By Henk C. A. van Tilborg (auth.)

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Clearly 'I::; 1 2 ::; " 1/2/i }:. i=1 ••• ::; '" and " Pi = 1. ::; }:. 6. ) + 1. ) codes for a source S with probabil- ity distribution 2 bas a value L satisfying H(2)::; L < H(2) + 1. ) by one bit or more. 8 to N -tuples of source symbols, one gets in the same way an expected length L IN) per N -gram, satisfying N • H(2)::; LIN) < N • H(2) + 1. It follows that H (2)::; So Iim L IN)IN N ...... LIN) 1 li"" < H (p) + N· = H (2). 8) This confirms the third interpretation of the entropy function H, that was Huffman Codes 51 given at the beginning of Chapter 4.

382 With a simple substitution in an English text, one has bits, assuming that alI 26! possible substitutions are equally likely. 20n bits, one obtains a unicity dis- tance of 28. " These two numbers are in remarkable agreement Let X and Y be two random varlables. The joint distribution of X and Y is denoted by PrXY {X =x,Y =y} =Px,y(x,y). 7) = x, given Y = y, is denoted by PrXY {X = xl Y = Y } = Px IY (x I y). 8) Shannon Theory 43 H(X I Y = y) = -LPXIY (x I y). log2PxIY (x I y). 9) x It can be interpreted as the expected amount of information that a realization of X gives, when the occurrence of Y =y is already known.

Ocorrelation AC (k) has the same value for alI k. GI states that zeros and ones occur with roughly the same probability. G2 implies that after Ol the symbol O bas about the same probability as the symboll, etc.. So G2 says that certain n-grams occur with the right probabilities. The interpretation of G3 is more difficult It does say that counting the number of agreements between a sequence and a shifted version of that sequence does not give any information about the period of that sequence, unless one shifts over a multiple of the period.

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