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 $R_\mathrm{h}$ 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 low SNR values. The upper bound relies on a recently proposed channel-coding scheme that effectively conveys $R_\mathrm{h}$ bits essentially error-free and the rest of the rate - over the same AWGN channel without help, with the error-free bits 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 capacity of the same scenario when the helper knows only the noise but not the message.

翻译：本文考虑在加性白高斯噪声（AWGN）信道上传输参数值的问题，其中除发射机和接收机外，还存在一个辅助器，该辅助器非因果地观测噪声，并以有限速率 $R_\mathrm{h}$ 向发射机和/或接收机提供描述。我们推导了最优可达估计误差 $\alpha$ 阶矩的上界和下界，并证明在 $\alpha$ 值较小和信噪比较低时，这些界相重合。上界基于一种最近提出的信道编码方案，该方案以基本无错误的方式有效传输 $R_\mathrm{h}$ 比特，而其余速率则在同一无辅助的 AWGN 信道上传输，其中无错误比特被分配给量化参数的最高有效位。随后，我们专注于具有总发射能量约束的场景，并针对若干情形推导了信道编码和参数调制方面的可达性结果：辅助器仅协助发射机或仅协助接收机且已知噪声时，以及辅助器协助发射机和/或接收机且同时已知噪声和消息时。特别地，对于协助接收机和发射机的消息知情辅助器，我们证明信道编码任务中的错误概率呈双指数衰减。最后，我们将这些结果转化为连续时间、功率受限且带宽无约束的 AWGN 信道的情形。作为副产品，我们证明仅存在于发射机端的消息知情辅助器的容量，可超过该辅助器仅知噪声而不知消息的相同场景的容量。