Monday, April 28, 2008

[MT 6] Discrete relation between VAC and MSD

There are some subtle points to be considered when the continuous VAC-MSD formula is discretized. The following relation, however, should work:

\mbox{MSD}_x(n)= \Delta t^2 \sum_{m=-n}^{m=+n} C_{vv}(m)\left(  n-|m| \right)

A good test is to insert the VAC of differential white noise

C_{vv}(m)=\frac{\sigma^2}{\Delta t^2}\left\{ 2\delta_{m,0}-\delta_{m,-1}-\delta_{m,+1}  \right\}
into the above formula. One obtains for n>=1 the following result:

which is the correct flat plateau at a level proportional to the variance of the spatial white noise.

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