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Chapters

Physical Audio Signal Processing
Finite-Difference Schemes
Finite-Difference Schemes

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Finite-Difference Schemes

Finite-Difference Schemes (FDSs) aim to solve differential equations by means of finite differences. For example, as discussed in §C.2, if denotes the displacement in meters of a vibrating string at time seconds and position meters, we may approximate the first- and second-order partial derivatives by

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

where

denotes the time sampling interval and

denotes the spatial sampling interval. Other types of finite-difference schemes were derived in Chapter 7 (§7.3.1), including a look at frequency-domain properties. These finite-difference approximations to the partial derivatives may be used to compute solutions of differential equations on a discrete grid:

$\begin{displaymath} \begin{array}{rclcl} x &\to& x_m&=& mX\nonumber \\ t &\to& t_n&=& nT\nonumber \end{array}\end{displaymath}$

Let us define an abbreviated notation for the grid variables

$\displaystyle y_{n,m}\isdef y(nT,mX)$

and consider the ideal string wave equation (cf, §C.1):

$\displaystyle y''= \frac{1}{c^2}{\ddot y} \protect$

(D.2)

where

is a positive real constant (which turns out to be wave propagation speed). Then, as derived in §C.2, setting

and substituting the finite-difference approximations into the ideal wave equation leads to the relation

$\displaystyle y_{n+1,m}+ y_{n-1,m}= y_{n,m+1}+ y_{n,m-1}$

everywhere on the time-space grid (i.e., for all

and

). Solving for $y_{n+1,m}$ in terms of displacement samples at earlier times yields an explicit finite-difference scheme for string displacement:

$\displaystyle y_{n+1,m}= y_{n,m+1}+ y_{n,m-1}- y_{n-1,m} \protect$

(D.3)

The FDS is called explicit because it was possible to solve for the state at time

as a function of the state at earlier times (and any other positions

). This allows it to be implemented as a time recursion (or ``digital filter'') which computes a solution at time

from solution samples at earlier times (and any spatial positions). When an explicit FDS is not possible (e.g., a non-causal filter is derived), the discretized differential equation is said to define an implicit FDS. An implicit FDS can often be converted to an explicit FDS by a rotation of coordinates [55,481].

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Next: Convergence

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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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