Appendix B: Definitions for Random Data Analysis
Julius S. Bendat, Allan G. Piersol · Wiley series in probability and statistics · 2010
The autocorrelation function R^ir) of a quantity x(t) is the average of the product of the quantity at time t with the quantity at time (ί + τ) for an appropriate averaging time T:The delay τ can be either positive or negative.For an ergodic process, Τ should approach infinity, but, in practice, Γ must be finite.The total mean square value x 2 can be estimated by Autospectral Density FunctionBy finite Fourier transform techniques, the autospectral (also called power spectral) density function G^(/) is defined for 0 < / < oo by where E[] is an ensemble average, for fixed /, over n d available sample records of \X(f, T)\ 2 .The quantity Xif, T) is a finite Fourier transform of x(t) of length T. The quantity G«(/) = 0 for/< 0.