Characterization of Nonlinear Systems with Memory by Means of Volterra Expansions with Frequency Partitioning: Application to a Cicada Mating Call
NAVAL UNDERSEA WARFARE CENTER DIV NEWPORT RI
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This report introduces several new concepts to achieve some alleviation of the curse of dimensionality in determining the higher order kernels in a Volterra expansion. These concepts include partitioning of the frequency scales, the fundamental region of formation, the diagonal strips in the two dimensional frequency space, the second- and third-order basis functions, and filtering of the measured nonlinear output to the particular frequency band under investigation. The equations for the real basis functions at second- and third-order take some unexpected forms. These results are based on taking full advantage of symmetry and conjugate-symmetry frequency-domain relations that exist for real, symmetric, second-order and third-order, time-domain kernels. The inability to construct ideal bandpass filters requires the use of a frequency-overlap procedure, followed by discarding of the edge estimates with inherent errors, and retaining only the interior estimates of higher accuracy. Application to a first- and second-order noise-free control example with a white broadband excitation gives excellent estimates of all the first- and second-order properties, such as the individual kernels, the individual Volterra waveforms, and the total estimated output waveform. Application to a cicada mating call with a distinctly non-white and non-Gaussian excitation gives good results for the estimated first- and second-order kernels and waveforms, considering the non-optimality of this type of excitation.
- Numerical Mathematics