## 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:

$C_{ff}(\tau)=\left<&space;f(t)\;f(t+\tau)&space;\right>_t.$

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

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:

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

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

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

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.