Assuming that one sample of “look-ahead” is available, another finite difference scheme can be defined as:
and
These equations represent zero phase filters, producing no delay at any frequency.
Note, however, that the second derivative approximation is not formed by applying the first derivative approximation to itself. Thus, the s- to z-plane mapping is different for the two approximations. Nor is it straight-forward to evaluate the s- to z-plane mapping in either case.
For the first derivative approximation, we can find:
For frequencies above
(or frequencies above ), the resulting values of z are outside the unit circle, and thus unstable.
Thus, while the centered finite difference approximation has desirable qualities, it is limited by frequency range considerations.