Stiff String Synthesis Models

Figure 6.13: Commuted simulation of a rigidly terminated, stiff string (no damping).

An ideal stiff-string synthesis model is drawn in Fig. 6.13 [10]. See §C.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. 6.12 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.C.9 corresponds to filter $H_s(z)$ in Fig.9.2 for the Extended Karplus-Strong algorithm. In practice, losses are also included for realistic string behavior (filter $H_d(z)$ in Fig.9.2).

Allpass filters were introduced in §2.8, and a fairly comprehensive summary is given in Book II of this series [452, Appendix C].^7.8The 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

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

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 6.11 below discusses some methods for designing stiffness allpass filters $H_s(z)$ from measurements of stiff vibrating strings, and §9.4.1 gives further details for the case of piano string modeling.