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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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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