Subchannel Energy-Domain Matched Filtering

• Subchannel Energy-Domain Matched Filtering is a method using the Complex Teager–Kaiser Energy Operator (CTKEO) to capture joint amplitude and frequency dynamics.
• It leverages a quadratic, real-valued operator formulation to provide instantaneous frequency estimation and robust matched filtering in multipath environments.
• Applications in power systems and communications enable precise synchronization energy measurement and effective SNR-based signal decomposition.

The Complex Teager–Kaiser Energy Operator (CTKEO) generalizes the classical Teager–Kaiser energy operator to complex analytic signals, providing a quadratic, real-valued, instantaneous “energy” measure that jointly encodes amplitude and frequency dynamics. CTKEO retains the structure and key properties of the real-valued operator while extending its applicability to systems and signals where phase and frequency modulation are prominent, including power systems, time–frequency analysis, communications, and signal separation. Its recent adoption as a metric for synchronization, instantaneous frequency estimation, and matched filtering demonstrates its versatility and significance in both theoretical and practical contexts (Pinheiro et al., 9 Mar 2025, Montillet, 2016, Vaca et al., 21 Jan 2026).

1. Formal Definition and Mathematical Structure

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

$$\Psi_c\{x(t)\} = |x'(t)|^2 - \Re \left\{ x(t)\, x''(t)^* \right\}$$

where $(\cdot)^*$ denotes complex conjugation and $\Re\{\cdot\}$ denotes the real part. This generalizes the classical (real) TKEO, $\Psi\{f\} = (f')^2 - f f''$, and reduces to it when $x$ is real-valued (Montillet, 2016, Vaca et al., 21 Jan 2026).

In alternative form and applications, particularly for complex instantaneous power in power systems, a symmetrized variant appears: $$\psi_c(\bar x) = \dot{\bar x}^*\, \dot{\bar x} - \frac12 ( \ddot{\bar x}\, \bar x^* + \bar x\, \ddot{\bar x}^* )$$ This operator is real-valued and conjugate-symmetric, capturing the joint effect of magnitude and phase/frequency modulations.

2. Theoretical Properties and Functional Space

CTKEO is a quadratic form on a suitable function space. For well-posedness in analysis and communications, signals are often constrained to subspaces such as $\mathcal{S}(\mathbb{R})$, a Schwartz-type space of functions closed under differentiation and possessing finite energy integrals (Montillet, 2016). Properties include:

• Bilinearity and Conjugate Symmetry: CTKEO is quadratic in its argument and satisfies $$\overline{\Psi_{\text{cTK}}[x]} = \Psi_{\text{cTK}}[x]$$.
• Derivative Chain Rule: For higher-order extensions $\Psi_k^{\pm}(\cdot)$, a chain rule expresses the derivative as a sum of higher and lower order energy-operator terms.
• Operator Decomposition: Derivatives of analytic signals can be decomposed into finite linear combinations of CTKEOs acting on derivatives, yielding closed forms for pulse expansions and energy integrations.

These properties facilitate signal decomposition in multipath channels and stable numerical implementations.

3. Instantaneous Frequency and Dynamic Signal Identity

CTKEO provides a means for direct instantaneous frequency (IF) estimation. For $$\Psi_c\{x(t)\} = |x'(t)|^2 - \Re \left\{ x(t)\, x''(t)^* \right\}$$0, with $$\Psi_c\{x(t)\} = |x'(t)|^2 - \Re \left\{ x(t)\, x''(t)^* \right\}$$1, the dynamic-signal identity states: $$\Psi_c\{x(t)\} = |x'(t)|^2 - \Re \left\{ x(t)\, x''(t)^* \right\}$$2 where $$\Psi_c\{x(t)\} = |x'(t)|^2 - \Re \left\{ x(t)\, x''(t)^* \right\}$$3 is the angular frequency, $$\Psi_c\{x(t)\} = |x'(t)|^2 - \Re \left\{ x(t)\, x''(t)^* \right\}$$4 is the logarithmic-magnitude rate, and $$\Psi_c\{x(t)\} = |x'(t)|^2 - \Re \left\{ x(t)\, x''(t)^* \right\}$$5 is its derivative.

