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

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

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 3

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. 4 [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

5 Decomposition of LT Code Recall: S 1 : k/2 information symbols S 2 : k/2 information symbols T : k information symbols Distributed LT LT Deconvolved Soliton Distribution Robust Soliton Distribution N

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 S 1 and S 2. 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 one of the received symbols.

7 Decomposition of LT Code Encoding procedure: D 1 : k/2 information symbols DSD : p(·) encoded symbols: Y Robust Soliton Distribution N D 2 : k/2 information symbols S1S1 S2S2 DSD p(·) encoded symbols: X 1 encoded symbols: X 2 Node N knows the degrees and neighbors of X 1 and X 2 Y = X 1  X 2 or Y = X 1 or Y = X 2 T DLT-2 Code

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 s 1 and s 2, and data block lengths ρk and k. 0 < ρ ≤ 1. s 1 and s 2 encode their input symbols with degree distributions Ω(x) and ϕ(x) with the largest degrees B 1 and B 2, respectively, and forward them to the relay. 8

DU-rateless codes Relay R receives output symbols from two sources and performs as follows – 1) With probabilities p 1 and p 2 it relays the first and the second source’s output symbol to the destination D, respectively. – 2) With probability p 3 = 1 − p 1 − p 2 it combines two incoming symbols and forwards the combined symbol to the destination. Fig. 1. Adopted model for DU-rateless codes. s 1, s 2, R, and D represent distributed sources, relay, and destination, respectively. 9

DU-rateless codes In DU-rateless coding, the corresponding bipartite graph at the receiver has two types of variable nodes (input symbols from s 1 and s 2 ), and three types of check nodes generated by the relay Fig. 2. The bi-partite graph representing input and output symbols for r = 2. 10

DU-rateless codes The check nodes in the first group are generated based on Ω(x) and are only connected to input symbols of s 1. The check nodes in the second group are generated based on ϕ(x) and are only connected to input symbols of s 2. The check nodes in the third group are generated using input symbols from both s 1 and s 2 with a degree distribution equal to Ω(x)×ϕ(x). A check node belongs to the first, second, and third group with probabilities p 1, p 2, and p 3, respectively. (ρk, k, Ω(x), ϕ(x), p 1, p 2, p 3, γ) 11

And-Or Tree analysis Two And-Or trees [10] T l,1 and T l,2 with depth 2l. Assume that T l,1 and T l,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 T l,1 is a Type-X OR-node, and the root of T l,2 is a Type-Y OR-node. 12 [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. T l,1 And-Or tree with two types of OR-nodes and three types of AND-nodes with a Type-X OR-node root.

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

And-Or Tree analysis We assume that in both T l,1 and T l,2 – Type-X OR-nodes choose i ∈ {0,...,A 1 } and j ∈ {0,...,A 1 } children from Type-I and Type-III AND- nodes with probabilities δ i,1 and δ j,1, respectively. – Type-Y OR-nodes choose i ∈ {0,...,A 2 } and j ∈ {0,...,A 2 } children from Type-II and Type-III AND- nodes with probabilities δ i,2 and δ j,2, respectively. – Type-I AND-nodes choose i ∈ {0,...,B 1 − 1} children from Type-X OR-nodes with probability β i,1 – Type-II AND-nodes choose i ∈ {0,...,B 2 − 1} children from Type-Y OR-nodes with probability β i,2. 15

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

And-Or Tree analysis We assume that in both T l,1 and T l,2 the ratio of the number of AND-nodes of Type-I, Type-II, and Type-III is p 1, p 2, and p 3 = 1− p 1 − p 2, where 0 ≤ pi ≤ 1, ∀ i. Type-X and Type-Y OR-nodes at depth 2l are independently assigned a value of 0 with probabilities y 0,1 and y 0,2. 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 y l,1 and y l,2, the probabilities that the root nodes of T l,1 and T l,2 evaluate to 0, respectively, if we treat the trees as a Boolean circuits. 17

