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Length 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 $ B$ and sustain $ S$ according to the formulas

$\displaystyle g_0$ $\displaystyle =$ $\displaystyle \exp( - 6.91 P / S)$ (10.1)
$\displaystyle {\hat g}(0)$ $\displaystyle =$ $\displaystyle g_0 (1 + B)/2$ (10.2)
$\displaystyle {\hat g}(1)$ $\displaystyle =$ $\displaystyle g_0 (1 - B)/4
\protect$ (10.3)

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 by $ -60$ dB (or $ \approx 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).

A Faust function implementing this FIR filter as the damping filter in the Extended Karplus Strong (EKS) algorithm is described in [454].

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About the Author: 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 (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 for details.


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