The fractional Fourier transform is an extension of the ordinary Fourier transform and depends on a parameter a that can be interpreted as a rotation angle in the time-frequency plane . The Radon-Wigner transform (RWT) of a signal is the squared modulus of its fractional Fourier transform. This RWT is invertible, i.e., the amplitude and phase of the signal can be derived from its RWT for angles a in the region [0, p ].
We have found that the translational, scaling and rotational time-frequency symmetries of deterministic signals, as well as of random signals, result in the mutual similarity of the Radon-Wigner spectra for certain angles which are determined by the type of the symmetry. Several characteristic parameters of certain fractal (scale-invariant) signals, like the Hurst exponent and the scaling factor, as well as the hierarchical structure, can be extracted from the Radon-Wigner map.
Special attention was given to the analysis of the fractional Fourier transform of periodic signals. It was found that the fractional Fourier transform of a periodic signal at some angles is the superposition of some of its scaled, weighted and shifted replicas, with an additional quadratic phase factor. The Radon-Wigner spectra for a certain set of angles are affine.
Another part of the research dealt with eigenfunctions and eigenvalues of the
fractional Fourier transform for a given angle: the so-called self-fractional
Fourier functions (SFFFs). Signals which are SFFFs exhibit rotational
symmetry in the time-frequency plane. It has been shown that any signal can
be represented as a linear combination of M SFFFs for angle 2p/M, which are
orthogonal to each other. Each of them contains a selection of Hermite-Gauss
modes of the generator function of the SFFFs. A procedure for the
construction of a SFFF with a given eigenvalue has been proposed. It
corresponds to Hermite-Gauss mode filtering of the related generator
function, which in the limit case 
reduces to a single mode selection.