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Unformatted text preview: Module 6, Lecture 2 Channel Coding: The Channel Coding Theorem G.L. Heileman Module 6, Lecture 2 Definitions encoder X n Y n W W channel p ( ) y  x decoder A discrete channnel is given by the triple ( X , p ( y  x ) , Y ), where X and Y are finite sets corresponding to the input and output; p ( y  x ) is a collection of probability mass functions, one for each x ∈ X such that for every x ∈ X and y ∈ Y , p ( y  x ) ≥ 0, and for every x , ∑ y p ( y  x ) = 1. The nth extension of a discrete memoryless channel (DMC) is the triple ( X n , p ( y n  x n ) , Y n ), where p ( y k  x k , y k 1 ) = p ( y k  x k ) , k = 1 , . . . , n (this is why it’s called “memoryless”). G.L. Heileman Module 6, Lecture 2 Definitions An ( M , n ) block code for the channel ( X , p ( y  x ) , Y ) consists of: 1 An index set W = { 1 , . . . , M } . 2 An encoding function X n : W → X n , yielding codewords X n (1) , . . . , X n ( M ). This codewords form the codebook . 3 A decoding function g : Y n → W which is a deterministic rule assigning a “guess” to each possible received vector. Note: if the encoder and decoder are fixed, and we assume the index is selected according to RV W , then W → X n → Y n → g ( Y n ) = ˆ W forms a Markov chain. G.L. Heileman Module 6, Lecture 2 Definitions The probability of a block error for an ( M , n ) block code and decoder, given channel ( X , p ( y  x ) , Y ), assuming index w ∈ W was sent is: λ w = Pr { g ( Y n ) 6 = w  X n = X n ( w ) } = X y n p ( y n  x n ( w )) · I ( g ( y n ) 6 = w ) where I ( · ) is the indicator function. Thus, the maximal probability of a block error for an ( M , n ) block code is λ ( n ) = max w ∈{ 1 ,..., M } { λ w } , and the (arithmetic) average probability of a block error for an ( M , n ) block code is defined as: P ( n ) e = 1 M M X w =1 λ w . G.L. Heileman Module 6, Lecture 2 Definitions Another way of calculating the average probability of a block error: P ( n ) e = M X w =1 p ( w ) · Pr { w 6 = ˆ w } . If W is uniformly distributed, (i.e., index w ∈ W is chosen according to the uniform distribution), and we assume X n ( w ) is sent, then P ( n ) e = M X w =1 1 M · Pr { w 6 = g ( Y n ) } , and if the probability of error is the same for any index w , then P ( n ) e = Pr { w 6 = g ( Y n ) } . The rate of an ( M , n ) block code is R = log M n bits/transmission . G.L. Heileman Module 6, Lecture 2 Channel Coding Theorem Let’s look at the theorem statement again: Theorem (Shannon’s channel coding theorem) Associated with each discrete memoryless channel, there is a nonnegative number C (the channel capacity) that determines the limits of the channel as follows: (1) For any > and R < C, and large enough n, there exists a block code of length n with rate ≥ R along with a decoding algorithm such that the maximal probability of block error is < ....
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 Spring '10
 G.L.Heileman
 Information Theory, Probability theory, Channel Coding Theorem, G.L. Heileman

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