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Chapters

Physical Audio Signal Processing
    Passive Impedances
       The Guitar Bridge
          Building a Synthetic Guitar Bridge Admittance
             Method 1

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Method 1

The first method is based on constructing a passive reflectance $ S(z)$ having the desired poles, and then converting to an admittance via the fundamental relation

$\displaystyle \Gamma(z) = \Gamma_0 \frac{1-S(z)}{1+S(z)} $

where $ \Gamma_0$ is an arbitrary real, positive number which can be interpreted as the wave admittance of the string on which waves enter and return from the bridge.

As we saw in Section M.1, every passive impedance corresponds to a passive reflectance which is a Schur function (stable and having gain not exceeding $ 1$ around the unit circle). Since damping is light in a guitar bridge impedance (otherwise the strings would not vibrate very long, and sustain is a highly prized feature of real guitars), we can expect the bridge reflectance to be close to an allpass transfer function $ H_A(z)$.

It is well known that every allpass transfer function can be expressed as

$\displaystyle H_A(z) \isdef \frac{\tilde{A}(z)}{A(z)} $

where

\begin{eqnarray*} A(z) &=& 1 + a_1 z^{-1} + a_2 z^{-2} + \cdots + a_{M-1} z^{-(M... ... z^{-M} \\ &=& z^{-M} A\left(z^{-1}\right) = \mbox{Flip}(A)(z) \end{eqnarray*}

We will then construct a Schur function as