Multi-Edge Framework for Unequal Error Protecting LT Codes H. V. Beltr˜ao Neto, W. Henkel, V. C. da...

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Multi-Edge Framework for Unequal Error Protecting LT Codes

H. V. Beltr˜ao Neto, W. Henkel, V. C. da Rocha Jr.

Jacobs University Bremen, Germany

IEEE ITW(Information Theory Workshop) 2012

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Outlines

• Introduction• Construction algorithms for UEP LT Codes• Simulation results• Conclusions

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Introduction

• Rateless codes– LT codes, Raptor codes, Online codes, …– can adapt their rate on the fly to suit different

channel conditions– are interesting for multicast transmission since

they eliminate the requirement for retransmission

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Introduction

• The multi-edge framework was originally derived for Low Density Parity-Check codes (LDPC) in [7].

• Several edge classes can be defined within the bipartite graph induced by the encoding

• Every node is characterized by the number of connections to edges of each class.

[7] T. Richardson and R. Urbanke, “Multi-Edge Type LDPC Codes,” Tech. Rep., 2004, submitted to IEEE Transaction on Information Theory.

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Introduction

• me : the number of edge types used to represent the bipartite graph

• Each node in the bipartite graph has 2 vectors– x = (x1, ..., xme) that indicates the different types of edges

connected to it

– d = (d1, ..., dme) referred to as edge degree vector which denotes the number of connections of a node to edges of type i,

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Introduction

• xd = • Ldk : the number of variable nodes (input

symbol) of type(degree) d• Rdk : the number of check nodes (output

symbol) of type(degree) d

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Introduction

• type-1 edge (solid lines)• type-2 edges (dashed lines)

3 4 4 2 2 2 3 2

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Introduction

• The multi-edge degree distributions for the code depicted in Fig. 1 are:

• Divide the variable nodes into me protection classes (C1,C2, . . . ,Cme) with monotonically decreasing levels of protection

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Introduction

• The degree of an output symbol corresponds to the number of edges connected to it

• dj : the number of edges of type j connected to a check node

• Ωi : the probability of an output symbol having degree i • Ω(x) : the overall output symbol degree distribution

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Introduction

• Asymptotically (as k increases to infinity), we can approximate Eq. (6) by

• Ωj : the probability of an input symbol of the class Cj being chosen among the k input symbols

• The check node degree distribution is

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Introduction

• Codes with only 2 protection classes, i.e., codes with me = 2 Eqs. (8) and (10) are reduced to

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Construction algorithms for UEP LT Codes

• Weighted approach[5]– Partition of the k variable nodes into me sets of sizes α1k,

α2k,…, αmek, such that – The probability of an edge being connected to a particular

variable node within the set j being qj

– The selection probabilities are defined as ω1 = αkM and ω2 = (1 −α)KL • α is the fraction of input symbols that belong to the first class and,

kM > 1 and 0 < kL < 1 are assigned to the set of more important bits (MIB) and less important bits (LIB)

[5] N. Rahnavard, B. N. Vellambi, and F. Fekri, “Rateless codes with unequal error protection property,” IEEE Transactions on Information Theory, vol. 53, no. 4, pp. 1521–1532, April 2007.

13[5] N. Rahnavard, B. N. Vellambi, and F. Fekri, “Rateless codes with unequal error protection property,” IEEE Transactions on Information Theory, vol. 53, no. 4, pp. 1521–1532, April 2007.

UEP at the Rateless Encoding Stage

• Type-1 Codes

• Weakness– Change of degree distribution (input nodes)– It is likely that d1 = 0 for low-degree encoding

nodes

…

… …d1 = min([(K1/K)dkM,K1] d2 = d-d1

…K1 K2

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N. Rahnavard and F. Fekri, “Finite-length unequal error protection rateless codes: Design and analysis,” in IEEE GLOBECOM 2005.

http://www.powercam.cc/show.php?ch=12&fid=74&id=238

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Construction algorithms for UEP LT Codes

• Windowed approach[6]– Partition the input symbols into protection classes

of k1, k2,. . . , kme symbols such that k1+k2+. . . +kme = k

– The i-th window is defined as the set of the first input symbols– The most important symbols form the first

window while the whole block comprises the final meth window.

[6] D. Sejdinovi´c, D. Vukobratovi´c, A. Doufexi, V. ˇSenk, and R. Piechocki, “Expanding window fountain codes for unequal error protection,” IEEE Transactions on Communications, vol. 57, no. 9, pp. 2510–2516, Sep. 2009.

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Construction algorithms for UEP LT Codes

• Each output symbol is encoded first selecting a window i, with each window having associated to it a probability Гi of being chosen

EWF(Expanding Window Fountain) Codes

• Notation– EWF codes are applied on consecutive source blocks of k

symbols (data packets).– The sequence of r expanding windows, where each window is

contained in the next window in the sequence.• The number r of expanding windows is equal to the number of

importance classes of the source block.

– the size of the i-th window as ki, where k1<…<kr=k• s1=k1

• s1+s2+…+sr = k

• si = ki – ki-1

– the division of the source block into importance classes

17http://www.powercam.cc/slide/19578

EWF Codes

•

18http://www.powercam.cc/slide/19578

Fig. 1. Expanding window fountain (EWF) codes.

