Consider a one-dimensional signal x(n), and it's Fourier transform F{x(n)}=X(w). It is straight-forward to show that applying the Fourier operator twice yields a time reversed copy of the original. Of course, applying the Fourier operator a third time yields the Fourier transform of the time reversed signal, and applying it a fourth time yields the original signal. Denoting superscripts as the number of applications of the Fourier operator, we have:

F¹{x(n)}=X(w)   |   F²{x(n)} = x(-n)   |   F³{x(n)}=X(-w)   |   F⁴{x(n)}=x(n)

We need not restrict ourselves to integer mulitple applications of the Fourier operator. The Fourier operator can be applied in fractional increments. The easiest way to see this is to think of finite length signals and the Fourier transform as a matrix operation Mx=X, where the rows of the matrix M contain the Fourier basis. Then the integer and fractional applications of the Fourier operator amount to simply raising this matrix to the desired power.

Shown below is a fractal signal (0.00), its first (1.00) and second-order (2.00) Fourier transform (magnitude) and the intermediate fractional Fourier transforms in increments of 0.2. The last panel illustrates a continuum of Fourier transforms from first to fourth (top to bottom) "stacked" on top of one another.



Matlab routine for computing fractional Fourier transform.