Sequential Soft Decision Decoding of Reed Solomon Codes

1
Sequential Soft Decision Decoding of Reed Solomon
Codes

• Hari Palaiyanur
• Cornell University
• Prof. John Komo
• Clemson University
• 2003 SURE Program

2
Outline

• Background
• System Model
• Stack and Bucket Algorithms
• Results
• Conclusions

3
Background

• Error Control Coding adding redundancy to
  improve reliability over noisy channel
• Reed Solomon Codes (n,k) cyclic block codes
  over GF(qm), n qm 1
• (n1)k possible code words
• Erasure if symbol is unreliable, denote it as
  an erasure

4
Background

• Soft Decision Decoding takes advantage of side
  information, i.e. quality of received signal
• Sequential decoding searches through tree of
  possible code words

5
Motivation and History

• Soft decision decoding more reliable
• Soft decision decoding more decoding time
• Good, efficient errors only and errors and
  erasures decoders available
• Still need good, efficient soft decision decoders
• Stack/Bucket Algorithm - 1969

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System Model

• MFSK over AWGN channel
• For each symbol, detector outputs
• mi As ni (i sent)
• mi ni (i not sent)
• ni indep. G. R. V.

RS Encoder
MFSK Modulator
Data
AWGN Channel
Coherent MFSK Detector
Sequential RS Decoder
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Stack Algorithm
Start and Load stack with 2m initial nodes
P is Head of Stack
Len P k-1?
No
Yes
Remove P Search 2m forward nodes Update
Metrics Push onto Stack
Remove P Search 2m forward nodes and
encode Update Metrics Push onto Stack
Len P n?
No
Yes
Done. Decoded code word is P
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Metrics and RS (7,3) Example

• Metrics provide information about quality of
  symbol
• Forward metric is sum of previous metric and
  metric for added symbol
• Good metric
• mi max(mj)

Symbol 0 1 2 3 4 5 6
7
Mi 3.3 0.2 1.5 0.1 1.1 0.7 2.0
0.8
Node 0 0.0
Node 1 -3.1
Node 2 -4.8
Node 3 -3.4
Start
Node 4 -2.2
Node 5 -2.6
Node 6 -5.3
Node 7 -2.5
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Stack Algorithm

• Algorithm goes through many unnecessary code
  words
• Perform quick erasure decoding
• Let threshold be this code words metric
• Only push nodes onto stack if their metric is at
  least threshold

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Bucket Algorithm

• Instead of one sorted stack, use many buckets
• Buckets have certain metric gradations
• No need to keep code words sorted, if gradations
  are fine enough
• Sacrifice memory for speed

Metrics 0.0 to -5.0
Metrics -5.0 to -10.0
Metrics -10.0 to -15.0
Metrics Below -15.0
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Results RS (15,9)
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Results Number of Searches
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Results RS (31,21)
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Conclusions

• Soft decision decoding gives better reliability
  over hard decision decoding
• Stack algorithm adapted to RS codes for
  sequential soft decision decoding

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Future Work

• Other transmissions schemes including
  non-coherent MFSK
• Other channel models
• Quantize metrics to a limited number of bits

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Acknowledgements

• Prof. Komo for the project ideas and guidance
• Prof. Russell and Fred Block
• Profs. Noneaker and Xu for the program overall

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Questions?

• ?

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References

• 1 S.B. Wicker, Error Control Systems for
  Digital Communication and Storage, Englewood
  Cliffs, NJ Prentice Hall, 1995.
• 2 Komo, J.J. and L.L. Joiner, "Fast Error
  Magnitude Evaluations for Reed-Solomon Codes,"
  Proc. 1995 IEEE International Symposium on
  Information Theory, p. 416, Sep. 1995.
• 3 F. Jelinek, A Fast Sequential Decoding
  Algorithm Using a Stack, IBM Journal of
  Research and Development, Vol. 13, pp. 675-685,
  Nov. 1969

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Stack Algorithm RS(7,3) Code