4

Spectrally Correlated Processes

In this chapter, the class of the spectrally correlated (SC) stochastic processes is characterized and the problem of its statistical function estimation addressed. Processes belonging to this class exhibit a Loève 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 that of 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. SC processes properly model the output of Doppler channels when the product of the signal bandwidth and the data-record length is not too smaller than the ratio of the medium propagation speed and the radial speed between transmitter and receiver. Thus they find application in wide-band or ultra-wideband mobile communications. For SC 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 Loève bifrequency spectrum on its support curves. When the location of the support curves is unknown, the time-smoothed bifrequency cross-periodogram provides 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, a trade-off exists 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. If the location of a support curve is known, the cross-periodogram frequency smoothed along the given support curve and properly normalized provides a mean-square consistent and asymptotically Normal estimator of the density of the Loève bifrequency spectrum along that curve. Furthermore, well-known consistency results for ACS processes can be obtained by specializing the results for SC processes. It is shown that uniformly sampling a continuous-time SC process leads to a discrete-time SC process. Conditions on the sampling frequency and the process bandwidth are provided to avoid aliasing in the Loève bifrequency spectrum, and its density, of the discrete-time sampled signal. Finally, multirate processing of SC processes is treated.