Performance Analysis of LT Codes With Different Degree

7/21/2019 Performance Analysis of LT Codes With Different Degree

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Zhu Zhiliang, Liu Sha, Zhang Jiawei,

Zhao Yuli, Yu Hai

Performance analysis of LT codes

with different degree distribution

Software College, Northeastern University, Shenyang, Liaoning, China.

College of Information Science and Engineering, Northeastern University,

Shenyang, Liaoning, China


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Outline Introduction

Degree distribution of LT codesAnalysis of LT codes

Average degree

Degree release probability

Average overhead factor


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Introduction

The encoding/decoding complexity and error

performance are governed by the degree distribution of

LT code.

Designing a good degree distribution of encoded

symbols [7]

To improve the encoding/decoding complexity and errorperformance

In this paper , we analysis

Ideal soliton distribution Robust soliton distribution

Suboptimal degree distribution

Scale-free Luby distribution


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LT process

4

covered = { }

processed = { }

ripple = { }

released = { }

a1

a2

a3

a4

a5

c1

c2

c3

c4c5

c6

STATE:

ACTION: Init: Release c2, c4, c6

http://www.powercam.cc/slide/21817


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LT process

5

released = {c2,c4,c6}

covered = {a1,a3,a5}

processed = { }

ripple = {a1,a3,a5}

c1

c2

c3

c4c5

c6

STATE:

ACTION: Process a1

a1

a2

a3

a4

a5


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LT process

6

released = {c2,c4,c6,c1}


processed = {a1}

ripple = {a3,a5}

STATE:

ACTION: Process a3

a1

a2

a3

a4

a5

c1

c2

c3

c4c5

c6


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LT process

7

released = {c2,c4,c6,c1}


processed = {a1,a3}

ripple = {a5}

STATE:

ACTION: Process a5

a1

a2

a3

a4

a5

c1

c2

c3

c4c5

c6


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LT process

8

released = {c2,c4,c6,c1,c5}

covered = {a1,a3,a5,a4}

processed = {a1,a3,a5}

ripple = {a4}

STATE:

ACTION: Process a4

a1

a2

a3

a4

a5

c1

c2

c3

c4c5

c6


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LT process

9

released = {c2,c4,c6,c1,c5,c3}

covered = {a1,a3,a5,a4,a2}

processed = {a1,a3,a5,a4}

ripple = {a2}

STATE:

ACTION: Process a2

a1

a2

a3

a4

a5

c1

c2

c3

c4c5

c6


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Ideal soliton distribution [6]

Works poor

Due to the randomness in the encoding process,

Ripple would disappear at some point, and the

whole decoding process failed.

[6] M. Luby, LT codes, Proc.Annu.

Symp. Found. Comput. Sci.(Vancouver, Canada), 2002, pp.


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Robust soliton distribution [6]

Degree distribution of Ideal Soliton

Distribution

Maximum failure probability of the

decoder

when encoded

symbols are received


Symp. Found. Comput. Sci.(Vancouver, Canada), 2002, pp.


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Optimal degree distribution is proposed[12]

When kis large, the coefficient matrix of optimal degree

distribution is too sick.

No solution.

Suboptimal degree distribution:

[12] Zhu H P, Zhang G X, Xie Z D, "Suboptimaldegree distribution of LT codes". Journal of Applied

Sciences-Electronics and Information Engineering.

Jan 2009, Vol. 27, No. 1, pp. 6-11.

Ris initial ripple size

Eis the expected number of encoded

symbols required to recovery the input


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Scale-free Luby distribution [13]

Based on modified power-law distribution

Presenting that scale-free property have a higher chance tobe decoded correctly.

A large number of nodes with low degree

A little number of nodes with high degree

P1 : the fraction of encoded symbols withdegree-1

r: the characteristic exponent

A : the normalizing coefficient to ensure

[13] Yuli Zhao, Francis C. M. Lau, "Scale-

free Luby transform codes", International

Journal of Bifurcation and Chaos, Vol. 22,


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Analysis of LT codes

The encoding/decoding efficiency is evaluated by the

average degree of encoded symbols. Less average degree

Fewer times of XOR operations

Encoded symbol should be released until the decodingprocess finished

Degree release probability is very important

Less number of encoded symbols required to recovery

the input symbols means less cost of transmitting the

original data information.

The overhead should be considered


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Average degree

Ideal soliton distribution

Can be calculated based on the summation formula ofharmonic progression

r: Euler's constant which is similar to 0.58

Average degree of ideal soliton degree distribution is


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Average degree

Robust soliton distribution

The complexity of its average degree is


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Average degree


The complexity of its average degree is


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Average degree


Based on the properties of Scale-free

The average degree of Scale-free Luby Distribution will

be small

(r-1) is the sum of ap-progression

It is obvious that the average degree ofSF-LT codes is

smallerEncodin /decodin com lexit of SF-LT code is much


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[6]

In general, r(L) should be larger than 1

At least 1 encoded symbol is released when an input

symbol is processed.


Symp. Found. Comput. Sci.

(Vancouver, Canada), 2002, pp.271-282.

1

11

(+1)

2


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Ideal soliton distribution [6]

Degree release probabilityRobust soliton distribution


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Using limit theory, it can be expressed as ,where

Suppose Eencoded symbols is sufficient to recovery the

koriginal input symbols.

At each decoding step, larger than 1 encoded symbol is


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Initial ripple size must be bigger than Robust SolitonDistributions

kP1 is bigger than 1

The complexity is


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Degree release probability Suboptimal degree distribution's degree release

probability is bigger than the others

A h d f


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A decreasing ripple size provides a better trade-off

between robustness and the overhead factor[14]

The theoretical evolution of the ripple size :

Assuming that at each decoding iteration, the input symbolscan be added in to the ripple set without repetition

: the number of degree-iinput symbols leftL : the size of unprocessed input symbols

[14] Sorensen J. H., Popovski. P.,

Ostergaard J., "On LT codes with

decreasing ripple size", Arxiv preprint

PScache/1011.2078v1.

A h d f t


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Conclusion Robust LT codes, suboptimal LT code and SF-LT code

are capable to recovery the input symbols efficiently. From the overhead factor, SF-LT codes and suboptimal

LT codes need much less number of encoded symbols

to recovery given number of input symbols.

The average degree of SF-LT code is smaller than theothers.

SF-LT code performs much better probability of

successful decoding and enhanced encoding/decoding

complexity

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Performance Analysis of LT Codes With Different Degree

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