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

The Bertrand (1991) distribution, which is a member of the affine class of time-frequency distributions (TFDs), and the Altes (1970, 1990) distribution, which does not belong to any known class of TFDs, are studied. It is shown that both TFDs are closely related to a hyperbolic time-frequency geometry. Based on this geometry, a new hyperbolic class of TFDs which contains both the Bertrand distribution and the Altes distribution is defined and studied. It is shown that the hyperbolic class can be derived from the Cohen's (1966) class of TFDs by a frequency-warping procedure that potentially results in a constant-Q time-frequency analysis.<
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