Complex Frequency (CF): Kinetics & Applications

• Complex Frequency (CF) is a complex-valued rate that combines amplitude (radial) evolution and phase (rotational) dynamics to characterize signal behavior.
• CF links time-domain signal analysis with geometric interpretations, equating decoupled modal frequencies in diagonalizable LTI systems to eigenvalues of the state matrix.
• In power systems and wave physics, CF formulations improve transient stability assessments and enable robust frequency estimation during dynamic perturbations.

Searching arXiv for the specified papers to ground the article in the cited literature. Complex Frequency (CF) denotes a complex-valued rate that combines amplitude evolution and rotational evolution into a single object. For a complex signal or planar vector written as $x(t)=|x(t)|e^{j\theta(t)}$, one has

$$x'(t)=\eta(t)\,x(t),\qquad \eta(t)=\rho(t)+j\,\omega(t),$$

with

$$\rho(t)=\frac{d}{dt}\ln|x(t)|,\qquad \omega(t)=\frac{d}{dt}\theta(t).$$

In this form, the real part is a radial or dilatational rate and the imaginary part is the usual angular or rotational rate (García-Veloso et al., 10 Dec 2025). In geometric treatments, CF is the two-dimensional restriction of a higher-dimensional geometric frequency multivector, and for diagonalizable linear time-invariant (LTI) systems the CF of decoupled modal trajectories coincides with the eigenvalues of the state matrix (Sofos et al., 20 Mar 2026). In optics, mechanics, and wave physics, the same terminology is also used for exponentially decaying harmonic excitations that probe a system at a complex frequency $\tilde\omega=\omega_0-i\alpha$ (Khurgin et al., 2 Jul 2026). These uses share the same formal decomposition into growth or decay and oscillation, but they arise from different representational settings.

1. Canonical definitions and notation

A standard signal-theoretic formulation begins with a complex representation of a real or vector signal and takes its logarithmic time derivative. In the analytic-signal formulation associated with Hahn, a real signal $v(t)$ is embedded as

$$\tilde v(t)=v(t)+j\,\mathcal H\{v(t)\}=u(t)e^{j\theta(t)},$$

and the instantaneous complex phase is

$\bar\phi_{\mathfrak h}(t)=\Ln(\tilde v(t))=\Ln(u(t))+j\,\theta(t).$

Differentiation yields the instantaneous complex frequency

$$\bar s_{\mathfrak h}(t)=\frac{d}{dt}\bar\phi_{\mathfrak h}(t)=\frac{\dot u(t)}{u(t)}+j\,\dot\theta(t) =\varrho_{\mathfrak h}(t)+j\,\omega_{\mathfrak h}(t),$$

where $\varrho_{\mathfrak h}(t)$ is the radial (dilatation) frequency and $\omega_{\mathfrak h}(t)$ is the angular (rotational) frequency (García-Veloso et al., 10 Dec 2025).

A second formulation, associated in the recent literature with Milano and with Lei et al., uses a complex space vector such as a Clarke or Park transform. For a Park vector

$$x'(t)=\eta(t)\,x(t),\qquad \eta(t)=\rho(t)+j\,\omega(t),$$0

one defines

$$x'(t)=\eta(t)\,x(t),\qquad \eta(t)=\rho(t)+j\,\omega(t),$$1

García-Veloso et al. propose the unified notation ICP for instantaneous complex phase and ICF for instantaneous complex frequency, with $$x'(t)=\eta(t)\,x(t),\qquad \eta(t)=\rho(t)+j\,\omega(t),$$2 as the consolidated form (García-Veloso et al., 10 Dec 2025).

