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By Hideo Kusuoka and Julien I.E. Hoffman

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6 Arithmetic Codes 35 An arithmetic code is based on two ideas: 1. the code is a cumulative probabihty on strings of length n, sorted alphabetically, and computed recursively in n. and 2. for practical implementation all calculations are to be done in fixed-size registers. The first idea has been credited to Shannon, presumably without recursive calculation of the cumulative probabilities, but the solution to the crucial second problem is the key and the foundation of arithmetic coding. We consider first the case where no restriction is placed on the precision, and for the sake of simplicity we consider first the binary alphabet.

3) Xi What this means is that in terms of the random variables ^Pr{Xi = X i , X 2 = X 2 , . . , X n =Xn} = Pr{Xi = X 2 , . . , X n - l =Xn} , Xi which is just the statement of shift invariance. ,Xn) = l[P{x^). 2 Entropy of Stationary Processes There are two ways to define the entropy of a random process: 2. ,X,) provided, of course, that the limits exist. Theorem 13 For stationary processes Hi{X) = H2{X). Proof. 2, H{Xn-\-l\Xnj . . , X i ) < H{Xn-\-l\Xn, = H{Xn\Xn-lj - - • ,-^2) . . ,-^1), where the equality follows from stationarity.

0. 1. Read next symbol. If none exists, shift contents of C left w positions to obtain the final code string. 2. If the next symbol is the low-probability symbol, say, x, replace the contents of A by p[x) {p{x) < 1/2) and go to 4. 3. If the next symbol is the high probability symbol, add to register C the number p{x) and subtract from A the same number. 4. 5). Go to 1. 0. 1. Form an auxiliary quantity T by subtracting p{x) from the contents of C Test if T < 0. 2. If T < 0, decode the symbol as x.

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