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Lecture13

# Lecture13 - Colorado State University Ft Collins ECE 516...

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1 Colorado State University, Ft. Collins Fall 2008 ECE 516: Information Theory Lecture 13 October 9, 2008 Recap: Channel capacity () ( ) Y X I C x p ; max = 7.2 Symmetric channels Given X Y p | which is a matrix ( ) = i j i j x y p x y p x y p x y p X Y p | | | | | 1 1 1 1 If all rows are permutations of each other and all columns are permutations of each other. Weakly Symmetric channels Given X Y p | If all rows are permutations of each other and all column sums are equal ( ) r H Y X I C x p = = Y log ; max 7.3 Properties of channel capacity 1. 0 C , since 0 ; Y X I 2. X log C , since ( ) ( ) Y X H X H Y X I | ; = 3. Y log C , since ( ) ( ) X Y H Y H Y X I | ; = 4. Y X I ; is a continuous function of ( ) X p

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2 5. () Y X I ; is a concave function of ( ) X p Systematic calculation of C q h q P H T T r r r max s.t. 1 1 1 = = = i i T q q r r 0 i q , i 7.4 Channel Coding Theorem Defintions For a discrete channel, let Y X I C X p ; max = 1. Achievability: All rates below C are available. That is, if C R < , then one can design a code ( ) n nR , 2 such that ( ) ( ) 0 = n e n P λ as n . 2. Converse: No rate above C is achievable. That is, if C R > , then for all codes n e P is bounded away from zero. source W message encoder X n channel p ( y | x ) Y n decoder W ˆ Message W is drawn from a set { } M , , 1 L . It is mapped to a signal ( ) W X n , to be set by n channel uses. It is sent through the channel ( ) n n X Y p | , and received as n Y . Decoder applies a deterministic function ( ) g onto n Y and guesses W , ( ) n Y g W = ˆ An error occurs if W W ˆ .
3 Definition: A discrete memoryless channel is denoted by () Y X , | , X Y p , where X and Y are finite sets, ( ) x y p | for all X x and Y y , and 1 | = y x y p for all X x . Memoryless ( ) ( ) k k k k k x y p y x y p | , | 1 = Assume no feedback ( ) ( ) 1 1 1 | , | = k k k k k x x p y x x p DMC: () () = = n i i i n n x y p x y p 1 | | (If you can get channel feedback, the above expression still holds, but it will not be useful any more.) Definition: An n M , code for the channel ( ) ( ) Y X , | , x y p , consists of 1. An index set {} M , , 1 L 2. An encoding function: { } n M X , , 1 L ( )( ) ( ) ( ) () () () () M x M x M x M X M x x x X x x x X n n n n n n , , , 2 , , 2 , 2 2 2 1 , , 1 , 1 1 1 2

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Lecture13 - Colorado State University Ft Collins ECE 516...

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