Wiener–Khinchin theorem

In applied mathematics, the Wiener–Khinchin theorem, also known as the Wiener–Khintchine theorem and sometimes as the Wiener–Khinchin–Einstein theorem or the Khinchin–Kolmogorov theorem, states that the function of a wide-sense-stationary random process has a spectral decomposition given by the power spectrum of that process.

Norbert Wiener proved this theorem for the case of a deterministic function in 1930;Aleksandr Khinchin later formulated an analogous result for stationary stochastic processes and published that probabilistic analogue in 1934.Albert Einstein explained, without proofs, the idea in a brief two-page memo in 1914.

For continuous time, the Wiener–Khinchin theorem says that if ${\displaystyle x}$ is a wide-sense stationary process such that its (sometimes called ) defined in terms of statistical expected value E, ${\displaystyle r_{xx}(\tau )=\operatorname {E} {\big [}\,x(t)x^{*}(t-\tau )\,{\big ]}\ }$ exists and is finite at every lag ${\displaystyle \tau }$, then there exists a monotone function ${\displaystyle F(f)}$ in the frequency domain ${\displaystyle -\infty <f<\infty }$ such that

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