Presentation on theme: "Performance analysis of LT codes with different degree distribution"— Presentation transcript:
1Performance analysis of LT codes with different degree distribution Zhu Zhiliang, Liu Sha, Zhang Jiawei,Zhao Yuli, Yu HaiSoftware College, Northeastern University, Shenyang, Liaoning, China.College of Information Science and Engineering, Northeastern University, Shenyang, Liaoning, China
2Outline Introduction Degree distribution of LT codes Analysis of LT codesAverage degreeDegree release probabilityAverage overhead factor
3IntroductionThe encoding/decoding complexity and error performance are governed by the degree distribution of LT code.Designing a good degree distribution of encoded symbols To improve the encoding/decoding complexity and error performanceIn this paper , we analysisIdeal soliton distributionRobust soliton distributionSuboptimal degree distributionScale-free Luby distributionAverage degreeDegree release probabilityAverage overhead factor
11Ideal soliton distribution  Works poorDue to the randomness in the encoding process,Ripple would disappear at some point, and the whole decoding process failed. M. Luby, “LT codes”, Proc. Annu. Symp. Found. Comput. Sci. (Vancouver, Canada), 2002, pp
12Robust soliton distribution  Maximum failure probability of the decoderwhen encoded symbols are receivedDegree distribution of Ideal Soliton Distribution M. Luby, “LT codes”, Proc. Annu. Symp. Found. Comput. Sci. (Vancouver, Canada), 2002, pp
13Suboptimal degree distribution Optimal degree distribution is proposedWhen k is large, the coefficient matrix of optimal degree distribution is too sick.No solution.Suboptimal degree distribution: 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, ppR is initial ripple sizeE is the expected number of encoded symbols required to recovery the input symbols.
14Scale-free Luby distribution  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 degreeP1 : the fraction of encoded symbols with degree-1r : the characteristic exponentA : the normalizing coefficient to ensure Yuli Zhao, Francis C. M. Lau, "Scale-free Luby transform codes", International Journal of Bifurcation and Chaos, Vol. 22, No. 4, 2012.
15Analysis of LT codesThe encoding/decoding efficiency is evaluated by the average degree of encoded symbols.Less average degreeFewer times of XOR operationsEncoded symbol should be released until the decoding process finishedDegree release probability is very importantLess 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
16Average degree Ideal soliton distribution Can be calculated based on the summation formula of harmonic progressionr : Euler's constant which is similar to 0.58Average degree of ideal soliton degree distribution is
17Average degree Robust soliton distribution The complexity of its average degree is
18Average degree Suboptimal degree distribution The complexity of its average degree is
19Average degree Scale-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-progressionIt is obvious that the average degree of SF-LT codes is smallerEncoding/decoding complexity of SF-LT code is much lower than the others
20Degree release probability  M. Luby, “LT codes”, Proc. Annu. Symp. Found. Comput. Sci. (Vancouver, Canada), 2002, ppIn general, r(L) should be larger than 1At least 1 encoded symbol is released when an input symbol is processed.𝐿 𝑘−(𝐿+1) 𝑖−2 𝑘 𝑖
21Degree release probability Ideal soliton distribution  Degree release probability Robust soliton distribution
22Degree release probability Suboptimal degree distribution Using limit theory, it can be expressed as , whereSuppose E encoded symbols is sufficient to recovery the k original input symbols.At each decoding step, larger than 1 encoded symbol is released.
23Degree release probability Scale-free Luby distribution Initial ripple size must be bigger than Robust Soliton Distribution’sk·P1 is bigger than 1The complexity is
24Degree release probability Suboptimal degree distribution's degree release probability is bigger than the others
25Average overhead factor A decreasing ripple size provides a better trade-off between robustness and the overhead factor 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 Sorensen J. H., Popovski. P., Ostergaard J., "On LT codes with decreasing ripple size", Arxiv preprint PScache/ v1.: the number of degree-i input symbols leftL : the size of unprocessed input symbols
27ConclusionRobust 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