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75 bits, which is exactly equal to the entropy. 2 Huffman coding. (a) The process of building a binary tree; (b) the designed Huffman codes. 2 Source Coding 9 they cannot adjust codeword lengths at fractional bit precision. In this case, Huffman codes are not optimal. One alternative is to jointly code a source sequence instead of the individual source. According to the AEP, we can achieve an expected length per every source close to the entropy of H(S). Therefore, it is desirable to have an efficient coding procedure that works for a long block of source letters.

In other words, the entropy reduction is equal to what we get from the channel. An intuitive idea is described here about why we can transmit C bits of information over a channel. The basic idea is that, for large block lengths, every channel looks like the noisy typewriter channel and the channel has a subset of inputs that produce essentially disjointed sequences at the output. For each input n-sequence, we wish to ensure that no two Y sequences produce the same Yˆ output sequence. Otherwise, we will not be able to decide which Y sequences was sent.

In information theory, the analog of the law of large numbers is the asymptotic equipartition property (AEP). d. p(s), the AEP indicates 1 log p(S1 , S2 , n , Sn ) ! 12) in probability. It is a direct consequence of the weak law of large numbers. d. variables, 1/n ∑i Si is close to its expected value E(S) for large n. The AEP states that 1/n log p(S1 , S2 , , Sn ) is close to the entropy H(S), where p(S1 , S2 , , Sn ) is the probability of observing the sequence S1 , S2 , , Sn . Thus, the probability p(S1 , S2 , , Sn ) will be close to 2 nH(S) .

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