By Claude E. Shannon
Read or Download A Mathematical Theory of Communication PDF
Similar number theory books
This e-book furthers new and fascinating advancements in experimental designs, multivariate research, biostatistics, version choice and similar topics. It good points articles contributed by way of many widespread and lively figures of their fields. those articles conceal a big selection of vital matters in smooth statistical thought, tools and their purposes.
P-adic numbers are of serious theoretical value in quantity concept, on account that they enable using the language of research to review difficulties bearing on toprime numbers and diophantine equations. extra, they give a realm the place you'll do issues which are similar to classical research, yet with effects which are particularly strange.
- Local Fields
- Lectures on p-adic L-functions
- P-adic function analysis
- Observations on a theorem of Fermat and others on looking at prime numbers
- Siegel's Modular Forms and Dirichlet Series
Additional info for A Mathematical Theory of Communication
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.