Noise-Modulated Spectral Bias

• Noise-modulated spectral bias is the systematic alteration of a signal’s frequency content due to noise interactions, impacting fields like signal processing and machine learning.
• It employs mathematical techniques such as FIR NTF synthesis, Fourier analysis, and Lipschitz norm constraints to intentionally shape or correct noise in various applications.
• Practical insights include using deconvolution methods, adaptive filtering, and unbiased estimators to improve system performance and reduce measurement errors.

Noise-modulated spectral bias refers to the alteration, shaping, or selective filtering of spectral content (frequency components) of a signal or system response due to the interplay between noise characteristics and system design, estimation, or learning procedures. The concept arises across engineering, signal processing, neuroscience, quantum information, and machine learning domains, encapsulating how noise can systematically modulate, induce, or interact with inherent spectral biases—preferences for certain frequency ranges—within algorithms, physical processes, and devices.

1. Mathematical Formalism of Noise-Modulated Spectral Bias

The quantitative description of noise-modulated spectral bias varies by context, but uniformly centers on how noise or its weighting affects the spectral composition of signals after system, algorithmic, or estimator processing. In delta-sigma (ΔΣ) modulation, noise is shaped via intentional design to yield frequency-dependent bias:

$\nu = \int_0^{\frac{1}{2} \Psi_n(f)\, w(f)\ df$

where $\Psi_n(f)$ is the output noise power spectral density and $w(f)$ is a user-specified importance or tolerance function encoding frequency preference (Callegari et al., 2013). The minimization of $\nu$—subject to stability and delay constraints—guides the design of discrete FIR Noise Transfer Functions (NTFs) to modulate quantization noise spectral bias.

In neural network learning, spectral bias refers to the intrinsic tendency of networks to fit low-frequency target functions first, with high-frequency components only learned as optimization progresses. The spectral signature of noise memorization and spectral bias is captured by local Fourier transform metrics and their temporal evolution (Zhang et al., 2020):

$R_k = \log \frac{A_k}{\sum_j A_j}$

$$A_k = \frac{1}{C} \sum_{c=1}^C \mathbb{E}_{\boldsymbol{x} \sim \mathcal{D}} |\tilde{f}_{c, \boldsymbol{x}}(k)|^2$$

where $A_k$ quantify energy in Fourier band $k$ of the output logits.

In spectral estimation theory, the bias in measured power spectra due to noise is often expressed as a convolution:

$S_o(f) = S_u(f) * S_g(f)$

with $S_g(f)$ capturing the impact of the sampling function, including dead time, randomness, or nonlinearities (Buchhave et al., 2019). Correction is typically effected through deconvolution using the measured autocovariance of the sampling process.

2. Origins and Mechanisms of Spectral Bias Induced or Modulated by Noise

Quantization Noise Shaping and Weighting

In ΔΣ modulator design, noise-modulated spectral bias is implemented by deliberately shaping the quantization noise PSD via optimal NTF design based on application-specific weighting functions $w(f)$ (Callegari et al., 2013). This enables prioritization of frequency bands critical to the intended use (e.g., minimizing audible noise in psychoacoustic coders, suppressing DC noise for sensors).

Noise in Neural Network Generalization and Memorization

Deep neural networks exhibit a learning spectral bias toward low-frequency function components (Rahaman et al., 2018). Upon exposure to noisy labels or inputs, memorization induces high-frequency content only on-manifold (i.e., for training data points), but this bias is not strictly monotonic: continued training causes these high-frequency components to recede, restoring global low-frequency bias and generalization (Zhang et al., 2020). This demonstrates noise-induced, temporally modulated spectral bias.

Sampling and Spectral Estimation

Random sampling, dead time, and velocity-dependent sample rate can modulate the observed spectral content and introduce frequency-selective bias in data acquired via instruments such as laser Doppler anemometers (Buchhave et al., 2019). The sampling function imprints its own spectral structure, convoluting the true signal's spectrum and biasing spectral estimates until corrected.

Intermittent Noiselike Emission in Astrophysics

Intermittency in source emission—occurring, for instance, in pulsars—modulates spectral bias by introducing correlations among spectral channel noise, without affecting the mean spectrum (Gwinn et al., 2011). While the average spectrum is unaltered, the spectral covariance (higher-order statistics) carries the signature of emission intermittency.

3. Methodologies for Modulating, Measuring, and Correcting Spectral Bias

FIR NTF Synthesis with Arbitrary Weighting

Delta-sigma modulator designers solve a semidefinite program to minimize weighted quantization noise via:

$$\nu = \frac{\Delta^2}{3} \vec{a}^T \mathbf{Q} \vec{a}$$

where $\mathbf{Q}$ is the Toeplitz matrix encoding $w(f)$. Constraints on $\|NTF\|_\infty$ (Lee criterion) ensure loop stability. Multiband, DC-avoidant, and psychoacoustic noise distributions are explicitly realizable (Callegari et al., 2013).

Neural Network Spectral Control

In self-supervised image denoising, spectral bias is both measured and controlled. The Image Pair Frequency-Band Similarity metric tracks how network outputs absorb high-frequency structure vs. noise (Zhang et al., 1 Oct 2025). Spectral control is achieved via:

• Frequency selection (feeding the noisier image to accelerate high-frequency learning)
• Lipschitz norm-based optimization (spectral norm constraints on kernels restrict high-frequency propagation)
• Low-rank spectral separation and reconstruction modules to disentangle genuine detail from high-frequency noise.

