### Further Reading in Nonlinear Methods

Other well known numerical integration methods for ODEs include
second-order backward difference formulas (commonly used in circuit
simulation [555]), the fourth-order Runge-Kutta method
[99], and their various explicit, implicit, and semi-implicit
variations. See [555] for further discussion of these and
related finite-difference schemes, and for application examples in the
*virtual analog* area (digitization of musically useful analog
circuits). Specific digitization problems addressed in [555]
include electric-guitar distortion devices
[553,556], the classic ``tone stack''
[552] (an often-used bass, midrange, and treble
control circuit in guitar amplifiers), the Moog VCF, and other
electronic components of amplifiers and effects. Also discussed in
[555] is the ``K Method'' for nonlinear system digitization,
with comparison to nonlinear wave digital filters (see Appendix F for
an introduction to *linear* wave digital filters).

The topic of *real-time finite difference schemes* for virtual
analog systems remains a lively research topic
[554,338,293,84,264,364,397].

For *Partial Differential Equations* (PDEs), in which spatial derivatives are mixed with time
derivatives, the finite-difference approach remains fundamental. An
introduction and summary for the LTI case appear in Appendix D. See
[53] for a detailed development of finite difference
schemes for solving PDEs, both linear and nonlinear, applied to
digital sound synthesis. Physical systems considered in
[53] include bars, stiff strings, bow coupling, hammers
and mallets, coupled strings and bars, nonlinear strings and plates,
and acoustic tubes (voice, wind instruments). In addition to numerous
finite-difference schemes, there are chapters on finite-element
methods and spectral methods.

#### Outline

In this chapter, we will look at a variety of ways to digitize macroscopic point-to-point transfer functions corresponding to a desired impulse response :

- Sampling to get
- Pole mappings (such as followed by Prony's method)
- Modal expansion
- Frequency-response matching using digital filter design methods

Next, we'll look at the more specialized technique known as
*commuted synthesis*, in which computational efficiency may be
greatly increased by interchanging (``commuting'') the series order of
component transfer functions. Commuted synthesis delivers large gains
in efficiency for systems with a short excitation and high-order
resonators, such plucked and struck strings. In Chapter 9,
commuted synthesis is applied to piano modeling.

Extracting the least-damped modes of a transfer function for separate parametric implementation is often used in commuted synthesis. We look at a number of ways to accomplish this goal toward the end of this chapter.

We close the chapter with a simple example of transfer-function
modeling applied to the digital *phase shifter* audio effect.
This example classifies as *virtual analog* modeling, in which a
valued analog device is converted to digital form in a way that
preserves all valued features of the original. Further examples of
transfer-function models appear in Chapter 9.

**Next Section:**

Relation to Finite Difference Approximation

**Previous Section:**

Summary