Example Allpass Filters

  • The simplest allpass filter is a unit-modulus gain

    $\displaystyle H(z) = e^{j\phi}

    where $ \phi$ can be any phase value. In the real case $ \phi$ can only be 0 or $ \pi$, in which case $ H(z)=\pm 1$.

  • A lossless FIR filter can consist only of a single nonzero tap:

    $\displaystyle H(z) = e^{j\phi} z^{-K}

    for some fixed integer $ K$, where $ \phi$ is again some constant phase, constrained to be 0 or $ \pi$ in the real-filter case. Since we are considering only causal filters here, $ K\geq 0$. As a special case of this example, a unit delay $ H(z)=z^{-1}$ is a simple FIR allpass filter.

  • The transfer function of every finite-order, causal, lossless IIR digital filter (recursive allpass filter) can be written as

    $\displaystyle H(z) = e^{j\phi} z^{-K} \frac{\tilde{A}(z)}{A(z)} \protect$ (3.16)

    where $ K\geq 0$, $ A(z) = 1 + a_1 z^{-1}+ a_2 z^{-2} + \cdots + a_N
z^{-N}$, and $ \tilde{A}(z)\isdef z^{-N}\overline{A}(z^{-1})$. The polynomial $ \tilde{A}(z)$ can be obtained by reversing the order of the coefficients in $ A(z)$ and conjugating them. (The factor $ z^{-N}$ serves to restore negative powers of $ z$ and hence causality.)

In summary, every SISO allpass filter can be expressed as the product of a unit-modulus gain factor, a pure delay, and an IIR transfer function in which the numerator is the ``flip'' of the denominator, as in Eq.$ \,$(2.16).

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More General Allpass Filters