Degree Distribution of XORed Fountain codes

1. Degree Distribution of XORed Fountain codes Theoretical derivation and Analysis Lucie Nodin, Anya Apavatjrut, Claire Goursaud, Jean-Marie Gorce

2. Planning • Part I : Overview • Wireless sensor network • Fountain codes • Network coding • Part II : Contribution • Theoretical analysis of the degree distribution of the XORed Fountain code • Theoretical approach to preserve the degree distribution • Application to LT and Raptor Codes • Conclusion

3. Part I: Overview • An approach to network coding of fountain code in a wireless sensor network Wireless Sensor Network Fountain codes Network Coding

4. Wireless Sensor Network • Overview: A set of independent sensor nodes spatially distributed in a large area • Limitation: Battery life, limited computational capability, limited resource • Requirement: Energy awareness, Robustness, Reliability

5. Fountain Codes • Characteristics: erasure block code • Benefits: rateless, universal, limited feedback channel is required • Limitation: overhead, additional computational complexity, redundancy • Choices of fountain codes: LT, Raptor (due to its low decoding complexity of the Belief Propagation algorithm)

6. Fountain Codes • Principle of Fountain Codes : LT • Encoding Process • Randomly choose the degree d of the packet from the Robust Soliton Distribution • Uniformly select ddistinct fragments among K and apply a bitwise sum (XOR) between these d fragments. Source packet f1 f2 f3 K = block length d = degree of the packet f1 f2 f2 f1 f2 f2 f1 f2 f3 f2 f3 f3 p1 p3 p6 p7 p2 p5 p4

7. Fountain Codes • Principle of Fountain Codes : LT • Decoding Process: Belief Propagation • Find the encoded packet that have degree one. Degree one packet is considered as a decoded fragment of information. If none exists the decoding process halts at this step. • Remove the combination of this decoded fragments from other un-decoded packets. • Repeat these steps iteratively until all the packets are decoded successfully or until the decoding process halts due to the lack of degree one packet.

8. Fountain Codes • Principle of Fountain Codes : LT • Decoding Process: Belief Propagation good degree distribution : large-> encoded packets cover all initial fragments small-> ensure decoding capability f2 f3 f1 f2 f2 f1 f2 f2 f1 f2 f3 f2 f3 f3 p1 p3 p6 p7 p2 p5 p4 K = block length d = degree of the packet f1 f2 f3

9. Fountain Codes • Degree Distribution • Robust Soliton Distribution is the optimal distribution for the BP decoding [Luby2002] • Ideal Soliton Distribution • Robust Soliton Distribution where , and

10. Fountain Code • Degree Distribution • Degree distribution of Raptor code – precode+weakened LT code [Shokrollahi2006] where and is the overhead which allows to recover the initial data

11. Network Coding • Network Coding • Overview: processing of information at intermediate nodes • Benefits: redundancy optimization, packet diversity Question : How to properly apply XOR operations among the encoded packets at relay nodes R? ? Packet 1 R Packet XORed Packet 2

12. Part II: Contribution • Relatedwork • Decode and Reencode • Successive encoding by relaynodes[Gummadi et al.2008] • XORingalgorithms are implementedat the relaynodes in order to preserve the targetdegree distribution [Apavatjrut et al.2010, Champel2009] • In thiswork… • Whereas the previousworks focus on algorithmimplementation, thisworkfocuses on theoreticalanalysis.

13. Part II: Contribution • Theoretical analysis of the degree distribution of the XORed Fountain code • Theoretical approach to preserve the degree distribution • Application to LT and Raptor Codes

14. XORing Fountain Codes • Insight of XORing packets encoded with fountain codes • Packet Header f1 f2 f3 Ex. K=3 f1 f2 f2 f1 f2 f2 f1 f2 f3 f2 f3 f3 p1 p3 p6 p7 p2 p5 p4 1 1 0 1 0 0 0 1 0

15. XORing Fountain Codes • Insight of XORing packets encoded with fountain codes • example no overlap with overlap 0 1 0 1 0 1 0 1 1 0 0 0 1 1 0 0 1 1 0 1 1 0 0 1 dR = degree of the resulting packet after a XOR operation d1 = degree of the first packet d2 = degree of the second packet o = number of degree overlap between the two packets

