Instantaneous power spectra
O. D. Grace · The Journal of the Acoustical Society of America · 1981
Conventional spectral analysis methods do not describe the distribution of signal power in both time and frequency simultaneously. Time dependent Fourier transforms are defined to extend the conventional theory to include instantaneous energy and power spectra for the past and future signal. These spectra are combined to define an instantaneous power spectrum (IPS) of the entire signal. The IPS is a noncausal mathematical property of the signal from which observable properties can be derived, e.g., the lagged window Fourier transform estimate of the time-frequency variations of the signal. The IPS for a nonstationary process is shown to be given by the Fourier transform of Loeve’s generalized power spectrum of the process and to be a generalization of the Wiener–Kintchine theorem. It is also shown to represent the special cases of the stationary process, the locally stationary process and the deterministic process.