WienerKhinchinteoremet

WienerKhinchinteoremet, commonly known as the Wiener-Khinchin theorem, relates the autocorrelation function of a wide-sense stationary random process to its power spectral density. For a process X(t) with mean m and autocorrelation R_X(τ) = E[(X(t)−m)(X(t+τ)−m)], if R_X(τ) is absolutely integrable, the power spectral density S_X(ω) is defined as the Fourier transform S_X(ω) = ∫_{−∞}^{∞} R_X(τ) e^{−iωτ} dτ. Conversely, under suitable conditions, R_X(τ) = (1/2π) ∫_{−∞}^{∞} S_X(ω) e^{iωτ} dω. Thus S_X(ω) and R_X(τ) form a Fourier transform pair. In discrete time, the analogous relation uses the discrete-time Fourier transform (DTFT) for the autocovariance sequence.

The theorem assumes the process is wide-sense stationary, meaning its mean is constant and its autocovariance depends only on lag, not on absolute time. It is typically stated for the existence of the spectral density as a nonnegative function; when integrability conditions fail, the transforms exist in a generalized sense. The PSD S_X(ω) represents the distribution of power across frequency, while R_X(τ) describes how values at different times co-vary.

Applications of the Wiener-Khinchin theorem are widespread in signal processing, communications, and time-series analysis. It underpins spectral estimation, filtering, and noise characterization. A canonical example is white noise, for which R_X(τ) = σ^2 δ(τ) and S_X(ω) is constant across frequencies, reflecting flat power distribution. The theorem, named after Norbert Wiener and Aleksandr Khinchin, was developed in the 1930s–1940s and remains fundamental to stochastic signal analysis.