Wiener–Khinchin theorem

The Wiener- Khinchin theorem, also known as the Wiener- Khinchin Chintchin criterion or Kolmogorov 's theorem, stating that the power spectral density of a stationary random process is the Fourier transform of the corresponding autocorrelation function. The theorem is also true trivially, that is, by simply inserting the Fourier transform, in contrast to random process signals exist in this case, for continuous functions of periodic signals and can thus be applied to a disturbed by noise periodic signal.

The set is named after Alexander Khinchin and Norbert Wiener ( sometimes even after Andrei Nikolaevich Kolmogorov )

Formulation

For continuous-time signals, the theorem has the form ( j is the imaginary unit, f is the frequency):

The auto-correlation function:

Where E is the expected value of the product.

The power spectral density function is further defined in the existence of the Fourier transform of the signal as:

For " noise signals " there is the Fourier transform, however, generally do not. The name of power spectral density (PSD, power spectral density ) is because the signal is a common voltage, and the auto-correlation function then provides an energy. " Spectral density " means that the power is given as a function of frequency per frequency interval. The PSD allows statements about the existence of periodicities in noisy signals. After the Vienna - Chintchin theorem, the PSD can be obtained from the autocorrelation function. For the detection of periodic signals in background noise, the autocorrelation function was, however, applied earlier, eg Yule in the 1920s.

Conversely, there is also the auto correlation function of a Fourier inverse transform of the power spectral density:

Note: when formulated with angular frequency are the corresponding formulas:

This is actually the common form of the Fourier transformation, in this case, a formulation is as in the signal theory selected without angular frequency (see Fourier Transform).

Calculations in the frequency domain are interchangeable on this theorem against those in the period, similar to the ergodic theorem and the ergodic hypothesis, the interchangeability of time and ensemble averages says in typical systems of statistical mechanics.

In the case of discrete-time signals ( a time series with N terms ), the Wiener- Khinchin theorem, a similar form:

The sum is limited to a finite number of applications on ( ) terms.

Further, the auto-correlation function and the power density spectrum of.

Mathematical formulation

Is the characteristic function of a probability distribution density function if and only if there is a function so that

The probability distribution is given by F, with the characteristic function (except for the pre-factors Fourier Transform) of. The theorem is a special case of the Plancherel formula (also called set of Plancherel ).

Or in the original formulation of Chintchin:

If and only if the correlation function of a stationary random process, if

With a distribution function.

Application in systems analysis

The theorem allows linear time-invariant systems (LTI ) systems, such as to investigate electrical circuits with linear components, if their input and output signals are not square integrable and thus there are no Fourier transform, as in the case of random signals ( noise). The Fourier transform of the autocorrelation function of the output signal according to the theory of LTI systems is equal to the input signal multiplied by the square value of the frequency response, that is the Fourier transform of the impulse response of the system. After Wiener Chintchin Theorem, the Fourier transform of the autocorrelation function is equal to the power spectral density and the power density of the output signal is thus equal to that of the input signal multiplied by the power transfer function analogous to the case of periodic signals with LTI.