Lecture16 - Colorado State University, Ft. Collins ECE 516:...

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1 Colorado State University, Ft. Collins Fall 2008 ECE 516: Information Theory Lecture 16 October 23, 2008 Recap: 7.13 Joint Source-channel Coding Theorem 1. C H < is sufficient and necessary for reliable information transmission. 2. We do not lose anything by separating source and channel coding. 7.14 Channel Coding and Error Exponent () ()( ) R E x y p x p R P n r yx n e = ∑∑ + + < YX ρ 1 1 1 1 0 | log max log 1 which is known as the random coding exponent. ( ) R nE n e r e P
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2 Other results: 1. Error exponent can be improved at low rate. 2. For crit R R , it can be shown that () ( ) () R E P n r n e n log 1 lim . Therefore, R E r is the maximum achievable error exponent for crit R R . (How can we possibly derive an upper bound for the error exponent? Basic idea, channel distortions are random, hence there is certain probability that the channel behaves like another channel with low capacity.) 3. If message error rate satisfies ( ) ( ) R nE n e e P , the bit error rate also satisfies ( ) R nE n b e P . (Consider R nE n b b e P , ( ) ( ) R E R E b . Message error probability must be less than the union bound, i.e., ( ) ( ) ( ) n nE n b n e b ne nP P . This contradicts with R nE n e e P .) 8 Differential Entropy (Chapter 8) Aim: generalization of entropy, AEP to continuous r.v.s Motivation: to study continuous channels (e.g., Gaussian) in the next chapter X is a continuous r.v. with CDF ( ) [ ] x X P x F = and PDF ( )( ) x F x f =
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This note was uploaded on 03/17/2010 for the course ECE 516 taught by Professor Rocky during the Spring '08 term at Colorado State.

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Lecture16 - Colorado State University, Ft. Collins ECE 516:...

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