Bias-Corrected Instantaneous Frequency Estimation

Updated 11 June 2026

The paper introduces CTKEO and innovative FIR/IIR and kernel smoothing approaches that significantly reduce estimation bias under strong amplitude modulation.
It demonstrates that incorporating envelope-curvature terms and higher-order phase derivatives efficiently corrects bias, achieving performance near the Cramér-Rao bound.
Empirical evaluations in power systems, radar, and biomedical signals confirm robust IF tracking and minimal error in transient, rapidly changing conditions.

Bias-corrected instantaneous frequency estimation encompasses a class of methodologies designed to accurately extract the instantaneous frequency (IF) of time-varying signals while systematically removing or mitigating the bias intrinsic to conventional IF estimators. This is essential in scenarios where amplitude modulation, rapid envelope changes, or higher-order phase effects induce significant systematic error in standard IF measurement pipelines, particularly in power systems, communication, radar, and biomedical signal analysis. A growing body of techniques—spanning local nonlinear energy operators, optimized filter design, high-order time-frequency analysis, and kernel smoothing—forms the basis for state-of-the-art bias-corrected IF estimation.

1. Theoretical Foundations and Bias Mechanisms

Bias in instantaneous frequency estimation arises primarily from simplified signal models and the local breakdown of their assumptions. Standard approaches, such as the energy separation algorithm (ESA) paired with the Teager–Kaiser energy operator (TKEO), assume a narrowband analytic signal with a slowly varying envelope. For a real signal of the form $x(t) = a(t) \cos(\phi(t))$ , the real TKEO $\Psi[x] = (x')^2 - x x''$ yields IF approximations under a slow-envelope hypothesis. When directly extended to complex phasors or more general AM–FM components, neglecting envelope-curvature and higher-order phase terms introduces systematic bias.

For a general analytic phasor $\bar{x}(t)=|\bar{x}|e^{j\phi}$ with possibly rapid amplitude and phase dynamics, the bias in classical IF estimators is governed by omitted terms such as $\rho' = (|\bar{x}|''/|\bar{x}|) - (\frac{|\bar{x}|'}{|\bar{x}|})^2$ (envelope curvature) and unmodeled higher-order phase derivatives. These yield a persistent error floor, particularly during transients and under strong amplitude modulation (Vaca et al., 21 Jan 2026).

2. Bias Correction via the Complex Teager-Kaiser Energy Operator

A rigorous bias correction in power-system applications is achieved by introducing the complex TKEO (CTKEO). CTKEO is defined as

$\Psi[\bar{x}] = |\bar{x}'|^2 - \mathrm{Re}\{\bar{x} \bar{x}''^*\}$

and, when combined with the dynamic-signal identity $\bar{x}' = (\rho + j\omega)\bar{x}$ , leads to the following exact decomposition:

$2\omega^2 = \frac{\Psi[\bar{x}]}{|\bar{x}|^2} + \rho' = \frac{\Psi[\bar{x}]}{|\bar{x}|^2} - \left(\frac{|\bar{x}|'}{|\bar{x}|}\right)^2 + \frac{|\bar{x}|''}{|\bar{x}|}$

This yields the time-varying (TV) bias-corrected IF estimator:

$\omega_\mathrm{TV} = \sqrt{\tfrac12\left[\frac{\Psi[\bar{x}]}{|\bar{x}|^2} - \left(\frac{|\bar{x}|'}{|\bar{x}|}\right)^2 + \frac{|\bar{x}|''}{|\bar{x}|}\right]}$

The explicit envelope-curvature terms ( $|\bar{x}|', |\bar{x}|''$ ) remove bias due to amplitude dynamics, guaranteeing exactness for pure analytic phasors undergoing rapid magnitude changes. Notably, CTKEO is local in time, does not require phase unwrapping, and remains robust under unbalanced conditions (Vaca et al., 21 Jan 2026).

3. Bias Correction through Filter Design: FIR/IIR and Regression-based Approaches

Discrete-time bias correction often exploits optimal fixed-lag filter design in both FIR and IIR domains. For a complex exponential $x[n] = A e^{i\theta[n]}$ in noise, the IF is the first difference of the phase. FIR and IIR smoothers are constructed to enforce exact tracking (zero steady-state bias) for signals with polynomial phase progression up to a specified degree $\Psi[x] = (x')^2 - x x''$ 0. This is achieved by constraining the filter moments:

$\Psi[x] = (x')^2 - x x''$ 1

where $\Psi[x] = (x')^2 - x x''$ 2 is the fixed lag and $\Psi[x] = (x')^2 - x x''$ 3 depends on the number of differentiators and polynomial order. By optimizing filter coefficients to minimize the noise gain subject to these constraints, the estimator remains unbiased for all polynomial-phase signals of degree up to $\Psi[x] = (x')^2 - x x''$ 4—removing the bias seen in "naive" smoothers (Kennedy, 2023, Kennedy, 2023).

The classical Kay weights, Savitzky-Golay regression filters, and their recursive (IIR) analogues are prominent realizations. Recursive designs achieve the FIR lower bound on variance and bias with reduced computational cost (Kennedy, 2023).

4. Kernel and High-order Time-Frequency Methods for Bias Correction

Kernel smoothing approaches employ derivative estimators with adaptive bandwidth to balance bias (from truncating Taylor expansions) and variance (from noise). For instantaneous frequency, the "plug-in" strategy involves:

Pilot estimation of higher-order derivatives (e.g., third derivative via a high-order kernel),
Adaptive selection of the kernel halfwidth $\Psi[x] = (x')^2 - x x''$ 5 to minimize MSE,
Final estimation of IF via

$\Psi[x] = (x')^2 - x x''$ 6

where $\Psi[x] = (x')^2 - x x''$ 7 and $\Psi[x] = (x')^2 - x x''$ 8 are kernel estimates of $\Psi[x] = (x')^2 - x x''$ 9 and its time derivative, respectively.

