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

More documents

Recommendations

Info

Chapter 4: Unequal Erasure Protection (UEP) in Network Coding 49 quality among receiving nodes, whereas that of Section 4.10 is fair competition among the nodes, i.e., the node who pays more receives better quality. 4.9 Minimax Assignment of Global Encoding Kernels (GEKs) In this section, we propose a strategy for assigning a GEK to each subtree. The ultimate goal is equal data quality for every receiving node. To derive a solution that is as close as possible to that goal, we use a minimax criterion, i.e., minimizing the maximum expected degradation, which is equivalent to maximizing the expected data quality of the poorest sink [5]. Since our example concerns allocating GEKs to five subtrees from a set of seven available GEKs, as shown in Table I, the attempt to find the optimal solution at once is impractical because it results in the search within the space of the size 7 5 . Therefore, we may adopt an iterative approach, in which at the iteration t, n(t) GEKs are allocated to n(t) subtrees, resulting in the search within the space of the size 7 n(t) [5]. The assignment of n(t) GEKs to n(t) subtrees at the t th iteration does not have the same effect on every sink node. An individual sink may either 1) not be connected to any of those subtrees and thus gain nothing more than the previous iteration, or 2) be connected to certain subtrees but still cannot recover any other data than that received in the previous iteration, or 3) be connected to certain subtrees and can recover some more data. In the first and the second cases, the expected data quality of the sink does not improve at the t th iteration, whereas, in the third case, it does. In each iteration, we aim to ensure that the temporary minimax criterion is satisfied. To do so, we first have to evaluate the temporary expected utility E t−1 [U i ] from the previous t − 1 th iteration for the sink i. After that, we find the minimum E t [U i ] among all i for every possible GEK assignment at the current t th iteration, and then select the assignment that gives the maximum value of min E t [U i ] [5].
50 Chapter 4: Unequal Erasure Protection (UEP) in Network Coding The temporary expected utility E t [U i ] of the i th sink at the t th iteration is given by E t [U i ] = ω∑ φ t,i,j · ∆ j , (4.14) j=1 where φ t,i,j represents the probability that the prefix P j can be recovered at the sink i after the GEK assignment at the iteration t. φ t,i,j equals ρ i,j in Eq. (4.8) if P j can be recovered by the GEKs assigned so far up to the iteration t. Otherwise, it equals zero. ∆ j denotes the incremental utility if P j is recovered [5]. We suggest that the number n(t 1 ) of GEKs assigned at the iteration t 1 should be greater than or equal to the number n(t 2 ) at the iteration t 2 if t 1 < t 2 , i.e., it is better to take larger search spaces into consideration during some first iterations, after which things do not improve much. In our network example, we assume that every edge-disjoint path has an erasure probability of 0.1 and the utility vector ∆U k = [∆ 1 , ∆ 2 , ∆ 3 ] = [1, 0.5, 0.25] for every sink node R k . If we let n(1) = 3 and n(2) = n(3) = 1, then the minimum temporary expected data quality in the first iteration is maximized by allocating the GEKs [1 0 0] T to T 3 , [0 1 0] T to T 1 , and [1 1 0] T to T 5 . The sink R 1 now has the temporary expected utility of E 1 [U 1 ] = (0.9)(1) + (0.81)(0.5) = 1.305, since, by receiving [1 0 0] T from T 3 with the probability of 0.9, R 1 can recover the first prefix, and by receiving both [1 0 0] T from T 3 and [0 1 0] T from T 1 with the probability of 0.81, it can recover the second prefix. In the same manner, the temporary expected data qualities E 1 [U 2 ], E 1 [U 3 ], and E 1 [U 4 ] of R 2 , R 3 , and R 4 become 1.305, 1.305, and 1.215, respectively. This gives the minimum temporary expected data quality, min E 1 [U i ], of 1.215 at the sink R 4 , which is the maximum that one can find at this iteration. Table 4.2 shows the resulting expected data qualities at all sink nodes and the minimum one, after the third iteration finishes and all subtrees have obtained their GEKs. Table 4.3 shows that, in our network example, the suggested strategy achieves the global optimum. In addition, on average, a random GEK assignment results in the minimum expected quality that is closer to that of the worst case than the best one. Thus,
Page 1:
Network Coding and Wireless Physica
Page 4 and 5:
Abstract The general abstraction of
Page 6 and 7:
Contents Abstract i Abstract ii Dec
Page 8 and 9:
vi CONTENTS 5.4 Degree Distribution
Page 10 and 11:
List of Figures 2.1 A basic digital
Page 12 and 13: x LIST OF FIGURES 5.11 Comparison o
Page 14 and 15: Part I Motivation, Overview, and In
Page 16 and 17: Chapter 1: Motivation and Overview
Page 18 and 19: Chapter 2 Introduction to Digital C
Page 20 and 21: Chapter 2: Introduction to Digital
Page 30 and 31: Chapter 3: Introduction to Graphs a
Page 46 and 47: 33 The emergence of scalable video
Page 48 and 49: Chapter 4: Unequal Erasure Protecti
Page 72 and 73: Chapter 5: Network Coding with LT C
Page 88 and 89: Part III Wireless Physical-layer Se
Page 90 and 91: Chapter 6 Wireless Physical-layer S
Page 92 and 93: Chapter 6: Wireless Physical-layer
Page 104 and 105: Chapter 7 Physical-layer Key Encodi
Page 106 and 107: Chapter 7: Physical-layer Key Encod
Page 112 and 113:
Chapter 7: Physical-layer Key Encod
Page 114 and 115:
Page 116 and 117:
Page 118 and 119:
Page 120 and 121:
Page 122 and 123:
Page 124 and 125:
Chapter 8: Summary, Conclusion, and
Page 126 and 127:
Page 128 and 129:
Page 130 and 131:
Page 132 and 133:
Bibliography [1] 3GPP, “3GPP; Tec
Page 134 and 135:
BIBLIOGRAPHY 121 [21] F.A. Hayek,
Page 136 and 137:
BIBLIOGRAPHY 123 [43] M. Friedman a
Page 138 and 139:
BIBLIOGRAPHY 125 [63] S. Jaggi, P.
Page 140:
BIBLIOGRAPHY 127 [81] X. Sun, X. Wu
show all

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

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?