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Length $ 3$ FIR Loop Filter Controlled by ``Brightness'' and ``Sustain''

Another convenient parametrization of the second-order symmetric FIR case is when the dc normalization is relaxed so that two degrees of freedom are retained. It is then convenient to control them as brightness and sustain according to the formulas

\begin{eqnarray*}
g_0 &=& \exp( - 6.91 P / S) \\
{\hat g}(0) &=& g_0 (1 + B)/2 \\
{\hat g}(1) &=& g_0 (1 - B)/4
\end{eqnarray*}

where $ P$ is the period in seconds (total loop delay), $ S$ is the desired sustain time in seconds, and $ B$ is the brightness parameter in the interval $ [0,1]$. The sustain parameter $ S$ is defined here as the time to decay $ 60$ dB (or $ 6.91$ time-constants) when brightness $ B$ is maximum ($ B=1$) in which case the loop gain is $ g_0$ at all frequencies, or $ {\hat G}(e^{j\omega T}) = g_0$. As the brightness is lowered, the dc gain remains fixed at $ g_0$ while higher frequencies decay faster. At the minimum brightness, the gain at half the sampling rate reaches zero, and the loop-filter amplitude-response assumes the form

$\displaystyle {\hat G}(e^{j\omega T}) = g_0\frac{1 + \cos(\omega T)}{2} = g_0 \cos^2\left(\frac{\omega T}{2}\right).
$


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[How to cite and copy this work] 
``Physical Audio Signal Processing for Virtual Musical Instruments and Digital Audio Effects'', by Julius O. Smith III, (December 2005 Edition).
Copyright © 2006-07-01 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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