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Near Shannon Limit Performance of Low Density Parity Check Codes

Near Shannon Limit Performance of Low Density Parity Check Codes. David J.C. MacKay and Radford M. Neal , Electronics Letters 29th Vol. 32 No. 28, August 1996. Outline. Features of LDPC Codes History of LDPC Codes Some Fundamentals A Simple Example Properties of LDPC Codes

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Near Shannon Limit Performance of Low Density Parity Check Codes

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  1. Near Shannon Limit Performance of Low Density Parity Check Codes David J.C. MacKay and Radford M. Neal, Electronics Letters 29th Vol. 32 No. 28, August 1996.

  2. Outline • Features of LDPC Codes • History of LDPC Codes • Some Fundamentals • A Simple Example • Properties of LDPC Codes • How to construct? • Decoding • Concept of Message Passing • Sum Product Algorithm • Concept of Iterative Decoding • Channel Transmission • Decoding Algorithm • Decoding Example • Performance • Cost

  3. Shannon Limit (1/2) • Shannon Limit: • Describes the theoretical maximum information transfer rate of the communication channel (channel capacity), for a particular levels of noise. • Given: • A noisy channel with channel capacity C • Information transmitted at a rate R

  4. Shannon Limit (2/2) • If R < C, there exist codes that allow the probability of error at the receiver to be made arbitrarily small. • Theoretically, it is possible to transmit information nearly without error at any rate below a limiting rate, C. • If R > C, all codes will have a probability of error greater than a certain positive minimal level. • This level increases as the rate increases. • Information cannot be guaranteed to be transmitted reliably across a channel at rates beyond the channel capacity.

  5. Features of LDPC Codes • Low-Density Parity-Check Codes (LDPC Codes) is an error correcting code. • A method of transmitting a message over a noisy transmission channel. • Approaching Shannon capacity • For example, 0.3 dB from Shannon limit (1999). • An closer design from (Chung:2001), 0.0045 dB away from capacity. • Linear decoding complexity in time. • Suitable for parallel implementation.

  6. History of LDPC Codes • Also known as Gallager codes, by whom developed the LDPC concept in his doctoral dissertation at MIT in 1960. • Long time being ignored due to requirement of high complexity computation. • In 90’s, Rediscovered by MacKay and Richardson/Urbanke.

  7. Some Fundamentals • The structure of a linear block code is described by the generator matrix G or the parity-check matrix H. • H is sparse. • Very few 1's in each row and column. • Regular LDPC codes: • Each column of H had a small weight (e.g. 3), and the weight per row was also uniform. • H was constructed at random subject to these constraints. • Irregular LDPC codes: • The number of 1's per column or row is not constant. • Usually irregular LDPC codes outperforms regular LDPC codes.

  8. A Simple Example (1/6) www.wikipedia.org • Variable Node: box with an '=' sign. • Check Node: box with a '+' sign. • Constraints: • All lines connecting to a variable node have the same value, and all values connecting to a check node must sum, modulo two, to zero (they must sum to an even number).

  9. A Simple Example (2/6) • There are 8 possible 6 bit strings which correspond to valid codewords: • 000000, 011001, 110010, 111100, 101011, 100101, 001110, 010111. • This LDPC code fragment represents a 3-bit message encoded as 6 bits. • The redundancy is been used to aid in recovering from channel errors.

  10. A Simple Example (3/6) • The parity-check matrix representing this graph fragment is:

  11. A Simple Example (4/6) • Consider the valid codeword: 101011 • Been transmitted across a binary erasure channel, and received with the 1st and 4th bit erased: ?01?11 • Belief Propagation is particularly simple for the binary erasure channel. • Consists of iterative constraint satisfaction.

  12. A Simple Example (5/6) • Consider the erased codeword: ?01?11 • In this case: • The first step of belief propagation is to realize that the 4th bit must be 0 to satisfy the middle constraint. • Now that we have decoded the 4th bit, we realize that the 1st bit must be a 1 to satisfy the leftmost constraint.

