Evaluation of Harmonic Coupling Weights in Nonlinear Periodicity Preservation Systems

When stimulated by a periodic stimulus, every component of a possibly nonlinear periodicity preservation system oscillates periodically with the same period as the stimulus. Changes in the harmonic structure of the the stimulus couples each input harmonic to, in general, every response harmonic. Knowledge of harmonic coupling weights (HCWs) allows model free characterization of effects of all small stimulus perturbation. This paper develops foundational methodology for experimentally measuring the HCWs. The on-frequency method uses small tickle tones at harmonic frequencies to do this. The off-frequency method places tickle tones adjacent to harmonic frequencies and is applicable to PP systems that are locally frequency invariant. Memoryless nonlinear PP systems are an example where the approach works exactly. The off-frequency method also requires fewer experiments. In some cases, the off-frequency method can be extended to measure numerous HCWs in a single experiment. Some system types such as the memoryless nonlinearity, require fewer experiments than general PP systems. Methods to experimentally measure the Hessian are also presented.