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By Claude E. Shannon

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

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Consider a probability measure space whose elements are ordered pairs x y. The variables x, y are to be identified as the possible transmitted and received signals of some long duration T . Let us call the set of all points whose x belongs to a subset S1 of x points the strip over S1 , and similarly the set whose y belong to S2 the strip over S2 . We divide x and y into a collection of non-overlapping measurable subsets Xi and Yi approximate to the rate of transmission R by ; R1 = 1 T ; ∑ PXi ; Yi  log PXi PYi  PXi Yi  i where PXi  is the probability measure of the strip over Xi PYi  is the probability measure of the strip over Yi PXi Yi  is the probability measure of the intersection of the strips ; : A further subdivision can never decrease R1 .

23. ” Suppose the first ensemble has the probability density function px1 xn  and the second qx1 xn . Then the ;:::; 40 ;:::; density function for the sum is given by the convolution: Z rx1 ;:::; xn =  Z py1 ;:::; yn qx1 , y1;:::; xn , yn dy1  dyn: Physically this corresponds to adding the noises or signals represented by the original ensembles of functions. The following result is derived in Appendix 6. Theorem 15: Let the average power of two ensembles be N1 and N2 and let their entropy powers be N 1 and N 2 .

To a considerable extent the continuous case can be obtained through a limiting process from the discrete case by dividing the continuum of messages and signals into a large but finite number of small regions and calculating the various parameters involved on a discrete basis. As the size of the regions is decreased these parameters in general approach as limits the proper values for the continuous case. There are, however, a few new effects that appear and also a general change of emphasis in the direction of specialization of the general results to particular cases.

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A Mathematical Theory of Communication by Claude E. Shannon

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