By Claude E. Shannon
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Extra info for A Mathematical Theory of Communication
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.
A Mathematical Theory of Communication by Claude E. Shannon