Soft Decision Decoding of RS Codes Using Adaptive Parity Check Matrices

1
Soft Decision Decoding of RS Codes Using Adaptive
Parity Check Matrices

• Jing Jiang and Krishna R. Narayanan
• Wireless Communication Group
• Department of Electrical Engineering
• Texas AM University

2
Reed Solomon Codes

• Consider an (n,k) RS code over GF(2m), n 2m-1
• Linear block code e.g. (7,5) RS code over GF(8)
• ? be a primitive element in GF(8)

• Cyclic shift of any codeword is also a valid
  codeword
• RS codes are MDS (dmin n-k1)
• The dual code is also MDS

3
Introduction

• Advantages
• Guaranteed minimum distance
• Efficient bounded distance hard decision decoder
  (HDD)
• Decoder can handle errors and erasures

• Drawback
• Performance loss due to bounded distance decoding
• Soft input soft output (SISO) decoding is not
  easy!

4
Presentation Outline

• Existing soft decision decoding techniques

• Iterative decoding based on adaptive parity check
  matrices

• Variations of the generic algorithm

• Applications over various channels

• Conclusion and future work

5
Existing Soft Decoding Techniques
6
Enhanced Algebraic Hard Decision Decoding

• Generalized Minimum Distance (GMD) Decoding
  (Forney 1966)
• Basic Idea
• Erase some of the least reliable symbols
• Run algebraic hard decision decoding several
  times
• Drawback GMD has a limited performance gain

• Chase decoding (Chase 1972)
• Exhaustively flip some of the least reliable
  symbols
• Running algebraic hard decision decoding several
  times
• Drawback Has an exponentially increasing
  complexity

• Combined Chase GMD(Tang et al. 2001).

7
Algebraic Soft Input Hard Output Decoding

• Algebraic SIHO decoding
• Algebraic interpolation based decoding (Koetter
  Vardy 2003)
• Reduced complexity KV algorithm (Gross et al.
  submitted 2003)

• Basic ideas
• Based on Guruswami and Sudans algebraic list
  decoding
• Convert the reliability information into a set of
  interpolation points
• Generate a list of candidate codewords
• Pick up the most likely codeword from the
  codeword list

8
Reliability based Ordered Statistic Decoding

• Reliability based decoding
• Ordered Statistic Decoding (OSD) (Fossorier
  Lin 1995)
• Box Match Algorithm(BMA) (Valembois
  Fossorier to appear 2004)

• Basic ideas
• Order the received bits according to their
  reliabilities
• Make hard decisions on a set of independent
  reliable bits (MR Basis)
• Re encode to obtain a list of candidate codewords

• Drawback
• The complexity increases exponentially with the
  reprocessing order
• BMA must trade memory for complexity

9
Trellis based Decoding using the Binary Image
Expansion

• Maximum-likelihood decoding and variations
• Trellis based decoding using binary image
  expansion (Vardy Beery 91)
• Reduced complexity version (Ponnampalam
  Vucetic 2002)

• Basic ideas
• Binary image expansion of RS
• Trellis structure construction using the binary
  image expansion

• Drawback
• Exponentially increasing complexity
• Work only for very short codes or codes with very
  small distance

10
Binary Image Expansion of RS Codes
11

• Consider the (7,5) RS code

12
Recent Iterative Techniques

• Sub-trellis based iterative decoding (Ungerboeck
  2003)

• Self-concatenation structure based on
  sub-trellis constructed from the parity check
  matrix

• Drawbacks
• Performance deteriorates due to large number of
  short cycles
• Work for short codes with small minimum
  distances
• Potential error floor problem in high SNR region

13
Recent Iterative Techniques (contd)

• Stochastic shifting based iterative decoding
  (Jing Narayanan, to appear 2004)

• Due to the irregularity in the H matrix,
  iterative decoding favors some bits
• Taking advantage of the cyclic structure of RS
  codes

Shift by 2

• Stochastic shift prevent iterative procedure
  from getting stuck

• Best result RS(63,55) about 0.5dB gain from HDD
• However, for long codes, this algorithm still
  doesnt provide good improvement

14
Remarks on Existing Techniques

• Most SIHO algorithms are either too complex to
  implement or having only marginal gain

• Moreover, SIHO decoders cannot generate soft
  output directly

• Trellis-based decoders have exponentially
  increasing complexity

• Iterative decoding algorithms do not work for
  long codes, since the parity check matrices of RS
  codes are not sparse

• Soft decoding of large RS codes as employed in
  many standard transmission systems, e.g.,
  RS(255,239), with affordable complexity remains
  an open problem (Ungerboeck, ISTC2003)

15
Questions

• Q Why doesnt iterative decoding work for codes
  with non-sparse parity check matrices?

• Q Can we get some idea from the failure of
  iterative decoder?

16
How does standard message passing algorithm work?

• If two or more of the incoming messages are
  erasures the check is erased
• Otherwise, check to bit message is the value of
  the bit that will satisfy the check

17
How does standard message passing algorithm work?
Small values of vj can be thought of as erasures
and hence more than two edges with small vjs
saturate the check
18
A Few Unreliable Bits Saturate the Non-sparse
Parity Check Matrix

• Consider RS(7, 5) over GF(23)

• Iterative decoding is stuck due to only a few
  unreliable bits saturating the whole non-sparse
  parity check matrix

19
Sparse Parity Check Matrices for RS Codes

• Can we find an equivalent binary parity check
  matrix that is sparse?

