#### Delay Operator Notation

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

*shift theorem*for transforms, is the transform of delayed (right shifted) by samples. 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.

**Next Section:**

Finite Difference Approximation vs. Bilinear Transform

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FDA in the Frequency Domain