New higher order affine time-frequency representations

In the same spirit that the quadratic Wigner distribution and the Altes-Marinovic (1986, 1990) Q-distribution were extended to higher order time-frequency representations, we propose the new higher order Bertrand (1992) P₀-distribution (HO-P₀D), as an extension of the quadratic Bertrand P₀-distribution. We show that the new HO-P₀D preserves scale changes, and up to a known sign, constant time and hyperbolic time shifts on the signal. We also discuss the importance of the new HO-P₀D, derive some of its desirable properties, discuss its limitations, and derive a higher order class consisting of smoothed HO-P₀Ds. Finally, we propose a formulation for a higher order extension of the quadratic affine class which preserves scale changes and constant time shifts