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Distributed Rateless Codes with UEP Property

Distributed Rateless Codes with UEP Property. Ali Talari , Nazanin Rahnavard 2010 IEEE ISIT(International Symposium on Information Theory) & IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 60, NO. 8, AUGUST 2012. Outline. Introduction Distributed UEP rateless codes

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Distributed Rateless Codes with UEP Property

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  1. Distributed Rateless Codes with UEP Property Ali Talari, NazaninRahnavard 2010 IEEE ISIT(International Symposium on Information Theory) & IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 60, NO. 8, AUGUST 2012

  2. Outline • Introduction • Distributed UEP rateless codes • Distributed UEP rateless codes design • Simulation Results • Conclusions

  3. Introduction • Fountain codes - efficient and robust solution for data transmission over packet erasure networks • Rateless codes - can adapt their rate on the fly to suit different (or unknown) channel conditions • A kind of sparse graph codes; typically decoded by belief propagation algorithm (low computational complexity) • Increasingly commercialized, standardized by 3GPP and DVB

  4. Distributed LT codes • Introduced in [2](Puducheri et al, 2007) for encoding data from multiple sources independently • Common relay combines encoded packets from multiple sources • Resulting bit-stream approximates that of an LT code (deconvolution of Robust Soliton) • Substantial benefits in comparison with independent LT encoders and forwarding at the relay. [2] S. Puducheri, J. Kliewer, and T. Fuja, “The design and performance of distributed LT codes,” IEEE Trans. Inf. Theory, vol. 53, pp. 3740–3754, Oct. 2007.

  5. Robust Soliton Distribution Decomposition of LT Code • Recall: S2:k/2 information symbols S1: k/2 information symbols Deconvolved Soliton Distribution Distributed LT Distributed LT Deconvolved Soliton Distribution N T: k information symbols LT http://www.powercam.cc/slide/911

  6. Decomposition of LT Code • Proposed scenario: • Each data set contains k/2 information symbols. • Node N only performs one of two operations: • Transmits the bitwise XOR of the symbols receives from S1 and S2. • Transmits just one of them. • The resulting encoded symbol received by T: • The degree of the resulting symbol has two cases: • The sum of the degrees (up to i = k). • The individual degree of oneof the received symbols. http://www.powercam.cc/slide/911

  7. Robust Soliton Distribution Decomposition of LT Code • Encoding procedure: D2:k/2 information symbols D1: k/2 information symbols S2 DSD p(·) DSD : p(·) S1 encoded symbols: X2 encoded symbols: X1 Y= X1X2 or Y = X1 or Y = X2 N DLT-2Code encoded symbols: Y Node N knows the degrees and neighbors of X1 and X2 T

  8. DU-rateless codes • In DU-rateless coding, each source performs rateless coding with a distinct degree distribution on its data block and forwards its output symbols to the relay. • For the sake of simplicity, r = 2. • Consider a distributed data transmission with two sources s1 and s2, and data block lengths ρk and k. • 0 < ρ ≤ 1. • s1 and s2 encode their input symbols with degreedistributions Ω(x) and ϕ(x) with the largest degrees B1 andB2, respectively, and forward them to the relay.

  9. DU-rateless codes • Relay Rreceives output symbols from two sources andperforms as follows • 1) With probabilities p1 and p2 it relays the first and thesecond source’s output symbol to the destination D,respectively. • 2) With probability p3 = 1 − p1 − p2 it combines twoincoming symbols and forwards the combined symbolto the destination. Fig. 1. Adopted model for DU-rateless codes. s1, s2, R, and Drepresent distributed sources, relay, and destination, respectively.

  10. DU-rateless codes • In DU-rateless coding, the corresponding bipartite graph at the receiver has two types of variable nodes (input symbols from s1 and s2), and three types of check nodes generated by the relay Fig. 2. The bi-partite graph representing input and output symbols for r = 2.

  11. DU-rateless codes • The check nodes in the first group are generated based on Ω(x) and are only connected to input symbols of s1. • The check nodes in the second group are generated based on ϕ(x) and are only connected to input symbols of s2. • The check nodes in the third group are generated using input symbols from both s1 and s2with a degree distribution equal to Ω(x)×ϕ(x). • A check node belongs to the first, second, and third group with probabilities p1, p2, and p3, respectively. • (ρk, k,Ω(x), ϕ(x), p1, p2, p3, γ)

  12. And-Or Tree analysis • Two And-Or trees [10] Tl,1 and Tl,2 with depth 2l. • Assume that Tl,1 and Tl,2 have Type-X and Type-Y OR-nodes and Type-I, Type-II, and Type-III AND-nodes. • For each tree, the root of the tree is at depth 0, its children are at depth 1, their children at depth 2. • Each node at depth 0, 2, 4, . . . , 2l − 2 is an OR-node. • Each node at depth 1, 3, 5, . . . , 2l − 1 is called an AND-node. • The root of Tl,1 is a Type-X OR-node, and the root of Tl,2 is a Type-Y OR-node. [10] M. G. Luby, M. Mitzenmacher, and M. A. Shokrollahi, “Analysis of random processes via And-Or tree evaluation,” (Philadelphia, PA, USA), pp. 364–373, Society for Industrial and Applied Mathematics, 1998.

