Introduction:LDPC Codes.

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IntroductionLDPC Codes.
KIMOON LEE, DEPT of MATH, MSU leekimoo_at_msu.edu
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Contents.

• Communication System.
• 1.1 Channel Coding.
• Channel Models.
• 2.1. Binary Symmetric Channels (BSC).
• 2.2. Binary Erasure Channels (BEC).
• Low-Density Parity-Check (LDPC) Codes.
• 3.1. Definition.
• 3.2. Sparse Graph Expression.
• 3.3. Decoding Algorithms Message Passing
  Algorithms.
• 3.3.1. Hard-Decoding AlgorithmsBSC.
• 3.3.2. Error Probability EvolutionBSC.
• 3.3.3. Message Passing Algorithms on BEC.
• 3.3.4. Error Probability EvolutionBEC.
• 3.4. Soft-Decoding Algorithms (SDA) on
  BI-AWGNC.
• 3.5. Encoding Algorithms
• 3.4.1. Gaussian Elimination Method
• 3.4.2. Approximated Lower Triangular Matrix
  Method.

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1. Communication Systems.
MODEM
Information Source
Channel Encoder
Modulator
Source Encoder
Source Coding
Channel
Channel Coding
Source Decoder
Demodulator
Channel Decoder
Output Transducer
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1.1 Channel Coding
Channel
S. Encoder
S. Decoder
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2. Channel Models.
Shannons Channel Coding Theorem
1. A channel is characterized by its channel
capacity C(p)
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2.1 Binary Symmetric Channels C(pe)1-H(pe)
Example of BSC Radio Frequency (RF) based
communication over electro-magnetic space,
Compact discs.
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2.2. Binary Erasure Channels (BEC) C(p)1-p,
ploss rate.

• Example of BEC The Internet
• Symmetric error rate is small and corrupted
  packets are dropped, symmetric error is not a
  serious problem.
• Packets are lost due to network congestion
  control.

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3.1. Definitions and Backgrounds
3. Low-Density Parity-Check Codes.
(1). Definition An n,k-Binary linear code C
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(2). Definition An n,k-LDPC code C(H)
Perspectives of LDPC Codes
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In LDPC Code studies

• The design of a code is started by first
  designing a good H matrix. This unlike the
  typical approach of starting a code design by
  first designing a G matrix.
• Once H matrix is designed a corresponding G
  matrix can be obtained as follows
• The above G can then be used for channel
  encoding. In general the G obtained by above
  method is not sparse and thus the channel
  encoding may not have linear time complexity.
  This is not a major drawback as often the
  encoding can be carried out in advance.
• In addition, it should be mentioned that
  algorithms based on encoding directly from the H
  matrix have been designed. These encoding
  algorithms have been shown to have time
  complexity that is linear with respect to n. We
  describe these algorithms in more detail at a
  later stage.

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3.2. Sparse Graph Expression.
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Example
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3.3. Decoding Algorithms (Message Passing
Algorithms).
1. The first demodulator outputs are translated
into initial messages (bits/probabilities) by
some arguments
2. Initial messages (or estimates) are copied to
variable nodes for the first time

• Once these initial messages are obtained, the
  decoding algorithm of the code is carried out by
  an iterative algorithm on the sparse graph A
  node receives messages from its neighboring
  nodes, (1) updates the messages by some
  arguments, (2) then sends back the messages to
  its neighboring nodes. This procedure is
  repeated for several rounds.

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Notations.
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Decoding Procedures.
(a). Check Side
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(b). Variable Side
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3.3.1. BSCHDA/PFA.
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3.3.1-1. Hard-Decoding Algorithm (HDA)
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(a). Variable side
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(b). Check side
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3.3.1-2. Parallel Flipping Algorithm (PFA)
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3.3.2. Error Probability EvolutionBSC.
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3.3.3. Message Passing Algorithm on BEC.
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Example of MPA on BEC.
0. Initialization
1. Direct Recovery.
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Example of MPA on BEC.
2. Substitution Recovery.
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3.3.3. Message Passing Decoder on BEC.
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3.3.4. Error Probability EvolutionBEC.
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3.4. Soft-Decoding Algorithms on
BI-AWGNC.
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3.4.1. MODEM
Channel Decoder
Channel. Encoder
Modulator
Demodulator

• Modulator converts a codeword into a continuous
  signal.

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Example Binary Phase-Shift Keying (BPSK)
Example
BPSK
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3.4.1. MODEM
2. Demodulator converts a received signal to a
message estimate.
Channel Decoders
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Steps for Demodulation
1. Equalization.
2. Message Decision.
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3.4.2. Channel Capacity of BI-AWGNC
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3.4.4. Soft-Decoding Algorithms (SDA).
The type of messages in SDA
Initial Message
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3.4.4-1. Notations and Backgrounds

• Notations.

2. Bilinear transform
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3.4.4-2. SDA with Likelihood Ratios.
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Last round
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3.4.4-3. SDA with Log-likelihood Ratios.
1. The message Rij from cj to vi
2. The message Lij from vi to cj
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3.4.4-4. Min-Sum Algorithm.
1. Observation.
2. The message Rij from cj to vi
3. The message Lij from vi to cj
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3.4.4-5. SDA with A Posteriori Probabilities.
1. Observation.
2. The message mij from cj to vi
1. The message aij from vi to ci
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Last Round.
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3.5. Encoding Algorithms.