Time Invariance, Circle Symmetry, and the Completeness of the Cardiff Behavioral Model

Abstract

Frequency-domain behavioral models of nonlinear microwave devices (the Cardiff model, X-parameters, higher-order sinusoidal describing functions, and the baseband models of power-amplifier predistortion) share one functional form, and each modeling framework justifies the form by its own argument, time invariance among them. Here we give the mathematical argument by which time invariance alone determines the model form, and state the result as a theorem for two-port devices, with its extension to any number of ports. Under a shift of the time origin each harmonic phasor rotates by a multiple of the fundamental's angle, so the spectral map of a time-invariant device is equivariant under a weighted circle action, and classical invariant theory gives its general form. The result is a completeness theorem for the form already in use: every time-invariant two-port response in single-tone periodic steady state is a phase factor attached to the fundamental multiplying a function of the wave magnitudes and their relative phase. The Cardiff exponents become counters: $m$ is the order of the load-side nonlinearity, $n$ is set by the drive-side harmonic, and $r$ is the number of conjugate pairs. The bound $r_{\max }=\lfloor K/2\rfloor$ for a load-side nonlinearity of degree $K$ makes the familiar restriction $r\le 1$ a hypothesis about the device, exact for a cubic nonlinearity; for a loaded device the bound becomes a measurable decay in $r$. The results are illustrated with a published measurement and two simulations.

10 pages, 4 figures, 5 tables. v3: one sentence in Sec. VIII corrected (the multi-harmonic extension uses the single circle, not a torus); running head corrected. v2: prior work attributed; Remark 1 (N ports) added; results unchanged. Companion papers: arXiv:2609.38771, arXiv:2610.01031. Toolkit: doi:10.5281/zenodo.22816760