1. Geology 5600/6600 Signal Analysis 14 Sep 2015 © A.R. Lowry 2015 Last time: A stationary process has statistical properties that are time-invariant; a wide-sense stationary process has constant mean and autocorrelation dependent only on lag . Properties of Autocorrelation include: Wide-sense stationary white noise has time- invariant mean and variance given by: The Cross-Correlation for two wide-sense stationary processes,, is even and bounded: and widely used in geophysics! R xy (  ) = R yx (–  )R xy 2 (  ) ≤ R xx (0)R yy (0)

2. Discrete-time Processes can be expressed similarly: We’ll use the notation: for sampled Mean: Autocorrelation: Autocovariance: For a wide-sense stationary (WSS) process (here l = n 1 – n 2 is lag, analogous to continuous  ): And for a white noise process: if  = 0

3. Ergodic Processes: Ergodicity requires that time averages equal ensemble averages (e.g., expected values across time for one member of an ensemble equal expected values at one given time across ensemble functions). Weakly WSS ergodic processes have properties: 1) 2) 3) Ensemble Average Time Average Any Realization

4. Example of a non-ergodic process: let Then: So the first condition is satisfied. However does not go to zero as T , so the third condition is not satisfied.

5. We now have the basic working formulae: So in the case of ergodic signals, the auto- and cross-correlation functions can be expressed as convolutions of x with itself and with y respectively: Hence:

6. So how does all of this relate to the power spectrum ? (Hint: Convolution in the “spectral domain”— I.e., after Fourier transformation— is a simple multiplication…) The Auto-Power Spectrum of a random variable is given by the Wiener-Khinchin relation : Hence the power spectrum is the Fourier Transform of the correlation function, and a direct representation of the statistics of the random variable!