Download A Mathematical Theory of Communication by Claude E. Shannon PDF

By Claude E. Shannon

Scientific wisdom grows at a stupendous pace--but few books have had as lasting an effect or performed as very important a job in our smooth international as The Mathematical thought of Communication, released initially as a paper on communique thought within the Bell procedure Technical Journal greater than fifty years in the past. Republished in publication shape presently thereafter, it has for the reason that passed through 4 hardcover and 16 paperback printings. it's a innovative paintings, surprising 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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Therefore   R = W1 log2 eQ , W1 log 2 eN Q = W1 log N where Q is the average message power. S. measure of fidelity is Q R = W1 log N where N is the allowed mean square error between original and recovered messages. More generally with any message source we can obtain inequalities bounding the rate relative to a mean square error criterion. Theorem 23: The rate for any source of band W1 is bounded by W1 log Q1 N  R  W1 log QN where Q is the average power of the source, Q1 its entropy power and N the allowed mean square error.

This means that a criterion of fidelity can be represented by a numerically valued function: ; ; ; , ; v Px y ;  ; whose argument ranges over possible probability functions Px y. ,  We will now show that under very general and reasonable assumptions the function v Px y can be written in a seemingly much more specialized form, namely as an average of a function x y over the set of possible values of x and y: ZZ ; , ; v Px y  ; = ; ; : Px y x y dx dy To obtain this we need only assume (1) that the source and system are ergodic so that a very long sample will be, with probability nearly 1, typical of the ensemble, and (2) that the evaluation is “reasonable” in the sense that it is possible, by observing a typical input and output x1 and y1 , to form a tentative evaluation on the basis of these samples; and if these samples are increased in duration the tentative evaluation will, with probability 1, approach the exact evaluation based on a full knowledge of Px y.

If we change coordinates the entropy will in general change. In fact if we change to coordinates y1  yn the new entropy is given by Z H y =  Z px1 ;:::; xn J x y log px1 ;:::; xn J x y dy1  dyn ,  where J xy is the Jacobian of the coordinate transformation. On expanding the logarithm and changing the variables to x1  xn , we obtain: H y = H x , Z  Z px1 37 ;:::; xn  logJ x y dx1 ::: dxn : Thus the new entropy is the old entropy less the expected logarithm of the Jacobian.

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