Efficient Multihop Broadcast for
Wideband Systems
Ivana Maric and Roy Yates
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
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
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
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Accumulative broadcast
• Allow nodes to collect energy of unreliably received signals
Wireless advantage
Accumulative broadcast
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?
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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
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Relays Use Repetition Coding
• 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
P1
X
• After K nodes retransmit a codeword X:
source
P2
• Received signal for a symbol x:
Y = hx + n
X
P3
Y
X …
PK
m
• Maximum rate:
K
I(x;y) = W log2(1 + Σ hmkPk/NoW)
k=1
•
X
Upper bound:K
CMAC = W Σ log2(1 + hmkPk/NoW)
k=1
Repetition is OK for Large W
• Given fixed powers {P1,…PK } 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.
• As W  ∞,
I(x;y)  Σ hmkPk/Noln2
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P1
k
P2
• MAC Upper bound:
CMAC = W Σ log2(1 + hmkPk/NoW)
 Σ hmkPk/Noln2
k
P3
m
…
PK
• In such a network, how do we solve the broadcast problem?
Difference from Min-Energy Broadcast Tree
• The total transmit power of the minimum-energy broadcast tree upper bounds
the total transmit power of accumulative broadcasting
•
•
In the MBT problem, knowing the broadcast tree solves the problem completely:
power levels are uniquely determined
For accumulative broadcast, tree is not meaningful
•
Different total power for orders: 1-2-3 and 1-3-2
5
3
2
1
4
source
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Approach
•
Divide the problem into two subproblems:
• Choose a reliability schedule
•
•
•
•
An order in which nodes become reliable
Also, 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 decoding
Given a schedule, find the best power levels
•
Can be formulated as LP
LP for Transmit Powers
5
4
2
1
3
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• Fix a schedule: 1 2 3 4 5
• We know which nodes contribute to the energy
collected at a node
min (p1 + p2 + p3 + p4)
h21p1
≥ PT
h31p1 + h32p2
≥ PT
h41p1 + h42p2 + h43p3
≥ PT
h51p1 + h52p2 + h53p3 + h54p4 ≥ PT
p1, p2, p3, p4
≥0
• But, finding the optimal schedule is NP-complete.
Greedy Filling Heuristic
•
•
Given S={reliable nodes}, U={unreliable nodes}
Choose node k to maximize “filling rate” of the unreliable
nodes
k = arg max Σ hji
iєS
jєU
•
Choose power Pk to make one more node reliable
•
Offline Optimization:
• If node k transmits multiple times, Pk1, Pk2…, set Pk= Σ
Pki and transmit once
• Readjust schedule
Simple Experiments
• Throw N nodes in a square (100 trials)
• Propagation exponent=2
• For small N:
• Enumerate all schedules, find optimal AB powers
• Compare with
– greedy heuristic
– BIP [Wieselthier, Nguyen, Ephremides]
(ignores unreliable overheard messages)
• For large N:
• Compare greedy heuristic, BIP
Performance results
Note: Powers
normalized by
optimal sol’n
Performance results
Conclusion
•
Accumulative broadcast:
Nodes collect energy of unreliably received signals
•
Only reliable forwarding is allowed
•
Large BW, relays can use repetition coding
•
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
•
Proposed a heuristics to find a good schedule
•
Compared the algorithm performance with the optimum solution
and BIP performance