A graph-based framework for transmission of correlated information sources over multiuser channels

1
A graph-based framework for transmission of
correlated information sources over multiuser
channels

• S. Sandeep Pradhan
• University of Michigan,
• Ann Arbor, MI

2
Acknowledgements

• Suhan Choi
• Kannan Ramchandran
• David Neuhoff

3
Outline

• Introduction
• Motivation
• Problem Formulation
• Main Result
• Conclusions

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Multiuser Communication
5
Multiuser Communication
Many-To-One Communications
One-To-Many Communications

• Practical Applications
• Sensor Networks
• Wireless Cellular Systems, Wireless LAN
• Broadcasting Systems

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Motivation (1)
Near lossless transmission of correlated sources
over multiuser channels
Channel
Decoder/ Decoders
Encoder/ Encoders
Source Discrete Memoryless Vector Channel
Discrete Memoryless (without feedback)
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Motivation (2) Example
Channel
Encoder
Decoder
Encoder
S temparature readings in Ann Arbor
T temparature readings in
Detroit Channel wireless channel to Lansing.
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Motivation (3)Point-to-point Communication
Near lossless transmission of a source over a
channel
Channel
Decoder
Encoder
Separation Approach Shannon 1959
Channel
Source Encoder
Channel Encoder
Channel Decoder
Source Decoder
Reliable transmission ? Entropy of source lt
Capacity of channel
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Motivation (4)

• Separation Approach source codingchannel coding
• Source Coding (compression) Removal of
  redundancy
• Example Distributed source coding.
  
• Channel Coding Structured reintroduction of
  redundancy
• Example CDMA (uplink) with multiuser detection.
• This approach is modular.
• Source coding and channel coding optimization can
  be done separately.
• The Alternative Joint source-channel coding.

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Motivation (5) Example
C H A N N E L
Source Encoder
Channel Encoder
Source Decoder
Channel Decoder
Channel Encoder
Source Encoder
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Motivation (5) Example
C H A N N E L
Source Encoder
Channel Encoder
Source Decoder
Channel Decoder
Channel Encoder
Source Encoder
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Motivation (6)

• Indexes (bits) at multiple channel encoders are
  independent.
• Distributed information is represented as
  multiple independent bit streams.
• Unfortunately this scheme is not optimal

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Motivation (7) Example Cover, El Gamal, Salehi,
1980
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Motivation (8)

• Essence conventional separation-approach is not
  optimal for multiuser communication. This
  approach is modular but not optimal.
• Shannon showed that separation-approach is
  optimal for point-to-point communication.
• We have built the telephone-network and the
  Internet using this principle.
• Why does it work in point-to-point case and not
  in multiuser case?
• In other words how can we inject modularity in
  multiuser communication without losing optimality?

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Motivation (9)
Q What makes separation work in point-to-point
setting? A Typicality.
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Motivation (10) Example

• Bernoulli source with Pr(S1)0.2.
• Typical sequences are binary sequences with
  fraction of heads nearly equal to 0.2.
• If you toss a biased coin (bias0.2) many many
  times, you will most likely see a sequence which
  is typical.

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Motivation (11)
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Motivation (12)
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Motivation (13)

• Not all pairs of S-typical and T-typical
• sequences are jointly typical.
• Because H(S,T)ltH(S)H(T).

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Motivation (14)

• Joint typicality can be captured by a graph

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Motivation (15)

• Could nearly semi-regular bipartite graphs be
  used as discrete interface for multiterminal
  communication?

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Graph-based separation Approach ?
C H A N N E L
Source Encoder
Channel Encoder
Source Decoder
Channel Decoder
Channel Encoder
Source Encoder
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Graph-based separation Approach ?
Edges of A graph
C H A N N E L
Source Encoder
Channel Encoder
Source Decoder
Channel Decoder
Channel Encoder
Source Encoder
Related Work Slepian, Wolf, 73, BSTJ,
Ahlswede, Han, 83, IT
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Big Picture

• Extended source coding Structured way to retain
  redundancy in the source representation.
• Extended channel coding Structured way to
  reintroduce redundancy into this representation.

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Definitions Bipartite Graphs
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Definition Nearly Semi-Regular Bipartite Graphs
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Equivalence Classes of Graphs

• Consider
• can be partitioned into equivalence
  classes
• Two graphs belong to the same classes if one can
• be obtained from the other by relabeling the
  vertices.

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Examples
Two graphs that belong to the same equivalence
class
Two graphs that belong to different equivalence
classes
1 2 3 4
1 2 3 4
1 2 3 4
1 2 3 4
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Today

• A characterization of the set of nearly
  semi-regular graphs
• whose edges can be transmitted over a
  multiple-access
• channel.

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Multiple-Access Channel
C H A N N E L
Channel Encoder
Channel Decoder
Channel Encoder

• Input Alphabets
• Output Alphabet
• Stationary Discrete Memoryless Channel without
  feedback
• An ordered tuple

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Multiple-Access Channel

• This channel was introduced in 1971 by Ahlswede
  Liao.
• The capacity region is known.
• Literature on this is too exhaustive to list
  here.

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Multiple-Access Channel Capacity Ahlswede,
Liao, 1971
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Problem Formulation Transmission System
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Example
100100000 010100010 010100010
010101010 100000101 100010100
(2,2)
(1,1)
(2,3)
(3,3)
(1,2)
(3,1)
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In other words

• The messages have the distribution

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Definition of Achievable Rates
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Remark on Achievable Rates

• Find a sequence of nearly semi-regular graphs
• The number of vertices the degrees are
  increasing exponentially with given rates
• Edges from these graphs are reliably transmitted
• Rates are achievable
• Definition Rate region
• The set of all achievable tuple of rates
• Goal Find the rate region
• Note the distribution of the message pair is
  changing with blocklength n.

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Main Result
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Remark on Theorem 1
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Sketch of the Proof of Theorem 1 (1)
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Sketch of the Proof of Theorem 1 (2)
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Sketch of the Proof of Theorem 1 (3)
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Gaussian Example
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Gaussian Example Contd.
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Source Coding Module

• Similarly a problem formulation for representing
  a pair of correlated sources into nearly
  semi-regular bipartite graphs can be done.
• One can then obtain a characterization of a set
  of nearly semi-regular bipartite graphs
  which can reliably represent the source pair.

Edges of a graph
Edges of a graph
Channel
Source Encoder
Channel Encoder
Channel Decoder
Source Decoder
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Transmission of sources over channels

• Given a source-pair and a multiple-access
  channel.
• What if
• Q Does it mean that we can reliably transmit the
  pair over the multiple-access channel?
• A Not in general.
• Because the graph for the source and that for the
  channel may belong to different equivalence
  classes.

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Conclusions

• A graph-based framework for transmission of
  correlated sources over multiple-access channels.
• A characterization of a set of nearly
  semi-regular bipartite graphs whose edges can be
  transmitted over a multiple-access channel.