LDPC-2 - UCLA Extension Short Course on LDPC Codes...

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UCLA Extension Short Course on LDPC Codes Low-Density Parity-Check Codes Part II - The Iterative Decoder William Ryan, Professor Electrical and Computer Engineering Department The University of Arizona Box 210104 Tucson, AZ 85721 [email protected] July 18-19, 2007
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1 Decoding Overview in addition to presenting LDPC codes in his seminal work in 1960, Gallager also provided a decoding algorithm that is effectively optimal since that time, other researchers have independently discovered that algorithm and related algorithms, albeit sometimes for different applications the algorithm iteratively computes the distributions of variables in graph-based models and comes under different names, depending on the context: - sum-product algorithm - min-sum algorithm (approximation) - forward-backward algorithm, BCJR algorithm (trellis-based graphical models) - belief-propagation algorithm, message-passing algorithm (machine learning, AI, Bayesian networks)
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2 the iterative decoding algorithm for turbo codes has been shown by McEliece (1998) and others to be a specific instance of the sum-product/belief- propagation algorithm the “sum-product," "belief propagation," and "message passing" all seem to be commonly used for the algorithm applied to the decoding of LDPC codes
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3 Example: Distributed Soldier Counting (Pearl, 1988) A. Soldiers in a line . Counting rule: Each soldier receives a number from his right (left), adds one for himself, and passes the sum to his left (right). Total number of soldiers = (incoming number) + (outgoing number) * * the incoming and outgoing numbers are taken from the same side of a soldier
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4 B. Soldiers in a Y Formation Counting rule: The “message” that soldier X passes to soldier Y is the sum of all incoming messages, plus one for soldier X, minus soldier Y’s message X Y X n Z X Z X X n Z X Y X Z Y X I I I I I I + = + = \ ) ( ) ( Total number of soldiers = X X n Z X Z I I + ) ( (counted at arbitrary soldier X)
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5 C. Formation Contains a Cycle The situation is untenable: No viable counting strategy exists; there is also a positive feedback effect within the cycle and the count tends to infinity. Conclusion: message-passing decoding cannot be optimal when the codes graph contains a cycle
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6 The Turbo Principle total Y I X Y I YX I total X I X Y I I total X I I X Y I total Y I X Y X Y X Y neighbors of X neighbors of X except Y neighbors of X neighbors of Y except X neighbors of Y neighbors of Y I X I X I X I Y I Y I Y
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7 The Turbo Principle Applied to LDPC Decoding the variable nodes in a Tanner graph represent repetition (REP) codes the check nodes represent single-parity-check (SPC) codes Π
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8 () + + = + = x yx y z zx yx xy I I I sign I I sign x ˆ I xy = function of I zx (except I yx ) and I x (developed below) X I yx I xy from channel: I x = 2y x / σ 2 decision is VN (repetition code) decoder CN (SPC) decoder X I yx I xy I xy = function of inputs, excluding I yx (developed below) Î
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This note was uploaded on 10/10/2009 for the course ECE 637 taught by Professor Staff during the Spring '08 term at Arizona.

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LDPC-2 - UCLA Extension Short Course on LDPC Codes...

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