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Modulation and Estimation with a Helper

Anatoly Khina, Neri Merhav

TL;DR

The problem of transmitting a parameter value over an additive white Gaussian noise (AWGN) channel is considered, where, in addition to the transmitter and the receiver, there is a helper that observes the noise non-causally and provides a description of limited rate to the transmitter and/or the receiver.

Abstract

The problem of transmitting a parameter value over an additive white Gaussian noise (AWGN) channel is considered, where, in addition to the transmitter and the receiver, there is a helper that observes the noise non-causally and provides a description of limited rate $R_h$ to the transmitter and/or the receiver. We derive upper and lower bounds on the optimal achievable $α$-th moment of the estimation error and show that they coincide for small values of $α$ and for high values of $R_h$. The upper bound relies on a recently proposed channel-coding scheme that effectively conveys $R_h$ bits essentially error-free and the rest of the rate - over the same AWGN channel without help, with the error-free bits being allocated to the most significant bits of the quantized parameter. We then concentrate on the setting with a total transmit energy constraint, for which we derive achievability results for both channel coding and parameter modulation for several scenarios: when the helper assists only the transmitter or only the receiver and knows the noise, and when the helper assists the transmitter and/or the receiver and knows both the noise and the message. In particular, for the message-informed helper that assists both the receiver and the transmitter, it is shown that the error probability in the channel-coding task decays doubly exponentially. Finally, we translate these results to those for continuous-time power-limited AWGN channels with unconstrained bandwidth. As a byproduct, we show that the capacity with a message-informed helper that is available only at the transmitter can exceed the sum of the capacity without help and the help rate $R_h$, when the helper knows only the noise but not the message.

Modulation and Estimation with a Helper

TL;DR

The problem of transmitting a parameter value over an additive white Gaussian noise (AWGN) channel is considered, where, in addition to the transmitter and the receiver, there is a helper that observes the noise non-causally and provides a description of limited rate to the transmitter and/or the receiver.

Abstract

The problem of transmitting a parameter value over an additive white Gaussian noise (AWGN) channel is considered, where, in addition to the transmitter and the receiver, there is a helper that observes the noise non-causally and provides a description of limited rate to the transmitter and/or the receiver. We derive upper and lower bounds on the optimal achievable -th moment of the estimation error and show that they coincide for small values of and for high values of . The upper bound relies on a recently proposed channel-coding scheme that effectively conveys bits essentially error-free and the rest of the rate - over the same AWGN channel without help, with the error-free bits being allocated to the most significant bits of the quantized parameter. We then concentrate on the setting with a total transmit energy constraint, for which we derive achievability results for both channel coding and parameter modulation for several scenarios: when the helper assists only the transmitter or only the receiver and knows the noise, and when the helper assists the transmitter and/or the receiver and knows both the noise and the message. In particular, for the message-informed helper that assists both the receiver and the transmitter, it is shown that the error probability in the channel-coding task decays doubly exponentially. Finally, we translate these results to those for continuous-time power-limited AWGN channels with unconstrained bandwidth. As a byproduct, we show that the capacity with a message-informed helper that is available only at the transmitter can exceed the sum of the capacity without help and the help rate , when the helper knows only the noise but not the message.
Paper Structure (22 sections, 28 theorems, 98 equations, 6 figures)

This paper contains 22 sections, 28 theorems, 98 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Problem Setup
  3. Background: Channel Coding over the AWGN Channel with a Helper
  4. Power-Limited Input
  5. MP$\alpha$E Exponent Upper Bounds
  6. MP$\alpha$E Exponent Lower Bounds
  7. Vector Parameter
  8. Numerical Comparison of Upper and Lower Bounds
  9. Energy-Limited Input
  10. Helper with a Fixed Rate
  11. Transmitter with a Cribbed Helper
  12. Two-Sided Cribbed Helper
  13. Side channel
  14. Assisted noisy-channel coding
  15. Hybrid
  16. ...and 7 more sections

Key Result

Theorem 3.1

The capacity of an AWGN channel with SNR $S$ and help rate $R_\mathrm{h}$ is

Figures (6)

  • Figure 1: Parameter estimation--modulation over an AWGN channel with an assisted transmitter.
  • Figure 2: Graphs of the converse and achievability bounds as functions of $S$, for $R_\mathrm{h}=1$ and $\alpha=0.3, 2$: The DPT impossibility bound \ref{['eq:DPT:exp']}, the transmitter- and receiver-assisted Ziv--Zakai based impossibility bounds of Theorems \ref{['thm:Ziv-Zakai-based-LB']} and \ref{['thm:Ziv-Zakai-based-LB:Rx-helper']}, respectively, and the achievability bound Corollary \ref{['cor:expurgated']}.
  • Figure 3: Graphs of the converse and achievability bounds as functions of $\alpha$, for $R_\mathrm{h}=1$ and $S=0.1, 10$: The DPT impossibility bound \ref{['eq:DPT:exp']}, the transmitter- and receiver-assisted Ziv--Zakai based impossibility bounds of Theorems \ref{['thm:Ziv-Zakai-based-LB']} and \ref{['thm:Ziv-Zakai-based-LB:Rx-helper']}, respectively, and the achievability bound Corollary \ref{['cor:expurgated']}.
  • Figure 4: Achievable transmitter-assisted channel-coding error exponents as a function of the rate $\mathrm{R}$ for various values of the help rate $\mathrm{R_h}$ and the capacity without assistance $\mathrm{C}_0$: $\mathrm{E}_a(\mathrm{R})$---the achievable error exponent of \ref{['eq:CT:Neri:EE']} of a helper that knows only the noise, $\mathrm{E}_C(\mathrm{R})$---the achievable of Corollary \ref{['cor:CT:power-limited-bounds']} of a helper that knows both the noise and the message. The blue vertical arrows denote an arbitrarily large error exponent $\mathrm{E}_a(\mathrm{R})$ for $\mathrm{R} < \mathrm{R_h}$.
  • Figure 5: Achievable MP$\alpha$E exponents as a function of the capacity without assistance $\mathrm{C}_0$ for help rate $\mathrm{R_h} = 1$ and $\alpha = 2, 0.3$: "Cribless: converse" and "Cribless: achievable" denote the upper and lower bounds on the MP$\alpha$E exponent of Corollary \ref{['cor:CT:power-limited-bounds']} for the scenario of a helper that knows only the noise and assists only the transmitter, "Cribbed: achiev." denotes the achievable error exponent of Corollary \ref{['cor:CT:cribbed-helper:PPM:D']} for the scenario of a cribbed helper that assists only the transmitter, and "Cribbed two-sided: achiev." denotes the achievable error exponent of Corollary \ref{['cor:CT:2sided:PPM:D']} for the scenario of a cribbed helper that assists both the transmitter and the receiver.
  • ...and 1 more figures

Theorems & Definitions (35)

  • Theorem 3.1: Lapidoth-Marti:Tx-assisted:TIT2020
  • Theorem 3.2: Merhav:Tx-assisted-EE:TIT2021
  • Theorem 3.3: Merhav:Tx-assisted-EE:TIT2021
  • Remark 3.1
  • Theorem 4.1
  • Remark 4.1: DPT-based upper bound on the MP$\alpha$E exponent
  • Theorem 4.2
  • Theorem 4.3
  • Remark 4.2
  • Corollary 4.1
  • ...and 25 more