In power-system notation, the same structure is written at bus $$x'(t)=\eta(t)\,x(t),\qquad \eta(t)=\rho(t)+j\,\omega(t),$$3 as

$$x'(t)=\eta(t)\,x(t),\qquad \eta(t)=\rho(t)+j\,\omega(t),$$4

with $$x'(t)=\eta(t)\,x(t),\qquad \eta(t)=\rho(t)+j\,\omega(t),$$5 and $$x'(t)=\eta(t)\,x(t),\qquad \eta(t)=\rho(t)+j\,\omega(t),$$6 the local angular-frequency deviation relative to the rotating Park reference (Milano, 2021). This notation makes explicit that CF is not merely a phase derivative.

2. Geometric frequency and the generalized-eigenvalue interpretation

In the geometric formulation, one starts from a time-dependent generalized velocity $$x'(t)=\eta(t)\,x(t),\qquad \eta(t)=\rho(t)+j\,\omega(t),$$7 and defines the geometric frequency multivector as

$$x'(t)=\eta(t)\,x(t),\qquad \eta(t)=\rho(t)+j\,\omega(t),$$8

Here $$x'(t)=\eta(t)\,x(t),\qquad \eta(t)=\rho(t)+j\,\omega(t),$$9 is the inner product, $$\rho(t)=\frac{d}{dt}\ln|x(t)|,\qquad \omega(t)=\frac{d}{dt}\theta(t).$$0 is a bivector, $$\rho(t)=\frac{d}{dt}\ln|x(t)|,\qquad \omega(t)=\frac{d}{dt}\theta(t).$$1 is the symmetric radial part, and $$\rho(t)=\frac{d}{dt}\ln|x(t)|,\qquad \omega(t)=\frac{d}{dt}\theta(t).$$2 is the antisymmetric rotational part. Geometrically, $$\rho(t)=\frac{d}{dt}\ln|x(t)|,\qquad \omega(t)=\frac{d}{dt}\theta(t).$$3 measures instantaneous radial growth or decay, while $$\rho(t)=\frac{d}{dt}\ln|x(t)|,\qquad \omega(t)=\frac{d}{dt}\theta(t).$$4 measures instantaneous rotation or twist (Sofos et al., 20 Mar 2026).

When the construction is restricted to $$\rho(t)=\frac{d}{dt}\ln|x(t)|,\qquad \omega(t)=\frac{d}{dt}\theta(t).$$5, the bivector part becomes identifiable with the complex unit, and the geometric frequency becomes an ordinary complex number,

$$\rho(t)=\frac{d}{dt}\ln|x(t)|,\qquad \omega(t)=\frac{d}{dt}\theta(t).$$6

In this sense, CF is the planar specialization of a more general geometric invariant of curves (Sofos et al., 20 Mar 2026).

The generalized-eigenvalue result concerns diagonalizable LTI systems

$$\rho(t)=\frac{d}{dt}\ln|x(t)|,\qquad \omega(t)=\frac{d}{dt}\theta(t).$$7

If $$\rho(t)=\frac{d}{dt}\ln|x(t)|,\qquad \omega(t)=\frac{d}{dt}\theta(t).$$8 is diagonalized by a similarity transform into real $$\rho(t)=\frac{d}{dt}\ln|x(t)|,\qquad \omega(t)=\frac{d}{dt}\theta(t).$$9 blocks $\tilde\omega=\omega_0-i\alpha$0 and real $\tilde\omega=\omega_0-i\alpha$1 blocks

$\tilde\omega=\omega_0-i\alpha$2

then, after setting $\tilde\omega=\omega_0-i\alpha$3 and transforming $\tilde\omega=\omega_0-i\alpha$4, the transformed dynamics decouple into scalar modes and planar modes. For each scalar mode, the geometric frequency is exactly the real eigenvalue $\tilde\omega=\omega_0-i\alpha$5. For each planar mode, direct computation gives $\tilde\omega=\omega_0-i\alpha$6 and $\tilde\omega=\omega_0-i\alpha$7. The proposition stated in the paper is that, for any diagonalizable LTI system, the complex frequencies of the decoupled state modes coincide with the eigenvalues of $\tilde\omega=\omega_0-i\alpha$8 (Sofos et al., 20 Mar 2026).

