Lecture8 - Colorado State University Ft Collins ECE 516...

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1 Colorado State University, Ft. Collins Fall 2008 ECE 516: Information Theory Lecture 8 September 23, 2008 Recap: Theorem: Let n X X , , 1 L be iid with ( ) x p and ( ) X H . Let 0 > ε . Then, there exists a code which maps sequences ( ) n x x , , 1 L of length n into binary strings (of variable lengths), such that the mapping is one-to-one (and therefore invertible) and ( ) ( ) ε + X H n X l E n 4 Entropy Rates of Stochastic Processes (Chp. 4) Definition: A stochastic process { } n X is stationary if the joint PMF of any k samples is invariant with respect to any amount of time shift, i.e., [ ] [ ] k l nk l n l n k nk n n x X x X x X P x X x X x X P = = = = = = = + + + , , , , , , 2 2 1 1 2 2 1 1 L L for any k n n , , 1 L , any k , any l , and X k x x , , 1 L . Definition: A stochastic process { } n X is ergodic if its time average (sample mean) is equal to its actual mean (ensemble mean), i.e., x n i i n X n μ = 1 1 lim w.p. 1 Definition: The entropy rate of a stochastic process { } n X is defined by ( ) ( ) n n X X H n H , , 1 lim 1 L = X when limit exists. Theorem: For a stationary stochastic process { } n X , the two limits ( ) X H and ( ) X H exist, and are equal, i.e., ( ) ( ) X X H H = Lemma: (Cesaro Mean) If a a n n lim and = = n i i n a n b 1 1 , then a b n n lim .
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