Solving for the IF yields the unbiased estimator: $$\Psi_c\{x(t)\} = |x'(t)|^2 - \Re \left\{ x(t)\, x''(t)^* \right\}$$6 The term $$\Psi_c\{x(t)\} = |x'(t)|^2 - \Re \left\{ x(t)\, x''(t)^* \right\}$$7 acts as an envelope-curvature correction, compensating for amplitude dynamics ignored by narrowband TKEO estimators (Vaca et al., 21 Jan 2026). For constant-magnitude signals, this reduces to the classical TKEO-based IF, $$\Psi_c\{x(t)\} = |x'(t)|^2 - \Re \left\{ x(t)\, x''(t)^* \right\}$$8.

4. Synchronization Energy and Power Systems Applications

CTKEO underpins the definition of "Synchronization Energy" (SE) in power system device analysis (Pinheiro et al., 9 Mar 2025). For the instantaneous complex power $$\Psi_c\{x(t)\} = |x'(t)|^2 - \Re \left\{ x(t)\, x''(t)^* \right\}$$9, comprising Park-vector voltages ($(\cdot)^*$0) and currents ($(\cdot)^*$1), the SE is

$(\cdot)^*$2

Here, $(\cdot)^*$3 and $(\cdot)^*$4 are the instantaneous frequencies of voltage and current; $(\cdot)^*$5 are local time–frequency variances associated with amplitude nonstationarity.

SE quantifies the energy required for a device to maintain or restore synchronism:

• $(\cdot)^*$6 reflects isofrequential deviation,
• Nonzero $(\cdot)^*$7 indicate nonstationary amplitude.

Local synchronization is characterized by $(\cdot)^*$8, specifically when $(\cdot)^*$9 and amplitude derivatives vanish. Case studies in SMIB, Kundur Two-Area, IEEE 14-bus, and IBR scenarios confirm the power of CTKEO-based SE as a metric capturing both frequency slip and amplitude dynamics (Pinheiro et al., 9 Mar 2025).

5. Matched Filtering and Communications

CTKEO has been incorporated into matched filter design for detection in multipath fading channels (Montillet, 2016). By working within $\Re\{\cdot\}$0, finite-impulse-response representations allow the received signal to be decomposed into subchannels, each associated with path-specific delays and derivatives. Each path’s instantaneous energy is given by integrating CTKEO over the corresponding interval: $\Re\{\cdot\}$1 This decomposition supports signal-to-noise ratio (SNR) assessments per subchannel and enables robust detection in the presence of amplitude and frequency distortion. The CTKEO framework is robust to amplitude–phase coupling, and it supports further extension to higher-order and multidimensional signal processing.

6. Numerical Implementation and Illustrative Use Cases

For sampled signals $\Re\{\cdot\}$2, first and second derivatives are estimated via (possibly smoothed) finite differences. CTKEO is evaluated as

$\Re\{\cdot\}$3

Envelope and derivatives are computed, and the IF is estimated following the CTKEO dynamic-signal identity. Stability requires avoidance of amplitude zeros and possibly low-pass filtering of frequency estimates.

Numerous examples have established:

• Exactness for pure sinusoids, with both classical and corrected estimators converging.
• Bias correction by the CTKEO-based IF estimator in amplitude-modulated or unbalanced signals.
• Alignment with geometric “curvature” frequency in phase-plane representations, crucial in both simulated and field-measured transient events (Vaca et al., 21 Jan 2026).

Performance as a synchronization or energy metric is demonstrated for a range of scenarios, including transient faults, oscillatory instability, and inverter control sensitivity (Pinheiro et al., 9 Mar 2025).

7. Advantages, Limitations, and Extension Prospects

Advantages of CTKEO include:

• Real-valued, robust capture of amplitude and phase/frequency information for complex analytic signals.
• Direct applicability in local IF estimation, synchronization metrics, and SNR-based signal decomposition.
• Operator structure that supports closed-form decompositions and stability in Schwartz-type functional spaces.

Limitations are imposed by the requirement for high differentiability and decay (belonging to $\Re\{\cdot\}$4), limited addressal of Doppler/time-varying channels, and the need for regularization in dense or low-delay-separation multipath scenarios (Montillet, 2016).

Potential extensions comprise:

• Adaptive or time-varying order implementation for fast channel variations.
• Sparse-reconstruction-aligned signal decomposition for improved path separation.
• Higher-order and multi-antenna generalizations for broader signal domains and applications.

This suggests CTKEO provides a unifying, compact mathematical tool for quantifying instantaneous energy, frequency, and synchronization in diverse contexts, with significant demonstrated performance and scope for further research-driven generalizations (Pinheiro et al., 9 Mar 2025, Montillet, 2016, Vaca et al., 21 Jan 2026).