Information-Theoretic Extensions of the Shannon-Nyquist Sampling Theorem

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 classical 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 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 white Gaussian channel under the Brownian motion formulation and its discrete-time counterparts has long been missing. This paper is to fill this gap by establishing information-theoretic extensions of the Shannon-Nyquist theorem, which naturally yield causality-preserving connections between continuous-time Gaussian feedback channels and their associated discrete-time versions in the forms of sampling and approximation theorems. As an example of the possible applications of the extensions, we use the above-mentioned connections to analyze the capacity of a continuous-time white Gaussian feedback channel.