Stiff String Synthesis Models

Figure 4.14: Commuted simulation of a rigidly terminated, stiff string (no damping).

An ideal stiff-string synthesis model is drawn in Fig. 4.14 [13]. See §G.6 for a detailed derivation. The delay-line length $N$ is the number of samples in $K$ periods at frequency $f_K$, where $K$ is the number of the highest partial supported (normally the last one before $f_s/2$). This is the counterpart of Fig. 4.11 which depicted ideal-string damping which was lumped at a single point in the delay-line loop. For the ideal stiff string, however, (no damping), it is dispersion filtering that is lumped at a single point of the loop. Dispersion can be lumped like damping because it, too, is a linear, time-invariant (LTI) filtering of a propagating wave. Because it is LTI, dispersion-filtering commutes with other LTI systems in series, such as delay elements. The allpass filter in Fig. G.9 corresponds to filter $H_s(z)$ in Fig. 4.13 for the Extended Karplus-Strong algorithm. In practice, losses are also included for realistic string behavior (filter $H_d(z)$ in Fig. 4.13).

Allpass filters were introduced in §1.8, and a fairly comprehensive summary is given in Book II of this series [426, Appendix D].^5.5The general transfer function for an allpass filter is given (in the real, single-input, single-output case) by

$$H_s(z) = z^{-K} \frac{\tilde{A}(z)}{A(z)}$$

where $K\geq 0$ is an integer pure-delay in samples (all delay lines are allpass filters),

$$A(z) \isdef 1 + a_1 z^{-1}+ a_2 z^{-2} + \cdots + a_Nz^{-N},$$

and

$$\tilde{A}(z)\isdef z^{-N}\overline{A}(z^{-1}) = 1 + \overline{a_1} z^{-1}+ \overline{a_2} z^{-2} + \cdots + \overline{a_N}z^{-N}.$$

We may think of $\tilde{A}(z)$ as the flip of $A(z)$. For example, if $A(z)=1+1.4z^{-1}+0.49z^{-2}$, we have $\tilde{A}(z)=0.49+1.4z^{-1}+z^{-2}$. Thus, $\tilde{A}(z)$ is obtained from $A(z)$ by simply reversing the order of the coefficients (and conjugating them if they are complex, but normally they are real in practice). For an allpass filter $H_s(z)$ simulating stiffness, we would normally have $K=0$, since the filter is already in series with a delay line.

Section 4.10 below discusses some methods for designing stiffness allpass filters $H_s(z)$ from measurements of stiff vibrating strings, and §5.2 gives further details for the case of piano string modeling.