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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