APPENDIX B

Definitions for Random Data Analysis

Autocorrelation Function

The autocorrelation function *R _{xx}*(

*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 *G _{xx}*(

*f*) is defined for 0<

*f*<1 by

where *E*[] is an ensemble average, for fixed *f*, over *n _{d}* 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

*G*(

_{xx}*f*) = 0 for

*f*< 0.

For theoretical studies, a two-sided autospectral density function *S _{xx}*(

*f*) can be defined for − ∞ <

*f*< ∞ by setting

For stationary random data, the autospectral density function *G _{xx}*(

*f*) is twice ...