Complex Teager-Kaiser Energy Operator (CTKEO)

• CTKEO is a complex energy operator that uses local derivatives to compute instantaneous frequency and amplitude, offering a clear definition and robust analysis.
• It possesses bilinearity, symmetry, and chain-rule properties, enabling reliable performance in non-narrowband, rapidly-varying, and multicomponent environments.
• CTKEO has practical applications in power system phasor analysis, synchronization metrics, and matched filtering in communications through bias-corrected estimators.

The Complex Teager–Kaiser Energy Operator (CTKEO) generalizes the classic Teager–Kaiser Energy Operator (TKEO) from real to complex-valued analytic signals, providing a strictly local, derivative-based measure of instantaneous frequency and amplitude dynamics. CTKEO admits rigorous definitions in both continuous and discrete time and possesses key symmetry, invariance, and bilinearity properties, making it robust in non-narrowband, rapidly time-varying, or multicomponent settings. Its applications span power system phasor analysis, local synchronization metrics, and subchannel energy-domain matched filtering in communication systems, offering bias-corrected, bias-revealing, and geometrically interpretable formulations. CTKEO enables direct connection between local phasor kinematics and observable signal properties such as curvature, torsion, and expansion/contraction rates.

1. Mathematical Formulation and Properties

Let $x(t)$ be a real, twice differentiable signal. The original (real) TKEO is given by

$\Psi\{x\} = (x')^2 - x\,x''.$

For complex analytic signals $\bar x(t)\in\mathbb{C}$, the CTKEO is defined as the (real, phase-invariant) symmetric bilinear form

$$\Psi\{\bar x,\bar y\} = \tfrac12\bigl(\bar x'{}^*\,\bar y' + \bar x'\,\bar y'{}^*\bigr) - \tfrac14\bigl(\bar x\,\bar y''{}^* + \bar x^*\,\bar y'' + \bar y\,\bar x''{}^* + \bar y^*\,\bar x''\bigr),$$

with the quadratic specialization

$$\Psi\{\bar x\} \equiv \Psi\{\bar x,\bar x\} = |\bar x'|^2 - \Re\{\bar x\,\bar x''{}^*\}.$$

In discrete time, one representative extension is

$$\Psi_c\{x[n]\} = x[n]\,x^*[n] - \tfrac12\left(x[n-1]\,x^*[n+1] + x[n+1]\,x^*[n-1]\right).$$

The operator possesses crucial Hermitian–quadratic structure, bilinearity, and chain-rule/differentiation closure properties; for example, for analytic signals in the Schwartz subclass $\mathcal{S}(\mathbb{R})$, the operator is well-defined, and all formal series in derivatives terminate by rapid decay. These properties enable decompositions of derivative chains and powers, facilitating energy-domain analysis even in multipath or nonlinear superpositions (Montillet, 2016).

2. CTKEO and Dynamic-Signal (Complex-Frequency) Identity

Given a complex analytic phasor $\bar x(t) = |\bar x(t)|e^{j\phi(t)}$, its derivative encodes the local complex frequency: $$\frac{\bar x'}{\bar x} = \frac{|\bar x|' }{|\bar x|} + j\,\phi' \equiv \rho + j\,\omega,$$ where $\rho$ is the instantaneous logarithmic-magnitude (radial) rate and $\Psi\{x\} = (x')^2 - x\,x''.$0 is instantaneous (angular) frequency.

From this, one finds $\Psi\{x\} = (x')^2 - x\,x''.$1, $\Psi\{x\} = (x')^2 - x\,x''.$2, and the CTKEO identity collapses, after expansion, to

$\Psi\{x\} = (x')^2 - x\,x''.$3

This exposes the operator as a strictly local, physically interpretable function of phasor velocity and acceleration, directly encoding rotation (curvature), expansion/contraction (envelope), and their rates of change (Vaca et al., 21 Jan 2026).

3. Bias-Corrected Instantaneous Frequency Estimation

An exact, bias-corrected estimator of instantaneous frequency (IF) based on CTKEO is obtained by algebraic manipulation: $\Psi\{x\} = (x')^2 - x\,x''.$4 where the terms correspond to raw local TKEO energy, a first-order envelope-curvature correction, and a second-order envelope-curvature correction.

