Frequency multiplexing tickle tones to determine harmonic coupling weights in nonlinear systems

Abstract

Obtaining the Jacobian parameters for a nonlinear system, such as in X-parameter or S-function measurement, requires excitation of tickle tones either on or slightly off of the main harmonic frequencies. Traditional off-frequency measurements involve simultaneous excitation with tickle tones both slightly above and/or below a single harmonic frequency at a time. This paper describes a multiplexing technique for the excitation of the system with perturbations at all harmonics simultaneously. This technique requires both frequency and cross-frequency noninterference for successful results from the measurements. If these conditions exist for the nonlinear system, the off-frequency multiplexing technique for tickle tones can reduce the measurements necessary to find the Jacobian matrices by a factor equal to the number of harmonics considered in the polyharmonic distortion. These methods provide exact results for memoryless polynomial nonlinearities.In making nonlinear network parameter measurements, such as the X-parameters 1 [1] and the S-functions [2-4], it is necessary to apply a large-signal "operating" condition, followed by small perturbations to this condition known as "tickle" tones. The authors of this paper recently have written a survey paper on nonlinear network parameters that briefly describes measurement options from a practical stance [5]. Two options exist for the tickle-tone measurements: (1) onfrequency tickle tones, where the perturbation is added at the same frequency as the desired stimulus harmonic and (2) offfrequency tickle tones [6], where the perturbation is added at a slightly offset frequency from the appropriate fundamental or harmonic. The excitation approaches have been significantly considered by Betts in recent publication [7]. While a significant percentage of modern nonlinear network analyzers use the on-frequency method, an advantage of the offfrequency method is that the results of the tickle-tone perturbations can be isolated away from large signals overlapping in frequency.This paper demonstrates that the off-frequency technique can be extended to allow simultaneous application of 1 X-parameters is a registered trademark of Agilent Technologies. perturbations at multiple harmonics, a technique herein labeled frequency multiplexing of tickle tones. This can significantly reduce the number of measurement iterations necessary to fully determine the matrix of nonlinear network parameters.We examine the frequency multiplexing approach for periodicity preservation (PP) systems approach in finding the derivatives composing the system Jacobian in the Fourier series coefficient domain.The PP and time-invariant periodicity preservation (TIPP) systems have been introduced in a recently authored paper in process of publication [8], and the limitations of use on some nonlinear systems examined quantitatively in [9]. This theory forms the foundation for experimental nonlinear network analysis.A basic discussion of the system theory describing the context of this measurement is useful. A mor...

Evaluation of Harmonic Coupling Weights in Nonlinear Periodicity Preservation Systems

“…Of specific interest in coupling among harmonics imposed by the perturbations around nonlinearities [1]- [24]. In previous papers [3], we have outlined the affine approximation and the relationships among time, frequency, and mixed time frequency characterizations of affine approximations of nonlinear periodicity preservation (PP) systems.…”