## 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:

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

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

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

$\mbox{MSD}_x(n\!>\!0)=2\sigma^2$
which is the correct flat plateau at a level proportional to the variance of the spatial white noise.