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An Efficient Class of Loss Filters

A first-order IIR filter that, when cascaded with a delay line, simulates wave propagation in a lossy resonator of length $L_R$, can take the form

$\displaystyle G_L(z)$ $\textstyle =$ $\displaystyle \left( \alpha \frac{1-r}{1-r z^{-1}} + (1 - \alpha) \right) z^{-L_R F_s / c}$  
    $\displaystyle \stackrel{\triangle}{=}H_L(z) z^{-L_R F_s / c} \: .$ (49)

At the Nyquist frequency, and for $r\simeq 1$, such filter gains $G_L(e^{j \pi}) \simeq 1 -
\alpha$ and we have to use $\alpha = 1 - e^{-\frac{1}{2}\Upsilon c L_R}$ to have the correct attenuation at high frequency. Fig. 5 shows the magnitude and phase delay (in seconds) obtained with the first-order filter $G_L$ for three values of its parameter $r$.

By comparison of the curves of Fig. 4 with the responses of Fig. 5, we see how the latter can be used to represent the losses in a section of one-dimensional waveguide section. Therefore, the simulation scheme turns out to be that of Fig. 6. Of course, better approximations of the curves of Fig. 4 can be obtained by increasing the filter order or, at least, by controlling the zero position of a first-order filter. However, the form (50) is particularly attractive because its low-frequency behavior is controlled by the single parameter $r$.

Figure 5: Magnitude and phase delay of a first-order IIR filter, for different values of the coefficient $r$, $\alpha $ is set to $1 - e^{-\frac{1}{2}\Upsilon c L_R }$ with the same values of $\Upsilon $ used for the curves in fig. 4
\begin{figure}\centerline{\epsfysize=180pt \epsfbox{figure/iir1.eps}}

Figure: Length-$L_R$ one-variable waveguide section with small losses
\begin{figure}\centerline{\epsfxsize=230pt \epsfbox{figure/DelayH.eps}}\end{figure}

As far as the wave impedance is concerned, in the discrete-time domain, it can be represented by a digital filter obtained from (49) by bilinear transformation, which leads to

R^+ = \frac{2 L F_s}{G_0} \frac{1 - z^{-1}}{2 L F_s + 1 - (2 L F_s - 1)z^{-1}} \; ,
\end{displaymath} (50)

that is a first-order high-pass filter. The discretization by impulse invariance can not be applied in this case because the impedance has a high-frequency response that would alias heavily.

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Download gdwn.pdf

``Generalized Digital Waveguide Networks'', by Julius O. Smith III and Davide Rocchesso, preprint submitted for publication, Summer 2001.
Copyright © 2008-03-12 by Julius O. Smith III and Davide Rocchesso
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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