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By Giuseppe Longo

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Extra info for Coding for Markov Sources: Course Held at the Department for Automation and Information June 1971

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F1 s UH, -en>,-.. · : E(n)) - }\2. € Ell:) ~{y) ~ \tEE -E• (n)( ) suP. __ Jlt (n) ? = (n)(t7) Jnl }lt - ? (n)(E(n) E(n))& - (n) ~ :;;,.. Cn>r. €E,. fli '(y) ~ (~( )l1 11) - (n)(E(n) P2 ~ E(n)) t- being the inequality marked by & a consequence of eq. 58) and the theorem. In quite a similar way the following theorem 50 Alternative Proof of Shannon Theorem can be proved: (n) Theorem 2. _ . 00) J12. 2. Application to coding for discrete memoryless sources. 1. Consider theorem 2. ::{l, ... l}insteadof k k with the usual meaning of the notations.

Since in force of eq. 73), still valid, we have: as L we can say that the set ). l. I. in f : ~cc"+G;;,:. 8 we obtain the following Theorem 2. tS, then the same notations as in theorem 2. 1. Description of the Markov source. e. p. ~. for some i ( 1 < " < k ~ ~d(J' (Kronecker index). (*) = . 59 Description of Markov Sources and by the transition probability matrix ( 3. 2) 't a source ~ is associated having To the chain an alphabet A of K letters: A ={a~ , a2, ... , and working as follows: when puts the letter a~ ce ak } goes to state ~i.

9) N~ the , whose length N ~ (L) u~ (L) = N~ (u·~ ). 8), it outputs a sequence (L) ( 2. ) ) q, ( y 2 ) = is the ''code mapping", which defines the rule bywhich a codeword is associated to a Source sequence. If we wish to decrease the transmission cost, we should decrease the average length of the codewords, which can be done, roughly, by choosing such a + which associates probable Source sequences with short codewords and unprobable Source sequences with long codewords, while respecting some specific rules for granting unique decipherability (such as the prefix property and the like}.

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