On the Concept of an Instantaneous Power Spectrum, and Its Relationship to the Autocorrelation Function

A definition sufficient to define a function of frequency ρ(t,f) which could be regarded as the instantaneous power spectrum was given by Page. This function is not unique, since it may have added to it a complementary function of frequency ρ_c(t,f) satisfying $${\displaystyle {\int }_{\mathrm{-\infty }}^{\infty }}{\rho }_c\text{(T,f)df=0}$$ without changing the original signal in any way. It is shown that when a signal is observed by different observers, starting their observations at different times, they will not derive the same ``instantaneous power spectral function,'' and the differences between their results will be complementary functions as defined above. It is well known that the Fourier transform of the autocorrelation function is equal to the square of the magnitude of the spectral function. Analogously, a ``running autocorrelation function'' is defined and it is shown that the Fourier transform of its partial differential coefficient with respect to time is an instantaneous power spectral function.