Power-preserving nonlinear modal coupling, feedback frequency/phase modulation, and the stretched allpass filter

Abstract

Motivated by the sound produced by nonlinear modal coupling in acoustic systems, e.g., changes of sounding frequency in the form of pitch glides/chirps and the generation of harmonics, we model a nonlinear coupling between two modes of oscillation and derive expressions mapping "musical" parameters to the model's "synthesis" parameters. We begin by showing that a unitary power-preserving matrix formulation of the coupled modes leads to a feedback frequency (FM) equation, the instantaneous frequency of which is integrated over time to yield the phase in the corresponding—and preferred—phase modulation (PM) representation. Though the integration is made difficult by the feedback term, an analytic solution is presented by first observing that the real/imaginary parts of the PM equation are equivalent to the real/imaginary parts of the transfer function for a "stretched" allpass filter—one where the delay is not constrained to be a single sample and where the real part produces a periodic comb-like signal, that if made to vary over time (rather than frequency), produces the sounding frequency (a "musical" parameter) resulting from the nonlinear coupling. Ultimately, it is hoped this work may provide computationally efficient models of percussion instruments with rich nonlinear behaviour at little additional computational cost.