Unitary and Hermitian fractional operators and their relation to the fractional Fourier transform

Inspired by the recent popularity of the fractional Fourier transform (FRFT ) and motivated by the use of Hermitian and unitary operator methods in sig nal analysis. We introduce a new unitary fractional operator associated wit h the FRFT, The new operator generalizes the unitary time-shift and frequen cy-shift operators by describing shifts at arbitrary orientations in the ti me-frequency (t-f) plane. We establish the connection with the FRFT by deri ving two signal transformations, one invariant and one covariant, to the ne wly introduced unitary fractional operator. By using Stone's theorem and th e duality concept, we derive what we call the Hermitian fractional operator which also generalizes the well-known Hermitian time and frequency operato rs.