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 signal analysis, we introduce a new unitary fractional operator associated with the FRFT. The new operator generalizes the unitary time-shift and frequency-shift operators by describing shifts at arbitrary orientations in the time-frequency (t-f) plane. We establish the connection with the FRFT by deriving two signal transformations, one invariant and one covariant, to the newly introduced unitary fractional operator. By using Stone´s theorem and the duality concept, we derive what we call the Hermitian fractional operator which also generalizes the well-known Hermitian time and frequency operators.