Stationary Processes and the Power Spectrum

A wide-sense stationary Gaussian process is one in which the second-order statistics, that is the expected value of the product of two samples, depends only upon the time difference between the samples and not upon the absolute times. The ACF coefficient of lag $\tau$, denoted by $r_\tau$, is an estimate of the mean or expected value of the product $x_t x_{t-\tau}$,

$$r_\tau = {\cal E}(x_t x_{t-\tau}),$$

which for stationary time-series, is independent of $t$. The ACF is the fundamental feature extraction behind many stationary time-series analysis models including autoregressive (AR), moving average (MA), and autoregressive moving average (ARMA) processes are popular models.

Such a process is said to be wide-sense stationary and is fully defined by its power spectrum representation. The power spectrum is defined by the Fourier transform pair with the ACF (Wiener-Kinchine theorem)

$$P_x(\omega) \stackrel{\mbox{\tiny$\Delta$}}{=}\sum_{t= -\infty}^{... ...ac{1}{2\pi} \; \int_{\omega= 0}^{2\pi} P_x(\omega) \; e^{i\omega t} \; d\omega.$$ (10.1)

A stationary Gaussian process is thus defined by either the power spectrum or the ACF. Since the power spectrum is the Fourier transform of the autocorrelation function (ACF) with integration taken over all time, the existence of the power spectrum means that the ACF must die down to zero. This, of course, assumes there are no sinusiodal components which result in Dirac delta functions in the power spectrum.