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Digital Filter Models of Damped Strings

In an efficient digital simulation, lumped loss factors of the form $ G^k(\omega)$ are approximated by a rational frequency response $ {\hat G}_k(e^{j\omega T})$. In general, the coefficients of the optimal rational loss filter are obtained by minimizing $ \vert\vert\,G^k(\omega) -
{\hat G}_k(e^{j\omega T})\,\vert\vert $ with respect to the filter coefficients or the poles and zeros of the filter. To avoid introducing frequency-dependent delay, the loss filter should be a zero-phase, finite-impulse-response (FIR) filter [362]. Restriction to zero phase requires the impulse response $ {\hat g}_k(n)$ to be finite in length (i.e., an FIR filter) and it must be symmetric about time zero, i.e., $ {\hat g}_k(-n)={\hat g}_k(n)$. In most implementations, the zero-phase FIR filter can be converted into a causal, linear phase filter by reducing an adjacent delay line by half of the impulse-response duration.

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Well Posed PDEs for Modeling Damped Strings