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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 s 1. Call the variable node v the root of G l,1. Subgraph G l,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 G l,1 is a tree asymptotically [10]. We can map encoded symbols from s 1, encoded symbols from s 2, and combined encoded symbols in G l to Type-I, Type-II, and Type-III AND-nodes in T l,1, respectively. Variable nodes of s 1 and s 2 in G l,1 can be mapped to Type-X and Type-Y OR-nodes in T l,1. G l,2 can be mapped to T l,2 in the same way that G l,1 is mapped to T l,1. 20 [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.

And-Or Tree analysis To complete DU-rateless codes analysis, we only need to compute the probabilities β i,1, β i,2, β i,3, β i,4, and functions δ 1 (x) and δ 2 (x) 21

Distributed UEP rateless codes design For DU-rateless coding with r = 2, two error rates BER 1 and BER 2 are 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. 23 Fig. 5. Concept of pareto optimality, pareto front, and domination for a two-objective minimization problem with two decision variables, x 1 and x 2.

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 γ succ = 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 p 1, p 2, and p 3 that minimize BER 1 and BER 2 for various values of η = BER 2 / BER 1, and ρ ∈ {0.3, 0.5, 1}. Two objective functions, BER 1 and BER 2, with 202 independent decision variables, i.e. X = {Ω 1,Ω 2,...,Ω 100, ϕ 1, ϕ 2,..., ϕ 100, p 1, p 2 }. 24 [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.

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Simulation Results Two DU-rateless codes for η ∈ {10, 10 2 }, ρ = 1, and γ succ = η = 1 corresponds to equal error protection (EEP) case where data from s 1 and s 2 are equally protected. – A larger η shows a higher recovery rate of S 1 input symbols at D or equivalently a higher level of protection compared to S 2. γ succ is called the coding overhead and asymptotically (k →∞) approaches 1. – For practical finite values of k, γ succ may be much larger than 1. 27

Simulation Results 28

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Simulation Results The parameters of a DU-rateless code for ρ = 1, η = 10, and γ succ = 1.05, with p 1 = , and p 2 = , which gives p 3 = We can see that to achieve an optimum distributed coding 40.05% of the generated output symbols at s 1 and s 2 should be combined at the relay. 30

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Performance Evaluation for Finite-length 32 Fig. 3. The resulting BERs for asymptotic case and finite length case (k = 10 4 ) for DU-rateless codes optimized for γ succ = 1.05 and γ succ = 1.02 with parameters η = 10 and ρ = 1.

33 Performance Evaluation for Finite-length Fig. 3. The resulting BERs for asymptotic case and finite length case (k = 10 4 ) for DU-rateless codes optimized for γ succ = 1.05 and γ succ = 1.02 with parameters η = 10 and ρ = 1.

Performance Comparison with LT Codes 34 Fig. 4. Performance comparison of the employed DU-rateless code and the equivalent optimal separate LT codes. As shown, the overhead for achieving BER 1 = 5 × 10 −7 reduces from 1.25 to 1.15 if we employ a DU-rateless code instead of two separate LT codes.

35 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 for k = Performance Comparison with DLT codes DU-rateless codes are capable of providing UEP and also support sources with unequal block sizes.

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. 36

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 [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 [4] A. Talari and N. Rahnavard, “Distributed rateless codes with UEP property,” in Proc 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 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, [7] R. Gummadi and R. Sreenivas, “Relaying a fountain code across multiple nodes,” in Proc 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 [9] N. Rahnavard and F. Fekri, “Generalization of rateless codes for unequal error protection and recovery time: asymptotic analysis,” in Proc IEEE International Symp. Inf. Theory, pp. 523– 527. [13] N. Rahnavard and F. Fekri, “Finite-length unequal error protection rateless codes: design and analysis,” in Proc IEEE Global Telecommun. Conf., vol. 3, p