Affine smoothing of the Wigner-Ville distribution

A formalism of signal energy representations depending on time and scale is presented as an alternative method to time-frequency representations. Precise links between time-frequency and time-scale energy distributions are provided. It is known that a full description of the former is given by Cohen's class, which can be described as a generalization of the spectrogram appropriately parameterized by a smoothing function acting on the Wigner-Ville distribution. A full description of the latter is given, resulting in a class of representations in which the smoothing of the Wigner-Ville distribution is scale-dependent. Through proper choice of the smoothing function, interesting properties may be imposed on the representation, which makes it a versatile tool for the analysis of nonstationary signals. Also, specific choices allow known definitions to be recovered (including the Bertrands' and the energetic version of the wavelet transform, referred to as the scalogram). Another very flexible choice uses separable smoothing functions. It is shown, in particular, that Gaussian kernels provide a continuous transition between spectrograms and scalograms by means of the Wigner-Ville distribution