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

$$g_0 = \exp( - 6.91 P / S)$$ (10.1)

$${\hat g}(0) = g_0 (1 + B)/2$$ (10.2)

$${\hat g}(1) = g_0 (1 - B)/4 \protect$$ (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

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

A Faust function implementing this FIR filter as the damping filter in the Extended Karplus Strong (EKS) algorithm is described in [456].