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Embedded SystemsFPGAElectronics

Physical Audio Signal Processing
    Equivalence of Digital Waveguide and Finite Difference Schemes
       Introduction
          Finite Difference Time Domain (FDTD) Scheme

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Finite Difference Time Domain (FDTD) Scheme

As discussed in §C.2, we may use centered finite difference approximations (FDA) for the second-order partial derivatives in the wave equation to obtain a finite difference scheme for numerically integrating the ideal wave equation [481,311]:

$\displaystyle {\ddot y}(t,x)$ $\displaystyle \approx$ $\displaystyle \frac{y(t+T,x) - 2 y(t,x) + y(t-T,x) }{T^2}$ (E.1)
$\displaystyle y''(t,x)$ $\displaystyle \approx$ $\displaystyle \frac{y(t,x+X) - 2 y(t,x) + y(t,x-X) }{X^2} \protect$ (E.2)

where $ T$ is the time sampling interval, and $ X$ is a spatial sampling interval.

Substituting the FDA into the wave equation, choosing $ X=cT$, where $ c \isdeftext \sqrt{K/\epsilon }$ is sound speed (normalized to $ c=1$ below), and sampling at times $ t=nT$ and positions $ x=mX$, we obtain the following explicit finite difference scheme for the string displacement:

$\displaystyle y(n+1,m) = y(n,m+1) + y(n,m-1) - y(n-1,m)$ (E.3)

where the sampling intervals $ T$ and $ X$ have been normalized to 1. To initialize the recursion at time $ n=0$, past values are needed for all $ m$ (all points along the string) at time instants $ n=-1$ and $ n=-2$. Then the string position may be computed for all $ m$ by Eq.$ \,$(E.3) for $ n=0,1,2,\ldots\,$. This has been called the FDTD or leapfrog finite difference scheme [127].

Previous: Introduction
Next: Digital Waveguide (DW) Scheme

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About the Author: 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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