Joint fractional signal representations

Using the recently introduced Hermitian fractional operator within the characteristic function operator method, we derive joint fractional representations (JFRs) of signals. JFRs are functions of fractional variables defined by the fractional Fourier transform (FRFT). The JFRs generalize the conventional time–frequency representations in the same manner as the FRFT generalizes the conventional Fourier transform. We derive the fractional counterparts of the well-known time–frequency analysis tools such as the ambiguity function (AF) and the Wigner distribution (WD) and present some of their properties. We also analytically compute the fractional AF and the fractional WD of some simple functions and provide plots for a Gaussian amplitude-modulated chirp (linear FM) and a rectangular function.