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Embedded SystemsFPGAElectronics

Physical Audio Signal Processing
    Acoustic Modeling with Digital Delay
       Allpass Filters
          Example Allpass Filters

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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).

Previous: More General Allpass Filters
Next: Gerzon Nested MIMO Allpass

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About the Author: 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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