Wednesday, May 28, 2008

[MT 9] Wiener–Khinchine theorem

Reminder: We consider a real-valued, stationary random signal f(t) with zero mean (the average has been substracted, in order to yield the pure fluctuations).

The autocorrelation function (ACF) of f(t) is defined as:

The power spectral density (PSD, power spectrum) of f(t) is defined as:
S_f(\omega) = \lim_{T\rightarrow\infty} \frac{1}{T} \left| f_T(\omega) \right|^2.

where f_T(t) is the function f(t) restricted to the intervall [-T/2..+T/2] (i.e. all values outside the intervall are set to zero) and f_T(\omega) is the Fourier trafo of f_T(t).

According to the W.K. theorem, the ACF and the PSD form a Fourier pair:

S_f(\omega)={\cal F}\left\{ C_{ff}(\tau) \right\}=C_{ff}(\omega).

The following properties are obvious:

  • The autocorrelation function C(\tau) of f(t) must be real.
  • C(\tau) should be an even function of lag time: C(-\tau)=C(\tau), because stationary signals should not behave different when we go forward or backward in time.
  • The Fourier transform of an even real function is, again, an even real function, so S(\omega) is even and real. The reality is clear, anyway, since S_f(\omega) is the modulus squared of f_T(\omega).
  • The value C(\tau=0) gives the statistical variance of f(t).
  • The value S(\omega=0) is proportional to the mean value of f(t).

Two cases are or special importance:

Exponential correlation:
C(\tau) = C_0 e^{-\tau/\tau_c}  \;\Leftrightarrow\;  S(\omega)=\frac{2C_0\tau_c}{1+(\omega\tau_c)^2}
Here, the variance is C_0.

For powerlaw correlations
where \tau is a dimensionless time and \gamma is between 0 and 1 one obtains:
S(\omega)=\left[ 2 \Gamma(1\!-\!\gamma)\cos(\pi(1\!-\!\gamma)/2)\right]  \cdot |\omega|^{\gamma-1}

Here, the variance C(\tau=0) and S(\omega=0) are divergent.

See for reference this and that link. These two references also contains interesting hints about the strange things that happen when \gamma approaches or even exceeds 1.

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