BSTJ : A Mathematical Theory of Communication (Shannon, C.E.) by Unknown

Author:Unknown
Language: eng
Format: epub
Published: 1948-10-03T16:00:00+00:00

Tl

which requires A i, = - - C,/.

as

TT

In this case /I,-y = ^r ^<i ^^id both equations reduce to identities.

APPENDIX 7

The following will indicate a more general and more rigorous approach to the central definitions of communication theory. 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 5i of X points the strip over .^i, and similarly the set whose y belongs to ^2 the strip over 52. We divide x and y into a collection of non-overlapping measurable subsets Xi and Y; approximate to the rate of transmission R by

where

P(Xi) is the probability measure of the strip over Xi P(Yi) is the probability measure of the strip over I",-P{Xi, I.) is the probability measure of the intersection of the strips.

A further subdivision can never decrease Ri. For let Xi be divided into Xi = Xl + X[' and let

PiW) = a P(Xi) = b + c

P{X[) ^-b P{X[, ]\) - d

P{X") = c P{X'l, I",) - e

PiX,,\\) = d+e - ■■ -

Then in the sum we have replaced (for the A'l, I'l intersection)

((/ + e) log ———- by rf log — + e log — . a{o + c) ao ac

It is easily shown that with the limitation we have on b, c, d, e,

d + e

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