By Claude E. Shannon

*The Mathematical idea 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 ebook shape presently thereafter, it has on account that passed through 4 hardcover and 16 paperback printings. it's a progressive paintings, striking 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. The set of all functions limited in band to W and in amplitude to A. 4. The set of all English speech signals as functions of time. 1 For example with the set, f t = sint + ; we may give a probability distribution for , P . The set then becomes an ensemble. Some further examples of ensembles of functions are: ; ;:::; n) with the probability of fk being pk . 1. A finite set of functions fk t (k = 1 2 2. t + i i=1 with the amplitudes ai distributed normally and independently, and the phases (from 0 to 2 ) and independently.

The probability of a certain subset of the g t functions is equal to that of the subset of the f t functions which produce members of the given subset of g functions under the operation T . Physically this corresponds to passing the ensemble through some device, for example, a filter, a rectifier or a modulator. The output functions of the device form the ensemble g t . , if g t = T f t implies g t + t1 = T f t + t1 for all f t and all t1 . It is easily shown (see Appendix 5 that if T is invariant and the input ensemble is stationary then the output ensemble is stationary.

E NTROPY L OSS IN L INEAR F ILTERS Theorem 14: If an ensemble having an entropy H1 per degree of freedom in band W is passed through a filter with characteristic Y f the output ensemble has an entropy H2 = H1 + 1 W Z W log jY f j2 d f : The operation of the filter is essentially a linear transformation of coordinates. If we think of the different frequency components as the original coordinate system, the new frequency components are merely the old ones multiplied by factors. The coordinate transformation matrix is thus essentially diagonalized in terms of these coordinates.