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

Nicholas B. Tufillaro

arXiv:2609.35828v3eess.SP

TL;DR

Frequency-domain behavioral models share a functional form, but their usual justifications are framework-specific. The paper derives that form from time invariance using a weighted circle action and invariant theory, proving completeness for smooth two-port responses in single-tone periodic steady state. It interprets the Cardiff exponents in terms of nonlinearity and bounds, with illustrations from measurement and simulation.

  • Problem

    Several behavioral-modeling frameworks use the same phase-factor and magnitude/relative-phase form, while the paper seeks a unified mathematical account of why time invariance determines it.

    Source: §I · p. 1

  • Method

    The paper derives equivariance from time-origin shifts and applies invariant theory to characterize the resulting spectral maps and their Cardiff exponents.

    Source: §I · p. 1, §II · p. 2

  • Results

    The Cardiff form is complete for smooth time-invariant two-port responses in single-tone periodic steady state, and the r ≤ 1 restriction is exact for at-most-cubic load-side nonlinearity.

    Source: §I · pp. 1–2

  • Takeaways & Limitations

    For loaded devices, coefficient decay in r can be measured from a fit and used as a stopping rule.

    Source: §I · p. 2

    The stated scope is two-port fundamental load-pull under single-tone CW excitation in periodic steady state; Theorem 2 assumes a memoryless load-side nonlinearity.

    Source: §VIII · p. 8

Abstract

from arXiv · show

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 $r\le1$ 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.

I. INTRODUCTION

The paper places the Cardiff, X-parameter, describing-function, and predistortion forms under a shared account: time invariance constrains the spectral map through a weighted circle action. This makes the familiar two-port form complete and gives physical interpretations to its exponents.

I. INTRODUCTION

  • For a smooth two-port response, the Cardiff form is complete, with m = |n| + 2r imposed by smoothness at zero load wave.A residual that a single-tone CW fit cannot remove points to another tone, port, harmonic injection, or a measurement outside steady state.Source: §I · p. 1
  • The Cardiff exponents encode load-side nonlinearity order m, drive-side harmonic n, and conjugate-pair count r.For a load-side polynomial of degree K, r_max = floor(K/2); the mixing order is m + |n − h|.Source: §I · p. 1
  • The restriction r ≤ 1 is exact for at-most-cubic load-side nonlinearity, but higher-order and loaded devices can require larger r.For a loaded device, coefficient decay in r can be measured from the fit and used as a stopping rule.Source: §I · p. 2
  • The account connects prior Cardiff, X-parameter, describing-function, and predistortion results through a shared mathematical correspondence.The paper confines its analysis to single-tone periodic steady state; memory under modulation is treated separately.Source: §I · pp. 1–2

II. TIME INVARIANCE AS A CIRCLE ACTION

  • Time invariance makes the spectral map equivariant under a circle action in which each harmonic rotates at a rate set by its index.The time shift acts on the input and output phasors with integer weights equal to their harmonic indices.Source: §II · p. 2

III. INVARIANTS, EQUIVARIANTS AND COMPLETENESS

For two fundamental incident waves, the circle-action invariants reduce to their magnitudes and relative phase. Classical invariant theory establishes that these quantities suffice, including for smooth responses.

A. Invariants of the circle action

  • The two fundamental incident waves have invariant magnitudes and a weight-balanced product whose phase gives their relative phase.The product A2,1 times the conjugate of A1,1 cancels the common phase rotation.Source: A. Invariants of the circle action · pp. 2–3
  • Hilbert’s finite-generation result and Schwarz’s smooth extension show that every smooth invariant is a smooth function of the magnitude and relative-phase generators.For two fundamental waves, the Hilbert basis comprises the squared magnitudes and the real and imaginary parts of their balanced product, subject to one relation.Source: A. Invariants of the circle action · p. 3

B. Equivariants and the completeness theorem

The equivariant output is a phase carrier multiplied by an invariant function, yielding the complete Cardiff representation for a two-port in single-tone periodic steady state. The same construction extends to N ports, with smoothness imposing port-by-port exponent constraints.

