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F

OUNTAIN

C

ODES

, LT C

ODES AND

R

APTOR

C

ODES

Susmita Adhikari Eduard Mustafin G ökhan Gül

1 . Motivation

O UTLINE

2. Fountain Codes 3. Degree Distribution 4. LT Codes 5 . Raptor Codes 7 . Conclusion

Kiel, February 2008 2

M OTIVATION



Binary Erasure Channel

1-p 0 1 p e p 1-p Capacity = (1 - p) 1 0 Kiel, February 2008 3

M OTIVATION



Automatic Repeat Request (ARQ)

 Wasteful usage of bandwidth, network overloads and intolerable delays.



Forward Error Correcting (FEC) codes

 Reed-Solomon codes, LDPC codes, Tornado codes.

 Rate should be determined in compliance with the erasure probability

p.

 High computational cost of overall encoding and decoding.

Kiel, February 2008 4

FOUNTAIN CODES



Rateless

: Number of code symbols, which can be generated from input info symbols is potentially unlimited.



Capacity achieving

: Decoder can recover info symbols from any set of code symbols, which is only slightly longer than input message length.



Universal

: Fountain Codes are near optimal for any BEC.

Kiel, February 2008 5

F OUNTAIN C ODES

For info symbols

s

1 ,

s 2 …s K ,

code symbols

t

1 , and random generator matrix

G, t 2 …

 Encoding: Example: Kiel, February 2008 6

FOUNTAIN CODES

For info symbols

s

1 ,

s 2 …s K ,

code symbols

t

1 , and random generator matrix

G, t 2 …

 Decoding: Example: Easier to calculate using

Tanner Graph

Kiel, February 2008 7

F OUNTAIN C ODES

[ ] [ 1 1 0 Kiel, February 2008

.

.

.

.

.

.

] 8

F OUNTAIN C ODES

.

.

.

 Encoding complexity of and decoding complexity of , which makes them impractical.

Kiel, February 2008 9

D EGREE D ISTRIBUTION



Degree

is the number of edges connecting an encoded symbol to the input symbols.



Degree distribution

is the probability density function of all degrees used for encoding.

 Average degree of the encoded symbols should be at least for a realiable decoding.

 Degree distribution affects the encoding and decoding costs. 10

LT C ODES

 First practical implementation of Fountain Codes, proposed by Michael Luby.

 LT codes are rateless and universal.

 Computational complexity of both the encoding and the decoding process increase logarithmically with the increase of the data length.

 Therefore, compared to Fountain Codes, LT codes provide a considerably reduced computational cost while achieving the capacity.

Kiel, February 2008 11

LT C ODES -

“Encoding”



Encoding Algorithm

 Divide the message

M

into equi-length parts of

k

bits resulting in

K

number of symbols.

M

0 0 1 1 0 0 0 1 0 1 1 0

k Kiel, February 2008 12

LT C ODES -

“Encoding”



Encoding Algorithm

 Randomly choose the degree d of the encoding symbol from a degree distribution . d=3 Kiel, February 2008 13

LT C ODES -

“Encoding”



Encoding Algorithm

 Choose randomly distinct input symbols as the edges of the encoding symbol in the tanner graph.

0 0 1 1 0 0 0 1 0 1 1 0

2 3 5 d=3 Kiel, February 2008 14

LT C ODES -

“Encoding”



Encoding Algorithm

 Determine the encoding symbol as bitwise modulo 2 sum of the edge symbols.

0 0 1 1 0 0 0 1 0 1 1 0

1 2 3 4 5 6 d=3

1 1

Kiel, February 2008 15

LT C ODES -

“Encoding”

1 2 3 4 5 6 d v 2 (1,3) 2 (1,2) 2 (5,6) 1 (3) 1 (4) 2 (4,6) 1 (2) 1 (5) 3 (1,4,6) 1 (3) Kiel, February 2008 16

LT C ODES -

“Decoding”



Decoding Algorithm

 Find a check node with degree one and assign its value to the corresponding input symbol. If there is no such a node halt the decoding and report the failure of decoding.

 Add this value to all check nodes connected to this input symbol.

 Remove all edges from the graph, which are connected to the related input symbol.

 Repeat the first three steps until all input symbols are recovered.

Kiel, February 2008 17

LT C ODES -

“Decoding”

Decoding Failure!

.

.

.

d v 2 2 (1,3) (1,2) 1 (3) 2 (4,6) 1 (5) 3 (1,4,6) Kiel, February 2008

.

18

LT C ODES -

“Decoding”

Decoding Successful!

.

.

.

d v 2 2 (1,3) (1,2) 1 (3) 1 (4) 2 (4,6) 1 (5) 3 (1,4,6) Kiel, February 2008 19

R APTOR C ODES

     An extension of LT codes, introduced by Shokrollahi.

Core idea “To relax the condition of recovering all input symbols and to require only a constant fraction of input symbols be recoverable.”  Idea achieved by concatenation of an LT code and a precode.

LT code recovers a large proportion of input symbols.

Precode recovers the fraction unrecovered by LT code.

Encoding and decoding complexity increases linearly with K.

Kiel, February 2008 20

R APTOR C ODES

“Encoding”

Kiel, February 2008 21

R APTOR C ODES

“Decoding”

Recovered Message Symbols Decoding Successful!

Unrecovered Symbols Received Symbols Kiel, February 2008 Erased Symbol 22

RAPTOR CODES

on Noisy Channels

 Raptor Codes can efficiently be used over noisy channels with the same encoding scheme that we have previously described and BP algorithm using soft inputs.

 AWGN Channel with E s /N 0 =-2.83 dB and K=9500 info bits, N av =20737 code-bits, Cap=0.5bit/symbol, R av =0.458bit/symbol.

 Tends to approach the capacity with the increase of message length on both AWGN and the fading channel with Rayleight distribution.

Kiel, February 2008 23

RAPTOR CODES

on Noisy Channels

 Capacity achieving Raptor Codes haven’t been proven yet for other symetric channels. However, it is proven that Raptor Codes are not universal for all rates for symetric channels other than BEC.

 Generalized Raptor Codes outperform ordinary Raptor Codes using rate-compatible distribution arrangement on BSM and AWGN channels.

Kiel, February 2008 24

C ONCLUSION

 Fountain Codes  Advantage: Rateless, universal and capacity achieving.

 Disadvantage: Higher encoding and decoding complexity.

 LT Codes  Advantage: Lower complexity than Fountain Codes.

 Disadvantage: Complexity increases logarithmically with the message length.

 Raptor Codes   Advantage: The lowest complexity achievable Can be applied to arbitrary channels efficiently.

Kiel, February 2008 25

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