degree distribution of xored fountain codes 1 lucie nodin, anya apavatjrut, claire goursaud,...
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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.
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Thank you
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