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Few-Shot Channel-Agnostic Analog Coding: A Near-Optimal Scheme

Mohammad Ali Maddah-Ali, Soheil Mohajer

TL;DR

The work addresses few-shot lossy joint source-channel coding for a scalar analog source over an AWGN channel with extremely small $N$. It introduces a channel-agnostic coding scheme based on a novel $S$-progressive expansion of the source, followed by partitioning digits into $N$ sets and applying reverse progressive expansion with shielding to suppress carry-over of noise. Theoretical results establish an achievable distortion scaling $D = c_1 \mathsf{SNR}^{-N} (\log \mathsf{SNR})^{10N} + c_2 \mathsf{SNR}^{-N} + c_3$ and show the corresponding SDR obeys $N\log(\mathsf{SNR})-10N\log\log(\mathsf{SNR})+o(\log\log(\mathsf{SNR})) \le \log(\mathsf{SDR}^*) \le N\log(1+\mathsf{SNR})$. The scheme is channel-agnostic, near-optimal across all $\mathsf{SNR}$, and highlights progress toward robust, delay-sensitive analog coding with minimal channel knowledge at the transmitter.

Abstract

In this paper, we investigate the problem of transmitting an analog source to a destination over $N$ uses of an additive-white-Gaussian-noise (AWGN) channel, where $N$ is very small (in the order of 10 or even less). The proposed coding scheme is based on representing the source symbol using a novel progressive expansion technique, partitioning the digits of expansion into $N$ ordered sets, and finally mapping the symbols in each set to a real number by applying the reverse progressive expansion. In the last step, we introduce some gaps between the signal levels to prevent the carry-over of the additive noise from propagation to other levels. This shields the most significant levels of the signal from an additive noise, hitting the signal at a less significant level. The parameters of the progressive expansion and the shielding procedure are opportunistically independent of the $\SNR$ so that the proposed scheme achieves a distortion $D$, where $-\log(D)$ is within $O(\log\log(\SNR))$ of the optimal performance for all values of $\SNR$, leading to a channel-agnostic scheme.

Few-Shot Channel-Agnostic Analog Coding: A Near-Optimal Scheme

TL;DR

The work addresses few-shot lossy joint source-channel coding for a scalar analog source over an AWGN channel with extremely small . It introduces a channel-agnostic coding scheme based on a novel -progressive expansion of the source, followed by partitioning digits into sets and applying reverse progressive expansion with shielding to suppress carry-over of noise. Theoretical results establish an achievable distortion scaling and show the corresponding SDR obeys . The scheme is channel-agnostic, near-optimal across all , and highlights progress toward robust, delay-sensitive analog coding with minimal channel knowledge at the transmitter.

Abstract

In this paper, we investigate the problem of transmitting an analog source to a destination over uses of an additive-white-Gaussian-noise (AWGN) channel, where is very small (in the order of 10 or even less). The proposed coding scheme is based on representing the source symbol using a novel progressive expansion technique, partitioning the digits of expansion into ordered sets, and finally mapping the symbols in each set to a real number by applying the reverse progressive expansion. In the last step, we introduce some gaps between the signal levels to prevent the carry-over of the additive noise from propagation to other levels. This shields the most significant levels of the signal from an additive noise, hitting the signal at a less significant level. The parameters of the progressive expansion and the shielding procedure are opportunistically independent of the so that the proposed scheme achieves a distortion , where is within of the optimal performance for all values of , leading to a channel-agnostic scheme.
Paper Structure (8 sections, 11 theorems, 71 equations, 3 figures)

This paper contains 8 sections, 11 theorems, 71 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. The Fundamental Limits
  4. The Main Results
  5. An Achievable Scheme for $M=1$
  6. Preliminaries: A Progressive Expansion
  7. An Achievable Scheme
  8. Performance Analysis

Key Result

Theorem 1

For $M=1$ source symbol distributed as ${U\sim \mathsf{Unif}([-1/2,1/2])}$, the average distortion is achievable for some constants $(c_1,c_2)$ that do not depend on $\mathsf{SNR}$ or $D$, (but may depend on $N$), and $c_3$ is order-wise smaller than $1/\mathsf{SNR}^N$.

Figures (3)

  • Figure 1: An example of progressive expansion with $S=2$.
  • Figure 2: The cumulative distribution function of $\tilde{X}_{}(n)$ generated from uniformly distributed $U$.
  • Figure 3: The block diagram of the encoder and decoder. The $\mathsf{PE}(S)$ blocks represent the $S$-progressive expansion, while $\mathsf{PE}^{-1}(S)$ shows its inverse (progressive expansion to decimal conversion).

Theorems & Definitions (24)

  • Remark 1
  • Theorem 1
  • Corollary 1
  • Remark 2: Fixed-Based vs. progressive expansion
  • Definition 1
  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • Lemma 5
  • ...and 14 more