Wiener–Khinchin theorem

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 power spectral density of a wide-sense-stationary random process is the Fourier transform of the corresponding autocorrelation function.

History

Norbert Wiener first published this theorem in 1930, and Aleksandr Khinchin did so independently in 1934. Albert Einstein had probably anticipated the idea in a brief two-page memo in 1914.

Continuous case

For the continuous case, the power spectral density of $x(t)\,$ is:

$S_{xx}(f)=\int _{-\infty }^{\infty }r_{xx}(\tau )e^{-2\pi if\tau }\ d\tau$

where

$r_{xx}(\tau )=\operatorname {E} {\big [}\,x(t)x^{*}(t-\tau )\,{\big ]}\$

is the autocorrelation function (sometimes called autocovariance) defined in terms of statistical expected value. Note that the autocorrelation function is defined in terms of the expected value of a product, and that the Fourier transform of $x(t)\,$ does not exist in general, because stationary random functions are not genearally square integrable.

The asterisk denotes complex conjugate, and of course it can be omitted if the random process is real-valued.

Discrete case

For the discrete case, the power spectral density of the function with discrete values $x[n]\,$ is:

$S_{xx}(f)=\sum _{k=-\infty }^{\infty }r_{xx}[k]e^{-2\pi ikf}$

where

$r_{xx}[k]=\operatorname {E} {\big [}\,x[n]x^{*}[n-k]\,{\big ]}\$

is the discrete autocorrelation function of $x[n]\,$. Being a sampled and discrete-time sequence, the spectral density is periodic in the frequency domain.

Application

The theorem is useful for analyzing linear time-invariant systems, LTI systems, when the inputs and outputs are not square integrable, so their Fourier transforms do not exist. A corollary is that the Fourier transform of the autocorrelation function of the output of an LTI system is equal to the product of the Fourier transform of the autocorrelation function of the input of the system times the squared magnitude of the Fourier transform of the system impulse response. This works even when the Fourier transforms of the input and output signals do not exist because these signals are not square integrable, so the system inputs and outputs cannot be directly related by the Fourier transform of the impulse response.

Since the Fourier transform of the autocorrelation function of a signal is the power spectrum of the signal, this corollary is equivalent to saying that the power spectrum of the output is equal to the power spectrum of the input times the power transfer function.

This corollary is used in the parametric method for power spectrum estimation.

Discrepancy of definition

By the definitions involving infinite integrals in the articles on spectral density and autocorrelation, the Wiener–Khinchin Theorem is a simple Fourier transform pair, that exists for any square-integrable function, i.e. for a large family of functions for which their Fourier transforms exist. More usefully, and historically, the theorem applies to wide-sense-stationary random processes, signals whose Fourier transforms do not exist. This uses the definition of autocorrelation function in terms of the expected value function rather than an infinite integral. This version of the Wiener–Khinchin Theorem is common in modern technical literature, and it seemingly obscures some of the contributions of Norbert Wiener, Aleksandr Khinchin, and Andrei Kolmogorov.

However, using the Wiener-Khinchin Theorem in this does avoid a good deal of unjustified mathematical hand-waving.