• For RS codes, this is not possible!

• The H matrix is the G matrix of the dual code
• The dual of an RS code is also an MDS Code
• Every row has weight at least (N-K)!

20
Iterative Decoding Based on Adaptive Parity Check
Matrix

• Idea reduce the sub-matrix corresponding to the
  unreliable positions to a sparse nature.

• For example, consider (7,4) Hamming code

• After the adaptive update, iterative decoding
  can proceed.

21
Adaptive Decoding Procedure
22
More Details about the Matrix Adaptive Scheme

• Consider the previous example (7,4)Hamming code

parity check matrix

• We can guaranteed reduce some (n-k)m columns to
  degree 1
• We attempt to chose these to be the least
  reliable independent bits
• Least Reliable Basis

23
Interpretation as an Optimization Procedure

• Standard iterative decoding procedure is
  interpreted as gradient descent optimization
  (Lucas et al. 1998).

• Proposed algorithm is a generalization, two-stage
  optimization procedure

• Parity check matrix update (change direction)
• All bit-level reliabilities are sorted by their
  absolute values
• Systemize the sub-matrix corresponding to LRB in
  the parity check matrix

• The damping coefficient serves to control the
  convergent dynamics.

24
A Hypothesis
Adaption help gradient descent to converge
25
Complexity Analysis

• Complexity can be even reduced when implemented
  in parallel

26
Complexity Comparison
Method Dominant Complexity
GMD
Chase
KV
OSD
Trellis
ADP
27
Variation1 Symbol-level Adaptive Scheme

• Systemizing the sub-matrix involves undesirable
  Gaussian elimination.

• This problem can be detoured via utilizing the
  structure of RS codes.

• We implement Symbol-level adaptive scheme.

This step can be efficiently realized using
Forneys algorithm (Forney 1965)
28
Variation2 Degree-2 sub-graph in the unreliable
part

• Reduce the unreliable sub-matrix to a sparse
  sub-graph rather than an identity to improve the
  asymptotic performance.

29
Variation2 Degree-2 sub-graph in the unreliable
part (contd)

• Q How to adapt the parity check matrix?

30
Variation3 Different grouping of unreliable bits
(contd)

• Some bits at the boundary part may also have the
  wrong sign.

• Run the proposed algorithm several times, each
  time with an exchange of some reliable and
  unreliable bits at the boundary.

• Consider the received LLR of an RS(7,5) code

.

• A list of candidate codewords are generated using
  different groups. Pick up the most likely from
  the list.

31
Variation4 Partial updating scheme (contd)

• The main complexity comes from updating the bits
  in the high density part, however, only few bits
  at the boundary part will be affected.

• In variable node updating stage update only the
  unreliable bits in the sparse sub-matrix and a
  few reliable bits at the boundary part.

• In check node updating stage make an
  approximation of the check sum via taking
  advantage of the ordered reliabilities.

32
Applications

• Q How do the proposed algorithm and its
  variations perform?

• Simulation results
• Proposed algorithm and variations over AWGN
  channel
• Performance over symbol level fully interleaved
  slow fading channel
• RS coded turbo equalization (TE) system over EPR4
  channel
• RS coded modulation over fast fading channel

• Simulation setups
• A genie aided HDD is assumed for AWGN and
  fading channel.
• In the TE system, all coded bits are interleaved
  at random. A genie aided stopping rule is
  applied.

33
Additive White Gaussian Noise (AWGN) Channel
34
AWGN Channels
35
AWGN Channels (contd)
36
AWGN Channels (contd)
37
AWGN Channels (contd)
38
Remarks

• Proposed scheme performs near ML for medium
  length codes.

• Symbol-level adaptive updating scheme provides
  non-trivial gain.

• Partial updating incurs little penalty with great
  reduction in complexity.

• For long codes, proposed scheme is still away
  from ML decoding.

• Q How does it work over other channels?

39
Interleaved Slow Fading Channel
40
Fully Interleaved Slow Fading Channels
41
Fully Interleaved Slow Fading Channels (cont.)
42
Turbo Equalization Systems
43
Embed the Proposed Algorithm in the Turbo
Equalization System
44
Turbo Equalization over EPR4 Channels
45
Turbo Equalization over EPR4 Channels
46
RS Coded Modulation
47
RS Coded Modulation over Fast Rayleigh Fading
Channels
48
RS Coded Modulation over Fast Rayleigh Fading
Channels (contd)
49
Remarks

• More noticeable gain is observed for fading
  channels, especially for symbol-level adaptive
  scheme.

• In RS coded modulation scheme, utilizing
  bit-level soft information seems provide more
  gain.

• The proposed TE scheme can combat ISI and
  performs almost identically as the performance
  over AWGN channels.

• The proposed algorithm has a potential error
  floor problem.
• However, simulation down to even lower FER is
  impossible.
• Asymptotic performance analysis is still under
  investigation.

50
Conclusion and Future work

• Iterative decoding of RS codes based on adaptive
  parity check matrix works favorably for practical
  codes over various channels.

• The proposed algorithm and its variations provide
  a wide range of complexity-performance tradeoff
  for different applications.

• More works under investigation
• Asymptotic performance bound.
• Understanding how this algorithm works from an
  information theoretic perspective, e.g., entropy
  of ordered statistics.
• Improving the generic algorithm using more
  sophisticated optimization schemes, e.g.,
  conjugate gradient method.

51
Thank you!