1. Multi-Edge Framework for Unequal Error Protecting LT Codes H. V. Beltr˜ao Neto, W. Henkel, V. C. da Rocha Jr. Jacobs University Bremen, Germany IEEE ITW(Information Theory Workshop) 2012 1

2. Outlines Introduction Construction algorithms for UEP LT Codes Simulation results Conclusions 2

3. Introduction Rateless codes – LT codes, Raptor codes, Online codes, … – can adapt their rate on the fly to suit different channel conditions – are interesting for multicast transmission since they eliminate the requirement for retransmission 3

4. Introduction The multi-edge framework was originally derived for Low Density Parity-Check codes (LDPC) in [7]. Several edge classes can be defined within the bipartite graph induced by the encoding Every node is characterized by the number of connections to edges of each class. 4 [7] T. Richardson and R. Urbanke, “Multi-Edge Type LDPC Codes,” Tech. Rep., 2004, submitted to IEEE Transaction on Information Theory.

5. Introduction m e : the number of edge types used to represent the bipartite graph Each node in the bipartite graph has 2 vectors – x = (x 1,..., x me ) that indicates the different types of edges connected to it – d = (d 1,..., d me ) referred to as edge degree vector which denotes the number of connections of a node to edges of type i, 5

6. Introduction x d = L d k : the number of variable nodes (input symbol) of type(degree) d R d k : the number of check nodes (output symbol) of type(degree) d 6

7. Introduction type-1 edge (solid lines) type-2 edges (dashed lines) 7 34422232

8. Introduction The multi-edge degree distributions for the code depicted in Fig. 1 are: Divide the variable nodes into m e protection classes (C 1,C 2,...,C me ) with monotonically decreasing levels of protection 8

9. Introduction The degree of an output symbol corresponds to the number of edges connected to it d j : the number of edges of type j connected to a check node Ω i : the probability of an output symbol having degree i Ω(x) : the overall output symbol degree distribution 9

10. Introduction Asymptotically (as k increases to infinity), we can approximate Eq. (6) by Ω j : the probability of an input symbol of the class C j being chosen among the k input symbols The check node degree distribution is 10

11. Introduction Codes with only 2 protection classes, i.e., codes with m e = 2 Eqs. (8) and (10) are reduced to 11

12. Construction algorithms for UEP LT Codes Weighted approach[5] – Partition of the k variable nodes into m e sets of sizes α 1 k, α 2 k,…, α me k, such that – The probability of an edge being connected to a particular variable node within the set j being q j – The selection probabilities are defined as ω 1 = αk M and ω 2 = (1 −α)K L α is the fraction of input symbols that belong to the first class and, k M > 1 and 0 < k L < 1 are assigned to the set of more important bits (MIB) and less important bits (LIB) 12 [5] N. Rahnavard, B. N. Vellambi, and F. Fekri, “Rateless codes with unequal error protection property,” IEEE Transactions on Information Theory, vol. 53, no. 4, pp. 1521–1532, April 2007.

13. 13 [5] N. Rahnavard, B. N. Vellambi, and F. Fekri, “Rateless codes with unequal error protection property,” IEEE Transactions on Information Theory, vol. 53, no. 4, pp. 1521–1532, April 2007.

14. UEP at the Rateless Encoding Stage Type-1 Codes Weakness – Change of degree distribution (input nodes) – It is likely that d 1 = 0 for low-degree encoding nodes … … … d 1 = min([(K 1 /K)dk M,K 1 ]d 2 = d-d 1 … K1K1 K2K2 14 N. Rahnavard and F. Fekri, “Finite-length unequal error protection rateless codes: Design and analysis,” in IEEE GLOBECOM 2005. http://www.powercam.cc/show.php?ch=12&fid=74&id=238

15. Construction algorithms for UEP LT Codes Windowed approach[6] – Partition the input symbols into protection classes of k 1, k 2,..., k me symbols such that k 1 +k 2 +... +k me = k – The i-th window is defined as the set of the first input symbols – The most important symbols form the first window while the whole block comprises the final m e th window. 15 [6] D. Sejdinovi´c, D. Vukobratovi´c, A. Doufexi, V. ˇSenk, and R. Piechocki, “Expanding window fountain codes for unequal error protection,” IEEE Transactions on Communications, vol. 57, no. 9, pp. 2510–2516, Sep. 2009.

16. Construction algorithms for UEP LT Codes 16 Each output symbol is encoded first selecting a window i, with each window having associated to it a probability Г i of being chosen

17. EWF(Expanding Window Fountain) Codes Notation – EWF codes are applied on consecutive source blocks of k symbols (data packets). – The sequence of r expanding windows, where each window is contained in the next window in the sequence. The number r of expanding windows is equal to the number of importance classes of the source block. – the size of the i-th window as k i, where k 1 <…<k r =k s 1 =k 1 s 1 +s 2 +…+s r = k s i = k i – k i-1 – the division of the source block into importance classes 17 http://www.powercam.cc/slide/19578

18. EWF Codes 18 http://www.powercam.cc/slide/19578

19. Fig. 1. Expanding window fountain (EWF) codes. 19 http://www.powercam.cc/slide/19578

20. EWF Codes 2 importance classes – The expressions for the erasure probabilities of Most Important Bit (MIB) class and Least Important Bit (LIB) class after l iterations 20 http://www.powercam.cc/slide/19578

