#### 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

Thus, the first-order difference (derivative approximation) is represented in the time domain by . We can think of as since, by the shift theorem for transforms, is the transform of delayed (right shifted) by samples.

The obvious definition for the second derivative is

 (8.4)

However, a better definition is the centered finite difference

 (8.5)

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.

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FDA in the Frequency Domain