Decay-time Shortening

To shorten the decay time, a loss factor of $\rho$ can be introduced in the feedback loop, yielding

$$y(n) = x(n) + \rho\frac{y(n-N) + y(n-(N+1))}{2}$$

The amplitude envelope of a sinusoid at frequency $f$, is now proportional to

$$\alpha_f(t,\rho) = \vert\rho \cos(\pi fT_s)\vert^{t f_1} = \vert\rho\vert^{t f_1}\alpha_f(t).$$

and the decay-time constant for the fundamental frequency becomes

$$\tau_1(\rho) = -\frac{1}{f_1 \ln \vert \rho \cos(\pi f_1T_s) \vert}.$$

Note that $\rho$ cannot be used to lengthen the decay time, since the amplitude at 0 Hz would increase exponentially.

$\vert \rho \vert \le 1$ if the string is to be stable.

$\rho$ is used to shorten the low-pitch notes.