By Claude E. Shannon
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Extra resources for A Mathematical Theory of Communication
When a message x is originated it will (with probability approaching 1 as T ! ∞) lie within at least one of the fans. The corresponding binary number is transmitted (or one of them chosen arbitrarily if there are several) over the channel by suitable coding means to give a small probability of error. Since R1 C this is possible. At the receiving point the corresponding y is reconstructed and used as the recovered message. The evaluation v01 for this system can be made arbitrarily close to v1 by taking T sufficiently large.
Some examples are: 1. The set of functions: : f t = sint + Each particular value of determines a particular function in the set. 2. The set of all functions of time containing no frequencies over W cycles per second. 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 .
The entropy for a continuous stochastic process has many properties analogous to that for discrete processes. In the discrete case the entropy was related to the logarithm of the probability of long sequences, and to the number of reasonably probable sequences of long length. In the continuous case it is related in a similar fashion to the logarithm of the probability density for a long series of samples, and the volume of reasonably high probability in the function space. More precisely, if we assume px1 xn continuous in all the xi for all n, then for sufficiently large n ;:::; log p , H0 n 38 ;:::; for all choices of x1 xn apart from a set whose total probability is less than , with and arbitrarily small.
A Mathematical Theory of Communication by Claude E. Shannon