Convolution of a discrete signal $f$ with $g$ computing the centered finite difference derivative of $f$

Question

I have the following discrete signal with 8 samples:

$f[x]=[1,2,1,0,1,0,-1,-2]$

I need to find a $g$ such that convolution of the discrete signal above with $g$ computes the centered finite difference derivative of $f$ with step length $1$.

However, I'm quite lost here, and need some help/hints.