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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.6.15, 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}}{1-H_l(z)\,z^{-(2M+2N)}}\\
&=& \left(1+z^{-2M}\right)\frac{z^{-N}}{1-H_l(z)\,z^{-(2M+2N)}}\\
\end{eqnarray*}

The final factored form above corresponds to the final equivalent form shown in Fig.6.17.


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``Physical Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2010, ISBN 978-0-9745607-2-4.
Copyright © 2014-03-23 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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