Complex Teager–Kaiser Energy Operator (CTKEO)

• CTKEO is a complex energy operator that quantifies instantaneous signal energy by capturing both amplitude and frequency modulations in analytic signals.
• It leverages differential methods, including central differencing and the derivative chain rule, to yield unbiased instantaneous frequency estimates and robust channel decomposition.
• CTKEO is applied in power systems, multipath detection, and signal processing to isolate synchronization energy and enhance dynamic signal analysis.

The Complex Teager–Kaiser Energy Operator (CTKEO) generalizes the classical Teager–Kaiser energy operator to complex-valued, analytic signals. It provides a robust, instantaneous measure of signal energy, capable of capturing both amplitude and frequency modulations in real-time. In advanced applications across power system dynamics, signal processing, and multipath detection, CTKEO serves as a cornerstone tool for extracting instantaneous features such as synchronization energy and unbiased instantaneous frequency, as well as for decomposing signal energy in complex channels.

1. Formal Definitions and Operator Structure

Let $x(t)$ be a complex-valued, twice-differentiable signal. The continuous-time CTKEO is defined as

$$\Psi_{\mathrm{cTK}}[x](t) = |x'(t)|^2 - \Re\{x(t) x''(t)^*\},$$

where $(\cdot)^*$ denotes complex conjugation and $\Re\{\cdot\}$ the real part (Vaca et al., 21 Jan 2026). In an alternative, algebraically equivalent formulation, the operator can be written as

$$\psi_c(\bar x) = \dot{\bar x}^*\,\dot{\bar x} - \tfrac12[\ddot{\bar x}\,\bar x^* + \bar x\,\ddot{\bar x}^*].$$

When restricted to real-valued signals, CTKEO reduces to the classical Teager–Kaiser operator: $$\Psi_{\mathrm{TK}}[f](t) = (f'(t))^2 - f(t) f''(t).$$ The discrete-time version utilizes central differences and maintains bilinearity and conjugate symmetry (Montillet, 2016).

2. Mathematical Properties and Functional Spaces

CTKEO is a real-valued quadratic form and exhibits conjugate symmetry. Signals are typically required to lie in a subspace $\mathcal{S}(\mathbb{R})$ of Schwartz-class functions, ensuring all derivatives exist, the energy integrals converge, and Taylor expansions are valid within intervals of interest (Montillet, 2016). This finiteness property enables stable numerical implementation and allows higher-order derivatives to be expressed as linear combinations of lower-order energy operators. Key operator identities include a derivative chain rule and the existence of a finite, linearly independent derivative basis for analytic signals, essential for robust channel decomposition tasks.

3. Dynamics and Instantaneous Frequency Analysis

By expressing a complex signal as $x(t) = |x(t)| e^{j\phi(t)}$ with logarithmic-magnitude rate $\rho(t) = \frac{d}{dt}\log|x(t)|$ and instantaneous angular frequency $\omega(t) = \phi'(t)$, one obtains the “dynamic-signal identity”: $$\Psi_{\mathrm{cTK}}[x](t) = |x'(t)|^2 - \Re\{x(t) x''(t)^*\},$$0 where $$\Psi_{\mathrm{cTK}}[x](t) = |x'(t)|^2 - \Re\{x(t) x''(t)^*\},$$1 denotes the time derivative of $$\Psi_{\mathrm{cTK}}[x](t) = |x'(t)|^2 - \Re\{x(t) x''(t)^*\},$$2 (Vaca et al., 21 Jan 2026). In the narrow-band case ($$\Psi_{\mathrm{cTK}}[x](t) = |x'(t)|^2 - \Re\{x(t) x''(t)^*\},$$3), CTKEO simplifies to $$\Psi_{\mathrm{cTK}}[x](t) = |x'(t)|^2 - \Re\{x(t) x''(t)^*\},$$4. Retaining the envelope-curvature correction $$\Psi_{\mathrm{cTK}}[x](t) = |x'(t)|^2 - \Re\{x(t) x''(t)^*\},$$5 removes the bias arising during rapid amplitude changes, a critical advance over classical TKEO estimators. The unbiased instantaneous frequency estimator is thus: $$\Psi_{\mathrm{cTK}}[x](t) = |x'(t)|^2 - \Re\{x(t) x''(t)^*\},$$6 offering improved accuracy in the presence of amplitude transients and unbalanced conditions (Vaca et al., 21 Jan 2026).

