Self Fourier functions and fractional Fourier transforms

It was shown [ 21 that any SFF can be decomposed in this manner. Thus, F(x) is an SFF if, and only if, it can be expressed as the sum of four functions in the form of the above equation. Additional SFF studies are reported in refs. [ 3-51. Another issue that has been recently investigated is the fractional Fourier transform [ 6-91. Two distinct definitions of the fractional Fourier transform have been given. In the first one [ 6-81, the fractional Fourier transform was defined physically, based on the propagation in quadratic graded index (GRIN) media. The second definition is based on Wigner distribution functions (WDF) [ 91. Here the fractional Fourier transform is calculated by finding the WDF of the input image, rotating it by an angle (~=arr/2, and performing the inverse Wigner transform. It was shown [ lo] that both definitions of the fractional Fourier transform are equivalent. In this communication, we first discuss fractional Fourier transforms of SFFs and then discuss self fractional Fourier functions ( SFFF’s). Assuming an SFF F(x) with a generating function g(x), one can rewrite eq. (4) as