21. Flexible UEP LT Codes An LT code with 2 protection classes. Its check node degree distribution can be written as for d = (d 1, d 2 ) and To keep the original overall output symbol degree distribution Ω(x), the coefficients R d must satisfy 21

22. Flexible UEP LT Codes In the two-class case, R (1,0) + R (0,1) = Ω 1, R (2,0) + R (1,1) + R (0,2) = Ω 2...... The idea of our proposed scheme : – Increase the probability of selection of the most important input symbols – Increase the occurrence probability of output symbols which are more connected to input symbols of the most sensitive class – Increase the values of the coefficients R (d1, d2) with d 1 > d 2 22

23. Flexible UEP LT Codes The encoding of the flexible UEP LT codes is similar to the traditional LT codes One must select an edge degree vector d according to R d : the probability of the edge degree vector d being chosen An output symbol with edge degree vector d = (d 1, d 2 ) is formed by selecting d 1 input symbols from C 1, d 2 input symbols from C 2 uniformly and at random, and performing a bitwise XOR operation between them. 23

24. Flexible UEP LT Codes LT code with degree distribution Ω(x) = 0.15x + 0.55x 2 + 0.30x 3 Construct a two-class UEP LT code where 10% of the input symbols belong to the most protected class, i.e., α = 0.1. Compute the coefficients R d for the non-UEP case by means of Eq. (11) with ω 1 = α and ω 2 = 1 − α. We will refer to the EEP LT codes check node coefficients as R d and the ones of UEP LT codes as R d UEP Increase the values of the coefficients R (d1,d2) with d 1 > d 2 24

25. Flexible UEP LT Codes A simple way of realizing this is to transfer a fraction f of a coefficient R (a,b) where a < b to the coefficient R (b,a) – If f = 0.1 In order to further refine the performance of the UEP LT codes, we can define different values of f – f = f 1 for symbols with d 2 = 0 and f = f 2 otherwise. 25

26. Asymptotic analysis of multi-edge LT codes Theorem 1: The erasure probability of an input symbol of class j of a multi-edge LT code with node perspective degree distribution pair (L(x),R(x)) at iteration l ≥ 0 is given by where y −1 = 1, denotes the fraction of type j (j = 1,...,m e ) edges connected to check nodes of type d. 26

27. Asymptotic analysis of multi-edge LT codes Multi-edge UEP LT codes with the overall output symbol degree distribution proposed in [2] 27 [2] A. Shokrollahi, “Raptor codes,” IEEE Transactions on Information Theory, vol. 52, no. 6, pp. 2551– 2567, June 2006.

28. Asymptotic analysis of multi-edge LT codes Parameters – The weighted approach (k m = 2.077) – The windowed approaches (Г 1 = 0.084) are optimized for an overhead γ = 1.05 according to [6] The flexible UEP LT performance was obtained for f 1 = 0.09 and f 2 = 0.13 28 [6] D. Sejdinovi´c, D. Vukobratovi´c, A. Doufexi, V. ˇSenk, and R. Piechocki, “Expanding window fountain codes for unequal error protection,” IEEE Transactions on Communications, vol. 57, no. 9, pp. 2510–2516, Sep. 2009.

29. Fig. 2. Asymptotic performance of the weighted, windowed and the proposed flexible UEP LT construction strategies. The parameters of the three approaches were optimized for an overhead γ = 1.05. 29 1.05

30. Simulation Results Settings – Assume the transmission of k = 5000 input symbols divided into 2 different levels of protection – The first protection class is composed of 10% of the input symbols (k 1 = 0.1k) – The second protection class contains the other k 2 = k − k 1 input symbols 30

31. 31 Fig. 3. Simulation results of the weighted and flexible schemes for k = 5000. 1.075 1.05

32. Conclusions We introduced a multi-edge type analysis of unequal error protecting LT codes. Our proposed scheme performed better than the weighted and the windowed schemes. 32

33. References [1] M. Luby, “LT codes,” in Proceedings of the 43rd Annual IEEE Symposium on Foundations of Computer Science, Nov. 2002, pp. 271–282. [2] A. Shokrollahi, “Raptor codes,” IEEE Transactions on Information Theory, vol. 52, no. 6, pp. 2551–2567, June 2006. [3] P. Maymounkov, “Online codes,” NYU, Tech. Rep. TR2003-883, Nov. 2002. [5] N. Rahnavard, B. N. Vellambi, and F. Fekri, “Rateless codes with unequal error protection property,” IEEE Transactions on Information Theory, vol. 53, no. 4, pp. 1521–1532, April 2007. [6] D. Sejdinovi´c, D. Vukobratovi´c, A. Doufexi, V. ˇSenk, and R. Piechocki, “Expanding window fountain codes for unequal error protection,” IEEE Transactions on Communications, vol. 57, no. 9, pp. 2510–2516, Sep. 2009. [7] T. Richardson and R. Urbanke, “Multi-Edge Type LDPC Codes,” Tech. Rep., 2004, submitted to IEEE Transaction on Information Theory. [8] ——, Modern Coding Theory. Cambridge University Press, 2008. [9] M. Luby, M. Mitzenmacher, and A. Shokrollahi, “Analysis of random processes via and-or tree evaluation,” in Proceedings of the 9th SIAM Symposium on Discrete Algorithms, Jan. 1998, pp. 364–373. 33