Time-scale energy distributions: a general class extending wavelet transforms

The theory of a new general class of signal energy representations depending on time and scale is developed. Time-scale analysis has been introduced recently as a powerful tool through linear representations called (continuous) wavelet transforms (WTs), a concept for which an exhaustive bilinear generalization is given. Although time scale is presented as an alternative method to time frequency, strong links relating the two are emphasized, thus combining both descriptions into a unified perspective. The authors provide a full characterization of the new class: the result is expressed as an affine smoothing of the Wigner-Ville distribution, on which interesting properties may be further imposed through proper choices of the smoothing function parameters. Not only do specific choices allow recovery of known definitions, but they also provide, via separable smoothing, a continuous transition from Wigner-Ville to either spectrograms or scalograms (squared modulus of the WT). This property makes time-scale representations a very flexible tool for nonstationary signal analysis