1. BASiCS Group University of California at Berkeley Generalized Coset Codes for Symmetric/Asymmetric Distributed Source Coding S. Sandeep Pradhan Kannan Ramchandran {pradhan5, kannanr}@eecs.berkeley.edu

2. University of California, Berkeley Outline Introduction and motivation Preliminaries Generalized coset codes for distributed source coding Simulation results Conclusions

3. University of California, Berkeley Application: Sensor Networks Scene Sensor 1 Encoder Sensor 2 Encoder Sensor 3 Encoder Channels are bandwidth or rate-constrained Joint Decoding

4. University of California, Berkeley Introduction and motivation Distributed source coding Information theoretic results (Slepian-Wolf ‘73, Wyner-Ziv, ‘76) Little is known about practical systems based on these elegant concepts Applications: Distributed sensor networks/web caching, ad-hoc networks, interactive comm. Goal: Propose a constructive approach (DISCUS) (Pradhan & Ramchandran, 1999)

5. University of California, Berkeley Source Coding with Side Information at Receiver (illustration) X and Y => length-3 binary data (equally likely), Correlation: Hamming distance between X and Y is at most 1. Example: When X=[0 1 0], Y => [0 1 0], [0 1 1], [0 0 0], [1 1 0]. EncoderDecoder X Y X and Y correlated Y at encoder and decoder System 1 X+Y= 0 0 0 0 0 1 0 1 0 1 0 0 Need 2 bits to index this.

6. University of California, Berkeley What is the best that one can do? X EncoderDecoder Y X and Y correlated Y at decoder System 2 The answer is still 2 bits! How? 0 0 0 1 1 1 Coset-1 000 001 010 100 111 110 101 011 X Y

7. University of California, Berkeley Encoder -> index of the coset containing X. Decoder -> X in given coset. Note: Coset-1 -> repetition code. Each coset -> unique “syndrome” DIstributed Source Coding Using Syndromes Coset-1 Coset-4 Coset-3Coset-2

8. University of California, Berkeley Symmetric Coding X and Y both encode partial information Example: X and Y -> length-7 equally likely binary data. Hamming distance between X and Y is at most 1. 1024 valid X,Y pairs Solution 1: Y sends its data with 7 bits. X sends syndromes with 3 bits. { (7,4) Hamming code } -> Total of 10 bits Can correct decoding be done if X and Y send 5 bits each ? EncoderDecoder Y

9. University of California, Berkeley Solution 2: Map valid (X,Y) pairs into a coset matrix 1 2 3... 32 32. 2 1 Coset Matrix Y X Construct 2 codes, assign them to encoders Encoders -> index of coset of codes containing the outcome

10. University of California, Berkeley 1 0 1 1 0 1 0 0 1 0 0 1 0 1 0 1 1 0 0 1 0 1 1 1 0 0 0 1 G = 1 0 1 1 0 1 0 0 1 0 0 1 0 1 0 1 1 0 0 1 0 1 1 1 0 0 0 1 G 1 = G 2 = Example Theorem 1: With (n,k,2t+1) code, X and Y -> rate pairs (R 1,R 2 ) : This concept can be generalized to Euclidean-space codes.

11. University of California, Berkeley Achievable Rate Region for the Problem The rate region is: 3 4 5 6 7 7654376543 All 5 optimal points can be constructively achieved with the same complexity. An alternative to source-splitting approach (Rimoldi-97)

12. University of California, Berkeley Generalized coset codes: (Forney, ’88) S = lattice S’=sublattice Construct sequences of cosets of S’ in S in n-dimensions S’ -5.5-4.5-3.5-2.5-1.5-0.50.51.52.53.54.55.5 Example: S -4.5-2.5-0.51.53.55.5

13. University of California, Berkeley C = 0 0 1 0 1 1 Example: Let n=4 c=1011 1 0 1 1 sequence coming from the above sets -> valid codeword sequence -2.5 2.5 -0.5 -4.5 4-d Euclidean space code -4.5-2.5-0.51.53.55.5 -5.5-3.5-1.50.52.54.5 -4.5-2.5-0.51.53.55.5 -4.5-2.5-0.51.53.55.5

14. University of California, Berkeley Generalized coset codes for distributed source coding xxxxxxxxxxxxxxxxxxxxx 13579 13 19 xx 25 -5 -17 -23-11 xx 17 13 1925 -5 -17 -11 1 19 -17 17 13 6 Two-level hierarchy of subcode construction: Subset -> encoder 1 Subset -> encoder 2

15. University of California, Berkeley Example 2:

16. University of California, Berkeley

17. is a sublattice of

18. University of California, Berkeley is the set of coset representatives of in

19. University of California, Berkeley Encoders -> index of subsets in dense lattice  containing quantized codewords 1 1 2 3 4 1 2 3

20. University of California, Berkeley Encoding: Encoders quantize with main lattice Index of the coset of subsets in the main lattice is sent Decoding: Decoder -> pair of codewords in the given coset pairs Estimate the source Similar subcode construction for generalized coset code Computationally efficient encoding and decoding Theorem 2: Decoding complexity = decoding a codeword in

21. University of California, Berkeley Correlation distance d c => second minimum distance between 2 codevectors in coset pairs i,j Decoding error => distance between quantized codewords > d c. Theorem 3: d min => min. distance of the code 1 1 2 3 4 1 2 3

22. University of California, Berkeley Simulation Results:Trellis codes Model: Source = X~ i.i.d. Gaussian, Observation= Y i = X+N i, where N i ~ i.i.d. Gaussian. Correlation SNR= ratio of variances of X and N. Effective Source Coding Rate = 2bit / sample/encoder. Quantizers: Fixed-length scalar quantizers with 8 levels. Trellis codes with 16- states based on 8 level root scalar quantizer

23. University of California, Berkeley Prob. of decoding error Results Same prob. of decoding error for all the rate pairs

24. University of California, Berkeley Distortion Performance : Attainable Bound: C-SNR=22 dB, Normalized distortion: -15.5 dB

25. University of California, Berkeley Special cases: 2. Lattice codes Encoder-1 Hexagonal Lattice

26. University of California, Berkeley Conclusions Proposed constructive framework for distributed source coding -> arbitrary achievable rates Generalized coset codes for framework Distance properties & complexity -> same for all achievable rate points Trellis & lattice codes -> special cases Simulations