Deconvolution with Measured Sampling Function

Spectral bias induced by random sampling and dead time is corrected by dividing the measured autocovariance by that of the sampling function, followed by Fourier transformation (Buchhave et al., 2019):

$$C_{o, \text{corrected}}(\tau) = \frac{C_o(\tau)}{\hat{C}_g(\tau)}$$

$$S_{o, \text{corr}}(f) = \mathcal{F}[C_{o, \text{corrected}}(\tau)]$$

This approach generalizes to any process where noise or bias is encoded in the sample times.

Multitaper and Slepian Filtering in Quantum Noise Spectroscopy

Slepian-based quantum multitaper protocols use discrete prolate spheroidal sequences to define near-ideal band-pass filter functions (Norris et al., 2018). This substantially reduces spectral leakage and associated bias compared to conventional dynamical decoupling techniques.

Bias-Free Estimators in Optical Interferometry

Bias in power spectrum and bispectrum due to combined Poisson and Gaussian noise is eliminated by estimators derived from the moments of the noise distributions, which incorporate cross-terms and work with arbitrary sampling patterns (Gordon et al., 2011):

$S_0 = |c|^2 - \sum_p (i_p + \sigma_p^2)|H_p|^2$

$$B_0^{ijk} = c^{ij} c^{jk} c^{ki} - \dots + \sum_p (2 i_p + 3\sigma_p^2) H_p^{ij} H_p^{jk} H_p^{ki}$$

4. Applications and Impact Across Domains

Domain	Manifestation of Noise-Modulated Spectral Bias	Correction/Exploitation
ΔΣ modulator/audio coding	Frequency-selective shaping of quantization noise	Weighted NTF synthesis (Callegari et al., 2013)
Neural networks/DNNs	Non-monotonic learning of high-frequency noise	Spectral analysis, Lipschitz optimization (Zhang et al., 2020, Zhang et al., 1 Oct 2025)
Instrumental PSD analysis	Bias from random sampling and dead time	Sampling function deconvolution (Buchhave et al., 2019)
Astrophysics/pulsars	Correlated noise in spectral channels due to intermittency	Higher-order covariance statistics (Gwinn et al., 2011)
Optical interferometry	Bias in (bi)spectrum due to detection noise	Bias-free estimators (Gordon et al., 2011)
Quantum noise spectroscopy	Spectral leakage/aliasing bias in noise estimation	DPSS-based multitaper protocols (Norris et al., 2018)

A plausible implication is that as statistical and physical systems grow in complexity, precise measurement, shaping, or correction of noise-modulated spectral bias becomes both technically feasible and increasingly crucial. The ability to match the spectral distribution of noise to application-driven tolerances enables superior system optimization and diagnostic power in communications, sensing, imaging, and learning.

5. Conceptual Significance and Contemporary Perspectives

Noise-modulated spectral bias clarifies and quantifies the intricate role of noise across signal processing, physical, and learning systems. Rather than being an irreducible or merely detrimental factor, noise is subject to selective design, measurement, and control—its spectral footprint can be distributed, suppressed, or harnessed in accordance with practical or theoretical priorities. Notably,

• In modern DNNs, memorization of noise does not inexorably degrade generalization as previously thought; spectral bias may revert toward low frequencies with adept regularization and training schedules (Zhang et al., 2020).
• In measurement theory, virtually all spectral bias induced by sampling artifacts can be compensated using real-time arrival statistics (Buchhave et al., 2019).
• In optical interferometry, full bias elimination is feasible under arbitrary sampling and multiplexed noise conditions, preventing systematic errors in astronomical parameter estimation (Gordon et al., 2011).
• In quantum information, leveraging noise bias (e.g., via cat qubits) ensures circuit-level error correction and reliable benchmarking at scales previously inaccessible (Fellous-Asiani et al., 2023).

This suggests ongoing convergence between theoretical insight, algorithmic advances, and practical system design in the explicit control and exploitation of noise-modulated spectral bias.

6. Representative Mathematical Expressions and Tables

System/Class	Bias Expression	Correction/Control Approach
ΔΣ Modulator	$$\nu = \frac{\Delta^2}{3} \vec{a}^T \mathbf{Q} \vec{a}$$	Optimal FIR NTF synthesis, SDP (Callegari et al., 2013)
Neural Networks	$R_k = \log \frac{A_k}{\sum_j A_j}$	Spectral monitoring, Lipschitz constraints (Zhang et al., 1 Oct 2025)
Power Spectrum	$S_o(f) = S_u(f) * S_g(f)$	Deconvolution by measured $C_g(\tau)$ (Buchhave et al., 2019)
Interferometry	$S_0 = |c|^2 - \sum (i_p + \sigma_p^2)|H_p|^2$	Bias-free estimator (Gordon et al., 2011)
Quantum Sensing	$\mathcal{B}_{BB},\; \mathcal{B}_{LB}$ (band/local bias)	DPSS multitaper, adaptive weighting (Norris et al., 2018)

7. Perspectives for Future Research and Practice

The explicit modeling and correction of noise-modulated spectral bias elevate both the fidelity and utility of measurement and learning systems. Further research will likely explore:

• Automated, data-driven optimization of noise weighting functions for arbitrary signal or sensor architectures.
• Dynamic, training-schedule-aware modulation of spectral bias in neural networks.
• Extension of unbiased estimation and spectral correction protocols across multimodal, nonstationary, and adaptive systems.
• Harnessing favorable noise bias for quantum error correction and benchmarking at circuit scale.

In summary, noise-modulated spectral bias constitutes a unified and actionable framework for understanding, measuring, regulating, and exploiting the frequency-selective behavior of noise across technical fields. Its rigorous incorporation into system design and analysis is both theoretically justified and demonstrably impactful.