Performance analysis of LT codes with different degree distribution

Zhu Zhiliang, Liu Sha, Zhang Jiawei, Zhao Yuli, Yu HaiPerformance analysis of LT codes with different degree distributionSoftware College, Northeastern University, Shenyang, Liaoning, China.College of Information Science and Engineering, Northeastern University, Shenyang, Liaoning, ChinaOutline IntroductionDegree distribution of LT codesAnalysis of LT codesAverage degreeDegree release probabilityAverage overhead factor 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 error performance

In this paper , we analysis Ideal soliton distributionRobust soliton distributionSuboptimal degree distributionScale-free Luby distribution

Average degreeDegree release probabilityAverage overhead factor

LT process4covered = { }processed = { }ripple = { }released = { }a1a2a3a4a5c1c2c3c4c5c6STATE:ACTION:Init: Release c2, c4, c6http://www.powercam.cc/slide/21817LT process5released = {c2,c4,c6} covered = {a1,a3,a5}processed = { }ripple = {a1,a3,a5}c1c2c3c4c5c6STATE:ACTION:Process a1a1a2a3a4a5LT process6released = {c2,c4,c6,c1}covered = {a1,a3,a5}processed = {a1}ripple = {a3,a5}STATE:ACTION:Process a3a1a2a3a4a5c1c2c3c4c5c6LT process7released = {c2,c4,c6,c1} covered = {a1,a3,a5}processed = {a1,a3}ripple = {a5}STATE:ACTION:Process a5a1a2a3a4a5c1c2c3c4c5c6LT process8released = {c2,c4,c6,c1,c5}covered = {a1,a3,a5,a4}processed = {a1,a3,a5}ripple = {a4}STATE:ACTION:Process a4a1a2a3a4a5c1c2c3c4c5c6LT process9released = {c2,c4,c6,c1,c5,c3}covered = {a1,a3,a5,a4,a2}processed = {a1,a3,a5,a4}ripple = {a2}STATE:ACTION:Process a2a1a2a3a4a5c1c2c3c4c5c6LT process10released = {c2,c4,c6,c1,c5,c3} covered = {a1,a3,a5,a4,a2}processed = {a1,a3,a5,a4,a2}ripple = { }STATE:ACTION:Success!a1a2a3a4a5c1c2c3c4c5c6Ideal 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. 271-282.

Robust soliton distribution [6]

Degree distribution of Ideal Soliton DistributionMaximum failure probability of the decoderwhen encoded symbols are received

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

Suboptimal degree distributionOptimal 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 sizeE is the expected number of encoded symbols required to recovery the input symbols.

Scale-free Luby distribution [13]Based on modified power-law distributionPresenting that scale-free property have a higher chance to be decoded correctly.A large number of nodes with low degreeA little number of nodes with high degree

P1 : the fraction of encoded symbols with degree-1r : the characteristic exponentA : 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 codesThe 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 finishedDegree 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 degreeIdeal soliton distributionCan be calculated based on the summation formula of harmonic progression

r : Euler's constant which is similar to 0.58Average degree of ideal soliton degree distribution is

Average degreeRobust soliton distribution

The complexity of its average degree is

Average degreeSuboptimal degree distribution

The complexity of its average degree is

Average degreeScale-free Luby distribution Based on the properties of Scale-freeThe 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 probability[6]

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

[6] M. Luby, LT codes, Proc. Annu. Symp. Found. Comput. Sci. (Vancouver, Canada), 2002, pp. 271-282.Degree release probabilityIdeal soliton distribution [6]

Degree release probabilityRobust soliton distribution

Degree release probabilitySuboptimal 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 probabilityScale-free Luby distribution

Initial ripple size must be bigger than Robust Soliton DistributionskP1 is bigger than 1 The complexity is

Degree release probabilitySuboptimal degree distribution's degree release probability is bigger than the othersAverage overhead factor

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: the number of degree-i input symbols left

L : 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.Average overhead factor

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