Network Coding and Wireless Physical-layer ... - Jacobs University

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Chapter 5: Network Coding with LT Codes as Erasure Codes 69 5.6 The Cooperative Solution Performed by the Source and the Relay As an example, consider the case when both the edges BD and CE are erasure-prone. In this case, we cannot find a solution from the previous section that satisfies both receivers. If A chooses to transmit (b 1 + b 2 , b 2 ) to (AB, AC), the receiver D is satisfied but E still suffers from the degree distribution distortion. It can be imagined that the ideal solution can be found such that, instead of transmitting b 1 and b 2 with the robust soliton distribution, we use another degree distribution ν(i) which, after being distorted by erasures, becomes the robust soliton distribution. However, such a distribution is very difficult, if not impossible, to be found. Indeed, it is proven in a similar work [67] that one cannot find a degree distribution ν(i) for b 1 and b 2 such that b 1 + b 2 follows the robust soliton distribution. Since we cannot achieve this ideal solution, we will offer a cooperative scheme performed by the source node A and the relay node F that corrects the degree distribution distortion only in the positions that most affect the performance. Those positions are at degrees 2, 4, 44, and 84. The severe distortion at 4, 44, and 84 is due to the high probability that b 1 and b 2 have the degrees of 2 or 42, causing b 1 + b 2 to have the degree of 4, 44, or 84 with higher probability than what is required, whereas the distortion at 2 is due to the fact that b 1 +b 2 cannot have the degree 2 unless both b 1 and b 2 have the degree of 1, which is unlikely. According to our cooperative scheme, the distortion at 44 and 84 is reduced by applying a buffering scheme at the source. The scheme arranges the LT-encoder output in such an order that b 1 and b 2 with degrees of 2 and 42, 42 and 2, or 42 and 42 are not allowed to be simultaneously transmitted and mixed at the relay thereafter, as implemented in Step 4) of Algorithm 5.1. In addition, the distortion at 2 and 4 is corrected by selectively discarding some symbols at the relay, i.e., when both b 1 and b 2 have the degree of 2, there is a probability p c that the relay performs network coding and a probability 1 − p c that
70 Chapter 5: Network Coding with LT Codes as Erasure Codes either b 1 or b 2 is transmitted while the other is discarded. This will relieve the excess of symbols with degree 4 and the shortage of those with degree 2. The scheme is implemented in Algorithm 5.2. Algorithm 5.1 The buffering scheme performed by the source Let ⎧ ⎪⎨ 1 if the switch is pushed to the left buffer • S l = ⎪⎩ 0 if the switch is pushed to the right buffer , • λ n be the LT-encoder output at the discrete time n = 0, 1, 2, ..., N max , • d(λ n ) be the degree of λ n , • N B be the size of each buffer, • B l (m), m = 0, 1, 2, ..., N B −1 be the m th element in the left buffer such that B l (m) precedes B l (m + 1), • B r (q), q = 0, 1, 2, ..., N B − 1 be the q th element in the right buffer such that B r (q) precedes B r (q + 1), • π l , π r be the pointers of the left and the right buffer, respectively, • and b 1 , b 2 be the current symbol to be transmitted to the edges AB and AC, respectively, 1) Initialize n := 0 S l := 1 B l (0) := λ 0 π l := 1 π r := 0 {B l (m)|m = 1, 2, ..., N B − 1} := ∅ {B r (q)|q = 0, 1, 2, ..., N B − 1} := ∅ 2) n := n + 1 3) if S l = 0 {
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Network Coding and Wireless Physica
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Abstract The general abstraction of
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Contents Abstract i Abstract ii Dec
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vi CONTENTS 5.4 Degree Distribution
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List of Figures 2.1 A basic digital
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x LIST OF FIGURES 5.11 Comparison o
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Part I Motivation, Overview, and In
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Chapter 1: Motivation and Overview
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Chapter 2 Introduction to Digital C
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Chapter 2: Introduction to Digital
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Chapter 3: Introduction to Graphs a
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Page 46 and 47: 33 The emergence of scalable video
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Page 124 and 125: Chapter 8: Summary, Conclusion, and
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Bibliography [1] 3GPP, “3GPP; Tec
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BIBLIOGRAPHY 121 [21] F.A. Hayek,
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BIBLIOGRAPHY 123 [43] M. Friedman a
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BIBLIOGRAPHY 125 [63] S. Jaggi, P.
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BIBLIOGRAPHY 127 [81] X. Sun, X. Wu
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