Abstract
Many systems featuring nonlinearities and delay-free loops are of interest in digital audio, particularly in virtual analog and physical modeling applications. Many of these systems can be posed as systems of implicitly related ordinary differential equations. Provided each equation in the network is itself an explicit one, straightforward numerical solvers may be employed to compute the output of such systems without resorting to linearization or matrix inversions for every parameter change. This is a cheap and effective means for synthesizing delay-free, nonlinear systems without resorting to large lookup tables, iterative methods, or the insertion of fictitious delay and is therefor suitable for real-time applications. Several examples are shown to illustrate the efficacy of this approach.
Highlights
There are certain continuous-time systems that are of interest to computer musicians, that contain a delay-free loop—instantaneous feedback from the output to the input
Simple harmonic motion can be modeled as a spring-mass mechanism
The acceleration of the mass is proportional to its position
Summary
There are certain continuous-time systems that are of interest to computer musicians, that contain a delay-free loop—instantaneous feedback from the output to the input. WDF theory was not sufficient for representing systems involving delay-free loops To this end, the K-method [10] has been suggested as a means of solving coupled, multivariate nonlinearities in WDFs [11,12]. The K-method was originally proposed in [10] as a means of eliminating delay-free loops in virtual acoustic models It and its variants are frequently used in circuit simulation applications [13,14,15]. We present several examples from a class of nonlinear dynamical systems with delay-free loops that can be solved explicitly without resorting to iteration, tabulation or frequent re-calculation of filter coefficients. Since numerical solvers to ODEs can compute both x and y, the fact that both variables are fed back to the nonlinear functions f 1 and f 2 in a delay-free loop does not doom us to non-computability
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