On the Behavior of the Continuous-Time Spectrogram for Arbitrarily Narrow Windows

The behavior of the continuous-time (CT) spectrogram, in particular of its first frequency moment (FFM) and instantaneous bandwidth (IB), in the limit of infinitely thin windows is revisited: new insight is gained and novel quantitative results are presented. Power series in the width of the short-time Fourier transform (STFT) window are derived for the CT spectrogram and its moments, which are valid for any reasonably regular narrow window, and which universally yield the instantaneous frequency (IF) as the first term in the FFM expansion. The well-known result, for CT signals, according to which the spectrogram FFM tends to the IF when the square of the STFT window approaches a Dirac delta (delta) function is thus extended. Indeed, upon introduction of functional sequences depending on a small width parameter, it is shown that the spectrogram FFM always tends to the IF when the STFT window becomes arbitrarily narrow (with no actual need for the latter, or any of its powers, to converge to a delta function). It is also checked, within the framework here developed, that the spectrogram IB behaves as expected, going to infinity when the FFM approaches the IF. It is argued that, when estimating the IF of CT signals with the spectrogram, it may be eventually more convenient to work with functions that become arbitrarily close to a delta function than to its square root (whatever mathematical object this might be!). In addition, the results of the analysis on the IF and the IB are illustrated with a few examples of standard signal and window forms