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On Continuous-Time Gaussian Channels

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
HKU Shenzhen Institute of Research and Innovation, Shenzhen 518057, China
Author to whom correspondence should be addressed.
Results in this paper have been partially presented in the Proceedings of the 2014 IEEE International Symposium on Information Theory, Honolulu, HI, USA, 29 June–4 July 2014.
Entropy 2019, 21(1), 67;
Received: 8 December 2018 / Revised: 1 January 2019 / Accepted: 12 January 2019 / Published: 14 January 2019
(This article belongs to the Special Issue Multiuser Information Theory II)
PDF [569 KB, uploaded 24 January 2019]


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. View Full-Text
Keywords: continuous-time channel; Gaussian channel; feedback; memory; sampling theorem; network information theory; capacity; capacity region; mutual information continuous-time channel; Gaussian channel; feedback; memory; sampling theorem; network information theory; capacity; capacity region; mutual information
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited (CC BY 4.0).

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Liu, X.; Han, G. On Continuous-Time Gaussian Channels. Entropy 2019, 21, 67.

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