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Rateless Codes with Optimum Intermediate Performance

Rateless Codes with Optimum Intermediate Performance. Ali Talari and Nazanin Rahnavard Oklahoma State University, USA IEEE GLOBECOM 2009 & IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 60, NO. 5, MAY 2012. Outlines. Introduction Rateless codes design Evaluation results Conclusion.

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Rateless Codes with Optimum Intermediate Performance

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  1. Rateless Codes with OptimumIntermediatePerformance Ali Talari and Nazanin Rahnavard Oklahoma State University, USA IEEE GLOBECOM 2009 & IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 60, NO. 5, MAY 2012

  2. Outlines • Introduction • Rateless codes design • Evaluation results • Conclusion

  3. Introduction • Intermediate recovery rate is important in video or voice transmission applications where partial recovery of the source packets from received encoded packets is beneficial. • This motivates the design of forward error correction (FEC)codes with high intermediate performance. • In rateless coding, the employed degree distribution significantly affects the packet recovery rate.

  4. Intermediate Performance of Rateless Codes SujaySanghavi LIDS, MIT IEEE ITW(Information Theory Workshop) 2007

  5. Introduction • Sanghavi in [4] • z∈ [0, ½] • the optimum degree distribution has degree one packets only, • the optimum degree distribution has degree two packets only, [4] S. Sanghavi, “Intermediate performance of rateless codes,” IEEEInformation Theory Workshop (ITW), pp. 478–482, Sept. 2007.

  6. Introduction • For an integer m, where • It is shown that the optimum degree distribution in this region is given by

  7. Introduction • The number of source packets : k • Number of received coded packets : n • Received overhead by γ , where γ=n / k • The ratio of number of recovered packets at the receiver to k by z • Finding degree distributions with maximal packet recovery rates in intermediate range, 0 < γ< 1. • We define packet recovery rates at 3 values of γ as our conflicting objective functions and employ NSGA-II multi-objective genetic algorithms optimization method to find several degree distributions with optimum packet recovery rates.

  8. d=2 d=3 d=4 d=1 R1 R2 R3 R4 Time -> Growth Codes • Degree of a codeword “grows” with time • At each timepointcodeword of a specific degree has the most utility for a decoder (on average) • This “most useful” degree grows monotonically with time • R : Number of decoded symbols sink has [6] A. Kamra, V. Misra, J. Feldman, and D. Rubenstein, “Growth codes: Maximizing sensor network data persistence,” SIGCOMM Computer Communication Rev., vol. 36, no. 4, pp. 255–266, 2006. http://www.powercam.cc/slide/17704

  9. Ideas of Proposed Method • Method: • Growth Codes: • Been designed for sensor networks in catastrophic or emergency scenarios. • To make new received encoded packet useful. • Can be decoded immediately. • To avoid new received encoded packet useless. • Cannot be decoded. http://www.powercam.cc/slide/17704

  10. x2 x1 x3x5x6 y4   y4 x6 x5 x3 x3 x5 Ideas of Proposed Method • Growth Codes: • A received encoded packet is immediately useful: • if d - 1 of the data used to form this encoded packet are already decoded/known. already decoded data: new received packets: d = 3 d – 1 data are already decoded. http://www.powercam.cc/slide/17704

  11. x2 x1 x1x3 y1 x3 x5 Ideas of Proposed Method • Growth Codes: • A received encoded packet is useless: • if all ddata used to form a encoded packet are already known. already decoded data: new received packets: d = 2 d data are already decoded. new received packet is useless. http://www.powercam.cc/slide/17704

  12. Importance of Immediately Decodable Packet : Low Degree : High Degree Number of decoded original data: r Ideas of Proposed Method • Consider the degree of an encoded packet: • Decoder has decoded r originaldata. • The probability that new received encoded packet is immediately decodable to the decoder: http://www.powercam.cc/slide/17704

  13. Rateless codes design • We propose a novel approach that finds degree distributions for high recovery rates throughout intermediate range • We select one γ from each region, i.e. γ ∈ {0.5, 0.75, 1}, and define 3 objective functions to be the packet recovery rates at these γ ’s • 2 approaches • 1)We consider the infinite asymptotic case similar to existing studies. We formulate packet recovery rates using a technique called And-Or tree analysis [1, 7-9] • 2)We consider finite-lengthrateless codes with k = 100 and k = 1000 and show how degree distributions vary with k [8] N. Rahnavard and F. Fekri, “Generalization of rateless codes for unequal error protection and recovery time: Asymptotic analysis,” IEEE International Symposium on Information Theory, 2006, pp. 523–527, July 2006. [9] N. Rahnavard, B. Vellambi, and F. Fekri, “Rateless codes with unequal error protection property,” IEEE Transactions on Information Theory, vol. 53, pp. 1521–1532, April 2007.

