Invertible time-frequency representations

In this paper, we present a new class of representations of signals in the time-frequency (TF) plane. These representations are complex valued, linear, and satisfy reconstruction conditions in which the signal and its complex spectrum may be uniquely reconstructed from their TF representation. These surfaces are generalizations of one-dimensional linear transforms with which they share many properties.. The primary advantage of these representations is that the phase of the surface may be used to recover signal information which is not contained in real TF surfaces. Linearity guarantees that cross-terms normally associated with TF distributions do not exist in these representations. Several examples of invertible surfaces are presented, and it is demonstrated that these surfaces agree with normal intuition. Finally, a method, based on the phase gradient, is proposed as a method of modifying Fourier surfaces to produce representations which are more focused or more concentrated in time and frequency.