A shattered survey of the Fractional Fourier Transform

In this survey paper we introduce the reader to the notion of the fractional Fourier transform, which may be considered as a fractional power of the classical Fourier transform. It has been intensely studied during the last decade, an attention it may have partially gained because of the vivid interest in time-frequency analysis methods of signal processing, like wavelets. Like the complex exponentials are the basic functions in Fourier analysis, the chirps (signals sweeping through all frequencies in a certain interval) are the building blocks in the fractional Fourier analysis. Part of its roots can be found in optics where the fractional Fourier transform can be physically realized. We give an introduction to the definition, the properties and computational aspects of both the continuous and discrete fractional Fourier transforms. We include some examples of applications and some possible generalizations.nrpages: 42status: publishe