  14. Rateless codes design • In And-Or tree analysis technique [1, 7–9] the error rate of iterative decoding of rateless codes is probabilistically formulated for k = ∞ • Consider a rateless code with parameters Ω(x) and γ. • Let ylbe the probability that a packet is not recovered after l decoding iterations [1] P. Maymounkov, “Online codes,” NYU Technical Report TR2003-883, 2002. [7] M. G. Luby, M. Mitzenmacher, and M. A. Shokrollahi, “Analysis of random processes via And-Or tree evaluation,” (Philadelphia, PA, USA), pp. 364–373, Society for Industrial and Applied Mathematics, 1998.

  15. Rateless codes design • Let F denote the corresponding fixed point • This fixed point is the final packet error rate of a rateless code with parameters Ω(x) and γ • We define 3 objective functions given as fixed points of (2) for γ= 0.5, γ= 0.75, and γ= 1

  16. Evaluation Results • The upper bound on rateless codes recovery rates at γ= 0.5, γ= 0.75, and γ= 1 are 0.393469, 0.5828 and 1 • We define F(Ω(x)) as where are the weights assigned to each objective function

  17. Fig. 4. Comparison of the performance of the rateless codes employing designed degree distributions for asymptotic case with the upper bound on rateless codes intermediate performance.

  18. The designed degree distributions show a high performance and perform close to upper bound. • These degree distributions are optimum in intermediate performance. • According to the selected weights the resulting codes have the highest recovery rate at the γ with the highest weight.

  19. When k = 100, decoder requires larger fraction of degree one packets and lower degree packets are preferred. • Degree two packets constitute a high percentage of encoded packets compared to packets of other degrees.

  20. K=100 Fig. 5. Comparison of the performance of the ratelesscodes employing designed degree distributions for k = 100 with the upper bound on rateless codes intermediate performance.

  21. K=1000 Fig. 5. Comparison of the performance of the ratelesscodes employing designed degree distributions for k = 1000 with the upper bound on rateless codes intermediate performance.

  22. Conclusions • In this paper, we studied the intermediate performance of rateless codes and proposed to employ multi-objective genetic algorithms to find several optimum degree distributions in intermediate range. • We used the state-of-the-art optimization algorithm NSGA-II to find the set of optimum degree distributions which are called pareto optimal.

  23. References • [1] P. Maymounkov, “Online codes,” NYU Technical Report TR2003-883, 2002. • [4] S. Sanghavi, “Intermediate performance of rateless codes,” Information Theory Workshop, 2007. ITW ’07. IEEE, pp. 478–482, Sept. 2007. • [5] K. Deb, A. Pratap, S. Agarwal, and T. Meyarivan, “A fast and elitist multiobjective genetic algorithm: NSGA-II,” IEEE Transactions on Evolutionary Computation, vol. 6, pp. 182–197, Apr 2002. • [6] A. Kamra, V. Misra, J. Feldman, and D. Rubenstein, “Growth codes: Maximizing sensor network data persistence,” SIGCOMM Computer Communication Rev., vol. 36, no. 4, pp. 255–266, 2006. • [7] M. G. Luby, M. Mitzenmacher, and M. A. Shokrollahi, “Analysis of random processes via And-Or tree evaluation,” (Philadelphia, PA, USA), pp. 364–373, Society for Industrial and Applied Mathematics, 1998. • [8] N. Rahnavard and F. Fekri, “Generalization of rateless codes for unequal error protection and recovery time: Asymptotic analysis,” IEEE International Symposium on Information Theory, 2006, pp. 523–527, July 2006. • [9] N. Rahnavard, B. Vellambi, and F. Fekri, “Rateless codes with unequal error protection property,” IEEE Transactions on Information Theory, vol. 53, pp. 1521–1532, April 2007. • [10] http://cwnlab.ece.okstate.edu/research • S. Kim and S. Lee, “Improved intermediate performance of ratelesscodes,” ICACT 2009, vol. 3, pp. 1682–1686, Feb. 2009. • Valerio Bioglio, Marco Grangetto, Rossano Gaeta, Matteo Sereno: An optimal partial decoding algorithm for rateless codes. IEEE ISIT 2011,pp 2731-2735.

  24. An Optimal Partial Decoding Algorithm for RatelessCodes V. Bioglio, M. Grangetto, R. Gaeta, M. Sereno Dipartimento di Informatica Universit`a di Torino IEEE ISIT(International Symposium on Information Theory) 2011

  25. Fig. 2. Partial decoding performance of LT codes.

  26. Fig. 3. Algorithm complexity vs. k.

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