FRACTIONAL FOURIER DOMAINS

It is customary to define the time-frequency plane such that time and frequency are mutually orthogonal coordinates. Representations of a si gnal in these domains are related by the Fourier transform. We conside r a continuum of ''fractional'' domains making arbitrary angles with t he time and frequency domains. Representations in these domains are re lated by the fractional Fourier transform. We derive transformation, c ommutation, and uncertainty relations among coordinate multiplication, differentiation, translation, and phase shift operators between domai ns making arbitrary angles with each other. These results have a simpl e geometric interpretation in time-frequency space.