B. Equivariants and the completeness theorem

  • Dividing the output by a weight-h carrier makes an invariant, and multiplying back by the drive-phase carrier gives the general equivariant form.The response is a phase factor times functions of the input magnitudes and relative phase.Source: B. Equivariants and the completeness… · p. 3
  • Smoothness at zero load wave requires m = |n| + 2r in the Cardiff expansion.The coefficient of the relative-phase mode Q^n contains |A2,1|^|n| times a smooth function of |A1,1| and |A2,1|^2.Source: B. Equivariants and the completeness… · p. 3
  • For N ports driven at the fundamental, the representation is a Fourier series in N − 1 relative phases, with smoothness constraints imposed port by port.For each port q, the load-magnitude exponent obeys m_q = |n_q| + 2r_q.Source: B. Equivariants and the completeness… · p. 3
  • The Cardiff representation is exhaustive for smooth responses, rather than an approximation, truncation, or modeling assumption.The X-parameter phase-reference argument proves the structure where the drive wave is nonzero; the theorem also addresses the behavior at zero load wave.Source: B. Equivariants and the completeness… · pp. 3–4

C. Counting the coefficients

The Molien series counts independent invariants and equivariants, while the Cardiff restriction r≤1 retains every monomial through degree 3 but removes an increasing share at higher degrees.

C. Counting the coefficients

  • The Molien series answers how many independent invariants and equivariants exist at each degree; for two weight-one variables, the invariant ring is a free rank-two module over three quadratic invariants.Its generators are 1 and Im(A2,1 conjugate(A1,1)); the quadratic invariants are the two wave magnitudes squared and the real and imaginary parts of their cross-product.Source: C. Counting the coefficients · p. 4

A. Cardiff terms as monomials

  • The restriction r≤1 contains every monomial through degree 3 and first excludes A2 conjugate(A2)=|A|^4.Here r counts conjugate pairs in a monomial.Source: A. Cardiff terms as monomials · p. 4
  • On the exponent lattice, lines of constant n and m run diagonally and anti-diagonally, while r measures distance from the nearer axis.The r≤1 restriction keeps an L-shaped region along the axes and excludes the interior where a≥2 and b≥2.Source: A. Cardiff terms as monomials · p. 4
  • At D=4, r≤1 retains 14 of 15 monomials, while at D=12 it retains 46 of 91.The restriction removes many coefficients at high order and almost none at low order.Source: A. Cardiff terms as monomials · p. 4

B. The degree theorem

For a memoryless device with polynomial load-side dependence of degree K, the theorem bounds the Cardiff conjugate-pair index by floor(K/2). The bound is exact for cubic dependence and remains valid for non-separable device maps.

B. The degree theorem

  • For a memoryless device with load-side polynomial degree K and voltage set by the incident wave, every response term satisfies r≤floor(K/2).The proof uses a+b=p and r=min(a,b), so r conjugate pairs require at least 2r factors and p≤K.Source: B. The degree theorem · pp. 4–5
  • The restriction r≤1 holds for every term exactly when K≤3; at fourth order, the first new term is |A|^4.The bound is attained when the drive side supplies the output harmonic, as occurs at the fundamental for a nonlinearity with a linear term.Source: B. The degree theorem · p. 5
  • The degree bound also applies to non-separable F(v1, v2), because it depends only on the degree in v2.The authors report brute-force checks for K=2 through 7, both separable and non-separable.Source: B. The degree theorem · p. 5
  • Changing the expansion center from A=0 to A=c cannot increase min(a,b), although centered terms can have a+b below the source order.The highest degree present in a centered expansion remains exactly K.Source: B. The degree theorem · p. 5

C. Physical meaning of the three indices

The three Cardiff indices encode load-side order, drive-side harmonic, and conjugate-pair count. Together they also recover the mixing order used to choose exponents.

C. Physical meaning of the three indices

V. LOADED DEVICES: DECAY IN r AND A STOPPING RULE

Loading makes the response a fixed point rather than a finite polynomial, so every r can occur. Coefficients decay with r, providing a measured stopping rule instead of a fixed bound.