  13. A Simple Example (6/6) • Thus we are able to iteratively decode the message encoded with our LDPC Code. • We can validate this result by multiplying the corrected codeword r by the parity-check matrix H: • Because the outcome z (the syndrome) of this operation is the 3 x 1 zero vector, we have successfully validated the resulting codeword r. (mod 2)

  14. Properties of LDPC Codes • The structure of a linear block code is completely described by the generator matrix G or the parity-check matrix H. • r = Gm • r: codeword, • G: generator matrix, • m: input message • HG = 0 • Hr = 0 • A Low-Density Parity-Check Code is a code which has a very sparse, random parity check matrix H. • Typically, column weight is around 3 or 4.

  15. How to construct? (1/4) • Construction 1A: • An Mby N matrix is created at random with: • weight per column t (e.g., t = 3). • weight per row as uniform as possible. • overlap between any two columns no greater than 1. • No cycle graph.

  16. How to construct? (2/4) • Construction 2A: • Up to M/2 of the columns: designated weight 2 columns • Such that there is zero overlap between any pair of columns. • The remaining columns: made at random with weight 3. • The weight per row: as uniform as possible. • Overlap between any two columns of the entire matrix no greater than 1.

  17. How to construct? (3/4) • Construction 1B and 2B: • A small number of columns are deleted from a matrix produced by Constructions 1A and 2A. • The bipartite graph has no short cycles of length less than some lengthl.

  18. How to construct? (4/4) • Above constructions do not ensure all the rows of the matrix are linearly independent. • The M by N matrix created is the parity matrix of a linear code with rate at least R = K/N, where K = N - M. • The generator matrix of the code can be created by Gaussian elimination.

  19. Decoding • Decoding problem is to find the most probable vector x iteratively, such that: Hx mod 2 = 0 • Gallager's algorithm may be viewed as an approximate Belief Propagation (BP) algorithm. • Message Passing (MP).

  20. Concept of Message Passing(1/4) • How to know the number of soldiers stand in line? • Each soldier plus 1 to the number hearing from the neighbor on one side, then pass the result to the neighbor on the opposite side.

  21. Concept of Message Passing (2/4) • In the beginning, every soldier knows that: • There exists at least one soldier (himself). • Intrinsic Information. Figure 1. Each node represent a soldier & Local rule: +1

  22. Concept of Message Passing (3/4) • Start from the leftmost and rightmost soldiers. • Extrinsic Information. Figure 2. Extrinsic Information Flow.

  23. Concept of Message Passing (4/4) • The total number = [left + right] + [oneself] • Overall Information = Extrinsic Information + Intrinsic Information. Figure 3. Overall Information Flow.

  24. Sum Product Algorithm (1/4) • During decoding, apply Sum Product Algorithm to derive: • Extrinsic Information (Extrinsic Probability) • Overall Information (Overall Probability) • In this paper, channel transmission: • BPSK through AWGN. • Binary Phase-Shift Keying. • Additive White Gaussian Noise.

  25. Sum Product Algorithm (2/4) • A simple example: • If local rule:

  26. Sum Product Algorithm (3/4) Form 1. valid codeword for rule: • (Overall Probability with m2 = 0): • P10P20P30 + P11P20P31 • (Extrinsic Probability with m2 = 0): • P10P30 + P11P31

  27. Sum Product Algorithm (4/4) • Likelihood Ratio: , where ( with m1 or m2 is similar )

  28. Concept of Iterative Decoding (1/6) • A simple example: • If received a codeword with Likelihood Ratio: ( “10110” is an invalid codeword )

  29. Concept of Iterative Decoding (2/6) • A simple example: • Calculate Extrinsic Probability by Check Nodes:

  30. Concept of Iterative Decoding (3/6) • A simple example: • We then obtain the Overall Probability of 1st. Round: ( “10010” is an valid codeword )

