The Karplus-Strong Algorithm

International Standard Book Number — Click for https://mathworld.wolfram.com/ISBN.html

Faust (Functional AUdio STream language) is a language for real-time audio signal processing that compiles into C++ for a variety of platforms. — Click for https://faust.grame.fr/

A lowpass filter rejects high frequencies and does not affect low frequencies. — Click for http://ccrma.stanford.edu/~jos/filters/Simplest_Lowpass_Filter_I.html

Noise (in signal processing) is a random signal. In the language of statistical signal processing, a noise signal is typically modeled as 'stochastic process', which is in turn defined as a sequence of random variables. — Click for https://ccrma.stanford.edu/~jos/sasp/What_Noise.html

The displacement of an object describes how far away and in what direction it has been moved, or displaced, from its rest position. Displacement is a commonly chosen wave variable in physical modeling. — Click for http://ccrma.stanford.edu/~jos/Mohonk05/Ideal_Plucked_String_Displacement.html

The velocity of an object is the time derivative of the object's displacement. Velocity is a commonly chosen wave variable in physical modeling. — Click for http://ccrma.stanford.edu/~jos/Mohonk05/Ideal_Struck_String_Velocity.html

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White noise is a random signal in which all frequencies are equally present over time. It is analogous to white light which, as Isaac Newton first realized, consists of the superposition of all visible light frequencies. — Click for https://ccrma.stanford.edu/~jos/sasp/White_Noise.html

An initial value problem consists of a differential equation and an initial condition. The solution to an initial value problem is a function y(t) solving the differential equation and satisfying the initial condition y(t₀)=y₀. — Click for http://en.wikipedia.org/wiki/Initial_value_problem

A delay line is used to delay a signal in time. Delay lines for digital signals are typically implemented in software using a circular buffer. — Click for https://ccrma.stanford.edu/~jos/pasp/Delay_Lines.html

Click for https://www.researchgate.net/publication/259597207_The_Loss_Factor_as_a_Measure_of_Mechanical_Damping

A filter in the audio signal processing context is any operation that accepts a signal as an input and produces a signal as an output. Most practical audio filters are linear and time invariant, in which case they can be characterized by their impulse response or their frequency response. — Click for https://ccrma.stanford.edu/~jos/filters/What_Filter.html

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The Karplus-Strong Algorithm

The simulation diagram for the ideal string with the simplest frequency-dependent loss filter is shown in Fig. 9.1. Readers of the computer music literature will recognize this as the structure of the Karplus-Strong algorithm [238,208,492].

**Figure 9.1:** Rigidly terminated string with the simplest frequency-dependent loss filter. All loss factors (possibly including losses due to yielding terminations) have been consolidated at a single point and replaced by a one-zero filter approximation.
$\includegraphics[width=\twidth]{eps/fkarplusstrong}$

The Karplus-Strong algorithm, per se, is obtained when the delay-line initial conditions used to ``pluck'' the string consist of random numbers, or ``white noise.'' We know the initial shape of the string is obtained by adding the upper and lower delay lines of Fig. 6.11, i.e., $y(t_n,x_m) = y^{+}(n-m) + y^{-}(n+m)$ . It is shown in §C.7.4 that the initial velocity distribution along the string is determined by the difference between the upper and lower delay lines. Thus, in the Karplus-Strong algorithm, the string is ``plucked'' by a random initial displacement and initial velocity distribution. This is a very energetic excitation, and usually in practice the white noise is lowpass filtered; the lowpass cut-off frequency gives an effective dynamic level control since natural stringed instruments are typically brighter at louder dynamic levels [432,208].

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

The Karplus-Strong Algorithm

``Physical Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2010, ISBN 978-0-9745607-2-4 Copyright © 2024-06-28 by Julius O. Smith III Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

``Physical Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2010, ISBN 978-0-9745607-2-4
Copyright © 2024-06-28 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA), Stanford University