A contribution to the theory of short-time spectral analysis with nonuniform bandwidth filters

The mathematical representation of the short-time spectral analysis is extended from the case of uniform bandpass filters (that is, filters having the same complex envelope) to the case of nonuniform filters (that is, filters whose complex envelope depends upon their center frequency). This leads to an integral transform, formally similar to the Fourier transform, where the signal taken up to the observation time appears weighted by a function (namely, the complex envelope) depending on the frequency of analysis. Of course, for every choice of such a complex envelope (or of the equivalent set of filters), one has a corresponding integral transform to deal with. The particular case of complex envelopes as functions of the time-frequency product is studied here because of its great physical interest (it applies, for instance, to many existing "real-time audio analyzers"). The corresponding integral transform is shown to have two remarkable properties: 1) it admits an inverse integral transform; 2) it is "form invariant" under linear time scaling of the signal, and no other integral transform (that is, no other class of complex envelopes, even frequency independent) shares this property. The physical significance of such results is discussed, together with some ideas for applications and further theoretical work.