Recursive Dominant Frequency Correction

• Recursive Dominant Frequency Correction is a family of algorithms that iteratively refine the primary frequency in signals, effectively countering noise, nonlinearity, and convolutional distortions.
• Techniques such as multi-layered cepstral recursion and recursive IIR/FIR filtering deliver measurable performance gains, with empirical improvements like F-score increases in pitch tracking.
• Practical implementations require careful tuning of parameters—including recursion depths, filter poles, and nonlinearity exponents—to balance computational efficiency against bias correction and artifact suppression.

Recursive dominant frequency correction encompasses a family of algorithms, estimation frameworks, and signal transformations designed to iteratively recover, track, or correct the primary (dominant) frequency components in a signal, especially under conditions of noise, nonlinearity, convolutional filtering, or temporal variability. Contemporary methods span recursive statistical filtering, multi-layered cepstral analysis, and perturbative corrections for nonlinear oscillators. These methods share a core goal: to enhance the saliency and accuracy of instantaneous or fundamental frequency estimates through stepwise, feedback-driven or multilayer recursion, robust both to additive and convolutional distortions and to parameter drift or higher-order interactions.

1. Recursive Formulations Across Signal Domains

Recursive dominant frequency correction arises in several key contexts:

• Digital IIR/FIR smoothing of phase differences: Used for instantaneous frequency estimation in analytic signals, recursive filters (e.g., CLI/Erlang, Kalman) provide latency- and complexity-efficient smoothing, achieving bias-free tracking of polynomial phase signals under colored or white noise (Kennedy, 2023, Kennedy, 2023).
• Multi-layered Cepstral Recursion: In multi-pitch or multi-F0 extraction (e.g., musical signals), stacking multiple frequency/quefrency-domain operations recursively suppresses slow-varying envelopes and enhances periodic (F0) structure, outperforming classical single-pass spectral or cepstral peak-picking, especially under convolutional noise (Yu et al., 2019).
• Perturbative Recursive Frequency Correction: In nonlinear dynamics, equivalent linearization applies an order-by-order (recursive) removal of secular terms from the response, yielding frequency corrections to arbitrary order in the nonlinearity (Chattopadhyay et al., 2016).

A recursive approach is critical whenever information about the dominant frequency is degraded or distorted (missing fundamentals, strong noise, nonlinearity) and requires explicit stepwise enhancement or bias correction unreachable through single-pass or naive averaging.

2. Mathematical Formalism and Algorithmic Structure

Digital Recursive Filtering (IIR/FIR)

Given a complex analytic signal $x[n] = v[n] + \varepsilon[n]$, with $v[n] = A e^{j\theta[n]}$ and $\theta[n]$ an $m$th-degree polynomial, recursive filters $H(z)$ are constructed as:

$$H(z) = \sum_{k=0}^{K_p-1} c_k P_k(z), \qquad P_k(z) = \frac{z^{-1}}{1 - p_k z^{-1}}$$

with pole locations $\{p_k\}$ dictating bandwidth and stability; coefficients $\{c_k\}$ are determined to ensure zero bias at DC up to order $m$ and minimized colored-noise gain (Kennedy, 2023). The coefficients are solved via Lagrange-constrained minimization, yielding optimal minimum-variance recursive smoothers or predictors.

Recursive smoothing of phase differences (e.g., via a CLI/Erlang filter):

$y_k[n] = p \cdot y_k[n-1] + (1-p) \cdot y_{k-1}[n]$

cascades over $K_{CLI}$ sections, supporting infinite impulse response smoothing with group delay and bandwidth controlled by $p$ and $K_{CLI}$ (Kennedy, 2023).

Multi-Layered Cepstral Recursion

A generalized recursive block alternates Fourier transform, domain-specific high-pass filtering, and nonlinear activation:

• Layer $l$:

$$Z^{(l)} = \sigma^{(l)}\big[ W^{(l)} F Z^{(l-1)} \big]$$

with $\sigma^{(l)}(x) = \max(x, 0)^{\gamma_l}$, and domain-appropriate diagonal $W^{(l)}$. Alternating between frequency and quefrency domains, layers recursively focus on periodic structures while suppressing smooth or aperiodic backgrounds.

• Fusion via Combined Frequency and Periodicity (CFP):

$$Y^{(L-1, L)}[k, n] = Z^{(L)}[k, n] \cdot Z^{(L-1)}[\mathrm{round}(N/k), n]$$

aligns the mapped periodic features to the true fundamental, forming a saliency map highly selective for actual F0s (Yu et al., 2019).

Recursive Analytical Frequency Correction for Nonlinear Oscillators

For systems $$\ddot{x} + \omega_0^2 x + \varepsilon f(x, \dot{x}) = 0$$, recursive equivalent linearization solves for frequency corrections order-by-order in $\varepsilon$:

$$\omega = \omega_0 + \sum_{n \geq 1} \varepsilon^n \omega_n$$

with

$$\Delta_n = -\frac{2}{A} \langle F_n(t), \cos(\omega_0 t) \rangle, \quad \omega_n = \frac{\Delta_n}{2\omega_0}$$

where $F_n(t)$ is the $\varepsilon^n$ coefficient in the Taylor expansion, and the recursion removes secular terms at each order (Chattopadhyay et al., 2016).

