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维纳-辛钦定理，又称维纳-辛钦-爱因斯坦定理或辛钦-柯尔莫哥洛夫定理。该定理指出：宽平稳随机过程的功率谱密度是其自相关函数的傅立叶变换。

对于连续随机过程$x(t)\$，其功率谱密度为

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

其中

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

是定义在数学期望意义上的自相关函数。

$S_{xx}(f)\$

是函数$x(t)\,$的功率谱密度。注意到自相关函数的定义是乘积的数学期望，而$x(t)\,$的傅立叶变换不存在，因为平稳随机函数不满足平方可积。

星号*表示复共轭，当随机过程是实过程时可以将其省去。

对于离散随机过程$x[n]\$ ，其功率谱密度为

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

其中

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

且

$S_{xx}(f)\$

是离散函数$x[n]\,$的功率谱密度。由于$x[n]\,$是采样得到的离散时间序列，其谱密度在频域上是周期函数。

应用

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.

不同的定义

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.

参见

• 随机过程
• 傅立叶变换
• 谱密度

参考文献

Category:傅里叶分析 Category:随机过程