The Extended Karplus-Strong Algorithm

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The Extended Karplus-Strong Algorithm

Figure 4.13 shows a block diagram of the Extended Karplus-Strong (EKS) algorithm described in the Computer Music Journal by Jaffe and Smith [213].

**Figure 4.13:** Extended Karplus-Strong (EKS) algorithm.
$\begin{figure}\input fig/eks.pstex_t \end{figure}$

The EKS adds the following features to the KS algorithm:

$\begin{eqnarray*} H_p(z) &=& \frac{1-p}{1 - p\,z^{-1}}\eqsp \mbox{pick-direction... ...1-R_L}{1 - R_L\,z^{-1}}\eqsp \mbox{dynamic-level lowpass filter} \end{eqnarray*}$

where

$\begin{eqnarray*} N &=& \mbox{pitch period ($2\times$\ string length) in samples... ...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 as above in order to avoid delays close to zero in the tuning allpass, since a zero delay is obtained by a pole-zero cancellation on the unit circle. First-order allpass interpolation of delay lines was discussed in §3.2.2.

A history of the Karplus-Strong algorithm and its extensions is given in §E.8. EKS sound examples are also available on the Web. Techniques for designing the string-damping filter and/or the string-stiffness allpass filter are summarized below in §4.11.

C++ software that can be quickly adapted to implement a synthesis algorithm along the above lines is given in Appendix C. More specifically, Appendix D contains a section (§D.2) on implementing the Extended Karplus-Strong algorithm in the Faust programming language.

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written by Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.

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Physical Audio Signal Processing
    Elementary String Instruments
       Frequency-Dependent Damping
          The Extended Karplus-Strong Algorithm