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By Claude E. Shannon

Scientific wisdom grows at a wonderful pace--but few books have had as lasting an impression or performed as very important a job in our smooth international as The Mathematical conception of Communication, released initially as a paper on conversation conception within the Bell method Technical Journal greater than fifty years in the past. Republished in ebook shape almost immediately thereafter, it has for the reason that passed through 4 hardcover and 16 paperback printings. it's a innovative 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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In this case all but 2TW of the coordinates will be zero. Thus functions limited to a band W and duration T correspond to points in a space of 2TW dimensions. A subset of the functions of band W and duration T corresponds to a region in this space. For example, the functions whose p total energy is less than or equal to E correspond to points in a 2TW dimensional sphere with radius r = 2W E. An ensemble of functions of limited duration and band will be represented by a probability distribution px1 xn  in the corresponding n dimensional space.

A particular case of changing coordinates is the linear transformation y j = ∑ ai j xi i : In this case the Jacobian is simply the determinant jai j j,1 and : H y = H x + log jai j j In the case of a rotation of coordinates (or any measure preserving transformation) J = 1 and H y = H x. 21. E NTROPY OF AN E NSEMBLE OF F UNCTIONS Consider an ergodic ensemble of functions limited to a certain band of width W cycles per second. Let ;:::; xn  be the density distribution function for amplitudes x1 ;:::; xn at n successive sample points.

51 ACKNOWLEDGMENTS The writer is indebted to his colleagues at the Laboratories, particularly to Dr. H. W. Bode, Dr. J. R. Pierce, Dr. B. McMillan, and Dr. B. M. Oliver for many helpful suggestions and criticisms during the course of this work. Credit should also be given to Professor N. Wiener, whose elegant solution of the problems of filtering and prediction of stationary ensembles has considerably influenced the writer’s thinking in this field. APPENDIX 5 Let S1 be any measurable subset of the g ensemble, and S2 the subset of the f ensemble which gives S1 under the operation T .

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