This furnishes a unified geometric interpretation of eigenvalues. In the same framework, eigenvalues appear as constant radial-and-curvature rates of modal trajectories, linking classical linear systems to differential geometry of curves. The paper also states the principal limitation: for nonlinear systems, no fixed similarity $\tilde\omega=\omega_0-i\alpha$9 diagonalizes the Jacobian globally, so the eigenvalues of the instantaneous Jacobian do not in general match the instantaneous geometric frequency of $v(t)$0. Nevertheless, the geometric frequency

$v(t)$1

always remains well defined and provides a geometric interpretation of the system flow (Sofos et al., 20 Mar 2026).

3. Power-system formulations, estimation, and control

In power systems, CF was introduced to unify voltage-magnitude and angle dynamics at each bus. Writing the complex bus powers as $v(t)$2, with $v(t)$3 denoting the Hadamard product, the transient link between injections and complex frequency is given by three equivalent relations: $v(t)$4

$v(t)$5

$v(t)$6

These equations follow from differentiating $v(t)$7 in the Park frame together with $v(t)$8. The stated interpretation is that the real part $v(t)$9 is indispensable to avoid spurious spikes in frequency estimates during transients and to capture voltage-magnitude-driven frequency coupling (Milano, 2021).

Estimation methods have been developed accordingly. One approach for inverter-based resources is a dual-PLL estimator: one loop tracks phase and delivers $$\tilde v(t)=v(t)+j\,\mathcal H\{v(t)\}=u(t)e^{j\theta(t)},$$0, and a second loop tracks $$\tilde v(t)=v(t)+j\,\mathcal H\{v(t)\}=u(t)e^{j\theta(t)},$$1 and delivers $$\tilde v(t)=v(t)+j\,\mathcal H\{v(t)\}=u(t)e^{j\theta(t)},$$2, so that $$\tilde v(t)=v(t)+j\,\mathcal H\{v(t)\}=u(t)e^{j\theta(t)},$$3 is available for control. The same paper sets the steady-state reference to

$$\tilde v(t)=v(t)+j\,\mathcal H\{v(t)\}=u(t)e^{j\theta(t)},$$4

thereby enforcing constant voltage magnitude and nominal grid angular speed. The CF-based current-reference law is then

