Emergent Behavior from Idiosyncratic Feedback Networks
Christopher G. Burns · University of Michigan Library Repository · 2003
Traditional waveguide networks are designed for stability and predictability. However, idiosyncratic variations to feedback network structures become possible when peak gain is controlled throughout the network by nonlinear waveshaping functions. These functions facilitate destabilizing changes to the network gain structure and topology, producing unpredictable behavior with interesting applications in composition and improvisation. These applications suggest further extensions to the waveguide network model, including nonstandard excitation functions, control parameters, and spatialization techniques. 1 Feedback networks and stability Traditional signal processing architectures which utilize feedback are constructed to ensure their stability and predictability. For instance, there are canonic limitations on the coefficients of IIR filters, and waveguide sections are explicitly designed to preserve system stability. However, it is possible to impose at least bounded amplitudes in signal processing systems which do not meed traditional criteria for stability. For instance, Xenakis’ GENDYN software uses “elastic-mirrors” to ensure that amplitudes generated by random walks will not exceed appropriate boundaries (Xenakis 1992, Hoffmann 2000). Peak-limiting compression algorithms are another method for achieving this result. A third, easily implemented technique for peak control is waveshaping with nonlinear functions that produce soft clipping. Charles Sullivan presented one such function in his work on physical models of the electric guitar (1990): { 2/3; x • 1 f(x) = { x – x / 3; -1 < x < 1 { -2/3; x • -1 Figure 1 presents a plot of this function over the range -1 • x • 1. Relatives of this function (of the form x – x / n, for odd n • 3) can also be used to produce different varieties of soft clipping. Nonlinear functions manage peak gain because they are inherently lossy. They also change the harmonic structure of their input, coloring the system output. (The effect will be particularly pronounced when nonlinearities are deployed at several different points in a system). While this timbral alteration may seem like a disadvantage, in a compositional context it can just as well be viewed as a desirable feature. After all, destabilized feedback networks are likely to be of interest precisely because of their unusual sonic characteristics. 2 Beyond standard architectures When the peak gain of a feedback network is controlled through nonlinear waveshaping or other limiting techniques, then a variety of extensions and modifications to traditional waveguide designs become possible. Variations tested include unusual excitation methods, network topologies, spatialization techniques, and control parameters. In general these extensions cannot be readily interpreted as models of physical or acoustic phenomena. Indeed, some of the network topologies demonstrated are so distant from traditional architectures that they no longer can properly be described as waveguides; hence the more general term “feedback network” used throughout