On Sampling Continuous-Time Gaussian Channels
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For a continuous-time Gaussian channel, it has been shown that as sampling gets infinitesimally fine, the mutual information of the corresponding discrete-time counterparts converges to that of the original continuous-time channel. We give in this paper more quantitative strengthenings of this result, which, among other implications, characterize how over-sampling approaches the true mutual information of a continuous-time Gaussian channel with bandwidth limit. Compared to the Shannon-Nyquist sampling theorem, a widely used tool to connect continuous-time Gaussian channels to their discrete-time counterparts that requires the band-limitedness of the channel input, our results only require some integrability conditions on the power spectral density function of the input.
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