Introduction of Low Density Parity Check Codes

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Introduction of Low Density Parity Check Codes

• Mong-kai Ku

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What is Coding?

• Coding is the conversion of information to
  another form for some purpose.
• Source Coding The purpose is lowering the
  redundancy in the information. (e.g. ZIP, JPEG,
  MPEG2)
• Channel Coding The purpose is to defeat channel
  noise.

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Block Diagram of a general Communication System
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Channel Coding

• Channel encoding The application of redundant
  symbols to correct data errors.
• Modulation Conversion of symbols to a waveform
  for transmission.
• Demodulatin Conversion of the waveform back to
  symbols, usually one at a time.
• Decoding Using the redundant symbols to correct
  errors.

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Why Channel Coding?

• Trade-off between Bandwidth, Energy and
  Complexity.
• Coding provides the means of patterning signals
  so as to reduce their energy or bandwidth
  consumption for a given error performance.

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Practical Example

• 1000 Base-T Ethernet use 4 pairs of cat-5
  cables, use PAM-5 modulation to increase the
  amount of data sent each symbol, and use 4D
  8-state trellis code to combat noise and
  interference.
• 802.11a use OFDM and concatenated code FEC

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Common Forward Error Correction Codes

• Convolutional Codes
• Block Codes (e.g. Reed-Solomon Code)
• Trellis-Coded-Modulation (TCM)
• Concatenated Codes

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How to evaluate Code Performance?

• Need to consider Code Rate (R), SNR (Eb/No), and
  Bit Error Rate (BER).
• Coding Gain is the saving in Eb/No required to
  achieve a given BER when coding is used vs. that
  with no coding.
• Generally the lower the code rate, the higher the
  coding gain.
• Better Codes provides better coding gains.
• Better Codes usually are more complicated and
  have higher complexity.

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State-of-the-art High Coding Gain Codes

• (15, ¼) Concatenated code constraint length 15,
  rate ¼ convolutional code
• Turbo Code
• These code can achieve coding gains close to
  Shannon bound, but the implementation cost is
  high!

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Low Density Parity Check Codes

• In 1962, Gallager reported work on binary codes
  defined in terms of low density parity check
  matrices.
• In 1996, LDPC codes are re-discovered by D.
  MacKay and R. Neal. They show it to have very
  good coding gain performance.
• Can achieve Turbo Code like performance, but the
  implementation cost is much lower

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LDPC Code Performance
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Linear Block Codes

• A Linear Code can be described by a generator
  matrix G or a parity check matrix H.
• A (N,K) block encoder accepts K-bit input and
  produces N-bit codeword
• x uG, and xHT 0 where x codeword, u
  information

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Encoding / Decoding of LDPC Codes

• Encoding is a matrix operation. However, large
  sparse LDPC matrix poses implementation problem
• Decoding problem is to find the most probable
  vector x such that Hx mod 2 0. Many possible
  implementation exists.

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Code Design for LDPC Codes

• To achieve good coding gain performance, good
  LDPC code design is essential.
• A code design based on density evolution is only
  0.0045dB away from the Shannon bound. However,
  its a rate-1/2 irregular code with maximum
  variable degree of 100 and blokc size of 107bits.
  It also requires an average of more than 1000
  iterations to achieve the result.

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Decoder Implementation of LDPC Codes

• Parallel Implementation Need to cope with
  complex inter-connect.
• JSSCC Mar 2002, 1Gb/s 1024b, rate ½
  implementation 690mW, 49mm2 in 0.16 um
• Not easy to scale to larger block size codes.

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Decoder Implementation of LDPC Codes (cont.)

• Serial Architecture Possible to handle larger
  block size, higher coding gain performance codes.
• Need fast implementation to achieve high
  throughput.
• Possibility Pipeline or SuperScalar Approach

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Application for LDPC Codes

• Wireless, Wired, and Optical Communications.
• Different throughput requirement
• Need to design codes that work with multi-level
  modulation (e.g. QAM or M-PSK)

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Code Design and Implementation

• Need to balance the code design requirement for
  highest coding gain with the ease of
  implementation.
• Trade-off between coding gain and implementation
  complexity.

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Research Opportunity

• Code Design
• Hardware Implementation
• LDPC Application for next generation
  communication systems (Wireless, OFDM, ADSL).