Sep 21, 2017 term paper 2

BEFORE GETTING TO THE QUESTION, WE NEED SOME OBSERVATIONS. (I) RECALL THE FORMULA FOR THE SUM OF…

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Before getting to the question, we need some observations.

(i) Recall the formula for the sum of a geometric series:

(1−ρ)−1, for |ρ|

(ii) Differentiating both sides of the equation in (i) with respect to ρ, we  (iii) If a symbol s from a perfect zeroth-order source occurs in the source text (randomly and independently of all other occurrences) with relative frequency f , then, starting from any point in the source text and going either forward or backward, assuming the source text extends infinitely in both directions, the probability of reading through exactly k letters before coming to the first occurrence of s (at the (k +1)st place scanned) is

using (ii). To put it another way, s occurs on average once every 1/f letters, which agrees with intuition, since  is the relative frequency of s.)

(iv) Suppose that S = {s1,…,sm } is the alphabet of a perfect zeroth-order source, with sj having relative frequency  j , 1 ≤ j ≤ m. Suppose the source text is encoded by the interval method, using some prefix-free set C = {ω­1,…} of code words. Then the average length of a code word Replacing sj will be  so the average length of a code word replacing a source letter will be  

Finally, the problem. Show that for any zeroth-order source, in interval encoding using C = {0,10,110,···}, the average length of a code word replacing a source letter will be m = |S|.


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