Positive local variances of time-frequency distributions and local uncertainty

Time-frequency distribution (TFD) theory suggests that the average frequency at each time and the spread about that average (the instantaneous frequency and bandwidth) are given by the first and second conditional moments in frequency of the TFD of the signal. However, these results are usually obtained from TFDs that are not distributions in the usual sense, and often-times yield peculiar results (e.g., complex-valued instantaneous bandwidth). We explore the conditional or local variances of some common TFDs, and determine for what signal types are the local variances positive. We also examine what bearing the uncertainty principle has on limiting the local resolution of TFDs, in terms of a bound on the local bandwidth-duration product.