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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

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Outlines

• Introduction• Rateless codes design• Evaluation results• Conclusion

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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.

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Intermediate Performance of Rateless Codes

Sujay SanghaviLIDS, MIT

IEEE ITW(Information Theory Workshop) 2007

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6

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,” IEEE Information Theory Workshop (ITW), pp. 478–482, Sept. 2007.

7

Introduction

– • For an integer m, where• • It is shown that the optimum degree distribution in this

region is given by

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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.

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Growth Codes• Degree of a codeword “grows” with time• At each timepoint codeword 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

R1 R3R2 R4

d=1 d=2 d=3 d=4

Time ->

http://www.powercam.cc/slide/17704

[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.

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

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.

y4 x3x5x6

already decoded data: new received packets:

x1 x2 x3 x5

x3 x5 y4 x6

d = 3

d – 1 data are already decoded.

http://www.powercam.cc/slide/17704

Ideas of Proposed Method• Growth Codes:

– A received encoded packet is useless:• if all d data used to form a encoded packet are already known.

y1 x1x3

already decoded data: new received packets:

x1 x2 x3 x5 d = 2

d data are already decoded.

new received packet is useless.

http://www.powercam.cc/slide/17704

Ideas of Proposed Method• Consider the degree of an encoded packet:

– Decoder has decoded r original data.– The probability that new received encoded packet is immediately

decodable to the decoder:

Number of decoded original data: r

Impo

rtanc

e of

Imm

edia

tely

D

ecod

able

Pac

ket

: Low Degree

: High Degree

http://www.powercam.cc/slide/17704

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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-length rateless 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.

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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 yl be 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.

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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

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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

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24Fig. 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.

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• 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.

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•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.

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29Fig. 5. Comparison of the performance of the rateless codes employing designed degree distributions for k = 100 with the upper bound on rateless codes intermediate performance.

K=100

30Fig. 5. Comparison of the performance of the rateless codes employing designed degree distributions for k = 1000 with the upper bound on rateless codes intermediate performance.

K=1000

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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.

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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 rateless codes,” 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.

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An Optimal Partial Decoding Algorithm for Rateless Codes

V. Bioglio, M. Grangetto, R. Gaeta, M. Sereno

Dipartimento di Informatica Universit`a di Torino

IEEE ISIT(International Symposium on Information Theory) 2011

35Fig. 2. Partial decoding performance of LT codes.

36Fig. 3. Algorithm complexity vs. k.