Quadratic time-frequency distributions: the new hyperbolic class and its intersection with the affine class

The proposed new class of quadratic time-frequency distributions is based on the 'hyperbolic time shift' and scale invariance properties that are important in the analysis of Doppler invariant signals used in bat and dolphin echolocation, and of 'locally self-similar' signals used in fractals and fractional Brownian motion. The hyperbolic class can be characterized by 2-D kernels, and kernel constraints are derived for some desirable TFD properties. The Bertrand distribution and the Altes distribution are members of the hyperbolic class. The authors define a 'localized' subclass and study the intersection between the affine class and the hyperbolic class.<>