Joint SourceChannel Coding for Correlated Senders Over MultipleAccess Channels

1

• Joint Source-Channel Coding for Correlated
  Senders Over Multiple-Access Channels
• Wei Zhong and Javier Garcia-Frias
• Department of Electrical and Computer Engineering
• University of Delaware

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Main Picture
N
coding
S1
Joint Decoder r S1S2SnN
coding
S2
coding
Sn
What is the capacity/limit? How to achieve? Still
an open problem
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System Model
S1...1010110
N
encoder
Joint Decoder
e
S2.0110111
encoder

• S1, S2 are binary sequences
• Correlated with an i.i.d. correlation
  characterized by Pr(e 1)p, i.e. S1 is different
  from S2 with prob. p

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Most Related Work

• Early work
• MAC with arbitrarily correlated sources (T. M.
  Cover et al., 1980)
• Separate source-channel coding not optimum
• Bounds, non-closed form
• Binary correlated sources
• Turbo-like codes for correlated sources over MAC
  (J. Garcia-Frias et al., 2003)
• Turbo codes
• Low Density Generator Matrix (LDGM) codes
• Interleaver design, exploiting correlation
• Correlated source and wireless channels (H. El
  Gamal et al., 2004)
• LDGM codes
• Not a pure MAC, need independent links for a
  small fraction of parity bits

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Theoretical Limits Assuming Separation between
Source and Channel Coding

• Theoretical limit unknown
• The separation limit is achieved by
• Slepian-Wolf source coding optimum channel
  coding

R1

• Ei Energy constraint for sender i (we assume
  E1E2)
• Ri Information rate for sender i (we assume
  R1R2R/2)

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Introduction of LDGM Codes

• Systematic linear codes with sparse generator
  matrix GI P, Ppml
• uu1uL systematic bits
• c uP coded (parity) bits
  
• LDGM codes are LDPC codes, since HGT I is also
  sparse
• Advantage over turbo codes Less decoding
  complexity
• Advantage over standard LDPC codes Less encoding
  complexity

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LDGM Codes in Channel Coding (BSC)

• Message length10,000
• Code rate Rc.5 with different degrees (X,Y)

• As noticed by MacKay, LDGM codes are bad (error
  floor does not decrease with the block length)

• Solution Concatenated scheme

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Serial Concatenated LDGM Codes
For BER10-5, 0.8 dB from theoretical limit,
comparable to LDPC and turbo codes
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LDGM Encoder for Correlated Senders over MAC
Single LDGM encoder per sender
u11 uL1
Sender 1
LDGM Encoder
Ok1
u12 uL2
Sender 2
LDGM Encoder
Ok2

• To exploit correlation, each sender encoded using
  the same LDGM code

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LDGM Encoder for Correlated Senders over MAC
Scheme A
Information bits
Parity bits
Sender 1
Sender 2
Information bits are correlated by
pPr(u1k?u2k) Parity bits are correlated by p
Pr(c1k?c2k)
Parity bits are generated as

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Drawback of Single LDGM Encoder Scheme

• Each sender is encoded by the same LDGM codebook.
• Decoder graph completely symmetric
• At the receiver, even if the decoder can recover
  the sum perfectly, there is no way to tell which
  sequence corresponds to sender 1 and which to
  sender 2
• Solution
• Introduce asymmetry in decoding graph
• Concatenated scheme with additional interleaved
  parity bits

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LDGM Encoder for Correlated Senders over MAC
Concatenated Scheme
u11 uL1
Ok1
Eouter
Einner
Sender 1
Encoder 1
u12 uL2
Channel Interleaver
Sender 2
Eouter
Einner
Ok2
Encoder 2

• Each sender is encoded by a serial concatenated
  LDGM code
• Sender 2s sequence is scrambled by a special
  channel interleaver
• Information bits are not interleaved (most
  correlation preserved).
• Inner coded bits are partially interleaved
  (trade-off between exploiting correlation and
  introducing asymmetry).
• Outer coded bits are totally interleaved (little
  correlation, introduce asymmetry).

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LDGM Decoder for Correlated Senders over MAC
Concatenated Scheme

• Detailed message passing expressions can be
  obtained by applying Belief Propagation over the
  graph

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Simulation Results Single LDGM Scheme

• Information sequences divided into blocks of
  length L10,000
• Rate 1/3 LDGM codes
• P0.01
• Error floor at 0.5p

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Simulation Results Single LDGM Scheme

• As SNR increases,
• Error due to channel noise fades away
• interference stays constant due to the ambiguity
  (symmetry in the decoder graph) explained before
• Comment single LDGM scheme is capable of
  transforming X1X2N into almost noise-free X1X2
  (leaving interference intact)

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Simulation Results Concatenated Scheme

• Trade-off between error floor and threshold,
  driven by fraction of interleaved inner parity
  bits

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Conclusion

• For correlated sources over MAC, code design
  should exploit correlation
• Joint source-channel coding using LDGM codes can
  indeed outperform separate-source-channel coding