Liu Xianming, Han Guangyue
School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan 430074, China.
Department of Mathematics, The University of Hong Kong, Hong Kong, China.
Entropy (Basel). 2019 Jan 14;21(1):67. doi: 10.3390/e21010067.
A continuous-time white Gaussian channel can be formulated using a white Gaussian noise, and a conventional way for examining such a channel is the sampling approach based on the Shannon-Nyquist sampling theorem, where the original continuous-time channel is converted to an equivalent discrete-time channel, to which a great variety of established tools and methodology can be applied. However, one of the key issues of this scheme is that continuous-time feedback and memory cannot be incorporated into the channel model. It turns out that this issue can be circumvented by considering the Brownian motion formulation of a continuous-time white Gaussian channel. Nevertheless, as opposed to the white Gaussian noise formulation, a link that establishes the information-theoretic connection between a continuous-time channel under the Brownian motion formulation and its discrete-time counterparts has long been missing. This paper is to fill this gap by establishing causality-preserving connections between continuous-time Gaussian feedback/memory channels and their associated discrete-time versions in the forms of sampling and approximation theorems, which we believe will play important roles in the long run for further developing continuous-time information theory. As an immediate application of the approximation theorem, we propose the so-called approximation approach to examine continuous-time white Gaussian channels in the point-to-point or multi-user setting. It turns out that the approximation approach, complemented by relevant tools from stochastic calculus, can enhance our understanding of continuous-time Gaussian channels in terms of giving alternative and strengthened interpretation to some long-held folklore, recovering "long-known" results from new perspectives, and rigorously establishing new results predicted by the intuition that the approximation approach carries. More specifically, using the approximation approach complemented by relevant tools from stochastic calculus, we first derive the capacity regions of continuous-time white Gaussian multiple access channels and broadcast channels, and we then analyze how feedback affects their capacity regions: feedback will increase the capacity regions of some continuous-time white Gaussian broadcast channels and interference channels, while it will not increase capacity regions of continuous-time white Gaussian multiple access channels.
连续时间高斯白信道可以用高斯白噪声来构建，而研究此类信道的传统方法是基于香农 - 奈奎斯特采样定理的采样方法，即将原始的连续时间信道转换为等效的离散时间信道，这样就可以应用各种已有的工具和方法。然而，该方案的一个关键问题是连续时间反馈和记忆无法纳入信道模型。事实证明，通过考虑连续时间高斯白信道的布朗运动公式可以规避这个问题。尽管如此，与高斯白噪声公式不同，长期以来一直缺少在布朗运动公式下的连续时间信道与其离散时间对应信道之间建立信息理论联系的纽带。本文旨在通过以采样定理和近似定理的形式在连续时间高斯反馈/记忆信道及其相关离散时间版本之间建立因果保持联系来填补这一空白，我们相信从长远来看，这将对进一步发展连续时间信息理论发挥重要作用。作为近似定理的直接应用，我们提出了所谓的近似方法来研究点对点或多用户设置下的连续时间高斯白信道。事实证明，由随机微积分的相关工具补充的近似方法，可以增强我们对连续时间高斯信道的理解，具体表现为对一些长期存在的经验法则给出替代且强化的解释，从新的角度恢复“早已熟知”的结果，以及严格建立近似方法所蕴含直觉预测的新结果。更具体地说，使用由随机微积分的相关工具补充的近似方法，我们首先推导连续时间高斯白多址接入信道和广播信道的容量区域，然后分析反馈如何影响它们的容量区域：反馈会增加一些连续时间高斯白广播信道和干扰信道的容量区域，而不会增加连续时间高斯白多址接入信道的容量区域。