V. LOADED DEVICES: DECAY IN r AND A STOPPING RULE

  • Loading makes the response a fixed point because the load-side voltage includes the device’s own output, leaving no finite K or strict r_max.Even a quadratic map’s fixed point has every order present.Source: §V · p. 5
  • The simulation closes the loop pointwise in time, samples B2,1 over |A| ≤ ρ, and fits Aa Āb monomials scaled to the measured region.The example uses a quadratic intrinsic dependence on load-side voltage, for which Theorem 2 gives r_max = 1.Source: §V · p. 5
  • Each increase in r reduces the largest closed-loop coefficient by one to one-and-a-half decades; halving load-wave amplitude reduces later step ratios by about four.The per-step decay is roughly (loop gain × |A|)^2, set by mismatch degree rather than the fit.Source: §V · p. 5
  • For a loaded device, retain terms until coefficient decay meets the measurement accuracy floor, rather than imposing a fixed r cutoff.Plot the largest coefficient in each r-class against r on a log axis; the decay rate is measurable from the fit.Source: §V · p. 6, Figure 3 · p. 6

VI. TAILORED A-PULL MEASURES THE DECOMPOSITION

The tailored A-pull method converts exponent terms into spectral clusters whose center and width encode the phase and magnitude exponents. The widest resolvable cluster identifies the load-side nonlinearity order and hence the intrinsic r bound.

VI. TAILORED A-PULL MEASURES THE DECOMPOSITION

  • A-pull steps the load-side wave around a spiral and Fourier-transforms the response; each exponent term contributes a cluster of spectral lines.The method expands around a chosen center load-side wave.Source: A. Clusters · p. 6
  • The spectral cluster center gives n and its half-width gives m, making the spectrum a picture of the exponent decomposition.Cluster lines are centered at nSpm with half-width mSam; the sampling rates are chosen to prevent neighboring clusters from overlapping.Source: A. Clusters · p. 6
  • The reported GAN measurement at 2.45 GHz identifies nine exponent pairs, with m, n, mixing order, and |K| listed for each.The caption states Sam = 5, Spm = 89, and N = 1069; r and the mixing order of Corollary 3 are computed from the listed quantities.Source: Table III · p. 6
  • The widest cluster’s half-width equals load-side nonlinearity order K, so the spectrum reveals K by inspection rather than by fitting.A cluster’s half-width is the lowest load-side order able to produce it.Source: A. Clusters · p. 6

B. The published data

The published coefficients support a cubic load-side order and the predicted r bound, while their measured decay suggests fourth-order content may sit between the spurious and noise floors.

B. The published data

  • Every reported pair has r ≤1 and m ≤3, implying detected load-side order K = 3 and matching the theorem’s r_max = 1 prediction row by row.The mixing-order formula also reproduces the tabulated order in all nine rows.Source: B. The published data · p. 6
  • At fixed n, the r = 1 to r = 0 coefficient ratios are 0.036, 0.058, and 0.20 for n = 0, 1, and −1, respectively.Extrapolating the n = 0 ratio predicts the (4, 0) term near −58 dBc, between the reported spurious and noise floors.Source: B. The published data · p. 6
  • If the spurious floor approaches the noise floor, (4, 0) is predicted to appear first near −58 dBc, followed by (5, 1); the magnitude is extrapolated, but the ordering is structural.If neither appears down to −60 dBc, the measurement would confirm r ≤1 for that device rather than impose it as a modeling assumption.Source: B. The published data · p. 7, Table IV · p. 7

VII. TWO SIMULATIONS

The simulations show Cardiff cluster structure in both a polynomial device and an ASM-HEMT model. Their detected load-side orders and higher-r content depend on the device and drive conditions.

