Cut Set Bounds in Network Information Theory

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Cut Set Bounds in NetworkInformation Theory

• Presentation in ECE 1528 Multiuser Information
  Theory
• April 12th 2007

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Overview

• Weve generally focused on single source and/or
  destination
• single source/single destination (ECE 1502)
• multiple sources/single receiver (MAC)
• single source/multiple receivers (BC)
• Here m source-destination pairs
• This presentation
• rate bounds
• specializations to MAC, relay and two-way
  channels
• stolen from Cover and Thomas

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General Multi-Terminal Network
What is the total flow across the cut?
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Multiple Sources

• m nodes
• Node i sends information to node j _at_ rate R(ij)
• message W(ij) independent and uniformly
  distributed over set 1, 2, 2nR(ij)
• Transmitted symbol _at_ node i x(i)
• Received symbol _at_ node j y(j)
• System p(y(1), y(2), , y(m) x(1) x(2) x(m))
• Encoder
• xk(i) Xk(i)(W(i1), W(i2), , W(im),
  Y1(i), Y2(i),,Y(k-1)(i))
• Decoder
• Wk(ij) (Y1(j), Y2(j),,Y(n)(j), W(j1),
  W(j2), , W(jm))

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Multiple Sources

• Divide the set of notes into two subsets S and
  its complement Sc.
• Rate R(ij) is achievable if Pe(n) (ij) ? 0 as n ?
  8
• Theorem If information rates R(ij) are
  achievable, then ? a distribution p(x(1) x(2)
  x(m)) such that

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Proof (Key Steps)
nen (Fano)
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Proof (Key Steps)
Conditioning reduces entropy
Depends on Y1Sc. Y1Sc and WTc only
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MAC
Cut 1

• Cut 1 S 1, Sc 2
• R1 I(X1 Y X2)
• Cut 2 S 1, Sc 2
• R2 I(X2 Y X1)
• Cut 3 S 1, 2 Sc Ø
• R1 R2 I(X1., X2 Y)

Cut 3
X1
Y
X2
Cut 2
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Relay Channel
Y1 X1
Cut 2
Cut 1
Y
X

• Cut 1 C I(X Y1, Y X1)
• Cut 2 C lt I(X, X1 Y )

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Two-Way Channel?
X1
X2
Y2
Y1
Cut 1
Cut 2

• Cut 1 R1 I(X1 Y1 X2)
• Cut 2 R2 lt I(X2, Y2 X1 )

Inner and outer bounds from independent or
cooperative sources
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Finally

• Important this bound may not be achievable
• Some differences from single user channels
• source-channel separation theorem is not valid
• Slepian-Wolf region need not intersect with the
  capacity region
• feedback can improve the capacity even for
  memory-less channels
• the feedback channel is a channel of its own