Uncertainty in measurements on spectrally correlated stochastic processes

In this paper, the class of the spectrally correlated stochastic processes is introduced. Processes belonging to this class exhibit a Loe`ve (1963) bifrequency spectrum with spectral masses concentrated on a countable set of support curves in the bifrequency plane. Thus, such processes have spectral components that are correlated. The introduced class generalizes the almost-cyclostationary (ACS) processes that are obtained as a special case when the separation between correlated spectral components assumes values only in a countable set. In such a case, the support curves are lines with unit slope. For the spectrally correlated processes, the amount of spectral correlation existing between two separate spectral components is characterized by the bifrequency spectral correlation density function, which is the density of the Loe`ve bifrequency spectrum on its support curves. It is shown that, in general, when the location of the support curves is unknown, the time-smoothed cross-periodogram can provide a reliable (low bias and variance) single sample-path-based estimate of the bifrequency spectral correlation density function in those points of the bifrequency plane where the slope of the support curves is not too far from unity. Moreover, there exists a tradeoff between the departure of the nonstationarity from the almost-cyclostationarity and the reliability of spectral correlation measurements obtainable by a single sample-path. Furthermore, in general, the estimate accuracy cannot be improved as wished by increasing the data-record length and the spectral resolution.