Gaussian Signals, Covariance Matrices, and Sample Path Properties

Abstract

In general, determination of 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

$$\displaystyle \begin{aligned}{\mathbf{P}}\Bigl( a_1<X(t_1)<b_1, \dots, a_n<X(t_n)<b_n\Bigr), \end{aligned}$$

for an arbitrary n, and arbitrary windows a₁ < b₁, …, a_n < b_n. But, usually, this information cannot be recovered if the only signal characteristic known is the autocorrelation function. The latter depends on the 2-point distributions but does not uniquely determine them. However, in the case of Gaussian signals, the autocovariances determine not only 2-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 paths properties of stationary signals such as continuity and differentiability; in the Gaussian case these issues have fairly complete answers.

Of course, faced with real-world data the proposition that they are distributed according to a Gaussian distribution must be tested rigorously. Many such tests have been developed by the statisticians.¹ In other cases, one can make an argument in favor of such a hypothesis based on the Central Limit Theorem (4.5.5) and (4.5.6).

¹See, e.g., M. Denker and W.A. Woyczyński’s book mentioned in previous chapters.