The capacity of the band-limited Gaussian channel

Shannon's celebrated formula W ln(1 + P
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/N
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W) for the capacity of a time-continuous communication channel with bandwidth W cps, average signal power P
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, and additive Gaussian noise with flat spectral density N
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has never been justified by a coding theorem (and “converse”). Such a theorem is necessary to establish W ln(1 + P
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/N
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W) as the supremum of those transmission rates at which one may communicate over this channel with arbitrarily high reliability as the coding and decoding delay becomes large. In this paper, a number of physically consistent models for this time-continuous channel are proposed. For each model the capacity is established as W ln(1 + P
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/N
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W) by means of a coding theorem and converse.