By Claude E. Shannon
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Extra resources for A Mathematical Theory of Communication
M. Oliver for many helpful suggestions and criticisms during the course of this work. Credit should also be given to Professor N. Wiener, whose elegant solution of the problems of filtering and prediction of stationary ensembles has considerably influenced the writer’s thinking in this field. APPENDIX 5 Let S1 be any measurable subset of the g ensemble, and S2 the subset of the f ensemble which gives S1 under the operation T . Then S1 = T S2 Let H be the operator which shifts all functions in a set by the time .
The form of px giving a maximum entropy subject to the condition that the standard deviation of x be fixed at is Gaussian. To show this we must maximize H x = , Z px log px dx Z with 2 Z pxx2 dx and 1 = = px dx as constraints. This requires, by the calculus of variations, maximizing Z , px log px + pxx2 + px dx: The condition for this is ,1 , log px + x2 + = 0 and consequently (adjusting the constants to satisfy the constraints) px = p1 2 e,x 2 =2 2 : Similarly in n dimensions, suppose the second order moments of px1 Z Ai j = Z xi x j px1 ;:::; xn are fixed at Ai j : ;:::; xn dx1 dxn : Then the maximum entropy occurs (by a similar calculation) when px1 Gaussian distribution with the second order moments Ai j .
This means that to transmit the output of a continuous source with exact recovery at the receiving point requires, 47 in general, a channel of infinite capacity (in bits per second). Since, ordinarily, channels have a certain amount of noise, and therefore a finite capacity, exact transmission is impossible. This, however, evades the real issue. Practically, we are not interested in exact transmission when we have a continuous source, but only in transmission to within a certain tolerance. The question is, can we assign a definite rate to a continuous source when we require only a certain fidelity of recovery, measured in a suitable way.