$$\tilde v(t)=v(t)+j\,\mathcal H\{v(t)\}=u(t)e^{j\theta(t)},$$5

On the WSCC 9-bus benchmark, with performance measured by $$\tilde v(t)=v(t)+j\,\mathcal H\{v(t)\}=u(t)e^{j\theta(t)},$$6 at bus 2 and network-wide $$\tilde v(t)=v(t)+j\,\mathcal H\{v(t)\}=u(t)e^{j\theta(t)},$$7, both evaluated at $$\tilde v(t)=v(t)+j\,\mathcal H\{v(t)\}=u(t)e^{j\theta(t)},$$8 and normalized so Standard control $$\tilde v(t)=v(t)+j\,\mathcal H\{v(t)\}=u(t)e^{j\theta(t)},$$9, the reported load-step results are Standard $\bar\phi_{\mathfrak h}(t)=\Ln(\tilde v(t))=\Ln(u(t))+j\,\theta(t).$0, $\bar\phi_{\mathfrak h}(t)=\Ln(\tilde v(t))=\Ln(u(t))+j\,\theta(t).$1; Virtual inertia $\bar\phi_{\mathfrak h}(t)=\Ln(\tilde v(t))=\Ln(u(t))+j\,\theta(t).$2, $\bar\phi_{\mathfrak h}(t)=\Ln(\tilde v(t))=\Ln(u(t))+j\,\theta(t).$3; and $\bar\phi_{\mathfrak h}(t)=\Ln(\tilde v(t))=\Ln(u(t))+j\,\theta(t).$4-control $\bar\phi_{\mathfrak h}(t)=\Ln(\tilde v(t))=\Ln(u(t))+j\,\theta(t).$5, $\bar\phi_{\mathfrak h}(t)=\Ln(\tilde v(t))=\Ln(u(t))+j\,\theta(t).$6. For the three-phase fault, the corresponding values are Standard $\bar\phi_{\mathfrak h}(t)=\Ln(\tilde v(t))=\Ln(u(t))+j\,\theta(t).$7, $\bar\phi_{\mathfrak h}(t)=\Ln(\tilde v(t))=\Ln(u(t))+j\,\theta(t).$8; Virtual inertia $\bar\phi_{\mathfrak h}(t)=\Ln(\tilde v(t))=\Ln(u(t))+j\,\theta(t).$9, $$\bar s_{\mathfrak h}(t)=\frac{d}{dt}\bar\phi_{\mathfrak h}(t)=\frac{\dot u(t)}{u(t)}+j\,\dot\theta(t) =\varrho_{\mathfrak h}(t)+j\,\omega_{\mathfrak h}(t),$$0; and $$\bar s_{\mathfrak h}(t)=\frac{d}{dt}\bar\phi_{\mathfrak h}(t)=\frac{\dot u(t)}{u(t)}+j\,\dot\theta(t) =\varrho_{\mathfrak h}(t)+j\,\omega_{\mathfrak h}(t),$$1-control $$\bar s_{\mathfrak h}(t)=\frac{d}{dt}\bar\phi_{\mathfrak h}(t)=\frac{\dot u(t)}{u(t)}+j\,\dot\theta(t) =\varrho_{\mathfrak h}(t)+j\,\omega_{\mathfrak h}(t),$$2, $$\bar s_{\mathfrak h}(t)=\frac{d}{dt}\bar\phi_{\mathfrak h}(t)=\frac{\dot u(t)}{u(t)}+j\,\dot\theta(t) =\varrho_{\mathfrak h}(t)+j\,\omega_{\mathfrak h}(t),$$3. On the 1,479-bus Irish grid model, the Center-of-Inertia frequency deviation under the loss of the 500 MW interconnector is reported as a $$\bar s_{\mathfrak h}(t)=\frac{d}{dt}\bar\phi_{\mathfrak h}(t)=\frac{\dot u(t)}{u(t)}+j\,\dot\theta(t) =\varrho_{\mathfrak h}(t)+j\,\omega_{\mathfrak h}(t),$$4 dip with slow oscillation ($$\bar s_{\mathfrak h}(t)=\frac{d}{dt}\bar\phi_{\mathfrak h}(t)=\frac{\dot u(t)}{u(t)}+j\,\dot\theta(t) =\varrho_{\mathfrak h}(t)+j\,\omega_{\mathfrak h}(t),$$5) for Standard control and a $$\bar s_{\mathfrak h}(t)=\frac{d}{dt}\bar\phi_{\mathfrak h}(t)=\frac{\dot u(t)}{u(t)}+j\,\dot\theta(t) =\varrho_{\mathfrak h}(t)+j\,\omega_{\mathfrak h}(t),$$6 dip with faster damping ($$\bar s_{\mathfrak h}(t)=\frac{d}{dt}\bar\phi_{\mathfrak h}(t)=\frac{\dot u(t)}{u(t)}+j\,\dot\theta(t) =\varrho_{\mathfrak h}(t)+j\,\omega_{\mathfrak h}(t),$$7 mode) for $$\bar s_{\mathfrak h}(t)=\frac{d}{dt}\bar\phi_{\mathfrak h}(t)=\frac{\dot u(t)}{u(t)}+j\,\dot\theta(t) =\varrho_{\mathfrak h}(t)+j\,\omega_{\mathfrak h}(t),$$8-control (Bernal et al., 2024).

