Distributed Compression For Binary Symetric Channels

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

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

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 Distributed Compression For Binary Symetric Channels

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

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

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

11. 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. 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 Distributed Compression For Binary Symetric Channels

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

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

19. 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