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Digitization of Lumped Models

Since lumped models are described by differential equations, they are digitized (brought into the digital-signal domain) by converting them to corresponding finite-difference equations (or simply ``difference equations''). General aspects of finite difference schemes are discussed in Appendix D. This chapter introduces a couple of elementary methods in common use:

  1. Finite difference approximation: $ sT= 1-z^{-1}$
  2. Bilinear transform: $ sT= 2\,(1-z^{-1})/(1+z^{-1})$
Note that digitization by the bilinear transform is closely related to the Wave Digital Filter (WDF) method introduced in Appendix F. Section 9.3.1 discusses a bilinearly transformed mass colliding with a digital waveguide string (an idealized struck-string example).

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``Physical Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2010, ISBN 978-0-9745607-2-4
Copyright © 2023-08-20 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University