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 $h(n)=\delta(n)$ has frequency response $H(e^{j\omega T})=1$ , which is zero phase for all $\omega$ .
Every second-order zero-phase FIR filter has an impulse response of the form

$\displaystyle h(n) \eqsp b_{1}\delta(n+1) + b_0\delta(n) + b_1 \delta(n-1),$
where the coefficients are assumed real. The transfer function of the general, second-order, real, zero-phase filter is

$\displaystyle H(z) \eqsp b_{1}z + b_0 + b_1 z^{-1}$
and the frequency response is

$\displaystyle H(e^{j\omega T}) \eqsp b_{1}e^{j\omega T}+ b_0 + b_1 e^{-j\omega T}\eqsp b_0 + 2 b_1 \cos(\omega T)$
which is real for all $\omega$ .
Extending the previous example, every order zero-phase real FIR filter has an impulse response of the form

$\begin{eqnarray*} \lefteqn{ h(n) \eqsp b_{N}\delta(n+N) \;+\; \cdots \;+\; b_{... ...ots & & \;+\; b_1 \delta(n-1) \;+\; \cdots \;+\; b_N\delta(n-N) \end{eqnarray*}$

and frequency response

$\displaystyle H(e^{j\omega T}) \eqsp b_0 \;+\; 2 \sum_{k=1}^N b_k \cos(k\omega T)$
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 $h(n)=\delta(n+1/2) + \delta(n-1/2)$ . Since the bandlimited digital impulse signal $\delta (n)$ is ideally interpolated using bandlimited interpolation [90,84], giving samples of sinc $(n)\isdeftext \sin(\pi n)/(\pi n)$ --the unit-amplitude sinc function having zero-crossings on the integers, we see that sampling on the integers yields an IIR filter:

$\displaystyle h(n) = \sum_{m=-\infty}^{\infty}$ sinc $\displaystyle (n-m-1/2) +$ sinc $\displaystyle (n-m+1/2)$
Similarly, there are no odd-order (even-length) zero-phase filters.

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written by Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.

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Introduction to Digital Filters
    Filters Preserving Phase

          Elementary Zero-Phase Filter Examples