Coding for Markov Sources: Course Held at the Department for by Giuseppe Longo

By Giuseppe Longo

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

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12). 1 Let it be given a DMS with entropy H (*) genera! 15) H +a fog D it is possible to encode these sequences into codewords of length N of symbols taken from an alphabet '& of size in such a way that for any neous decoding ~ >0 D the probability of erro- Pe satisfies provided L is large enough, whatever the positive constant ~ is. 16) N < L then (~~)The entropy of a (cf. 4)). k OMS is given by H = -1:. p~ togp~ 1 ~ Encoding Rate 37 for arbitrary the positive t > 0 o , provided L is large enough, whatever is.

L the necessary tools for invest! gating the asymptotic behaviour of the error probability Pe. Assume first that the encoding rate R satisfies eq. 68); since R > H , we already lmow that PQ ~ 0 as L- oo • Let ot" be the (unique) solution for eqn. (2. 73) f. ) >R>H(

36) Now we are in a position to state the following two pro>positions: Proposition 2. ;n). al"1 t y · · f y t h e 1nequ sat1s (n) ( ~1-) I (p1 II p2 (n) ) an n-th sample. -divergence computed on the basis of 45 n-length Samples (2. 37) (n) while the samples in E2 occur with probability less than 1j being , and "rl arbitrary positive numbers. t; Proof; This proposition is a direct consequence of the weak f1 law of large numbers. Actually if is true, the quantity (2. e. towards I (}It Proposition E 2 eona }Ji pt (x) (x) , whence the thesis.

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