Gaussian Signals, Covariance Matrices, and Sample Path Properties
Wojbor A. Woyczyński · Birkhäuser Boston eBooks · 2010
In general, determining the shape of the sample paths of a random signal X(t) requires knowledge of n-D (or, in the terminology of signal processing, n-point) probabilities $$P\left(a_{1} < X(t_{1})<b_{1},\ldots,a_{n} < X(t_{n})<b_{n}\right),$$ for an arbitrary n and arbitrary windows $$a_{1}\,<\,b_{1},\ldots,a_{n}\,<\,b_{n}$$ . for an But, usually, this information cannot be recovered if the only signal characteristic known is the autocorrelation function. The latter depends on the two-point distributions but does not uniquely determine them. However, in the case of Gaussian signals, the autocovariances determine not only two-point probability distributions but also all the n-point probability distributions, so that complete information is available within the second-order theory. In particular, that means that you only have to estimate means and covariances to obtain the complete model. Also, in the Gaussian universe, the weak stationarity implies the strict stationarity as defined in Chap. 4. For the sake of simplicity, all signals in this chapter are assumed to be real-valued. The chapter ends with a more subtle analysis of sample path properties of stationary signals such as continuity and differentiability; in the Gaussian case these issues have fairly complete answers.