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Elementary ZeroPhase
Filter Examples
A practical zerophase filter was illustrated
in Figures 10.1 and 10.2. Some simple general cases are as
follows:
 The trivial (non)filter
has frequency response
, which is zero phase for all
.
 Every secondorder zerophase FIR filter has an impulse
response of the form
where the coefficients
are assumed real. The transfer function
of the general, secondorder, real, zerophase filter is
and the frequency response is
which is real for all
.
 Extending the previous example, every order
zerophase real FIR
filter has an impulse response of the form
and frequency response
which is clearly real whenever the coefficients
are real.
 There is no firstorder (length 2) zerophase filter, because,
to be even, its impulse response would have to be proportional to
. Since the bandlimited digital
impulse signal
is ideally interpolated using bandlimited
interpolation [91,84], giving samples of
sinc
the
unitamplitude sinc function having zerocrossings on the
integers, we see that sampling
on the integers yields
an IIR filter:
 Similarly, there are no oddorder (evenlength) zerophase filters.
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