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

Abstract

The functional form shared by frequency-domain behavioral models of nonlinear microwave devices (the Cardiff model, X-parameters, higher-order sinusoidal describing functions) is usually presented as a modeling choice; here it is shown to follow from time invariance. Under a shift of the time origin the $k$-th harmonic phasor rotates by $k$ times the fundamental's angle, so the spectral map of a time-invariant device is equivariant under a weighted action of the circle group, and classical invariant theory gives the general form of every such map; the three communities' phase-normalization factors are its weight-carrying factor. The result is a completeness theorem: in single-tone periodic steady state no time-invariant two-port response lies outside the Cardiff form, and the index relation $m=|n|+2r$ is the smoothness condition at zero load wave. Writing each Cardiff term as a monomial $A^{a}\bar{A}^{b}$ in the load wave gives the exponents physical interpretations -- $m$ is the order of the load-side nonlinearity, $n$ set by the drive-side harmonic, $r=\min(a,b)$ the number of conjugate pairs -- and the bound $r_{\max}=\lfloor K/2\rfloor$ for load-side polynomial degree $K$, so the familiar restriction $r\le1$ is exact for a cubic load-side nonlinearity and fails at fourth order. For a loaded device the bound becomes a measurable decay in $r$. Tailored A-pull measurement displays the decomposition directly, and two simulations reproduce its published pattern of detected terms and place the first fourth-order term near $-58$ dBc, between that measurement's $-40$ dBc spurious floor and its $-60$ dBc noise floor.