Affine smoothing of the Wigner-Ville distribution

A formalism of signal energy representations depending on time and scale is presented. 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