4.14 Summary

In this chapter, the class of the spectrally correlated stochastic processes has been introduced and characterized. Processes belonging to this class have a Loève bifrequency spectrum with spectral masses concentrated on a countable set of curves in the bifrequency plane. The almost-cyclostationary processes are obtained as a special case when the separation between correlated spectral components can assume values only in a countable set. In such a case, the support curves of the Loève bifrequency spectrum are lines with unit slope. Spectrally correlated processes are an appropriate model for the output of multipath Doppler channels, when the input is a wide-band or ultra wide-band signal. Frequency-warped versions of ACS processes are SC.

The amount of spectral correlation existing between two separate spectral components of a SC process is characterized by the bifrequency spectral correlation density function which is the density of the Loève bifrequency spectrum on its support curves.

In the case of unknown support curves, the time-smoothed cross-periodogram is considered as estimator of the bifrequency spectral correlation density function. Bias and covariance are determined in the case of finite bandwidth of the spectral components and finite-time average for computing their time-correlation (Lemmas 4.5.5 and 4.5.6). The asymptotic bias (Theorem 4.5.7) and a bound for the asymptotic covariance (Theorem 4.5.9) are determined as the data-record length for the correlation estimate approaches infinite and the bandwidth of the spectral components approaches zero. The asymptotic covariance is of the order of the reciprocal of the smoothing product, that is, the product spectral-component-bandwidth times data-record length. Furthermore, at every point belonging to the support of the Loève bifrequency spectrum, the asymptotic expectation of the time-smoothed cross-periodogram is zero unless the slope of the curve is unit in a neighborhood of this point (as it happens for ACS processes). Consequently, in the case of SC processes that are not ACS, the variance cannot be made arbitrarily small (e.g., augmenting the data-record length) without obtaining a significant bias. Well known results of consistency of the time-smoothed cyclic periodogram of ACS processes are found as special cases of the results for SC processes.

The results on the asymptotic bias and variance of the time-smoothed cross-periodogram of SC processes imply that, in general, the spectral correlation density of SC processes can be estimated with some uncertainty. For SC processes that are not ACS, it can be estimated with some degree of reliability (small bias and variance) only if the departure of the nonstationarity from the almost-cyclostationarity is not too large. Specifically, if at a given point of the bifrequency plane the slope of the support curve is far from unity, then, to avoid a significant bias, the smoothing product cannot be made too large and, consequently, the variance cannot be too small (Corollary 4.5.8). Thus, a trade-off exists between the departure of the nonstationarity from the almost-cyclostationarity and the spectral correlation estimate accuracy obtainable by single sample-path measurements. Such a trade-off is not the usual bias-variance trade-off as for ACS processes. In fact, unlike the case of ACS processes, for SC processes the estimate accuracy cannot be improved as wished by increasing the data-record length and the spectral resolution.

In the case of known support curves, the cross-periodogram frequency smoothed along a support curve and properly normalized is proposed as an estimator of the spectral correlation density along that curve. Its expected value (Theorem 4.6.3) and covariance (Theorem 4.6.4) are determined in the case of finite data-record length and spectral resolution. Asymptotic expected value, rate of convergence to zero of the bias, and asymptotic covariance are determined as the data-record length approaches infinity and the spectral resolution approaches zero. It is shown that the asymptotic expected value equals the spectral correlation density times a function of frequency and of the slope of the support curve (Theorem 4.7.5), the rate of convergence of the bias is of the order of the reciprocal of the data-record length (Theorem 4.7.6), and the asymptotic covariance is of the order of the reciprocal of the smoothing product, that is the product data-record length times the frequency resolution (Theorem 4.7.7). Thus, the properly normalized frequency-smoothed cross-periodogram is a mean-square consistent estimator of the spectral correlation density. Moreover, frequency-smoothed cross-periodograms at different frequencies are shown to be asymptotically jointly complex Normal (Theorem 4.7.11). Well-known results for ACS processes are obtained by specializing the results for SC processes.

The problem of uniformly sampling SC processes is addressed. At first, strictly band-limited SC processes are characterized in terms of support curves and spectral correlation density functions (Theorem 4.9.2 and Corollary 4.9.3). Then, the class of the discrete-time SC processes is introduced and characterized (Theorem 4.8.3). It is shown that uniformly sampling a continuous-time SC process leads to a discrete-time SC process (Theorem 4.9.4). Moreover, aliasing issues are discussed. It is shown that for a strictly band-limited SC process, sampling at twice the bandwidth leads to non overlapping replicas in the Loève bifrequency spectrum of the SC discrete-time process (Theorem 4.9.5). However, a more stringent condition on the sampling frequency needs to be satisfied in order to assure that, for the discrete-time process, the spectral correlation densities on the support curves are scaled version of those of the continuous-time process for all values of frequencies in [− 1/2, 1/2] (Theorem 4.9.6). For the sake of generality, jointly SC processes and cross-statistical functions are considered.

The problem of linear time-variant processing of SC processes is addressed. The class of the SC processes is shown to be closed under linear time-variant transformations that are classified as deterministic in the FOT probability approach. For discrete-time processes, multirate processing is considered in detail. The effects of expansion and decimation operations on jointly SC processes are analyzed by considering input/output relationships of the Loève bifrequency cross-spectrum for expansors and decimators. The class of the jointly SC processes is shown to be closed under linear time-variant transformations realized by using expansors and decimators. It is proved that even if the input processes are jointly ACS, in general the output processes are jointly SC. Decimation and interpolation filters are considered. Sufficient conditions on interpolation and decimation factors and reconstruction filter bandwidths are derived to suppress imaging and aliasing replicas in the Loève bifrequency cross-spectra of interpolated or decimated processes. Sufficient conditions are derived to reconstruct a frequency-scaled image-and alias-free version of the Loève bifrequency cross-spectrum of two jointly SC processes starting from their fractionally sampled versions.

Finally, the problem of the spectral analysis with nonuniform frequency spacing is addressed when nonuniform frequency spacing is obtained by frequency-warping techniques. Such techniques are shown to modify the nonstationarity properties of the original stochastic process under analysis. Specifically, they are shown to transform ACS processes into SC processes.