Noisy-channel coding theorem: Difference between revisions

Revision as of 09:22, 27 June 2006

Revision as of 12:34, 13 July 2006

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:::<math>2^{-n(H(Y) + \epsilon)} \le p(Y_1^n) \le 2^{-n(H(Y)-\epsilon)}</math>

:::<math>{2^{-n(H(X,Y) + \epsilon)} }\le p(X_1^n, Y_1^n) \le 2^{-n(H(X,Y) -\epsilon)} \}</math>

We say that two sequences <math>{X_1^n}</math> andvà <math>Y_1^n</math> are ''jointly typical'' if they lie in the jointly typical set defined above.

'''Steps'''

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*From the joint AEP, we know that the probability that no jointly typical X exists goes to 0 as n grows large. We can bound this error probability by <math>\epsilon</math>.

*Also from the joint AEP, we know the probability that a particular <math>X_1^{n(i)}</math> and the <math>Y_1^n</math> resulting from W = 1 are jointly typical is <math>\le 2^{-n(I(X;Y) - 3\epsilon)}</math>.

Define: <math>E_i = \{(X_1^n(i), Y_1^n) \in A_\epsilon^{(n)}\}, i = 1, 2, ..., 2^{nR}</math>