The time-dependent weighted average instantaneous frequency

It is often claimed that instantaneous frequency, taken as the derivative of the phase of the signal, is 'appropriate' or 'meaningful' only for monocomponent signals, and that for multicomponent signals a weighted average of the individual instantaneous frequencies should be used. That is fine as a definition, but generally difficult to do in practice because there is no systematic and general method for determining the components of a signal (i.e., the individual amplitudes and phases), which is itself a challenging problem. We show that spectrograms can give this exact result for many signals, obtained via the conditional mean frequency of the spectrogram, and give the conditions on the window for that to be the case. Using this approach, one does not need to know the individual instantaneous amplitudes and phases of the components to obtain the weighted average instantaneous frequency of the signal.