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Physical Audio Signal Processing
    Elementary String Instruments
       The Externally Excited String
          Equivalent Forms
             Algebraic derivation

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Algebraic derivation

The above equivalent forms are readily verified by deriving the transfer function from the striking-force input $ f_i(n)$ to the output force signal $ f_o(n)$

Referring to Fig.4.16, denote the input hammer-strike $ z$ transform by $ F_i(z)$ and the output signal $ z$ transform by $ F_o(z)$. Also denote the loop-filter transfer function by $ H_l(z)$. By inspection of the figure, we can write

$\displaystyle F_o(z) = z^{-N} \left\{ F_i(z) + z^{-2M}\left[F_i(z) + z^{-N} H_l(z)F_o(z)\right]\right\}. $

Solving for the input-output transfer function yields

\begin{eqnarray*} H(z) \isdef \frac{F_o(z)}{F_i(z)} &=& z^{-N} \frac{1+z^{-2M}}... ...}}\\ &=& \left(1+z^{-2M}\right)\frac{z^{-N}}{1-z^{-(2M+2N)}}\\ \end{eqnarray*}

The final factored form above corresponds to the final equivalent form shown in Fig.4.18.

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written by 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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