Modulation and Estimation with a Helper

TL;DR

This paper investigates how a helper with non-causal noise knowledge can improve parameter transmission over AWGN channels by deriving bounds, proposing coding schemes, and analyzing scenarios with different helper capabilities and knowledge.

Contribution

It introduces bounds and coding schemes for parameter estimation with a helper, including novel results for helpers with message knowledge and noise awareness.

Findings

Bounds on estimation error moments are tight for small lpha and high helper rates.

A coding scheme effectively conveys helper information, improving estimation accuracy.

Error probability decays doubly exponentially with message-informed helpers.

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 $\alpha$-th moment of the estimation error and show that they coincide for small values of $\alpha$ 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.