User:Mygskr/WienerKhinchin

In statistical signal processing, the Wiener–Khinchin theorem (also known as the Einstein-Wiener–Khinchin theorem or the Khinchin–Kolmogorov theorem) states that the power spectral density of a wide-sense-stationary random process is equal to the Fourier transform of the its autocorrelation function.

Theorem

Any wide-sense stationary random process $x(t)$ has an autocorrelation function given by

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

where $\operatorname {E} (\cdot )$ is the expectation over the This function captures the statistical dependence between the realization of the process at different times. However, a realization of a wide-sense-stationary random process cannot be integrable, and thus does not possess a Fourier transform . However, it is possible to capture the frequency content of the signal using the power spectral density. Consider a Fourier transform of $x(t)$ truncated outside of the range $t\in [-T,T]$:

${\mathcal {F}}_{T}(f)=\int _{-T}^{T}f(t)\exp(-2\pi ift),dt$.

Then the power spectral density of $x(t)$ is given by the limit

$S_{x}(\omega )=\lim _{T\rightarrow \infty }\mathbf {E} \left[{\frac {1}{2T}}|{\mathcal {F}}_{T}(\omega )|^{2}\right].$

The Wiener-Khinchin theorem states that

$S_{x}(\omega )=\int _{-\infty }^{\infty }R_{x}(\tau )e^{-2\pi if\tau }\ d\tau$.

In other words, the power spectral density of $x(t)$ is the Fourier transform of its autocorrelation function.

History

Einstein first derived this result in a paper published in 1914. He never again returned to the study of time series, and his work was forgotten. Norbert Wiener published a similar result in 1930, and Khinchin (sometimes spelled Khintchine) independently later in 1934, and the result came to be known as the Wiener-Khinchin theorem. The three scientists used different techniques to obtain the same fundamental result: that the power of a stationary signal contained in a frequency band can be found by integrating the Fourier transform of the autocorrelation over that band. Since modern sources typically define the power spectral density as the Fourier transform of the autocorrelation, the significance of the theorem is not always appreciated.

Discrete version

A discrete version of the theorem states that the power spectral density of a discrete random process is given by the DTFT of its autocorrelation:

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

where

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

and where $S_{xx}(f)\$ is the power spectral density of the function with discrete values $x[n]\,$. 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, trivially provable for any square integrable function, i.e. for functions whose Fourier transforms exist. More usefully, and historically, the theorem applies to wide-sense-stationary random processes, signals whose Fourier transforms do not exist, using the definition of autocorrelation function in terms of expected value rather than an infinite integral. This trivialization of the Wiener–Khinchin theorem is commonplace in modern technical literature, and obscures the contributions of Aleksandr Yakovlevich Khinchin, Norbert Wiener, and Andrey Kolmogorov.

Notes

Category:Fourier analysis Category:Signal processing Category:Probability theorems