Definitions for Random Data Analysis
The autocorrelation function Rxx(τ) of a quantity x(t) is the average of the product of the quantity at time t with the quantity at time (t + τ) for an appropriate averaging time T:
The delay τ can be either positive or negative. For an ergodic process, T should approach infinity, but, in practice, T must be finite. The total mean square value can be estimated by
Autospectral Density Function
By finite Fourier transform techniques, the autospectral (also called power spectral) density function Gxx(f) is defined for 0<f<1 by
where E is an ensemble average, for fixed f, over nd available sample records of |X(f, T)|2. The quantity X(f, T) is a finite Fourier transform of x(t) of length T. The quantity Gxx(f) = 0 for f < 0.
For theoretical studies, a two-sided autospectral density function Sxx(f) can be defined for − ∞ < f < ∞ by setting
For stationary random data, the autospectral density function Gxx(f) is twice ...