A Mathematical Theory of Communication by Claude E. Shannon

By Claude E. Shannon

Scientific wisdom grows at a beautiful pace--but few books have had as lasting an influence or performed as very important a task in our smooth international as The Mathematical thought of Communication, released initially as a paper on conversation thought within the Bell process Technical Journal greater than fifty years in the past. Republished in ebook shape almost immediately thereafter, it has on the grounds that passed through 4 hardcover and 16 paperback printings. it's a progressive paintings, incredible in its foresight and contemporaneity. The college of Illinois Press is happy and commemorated to factor this commemorative reprinting of a classic.

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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.

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