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More General Finite-Difference Methods

The FDA and bilinear transform of the previous sections can be viewed as first-order conformal maps from the analog $ s$ plane to the digital $ z$ plane. These maps are one-to-one and therefore non-aliasing. The FDA performs well at low frequencies relative to the sampling rate, but it introduces artificial damping at high frequencies. The bilinear transform preserves the frequency axis exactly, but over a warped frequency scale. Being first order, both maps preserve the number of poles and zeros in the model.

We may only think in terms of mapping the $ s$ plane to the $ z$ plane for linear, time-invariant systems. This is because Laplace transform analysis is not defined for nonlinear and/or time-varying differential equations (no $ s$ plane). Therefore, such systems are instead digitized by some form of numerical integration to produce solutions that are ideally sampled versions of the continuous-time solutions. It is often necessary to work at sampling rates much higher than the desired audio sampling rate, due to the bandwidth-expanding effects of nonlinear elements in the continuous-time system.

A tutorial review of numerical solutions of Ordinary Differential Equations (ODE), including nonlinear systems, with examples in the realm of audio effects (such as a diode clipper), is given in [556]. Finite difference schemes specifically designed for nonlinear discrete-time simulation, such as the energy-preserving ``Kirchoff-Carrier nonlinear string model'' and ``von Karman nonlinear plate model'', are discussed in [53].

The remaining sections here summarize a few of the more elementary techniques discussed in [556].

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``Physical Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2010, ISBN 978-0-9745607-2-4.
Copyright © 2021-05-21 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University