Uncertainty and spectrogram geometry

Ultimate possibilities of localization for time-frequency representations are first reviewed from a joint perspective, evidencing that Heisenberg-type pointwise limits are not exclusive of sharp localization along trajectories in the plane. Spectrogram reassignment offers such a possibility and, in order to revisit its connection with uncertainty, geometrical properties of spectrograms are statistically investigated in the generic case of white Gaussian noise. Based on Voronoi tessellations and Delaunay triangulations attached to extrema, it is shown that, in a first approximation, local energy “patches” are distributed according to a randomized hexagonal lattice with a typical scale within a factor of a few that of minimum uncertainty Gabor logons.