Code and Decoder Design of LDPC Codes for Gbps Systems

1
Code and Decoder Design of LDPC Codes for Gbps
Systems

• Jeremy Thorpe
• Presented to Microsoft Research 2002.11.25

2
Talk Overview

• Introduction (13 slides)
• Wiring Complexity ( 9 slides)
• Logic Complexity (7 slides)

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Reliable Communication over Unreliable Channels

• Channel is the means by which information is
  communicated from sender to receiver
• Sender chooses X
• Channel generates Y from conditional probability
  distribution P(YX)
• Receiver observes Y

P(YX)
Y
X
channel
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Shannons Channel Coding Theorem

• Using the channel n times, we can communicate k
  bits of information with probability of error as
  small as we like as long as
• as long as n is large enough. C is a number that
  characterizes any channel.
• The same is impossible if RgtC.

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The Coding Strategy

• Encoder chooses the mth codeword in codebook C
  and transmits it across the channel
• Decoder observes the channel output y and
  generates m based on the knowledge of the
  codebook C and the channel statistics.

Encoder
Channel
Decoder
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Linear Codes

• A linear code C can be defined in terms of either
  a generator matrix or parity-check matrix.
• Generator matrix G (kn)
• Parity-check matrix H (n-kn)

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Regular LDPC Codes

• LDPC Codes linear codes defined in terms of H.
• The number of ones in each column of H is a fixed
  number ?.
• The number of ones in each row of H is a fixed
  number ?.
• Typical parameters for Regular LDPC codes are
  (?,?)(3,6).

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Graph Representation of LDPC Codes

• H is represented by a bipartite graph.
• nodes v (degree ?) on the left represent
  variables.
• Nodes c (degree ?)on the right represent
  equations

Variable nodes
. . .
. . .
Check nodes
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Message-Passing Decoding of LDPC Codes

• Message Passing (or Belief Propagation) decoding
  is a low-complexity algorithm which approximately
  answers the question what is the most likely x
  given y?
• MP recursively defines messages mv,c(i) and
  mc,v(i) from each node variable node v to each
  adjacent check node c, for iteration i0,1,...

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Two Types of Messages...

• Liklihood Ratio
• For y1,...yn independent conditionally on x

• Probability Difference
• For x1,...xn independent

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...Related by the Biliniear Transform

• Definition
• Properties

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Variable to Check Messages

• On any iteration i, the message from v to c is

v
c
. . .
. . .
13
Check to Variable Messages

• On any iteration, the message from c to v is

v
c
. . .
. . .
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Decision Rule

• After sufficiently many iterations, return the
  likelihood ratio

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Theorem about MP Algorithm

• If the algorithm stops after r iterations, then
  the algorithm returns the maximum a posteriori
  probability estimate of xv given y within radius
  r of v.
• However, the variables within a radius r of v
  must be dependent only by the equations within
  radius r of v,

r
...
v
...
...
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Wiring Complexity
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Physical Implementation (VLSI)

• We have seen that the MP decoding algorithm for
  LDPC codes is defined in terms of a graph
• Nodes perform local computation
• Edges carry messages from v to c, and c to v
• Instantiate this graph on a chip!
• Edges ?Wires
• Nodes ?Logic units

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Complexity vs. Performance

• Longer codes provide
• More efficient use of the channel (eg. less power
  used over the AWGN channel)
• Faster throughput for fixed technology and
  decoding parameters (number of iterations)

• Longer codes demand
• More logic resources
• Way more wiring resources

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The Wiring Problem

• The number of edges in the graph grows like the
  number of nodes n.
• The length of the edges in a random graph also
  grows like .

A random graph
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Graph Construction?

• Idea find a construction that has low
  wire-length and maintains good performance...
• Drawback it is difficult to construct any graph
  that has the performance of random graph.

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A Better Solution

• Use an algorithm which generates a graph at
  random, but with a preference for
• Short edge length
• Quantities related to code performance

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Conventional Graph Wisdom

• Short loops give rise to dependent messages
  (which are assumed to be independent) after a
  small number of iterations, and should be
  avoided.

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Simulated Annealing!

• Simulated annealing approximately minimizes an
  Energy Function over a Solution space.
• Requires a good way to traverse the solution
  space.

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Generating LDPC graphs with Simulated Annealing

• Define energy function with two components
• Wirelength
• Loopiness

• traverse the space by picking two edges at random
  and do

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Results of Simulated Annealing

• The graph on the right has nearly identical
  performance to the one shown previously

A graph generated by Simulated Annealing
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Logic Complexity
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Complexity of Classical Algorithm

• Original algorithm defines messages in terms of
  arithmetic operations over real numbers
• However, this implies floating-point addition,
  multiplication, and even division!

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A modified Algorithm

• We define a modified algorithm in which all
  messages are their logarithms in the original
  scheme
• The channel message ? is similarly replaced by
  it's logarithm.

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Quantization

• Replaced a product by a sum, but now we have a
  transcendental function f.
• However, if we quantize the messages, we can
  pre-compute f for all values!

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Quantized MP Performance

• The graph to the following page shows the bit
  error rate for a regular (3,6) of length n10,000
  code using between 2 and 4 bits of quantization.
• (Some error floors predicted by density
  evolution, some are not)

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Quantization Tradeoffs

• A quantizer is characterized by its range and
  granularity
• For fixed channel quantization
• A finely granulated quantizer (Q1) performs well
  at low SNR.
• However, the quantizer must be broadened (Q2) to
  avoid saturation, and resulting error floor.

Q1
Q2
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Conclusion

• There is a tradeoff between logic complexity and
  performance
• Nearly optimal performance (.1 dB 1.03
  power) is achievable with 4-bit messages.
• More work is needed to avoid error-floors due to
  quantization.