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By Giuseppe Longo

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Extra info for Source Coding Theory: Lectures Held at the Department for Automation and Information June 1970

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The adjective "suitable" is now given a rather precise meaning, since the typical sequences are sufficient to represent the source adequately, up to the error probability P8 • Conversely we want now to prove that the typical sequences are also necessary to describe the source output. In other words, if the ratio N / L is kept at some fixed value below 1-11 to~ 01. e does not approach zero as L tends to the Shannon Theorem on Source Coding 48 infinity (this is the so-called weak converse to the source coding theorem) • Actually P e tends to one as L goes to the infinity (this is the so-called strong converse to the source coding theorem).

So, since we can let a sufficiently small Pe enter our encoding-decoding scheme, block codes seem very promising. Now we look closer at block codes, illu~ trating a simple encoding scheme which will be very useful also in subsequent developments. If we give up the 39 Ambiguous set and error probability idea of having an invertible ~ function and provide only W ( W< k L) distinct codewords, then there will be codewords corning {a priori) from more than one sequence, which makes decoding ambiguous and introduces a nonzero 'Pe .

22) Proof. I. •. 2(x1) ... p. 24) E \") , we get : (E\")/ Prob h 1 ) ~ 2 n(I(,u. ). (E \") / h,). : 1 -11- <> > E~n) c. 25) Taking logarithms on both sides of eq. 20) we get the desired result expressed by eq. 21) in force of the arbitrariness of t. In the same way one can prove the follow- ing Theorem 7. 2. >! '( 0 ( 0 < "(0 < 1 ) of "(, the is described by the follow ... 27) or t ~m n-oo (fi! ,IIJ1i). 28) 8. The Neyman- Pearson Lemma. In this section we shall state and prove a particular case of the well-known Neyman-Pearson le~ rna, which gives a useful hint concerning how to choose the sets fima fi ~ E~n) and and "(!

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