A. A polynomial device

  • At a −60 dBc floor, the cubic polynomial device matches the measured result, while lowering the floor reveals fourth-order terms in the predicted order, beginning with (4, 0).Its (4, 0) line is −59.6 dBc; the measured-coefficient extrapolation is −58 dBc.Source: A. A polynomial device · p. 7, Figure 4 · p. 8
  • For the cubic device, opening the loop reduces (4, 0) to a numerical zero, whereas the K = 4 open-loop case gives −109 dBc, indicating that load feedback generates essentially all fourth-order content in the K = 3 case.The closed-loop cubic-device (4, 0) line is −59.6 dBc.Source: A. A polynomial device · p. 7, Figure 4 · p. 8

B. A compact transistor model in a circuit simulator

  • At a −60 dBc floor, ASM-HEMT yields K_eff = 2 and r_max = 1, with five detected pairs at either drive; its (4, 0) term is the strongest r = 2 term at both drives.The (4, 0) level is −71.8 dBc at the softer drive and −97.6 dBc at the harder drive.Source: Table V · p. 7, B. A compact transistor model in a… · p. 7
  • ASM-HEMT spectra place visible lines on Cardiff cluster positions, with off-cluster bins empty to −80 dBc and −107 dBc at the two drives.The model was run with self-heating and trapping switched off, making it memoryless as assumed for Theorem 2.Source: B. A compact transistor model in a… · p. 7
  • For ASM-HEMT, decay in r is not geometric: along n = 0, successive steps are −44, −28, and −14 dB at the softer drive and −52, −46, and −33 dB at the harder drive.A one-step extrapolation underestimates the (4, 0) levels by 16 dB and 6 dB, respectively.Source: B. A compact transistor model in a… · pp. 7–8

VIII. DISCUSSION

The discussion turns model-basis choices into measurable properties of the driven device and gives a practical identification procedure, while delimiting the theorem’s scope.

VIII. DISCUSSION

  • The coefficient count at each order and the truncation in r are quantified by the Molien series and by the intrinsic-order bound or, for loaded devices, measured coefficient decay.The harmonic count is likewise tied to measurable roll-off, while the number of memory states is left to a companion paper [30].Source: §VIII · p. 8
  • For identification, convert Cardiff exponents to (a, b), set r = min(a, b), infer load-side order from the largest resolved m, and for loaded devices truncate where coefficient decay reaches the floor.A-pull cluster widths give K by inspection; check off-cluster bins first, and treat a persistent CW residual as evidence of another tone, port, harmonic injection, or non-steady measurement.Source: §VIII · p. 8
  • The theorem’s stated scope is two-port, single-tone CW fundamental load-pull in periodic steady state; Theorem 2 additionally assumes a memoryless load-side nonlinearity.Multi-harmonic inputs extend the circle action, while incommensurate two-tone excitation requires a two-torus argument.Source: §VIII · p. 8
  • The quantitative evidence rests on simulation and one published table; the proposed measurement would sweep fundamental load-pull at one bias and drive with the spurious floor near the noise floor.Source: §VIII · pp. 8–9

APPENDIX A MOLIEN SERIES AND THE HIRONAKA DECOMPOSITION

The appendix describes the invariant and equivariant generators behind the model basis, then derives the bound on conjugate-pair count from load-side polynomial degree.

APPENDIX A MOLIEN SERIES AND THE HIRONAKA DECOMPOSITION

  • The Molien–Weyl constant-term operation counts invariant polynomials by degree; replacing the constant-term extraction with weight h counts weight-h equivariants.Source: Appendix A · p. 9
  • For the two fundamental incident waves, the invariant generators are their squared magnitudes and the real and imaginary parts of their cross-product.Every invariant is uniquely p + q Im(A2,1 conjugate(A1,1)), with p and q polynomials in the three primary invariants.Source: Appendix A · p. 9
  • The Cardiff basis (5), written in gauge P and found experimentally one harmonic at a time, is a generating set of this module.Numerical completeness at each degree means the design matrix of weight-h monomials evaluated at random sample points has rank equal to the Molien count, verified by the routines of.Source: Appendix A · p. 9

APPENDIX B AN ELEMENTARY DERIVATION OF THE BOUND ON r

  • For load-side degree K, a term A^a conjugate(A)^b has a + b ≤ K and r = min(a, b), so 2r ≤ K and r ≤ floor(K/2).Thus K = 3 gives r ≤1, while K = 4 allows r ≤2 and introduces |A|^4.Source: Appendix B · p. 9
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