A complementary line of work uses the complex Teager-Kaiser energy operator (CTKEO) to obtain a local instantaneous-frequency estimator aligned with CF kinematics. For a twice-differentiable complex signal $$\bar s_{\mathfrak h}(t)=\frac{d}{dt}\bar\phi_{\mathfrak h}(t)=\frac{\dot u(t)}{u(t)}+j\,\dot\theta(t) =\varrho_{\mathfrak h}(t)+j\,\omega_{\mathfrak h}(t),$$9, the operator

$\varrho_{\mathfrak h}(t)$0

leads to the exact estimator

$\varrho_{\mathfrak h}(t)$1

The paper denotes this expression by $\varrho_{\mathfrak h}(t)$2 and states that the explicit correction terms remove the bias affecting conventional estimators when the envelope varies rapidly, during faults, unbalances, or harmonic distortion. The method is fully local in time, uses derivative-based computation, and requires no phase unwrapping (Vaca et al., 21 Jan 2026).

4. Complex-frequency excitations in resonant and wave systems

In optics and related wave problems, a CF excitation is an exponentially decaying harmonic waveform,

$\varrho_{\mathfrak h}(t)$3

which corresponds to probing a system at the complex frequency

$\varrho_{\mathfrak h}(t)$4

For a lossy Lorentzian resonance with transfer function

$\varrho_{\mathfrak h}(t)$5

linearization around $\varrho_{\mathfrak h}(t)$6 shows that the effective linewidth is reduced from $\varrho_{\mathfrak h}(t)$7 to $\varrho_{\mathfrak h}(t)$8, and choosing $\varrho_{\mathfrak h}(t)$9 cancels the intrinsic decay to first order. Khurgin et al. distinguish real-time CF generation, post-detection synthesized CF reconstruction, and simple inverse filtering. Their stated conclusion is that real-time CF excitation is robust to shot noise and relative-intensity noise, whereas synthesized CF response shows only limited improvement once realistic detection and readout noise is included; in low-noise conditions, a simpler matched-filter procedure attains equal or better recovery than synthesized CF reconstruction (Khurgin et al., 2 Jul 2026).

In mechanical system identification, Li et al. analyze an underdamped single-degree-of-freedom spring-mass-damper system with transfer function

$\omega_{\mathfrak h}(t)$0

Applying the CF operator $\omega_{\mathfrak h}(t)$1 to the excitation and then defining the mapped output $\omega_{\mathfrak h}(t)$2 yields mapped dynamics

$\omega_{\mathfrak h}(t)$3

When

$\omega_{\mathfrak h}(t)$4

the effective damping vanishes in the mapped variables, producing an arbitrarily sharp response at $\omega_{\mathfrak h}(t)$5. The same paper derives closed-form Fisher-information expressions under Gaussian white and colored noise and states the optimal choice as

$\omega_{\mathfrak h}(t)$6

In an aluminum cantilever experiment with $\omega_{\mathfrak h}(t)$7 and $\omega_{\mathfrak h}(t)$8 at $\omega_{\mathfrak h}(t)$9, ten noise realizations yielded harmonic-excitation estimates with mean $$x'(t)=\eta(t)\,x(t),\qquad \eta(t)=\rho(t)+j\,\omega(t),$$00 and variance $$x'(t)=\eta(t)\,x(t),\qquad \eta(t)=\rho(t)+j\,\omega(t),$$01, while CF excitation yielded mean $$x'(t)=\eta(t)\,x(t),\qquad \eta(t)=\rho(t)+j\,\omega(t),$$02 and variance $$x'(t)=\eta(t)\,x(t),\qquad \eta(t)=\rho(t)+j\,\omega(t),$$03 (Li et al., 13 May 2026).

