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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.


In this chapter, we will look at a variety of ways to digitize macroscopic point-to-point transfer functions $ \Gamma (s)$ corresponding to a desired impulse response $ \gamma(t)$:
  1. Sampling $ \gamma(t)$ to get $ \gamma(nT), n = 0,1,2,\ldots$
  2. Pole mappings (such as $ z_i = e^{s_i T}$ followed by Prony's method)
  3. Modal expansion
  4. 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.
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Relation to Finite Difference Approximation
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