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  • Print publication year: 2013
  • Online publication date: October 2013

3 - Spectral linearization approximation


Simplification of basic X-parameters for small mismatch

The considerations used for the definition of X-parameters apply to the steady-state behavior of time-invariant nonlinear components with incident (and hence also scattered) waves on a harmonic frequency grid. The generality of the formalism comes at the cost of considerable complexity. Each spectral map is a nonlinear function of every applied DC-bias condition and all the magnitudes and phases of each spectral component of every signal at every port. Sampling such behavior in all variables for many ports and harmonics would be prohibitive in terms of data acquisition time, data file size, and model simulation speed.

Fortunately, in most cases of practical interest, only a few large-signal components need to be considered with complete generality, while most spectral components can be considered small and dealt with by methods of perturbation theory. Specifically, we can apply the following methodology.

Identify the few large tones that drive the main nonlinear behavior of the system.

Identify the nonlinear spectral map defined when only these few large tones drive the system (without any of the other tones identified as small-level signals). This nonlinear spectral map represents a specific steady state the system arrives at due to the large-signal stimulus.

Linearize the spectral maps around this specific steady state defined by the few important large tones.

Consider all the rest of the signals (the small signals) to be treated linearly based on the linearized map.

Horn, J., Verspecht, J., Gunyan, D., Betts, L., Root, D. E., and Eriksson, J., “X-parameter measurement and simulation of a GSM handset amplifier,” in EuMIC 2008, Amsterdam, Oct. 2008.
Skyworks, “SKY77329 PA module for quad-band GSM / EDGE,” SKY77329 Datasheet, Oct. 2005.
Additional reading
Root, D. E., Horn, J., Nielsen, T., et al., “X-parameters: the emerging paradigm for interoperable characterization, modeling, and design of nonlinear microwave and RF components and systems,” in IEEE Wamicon2011 Tutorial, Clearwater, FL, Apr. 2011.
Root, D. E., Verspecht, J., Sharrit, D., Wood, J., and Cognata, A., “Broad-band, poly-harmonic distortion (PHD) behavioral models from fast automated simulations and large-signal vectorial network measurements,” in IEEE Trans. Microw. Theory Tech., vol. 53, no. 11, pp. 3656–3664, Nov. 2005.
Verspecht, J., “Describing functions can better model hard nonlinearities in the frequency domain than the Volterra theory,” Ph.D. thesis annex, Vrije Univ. Brussel, Belgium, Nov. 1995; available at .
Verspecht, J. and Root, D. E., “Poly-harmonic distortion modeling,” IEEE Microwave, vol. 7, no. 3, pp. 44–57, June 2006.
Verspecht, J., Bossche, M. V., and Verbeyst, F., “Characterizing components under large signal excitation: defining sensible ‘large signal S-Parameters’?!,” in 49th ARFTG Conf. Dig., Denver, CO, 1997, pp. 109–117.