Find Correlation Length of Auto-Correlation

[Shaddyab]

How can I find the correlation length of an auto-correlation?

Is it the distance to the first zero of the auto-correlation?
Is it the distance to the first min. of the auto-correlation?
Is it the distance to the first max. of the auto-correlation?

or should I integrate?

Thank you

 

[marcusl]

It is often defined as the width between half-power points. Square the autocorrelation function and measure the width halfway down from the peak. You will also see this referred to as the full width at half max (FWHM).

(Equivalently you could measure the width between the sqrt(2) high points of the unsquared autocorrelation function.)

 

[marcusl]

I should point out that definitions differ depending on the application, and other definitions may be more useful to you. In optics where Gaussian beams are common, the coherence length is often defined as the standard deviation of the Gaussian spatial autocorrelation. The standard deviation is also taken as the coherence bandwidth of a Lorentzian line.

For exponentially decaying quantities (attenuation of light passing through a lossy medium, attenuation of waves penetrating a conductor, etc.), the coherence length is defined as the distance at which the field or current decays to 1/e of its original value. These lengths go by names like optical depth, skin depth, Debye length.

Woodward (who first applied information theory applied to radar detection) introduced a different definition of coherence time that has been widely used. Consider a function u and its Fourier transform U. If u is finite energy and normalized such that
$$\int{|u(t)|^2 dt}=1$$
then the coherence time tau may be defined in terms of the autocorrelation function rho of u by
$$\tau=\int{|\rho|^2 dt}$$
Since rho is the inverse transform of the power spectrum |U|^2 by the Wiener-Khinchin theorem, we have equivalently
$$\tau=\int{|U|^4 df}$$
This definition has also been adopted in optics and other fields as well. (Note that these results apply coherence length by substituting the variable x for t).