1. Efficient Multihop Broadcast for Wideband Systems Ivana Maric and Roy Yates

2. Wireless Broadcast Wireless network of N nodes Source transmits with rate R Messages are to be delivered to all the nodes Nodes can choose a power level for each transmission Problem: Broadcast at rate R to all nodes with minimum total power Comment: N=3 nodes is a single relay channel How many simplifications are needed? source

3. System Model: Orthogonal Channels A link: AWGN channel with bandwidth W Large bandwidth resources Each transmission in an orthogonal channel Nodes can listen to all the channels Motivation: Sensor networks Low-powered nodes, very low data rates

4. Minimum-energy broadcast problem Min-energy broadcast tree problem [J. Wieselthier, G. Nguyen, A. Ephremides] Wired network: Min-cost spanning tree problem ‘Wireless multicast advantage’: all the nodes in the transmission range will benefit from a transmission Problem is NP-complete [M. Čagalj et al., Ahluwalia et al., W. Liang] source

5. Accumulative broadcast Wireless advantage Wireless advantage Accumulative broadcast Accumulative broadcast Allow nodes to collect energy of unreliably received signals

6. Accumulative broadcast As the message is forwarded, a node has multiple opportunities to receive energy needed for reliable reception of that message Key issue: Who do you listen to? source

7. Reliable forwarding A node can forward a message only after reliable decoding Disadvantage: suboptimal source Benefits: Simplifies the system architecture Still allows for unreliable overheard information Imposes an ordering on the node transmissions

8. Relays Use Repetition Coding P1P1 P2P2 P3P3 PKPK source … Relays resend the same codeword A node m will decode a codeword using transmissions of a subset of nodes that became reliable prior to node m Y X X X X m k=1 K After K nodes retransmit a codeword X: Received signal for a symbol x: Y = hx + n Maximum rate: I(x;y) = W log 2 (1 + Σ h mk P k /NoW) Upper bound: C MAC = W Σ log 2 (1 + h mk P k /NoW) k=1 K

9. Repetition is OK for Large W source Given fixed powers {P 1,…P K } and reliable forwarding, the maximum rate achievable from the source to any destination is achieved by the repetition coding in the limit of large W.Given fixed powers {P 1,…P K } and reliable forwarding, the maximum rate achievable from the source to any destination is achieved by the repetition coding in the limit of large W. In such a network, how do we solve the broadcast problem?In such a network, how do we solve the broadcast problem? As W  ∞, I(x;y)  Σ h mk P k /Noln2As W  ∞, I(x;y)  Σ h mk P k /Noln2 MAC Upper bound:MAC Upper bound: C MAC = W Σ log2(1 + h mk P k /NoW)  Σ h mk P k /Noln2 C MAC = W Σ log2(1 + h mk P k /NoW)  Σ h mk P k /Noln2 m P1P1 P2P2 P3P3 PKPK … k k

10. Difference from Min-Energy Broadcast Tree In the MBT problem, knowing the broadcast tree solves the problem completely: power levels are uniquely determinedIn the MBT problem, knowing the broadcast tree solves the problem completely: power levels are uniquely determined For accumulative broadcast, tree is not meaningfulFor accumulative broadcast, tree is not meaningful Different total power for orders: 1-2-3 and 1-3-2Different total power for orders: 1-2-3 and 1-3-2 source 3 2 1 4 5 The total transmit power of the minimum-energy broadcast tree upper bounds the total transmit power of accumulative broadcastingThe total transmit power of the minimum-energy broadcast tree upper bounds the total transmit power of accumulative broadcasting source

11. Approach Divide the problem into two subproblems:Divide the problem into two subproblems: Choose a reliability scheduleChoose a reliability schedule An order in which nodes become reliableAn order in which nodes become reliable Also, an order in which nodes are given a chance to transmitAlso, an order in which nodes are given a chance to transmit For each node, schedule specifies a subset of nodes that contribute to its reliable decodingFor each node, schedule specifies a subset of nodes that contribute to its reliable decoding Given a schedule, find the best power levelsGiven a schedule, find the best power levels Can be formulated as LPCan be formulated as LP

12. LP for Transmit Powers 5 source 1 2 4 3 min (p 1 + p 2 + p 3 + p 4 ) h 21 p 1 ≥ P T h 31 p 1 + h 32 p 2 ≥ P T h 41 p 1 + h 42 p 2 + h 43 p 3 ≥ P T h 51 p 1 + h 52 p 2 + h 53 p 3 + h 54 p 4 ≥ P T p 1, p 2, p 3, p 4 ≥ 0 But, finding the optimal schedule is NP-complete.But, finding the optimal schedule is NP-complete. Fix a schedule: 1 2 3 4 5Fix a schedule: 1 2 3 4 5 We know which nodes contribute to the energy collected at a nodeWe know which nodes contribute to the energy collected at a node

13. Greedy Filling Heuristic Given S={reliable nodes}, U={unreliable nodes}Given S={reliable nodes}, U={unreliable nodes} Choose node k to maximize “filling rate” of the unreliable nodesChoose node k to maximize “filling rate” of the unreliable nodes iєSiєSiєSiєS jєUjєUjєUjєU k = arg max Σ h ji Choose power P k to make one more node reliableChoose power P k to make one more node reliable Offline Optimization:Offline Optimization: If node k transmits multiple times, P k1, P k2 …, set P k = Σ P ki and transmit onceIf node k transmits multiple times, P k1, P k2 …, set P k = Σ P ki and transmit once Readjust scheduleReadjust schedule

14. Simple Experiments Throw N nodes in a square (100 trials)Throw N nodes in a square (100 trials) Propagation exponent=2Propagation exponent=2 For small N:For small N: Enumerate all schedules, find optimal AB powersEnumerate all schedules, find optimal AB powers Compare withCompare with –greedy heuristic –BIP [Wieselthier, Nguyen, Ephremides] (ignores unreliable overheard messages) For large N:For large N: Compare greedy heuristic, BIPCompare greedy heuristic, BIP

15. Performance results Note: Powers normalized by optimal sol’n

16. Performance results

17. Conclusion Accumulative broadcast: Nodes collect energy of unreliably received signalsAccumulative broadcast: Nodes collect energy of unreliably received signals Only reliable forwarding is allowedOnly reliable forwarding is allowed Large BW, relays can use repetition codingLarge BW, relays can use repetition coding Formulate problem as two subproblems:Formulate problem as two subproblems: 1. Find a reliable schedule 2. LP to find the optimum power levels for a given schedule Finding the optimum schedule is NP-complete Finding the optimum schedule is NP-complete Proposed a heuristics to find a good schedule Proposed a heuristics to find a good schedule Compared the algorithm performance with the optimum solution and BIP performanceCompared the algorithm performance with the optimum solution and BIP performance