WienerKhinchinTheorem

The Wiener-Khinchin theorem, named after Norbert Wiener and Aleksandr Khinchin, is a foundational result in the theory of random processes. It relates the time-domain structure of a wide-sense stationary process to its frequency-domain representation, providing a bridge between autocorrelation and power spectral density.

For a wide-sense stationary stochastic process x(t) with autocorrelation function R_xx(τ) = E[(x(t) − μ)(x(t+τ) − μ)], the theorem states that the power spectral density S_xx(f) is the Fourier transform of R_xx(τ): S_xx(f) = ∫_{−∞}^{∞} R_xx(τ) e^{−i 2π f τ} dτ. The inverse relation holds as R_xx(τ) = ∫_{−∞}^{∞} S_xx(f) e^{i 2π f τ} df. When the mean μ is zero, R_xx equals the autocovariance; in general, R_xx(τ) describes the covariance structure as a function of lag τ.

Key properties include that S_xx(f) is nonnegative and, for real-valued processes, even; for complex-valued processes, S_xx(f) is Hermitian. The total power of the process satisfies ∫_{−∞}^{∞} S_xx(f) df = R_xx(0) = Var(x). The theorem requires the autocorrelation to be well-defined and integrable in the appropriate sense, which holds for many stationary processes encountered in practice.

Applications of the Wiener-Khinchin theorem include spectral estimation, noise analysis, communication theory, and system identification, where one seeks to understand how time-domain dependencies manifest as frequency-domain energy distribution.