The Extended Karplus-Strong Algorithm

Figure 9.2 shows a block diagram of the Extended Karplus-Strong (EKS) algorithm [208].

Figure 9.2: Extended Karplus-Strong (EKS) algorithm.

The EKS adds the following features to the KS algorithm:

$$\begin{eqnarray*} H_p(z) &=& \frac{1-p}{1 - p\,z^{-1}}\eqsp \mbox{pick-direction lowpass filter}\\ [5pt] H_\beta(z) &=& 1 - z^{-\lfloor\beta N+1/2\rfloor}\eqsp \mbox{pick-position comb filter, $\beta\in(0,1)$}\\ [5pt] H_d(z) &=& \mbox{string-damping filter (one/two poles/zeros typical)}\\ [5pt] H_s(z) &=& \mbox{string-stiffness allpass filter (several poles and zeros)}\\ [5pt] H_\eta(z) &=& -\frac{\eta(N)-z^{-1}}{1 - \eta(N)\,z^{-1}}\eqsp \mbox{first-order string-tuning allpass filter}\\ [5pt] H_L(z) &=& \frac{1-R_L}{1 - R_L\,z^{-1}}\eqsp \mbox{dynamic-level lowpass filter} \end{eqnarray*}$$

where

$$\begin{eqnarray*} N &=& \mbox{pitch period ($2\times$\ string length) in samples}\\ \beta &=& \mbox{normalized pick position} \in (0,1)\\ p &=& \mbox{0 for one pick direction, $0<d<1$\ for opposite direction}\\ R_L &=& e^{-\pi LT}, \mbox{ where $L$\ is \lq\lq desired bandwidth'' in Hz}\\ T &=& \mbox{Sampling interval (Hz)}\\ \eta &\in& [-1/11,2/3] \mbox{ for tuning-delays in the range $[0.2,1.2]$\ samples}\\ \left\vert H_d(e^{j\omega T})\right\vert &\le& 1 \mbox{ required for stability} \end{eqnarray*}$$

Note that while $\eta\in[0,1)$ can be used in the tuning allpass, it is better to offset it to $[\epsilon,1+\epsilon)$ to avoid delays close to zero in the tuning allpass. (A zero delay is obtained by a pole-zero cancellation on the unit circle.) First-order allpass interpolation of delay lines was discussed in §4.1.2.

A history of the Karplus-Strong algorithm and its extensions is given in §A.8. EKS sound examples are also available on the Web. Techniques for designing the string-damping filter $H_d(z)$ and/or the string-stiffness allpass filter $H_s(z)$ are summarized below in §6.11.

An implementation of the Extended Karplus-Strong (EKS) algorithm in the Faust programming language is described (and provided) in [456].