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
  $$h(n) \eqsp b_{1}\delta(n+1) + b_0\delta(n) + b_1 \delta(n-1),$$
  where the coefficients $b_i$ are assumed real. The transfer function of the general, second-order, real, zero-phase filter is
  $$H(z) \eqsp b_{1}z + b_0 + b_1 z^{-1}$$
  and the frequency response is
  $$H(e^{j\omega T}) \eqsp b_{1}e^{j\omega T}+ b_0 + b_1 e^{-j\omega T}\eqsp b_0 + 2 b_1 \cos(\omega T)$$
  which is real for all $\omega$ .

• Extending the previous example, every order $2N$ zero-phase real FIR filter has an impulse response of the form
  $$\begin{eqnarray*} \lefteqn{ h(n) \eqsp b_{N}\delta(n+N) \;+\; \cdots \;+\; b_{1}\delta(n+1) \;+\; b_0\delta(n)}\quad\,\\ % trying to align cdots & & \;+\; b_1 \delta(n-1) \;+\; \cdots \;+\; b_N\delta(n-N) \end{eqnarray*}$$
  

  and frequency response

  $$H(e^{j\omega T}) \eqsp b_0 \;+\; 2 \sum_{k=1}^N b_k \cos(k\omega T)$$
  which is clearly real whenever the coefficients $b_k$ 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 [91,84], 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 $h$ on the integers yields an IIR filter:
  $$h(n) = \sum_{m=-\infty}^{\infty}$$   sinc$$(n-m-1/2) +$$   sinc$$(n-m+1/2)$$

• Similarly, there are no odd-order (even-length) zero-phase filters.