ch13 - Almanasrah, A.M., Abushagur, M. Fractional Fourier...

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Almanasrah, A.M . , Abushagur, M. Fractional Fourier Transform s .” The Transforms and Applications Handbook: Second Edition. Ed. Alexander D. Poularikas Boca Raton: CRC Press LLC, 2000
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© 2000 by CRC Press LLC 13 Fractional Order Fourier Transforms 13.1 Introduction 13.2 The Fractional Order Fourier Transform Definition Two Important Properties of the FRFT 13.3 Generalized Operational Calculus Multiplication Rule • Differentiation Rule • Shift Rule • Exponential Rule 13.4 Applications 13.5 FRFT Examples 13.1 Introduction Fourier transform (FT) is the most widely used transform. It has applications in many fields of science and engineering. The ordinary Fourier transform as it is known is a special case, of unity order, of the fractional order Fourier transform. It is natural that a mathematical operation, with a meaningful order of operation, originally be defined for integer orders. It is also true that the generalized operation becomes more powerful and enlarges the circle of applications that can be utilized by such an operation. A very simple yet basic example is the power operation. The example grows to be more complicated when considering derivatives. It is quite meaningful to talk about the first and the second derivative and so on. However, the definition can be expanded to cover the fractional orders. This can be made plausible by the fundamental definition of the FT of a derivative. Likewise the integration operation was treated to confront non–integer orders. This motivated mathematicians to apply the same argument to other operations. The FT has gotten special attention because of its unique role in mathematics and applied physics. V. Namias 1 introduced this concept to the FT under which the original FT is considered as being the fractional order Fourier transform (FRFT) of unity order. A review article of the subject has recently appeared in the literature in Reference 2. In this chapter we present some of the basic fundamentals of the FRFT. We also summarize the rigorous revision of the original derivation where a mathematical framework was added to the FRFT. Finally, applications and a number of examples of the FRFT are presented. 13.2 The Fractional Order Fourier Transform Definition The ordinary Fourier transform is mostly known through its famous pair of equations (Equations 13.1 and 13.2). However, in theoretical considerations, it is very useful to introduce the transform as a linear operator acting on some function. Since it is more convenient to work with one dimension, we present the following derivation in one dimension only, understanding that the two dimensional expansion is a straightforward process.
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ch13 - Almanasrah, A.M., Abushagur, M. Fractional Fourier...

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