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維納-辛欽定理，又稱維納-辛欽-愛因斯坦定理或辛欽-柯爾莫哥洛夫定理。該定理指出：寬平穩隨機過程的功率譜密度是其自相關函數的傅立葉變換。

對於連續隨機過程$x(t)\$，其功率譜密度為

$S_{xx}(f)=\int _{-\infty }^{\infty }r_{xx}(\tau )e^{-j2\pi f\tau }\ d\tau$

其中

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

是定義在數學期望意義上的自相關函數。

$S_{xx}(f)\$

是函數$x(t)\,$的功率譜密度。注意到自相關函數的定義是乘積的數學期望，而$x(t)\,$的傅立葉變換不存在，因為平穩隨機函數不滿足平方可積。

星號*表示復共軛，當隨機過程是實過程時可以將其省去。

對於離散隨機過程$x[n]\$ ，其功率譜密度為

$S_{xx}(f)=\sum _{k=-\infty }^{\infty }r_{xx}[k]e^{-j2\pi kf}$

其中

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

且

$S_{xx}(f)\$

是離散函數$x[n]\,$的功率譜密度。由於$x[n]\,$是採樣得到的離散時間序列，其譜密度在頻域上是周期函數。

應用

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 of estimating for the power spectrum estimation.

不同的定義

By the definitions involving infinite integrals in the articles on spectral density and autocorrelation, the Wiener–Khintchine 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–Khintchine theorem is commonplace in modern technical literature, and obscures the contributions of Aleksandr Yakovlevich Khinchin, Norbert Wiener, and Andrey Kolmogorov.

參見

• 隨機過程
• 傅立葉變換
• 譜密度

參考文獻

Category:傅立葉分析 Category:隨機過程