By Claude E. Shannon
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Extra resources for A Mathematical Theory of Communication
3 This is the famous ergodic theorem or rather one aspect of this theorem which was proved in somewhat different formulations by Birkoff, von Neumann, and Koopman, and subsequently generalized by Wiener, Hopf, Hurewicz and others. , E. Hopf, “Ergodentheorie,” Ergebnisse der Mathematik und ihrer Grenzgebiete, v. 5; “On Causality Statistics and Probability,” Journal of Mathematics and Physics, v. XIII, No. 1, 1934; N. Wiener, “The Ergodic Theorem,” Duke Mathematical Journal, v. 5, 1939. 33 Just as we may perform various operations on numbers or functions to obtain new numbers or functions, we can perform operations on ensembles to obtain new ensembles.
In this case the convolution of the noise and signal produce approximately a Gaussian distribution whose corresponding quadratic form is N + bii : The entropy power of this distribution is i1=n h ∏N + bii or approximately h n = N + : ∑ bii N n,1 =N+ 1 bii n∑ i1=n : The last term is the signal power, while the first is the noise power. PART IV: THE CONTINUOUS CHANNEL 24. T HE C APACITY OF A C ONTINUOUS C HANNEL In a continuous channel the input or transmitted signals will be continuous functions of time f t belonging to a certain set, and the output or received signals will be perturbed versions of these.
We are given Px for the source and an evaluation v determined by a distance function x y which will be assumed continuous in both x and y. With a particular system Px y the quality is measured by ; ; ZZ ; yPx; y dx dy: Furthermore the rate of flow of binary digits corresponding to Px; y is ZZ Px; y R= Px; y log dx dy: v= x PxPy We define the rate R1 of generating information for a given quality v1 of reproduction to be the minimum of R when we keep v fixed at v1 and vary Px y.