Below are various physical-model representations we will consider:

- Ordinary Differential Equations (ODE)
- Partial Differential Equations (PDE)
- Difference Equations (DE)
- Finite Difference Schemes (FDS)
- (Physical) State Space Models
- Transfer Functions (between physical signals)
- Modal Representations (Parallel Second-Order Filter Sections)
- Equivalent Circuits
- Impedance Networks
- Wave Digital Filters (WDF)
- Digital Waveguide (DW) Networks

ODEs and PDEs are purely mathematical descriptions (being differential
equations), but they can be readily ``digitized'' to obtain
computational physical models.^{2.5}*Difference equations* are simply
digitized differential equations. That is, digitizing ODEs and PDEs
produces DEs. A DE may also be called a *finite difference
scheme*. A discrete-time *state-space* model is a special
formulation of a DE in which a vector of *state variables* is
defined and propagated in a systematic way (as a vector first-order
finite-difference scheme). A linear difference equation with constant
coefficients--the Linear, Time-Invariant (LTI) case--can be reduced
to a collection of *transfer functions*, one for each pairing of
input and output signals (or a single *transfer function matrix*
can relate a vector of input signal *z* transforms to a vector of output signal
*z* transforms). An LTI state-space model can be *diagonalized* to
produce a so-called *modal representation*, yielding a
computational model consisting of a parallel bank of second-order
digital filters. *Impedance networks* and their associated
*equivalent circuits* are at the foundations of electrical
engineering, and analog circuits have been used extensively to model
linear systems and provide many useful functions. They are also
useful intermediate representations for developing computational
physical models in audio.
*Wave Digital Filters* (WDF) were introduced as a means of
digitizing analog circuits element by element, while preserving the
``topology'' of the original analog circuit (a very useful property
when parameters are time varying as they often are in audio effects).
*Digital waveguide networks* can be viewed as highly efficient
computational forms for propagating solutions to PDEs allowing wave
propagation. They can also be used to ``compress'' the computation
associated with a sum of quasi harmonically tuned second-order
resonators.

All of the above techniques are discussed to varying extents in this book. The following sections provide a bit more introduction before plunging into the chapters that follow.

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Center for Computer Research in Music and Acoustics (CCRMA), Stanford University