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USF Spectral Estimation: Prevalence of Gaussian Cramér-Rao Bounds Despite Modulo Folding

Ruiming Guo, Ayush Bhandari

TL;DR

This paper addresses spectral estimation under the Unlimited Sensing Framework (USF), where modulo folding occurs before sampling to overcome traditional digitization bottlenecks. It derives Cramér-Rao Bounds (CRBs) for USF-SpecEst and shows that the folded-sample CRBs are asymptotically scaled versions of the Gaussian CRB, with per-tone factors $\gamma=(1-\cos(\omega T))^{-1}$. The authors model the folding residue with a hybrid Gaussian-Bernoulli noise framework, prove that the approximation error vanishes as the number of samples grows, and validate the theory with Matrix Pencil experiments for single and multiple sinusoids. The results provide a concrete performance benchmark and practical guidance for deploying USF-based spectral estimation in high-dynamic-range, high-resolution sensing settings, highlighting the continued relevance of Gaussian CRBs even under non-linear folding.

Abstract

Spectral Estimation (SpecEst) is a core area of signal processing with a history spanning two centuries and applications across various fields. With the advent of digital acquisition, SpecEst algorithms have been widely applied to tasks like frequency super-resolution. However, conventional digital acquisition imposes a trade-off: for a fixed bit budget, one can optimize either signal dynamic range or digital resolution (noise floor), but not both simultaneously. The Unlimited Sensing Framework (USF) overcomes this limitation using modulo non-linearity in analog hardware, enabling a novel approach to SpecEst (USF-SpecEst). However, USF-SpecEst requires new theoretical and algorithmic developments to handle folded samples effectively. In this paper, we derive the Cramér-Rao Bounds (CRBs) for SpecEst with noisy modulo-folded samples and reveal a surprising result: the CRBs for USF-SpecEst are scaled versions of the Gaussian CRBs for conventional samples. Numerical experiments validate these bounds, providing a benchmark for USF-SpecEst and facilitating its practical deployment.

USF Spectral Estimation: Prevalence of Gaussian Cramér-Rao Bounds Despite Modulo Folding

TL;DR

This paper addresses spectral estimation under the Unlimited Sensing Framework (USF), where modulo folding occurs before sampling to overcome traditional digitization bottlenecks. It derives Cramér-Rao Bounds (CRBs) for USF-SpecEst and shows that the folded-sample CRBs are asymptotically scaled versions of the Gaussian CRB, with per-tone factors . The authors model the folding residue with a hybrid Gaussian-Bernoulli noise framework, prove that the approximation error vanishes as the number of samples grows, and validate the theory with Matrix Pencil experiments for single and multiple sinusoids. The results provide a concrete performance benchmark and practical guidance for deploying USF-based spectral estimation in high-dynamic-range, high-resolution sensing settings, highlighting the continued relevance of Gaussian CRBs even under non-linear folding.

Abstract

Spectral Estimation (SpecEst) is a core area of signal processing with a history spanning two centuries and applications across various fields. With the advent of digital acquisition, SpecEst algorithms have been widely applied to tasks like frequency super-resolution. However, conventional digital acquisition imposes a trade-off: for a fixed bit budget, one can optimize either signal dynamic range or digital resolution (noise floor), but not both simultaneously. The Unlimited Sensing Framework (USF) overcomes this limitation using modulo non-linearity in analog hardware, enabling a novel approach to SpecEst (USF-SpecEst). However, USF-SpecEst requires new theoretical and algorithmic developments to handle folded samples effectively. In this paper, we derive the Cramér-Rao Bounds (CRBs) for SpecEst with noisy modulo-folded samples and reveal a surprising result: the CRBs for USF-SpecEst are scaled versions of the Gaussian CRBs for conventional samples. Numerical experiments validate these bounds, providing a benchmark for USF-SpecEst and facilitating its practical deployment.
Paper Structure (5 sections, 2 theorems, 22 equations, 2 figures)

This paper contains 5 sections, 2 theorems, 22 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. Cramér-Rao Bounds
  4. Experiments
  5. Conclusion

Key Result

Lemma 1

Let $g(t)= \texttt{SoS}_K\left( \underline{\theta}\right)$, $\Delta^L = \Delta^{L-1}\circ\Delta$ denote the $L$-order finite-difference operator ($L\in\mathbb Z^{+}$) and $\kappa>0$ be an arbitrary constant. Then, where $\mathbf{a}=[a_1,\ldots,a_K]^{{\top}}$, ${\varepsilon}_s \stackrel{eq:moddec}{=} s - \mathscr{M}_\lambda ({s} )$ and $\Omega \stackrel{\rm{def}}{=} {\left\| \boldsymbol{\omega}

Figures (2)

  • Figure 1: CRB tests for single sinusoid (averaged over $10000$ random realizations). We consider $K=1$ sinusoid with parameters: $\omega_1 T = 1.05$, $a_1 = 1$, $\epsilon = 9.83 \times 10^{-6}$, and $N = 100$ samples, resulting in $M = 6$ spikes. (a) Scatterplot of the retrieved frequencies. (b) Performance evaluation compared to the CRBs in Theorem \ref{['thm:2']}.
  • Figure 2: CRB tests for multiple sinusoids (averaged over $10000$ random realizations). We consider $K=2$ sinusoids with the following parameters: $\{\omega_1 T, \omega_2 T\} = \{0.63, 1.00\}$, $\{a_1, a_2\} =\{1, 1\}$, $\epsilon = 2.51 \times 10^{-4}$, and $N = 100$ samples, resulting in $M = 2$ spikes. (a) Scatterplot of the retrieved frequencies. (b) Performance evaluation.

Theorems & Definitions (4)

  • Lemma 1
  • proof
  • Theorem 1
  • proof