Distributed Compression For Binary Symetric Channels

1
Distributed CompressionFor Binary Symetric
Channels

• Kivanc Ozonat

2
Introduction

• Description of the Problem
• Slepian-Wolf Theorem
• Prior Work
• Basic Encoder-Decoder Scheme
• Methodology
• Results

3
Problem Description

• Given two correlated data sets, a noisy version,
  X , at the decoder and the original, Y, at the
  encoder, how to transmit Y with the best coding
  efficiency?
• No communication of X and Y at the encoder

Encoder
Decoder
Y
X
4
Slepian-Wolf Theorem

• Slepian-Wolf Given the following scheme,

(X,Y)
(X,Y)
R1
Encode X
Encode Y
R2
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Slepian-Wolf Theorem

• Can transmit X and Y, if
• - R1 gt H(XY) , R2 gt H(YX), and
• - R1 R2 gt H(X,Y).

R2
H(Y)
H(YX)
R1
H(XY)
H(X)
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Slepian-Wolf Theorem

• Our problem is a special case of this

R2
H(Y)
H(YX)
H(YX)
R1
H(XY)
H(X)
H(X)
7
Prior Work
Bin 1
0 0 0 1 1 1
0 10 1 0 1
Bin 2
0 0 1 1 1 0
0 1 0
Bin 3
0 10 1 0 1
Bin 4
Y 0 1 1
0 1 1 1 0 0
Channel
Decoder
Encoder
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Prior Work

• How to maximally separate very long
• input sequences?
• Use error-correcting codes.

9
Prior Work

1-p
0
0
p
with EQUAL input probabilities of 0 and 1.
p
1
1
1-p
by Ramchandran, Pradhan.
10
Prior Work

• What if
• the input probabilities are NOT EQUAL?

11
Methodology

Plane 1
Bit Plane 1
Huffman Code The Input Sequence X
Decoder
Form the Bins using Error Correcting Codes
Bit Plane 2
Plane 2
Plane N
Bit Plane N
Y sequence
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Encoder

• Inputs 0 (with probability .7) and 1 (with
  probability .3)
• Huffman code 2-length sequences
• 00 ? 0 (with probability .49)
  
• 01 ? 10 (with probability .21)
• 10 ? 110 (with probability .21)
• 11 ? 111 (with probability .09)
• Bit-Plane 1 0, 1 , 1 ,1
• Bit-Plane 2 -, 0 , 1 ,1
• Bit-Plane 3 - , - , 0 ,1

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Encoder

001001
00, 10, 01
Error Control Coding To Form Bins
011
0, 110, 10
-10
-0-
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Decoder

• Decoder receives a BIN NUMBER, which corresponds
  to
• MULTIPLE CODEWORDS.
• How to select the correct codeword out of these
  multiple codewords?
• Use MAXIMUM LIKELIHOOD detection.

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Decoder

011
Decoder
Bin 4
011
110
This is what the decoder receives
Huffman codes for 2 length sequences z1 z2 z3

Assume Y 01, 11, 10 Compute the probability of
z1 z2 z3 given 01,11,10, using the channel
error probability.
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Parameters

Plane 1
Bit Plane 1
Huffman Code The Input Sequence X
Decoder
Form the Bins using Error Correcting Codes
Bit Plane 2
Plane 2
Plane N
Bit Plane N
Length 4
Use BCH (15,k)
Y sequence
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Bit Rate vs. Probability of Occurrence of 0s(at
the fixed error rate p of 0.06)
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Difference between the Actual Bit Rate and the
Slepian-Wolf Bound vsError Probability (p)
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Conclusions

• Huffman Code is not a very good choice
• Better error correcting codes can be selected.
• Gives good results for low error (p) cases
• and for cases in which the Huffman code
  gives nearly equal distribution of 0s and 1s.