In quantum control, CF pulses are designed from complex reflection zeros of a dissipative scattering system. For a qubit-waveguide architecture with reflection coefficient $$x'(t)=\eta(t)\,x(t),\qquad \eta(t)=\rho(t)+j\,\omega(t),$$04, a complex reflection zero $$x'(t)=\eta(t)\,x(t),\qquad \eta(t)=\rho(t)+j\,\omega(t),$$05 satisfies $$x'(t)=\eta(t)\,x(t),\qquad \eta(t)=\rho(t)+j\,\omega(t),$$06. The corresponding pulse is

$$x'(t)=\eta(t)\,x(t),\qquad \eta(t)=\rho(t)+j\,\omega(t),$$07

which is matched to the lossy system’s true response. For three coupled transmon qubits, the reported lossless comparison gives Gaussian pulses with $$x'(t)=\eta(t)\,x(t),\qquad \eta(t)=\rho(t)+j\,\omega(t),$$08 and neighbor excitation $$x'(t)=\eta(t)\,x(t),\qquad \eta(t)=\rho(t)+j\,\omega(t),$$09, implying $$x'(t)=\eta(t)\,x(t),\qquad \eta(t)=\rho(t)+j\,\omega(t),$$10 and $$x'(t)=\eta(t)\,x(t),\qquad \eta(t)=\rho(t)+j\,\omega(t),$$11, whereas CF pulses give $$x'(t)=\eta(t)\,x(t),\qquad \eta(t)=\rho(t)+j\,\omega(t),$$12, neighbor excitation $$x'(t)=\eta(t)\,x(t),\qquad \eta(t)=\rho(t)+j\,\omega(t),$$13, $$x'(t)=\eta(t)\,x(t),\qquad \eta(t)=\rho(t)+j\,\omega(t),$$14, and $$x'(t)=\eta(t)\,x(t),\qquad \eta(t)=\rho(t)+j\,\omega(t),$$15. In the lossy case with $$x'(t)=\eta(t)\,x(t),\qquad \eta(t)=\rho(t)+j\,\omega(t),$$16, a CF pulse at the true zero gives $$x'(t)=\eta(t)\,x(t),\qquad \eta(t)=\rho(t)+j\,\omega(t),$$17, $$x'(t)=\eta(t)\,x(t),\qquad \eta(t)=\rho(t)+j\,\omega(t),$$18, $$x'(t)=\eta(t)\,x(t),\qquad \eta(t)=\rho(t)+j\,\omega(t),$$19, and $$x'(t)=\eta(t)\,x(t),\qquad \eta(t)=\rho(t)+j\,\omega(t),$$20, while a pulse at the naive conjugate gives $$x'(t)=\eta(t)\,x(t),\qquad \eta(t)=\rho(t)+j\,\omega(t),$$21 and $$x'(t)=\eta(t)\,x(t),\qquad \eta(t)=\rho(t)+j\,\omega(t),$$22 (Trivedi et al., 3 Jun 2025).

5. Hypercomplex extensions for complex-valued signals

For signals that are themselves complex-valued, Le Bihan and Sangwine construct a quaternion-valued analogue of the analytic signal using a one-sided quaternion Fourier transform (QFT). For $$x'(t)=\eta(t)\,x(t),\qquad \eta(t)=\rho(t)+j\,\omega(t),$$23 and a pure-unit quaternion axis $$x'(t)=\eta(t)\,x(t),\qquad \eta(t)=\rho(t)+j\,\omega(t),$$24 orthogonal to the signal plane, the right-hand QFT is

$$x'(t)=\eta(t)\,x(t),\qquad \eta(t)=\rho(t)+j\,\omega(t),$$25

and the associated hypercomplex representation is

$$x'(t)=\eta(t)\,x(t),\qquad \eta(t)=\rho(t)+j\,\omega(t),$$26

whose QFT is one-sided: $$x'(t)=\eta(t)\,x(t),\qquad \eta(t)=\rho(t)+j\,\omega(t),$$27 The resulting quaternion signal admits a polar Cayley-Dickson form

$$x'(t)=\eta(t)\,x(t),\qquad \eta(t)=\rho(t)+j\,\omega(t),$$28

from which one identifies a complex envelope and an instantaneous phase (Bihan et al., 2012).

