Dynamic-Signal (Complex-Frequency) Identity

• Dynamic-Signal (Complex-Frequency) Identity is an algebraic relation linking a signal’s instantaneous amplitude, phase, and curvature with quadratic energy-like operators.
• It derives a closed-form estimator for instantaneous frequency using local first and second derivatives, offering bias-free analysis without phase unwrapping.
• The identity underpins applications in power grid control, resonator networks, and analytic signal theory, providing a unified framework for signal processing and dynamic analysis.

A dynamic-signal (complex-frequency) identity is an exact, local algebraic relation that connects the kinematic invariants of a complex-valued signal—most notably, its instantaneous frequency, amplitude variations, and curvature effects—with specific quadratic or differential energy-like operators. At its heart, this identity provides a bridge between geometric/differential descriptions of signal evolution and physically or computationally meaningful quantities, encompassing and generalizing classical notions such as the instantaneous (real) frequency, analytic signal theory, and resonance in dynamical systems.

1. Formal Development and Core Definitions

Let $\bar{x}(t)$ denote a twice-differentiable, non-vanishing complex-valued signal. The canonical amplitude–phase decomposition gives

$\bar{x}(t) = |\bar{x}(t)|\,e^{j\phi(t)},$

where $|\bar{x}(t)|$ is the instantaneous envelope and $\phi(t)$ the instantaneous phase. The “complex frequency” (alternatively, local complex rate) is defined as

$$\bar{\eta}(t) = \frac{\bar{x}'(t)}{\bar{x}(t)} = \underbrace{\frac{d}{dt}\ln|\bar{x}(t)|}_{\rho(t)} + j\,\underbrace{\frac{d\phi(t)}{dt}}_{\omega(t)} = \rho(t) + j\omega(t).$$

Here, $\omega(t)$ quantifies the instantaneous angular frequency and $\rho(t)$ the normalized log-envelope growth rate.

A central tool is the complex Teager–Kaiser energy operator (CTKEO), defined as

$$(\bar{x}) \equiv |\bar{x}'(t)|^2 - \Re\{\bar{x}(t)\,\bar{x}''^*(t)\},$$

which is real-valued and invariant under global phase shifts. Substitution of the amplitude–phase decomposition and subsequent algebraic manipulations yield the dynamic-signal identity: $$\boxed{ (\bar{x}) = |\bar{x}|^2\left(2\,\omega^2(t) - \rho'(t)\right) }$$ Solving for $\omega(t)$ gives the closed-form instantaneous-frequency estimator: $\bar{x}(t) = |\bar{x}(t)|\,e^{j\phi(t)},$0 Every constituent term is local; only first and second derivatives of $\bar{x}(t) = |\bar{x}(t)|\,e^{j\phi(t)},$1 and its modulus are required, and phase unwrapping is not needed (Vaca et al., 21 Jan 2026).

2. Geometric and Physical Interpretations

The dynamic-signal identity admits a differential geometric interpretation. For signals tracing trajectories in the complex plane (i.e., 2D real, or “planar phasor” signals), the imaginary part $\bar{x}(t) = |\bar{x}(t)|\,e^{j\phi(t)},$2 corresponds to the instantaneous curvature of $\bar{x}(t) = |\bar{x}(t)|\,e^{j\phi(t)},$3, while $\bar{x}(t) = |\bar{x}(t)|\,e^{j\phi(t)},$4 quantifies envelope expansion/contraction ("radial acceleration"). The identity decomposes local “energy” into:

• Rotational kinetic component (curvature, proportional to $\bar{x}(t) = |\bar{x}(t)|\,e^{j\phi(t)},$5),
• Envelope rate and curvature terms: centrifugal ($\bar{x}(t) = |\bar{x}(t)|\,e^{j\phi(t)},$6) and envelope acceleration ($\bar{x}(t) = |\bar{x}(t)|\,e^{j\phi(t)},$7) corrections,
• In higher phasor-space dimensions, torsion contributions may arise, but for 2D signals (typical in power systems and analytic-signal theory), torsion is zero (Vaca et al., 21 Jan 2026).

