power density spectrum

(Sometimes called power spectrum.) A measure of the contribution to the total variance from a given frequency band in the generalized Fourier representation of a random function.

If f(t) is a random function, the total energy

is infinite, so the Fourier integral representation is inadequate. If a transform is defined over a finite interval

under suitably restrictive conditions the power density spectrum may be defined as

The theorem, proved by N. Wiener, establishing the analogy between the analysis of random functions and ordinary Fourier analysis, is that the power density spectrum is the Fourier transform of the autocorrelation function, which is defined for random functions as