Orthogonal Matching Pursuit based Reconstruction for Modulo Hysteresis Operators

TL;DR

Problem: high dynamic range signals saturated by ADCs can be addressed by unlimited sampling with modulo nonlinearity, but standard modulo hysteresis may violate the ADC range. Approach: introduce a modified modulo hysteresis operator $\mathscr M_\lambda^{h,\alpha}$ guaranteeing $[-\lambda,\lambda]$ outputs and proving identifiability of $g \in \mathsf{PW}_\Omega$ from discrete samples under the oversampling condition $T < \frac{\pi}{\Omega}$; develop an SAOMP-based sparse recovery that reconstructs $g$ from $\mathscr M_\lambda^{h,\alpha} g(kT)$. Findings: identifiability result under smoothness constraints and a practical OMP/SAOMP reconstruction with competitive performance against a thresholding method, with instabilities when $\alpha$ greatly exceeds $T$. Significance: provides hardware-friendly, robust reconstruction for unlimited sampling in realistic modulo-hysteresis scenarios and yields a concrete algorithmic pipeline for bandlimited signal recovery.

Abstract

Unlimited sampling provides an acquisition scheme for high dynamic range signals by folding the signal into the dynamic range of the analog-to-digital converter (ADC) using modulo non-linearity prior to sampling to prevent saturation. Recently, a generalized scheme called modulo hysteresis was introduced to account for hardware non-idealities. The encoding operator, however, does not guarantee that the output signal is within the dynamic range of the ADC. To resolve this, we propose a modified modulo hysteresis operator and show identifiability of bandlimited signals from modulo hysteresis samples. We propose a recovery algorithm based on orthogonal matching pursuit and validate our theoretical results through numerical experiments.

Figures (3)

• Figure 1: Illustration of generalized modulo encoder $\mathscr M_H$ and modified modulo hysteresis operator $\mathscr M_\lambda^{h,\alpha}$ for random input $g \in \mathsf{PW}_\Omega$ with $\lambda = 0.2$, $h = 0.1$ and $\alpha = 0.3$, where $\mathscr M_\lambda^{h,\alpha}g \in [-\lambda,\lambda]$, but $|\mathscr M_H g(t)| > \lambda$ for some $t$.
• Figure 2: Illustration of SAOMP and TAlg reconstruction for random $g \in \mathsf{PW}_\Omega$ with $\Omega = 6.3 \frac{\text{rad}}{\text{s}}$ based on modulo hysteresis samples $\mathscr M_\lambda^{h,\alpha} g(n\mathrm T)$ with $\lambda = 0.1$, $h = 0.05$, $\alpha = 50 \text{ms}$ and $\mathrm T = 20.8 \text{ms}$.
• Figure 3: Success of SAOMP reconstruction for $\mathscr M_\lambda^{h,\alpha}g(n\mathrm T)$ with $\lambda = 0.1$, $\mathrm T = 20.8 \text{ms}$ and different values for $\alpha$ and $h$ showing how often the MSE is larger than $0.001$ for $50$ random functions $g \in \mathsf{PW}_\Omega$ with $\Omega = 6.3 \frac{\text{rad}}{\text{s}}$.

Theorems & Definitions (8)

• Definition 1: Modified Modulo Hysteresis
• Lemma 1
• proof
• Lemma 2
• proof
• Lemma 3
• proof
• Corollary 1