Project Presentation for ECE 1528

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Information Theory Of Large Networks
The Capacity of Wireless Networks Towards an
Information Theory of Large Networks An
Achievable Rate Region P. Gupta and P.R. Kumar
Project Presentation for ECE 1528 Multiuser
Information Theory Presented by Rani Daher
April 5, 2007
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Outline

• Bounds for arbitrary and random networks

• Achievable rate region for a single
  source-destination pair

• Application to the AWGN Channel

• Extension to multiple source-destination pairs

• A special configuration

• Conclusion and possible extensions

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Definitions (Networks, Models, Capacities)

• Arbitrary Networks v/s Random Networks

• Transport Capacity v/s Throughput Capacity

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Arbitrary Networks Upper Bound
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Arbitrary Networks Lower Bound

• Node placement to achieve Transport capacity of

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Random Networks Lower Bound

• Introduce a Voronoi Tessellation with some
  eccentricity properties

• Derive a bound on the number of interfering cells

• Choose sources and destinations i.i.d., routes
  are defined to be straight lines

• Prove that each cell contains one node whp using
  the Vapnik-Chervonenkis Theorem -gt routing is
  functional

• Bound the number of lines crossing each cell

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Random Networks Upper Bound
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Achievable Region for Source-Destination Pair

• Setting Network of n nodes over a V-DMC. Xji
  is the transmitted symbol, YjI the received
  symbol. U source-destination pairs (su,du). XF
  some side information.

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Achievable Rate Region

• Rate region translated into

for m M-1, M-2,,0
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Block Markov Encoding Generation of Codebooks

• Codebooks are generated randomly and the
  codebook of each level depends on the indexes of
  greater levels

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Random Partitions

• Use a nested set of random partitions in order
  to use the random binning argument

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Encoding

• The source sends the index wi it wishes to
  transmit

• Backward estimates are obtained by joint
  typicality
• Forward estimates are obtained by set inclusion

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Decoding

• An error is declared if, in either cases, no
  index is found, or more than one index is found.

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Error Probability (1)

• Three sources of error
• Transmissions not Jointly Typical
• Failure to estimate 1-step backward
• Failure to estimate k-steps backward

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Error Probability (2)

• First the probability of error for each block is
  bounded given that all previous blocks are
  decoded correctly

• 1) is bounded by e by joint typicality for b
  large enough

• 2) is bounded by e if the following holds

• 3) is bounded by e if the following holds

• Thus the total probability of error is e
  B(1n(M1))e

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AWGN Channel

• Modified codebook construction due to power
  constraint

• The total power constraints are satisfied and
  the achievable region for AWGN without fading is
  obtained by replacing the expression for the
  mutual information in the derived region by the
  corresponding value

• For the fading case with CSIR, the power
  expression is normalized by the channel gains and
  the expected value of each of the terms now
  define the new rate region
• For the CSIT case, similar normalization is
  required and the expected value is over the CSI
  matrix

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Multiple Source-Destination Pairs and Transport
Capacity

• Scheduling variable Q is introduced

• Each pair has a flowgraph
• Achievable region is the expected value over Q
  of the previous region. Note that the successive
  ordering should be preserved (similarly to a
  degraded channel)

• Practically, the transport capacity becomes the
  rates for each source-destination pair multiplied
  by the distance traversed by the bits.

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Feasibility of T(n) Transport Capacity

• Specific placement of nodes that can achieve a
  transport capacity on the order of n

• This configuration achieves T(n) bits-meter per
  second

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Conclusion and Future Projections

• Derived a rate region for a single
  source-destination pair which they apply to the
  Gaussian Channel and extend to multiple pairs

• This region includes that of many known channels

• Problem of applicability and the quantification
  of gains

• NETWORK CODING!!

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THANK YOU!