The central validity condition is a non-overlapping spectrum condition analogous to the Bedrosian property. If $$x'(t)=\eta(t)\,x(t),\qquad \eta(t)=\rho(t)+j\,\omega(t),$$29 is band-limited to $$x'(t)=\eta(t)\,x(t),\qquad \eta(t)=\rho(t)+j\,\omega(t),$$30 and $$x'(t)=\eta(t)\,x(t),\qquad \eta(t)=\rho(t)+j\,\omega(t),$$31, then for the orthonormal-carrier modulation

$$x'(t)=\eta(t)\,x(t),\qquad \eta(t)=\rho(t)+j\,\omega(t),$$32

the hypercomplex representation factorizes as

$$x'(t)=\eta(t)\,x(t),\qquad \eta(t)=\rho(t)+j\,\omega(t),$$33

The same framework also preserves the standard modulation-frequency-shift relation,

$$x'(t)=\eta(t)\,x(t),\qquad \eta(t)=\rho(t)+j\,\omega(t),$$34

This line of work does not merely extend notation: it supplies a one-sided spectral construction for amplitude-phase analysis of complex-valued signals themselves, rather than only of real signals embedded in the complex plane (Bihan et al., 2012).

6. Assumptions, limitations, and recurring points of confusion

A recurrent ambiguity is whether CF is simply a renamed instantaneous frequency. The literature does not support that reduction. In both the analytic-signal and power-system formulations, the real part is an explicit radial or magnitude-rate term, and several papers argue that omitting it obscures physically relevant dynamics during transients, unbalances, or fast envelope variation (García-Veloso et al., 10 Dec 2025).

A second point of confusion concerns equivalence between formulations. The analytic-signal and space-vector definitions are formally identical but coincide only under specific hypotheses: balanced, positive-sequence operation; absence of harmonics and interharmonics; Hilbert-pair orthogonality between Clarke components; and slow electromechanical transients such that Bedrosian’s theorem applies. Under those conditions, the space vector and the analytic signal differ only by the reference-frame rotation $$x'(t)=\eta(t)\,x(t),\qquad \eta(t)=\rho(t)+j\,\omega(t),$$35, and in the stationary Clarke frame the two ICFs coincide exactly (García-Veloso et al., 10 Dec 2025).

A third misconception is to treat the eigenvalue interpretation as universal. The geometric-frequency paper proves the equivalence only for diagonalizable LTI systems. For nonlinear systems, eigenvalues of the instantaneous Jacobian do not, in general, match the instantaneous geometric frequency of the state velocity, although the geometric frequency itself remains well defined and can still describe local amplitude and curvature dynamics, including limit cycles or chaos (Sofos et al., 20 Mar 2026).

In the excitation literature, CF methods are also subject to noise-dependent limits. Khurgin et al. state that post-detection synthesized CF response is noise-floor limited because the reconstructed noise grows exponentially as $$x'(t)=\eta(t)\,x(t),\qquad \eta(t)=\rho(t)+j\,\omega(t),$$36, whereas real-time CF retains spectral narrowing when shot noise and relative-intensity noise dominate but offers no benefit when signal-independent noise such as dark current or background dominates. They further state that, in low-noise conditions, a matched filter that multiplies the impulse response by $$x'(t)=\eta(t)\,x(t),\qquad \eta(t)=\rho(t)+j\,\omega(t),$$37 and then Fourier-transforms back is the most efficient post-processing route (Khurgin et al., 2 Jul 2026).

Finally, time-reversal arguments fail in dissipative multi-qubit systems. In the ideal lossless single-emitter case, poles and reflection zeros are complex conjugates, but intrinsic losses break this conjugation. The qubit-control literature therefore replaces naive time-reversed emission profiles with pulses targeted to the actual complex zeros of the full lossy scattering matrix (Trivedi et al., 3 Jun 2025). A plausible implication is that CF should be regarded less as a single domain-specific construct than as a common kinematic and spectral language for systems in which decay or growth and oscillation must be handled simultaneously.