SlepianWolf Coding over Broadcast Channels

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Slepian-Wolf Coding overBroadcast Channels

• Ertem Tuncel

UC San Diego, December 1, 2006
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Outline

• Problem definition and motivation
• Separate source-channel coding
• Optimal combination of source and channel codes.
• Joint source-channel coding
• Complete region of achievable rates.
• Operational separation.
• Comparison between separate and joint schemes
• No informational separation in general.

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Problem setting and motivation
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Problem setting and motivation
Local observation
Xt
Local observations
Y1t
Y2t
Y3t
Y4t
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Formal description
Discrete memoryless source and channel
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Questions

• What combination of coding schemes is optimal?

• How does the optimal joint scheme work?
• Is joint coding superior to combination of
  separate codes?

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PART I SEPARATESOURCE-CHANNELCODING
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The most general separate scheme
For K 3
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A suboptimal separate coding method

• Worst-case Slepian-Wolf coding over compound
  channels. Transmit only W(12K) .

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Trivial outer bound for source codes

• Is this outer bound exactly the achievable
  region?
• If so, what is the mechanism to achieve it?

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Slepian-Wolf coding revisited

• It is well-known that to achieve this trivial
  lower bound, we use binning.

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Multiple binning
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Error analysis

• Define X1 X2 XL X .

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Optimal source coding rates
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Why this is good news

• In other words To characterize the minimum
  achievable k in separate coding, it suffices to
  find the set of achievable total rates the
  broadcast channel can deliver to each receiver.

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Why this is good news
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Why this is good news

• Therefore, assuming WLOG that
• this implies that k is achievable in separate
  coding with K 2 if and only if

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Comparison
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Infinite gains possible
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Now the bad news

• What about K gt 2 ?

• Would the generalization of the degraded message
  sets scenario cover all the achievable total
  rates?

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Now the bad news

• The framework of degraded message sets is not
  sufficient for the analysis of minimum achievable
  k.

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PART II JOINTSOURCE-CHANNELCODING
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Joint source-channel coding

• Easy to prove that this is a convex region.

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Proof sketch for the converse
Fanos inequality
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Proof sketch for the direct result
Decoding Find i such that
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Probability of decoding error
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Is this separation?

• Not in the classical sense
• There are no stand-alone source or channel
  codes here.
• However, no interplay between (X,Y1,Y2,,YK) and
  (U,V1,V2,,VK) unlike in other studied joint
  source-channel coding scenarios.
• Operational separation!

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Virtual binning
Channel codebook
Typical X n
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Virtual binning
Channel codebook
Typical X n
For better channels, worse side information is OK
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Suboptimality of separate coding

• Consider a binary symmetric broadcast channel

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Suboptimality of separate coding
R2
R
R1
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Joint coding gains
R2
R
We observed gains of upto a ? 2.
R1
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Summary and conclusions

• Separate source-channel coding
• We have a single-letter characterization for K
  2.
• For K gt 2, it suffices to characterize the total
  channel coding rates deliverable to each
  receiver.
• Joint source-channel coding
• Single-letter characterization for all K.
• Effective capacity region R .
• Operational separation via virtual binning.
• No informational separation in general.

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Open questions

• Is there a single-letter characterization for the
  total channel coding rates?
• Any hint for how practical SW-coding over
  broadcast channels should work?
• Any other multi-terminal problem for which our
  simple joint coding technique works?
• Any implications for lossy coding (w or w/o side
  information)?