Two-Zero String Damping Filter

A disadvantage of the decay-stretching parameter is that it affects tuning, except when $S=0$. This can be alleviated by going to a second-order, symmetric, linear-phase FIR filter having a transfer function of the form [19]

$$H_d(z) = g_1 + g_0 z^{-1}+ g_1z^{-2}= z^{-1}\left[g_0 + g_1(z + z^{-1})\right].$$

Due to the symmetry of the impulse response $h_d=[g_1,g_0,g_1,0,0,\ldots]$ about time $n=1$, only two multiplies and two additions are needed per sample. The previous one-zero loop-filter required one multiply and two additions per sample. Since the delay is equal to one sample at all frequencies (in the needed coefficient range), we obtain tuning invariance for the price of one additional multiply per sample. We also obtain a bit more lowpass filtering. Listening to both cases, one might agree that the one-zero loop filter has a ``lighter, sweeter'' tone than the two-zero case. In general, the tone is quite sensitive to the details of all filtering in the feedback path of Fig.4.

See [14] for a derivation of the FIR filter coefficients h0,h1 as a function of brightness B. A Faust implementation may then be written as follows:

  t60 = hslider("decaytime_T60", 4, 0, 10, 0.01);  // sec
  B = hslider("brightness", 0.5, 0, 1, 0.01);        // 0-1

  rho = pow(0.001,1.0/(freq*t60));
  h0 = (1.0 + B)/2;
  h1 = (1.0 - B)/4;
  dampingfilter2(x) = rho * (h0 * x' + h1*(x+x''));