Convergence in Audio Applications

Because the range of human hearing is bounded (nominally between 20 and 20 kHz), spectral components of a signal outside this range are not audible. Therefore, when the solution to a differential equation is to be considered an audio signal, there are frequency regions over which convergence is not a requirement.

Instead of pointwise convergence, we may ask for the following two properties:

• Superposition holds.
• Convergence occurs within the frequency band of human hearing.

Superposition holds for all linear partial differential equations with constant coefficients (linear, shift-invariant systems [449]). We need this condition so that errors in the inaudible bands do not affect the audible bands. Inaudible errors are fine as long as they do not grow so large that they cause numerical overflow. An example in which this ``bandlimited design'' approach yields large practical dividends is in bandlimited interpolator design (see §4.4).

In many cases, such as in digital waveguide modeling of vibrating strings, we can do better than convergence. We can construct finite difference schemes which agree with the corresponding continuous solutions exactly at the sample points. (See §C.4.1.)

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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.