Loading in 5 sec....

Performance analysis of LT codes with different degree distributionPowerPoint Presentation

Performance analysis of LT codes with different degree distribution

- By
**chi** - Follow User

- 64 Views
- Uploaded on

Download Presentation
## PowerPoint Slideshow about ' Performance analysis of LT codes with different degree distribution' - chi

**An Image/Link below is provided (as is) to download presentation**

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.

- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -

Presentation Transcript

### Performance analysis of LT codes with different degree distribution

Zhu Zhiliang, Liu Sha, Zhang Jiawei,

Zhao Yuli, Yu Hai

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

College of Information Science and Engineering, Northeastern University, Shenyang, Liaoning, China

Outline distribution

- Introduction
- Degree distribution of LT codes
- Analysis of LT codes
- Average degree
- Degree release probability
- Average overhead factor

Introduction distribution

- 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 error performance

- In this paper , we analysis
- Ideal soliton distribution
- Robust soliton distribution
- Suboptimal degree distribution
- Scale-free Luby distribution
- Average degree
- Degree release probability
- Average overhead factor

LT process distribution

c1

a1

c2

a2

c3

a3

c4

a4

c5

a5

c6

covered = { }

processed = { }

ripple = { }

released = { }

STATE:

ACTION:

Init: Release c2, c4, c6

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

LT process distribution

c1

a1

c2

a2

c3

a3

c4

a4

c5

a5

c6

released = {c2,c4,c6}

covered = {a1,a3,a5}

processed = { }

ripple = {a1,a3,a5}

STATE:

ACTION:

Process a1

LT process distribution

c1

a1

c2

a2

c3

a3

c4

a4

c5

a5

c6

released = {c2,c4,c6,c1}

covered = {a1,a3,a5}

processed = {a1}

ripple = {a3,a5}

STATE:

ACTION:

Process a3

LT process distribution

c1

a1

c2

a2

c3

a3

c4

a4

c5

a5

c6

released = {c2,c4,c6,c1}

covered = {a1,a3,a5}

processed = {a1,a3}

ripple = {a5}

STATE:

ACTION:

Process a5

LT process distribution

c1

a1

c2

a2

c3

a3

c4

a4

c5

a5

c6

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

covered = {a1,a3,a5,a4}

processed = {a1,a3,a5}

ripple = {a4}

STATE:

ACTION:

Process a4

LT process distribution

c1

a1

c2

a2

c3

a3

c4

a4

c5

a5

c6

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

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

processed = {a1,a3,a5,a4}

ripple = {a2}

STATE:

ACTION:

Process a2

LT process distribution

c1

a1

c2

a2

c3

a3

c4

a4

c5

a5

c6

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

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

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

ripple = { }

STATE:

ACTION:

Success!

Ideal distributionsolitondistribution [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. 271-282.

Robust distributionsolitondistribution [6]

Maximum failure probability of the decoder

when encoded symbols are received

Degree distribution of Ideal Soliton Distribution

[6] M. Luby, “LT codes”, Proc. Annu. Symp. Found. Comput. Sci. (Vancouver, Canada), 2002, pp. 271-282.

Suboptimal degree distribution distribution

- Optimal degree distribution is proposed[12]
- When k is 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, "Suboptimal degree distribution of LT codes". Journal of Applied Sciences-Electronics and Information Engineering. Jan 2009, Vol. 27, No. 1, pp. 6-11.

R is initial ripple size

E is the expected number of encoded symbols required to recovery the input symbols.

Scale-free distributionLuby distribution [13]

- Based on modified power-law distribution
- Presenting that scale-free property have a higher chance to be decoded correctly.

- A large number of nodes with low degree
- A little number of nodes with high degree

P1: the fraction of encoded symbols with degree-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, No. 4, 2012.

Analysis of LT codes distribution

- 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 decoding process 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

- : degree distribution : average degree

Average degree distributionIdeal solitondistribution

- Can be calculated based on the summation formula of harmonic progression
- r : Euler's constant which is similar to 0.58
- Average degree of ideal soliton degree distribution is

Average distributiondegreeRobust solitondistribution

- The complexity of its average degree is

Average distributiondegreeSuboptimal degree distribution

- The complexity of its average degree is

Average degree distributionScale-free Luby distribution

- Based on the properties of Scale-free
- The average degree of Scale-free Luby Distribution will be small
- (r-1) is the sum of a p-progression
- It is obvious that the average degree of SF-LT codes is smaller
- Encoding/decoding complexity of SF-LT code is much lower than the others

Degree release distributionprobability

[6] M. Luby, “LT codes”, Proc. Annu. Symp. Found. Comput. Sci. (Vancouver, Canada), 2002, pp. 271-282.

- [6]
- In general, r(L) should be larger than 1
- At least 1 encoded symbol is released when an input symbol is processed.

Degree release distributionprobabilityIdeal solitondistribution [6]

Degree release probabilityRobust soliton distribution

Degree release distributionprobabilitySuboptimal degree distribution

- Using limit theory, it can be expressed as , where
- Suppose E encoded symbols is sufficient to recovery the k original input symbols.
- At each decoding step, larger than 1 encoded symbol is released.

Degree release distributionprobabilityScale-free Luby distribution

- Initial ripple size must be bigger than Robust Soliton Distribution’s
- k·P1is bigger than 1
- The complexity is

Degree release probability distribution

- Suboptimal degree distribution's degree release probability is bigger than the others

Average overhead factor distribution

- 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 symbols can be added in to the ripple set without repetition

[14] Sorensen J. H., Popovski. P., Ostergaard J., "On LT codes with decreasing ripple size", Arxiv preprint PScache/1011.2078v1.

: the number of degree-i input symbols left

L : the size of unprocessed input symbols

Average overhead factor distribution

Conclusion distribution

- 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 the others.
- SF-LT code performs much better probability of successful decoding and enhanced encoding/decoding complexity

Download Presentation

Connecting to Server..