16. XORing Fountain Codes • Overlap probability • Assuming that d1≤d2, the probability that o fragments overlap when XORing two packets with degree d1 and d2 can be expressed as K = block length d1 = degree of the first packet d2 = degree of the second packet O = number of degree overlap between the two packets f1 f2 f3 f4 f5 f6 f7 f k-1 fk

17. XORing Fountain Codes • Degree probability for a packet resulting from one XOR • Probability of getting resulting packet with degree by applying the total law of probabilities

18. XORing Fountain Codes • Degree probability for a packet resulting from several XORs • By XORing N+1 packets together, N XORs successive are done on two packets at each steps: Where pn is the degree distribution of the packet p1 once n XORs is done. The degree distribution are initialized as:

19. XORing Fountain Codes • Degree probability for a packet resulting from several XORs P(d) Degree (d)

20. XORing Fountain Codes • Degree probability for a packet resulting from several XORs When , Soliton Distribution Gaussian Distribution Randomlyapplying XOR operations -> decodinginefficiency

21. Preserving the Degree Distribution: Theoretical Approach • Question • How to select d1 and d2 in order to obtain the target degree dR • Solution • Find joint probability of picking (d1,d2) complex with 2xK unknown variables • Fixing degree d1 and find probability of picking d2 K unknown variables Pchoice = probability of picking d2

22. Preserving the Degree Distribution: Theoretical Approach • Matrix representation Such that represents the targeted resulting degree distribution represents a matrix of overlaps’ probabilities with coefficient represents the degree probability distribution of how to choose the second packets in order to obtain a specific

23. Preserving the Degree Distribution: Theoretical Approach • How to determined ? • Too difficult to be determined by matrix inversion • Estimation with the least square method and

24. Application to LT and Raptor Codes • By solving the system of equations for LT code : Pchoice can be determined as: Irregularity of Pchoice Degree Distribution Pchoice of the degrees to choose to recover Robust Soliton distribution for packets resulting from one XOR

25. Application to LT and Raptor Codes • By solving the system of equations for Raptor code : Pchoice can be determined as: Degree Distribution Pchoice of the degrees to choose to recover weaken Robust Soliton distribution for packets resulting from one XOR

26. Application to LT and Raptor Codes • Validation of the obtained results with simulations • Examples for LT codes : d1=1 Resulting degree distribution from one XOR between LT encoded packets when d1=1 and d2 is chosen according to Pchoicedistribution

27. Application to LT and Raptor Codes • Validation of the obtained results with simulations • Examples for LT codes : d1=2 Resulting degree distribution from one XOR between LT encoded packets when d1=2 and d2 is chosen according to Pchoicedistribution

28. Application to LT and Raptor Codes • Validation of the obtained results with simulations • Examples for LT codes : d1=98 Resulting degree distribution from one XOR between LT encoded packets when d1=98 and d2 is chosen according to Pchoicedistribution

29. Application to LT and Raptor Codes • Validation of the obtained results with simulations • Examples for LT codes : d1=99 Resulting degree distribution from one XOR between LT encoded packets when d1=99 and d2 is chosen according to Pchoicedistribution

30. Conclusion • Theoretical Analysis of the degree distribution of XORed fountain codes as well as a technique to preserve the degree has been proposed. • The theoretical derivation in this work can be used as a way to recover a given degree distribution after XOR operations. This can later be applied to all the network coding-like application with fountain codes. • Our theoretical and simulation results highlight that, under a certain conditions of packet selection, the target degree is reachable without the need to decode the packet entirely at the relay.

31. References • [Luby2002] M. Luby, “LT codes,” The 43rd Annual IEEE Symposium on Foundations of Computer Science, Proceedings., pp. 271 – 280, 2002. • [Shokrollahi2006] A. Shokrollahi, “Raptor codes,” IEEE Transactions on Information Theory, vol. 52, no. 6, pp. 2551 –2567, june 2006. • [Gummadi et al.2008] R. Gummadi and R. Sreenivas, “Relaying a fountain code across multiple nodes,” in IEEE Information Theory Workshop, 2008, pp. 149–153. • [Apavatjrut et al.2010] A. Apavatjrut, “Towards increasing diversity for the relaying of LT fountain codes in wireless sensor network”, to be published in IEEE Communications Letters. • [Champel2009] M.-L. Champel, “LT network codes,” INRIA, Tech. Rep., 2009.

32. Thank you