The systematic bias is dominated by the remainder of the Taylor expansion, yielding pilot-based "bias correction". Optimal scaling laws for kernel halfwidths follow $\bar{x}(t)=|\bar{x}|e^{j\phi}$ 0 (Riedel, 2018).

High-order synchrosqueezed transforms relax the local-linear-chirp assumption by considering Nth-order polynomial phase models locally. N-point identities are established for the moments of the wavelet–chirplet transform, yielding an Nth-order IF reassignment operator. The bias is bounded by $\bar{x}(t)=|\bar{x}|e^{j\phi}$ 1, with $\bar{x}(t)=|\bar{x}|e^{j\phi}$ 2, hence arbitrarily small bias for sufficiently large $\bar{x}(t)=|\bar{x}|e^{j\phi}$ 3 and small scale $\bar{x}(t)=|\bar{x}|e^{j\phi}$ 4 (Li et al., 1 Jun 2026).

5. Endpoint and Real-time Hilbert-based Bias Correction

For real-time, low-latency IF estimation, endpoint-corrected Hilbert transforms (ecHT) improve upon the standard analytic-signal approach by applying causal, narrow-band filtering to the analytic spectrum at each window endpoint. The endpoint output is:

$\bar{x}(t)=|\bar{x}|e^{j\phi}$ 5

This operation introduces a deterministic complex gain $\bar{x}(t)=|\bar{x}|e^{j\phi}$ 6, producing a static amplitude/phase bias. Scalar complex calibration $\bar{x}(t)=|\bar{x}|e^{j\phi}$ 7 corrects the mean bias, as shown by explicit analytic decomposition. Remaining residual error (“leakage”) is characterized and strictly bounded. The calibrated estimate achieves near-zero mean phase and frequency bias, with the residual variance set by leakage and the window SNR (Osmers et al., 20 Jan 2026).

6. Empirical Performance and Practical Implications

Comprehensive simulations and field data provide quantitative evidence for the efficacy of bias-corrected IF estimators. In power-system transient conditions—such as three-phase faults, voltage sags, and inverter ride-through events—bias-corrected CTKEO estimators maintain root-mean-square frequency error within 10 mHz and track geometric (curvature-defined) IF without spurious oscillations, especially when envelope dynamics are rapid (Vaca et al., 21 Jan 2026). FIR/IIR regression-based smoothers achieve the Cramér-Rao lower bound at moderate filter orders, outperforming naive or uniform averaging, and remain unbiased over polynomial-phase trajectories (Kennedy, 2023, Kennedy, 2023). High-order synchrosqueezed methods enable separation and retrieval of crossing IF curves in multicomponent signals with minimal systematic error, substantially reducing mode-retrieval ambiguity in the presence of strong frequency modulation (Li et al., 1 Jun 2026).

A table summarizing distinct families of bias-corrected IF estimators and their core bias-correction mechanisms:

Method/Class	Key Bias Correction Principle	Typical Problems Addressed
CTKEO-based	Envelope-curvature terms from kinematics	Power system, AM/FM signals
FIR/IIR regression	Filter moment constraints	Doppler, radar, FM demod.
Kernel estimator	Adaptive halfwidth, pilot smoothing	Biomedical, slowly-varying
High-order SST/HSWCT	Nth-order phase Taylor expansion	Multicomponent, chirps
Calibrated ecHT	Endpoint deterministic gain calibration	Real-time, windowed signals

7. Limitations, Assumptions, and Directions for Further Research

Bias-corrected IF estimation techniques are predicated on several assumptions:

The analytic signal is adequately band-limited and single-component; strong multi-tone or broadband distortion may require pre-processing or mode separation (e.g., synchrosqueezed transforms or empirical mode decomposition).
Accurate numerical differentiation (especially second derivatives) is necessary for curvature-based methods; this process is noise-sensitive and typically demands pre-filtering or smoothing.
Hilbert-based and time-frequency techniques depend on optimal parameter selection (e.g., window length, kernel order), and nonstationarity can degrade performance if not accounted for.
Many algorithms assume planar motion (zero torsion) in complex phasor spaces. Extending these to three-dimensional phase trajectories, incorporating explicit torsion, or accommodating nonanalytic signals are open research topics (Vaca et al., 21 Jan 2026, Li et al., 1 Jun 2026).

Future research directions include real-time hardware implementations (e.g., FPGA acceleration for CTKEO filters), integration with adaptive inverter controls for power electronics, and theoretical extension of bias-corrected estimators to high-noise, multimodal, or heavily corrupted measurement scenarios. Generalizing endpoint bias-correction and high-order reassignment operators to nonstationary, streaming, and missing-data regimes remains a priority for practical deployment.

In summary, bias-corrected instantaneous frequency estimation leverages envelope curvature, higher-order local signal models, optimal filter moment constraints, and endpoint calibration to systematically remove or sharply bound bias contributions that limit classical estimators, delivering robust, local, and physically interpretable IF estimates across a wide range of application domains (Vaca et al., 21 Jan 2026, Kennedy, 2023, Riedel, 2018, Kennedy, 2023, Osmers et al., 20 Jan 2026, Li et al., 1 Jun 2026).