For constant–modulus cases or slowly varying envelopes, the bias corrections vanish, yielding the classical TKEO approximation: $\Psi\{x\} = (x')^2 - x\,x''.$5 which otherwise exhibits bias during pronounced amplitude variation or multi-component mixing.

Validation in power system transients (IEEE 39-bus faults) and real-world PV plant voltage data demonstrates that $\Psi\{x\} = (x')^2 - x\,x''.$6 robustly tracks the true geometric IF even under rapid envelope changes, unbalance, noise, or harmonics, outperforming both the classical TKEO and real-signal energy separation algorithms (Vaca et al., 21 Jan 2026).

4. Geometric and Kinematic Interpretation

When a complex phasor is regarded as a planar trajectory (e.g., $\Psi\{x\} = (x')^2 - x\,x''.$7 for three-phase voltage), the CTKEO connects directly to geometric invariants:

• $\Psi\{x\} = (x')^2 - x\,x''.$8 is the local curvature (rotation per unit time),
• $\Psi\{x\} = (x')^2 - x\,x''.$9 is the expansion/contraction rate,
• $\bar x(t)\in\mathbb{C}$0 is the envelope curvature rate.

The derivative structure ensures no phase unwrapping or nonlocal integration is required: all quantities depend only on local time derivatives.

In such trajectory space, torsion (out-of-plane twisting) vanishes under normal operating conditions, and thus the CTKEO framework remains both interpretable and computationally tractable for non-narrowband, multi-component, or strongly nonstationary signals (Vaca et al., 21 Jan 2026, Pinheiro et al., 9 Mar 2025).

5. Synchronization Energy and Power Systems Applications

CTKEO underpins the Synchronization Energy (SE) metric, designed to quantify the instantaneous effort (in power oscillation energy) that a device must exert for local complex power synchronism: $\bar x(t)\in\mathbb{C}$1 where

• $\bar x(t)\in\mathbb{C}$2 is the device's complex power,
• $\bar x(t)\in\mathbb{C}$3 are its voltage and current angular frequencies,
• $\bar x(t)\in\mathbb{C}$4 are amplitude-variation (stationarity) terms.

Convergence of $\bar x(t)\in\mathbb{C}$5 characterizes local synchronization—isochronism and envelope stationarity—distinct from global system stability. Applications to SMIB, Kundur two-area, IEEE 14-bus, and grid-following IBR systems show sensitivity of SE to inertia, damping, grid impedance, and control gains, with SE ranking “dynamic strength” among devices. Examples show that global instability may arise even when local SE remains bounded (Pinheiro et al., 9 Mar 2025).

6. Signal Processing and Detection via CTKEO

In communication systems, CTKEO has been incorporated into matched-filter detection for signals subject to multipath fading. Given a multipath model

$\bar x(t)\in\mathbb{C}$6

the CTKEO enables decomposition into “subchannels," each with an instantaneous signal-to-noise ratio (SNR).

By writing the received signal in a basis of derivatives/powers and applying CTKEO, one decorrelates the multipath components in the energy domain. A solution analogous to a rake receiver is then implemented using linear combinations of delayed CTKEO samples, maximizing SNR on each subchannel and yielding improved peak-to-sidelobe ratios and aggregate detection performance (Montillet, 2016).

7. Practical Implementation and Summary of Robustness

Implementation of CTKEO in both continuous and discrete time is accomplished via local finite-difference (or finite-difference analog) operators, requiring only immediate neighbor samples. The operator is numerically robust owing to its smoothness and regularity properties (when restricted to appropriate function classes such as $\bar x(t)\in\mathbb{C}$7). Its scalar, local, and geometrically invariant form is advantageous for real-time online estimation, avoids phase unwrapping, and is resilient to non-narrowband, nonstationary, and multi-component conditions.

Extensive simulation and field data demonstrate the following:

• CTKEO removes bias due to envelope curvature unaccounted for in real-valued or classical TKEO approaches.
• It provides a unified, local metric that captures both frequency-matching and amplitude-stationarity for synchronization and signal analysis.
• Applications in power/energy systems, communications, and multi-component signal analysis confirm its technical robustness and interpretability (Vaca et al., 21 Jan 2026, Pinheiro et al., 9 Mar 2025, Montillet, 2016).