The marginals and time-frequency distributions

We address the question of the importance of satisfying the marginal conditions in a time-frequency distribution and discuss in detail the fundamental and practical issues involved. We also examine the marginals of the spectrogram. The spectrogram often gives reasonable results even though both marginals can never be satisfied exactly, but sometimes it can give very unreasonable results. We show by examples that the spectrogram gives the clearest characterization of the time-frequency properties of a signal when the combined error in both of its marginals is as small as possible. In contrast, when the error in either one of the marginals of the spectrogram is large, the time-frequency characterization degrades. Some different error measures are described. We discuss the issues involved both theoretically and by way of numerical simulations. We also address the issue of marginals for the random case.