This decomposition mirrors the split of geometric frequency into amplitude (radial) and rotational (curvature) components as formalized in the language of bivector calculus: $\bar{x}(t) = |\bar{x}(t)|\,e^{j\phi(t)},$8 where $\bar{x}(t) = |\bar{x}(t)|\,e^{j\phi(t)},$9 are the signal’s instantaneous velocity and acceleration vectors, and $|\bar{x}(t)|$0 denotes the wedge (bivector) product (Sofos et al., 20 Mar 2026).

3. Operator and Algebraic Generalizations

The dynamic-signal identity extends naturally to several operator and algebraic settings:

• Analytic and Hyperanalytic Signal Theory: For signals with complex or quaternionic structure, the dynamic-signal identity governs the evolution of the “analytic” (one-sided spectrum) extension. In the quaternionic analytic signal framework, unique polar representations yield unambiguous definitions of instantaneous complex phase and frequency (Huang et al., 2014, Bihan et al., 2012), allowing identities of the form:

$|\bar{x}(t)|$1

with $|\bar{x}(t)|$2 the instantaneous quaternionic complex frequency (Huang et al., 2014).

• Space-Vector Formalism and Unified Notation: Distinct formulations for instantaneous (complex) frequency—in analytic signal (Hilbert transform) and space-vector (planar phasor) contexts—are proven equivalent under planarity and non-vanishing envelope conditions. Both yield a canonical “dynamic-signal complex-frequency identity”:

$|\bar{x}(t)|$3

(García-Veloso et al., 10 Dec 2025).

• Power System and LTI Dynamics: In power network modeling, the dynamic-signal identity manifests as a relation between rates-of-change of active/reactive power and instantaneous “complex frequency” at each bus:

$|\bar{x}(t)|$4

where $|\bar{x}(t)|$5 and $|\bar{x}(t)|$6 are, respectively, the rate-of-change of log-voltage magnitude (envelope) and the instantaneous frequency deviation. This couples network algebraic variables and kinematic phase/amplitude changes without approximation (Milano, 2021, Domingo-Enrich et al., 2024).

• LTI Systems Spectral Theory: For a diagonalizable LTI system, the geometric frequency of system states under modal transformation coincides exactly with the eigenvalues of the system matrix, providing a concrete link between the algebraic spectrum and dynamic frequency content (Sofos et al., 20 Mar 2026). This correspondence does not hold in general for nonlinear or non-diagonalizable systems.

4. Domain-Specific Applications and Implications

The dynamic-signal (complex-frequency) identity underpins a spectrum of practical and theoretical applications:

• Robust Instantaneous Frequency Estimation: Advanced estimators in power system monitoring exploit the explicit separation of curvature and envelope-curvature corrections to reduce bias and avoid phase unwrapping. These estimators admit increased accuracy in the presence of strong amplitude modulation, extending classical Teager–Kaiser approaches to the complex domain (Vaca et al., 21 Jan 2026).
• Power Grid Synchronization and Control: Complex-frequency identities are foundational in the formulation of modern droop and virtual oscillator-based controllers, capturing the tight coupling between real/reactive power and kinematic voltage/frequency observables. The transition from static gain to dynamic transfer-function implementation in these controllers enables tailored dynamic responses and guarantees small-signal stability under passivity frameworks (Domingo-Enrich et al., 2024).
• Resonator Networks and Wave Scattering: In coupled-resonator systems, every mode is dynamically identified by its complex pole and reflection zero; selective complex-frequency driving can dynamically enforce zero reflection (“dynamic critical coupling”), yielding near-unity energy storage and crosstalk suppression significantly exceeding real-frequency excitation strategies (Trivedi et al., 2 Jun 2025).
• Signal Separation and Identity Encoding: In bioacoustics and pattern recognition, dynamic-signal identities justify and guide the use of time–frequency representations tracking instantaneous harmonic structure, revealing where distinctive identity information is encoded (e.g., in the early fundamental/harmonic content of corvid calls) (Stowell et al., 2016).

