are interested only in the frequency spectrum of a signal , we choose of the FRFT in the region is constant over the ?lter impulse response, , so that , and zero or of rapid decay outside that region. Passing , the output of the ?lter through the chirp multiplier, over . This is yields that part of the spectrum of clearly easier to implement than the one suggested in . which, in view of the inversion formula for the FRFT, can be reduced to

if if if if

ZAYED: FRACTIONAL FOURIER TRANSFORM

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Equation (7), which is the dual of (6), does not seem to have an immediate application in signal processing, but products of similar nature have proved to be useful in optics; see . REFERENCES
 L. B. Almeida, “Product and convolution theorems for the fractional Fourier transform,” IEEE Trans. Signal Processing Letters, vol. 4, pp. 15쭯17, 1997.  , “An introduction to the angular Fourier transform,” in Proc. IEEE Conf. Acoustics, Speech, Signal Processing , Minneapolis, MN. Apr. 1993. , “The fractional Fourier transform and time-frequency repre sentations,” IEEE Trans. Signal Processing, vol. 42, pp. 3084쭯3091, 1994.  T. Alieva, V. Lopez, F. Aguillo-Lopez, and L. B. Almeida, “The angular fourier transform in optical propagation problems,” J. Mod. Opt., vol. 41, pp. 1037쭯1040, 1994.  A. W. Lohmann, “Image rotation, Wigner rotation and the fractional fourier transform,” J. Opt. Soc. Amer. A, vol. 10, pp. 2181쭯2186, 1993.

 A. W. Lohmann and B. H. Soffer, “Relationships between the Radon―Wigner and fractional Fourier transforms,” J. Opt. Soc. Amer. A, vol. 11, pp. 1798쭯1801, 1994.  D. Mendlovic, H. M. Ozaktas, and A. Lohmann, “Fractional correlation,” Appl. Opt., vol. 34, pp. 303쭯309, 1995.  D. Mendlovic and H. M. Ozaktas, “Fractional Fourier transformations and their optical implementation I,” J. Opt. Soc. Amer. A, vol. 10, pp. 1875쭯1881, 1993.  X. Xia, “On bandlimited signals with fractional Fourier transform,” IEEE Signal Processing Lett., vol. 3, pp. 72쭯74, 1996.  H. M. Ozaktas, B. Barshan, D. Mendlovic, and L. Onural, “Convolution, ?ltering, and multiplexing in fractional Fourier domains and their relationship to chirp and wavelet transforms,” J. Opt. Soc. Amer. A, vol. 11, pp. 547쭯559, 1994.  H. M. Ozaktas and D. Mendlovic, “Fractional Fourier optics,” J. Opt. Soc. Amer. A, vol. 12, pp. 743쭯751, 1995. , “Fourier transforms of fractional order and their optical inter pretation,” Opt. Commun, vol. 101, pp. 163쭯169, 1993.  A. I. Zayed, Function and Generalized Function Transformations. Boca Raton, FL: CRC, 1996.

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