By Claude E. Shannon
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Prof. Helmut Koch ist Mathematiker an der Humboldt Universität Berlin.
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Extra resources for A Mathematical Theory of Communication
This becomes in the limit exp 1 W Z W log jY f j2 d f : Since J is constant its average value is the same quantity and applying the theorem on the change of entropy with a change of coordinates, the result follows. We may also phrase it in terms of the entropy power. Thus if the entropy power of the first ensemble is N1 that of the second is N1 exp 1 W Z W log jY f j2 d f 39 : TABLE I ENTROPY ENTROPY POWER POWER GAIN FACTOR IN DECIBELS GAIN IMPULSE RESPONSE 1 1,! 2 4 ! 3 ! 2 2 ,2 67 2 e !
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 .
Thus we are covering the space to within apart from a set of small measure . ∞ T log ;; ;; : This is a generalization of the measure type definitions of dimension in topology, and agrees with the intuitive dimension rate for simple ensembles where the desired result is obvious.