A unified framework for the Bertrand distribution and the Altes distribution: The new hyperbolic class of quadratic time-frequency distributions

We study the Bertrand distribution which is a member of the affine class of time-frequency distributions (TFDs), and the Altes distribution which does not belong to any known class of TFDs. We show that both TFDs are closely related to a hyperbolic time-frequency geometry. Based on this geometry, we define and study a new hyperbolic class of TFDs which contains both the Bertrand distribution and the Altes distribution. We show that the hyperbolic class can be derived from the well-known Cohen\u27s class of TFDs by a frequency-warping procedure that potentially results in a constant-Q time-frequency analysis