By Claude E. Shannon
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Extra info for A Mathematical Theory of Communication
The region ofp high probability is a sphere of radius nN. As n ! ∞ the probability of being outside a sphere of radius nN + approaches zero and 1n times the logarithm of the p volume of the sphere approaches log 2 eN. In the continuous case it is convenient to work not with the entropy H of an ensemble but with a derived quantity which we will call the entropy power. This is defined as the power in a white noise limited to the same band as the original ensemble and having the same entropy. In other words if H 0 is the entropy of an ensemble its entropy power is 1 N1 = exp2H 0 2 e In the geometrical picture this amounts to measuring the high probability volume by the squared radius of a sphere having the same volume.
There are, however, a few new effects that appear and also a general change of emphasis in the direction of specialization of the general results to particular cases. We will not attempt, in the continuous case, to obtain our results with the greatest generality, or with the extreme rigor of pure mathematics, since this would involve a great deal of abstract measure theory and would obscure the main thread of the analysis. A preliminary study, however, indicates that the theory can be formulated in a completely axiomatic and rigorous manner which includes both the continuous and discrete cases and many others.
The S most we have obtained for this case is a lower bound valid for all , an “asymptotic” upper bound (valid N S S for large ) and an asymptotic value of C for small. N N 45 Theorem 20: The channel capacity C for a band W perturbed by white thermal noise of power N is bounded by 2 S C W log 3 e N S where S is the peak allowed transmitter power. For sufficiently large N ; 2 S+N C W log e 1 + N where is arbitrarily small. As S N ! 0 (and provided the band W starts at 0) . S C W log 1 + N !