  13. Fig. 3. Tl,1And-Or tree with two types of OR-nodes and three types of AND-nodes with a Type-X OR-node root.

  14. Fig. 4. Tl,2And-Or tree with two types of OR-nodes and three types of AND-nodes with a Type-Y OR-node root.

  15. And-Or Tree analysis • We assume that in both Tl,1 and Tl,2 • Type-X OR-nodes choose i∈ {0, . . .,A1} and j ∈ {0, . . .,A1} children from Type-I and Type-III AND-nodes with probabilities δi,1 and δj,1, respectively. • Type-Y OR-nodes choose i∈ {0, . . .,A2} and j ∈ {0, . . .,A2} children from Type-II and Type-III AND-nodes with probabilities δi,2 and δj,2 , respectively. • Type-I AND-nodes choose i∈ {0, . . .,B1 − 1} children from Type-X OR-nodes with probability βi,1 • Type-II AND-nodes choose i∈ {0, . . .,B2 − 1} children from Type-Y OR-nodes with probability βi,2.

  16. And-Or Tree analysis • Type-III AND-nodes choose j ∈ {0, . . .,B1 − 1} and i∈ {1, . . .,B2} children from Type-X and Type-Y OR-nodes with probabilities βj,1 and βi,3. • Note that Type-III AND-nodes in Tl,1 should have at least one child from Type-Y OR-nodes, since otherwise it is a Type-I AND-node. • In Tl,2, Type-III AND-nodes can choose j ∈ {0, . . .,B2 − 1} and i∈ {1, . . .,B1} children from Type-Y and Type-X OR-nodes with probabilities βj,2 and βi,4 • Similar to Type-III AND-nodes in Tl,1, Type-III AND-nodes in Tl,2 need to have at least one child from Type-X OR-nodes to be distinguished from Type-II AND-nodes.

  17. And-Or Tree analysis • We assume that in both Tl,1 and Tl,2the ratio of the number of AND-nodes of Type-I, Type-II, and Type-III is p1, p2, and p3 = 1− p1 − p2, where 0 ≤ pi ≤ 1, ∀i. • Type-X and Type-Y OR-nodes at depth 2l are independently assigned a value of 0 with probabilities y0,1 and y0,2. • Also OR-nodes with no children are assumed to have a value 0, whereas AND-nodes with no children are assumed to have a value 1. • We are interested in finding yl,1 and yl,2, the probabilities that the root nodes of Tl,1 and Tl,2 evaluate to 0, respectively, if we treat the trees as a Boolean circuits.

  18. And-Or Tree analysis • Let G denote the bipartite graph corresponding to a DU-rateless code at the receiver. • Choose an edge (v,w) uniformly at random from all edges in G with one end among variable nodes of s1. Call the variable node v the root of Gl,1. • SubgraphGl,1 is the graph induced by v and all neighbors of v within distance 2l after removing the edge (v,w).

  19. And-Or Tree analysis • It can be shown that Gl,1 is a tree asymptotically [10]. • We can map encoded symbols from s1, encoded symbols from s2, and combined encoded symbols in Glto Type-I, Type-II, and Type-III AND-nodes in Tl,1, respectively. • Variable nodes of s1 and s2 in Gl,1 can be mapped to Type-X and Type-Y OR-nodes in Tl,1. • Gl,2 can be mapped to Tl,2 in the same way that Gl,1 is mapped to Tl,1. [10] M. G. Luby, M. Mitzenmacher, and M. A. Shokrollahi, “Analysis of random processes via And-Or tree evaluation,” (Philadelphia, PA, USA), pp. 364–373, Society for Industrial and Applied Mathematics, 1998.

  20. And-Or Tree analysis • To complete DU-rateless codes analysis, we only need to compute the probabilities βi,1, βi,2, βi,3, βi,4, and functions δ1(x) and δ2(x)

  21. Distributed UEP rateless codes design • For DU-rateless coding with r = 2, two error rates BER1 and BER2are defined. • The values of these two error rates are dependant, i.e. improving one error rate by modifying DU-rateless code parameters may result in degrading the other error rate. • Deal with two dependant error rates, as conflicting objective functions, we have a multi-objective optimization problem.

  22. Distributed UEP rateless codes design • More than one objective functions to minimize, we need to employ pareto optimality concept. Fig. 5. Concept of pareto optimality, pareto front, and domination for a two-objective minimization problem with two decision variables, x1 and x2.

