Wiener–Khinchin theorem

The Wiener–Khinchin theorem states that the power spectral density of a wide-sense stationary process is the Fourier transform of its autocorrelation function.

Autocorrelation: A measure of similarity between a signal and a shifted copy of itself. Its lag-dependent form is the time-domain quantity transformed by the theorem.
Random process: A collection of random variables indexed by time or another parameter. The theorem concerns statistical properties of an entire process, not one fixed waveform.
Periodogram: An estimate of a signal’s power spectrum formed from the squared magnitude of its Fourier transform. It motivates a practical route to estimating spectral content from finite data.
Norbert Wiener: An American mathematician whose work founded cybernetics and advanced harmonic analysis and stochastic processes. His work established one side of the theorem’s mathematical lineage.
Power spectral density: A function describing how a signal’s average power is distributed over frequency. It is the frequency-domain quantity obtained from the autocorrelation.
Expected value: The probability-weighted average value of a random variable. Autocorrelation is defined using an expectation over the process.
Welch's method: A spectral estimation method that averages periodograms from overlapping data segments. It uses the spectral viewpoint while reducing the variability of periodogram estimates.
Aleksandr Khinchin: A Russian mathematician known for foundational contributions to probability theory and statistical physics. His independent formulation supplies the other name attached to the theorem.
Fourier transform: A mathematical transform that represents a function in terms of its frequency components. It converts the autocorrelation function into the power spectrum.
Covariance function: A function measuring how pairs of random variables vary together across indexed locations. For a stationary process, autocorrelation and covariance differ only by the mean contribution.

Linked from 3 pages