Spectrographic measurement of instantaneous frequency and the time-dependent weighted average instantaneous frequency

Instantaneous frequency (IF) is an important signal characteristic arising in many fields. It is a concept intimately linked to time–frequency analysis, where it can be obtained from a time–frequency distribution (TFD) as the first conditional moment in frequency, suggesting that the IF is the average frequency at each time. However, this interpretation is questionable, since it is well known that the IF often ranges beyond the spectral support of the signal. In addition, to obtain the IF from a spectrogram (which is one possible TFD), a very wideband—and thus severely spectrally distorted—spectrogram must be used. More reasonable bandwidths are investigated, and, in particular, give the conditions by which the first conditional moment of a spectrogram can indeed be interpreted as the average frequency at each time. Under these conditions, namely when the spectrogram bandwidth is less than the frequency separation of the signal components in the time–frequency plane, the spectrogram yields not the usual IF, but a time-dependent weighted average instantaneous frequency (WAIF) of the signal. Also, while the IF and WAIF are generally different, sometimes they are the same (in particular, when there is symmetry in the time–frequency spectrum of the signal); in that case, the first conditional spectral moments of both wideband and narrow band spectrograms are the same and interpretable as the average frequency at each time.