The Capacity of Wireless Networks: from to

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The Capacity of Wireless Networksfrom
to

• Binbin Li
• Division of Systems Engineering
• Boston University

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Wireless Networks

• Communication networks formed by nodes with
  radios

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Wireless Networks

• No wired backbone
• No centralized control
• Nodes may cooperate in routing each others data
  packets
• At the Network Layer problems in routing,
  mobility of nodes and power constraints
• At the MAC layer problems with protocols such
  as TDMA, FDMA,CDMA
• At the Physical layer problems in power control

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Two Fundamental Questions

• How should nodes cooperate in maximizing
  information transfer in a wireless network?
• How much information can be transported in a
  wireless network?

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Random Network Model

• Assumptions
• node location
• source-destination pair
• signal power
• far field vs. near field
• Scaling Laws
• what if n ? ? ?
• dense network vs. extended network

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Previous Results

• Gupta and Kumar (IEEE-IT 2000)
• Under classical multihop transmission
  architecture, the total throughput of network can
  NOT be better than
• A scheme that uses only nearest neighbor
  communication can achieve a throughput that
  scales as

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Outline

• Close the Gap via Percolation Theory
• throughput is achievable, at least
  asymptotically
• Achieve the Optimum by Hierarchical Cooperation
• A nearly linear throughput can be
  achieved
• Explore the Limit through Physical Laws
• from Maxwell dominates

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Outline

• Close the Gap via Percolation Theory
• Achieve the Optimum by Hierarchical Cooperation
• Explore the Limit through Physical Laws

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Closing Gap

• Direct communication between any two node
• For dense network, let
  with , or ,

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4-Phase Communication Strategy

• Source nodes drain information to highway
  system
• Information is carried horizontally across the
  network through the highway
• Information is carried vertically across the
  network through the highway
• Information is delivered to destination nodes

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Highway System

• Percolation Theory (horizontal and vertical)
• disjoint sets of paths with each
  group crossing a rectangle of size

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9-TDMA Scheme

• A sequence of time slots

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Bounded SINR Const. Rate
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Bounded Dis. 4 Phase Rates
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Conclusions

• If , the overall per-node rate is
  limited by the highway phase only. Therefore,
  follows.
• For dense network, similarly, it can be shown
  , and therefore

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Outline

• Close the Gap via Percolation Theory
• Achieve the Optimum by Hierarchical Cooperation
• Explore the Limit through Physical Laws

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Motivation

• Multihop transmission architecture is simple, but
  possibly NOT best-of-all for any case.
• Mutually interfering signals can be turned into
  useful ones to be jointly decoded.

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A Physical-layer MIMO

• Both source nodes and destination nodes cooperate
  in clusters to form distributed transmit and
  receive antenna arrays, and then perform many
  simultaneous long-range communications
• MIMO channel model

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A Hierarchical and Recursive Scheme

• For a dense network with , assume there
  exists a scheme such that for each n, w.p. at
  least achieves an aggregate throughput
  for every
  source-destination pair.
• Then one can construct another scheme achieving a
  higher aggregate throughput
• w.p. for every
  source-destination pair.

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3-Phase Strategy

• Node s distributes its M bits among the M nodes
  in its cluster
• These nodes together form a distributed transmit
  antenna array, sending M bits simultaneously to
  the destination cluster containing d
• Each node in the destination cluster obtained one
  observation from MIMO transmission, and quantizes
  and ships it back to d, which can then do joint
  MIMO processing and decode the M transmitted bits

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Phase 1 Setting Up Transmit Cooperation

• Cluster work in parallel
• M sources, traffic
• Apply old scheme and finish in time slots

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Phase 2 MIMO Transmission

• Perform successive long-distance MIMO
  transmissions between source-destination pairs,
  one at a time, and finish in n time slots

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Phase 3 Cooperate to Decode

• Cluster work in parallel
• Nodes quantize each observation into fixed Q
  bits, at most traffic
• Apply old scheme and finish in time
  slots

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A New Throughput

• The 3-phase new scheme achieves a higher
  throughput of

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Multiscale Hierarchical Architecture Achieves
Nearly Linear Scaling

• TDMA works as the basic scheme, whose aggregate
  throughput is i.e. with failure
  prob. equal to zero.
• Starting from TDMA, recursively applying 3-Phase
  Strategy times, achieving a nearly
  linear throughput of

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Multiscale Hierarchical Architecture Achieves
Nearly Linear Scaling
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Results for Extended Network

• Hierarchical Architecture vs. Multihop
  Architecture

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A Cut-Set Upper Bound
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Which is Better?
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Outline

• Close the Gap via Percolation Theory
• Achieve the Optimum by Hierarchical Cooperation
• Explore the Limit through Physical Laws

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The Ultimate Limit of Capacity

• Far Field vs. Near Field
• Theoretical vs. Practical
• Physical laws dominate the universe.

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Still a Cut-Set Approach

• MIMO in matrix form

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Shannons Information Theory
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The Physical Limits

• By MISO channels between all nodes in D and each
  receiver in V
• There are at most independent channels,
  and the capacity of each is at most

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The Physical Limits

• Let be the matrix with entries
• By Maxwells
  physics of wave propagation
• There are at most independent
  channels, therefore

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The Physical Limits

• Therefore, the capacity of wireless
  communications is upper bounded by
• Therefore, claims of linear capacity scaling, and
  consequent constant per-node rate, are artifacts
  of unrealistic channel modeling assumptions.

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References

• M. Franceschetti, O. Dousse, D. N. C. Tse and P.
  Thiran, Closing the gap in the capacity of
  wireless networks via percolation theory, IEEE
  Trans. Inf. Theory, vol. 53, no. 3, pp.
  10091018, Mar. 2007.
• M. Franceschetti, M. D. Migliore and P. Minero,
  The capacity of wireless networks
  information-theoretic and physical limits,
  preprint, Nov. 2007.
• P. Gupta and P. R. Kumar, The capacity of
  wireless networks, IEEE Trans. Inf. Theory, vol.
  42, no. 2, pp. 388404, Mar. 2000.
• A. Özgür, O. Lévêqe and D. N. C. Tse,
  Hierarchical cooperation achieves optimal
  capacity scaling in ad hoc networks, IEEE Trans.
  Inf. Theory, vol. 53, no. 10, pp. 35493572, Oct.
  2007.

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Questions?

• Thank you!