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On the Identifiability from Modulo Measurements under DFT Sensing Matrix

Qi Zhang, Jiang Zhu, Fengzhong Qu, Zheng Zhu, De Wen Soh

TL;DR

The paper addresses identifiability in modulo-DFT sensing, where a DFT preprocessor precedes modulo sampling to extend ADC dynamic range. It derives a necessary-and-sufficient condition for unique recovery based on the factorization of x^N−1 over Gaussian rationals and the zero-support set V, with detailed analyses for N being a power of two, prime, or twice a prime. It extends the framework to periodic bandlimited signals under modulo sampling, revealing identifiability up to a DC ambiguity when the quantity H(N) meets P+1, and shows a universal oversampling threshold γ > 3(1+1/P) suffices for unique identification. A recovery algorithm based on solving integer linear equations is proposed, and simulations validate the theoretical results, demonstrating practical guidelines for choosing the measurement count N and solving the resulting ILP to recover the original signal. Together, these results offer a principled approach to reconstructing signals from modulo observations using a DFT frontend and ILP-based recovery, with implications for high-dynamic-range sensing and compressed sensing alike.

Abstract

Modulo sampling (MS) has been recently introduced to enhance the dynamic range of conventional ADCs by applying a modulo operator before sampling. This paper examines the identifiability of a measurement model where measurements are taken using a discrete Fourier transform (DFT) sensing matrix, followed by a modulo operator (modulo-DFT). Firstly, we derive a necessary and sufficient condition for the unique identification of the modulo-DFT sensing model based on the number of measurements and the indices of zero elements in the original signal. Then, we conduct a deeper analysis of three specific cases: when the number of measurements is a power of $2$, a prime number, and twice a prime number. Additionally, we investigate the identifiability of periodic bandlimited (PBL) signals under MS, which can be considered as the modulo-DFT sensing model with additional symmetric and conjugate constraints on the original signal. We also provide a necessary and sufficient condition based solely on the number of samples in one period for the unique identification of the PBL signal under MS, though with an ambiguity in the direct current (DC) component. Furthermore, we show that when the oversampling factor exceeds $3(1+1/P)$, the PBL signal can be uniquely identified with an ambiguity in the DC component, where $P$ is the number of harmonics, including the fundamental component, in the positive frequency part. Finally, we also present a recovery algorithm that estimates the original signal by solving integer linear equations, and we conduct simulations to validate our conclusions.

On the Identifiability from Modulo Measurements under DFT Sensing Matrix

TL;DR

The paper addresses identifiability in modulo-DFT sensing, where a DFT preprocessor precedes modulo sampling to extend ADC dynamic range. It derives a necessary-and-sufficient condition for unique recovery based on the factorization of x^N−1 over Gaussian rationals and the zero-support set V, with detailed analyses for N being a power of two, prime, or twice a prime. It extends the framework to periodic bandlimited signals under modulo sampling, revealing identifiability up to a DC ambiguity when the quantity H(N) meets P+1, and shows a universal oversampling threshold γ > 3(1+1/P) suffices for unique identification. A recovery algorithm based on solving integer linear equations is proposed, and simulations validate the theoretical results, demonstrating practical guidelines for choosing the measurement count N and solving the resulting ILP to recover the original signal. Together, these results offer a principled approach to reconstructing signals from modulo observations using a DFT frontend and ILP-based recovery, with implications for high-dynamic-range sensing and compressed sensing alike.

Abstract

Modulo sampling (MS) has been recently introduced to enhance the dynamic range of conventional ADCs by applying a modulo operator before sampling. This paper examines the identifiability of a measurement model where measurements are taken using a discrete Fourier transform (DFT) sensing matrix, followed by a modulo operator (modulo-DFT). Firstly, we derive a necessary and sufficient condition for the unique identification of the modulo-DFT sensing model based on the number of measurements and the indices of zero elements in the original signal. Then, we conduct a deeper analysis of three specific cases: when the number of measurements is a power of , a prime number, and twice a prime number. Additionally, we investigate the identifiability of periodic bandlimited (PBL) signals under MS, which can be considered as the modulo-DFT sensing model with additional symmetric and conjugate constraints on the original signal. We also provide a necessary and sufficient condition based solely on the number of samples in one period for the unique identification of the PBL signal under MS, though with an ambiguity in the direct current (DC) component. Furthermore, we show that when the oversampling factor exceeds , the PBL signal can be uniquely identified with an ambiguity in the DC component, where is the number of harmonics, including the fundamental component, in the positive frequency part. Finally, we also present a recovery algorithm that estimates the original signal by solving integer linear equations, and we conduct simulations to validate our conclusions.
Paper Structure (21 sections, 27 theorems, 44 equations, 3 figures, 1 table)

This paper contains 21 sections, 27 theorems, 44 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Notation
  3. Problem Setup
  4. Preliminaries
  5. Identifiability of Modulo-DFT Sensing Model
  6. Recovery algorithm
  7. Simulation
  8. A necessary condition for Vandermonde sensing matrices
  9. Proof of Theorem \ref{['TheoComplex']}
  10. Identifiability of Periodic Bandlimited Signals
  11. Problem setup
  12. Identifiability
  13. Simulation
  14. Proof of Theorem \ref{['theorem3']}
  15. Proof of Lemma \ref{['PBSlemma3']}
  16. ...and 6 more sections

Key Result

Proposition 1

The modulo-DFT sensing model (decomsimple) is identifiable with the constraint $\mathbf{s}\in\mathbb{C}^N_{\mathcal{V}}$ if and only if there does not exist a nonzero Gaussian integer polynomial $f(x)$ of maximum degree $N-1$ such that $f({\rm e}^{{\rm j}\frac{2\pi n}{N}}) = 0$ for all $n\in\mathcal

Figures (3)

  • Figure 1: Schematic architecture for the modulo-DFT sensing model. A complex-valued signal undergoes the delay and sample-and-hold circuits followed by the DFT sensing circuit and the modulo circuits whose outputs are then sampled by $N$ parallel ADCs.
  • Figure 2: Probabilities of successful recovery of the proposed recovery algorithm over $300$ Monte Carlo trials.
  • Figure 3: (a) The black part is the region that $H(N)<P+1$, indicating that the model (\ref{['periodicModel']}) is unidentifiable, and the white part is the region that $H(N)\geq P+1$, indicating that the model (\ref{['periodicModel']}) is identifiable. (b) The probabilities of successful recovery of PBL signals from modulo samples over $300$ Monte Carlo trials for different $P$ and $N$ of the proposed recovery algorithm.

Theorems & Definitions (44)

  • Proposition 1
  • Proof 1
  • Proof 2
  • Proof 3
  • Proposition 2
  • Proposition 3
  • Proposition 4
  • Proof 4
  • Theorem 1
  • Corollary 1
  • ...and 34 more