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Toward Optimal ANC: Establishing Mutual Information Lower Bound

François Derrida, Shahar Lutati, Eliya Nachmani

TL;DR

The paper addresses the lack of a universal theoretical limit for active noise cancellation (ANC) performance by introducing a unified lower bound on the normalized mean squared error (NMSE) that combines an information-theoretic term and a spectral-support term. The information-theoretic component links residual error to the fraction of disturbance entropy captured by the anti-noise, while the spectral component accounts for irreducible error in uncancelable frequency bands due to physical path limitations; the NMSE bound is given by the maximum of these two terms. The authors validate the bound empirically on the NOISEX dataset across varied reverberation times, showing the bound is tight and informative, with the dominant bound shifting from spectral constraints at low $t_{60}$ to information constraints at higher $t_{60}$. This framework provides a principled ceiling for ANC performance and offers actionable guidance on whether to improve algorithmic information processing or to modify the physical system to extend spectral coverage. The work has practical implications for designing more effective ANC systems and for benchmarking future deep-learning–based approaches against fundamental limits.

Abstract

Active Noise Cancellation (ANC) algorithms aim to suppress unwanted acoustic disturbances by generating anti-noise signals that destructively interfere with the original noise in real time. Although recent deep learning-based ANC algorithms have set new performance benchmarks, there remains a shortage of theoretical limits to rigorously assess their improvements. To address this, we derive a unified lower bound on cancellation performance composed of two components. The first component is information-theoretic: it links residual error power to the fraction of disturbance entropy captured by the anti-noise signal, thereby quantifying limits imposed by information-processing capacity. The second component is support-based: it measures the irreducible error arising in frequency bands that the cancellation path cannot address, reflecting fundamental physical constraints. By taking the maximum of these two terms, our bound establishes a theoretical ceiling on the Normalized Mean Squared Error (NMSE) attainable by any ANC algorithm. We validate its tightness empirically on the NOISEX dataset under varying reverberation times, demonstrating robustness across diverse acoustic conditions.

Toward Optimal ANC: Establishing Mutual Information Lower Bound

TL;DR

The paper addresses the lack of a universal theoretical limit for active noise cancellation (ANC) performance by introducing a unified lower bound on the normalized mean squared error (NMSE) that combines an information-theoretic term and a spectral-support term. The information-theoretic component links residual error to the fraction of disturbance entropy captured by the anti-noise, while the spectral component accounts for irreducible error in uncancelable frequency bands due to physical path limitations; the NMSE bound is given by the maximum of these two terms. The authors validate the bound empirically on the NOISEX dataset across varied reverberation times, showing the bound is tight and informative, with the dominant bound shifting from spectral constraints at low to information constraints at higher . This framework provides a principled ceiling for ANC performance and offers actionable guidance on whether to improve algorithmic information processing or to modify the physical system to extend spectral coverage. The work has practical implications for designing more effective ANC systems and for benchmarking future deep-learning–based approaches against fundamental limits.

Abstract

Active Noise Cancellation (ANC) algorithms aim to suppress unwanted acoustic disturbances by generating anti-noise signals that destructively interfere with the original noise in real time. Although recent deep learning-based ANC algorithms have set new performance benchmarks, there remains a shortage of theoretical limits to rigorously assess their improvements. To address this, we derive a unified lower bound on cancellation performance composed of two components. The first component is information-theoretic: it links residual error power to the fraction of disturbance entropy captured by the anti-noise signal, thereby quantifying limits imposed by information-processing capacity. The second component is support-based: it measures the irreducible error arising in frequency bands that the cancellation path cannot address, reflecting fundamental physical constraints. By taking the maximum of these two terms, our bound establishes a theoretical ceiling on the Normalized Mean Squared Error (NMSE) attainable by any ANC algorithm. We validate its tightness empirically on the NOISEX dataset under varying reverberation times, demonstrating robustness across diverse acoustic conditions.

Paper Structure

This paper contains 16 sections, 4 theorems, 23 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Related Work
  3. Background - Typical ANC setup
  4. Fundamental Limits on Cancellation Performance: Information-Theoretic Bounds
  5. System Support Constraints and Model-Independent Limits
  6. Unified Performance Bound
  7. Experiments
  8. Results and Discussions
  9. Boundaries hold NMSE performances over various reverberation times
  10. Boundary overview for a fixed reverberation time ($t_{60}=0.2s$)
  11. Limitations and Broader Impact
  12. Conclusions
  13. Technical Appendices and Supplementary Material
  14. Numerical calculation of the Information-Theoretic Boundary (Equation \ref{['eq:info-nmse_elaborated']})
  15. Numerical calculation of the System Support-Based Boundary (Equation \ref{['eq:support-nmse_elaborated']})
  16. ...and 1 more sections

Key Result

Theorem 1

Let $v(n)$ be a discrete-time WSS random process with Auto-Correlation Function (ACF) defined as $R_{vv}(\tau) = \mathop{\mathrm{\mathbb{E}}}\nolimits[v(n)v^*(n-\tau)]$. Its Power Spectral Density (PSD), $S_{vv}(e^{j\omega})$, is the Discrete-Time Fourier Transform (DTFT) of its ACF: Conversely, the ACF can be recovered from the PSD via the inverse DTFT: The average power of the process is the i

Figures (4)

  • Figure 1: High-level schematic of a feedforward ANC system with lower-bound calculation.
  • Figure 2: Model performances for various reverberation times
  • Figure 3: Time-Frequency analysis of the different noise
  • Figure 4: The lower bound and NMSE comparison over different datasets for $t_{60} =0.2s$

Theorems & Definitions (5)

  • Theorem 1: Wiener-Khinchin Theorem for Discrete-Time WSS Processes
  • Lemma 1: Information-Theoretic Bound on Cancellation Error
  • proof
  • Lemma 2: Support-Based Bound on Cancellation Error
  • Theorem 2: Unified Lower Bound on NMSE