### Elementary Zero-Phase Filter Examples

A practical zero-phase 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 second-order zero-phase FIR filter has an impulse
response of the form
- Extending the previous example, every order zero-phase real FIR
filter has an impulse response of the form

and frequency response

- There is no first-order (length 2) zero-phase 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
unit-amplitude
*sinc function*having zero-crossings on the integers, we see that sampling on the integers yields an*IIR*filter:sinc sinc - Similarly, there are no odd-order (even-length) zero-phase filters.

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Simple Linear-Phase Filter Examples

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Example Zero-Phase Filter Design