  31. Concept of Iterative Decoding (4/6) • A simple example: • 2nd. Round :

  32. Concept of Iterative Decoding (5/6) • A simple example: • 2nd. Round :

  33. Concept of Iterative Decoding (6/6) • A simple example: • We then obtain the Overall Probability of 2nd. Round: ( “10010” is an valid codeword )

  34. Channel Transmission • BPSK through AWGN: • A Gaussian channel with binary input ±a and additive noise of variance σ2 = 1. • t: BPSK-modulated signal, • AWGN channel: • Posterior probability:

  35. Decoding Algorithm (1/6) • We refer to the elements of x as bits, to the rows of H as checks, and denote: • the set of bits n that participate in check m by N(m): • the set of checks in which bit n participates by M(n): • a set N(m) with bit nexcluded by N(m)\n

  36. Decoding Algorithm (2/6)

  37. Decoding Algorithm (3/6) N(1) = {1, 2, 3, 6, 7, 10}, N(2) = {1, 3, 5, 6, 8, 9}, … etc M(1) = {1, 2, 5}, M(2) = {1, 4, 5}, … etc N(1)\1 = {2, 3, 6, 7, 10}, N(2)\3 = {1, 5, 6, 8, 9}, … etc M(1)\1 = {2, 5}, M(2)\4 = {1, 5}, … etc

  38. Decoding Algorithm (4/6) • The algorithm has two parts, in which quantities qmn and rmn associated with each non-zero element in the H matrix are iteratively updated: • qxmn:the probability that bit n of x is x, given the information obtained via checks other than check m. • rxmn:the probability of check m being satisfied if bit n of x is considered fixed at x, and other bits have a separable distribution given by the probabilities . • The algorithm would produce the exact posterior probabilities of all the bits of the bipartite graph defined by the matrix H.

  39. Decoding Algorithm (5/6) • Initialization: • q0mnand q1mn are initialized to the values f0n and f1n • Horizontal Step: • Define: • For each m, n: • Set:

  40. Decoding Algorithm (6/6) • Vertical Step: • For each n and m, and for x = 0, 1 we update: • We can also update the “pseudoposterior probabilities” q0n and q1n, given by: • , where αmnis chosen such that q0mn+ q1mn = 1

  41. Decoding Example (1/10)

  42. Decoding Example (2/10)

  43. Decoding Example (3/10) • BPSK through AWGN: • Simulated a Gaussian channel with binary input ±a and additive noise of variance σ2 = 1. • t: BPSK-modulated signal, • AWGN channel:

  44. Decoding Example (4/10) • BPSK through AWGN: • Posterior probability:

  45. Recall: Decoding Algorithm • Input: The Posterior Probabilities pn(x). • Initialization: Let qmn(x) = pn(x). 1. Horizontal Step: (a). Form the δq matrix from qmn(0) - qmn(1) (at sparse non-zero location). (b). For each nonzero location (m, n), let δrmn be the product of δqmatrix elements along its row, excluding the (m, n) position. (c). Let rmn(1) = (1 -δrmn ) / 2, rmn(0) = (1 +δrmn ) / 2 2. Vertical Step: For each nonzero location (m, n) let qmn(0) be the product along its column, excluding the (m, n) position, times pn(0). Similarly for qmn(1). Then normalize.

  46. Decoding Example (5/10) • Initialization: Let qmn(x) = pn(x).

  47. Decoding Example (6/10) • Iteration 1: Horizontal Step: (a) (b)

  48. Decoding Example (7/10) • Iteration 1: Horizontal Step: (c)

  49. Decoding Example (8/10) • Iteration 1: Vertical Step: (a) (b)

  50. Decoding Example (9/10) • Iteration 1: After Vertical Step: Hc mod 2 ≠ 0 Recall: update pseudoposterior probabilitiesqn(1) and qn(0), given by:

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