It is convenient to think of the FDA in terms of time-domain difference operators using a delay operator notation. The delay operator is defined by
The obvious definition for the second derivative is
However, a better definition is the centered finite difference
where denotes a unit-sample advance. This definition is preferable as long as one sample of look-ahead is available, since it avoids an operator delay of one sample. Equation (7.5) is a zero phase filter, meaning it has no delay at any frequency, while (7.4) is a linear phase filter having a delay of sample at all frequencies.
Finite Difference Approximation vs. Bilinear Transform
FDA in the Frequency Domain