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

   Introduction Degree distribution of LT codes Analysis of LT codes   Average degree Degree release probability  Average 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 distribution    Robust soliton distribution Suboptimal degree distribution Scale-free Luby distribution  Average degree  Degree release probability  Average overhead factor

LT process

STATE: a1 a2 a3 a4 a5 covered = { } processed = { } ripple = { } released = { } 4 ACTION: Init: Release c2, c4, c6 c1 c2 c3 c4 c5 c6 http://www.powercam.cc/slide/21817

LT process

STATE: a1 a2 a3 a4 a5 released = {c2,c4,c6} covered = {a1,a3,a5} processed = { } ripple = {a1,a3,a5} 5 ACTION: Process a1 c1 c2 c3 c4 c5 c6

LT process

STATE: a1 a2 a3 a4 a5 released = {c2,c4,c6,c1} covered = {a1,a3,a5} processed = {a1} ripple = {a3,a5} 6 ACTION: Process a3 c1 c2 c3 c4 c5 c6

LT process

STATE: a1 a2 a3 a4 a5 released = {c2,c4,c6,c1} covered = {a1,a3,a5} processed = {a1,a3} ripple = {a5} 7 ACTION: Process a5 c1 c2 c3 c4 c5 c6

LT process

STATE: a1 a2 a3 a4 a5 released = {c2,c4,c6,c1,c5} covered = {a1,a3,a5,a4} processed = {a1,a3,a5} ripple = {a4} 8 ACTION: Process a4 c1 c2 c3 c4 c5 c6

LT process

STATE: a1 a2 a3 a4 a5 released = {c2,c4,c6,c1,c5,c3} covered = {a1,a3,a5,a4,a2} processed = {a1,a3,a5,a4} ripple = {a2} 9 ACTION: Process a2 c1 c2 c3 c4 c5 c6

LT process

STATE: a1 a2 a3 a4 a5 released = {c2,c4,c6,c1,c5,c3} covered = {a1,a3,a5,a4,a2} processed = {a1,a3,a5,a4,a2} ripple = { } 10 ACTION: Success!

c1 c2 c3 c4 c5 c6

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. 271-282.

 Robust soliton distribution [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  Optimal degree distribution is proposed[12] [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.

•  When k is large, the coefficient matrix of optimal degree distribution is too sick.

 No solution.

Suboptimal degree distribution: R is initial ripple size E is the expected number of encoded symbols required to recovery the input symbols.

 Scale-free Luby 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

P 1

: 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

 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 Ideal soliton distribution  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 degree Robust soliton distribution  The complexity of its average degree is

 Average degree Suboptimal degree distribution  The complexity of its average degree is

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

𝐿 1 1 1 𝑘−(𝐿+1) 𝑖−2 𝑘 𝑖  In general, r(L) should be larger than 1  At least 1 encoded symbol is released when an input symbol is processed.

Degree release probability Ideal soliton distribution [6]  Degree release probability Robust soliton distribution 

Degree release probability Suboptimal 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 probability Scale-free Luby distribution  Initial ripple size must be bigger than Robust Soliton Distribution’s  k·P 1 is bigger than 1  The complexity is

Degree release probability

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

Average overhead factor  A decreasing ripple size provides a better trade-off between robustness and the overhead factor [14] [14] Sorensen J. H., Popovski. P., Ostergaard J., "On LT codes with decreasing ripple size", Arxiv preprint PScache/1011.2078v1.

 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

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