3. Empirical Performance and Robustness

The recursive correction framework increases estimation robustness and adaptability:

• Multi-layered cepstrum experiments demonstrate F-score improvement with depth; in polyphonic pitch tracking, increasing from 1 to 6 layers raises F-score from ≈80.5% to ≈86.5% under standard conditions, and more significantly under high-pass filtered corruption (up to ≈87.1% F-score at 100 Hz cut-off for 6 layers vs. ≈80.7% for 1 layer) (Yu et al., 2019).
• Recursive digital filters such as the CLI/Erlang achieve near-optimal mean squared error for frequency estimation at low computational cost. Monte Carlo simulation shows, for example, that in low noise, a three-stage Erlang filter with $p=0.8079$ achieves RMSE ≈ $2.5 \times 10^{-4}$ rad/sample—comparable to optimal (Kay) FIR realization but at a fraction of memory and processing requirements (Kennedy, 2023).
• Recursive equivalent linearization computes higher-order corrections that are essential when first-order corrections vanish (e.g., in the van der Pol oscillator), ensuring accurate frequency estimates even where perturbative drift exceeds leading-order bias (Chattopadhyay et al., 2016).

A plausible implication is that deeper or higher-order recursion provides diminishing marginal gains beyond a task-dependent threshold (typically 3–6 layers in MLC, or second to third-order in weakly nonlinear analysis).

4. Implementation Choices and Practical Guidelines

Implementation choices are tailored by noise level, computational constraints, and underlying signal model:

• Recursive Filters (Kennedy, 2023, Kennedy, 2023):
  • Select phase polynomial degree $m$, number of differentiators $K_o$, and transfer function order $K_p$.
  • Place poles using Bessel or Butterworth prototypes for desired bandwidth and phase linearity.
  • Solve for coefficients under DC-matching flatness constraints and colored noise minimization.
  • Optimize lag $q$ to trade off variance against temporal latency.
  • Pair main estimator (lag $q>0$) with a one-step predictor (lead $q=-1$) to mitigate unwrapping errors at low SNR.
• Multi-layered Cepstrum (Yu et al., 2019):
  • Typical settings: window $N\approx7939$, hop $H=10$ ms, Blackman–Harris window.
  • Cutoff frequency $f_c$ and quefrency $q_c$ tailored to musical range.
  • Initial nonlinearity exponents: $\gamma_0\approx0.25$, $\gamma_1\approx0.6$, $\gamma_{l\geq2}\approx1.0$. More layers and/or higher exponents enhance robustness at the cost of increased computational load.
  • Limit recursion depth to control cross-terms and aliasing artifacts.
• Equivalent Linearization (Chattopadhyay et al., 2016):
  • Expand both the solution and effective frequency recursively up to required order; typically second or third order suffices for practical accuracy in weak nonlinearity.
  • Project at each step onto the fundamental to ensure secular terms are suppressed.

Peak picking, suppression of cross-terms, and post-processing (e.g., Viterbi smoothing) are domain-specific but critical for final estimate stability.

5. Applications and Limitations

Applications encompass real-time pitch tracking in heavily filtered or mixed audio, embedded-frequency estimation in phase-locked loops or communications, estimation of Doppler shifts, and theoretical frequency corrections in nonlinear vibrational analysis. Recursive dominant frequency correction is especially effective under:

• Loss of low-frequency content due to system filtering.
• High levels of additive or colored noise.
• Time-varying frequency modulations or slow drifts.
• Nonlinear systems where traditional perturbative expansions are insufficient.

However, limitations include:

• Additional computational cost for deep recursions, especially for large input size or high sample rate scenarios.
• Susceptibility to cross-terms and artifacts in MLC variants when nonlinearity exponents are low or layers are too deep.
• In high-SNR breakdown, angle unwrapping must be managed (e.g., predictor pairing in IIR smoothers).
• For equivalent linearization, convergence is generally asymptotic, and expansions must be truncated judiciously for optimal accuracy (Chattopadhyay et al., 2016).

6. Summary Table of Recursive Dominant Frequency Correction Techniques

Methodology	Key Recursion Principle	Signal/Domain
Multi-layered Cepstrum	Iterated freq/queferency ops	Polyphonic audio/MF₀
CLI/Erlang IIR Filter	Cascade of leaky integrators	Analytic signal/phase
Kalman Filter	Recursive MMSE state estimator	Phase (linear/poly)
Equivalent Linearization	Power series in nonlinearity	Nonlinear oscillators

Each technique embodies the principle of recursively refining frequency estimates, either by stacking domain transforms (MLC), successively smoothing noisy observations (IIR/FIR), updating in state-space (Kalman), or order-by-order secular suppression (EL). The unifying aspect of recursive dominant frequency correction lies in its ability to progressively extract or restore dominant periodic features under adverse signal conditions, with system-appropriate bias and variance guarantees.

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Principal References: (Yu et al., 2019, Kennedy, 2023, Kennedy, 2023, Chattopadhyay et al., 2016)