By Giuseppe Longo
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Extra info for Source Coding Theory: Lectures Held at the Department for Automation and Information June 1970
7) and of the assumption jJ. 1 (E(n)):er 1- ~ 0 = p. 1 ( E ~n)) . = So the right-hand inequality in ( 8. e. eq. 2). The second part is proven in the same way. Fig. 1 gives an intuitive idea of the theorem we have just proved. 64 Proof of Shannon Theorem larrangement according. L1(E ~)') = foo quences of elements from n Fig. 1. Illustrating Neyman-Pearson lemma. a(E<;'l) = y: 9. Proof of Shannon Theorem on Source Coding Based on Neyman - Pearson LemmL When we are confronted with the source coding problem, it may happen we know the maximum probability of erroneous decoding, say Pe ,the receiver is prepared to tolerate.
Of codes capable of the performances foreseen by the theory • Moreover, nothing is said about the speed at which Pe goes to zero as L increases, in 53 Necessity of Further Investigation ca-se R > H , while such a parameter is very important for practical purposes the complexity of encoding and decoding apparatus grows with L, and an exact estimate of the L needed to achieve a desired probability of error is very useful. To evaluate such a speed, we now proceed to further investigation. 7. Testing a Simple Alternative.
I. •. 2(x1) ... p. 24) E \") , we get : (E\")/ Prob h 1 ) ~ 2 n(I(,u. ). (E \") / h,). : 1 -11- <> > E~n) c. 25) Taking logarithms on both sides of eq. 20) we get the desired result expressed by eq. 21) in force of the arbitrariness of t. In the same way one can prove the follow- ing Theorem 7. 2. >! '( 0 ( 0 < "(0 < 1 ) of "(, the is described by the follow ... 27) or t ~m n-oo (fi! ,IIJ1i). 28) 8. The Neyman- Pearson Lemma. In this section we shall state and prove a particular case of the well-known Neyman-Pearson le~ rna, which gives a useful hint concerning how to choose the sets fima fi ~ E~n) and and "(!
Source Coding Theory: Lectures Held at the Department for Automation and Information June 1970 by Giuseppe Longo