The FDA and bilinear transform of the previous sections can be viewed
as first-order conformal maps from the analog 
plane to the digital
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
plane to the
plane
for linear, time-invariant systems. This is because Laplace
transform analysis is not defined for nonlinear and/or time-varying
differential equations (no
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 [558]. 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 [558].
