### 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 where the coefficients are assumed real. The transfer function of the general, second-order, real, zero-phase filter is and the frequency response is which is real for all .
• Extending the previous example, every order zero-phase 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 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