19http://www.powercam.cc/slide/19578

EWF Codes

• 2 importance classes– The expressions for the erasure probabilities of

Most Important Bit (MIB) class and Least Important Bit (LIB) class after l iterations

20http://www.powercam.cc/slide/19578

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Flexible UEP LT Codes

• An LT code with 2 protection classes. Its check node degree distribution can be written as

• for d = (d1, d2) and

• To keep the original overall output symbol degree distribution Ω(x), the coefficients Rd must satisfy

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Flexible UEP LT Codes

• In the two-class case, R(1,0) + R(0,1) = Ω1, R(2,0) + R(1,1) + R(0,2) = Ω2 ......

• The idea of our proposed scheme :– Increase the probability of selection of the most

important input symbols – Increase the occurrence probability of output symbols

which are more connected to input symbols of the most sensitive class

– Increase the values of the coefficients R(d1, d2) with d1 > d2

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Flexible UEP LT Codes

• The encoding of the flexible UEP LT codes is similar to the traditional LT codes

• One must select an edge degree vector d according to

• Rd : the probability of the edge degree vector d being chosen

• An output symbol with edge degree vector d = (d1, d2) is formed by selecting d1 input symbols from C1, d2 input symbols from C2 uniformly and at random, and performing a bitwise XOR operation between them.

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Flexible UEP LT Codes

• LT code with degree distribution Ω(x) = 0.15x + 0.55x2 + 0.30x3

• Construct a two-class UEP LT code where 10% of the input symbols belong to the most protected class, i.e., α = 0.1.

• Compute the coefficients Rd for the non-UEP case by means of Eq. (11) with ω1 = α and ω2 = 1 − α.

• We will refer to the EEP LT codes check node coefficients as Rd and the ones of UEP LT codes as Rd

UEP

• Increase the values of the coefficients R(d1,d2) with d1 > d2

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Flexible UEP LT Codes

• A simple way of realizing this is to transfer a fraction f of a coefficient R(a,b) where a < b to the coefficient R(b,a)

– If f = 0.1

• In order to further refine the performance of the UEP LT codes, we can define different values of f– f = f1 for symbols with d2 = 0 and f = f2 otherwise.

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Asymptotic analysis of multi-edge LT codes

• Theorem 1: The erasure probability of an input symbol of class j of a multi-edge LT code with node perspective degree distribution pair (L(x),R(x)) at iteration l ≥ 0 is given by

where y−1 = 1,

denotes the fraction of type j (j = 1 , . . . ,me) edges connected to check nodes of type d.

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Asymptotic analysis of multi-edge LT codes

• Multi-edge UEP LT codes with the overall output symbol degree distribution proposed in [2]

[2] A. Shokrollahi, “Raptor codes,” IEEE Transactions on Information Theory, vol. 52, no. 6, pp. 2551–2567, June 2006.

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Asymptotic analysis of multi-edge LT codes

• Parameters– The weighted approach (km = 2.077)

– The windowed approaches (Г1 = 0.084)

are optimized for an overhead γ = 1.05 according to [6]

• The flexible UEP LT performance was obtained for f1 = 0.09 and f2 = 0.13

[6] D. Sejdinovi´c, D. Vukobratovi´c, A. Doufexi, V. ˇSenk, and R. Piechocki, “Expanding window fountain codes for unequal error protection,” IEEE Transactions on Communications, vol. 57, no. 9, pp. 2510–2516, Sep. 2009.

29Fig. 2. Asymptotic performance of the weighted, windowed and the proposed flexible UEP LT construction strategies. The parameters of the three approaches were optimized for an overhead γ = 1.05.

1.05

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Simulation Results

• Settings– Assume the transmission of k = 5000 input

symbols divided into 2 different levels of protection

– The first protection class is composed of 10% of the input symbols (k1 = 0.1k)

– The second protection class contains the other k2 = k − k1 input symbols

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Fig. 3. Simulation results of the weighted and flexible schemes for k = 5000.

1.0751.05

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Conclusions

• We introduced a multi-edge type analysis of unequal error protecting LT codes.

• Our proposed scheme performed better than the weighted and the windowed schemes.

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References• [1] M. Luby, “LT codes,” in Proceedings of the 43rd Annual IEEE Symposium on

Foundations of Computer Science, Nov. 2002, pp. 271–282.• [2] A. Shokrollahi, “Raptor codes,” IEEE Transactions on Information Theory, vol. 52,

no. 6, pp. 2551–2567, June 2006.• [3] P. Maymounkov, “Online codes,” NYU, Tech. Rep. TR2003-883, Nov. 2002.• [5] N. Rahnavard, B. N. Vellambi, and F. Fekri, “Rateless codes with unequal error

protection property,” IEEE Transactions on Information Theory, vol. 53, no. 4, pp. 1521–1532, April 2007.

• [6] D. Sejdinovi´c, D. Vukobratovi´c, A. Doufexi, V. ˇSenk, and R. Piechocki, “Expanding window fountain codes for unequal error protection,” IEEE Transactions on Communications, vol. 57, no. 9, pp. 2510–2516, Sep. 2009.

• [7] T. Richardson and R. Urbanke, “Multi-Edge Type LDPC Codes,” Tech. Rep., 2004, submitted to IEEE Transaction on Information Theory.

• [8] ——, Modern Coding Theory. Cambridge University Press, 2008.• [9] M. Luby, M. Mitzenmacher, and A. Shokrollahi, “Analysis of random processes

via and-or tree evaluation,” in Proceedings of the 9th SIAM Symposium on Discrete Algorithms, Jan. 1998, pp. 364–373.