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VARIABLE-LENGTH CODES FOR DISCRETE SOURCES 23 Before proving the theorem, we show how to represent codewords as base 2 expansions (the base 2 analog of base 10 decimals) in the binary number system. After understanding this representation, the theorem will be almost obvious. y1 y2 · · · yl represents � the rational number lm=1 ym 2−m . 011 represents 1/4 + 1/8. Ordinary decimals with l digits are frequently used to indicate an approximation of a real number to l places of accuracy. y1 y2 · · · yl is viewed as �l in the �l way, the 7 −m −m ‘covering’ the interval [ m=1 ym 2 , + 2−l ).

Xn has entropy H[X n ] = nH[X]. Such a block is a random symbol in its own right and can be encoded using a variable-length preﬁx-free code. This provides a ﬁxed-to-variable-length code, mapping n-tuples of source symbols to variablelength binary sequences. It will be shown that the expected number L of encoded bits per source symbol can be made as close to H[X] as desired. Surprisingly, this result is very simple. Let E[L(X n )] be the expected length of a variable-length preﬁx-free code for X n .

1 (Asymptotic equipartition property). Let Xn be a string of n iid discrete random symbols {Xk ; 1 ≤ k ≤ n} each with entropy H[X]. For all δ > 0 and all suﬃciently large n, Pr(Tεn ) ≥ 1 − δ and |Tεn | is bounded by (1 − δ)2n(H[X]−ε) < |Tεn | < 2n(H[X]+ε) . 27) Finally, note that the total number of diﬀerent strings of length n from a source with alphabet size M is M n . For non-equiprobable sources, namely sources with H[X] < log M , the ratio of the number of typical strings to total strings is approximately 2−n(log M −H[X]) , which approaches 0 exponentially with n.