By Claude E. Shannon
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Therefore R = W1 log2 eQ , W1 log 2 eN Q = W1 log N where Q is the average message power. S. measure of fidelity is Q R = W1 log N where N is the allowed mean square error between original and recovered messages. More generally with any message source we can obtain inequalities bounding the rate relative to a mean square error criterion. Theorem 23: The rate for any source of band W1 is bounded by W1 log Q1 N R W1 log QN where Q is the average power of the source, Q1 its entropy power and N the allowed mean square error.
This means that a criterion of fidelity can be represented by a numerically valued function: ; ; ; , ; v Px y ; ; whose argument ranges over possible probability functions Px y. , We will now show that under very general and reasonable assumptions the function v Px y can be written in a seemingly much more specialized form, namely as an average of a function x y over the set of possible values of x and y: ZZ ; , ; v Px y ; = ; ; : Px y x y dx dy To obtain this we need only assume (1) that the source and system are ergodic so that a very long sample will be, with probability nearly 1, typical of the ensemble, and (2) that the evaluation is “reasonable” in the sense that it is possible, by observing a typical input and output x1 and y1 , to form a tentative evaluation on the basis of these samples; and if these samples are increased in duration the tentative evaluation will, with probability 1, approach the exact evaluation based on a full knowledge of Px y.
If we change coordinates the entropy will in general change. In fact if we change to coordinates y1 yn the new entropy is given by Z H y = Z px1 ;:::; xn J x y log px1 ;:::; xn J x y dy1 dyn , where J xy is the Jacobian of the coordinate transformation. On expanding the logarithm and changing the variables to x1 xn , we obtain: H y = H x , Z Z px1 37 ;:::; xn logJ x y dx1 ::: dxn : Thus the new entropy is the old entropy less the expected logarithm of the Jacobian.