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