By Claude E. Shannon
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On the time of Professor Rademacher's loss of life early in 1969, there has been to be had a whole manuscript of the current paintings. The editors had simply to provide a couple of bibliographical references and to right a number of misprints and error. No great alterations have been made within the manu script other than in a single or locations the place references to extra fabric seemed; due to the fact that this fabric used to be no longer present in Rademacher's papers, those references have been deleted.
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Extra info for A Mathematical Theory of Communication
We have, therefore C = Max H y , H n W log 2 eP + N , W log 2 eN1 : This is the upper limit given in the theorem. The lower limit can be obtained by considering the rate if we make the transmitted signal a white noise, of power P. In this case the entropy power of the received signal must be at least as great as that of a white noise of power P + N1 since we have shown in in a previous theorem that the entropy power of the sum of two ensembles is greater than or equal to the sum of the individual entropy powers.
An average power limitation) of the form K = Px y x y dx dy. A partial solution of the general maximizing problem for determining the rate of a source can be given. Using Lagrange’s method we consider ; ; ZZ ; Px y log ; ; ; Px y + Px y x y + PxPy 50 xPx; y : dx dy ; The variational equation (when we take the first variation on Px y) leads to Py x = Bxe, where x;y is determined to give the required fidelity and Bx is chosen to satisfy Z Bxe, x;y dx = 1 : This shows that, with best encoding, the conditional probability of a certain cause for various received y, Py x will decline exponentially with the distance function x y between the x and y in question.
37, No. 5, May, 1949, pp. 468–78. 8 “Theoretical 44 and C W log2 eP + N1 , W log2 eN1 P + N1 = W log N1 : As P increases, the upper and lower bounds approach each other, so we have as an asymptotic rate W log : P+N N1 If the noise is itself white, N = N1 and the result reduces to the formula proved previously: C = W log 1 + P N : If the noise is Gaussian but with a spectrum which is not necessarily flat, N1 is the geometric mean of the noise power over the various frequencies in the band W .