# LDPC Codes - LDPC Codes Tom to edit Master subtitle style...

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Click to edit Master subtitle style 10/2/11 LDPC Codes Tom Emmons CJ Halabi Anirvan Mukherjee

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10/2/11 Outline Introduction to LDPC Parity Check LDPC Encoding LDPC Decoding Characteristics of LDPC codes Conclusion
10/2/11 An Introduction Linear codes with rate arbitrarily close to capacity ML decoding on BSC channel NP hard for most linear codes 2 ways forward: Look at subclasses for linear codes for which decoding is polynomial LDPC codes: Use sub-optimal decoding

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10/2/11 Why LDPC Codes? Proven to approach Shannon’s Limit for the BEC and BSC Performs well in practice Low Error Floor Linear encoding Pre-processing more complex, though Linear decoding Decoding can be implemented in
10/2/11 Parity Check Codes Parity check codes use extra parity check bits for error detection and/or correction Parity check bits are each the modulo 2 sum of a subset of the original set of bits Examples: ASCII: 7-bit symbols, 1 additional parity bit

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10/2/11 Parity Check Codes Each parity check condition can be written as a linear equation in the bits involved Ex: For the second Hamming parity bit: c1 + c3 + c4 = c6 Add c6 to each side for a more useful version: c1 + c3 + c4 + c6 = 0 A parity check code can be written as
10/2/11 Parity Check Matrix A parity check code can be written as a system of such linear equations. Hamming: In matrix form:

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10/2/11 Parity Check Matrix H: The parity check matrix A codeword c must be in the ker H, i.e. HcT = 0 H is not unique: Any matrix with the same rowspace is a valid parity
10/2/11 Generator Matrix The generator matrix: uG = c Constructs a codeword from the original bits Hamming Code:

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10/2/11 Bipartite Graph Representation Each row of H represents a constraint Create a bipartite graph G(L,R), where: Each code bit is represented by a “variable” node in L
10/2/11 Bipartite Graph Representation c1 c2 c3 c4 c5 c6 c7 r1 r2 r3

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10/2/11 LDPC Codes Uses “Sparseness” of matrix and graph descriptions Very few 1’s in each row and column Any two columns have an overlap of at most 1 Avoids overlapping Uses “Randomness” o Example LDPC Code
10/2/11 LDPC Codes 1960: Proposed by Gallagher in his Ph.D. dissertation After they were invented, they were largely forgotten and reinvented multiple times for the next 30 years (Notably MacKay, 1999 and Richardson/Urbanke 1998) Ignored due to:

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10/2/11 Gallagher’s LDPC Codes Also known as “regular” LDPC Codes Code given by: (n, j, k) n : Codeword Length j : Degree of each variable node (The number of parity-check equations involving each variable node) k : Degree of each check node (The number of variable nodes involved in
10/2/11 Gallagher’s Construction Example:

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## This note was uploaded on 10/02/2011 for the course ECE 5670 taught by Professor Scaglione during the Spring '11 term at Cornell University (Engineering School).

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LDPC Codes - LDPC Codes Tom to edit Master subtitle style...

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