5. Assumptions, Limitations, and Structural Generality

The validity of dynamic-signal (complex-frequency) identities depends on several structural assumptions:

• Single Component Analytic Signals: The signal must admit a non-vanishing, single-component amplitude–phase decomposition; overlapping components or amplitude zeros induce failure.
• Planarity and Smoothness: The geometric interpretation, and reduction to a two-dimensional curvature/rotation formalism, assumes planar phasor trajectories with sufficient differentiability. Torsional contributions arise only in higher dimensions (Vaca et al., 21 Jan 2026).
• Absence of Narrow-Band or Bedrosian Restrictions: The identities presented are exact for analytic components, regardless of the relative speeds of envelope and phase variations, although practical differentiation of noisy signals may require smoothing (Vaca et al., 21 Jan 2026, García-Veloso et al., 10 Dec 2025).
• Spectral Uniqueness in Quaternionic and Hypercomplex Extensions: Unique phase/frequency assignment in the quaternionic regime is guaranteed only after resolving sign and continuity ambiguities via phase unwrapping and zero-crossing tracking (Huang et al., 2014, Bihan et al., 2012).

6. Illustrative Examples

Instantaneous Frequency Estimation in a Modulated Phasor

Given $|\bar{x}(t)|$7, the explicit dynamic-signal identity recovers the true IF ($|\bar{x}(t)|$8) to within $|\bar{x}(t)|$9 relative error even in the presence of slow envelope variations, with the bias quantitatively reducing as envelope modulation is weakened (Vaca et al., 21 Jan 2026).

Power System Bus Dynamics

For an AC network under Park transformation,

$\phi(t)$0

directly couples any local power swing to instantaneous changes in both amplitude (voltage magnitude) and phase (frequency deviation) at the bus (Milano, 2021).

Resonator Energy Transfer

Complex-frequency driving at the dynamic reflection zero achieves theoretically perfect energy transfer: $\phi(t)$1 yielding efficiency $\phi(t)$2, tending to unity as $\phi(t)$3; for Gaussian pulses, $\phi(t)$4 even under optimal conditions (Trivedi et al., 2 Jun 2025).

7. Summary Table of Identity Forms and Contexts

Domain	Dynamic-Signal Identity Form	Key Context/Feature
Analytic Signal	$\phi(t)$5	Unified amplitude–phase kinematics (García-Veloso et al., 10 Dec 2025)
CTKEO (Power Sys.)	$\phi(t)$6	Bias-free IF estimation, geometry explicit (Vaca et al., 21 Jan 2026)
Power System Algebra	$\phi(t)$7	Voltage/frequency swings link to power flow (Milano, 2021)
Resonator Scattering	$\phi(t)$8	Dynamic critical coupling, mode selectivity (Trivedi et al., 2 Jun 2025)
Modal LTI Dynamics	Geometric freq. $\phi(t)$9 eigenvalue	Exact for diagonalizable systems (Sofos et al., 20 Mar 2026)
Hyperanalytic Signal	$$\bar{\eta}(t) = \frac{\bar{x}'(t)}{\bar{x}(t)} = \underbrace{\frac{d}{dt}\ln|\bar{x}(t)|}_{\rho(t)} + j\,\underbrace{\frac{d\phi(t)}{dt}}_{\omega(t)} = \rho(t) + j\omega(t).$$0, $$\bar{\eta}(t) = \frac{\bar{x}'(t)}{\bar{x}(t)} = \underbrace{\frac{d}{dt}\ln|\bar{x}(t)|}_{\rho(t)} + j\,\underbrace{\frac{d\phi(t)}{dt}}_{\omega(t)} = \rho(t) + j\omega(t).$$1	Unique polar/quaternion freq. (Huang et al., 2014)

In all contexts, the dynamic-signal (complex-frequency) identity provides a rigorous, unifying algebraic link between signal geometry, spectral evolution, and physically measurable quantities, enabling precise characterization, control, and interpretation of time-varying processes across disciplines.