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Chapters

Physical Audio Signal Processing
    Making Virtual Electric Guitars and Guitar Effects Using Faust and Octave
       Extended Karplus-Strong Algorithm
          Tuning the EKS String
             Tuning by Allpass Interpolation

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Tuning by Allpass Interpolation

To avoid affecting the damping of the string loop, allpass interpolation is a good choice. (See §3.2 for an overview of allpass interpolation.) The simplest case is first-order allpass interpolation, as used in the original EKS algorithm []:

$\displaystyle H_\eta(z) = -\frac{\eta - z^{-1}}{1 - \eta z^{-1}} $

Calculating the phase delay and looking at the low-frequency limit, we find that the low-frequency delay (in samples) approaches

$\displaystyle \Delta_\eta(0) = \frac{1+\eta}{1-\eta},$    or $\displaystyle \zbox {\eta = \frac{\Delta_\eta - 1}{\Delta_\eta + 1}.} \protect$ (D.8)

When $ \eta=0$, the delay is one sample, and when $ \eta=-1$, the delay is zero (due to a pole-zero cancellation in the allpass). The pole is at $ z=\eta$ and the zero is at $ z=1/\eta$. Since the pole should be well inside the unit circle to avoid stability problems, we use an offset one-sample delay range, as in the original EKS algorithm:

\begin{eqnarray*} \eta \in [-2/3,1/11] &\longleftrightarrow& \Delta_\eta \in [... ...}\\ &\iff& P-N-1 \in [0.2,1.2] \\ &\iff& N \in [P-1.2,P-2.2] \end{eqnarray*}

where $ P=f_s/F_0$ is the desired period in samples.

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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.