  23. Distributed UEP rateless codes design • Multi-objective optimization methods such as NSGA-II[3] search to find solutions that result in pareto front. • We fix the parameters γsucc= 1.05 and B1 = B2 = 100, and employ the state-of-the-art multi-objective genetic algorithm NSGA-II[3] to find the optimum value for Ω(x) and ϕ(x) along with relaying parameters p1, p2, and p3 that minimize BER1 and BER2for various values of η = BER2 / BER1, and ρ ∈ {0.3, 0.5, 1}. • Two objective functions, BER1 and BER2, with 202 independent decision variables, i.e. X = {Ω1,Ω2, . . . ,Ω100, ϕ1, ϕ2, . . . , ϕ100, p1, p2}. [3] K. Deb, A. Pratap, S. Agarwal, and T. Meyarivan, “A fast and elitist multiobjective genetic algorithm: NSGA-II,” IEEE Transactions on Evolutionary Computation, vol. 6, pp. 182–197, Apr 2002.

  24. Simulation Results • Two DU-rateless codes for η ∈ {10, 102},ρ = 1, and γsucc= 1.05. • η = 1 corresponds to equal error protection (EEP) case where data from s1 and s2 are equally protected. • A larger ηshows a higher recovery rate of S1input symbols at D or equivalently a higher level of protection compared to S2. • γsuccis called the coding overhead and asymptotically (k →∞) approaches 1. • For practical finite values of k, γsuccmay be much larger than 1.

  25. Simulation Results

  26. Simulation Results • The parameters of a DU-rateless code for ρ = 1, η = 10, and γsucc= 1.05, with p1 = 0.4822, and p2 = 0.1173, which gives p3 = 0.4005. • We can see that to achieve an optimum distributed coding 40.05% of the generated output symbols at s1 and s2 should be combined at the relay.

  27. Performance Evaluation for Finite-length Fig. 3. The resulting BERs for asymptotic case and finite length case (k = 104) for DU-rateless codes optimized for γsucc= 1.05 and γsucc= 1.02 with parameters η = 10 and ρ = 1.

  28. Performance Evaluation for Finite-length Fig. 3. The resulting BERs for asymptotic case and finite length case (k = 104) for DU-rateless codes optimized for γsucc= 1.05 and γsucc= 1.02 with parameters η = 10 and ρ = 1.

  29. Performance Comparison with LT Codes Fig. 4. Performance comparison of the employed DU-rateless code and the equivalent optimal separate LT codes. As shown, the overhead for achieving BER1 = 5 × 10−7 reduces from 1.25 to 1.15 if we employ a DU-rateless code instead of two separate LT codes.

  30. Performance Comparison with DLT codes Fig. 5. Performance comparison of the DU-rateless codes for designed for ρ = 1, η = 1, γ = 1.05 and the DLT codes with average output degree of 11.03 for k = 104. DU-rateless codes are capable of providing UEP and also support sources with unequal block sizes.

  31. Conclusions • We proposed DU-rateless codes, which are distributed rateless codes with Unequal Error Protection (UEP) property for two data sources with unequal data block lengths over erasure channels. • We analyzed DU-rateless codes employing And-Or tree analysis technique, and we designed several close to optimum sets of DU-rateless codes using multi-objective genetic algorithms.

  32. References • [2] S. Puducheri, J. Kliewer, and T. Fuja, “The design and performance of distributed LT codes,” IEEE Trans. Inf. Theory, vol. 53, pp. 3740–3754, Oct. 2007. • [3] K. Deb, A. Pratap, S. Agarwal, and T. Meyarivan, “A fast and elitist multiobjective genetic algorithm: NSGA-II,” IEEE Trans. Evolutionary Comput., vol. 6, pp. 182–197, Apr. 2002. • [4] A. Talari and N. Rahnavard, “Distributed rateless codes with UEP property,” in Proc. 2010 IEEE International Symp. Inf. Theory Proc., pp. 2453–2457. • [5] D. Sejdinovic, R. Piechocki, and A. Doufexi, “AND-OR tree analysis of distributed LT codes,” in Proc. 2009 IEEE Inf. Theory Workshop Netw. Inf. Theory, pp. 261–265. • [6] A. Liau, S. Yousefi, and I. Kim, “Binary soliton-like rateless coding for the y-network,” IEEE Trans. Commun., vol. PP, no. 99, pp. 1–6, 2011. • [7] R. Gummadi and R. Sreenivas, “Relaying a fountain code across multiple nodes,” in Proc. 2008 IEEE Inf. Theory Workshop, pp. 149–153. • [8] N. Rahnavard, B. Vellambi, and F. Fekri, “Rateless codes with unequal error protection property,” IEEE Trans. Inf. Theory, vol. 53, pp. 1521–1532, Apr. 2007. • [9] N. Rahnavard and F. Fekri, “Generalization of rateless codes for unequal error protection and recovery time: asymptotic analysis,” in Proc. 2006 IEEE International Symp. Inf. Theory, pp. 523–527. • [13] N. Rahnavard and F. Fekri, “Finite-length unequal error protection ratelesscodes: design and analysis,” in Proc. 2005 IEEE Global Telecommun. Conf., vol. 3, p. 5.

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