The fractional Fourier transform and time-frequency representations

The functional Fourier transform (FRFT), which is a generalization of the classical Fourier transform, was introduced a number of years ago in the mathematics literature but appears to have remained largely unknown to the signal processing community, to which it may, however, be potentially useful. The FRFT depends on a parameter α and can be interpreted as a rotation by an angle α in the time-frequency plane. An FRFT with α=π/2 corresponds to the classical Fourier transform, and an FRFT with α=0 corresponds to the identity operator. On the other hand, the angles of successively performed FRFTs simply add up, as do the angles of successive rotations. The FRFT of a signal can also be interpreted as a decomposition of the signal in terms of chirps. The authors briefly introduce the FRFT and a number of its properties and then present some new results: the interpretation as a rotation in the time-frequency plane, and the FRFT´s relationships with time-frequency representations such as the Wigner distribution, the ambiguity function, the short-time Fourier transform and the spectrogram. These relationships have a very simple and natural form and support the FRFT´s interpretation as a rotation operator. Examples of FRFTs of some simple signals are given. An example of the application of the FRFT is also given