Nonlinearities

Nonlinearities are extremely important in many musical instruments for generating a variable bandwidth over time. Examples include woodwinds, bowed strings, sitars, gongs, cymbals, and distorting electric guitars. In many more cases, sparing use of nonlinearity can serve to spice up the spectrum of any harmonic signal, as is used in so-called ``aural exciters.''

A general problem in the digital domain is that nonlinearities tend to cause aliasing. The simplest (weakest) nonlinearity is the squaring operation, and each time a signal is squared its bandwidth doubles. When a nonlinearity is used in a feedback loop, this bandwidth expansion happens over and over again until aliasing occurs. Even outside of feedback loops, large oversampling factors may be needed to avoid aliasing. Additionally, lowpass filters are often needed to push down the expanding bandwidth when it gets above a certain point. In general, there is very little practical theory for working with nonlinear elements in digital audio systems.

Another problem with nonlinearities in a physical modeling context is that they can effectively ``create'' or ``destroy'' signal energy. In a digital waveguide, the energy associated with a single signal sample is proportional to the square of that sample. Applying a nonlinear gain will change the signal energy, in general, and so some higher level framework must be introduced to ensure energy conservation in the presence of nonlinearities. Some recent work has been pursued on ``passive nonlinearities'' [69] which are developed based on analogous passive nonlinearities in continuous-time system (e.g., a nonlinear spring becomes a switching allpass filter in the digital world). However, in the discrete-time case, these analogies are not exact, and there remains the problem of how to achieve exactly lossless nonlinearities. A related problem is how to ``feed back'' round-off errors in otherwise lossless computations such that energy is exactly preserved; the solution is elementary, but applications do not yet seem to exist. Recent analytical work [45] has helped to characterize under what conditions feedback loops containing nonlinearities will at least be stable; for example, by restricting the class of nonlinearities to certain ratios of polynomials, stability can be guaranteed. A general treatment of the problem of stability of a waveguide network in the presence of nonlinearities may be found in [19].