The fractional Fourier transform

Researchers

• A. Bultheel
• H. Martinez

Description

integral kernel of FrFT for a = 0.75
Real and imaginary part

The fractional Fourier transform is an integral transform operator which can be considered as a fractional power of the classical Fourier transform. It has been intensely studied since early 1990. It can be realized naturally in optical systems. With a certain combination of optical components one obtains the classical Fourier transform of a signal at a certain distance from the lenses. At another distance a mirror image can be observed (that is twice the application of the classical Fourier operator), and continuing further, one obtains a mirror image of the Fourier image and then the original image will again emerge at a fourth position, andt this is cyclically repeated. What one obtains at the distances in between is an image that is a fractional Fourier transform of the original.

Mathematically, many other operators could be fractionalized as follows. Suppose one has a linear operator T on a Hilbert space with a complete set of orthonormal eigenvectors: T e_n = s_n e_n.

If A is the analysis operator A: f -> (c_n = < f,e_n>)
and S is the scaling operator S: (c_n) -> (d_n),
and A* is the synthesis operator: A^*: (d_n) -> g = sum_n d_n e_n

then T = A* S A is the spectral decomposition of T and the fractional version of the operator is the T^a = A* S^a A. If the set of eigenvectors is orthonormal, then the exponent acts as a genuine power of the operator.

Whereas the complex exponentials are the basic functions in classical Fourier analysis, the chirps (signals sweeping through a certain interval of frequencies) are the basic elements in in the fractional Fourier analysis.

It will be investigated what the good numerical computational procedures are to compute an approximation of the continuous fractional Fourier transform and of the discrete fractional Fourier transform. How much of the FFT techniques can be used or adapted in the computation?