A theorem stating that for a wide-sense stationary stochastic process the power spectral density is the Fourier transform of its (auto)correlation function, so spectral and temporal second-order statistics are Fourier pairs.

Principle

Stationarity of second-order moments makes the autocorrelation depend only on lag; linearity of the Fourier transform then converts convolutional/lag structure into multiplicative spectral structure, yielding a one-to-one transform relation between autocorrelation and spectral density.

Demonstration

For a real, zero-mean wide-sense stationary Gaussian process X(t) with autocorrelation R_X(τ), compute S_X(ω)=∫_{-∞}^{∞}R_X(τ)e^{-iωτ}dτ; for white noise R_X(τ)=N0/2·δ(τ) this yields constant S_X(ω) across ω, illustrating the transform relation.

Misapplication

Using the theorem for a non-stationary process, or treating sample autocorrelations from a single finite record as exact ensemble autocorrelations without bias/consistency corrections, leads to incorrect spectral conclusions.

Consequence

Enables spectral estimation from autocorrelation estimates and vice versa, underpins filter design and system identification methods that move analysis between time-domain covariance and frequency-domain power descriptions.

Reversal

The relation is invertible: the autocorrelation is the inverse Fourier transform of the power spectral density, so known spectra can reconstruct second-order time correlations.

Boundary

Requires wide-sense stationarity and that R_X be integrable (or interpreted in the distributional sense for generalized processes); discrete-time versions require analogous summability conditions and care with aliasing.

Semantic Tension

Competes with time-varying spectral representations (short-time Fourier, Wigner–Ville) that drop stationarity; those capture nonstationary energy distributions but sacrifice the simple autocorrelation–PSD Fourier duality.

Synthesis

The theorem formalizes that, under stationary second-order statistics, temporal autocorrelation and frequency-domain power are two representations of the same information, related by the Fourier transform and usable interchangeably for analysis and design.