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 $ b_i$ 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 $ 2N$ zero-phase real FIR filter has an impulse response of the form

h(n) \eqsp
\;+\; \cdots
\;+\; b_{...
& & \;+\; b_1 \delta(n-1)
\;+\; \cdots
\;+\; b_N\delta(n-N)

    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 $ 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:

    $\displaystyle h(n) = \sum_{m=-\infty}^{\infty}$   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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Simple Linear-Phase Filter Examples
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Example Zero-Phase Filter Design