By Jürgen Bierbrauer
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Extra resources for Information Theory / Data Compression [Lecture notes]
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4 Theorem. The BSC with error probability p has capacity 1 − h(p). It is reached when the input letters are equally probable. We can also understand the idea behind the definition of the channel capacity. Concentrate on the BSC. The AEP (see the end of Chapter 1 says that, for large n, the typical signals of length n emitted by a source of entropy H(X) can be thought of as a uniform space of size 2nH(X) . We saw that in the case of the BSC all conditional entropies H(Y |a) are = h(p). Each typical input signal determines 2nh(p) typical output signals.
And the edges (each edge connects two vertices) are indexed by letters. Starting point is always the root of the tree, which is numbered 0. The first text letter a leads to an edge starting at the root with endpoint 1. The second letter a of the text illustrates the principle: starting 47 from the root it is checked what is the longest section of text, starting from our second letter, which can be read following the edges. In our the a is found and that is all. The part of the compressed text we obtain so far is: (0, a) (1, b).
Then i ≤ 27 and ai /100 is the approximation to qi that the experiment produces. This is what Shannon did. In order to get an idea what F15 may be it needs to be bounded from above and below by expressions (15) involving the probabilities qi . 4 Theorem. i≥2 i · (qi (n) (n) − qi+1 )log(i) ≤ Fn ≤ H(qi ) Consider first the upper bound. Both expressions can be written as expectations (sums over all σ with weights p(σ)). It suffices to show the inequality for the second factor, for given σ. We have to show the following: − S p(S|σ)log(p(S|σ)) ≤ − (n) p(j(i)|σ)log(qi ).