Nonlinear Elements

Many musical instrument models require nonlinear elements, such as

• Amplifier distortion (electric guitar)
• Reed model (woodwinds)
• Bowed string contact friction

Since a nonlinear element generally expands signal bandwidth, it can cause aliasing in a discrete-time implementation. In the above examples, the nonlinearity also appears inside a feedback loop. This means the bandwidth expansion compounds over time, causing more and more aliasing.

The topic of nonlinear systems analysis is vast, in part because there is no single analytical approach which is comprehensive. The situation is somewhat analogous to an attempt to characterize ``all non-bacterial life''. As a result, the only practical approach is to identify useful classes of nonlinear systems which are amenable to certain kinds of analysis and characterization. In this chapter, we will look at certain classes of memoryless and passive nonlinear elements which are often used in digital waveguide modeling.

It is important to keep in mind that a nonlinear element may not be characterized by its impulse response, frequency response, transfer function, or the like. These concepts are only defined, in general, for linear time-invariant systems. However, it is possible to generalize these notions for nonlinear systems using constructs such as Volterra series expansions [52]. However, rather than getting involved with fairly general analysis tools, we will focus instead on approaching each class of nonlinear elements in the manner that best fits that class, with the main goal being to understand its audible effects on discrete-time signals.

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Subsections

• Memoryless Nonlinearities
  • Clipping Nonlinearity
  • Arctangent Nonlinearity
  • Cubic Soft Clipper
  • Series Expansions
  • Arctangent Series Expansion
  • Spectrum of a Memoryless Nonlinearities
  • Square Law Series Expansion
  • Power Law Spectrum
  • Arctangent Spectrum
  • Cubic Soft-Clipper Spectrum
  • Stability of Nonlinear Feedback Loops
  • Practical Advice

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Previous: Longitudinal Waves
Next: Memoryless Nonlinearities

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About the Author: Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.