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Lecture9

# Lecture9 - Colorado State University Ft Collins ECE 516...

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1 Colorado State University, Ft. Collins Fall 2008 ECE 516: Information Theory Lecture 9 September 25, 2008 Recap: Write ( ) = = X 1 0 i i i e p α , where { } i e are the set of linear independent eigenvectors, i e corresponding to eigenvalue i λ . Assume L > = 2 1 1 λ λ . Then, ( ) ( ) ( ) ( ) = = = = = X X 1 1 0 i i i i i i T i T n e e P p P p n n n λ α α As n , ( ) 1 1 e p α = n . (scaling is not important) If a Markov chain is irreducible and aperiodic, and if it is run for a long time, so ( ) μ p n , then it us a stationary ergodic process. ( ) ( ) ( ) ( ) ( ) = = = = = X X i ij j ij i n n n n n n n P P X X H X X H X X X H H H 1 log | lim | lim , , | lim 1 2 1 1 1 μ L X X ( ) ( ) X log n X H H X 5 Data Compression (Chp. 5) 5.1 Example of Codes Definition: A source code C for a r.v. X is a mapping from X to * D , the finite length strings of symbols from a D-ary alphabet. Definition: The expected length ( ) C L of the source code ( ) x C for a r.v. X with ( ) x p ( ) ( ) ( ) = X x x l x p C L Definition: A code is non-singular if every element of X maps into a different string in * D , i.e., ( ) ( ) j i j i x C x C x x

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