A New Interpretation of Bilinear Time-Frequency Distributions

Wigner´s theorem states that there exists no bilinear time-frequency distribution (TFD) that has correct marginals and is nonnegative everywhere. This means that any attempt to interpret a bilinear TFD as an energy or power distribution must be fraught with problems. In this paper, an alternative perspective is proposed, which allows a local interpretation at a point in the time-frequency plane. This approach is based on analyzing the properties of a chirping ellipse that, at a given time instant, gives the best local approximation of the signal from a given frequency. This chirping ellipse is described in terms of its mean shape, orientation, and direction of polarization (counterclockwise or clockwise). A time-frequency coherence measures the quality of the approximation that this ellipse presents. The ellipse parameters and the time-frequency coherence can be expressed in terms of the Rihaczek TFD.