Multiterminal Channel Capacity with State Information

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Multi-terminal Channel Capacity with State
Information

• Arak Sutivong, Young-Han Kim, and Mung Chiang
• Information Theory Group
• (Professor Tom Cover)

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Research Goals
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• To better understand the role of side information
  in multi-terminal Information Theory.
• Side information at the transmitter(s)
• Side information at the receiver(s)
• Side information at both
• Side information at the transmitter(s) allows for
  transmit optimization (e.g., water-filling,
  beam-forming, adaptive, etc.)
• Wish to quantify its effect on channel capacity.
• Consider primarily discrete memoryless channels
  (DMC).
• Extend existing results for point-to-point to
  multi-terminals.

Focus of this poster
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Causal vs. Noncausal Side Information
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Causal Side Information
Noncausal Side Information

• Fading coefficients
• Channel interference levels
• State of a Markov channel
• Channel gains

• Memory defect locations
• Host signal (watermarking)
• Crosstalk in DSL
• Known jamming sequence

Examples

• Adaptive rate/power control over Rayleigh fading
  channels
• MIMO beam-forming
• Precoding
• Multi-tone water-filling

• Storage over defective memory
• Digital watermarking
• DSL crosstalk cancellation
• Transmission in the presence of a known third
  party jamming

Scenarios Applications
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Causal Side Information (I)
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• First introduced by Shannon in 1958.
• Channel side information S is available causally.

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Causal Side Information (II)
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• Transform the channel with state into a channel
  without state, but with an exponentially larger
  alphabet size.
• Conceptually, the transmitted symbol at time i is
  a function of W and Si.

Same Capacity
Each symbol in a codeword comes from X.
Each symbol in a codeword is a vector of length
S, with each element coming from X (i.e., T
X S).
Pe ? 0 if R lt I(TY), i.e. R lt I(X(.)Y)
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Noncausal Side Information (I)
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• First studied by Gelfand and Pinsker in 1980.
• Channel side information S is available
  noncausally.

(where U is a finite alphabet auxiliary random
variable)
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Noncausal Side Information (II)
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• Generate 2n(RI(US)) Un codewords.
• Randomly color them into 2nR colors.
• Encoder
• For a given Sn, find a codeword Un such that
• (Un,Sn) is jointly strongly typical,
• Color of Un is Wthe message index.
• Based on Un and Sn, compute Xn and transmit it.
• Decoder
• Find Un such that (Un,Yn) is jointly strongly
  typical, and declare the color of this Un to be
  W.
• Pe ? 0 if R lt I(UY) I(US)

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Gaussian Channels with State Information
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Causal
Noncausal
Costas 1984 Writing on Dirty Paper
State Sn and its power Q affect the channel
capacity. However, it can be shown that the loss
is at most the shaping gain ( 0.25 bits).
State Sn, regardless of its power, can be made
irrelevant by the transmitter as long as the
transmitter has complete knowledge of Sn.
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Degraded Broadcast Channels
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Causal
Noncausal
Si may represent fading coefficients or
interference levels up to time i (feedback from
the receivers).
Sn may represent interference locally
generated, and is known noncausally (e.g.,
leakage of signals from other co-located transmit
antennas).
(Achievable Region)
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Gaussian Broadcast Channels
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Causal
Noncausal
The capacity region is unknown
Extra noise Sn is innocuous as long as the
transmitter knows about it!!
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Multiple Access Channels
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Causal
Noncausal
Si may represent the state of the channel that
is revealed to both transmitters just before
each transmission instant.
Sn may represent a jamming sequence introduced by
a third partyunknown to the receiver, but both
transmitters happen to know about it.
(Achievable Region)
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Gaussian Multiple Access Channels
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Causal
Noncausal
The capacity region is unknown
Once again, extra noise Sn is harmless as long as
both transmitters know about it!!
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Multi-terminal Communications
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(Bound)
Capacity region without state information is
still unknown. Existing upper-bounds are based
on a Max-Flow Min-Cut idea.
Causal
Noncausal
No known nontrivial bounds
(Bound)
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Concluding Remarks
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Causal Side Information
Noncausal Side Information

• Results exist primarily for discrete alphabets
• Utilizes extended alphabets by transforming
  channels with state into channels without state.
• Lacks theories for continuous alphabet channel,
  even for an additive Gaussian channel Y X S
  Z!!
• Open problems Capacity regions for most
  multi-terminal channels (e.g. multiple-access,
  broadcast, relay, interference, two-way, and
  multi-terminals, etc.) are still unknown.

• Many results exist for both discrete and
  continuous alphabets
• Utilizes auxiliary random variables and random
  binning
• Intuition from point-to-point extend well to
  multi-terminal
• Moral Compensate rather than Override the state
• Costas results extend to MAC, BC, Relay, etc.
• Open problems Capacity regions for interference,
  two-way, and multi-terminals in general are still
  unknown.