1. Distributed Compression For Binary Symetric Channels
Kivanc Ozonat
Distributed Compression For Binary Symetric Channels

2. Distributed Compression For Binary Symetric Channels
Introduction
Description of the Problem
Slepian-Wolf Theorem
Prior Work
Basic Encoder-Decoder Scheme
Methodology
Results
Distributed Compression For Binary Symetric Channels

3. Distributed Compression For Binary Symetric Channels
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
Distributed Compression For Binary Symetric Channels

4. Distributed Compression For Binary Symetric Channels
Slepian-Wolf Theorem
Slepian-Wolf : Given the following scheme,
(X,Y)
(X,Y) R1
Encode X
Encode Y R2
Distributed Compression For Binary Symetric Channels

5. Distributed Compression For Binary Symetric Channels
Slepian-Wolf Theorem
Can transmit X and Y, if:
- R1 > H(X|Y) , R2 > H(Y|X), and
- R1+ R2 > H(X,Y). R2
H(Y)
H(Y|X) R1
H(X|Y)
H(X)
Distributed Compression For Binary Symetric Channels

6. Distributed Compression For Binary Symetric Channels
Slepian-Wolf Theorem
Our problem is a special case of this: R2
H(Y)
H(Y|X)
H(Y|X) R1
H(X|Y)
H(X)
H(X)
Distributed Compression For Binary Symetric Channels

7. Distributed Compression For Binary Symetric Channels
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
Distributed Compression For Binary Symetric Channels

8. Distributed Compression For Binary Symetric Channels
Prior Work
How to maximally separate “very long”
input sequences?
Use error-correcting codes.
Distributed Compression For Binary Symetric Channels

9. Distributed Compression For Binary Symetric Channels
Prior Work
1-p p
with EQUAL input probabilities of 0 and 1. p 1 1
1-p
by Ramchandran, Pradhan.
Distributed Compression For Binary Symetric Channels

10. Distributed Compression For Binary Symetric Channels
Prior Work
What if
the input probabilities are NOT EQUAL?
Distributed Compression For Binary Symetric Channels

11. Distributed Compression For Binary Symetric Channels
Methodology
Plane 1
Bit Plane 1
Huffman
Code
The Input
Sequence X
Form the
Bins using
Error
Correcting
Codes
Decoder
Bit Plane 2
Plane 2
Plane N
Bit Plane N
Y sequence
Distributed Compression For Binary Symetric Channels

12. Distributed Compression For Binary Symetric Channels
Encoder
Inputs: 0 (with probability .7) and 1 (with probability .3)
Huffman code 2-length sequences:
00  (with probability .49)
01  (with probability .21)
10  (with probability .21)
11  (with probability .09)
Bit-Plane 1: 0, 1 , 1 ,1
Bit-Plane 2: -, 0 , 1 ,1
Bit-Plane 3: - , - , 0 ,1
Distributed Compression For Binary Symetric Channels

13. Distributed Compression For Binary Symetric Channels
Encoder
[001001]
[00], [10], [01]
Error
Control
Coding
To Form
Bins
011
[0], [110], [10]
-10
-0-
Distributed Compression For Binary Symetric Channels

14. Distributed Compression For Binary Symetric Channels
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.
Distributed Compression For Binary Symetric Channels

15. Distributed Compression For Binary Symetric Channels
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.
Distributed Compression For Binary Symetric Channels

16. Distributed Compression For Binary Symetric Channels
Parameters
Plane 1
Bit Plane 1
Huffman
Code
The Input
Sequence X
Form the
Bins using
Error
Correcting
Codes
Decoder
Bit Plane 2
Plane 2
Plane N
Bit Plane N
Length 4
Use BCH
(15,k)
Y sequence
Distributed Compression For Binary Symetric Channels

17. Distributed Compression For Binary Symetric Channels
Bit Rate vs. Probability of Occurrence of 0’s (at the fixed error rate p of 0.06)
Distributed Compression For Binary Symetric Channels

18. Distributed Compression For Binary Symetric Channels
Difference between the Actual Bit Rate and the Slepian-Wolf Bound vs Error Probability (p)
Distributed Compression For Binary Symetric Channels

19. Distributed Compression For Binary Symetric Channels
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.
Distributed Compression For Binary Symetric Channels