Degree Distribution of XORed Fountain codes

Theoretical derivation and Analysis

1

Lucie Nodin, Anya Apavatjrut, Claire Goursaud, Jean-Marie Gorce

Planning

2

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

Part I: Overview

3

An approach to network coding of fountain code in a wireless sensor network

Wireless Sensor Network

Fountain codes

Network Coding

Wireless Sensor Network

4

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

Fountain Codes

5

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)

Fountain Codes

6

Principle of Fountain Codes : LT Encoding Process

Randomly choose the degree d of the packet from the Robust Soliton Distribution

Uniformly select d distinct fragments among K and apply a bitwise sum (XOR) between these d fragments.

f2

f1f2

f2

f1

f1

f2

f3

f2

f1

f2 f

3f3

f2 f

3p1 p2 p3 p4 p5 p6 p7

K = block lengthd = degree of the packet

Source packet

21

2 degree

ff

Fountain Codes

7

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.

Fountain Codes

8

Principle of Fountain Codes : LT Decoding Process: Belief Propagation

good degree distribution : large-> encoded packets cover all initial fragments

small-> ensure decoding capability

f2

f1

f2

f3

f1f2p1

f1

p2

f2

p3

f2f3p4

f1f3p5

f2

p6

f2f3p7

f2

f3

K = block lengthd = degree of the packet

Fountain Codes

9

Degree Distribution Robust Soliton Distribution is the optimal

distribution for the BP decoding [Luby2002] Ideal Soliton Distribution

Robust Soliton Distribution

where , and

Kiii

iKi

2for)1(

1

1for1

)(

Z

iii

)()()(

i

iiZ )()( KK

cS

ln

S

Ki

S

Ki

K

SS

S

Ki

iK

S

i

for0

forln

11for

)(

Fountain Code

10

Degree Distribution Degree distribution of Raptor code –

precode+weakened LT code [Shokrollahi2006]

where and is the overhead which allows to recover the initial data

KiK

Kiii

i

i

for1

1

1

}1,...,2{for)1(

1

1

1

1for1

)(

2

22

)1(4

K

Network Coding

11

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?

R

Packet 1

Packet 2

Packet XORed

?

Part II: Contribution

12

Related work Decode and Reencode Successive encoding by relay nodes [Gummadi et

al.2008] XORing algorithms are implemented at the relay nodes

in order to preserve the target degree distribution [Apavatjrut et al.2010, Champel2009]

In this work… Whereas the previous works focus on algorithm

implementation, this work focuses on theoretical analysis.

Part II: Contribution

13

Theoretical analysis of the degree distribution of the XORed Fountain code

Theoretical approach to preserve the degree distribution

Application to LT and Raptor Codes

XORing Fountain Codes

14

Insight of XORing packets encoded with fountain codes Packet Header

f2

f1f2

f2

f1

f1

f2

f3

f2

f1

f2 f

3f3

f2 f

3p1 p2 p3 p4 p5 p6 p7

1 1 01p

1 0 02p

0 1 03p

Ex. K=3

XORing Fountain Codes

15

Insight of XORing packets encoded with fountain codes example

0 1 0 1

1 0 0 0

1 1 0 1

0 1 0 1

1 1 0 0

1 0 0 1

no overlap with overlap

21 dddR odddR 221

)()( 21 odod dR = degree of the resulting packet after a XOR operationd1 = degree of the first packetd2 = degree of the second packeto = number of degree overlap between the two packets

1p

2p

Rp

2p

Rp

1p

XORing Fountain Codes

16

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

2o

otherwise

),d(dif o

d

K

od

dK

o

d

)dp(o|d ,

0

min 21

2

2

11

21

K = block lengthd1 = degree of the first packetd2 = degree of the second packetO = number of degree overlap between the two packets

K

f1 f2 f3 f4 f5 f6 f7 fk-1

fk

31 d

52 d

XORing Fountain Codes

17

Degree probability for a packet resulting from one XOR Probability of getting resulting packet with degree

by applying the total law of probabilities

Rd

),|2

(),,|( 2121

21 ddddd

opodddp RR

otherwise 0),,|( 21 odddp R

positive andeven is )( if 21 dddR

K

d

K

d

RR dd

dddopdpdpdp

1 121

2121

1 2

),|2

()()()(

XORing Fountain Codes

18

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:

K

d

K

dnRn Kddopdpdpdp

1 121211

1 2

),,|()()()(

)()(

)()(

220

110

ddp

ddp

)(1 Rdp

)( Rn dp

)( 20 dp

)( 20 dp

)( 20 dp

)( 10 dp

XORing Fountain Codes

19

Degree probability for a packet resulting from several XORs

5.0,03.0,100 ck

P(d)

Degree (d)

Degree probability for a packet resulting from several XORs

When , Soliton Distribution Gaussian Distribution

XORing Fountain Codes

20

N

Randomly applying XOR operations -> decoding inefficiency

Preserving the Degree Distribution: Theoretical Approach

21

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

K

dchoiceR ddopdpddp

12121

2

),|()()|(

K

d

K

dR Kddopdpdpdp

1 12121

1 2

),,|()()()(

Preserving the Degree Distribution: Theoretical Approach

22

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

choiced PMP

dP

M

choiceP

dP

)?,|( 21, jddiopm ji

Preserving the Degree Distribution: Theoretical Approach

23

How to determined ? Too difficult to be determined by matrix inversion

Estimation with the least square method

and

choiceP

dt1t

choice PMM)(MXXP with

2minarg XMPX d

Application to LT and Raptor Codes

24

By solving the system of equations for LT code :

Pchoice can be determined as:

choiceLTPMμ

Degree Distribution Pchoice of the degrees to choose to recover Robust Soliton distribution for packets resulting from one XOR

Irregularity of Pchoice

By solving the system of equations for Raptor code :

Pchoice can be determined as:

Application to LT and Raptor Codes

25

orchoiceRaptPM

Degree Distribution Pchoice of the degrees to choose to recover weaken Robust Soliton distribution for packets resulting from one XOR

Application to LT and Raptor Codes

26

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 Pchoice distribution 5.0,03.0,100 ck

Application to LT and Raptor Codes

27

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 Pchoice distribution 5.0,03.0,100 ck

Application to LT and Raptor Codes

28

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 Pchoice distribution 5.0,03.0,100 ck

Application to LT and Raptor Codes

29

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 Pchoice distribution 5.0,03.0,100 ck

Conclusion

30

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.

References

31

[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