Page 290 - Introduction to Information Optics

5.7. Fractional Fourier Transform

Wigner distribution functions. The overall system performs a cascade of the
affine transformations of the Wigner distribution functions.

5.7. FRACTIONAL FOURIER TRANSFORM

The Fourier transform is among the most widely used transforms in
applications for science and engineering. The fractional Fourier transform has
been introduced in an attempt to get more power and a wider application circle
from the Fourier transform.

5.7.1. DEFINITION

In the framework of the fractional Fourier transform, the ordinary Fourier
transform is considered a special case of unity order of the generalized
fractional order Fourier transform. When defining such a transform, one wants
this fractional Fourier transform to be a linear transform and the regular
Fourier transform to a fractional Fourier transform of unity order. More
important, a succession of applications of the fractional Fourier transforms
should result in a fractional Fourier transform with a fractional order, which
is the addition of the fractional orders of all the applied fractional Fourier
transforms; i.e. in the case of two successive fractional Fourier transforms

where FT* denotes the fractional Fourier transform of a real valued order a.
The additive fractional order property can be obtained when one looks for
the eigenfunction of the transform. For this purpose, one considers an ordinary
differential equation
2 2
f"(x) + 4n [(2n + \}/2n - x \f(x) = 0. (5.25)
Taking the Fourier transform of Eq. (5.25) and using the properties of the
Fourier transforms of derivatives and moments one can show that
2
2
F"(u) + 4n [(2n + \)/2n - u \}F(u) = 0,
where F(u) =• FT[/(x)] is the Fourier transform of f(x). Hence, the Fourier
transform of the solution also is the solution of the same differential as Eq.
(5.25). Indeed, the solution of Eq. (5.25) is the Hermite -Gaussian function
[20], expressed as
2 1 /4
2
[j/ n(x) = —== H n(^/2n x) exp( - nx ),