4. Applications in Power Systems and Synchronization

CTKEO finds significant application in quantifying local synchronization energy (SE) of power system devices (Pinheiro et al., 9 Mar 2025). For a power-system bus, the instantaneous complex power injection is

$$\Psi_{\mathrm{cTK}}[x](t) = |x'(t)|^2 - \Re\{x(t) x''(t)^*\},$$7

where $$\Psi_{\mathrm{cTK}}[x](t) = |x'(t)|^2 - \Re\{x(t) x''(t)^*\},$$8 and $$\Psi_{\mathrm{cTK}}[x](t) = |x'(t)|^2 - \Re\{x(t) x''(t)^*\},$$9 are Park-vector representations. Applying CTKEO gives

$(\cdot)^*$0

with local variances $(\cdot)^*$1 defined in terms of voltage and current derivatives. The “Synchronization Energy” $(\cdot)^*$2 encapsulates both isofrequential mismatches ($(\cdot)^*$3) and local nonstationarity ($(\cdot)^*$4). Local synchronization is characterized by $(\cdot)^*$5, i.e., when the voltage and current frequencies align and all magnitude rates vanish (Pinheiro et al., 9 Mar 2025).

5. Signal Detection, Matched Filters, and Channel Decomposition

In communication theory and multipath channel analysis, CTKEO underpins energy-operator-based signal decomposition (Montillet, 2016). By expressing the received signal as a finite sum of delayed and differentiated basis functions, each corresponding to a multipath “finger,” the CTKEO isolates instantaneous energy contributions robust to amplitude and phase distortion. The operator’s bilinearity and derivative chain rule underpin closed-form SNR decompositions in matched-filter architectures. Integration of the CTKEO over time intervals associated with multipath delays enables discrimination of individual propagation paths in fading channels on the basis of instantaneous energy, with each path’s SNR extractable from the decomposition of CTKEO-applied derivative terms.

6. Numerical Implementation and Performance Characteristics

Efficient computation of CTKEO in discrete time proceeds by central differencing for first and second derivatives, with envelope and curvature corrections carefully evaluated to prevent numerical artifacts (Vaca et al., 21 Jan 2026). Robustness is ensured by enforcing high regularity in the analytic signals ($(\cdot)^*$6), smoothing of numerical derivatives, and constraint of envelope magnitude away from zero. In time-domain power system signals, CTKEO-based synchronization metrics closely match numerical estimations even under strong transients, amplitude modulations, or oscillatory regimes. In instantaneous frequency estimation, the envelope-curvature correction inherent in CTKEO eliminates systematic biases observed in classical approaches, especially during dynamic events, as validated in both simulation and field data from power grids.

7. Advantages, Limitations, and Prospective Directions

CTKEO unifies amplitude and frequency sensitivity, providing a succinct metric for both synchronization and instantaneous frequency without requiring phase unwrapping or cycle-averaged integrals (Pinheiro et al., 9 Mar 2025, Vaca et al., 21 Jan 2026). Its advantages include real-time interpretability, analytic tractability via bilinear and chain rule properties, and applicability across physical and communication domains. Limitations are anchored in the requirement for high regularity of input signals, and the need for sufficient separation among channel features in practical decompositions (Montillet, 2016). Potential extensions involve adaptation to time-varying channels, integration with sparse-reconstruction frameworks, and generalization to higher-order or multidimensional (MIMO) energy-operator families.

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Citations:

• (Pinheiro et al., 9 Mar 2025) Teager Energy Operator as a Metric to Evaluate Local Synchronization of Power System Devices
• (Vaca et al., 21 Jan 2026) Instantaneous Frequency in Power Systems using the Teager-Kaiser Energy Operator
• (Montillet, 2016) On The Use of Conjugate Teager-Kaiser Energy Operators with Matched Filters