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Delay Operator Notation
It is convenient to think of the FDA in terms of timedomain
difference operators using a delay operator notation. The
delay operator
is defined by
Thus, the firstorder 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 unitsample advance. This definition
is preferable as long as one sample of lookahead 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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