By Claude E. Shannon
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Additional resources for A Mathematical Theory of Communication
The rate of transmission in bits per second is calculated for each one and we choose that having the least rate. This latter rate is the rate we assign the source for the fidelity in question. The justification of this definition lies in the following result: Theorem 21: If a source has a rate R1 for a valuation v1 it is possible to encode the output of the source and transmit it over a channel of capacity C with fidelity as near v1 as desired provided R1 C. This is not possible if R1 C. The last statement in the theorem follows immediately from the definition of R1 and previous results.
These bounds are sufficiently close together in most practical cases to furnish a satisfactory solution to the problem. Theorem 18: The capacity of a channel of band W perturbed by an arbitrary noise is bounded by the inequalities P + N1 W log C W log P N+ N N1 1 where P = average transmitter power N = average noise power N1 = entropy power of the noise. Here again the average power of the perturbed signals will be P + N. The maximum entropy for this power would occur if the received signal were white noise and would be W log 2 eP + N .
LogM T T =W log P+N N ; so that no matter how small is chosen, we can, by taking T sufficiently large, transmit as near as we wish P+N binary digits in the time T . to TW log N P+N Formulas similar to C = W log for the white noise case have been developed independently N by several other writers, although with somewhat different interpretations. We may mention the work of N. Wiener,7 W. G. Tuller,8 and H. Sullivan in this connection. In the case of an arbitrary perturbing noise (not necessarily white thermal noise) it does not appear that the maximizing problem involved in determining the channel capacity C can be solved explicitly.