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Non-Linear Analog Processing Gains in Task-Based Quantization

Marian Temprana Alonso, Farhad Shirani, Neil Irwin Bernardo, Yonina C. Eldar

TL;DR

This work investigates non-linear analog processing prior to low-resolution ADCs in task-based quantization, aiming to minimize distortion of an unreconstructed task variable under a fixed ADC budget. It extends beyond linear analog processing to include polynomial operators, envelope detectors, and analog delay elements, deriving computable distortion expressions and linking them to the indirect rate-distortion function. The core finding is that nonlinear analog processing can substantially reduce task-reconstruction distortion, with trade-offs between design complexity and performance across scenarios. The results have practical implications for channel estimation, medical imaging, and object localization where power-limited sensing systems must extract task-relevant information efficiently.

Abstract

In task-based quantization, a multivariate analog signal is transformed into a digital signal using a limited number of low-resolution analog-to-digital converters (ADCs). This process aims to minimize a fidelity criterion, which is assessed against an unobserved task variable that is correlated with the analog signal. The scenario models various applications of interest such as channel estimation, medical imaging applications, and object localization. This work explores the integration of analog processing components -- such as analog delay elements, polynomial operators, and envelope detectors -- prior to ADC quantization. Specifically, four scenarios, involving different collections of analog processing operators are considered: (i) arbitrary polynomial operators with analog delay elements, (ii) limited-degree polynomial operators, excluding delay elements, (iii) sequences of envelope detectors, and (iv) a combination of analog delay elements and linear combiners. For each scenario, the minimum achievable distortion is quantified through derivation of computable expressions in various statistical settings. It is shown that analog processing can significantly reduce the distortion in task reconstruction. Numerical simulations in a Gaussian example are provided to give further insights into the aforementioned analog processing gains.

Non-Linear Analog Processing Gains in Task-Based Quantization

TL;DR

This work investigates non-linear analog processing prior to low-resolution ADCs in task-based quantization, aiming to minimize distortion of an unreconstructed task variable under a fixed ADC budget. It extends beyond linear analog processing to include polynomial operators, envelope detectors, and analog delay elements, deriving computable distortion expressions and linking them to the indirect rate-distortion function. The core finding is that nonlinear analog processing can substantially reduce task-reconstruction distortion, with trade-offs between design complexity and performance across scenarios. The results have practical implications for channel estimation, medical imaging, and object localization where power-limited sensing systems must extract task-relevant information efficiently.

Abstract

In task-based quantization, a multivariate analog signal is transformed into a digital signal using a limited number of low-resolution analog-to-digital converters (ADCs). This process aims to minimize a fidelity criterion, which is assessed against an unobserved task variable that is correlated with the analog signal. The scenario models various applications of interest such as channel estimation, medical imaging applications, and object localization. This work explores the integration of analog processing components -- such as analog delay elements, polynomial operators, and envelope detectors -- prior to ADC quantization. Specifically, four scenarios, involving different collections of analog processing operators are considered: (i) arbitrary polynomial operators with analog delay elements, (ii) limited-degree polynomial operators, excluding delay elements, (iii) sequences of envelope detectors, and (iv) a combination of analog delay elements and linear combiners. For each scenario, the minimum achievable distortion is quantified through derivation of computable expressions in various statistical settings. It is shown that analog processing can significantly reduce the distortion in task reconstruction. Numerical simulations in a Gaussian example are provided to give further insights into the aforementioned analog processing gains.
Paper Structure (16 sections, 4 theorems, 19 equations, 2 figures)

This paper contains 16 sections, 4 theorems, 19 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. An Illustrative Gaussian Example
  4. Scenario 1. Linear Analog Processing
  5. Scenario 2. Quadratic Analog Operators
  6. Scenario 3. Envelope Detectors
  7. Scenario 4. Analog Delay Elements
  8. Fundamental Performance Limits in Task-Based Quantization
  9. Finite-degree Polynomial Operators and Delay Elements
  10. Memoryless Finite-degree Polynomial Operators
  11. Low-Degree Polynomials without Delay Elements
  12. Envelope Detectors without Delay elements
  13. Simulation Results
  14. Conclusions
  15. Proof of Theorem \ref{['th:1']}
  16. ...and 1 more sections

Key Result

Theorem 1

Consider a task-based quantization setup parametrized by $(n,m,P_{S^n}, P_{X^m|S^n}, (\mathcal{F}_{a,j})_{j\in \mathbb{N}},n_q,\kappa,d(\cdot,\cdot))$ as described in the prequel. Assume that there exists $\mathbf{s}\in \mathbb{R}^{m}$ such that $\mathbb{E}(d(S^{m},\mathbf{s}))\leq \infty$. The mini where $P_{S^m,X^n,\widehat{S}^m}\triangleq P_{S^m,X^n}P_{\widehat{S}^m|X^n}$, i.e., the Markov chai

Figures (2)

  • Figure 1: The task-based quantization setup.
  • Figure 2: Comparison of MSE distortion for linear and nonlinear analog processing with and without analog delay elements for a jointly Gaussian scalar task $S$ and measurement $X$, with $n_q=\kappa=2$.

Theorems & Definitions (5)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Definition 1: Fully-Symmetric Vector
  • Theorem 4