A Graph-based Framework for A Graph-based Framework for Transmission of Correlated SourcesTransmission of Correlated Sources

over Multiuser Channelsover Multiuser Channels

Suhan ChoiMay 2006

Multiuser Communication ScenariosMultiuser Communication Scenarios

Multiuser Communication ScenariosMultiuser Communication Scenarios

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

Many-To-OneCommunications

One-To-ManyCommunications

Contents of DissertationContents of Dissertation

Many-to-One Communications (Multiple Access Channels)

Channel Coding Problem

Source Coding Problem

Examples and Interpretations

One-to-Many Communications (Broadcast Channels)

Channel Coding Problem

Source Coding Problem

Interpretation

Conclusion & Future Research Issues

Outline of the PresentationOutline of the Presentation

Many-to-One Communications Preliminaries

Channel Coding Problem

Source Coding Problem

Motivation & Remarks

Example

Conclusion & Future Research Issues

OutlineOutline

Many-to-One Communications Preliminaries

Channel Coding Problem

Source Coding Problem

Motivation & Remarks

Example

Conclusion & Future Research Issues

Definition of Bipartite GraphsDefinition of Bipartite Graphs

1

2

A

B

C

Semi-Regular Bipartite GraphsSemi-Regular Bipartite Graphs

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2

3

4

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2

3

5

4

6

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3

4

1

2

3

5

4

6

Nearly Semi-Regular Bipartite GraphsNearly Semi-Regular Bipartite Graphs

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4

1

2

3

5

4

6

Strongly Typical Sequences Strongly Typical Sequences

Non-typical set

Strongly Jointly Typical Sequences Strongly Jointly Typical Sequences

OutlineOutline

Many-to-One Communications Preliminaries

Channel Coding Problem

Source Coding Problem

Motivation & Remarks

Example

Conclusion & Future Research Issues

Problem Formulation:Problem Formulation:MAC with Correlated MessagesMAC with Correlated Messages

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2

3

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3

1

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3

1

2

3

Independent

Correlated

MACChannel

Encoder 1

ChannelEncoder 2

ChannelDecoder

Problem Formulation: Problem Formulation: Transmission SystemTransmission System

MACChannel

Encoder 1

ChannelEncoder 2

ChannelDecoder

Definition of Achievable RatesDefinition of Achievable Rates

Remark on Achievable Rates Remark on Achievable Rates & Capacity Region& Capacity Region

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: Capacity region, The set of all achievable tuple of rates

Goal: Find the capacity region

An Achievable Rate Region for the An Achievable Rate Region for the MAC with Correlated MessagesMAC with Correlated Messages

Remark on the Theorem 1Remark on the Theorem 1

Sketch of the Proof of Theorem 1 (1)Sketch of the Proof of Theorem 1 (1)

Sketch of the Proof of Theorem 1 (2)Sketch of the Proof of Theorem 1 (2)

Sketch of the Proof of Theorem 1 (3)Sketch of the Proof of Theorem 1 (3)

Sender 1Codewords

n n

Sender 2Codewords

Sketch of the Proof of Theorem 1 (4)Sketch of the Proof of Theorem 1 (4)

Sketch of the Proof of Theorem 1 (5)Sketch of the Proof of Theorem 1 (5)

n

Sender 1Codewords

n

Sender 2Codewords

graphgeneration

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2

3

4

1

2

3

4

Sketch of the Proof of Theorem 1 (6)Sketch of the Proof of Theorem 1 (6)

Converse Theorem for the Sum-Rate Converse Theorem for the Sum-Rate of the MAC with Correlated Messagesof the MAC with Correlated Messages

OutlineOutline

Many-to-One Communications Preliminaries

Channel Coding Problem

Source Coding Problem

Motivation & Remarks

Example

Conclusion & Future Research Issues

Source Coding ProblemSource Coding Problem(Representation of Correlated Sources using (Representation of Correlated Sources using nearly semi-regular bipartite graphs)nearly semi-regular bipartite graphs)

SourceEncoder 1

SourceEncoder 2

SourceDecoder

Problem Formulation: Problem Formulation: Transmission SystemTransmission System

Definition of Achievable RatesDefinition of Achievable Rates

Remark on Achievable Rates Remark on Achievable Rates & Our Goal& Our Goal

Find a sequence of nearly semi-regular graphs The number of vertices & the degrees are increasing

exponentially with given rates

Given sources are reliably represented by these graphs

→ Rates are achievable

The achievable rate region: The set of all achievable tuple of rates

Goal: Find the achievable region

The Achievable Rate RegionThe Achievable Rate Region

Sketch of the Proof of Theorem 3 (1) Sketch of the Proof of Theorem 3 (1) (Direct Part)(Direct Part)

Sketch of the Proof of Theorem 3 (2)Sketch of the Proof of Theorem 3 (2)

Sketch of the Proof of Theorem 3 (3)Sketch of the Proof of Theorem 3 (3)

graphgeneration

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4

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2

3

4

Sketch of the Proof of Theorem 3 (4)Sketch of the Proof of Theorem 3 (4)

Sketch of the Proof of Theorem 3 (5)Sketch of the Proof of Theorem 3 (5)

OutlineOutline

Many-to-One Communications Preliminaries

Channel Coding Problem

Source Coding Problem

Motivation & Remarks

Example

Conclusion & Future Research Issues

Motivation: Why we choose graphs?Motivation: Why we choose graphs?

Jointly Typicality can be captured by the graph

n

2

1

1

2

n

1

22

1

Nearly Semi-regularBipartite Graph

GraphTypicality Graph

Transmission of Correlated Sources over a Transmission of Correlated Sources over a Multiple Access Channel (MAC)Multiple Access Channel (MAC)

MACEncoder 1

Encoder 2

Decoder

MACChannel

Encoder 1

ChannelEncoder 2

ChannelDecoder

SourceEncoder 1

SourceEncoder 2

SourceDecoder

A Graph-Based Framework Modular approach in multiuser channels Fundamental Concept: Jointly typicality Encoding processes

Source coding: map correlated sources into edges of graphs Channel coding: send edges of these graphs reliably

OutlineOutline

Many-to-One Communications Preliminaries

Channel Coding Problem

Source Coding Problem

Motivation & Remarks

Example

Conclusion & Future Research Issues

Gaussian MAC with Jointly Gaussian Gaussian MAC with Jointly Gaussian Channel InputChannel Input

Gaussian MAC

Z

1X

2XY

A Special Case in the Gaussian MACA Special Case in the Gaussian MAC

A Special Case in the Gaussian MACA Special Case in the Gaussian MAC

Gaussian MAC with Correlated Gaussian MAC with Correlated MessagesMessages

Independent vs. Correlated Codewords

OutlineOutline

Many-to-One Communications Preliminaries

Channel Coding Problem

Source Coding Problem

Motivation & Remarks

Example

Conclusion & Future Research Issues

ConclusionConclusion

Many-to-One/One-to-Many Communication Problems Channel coding problem

→ Transmission of correlated messages (edges of graphs) over the

channel

Source Coding Problem

→ Representation of Correlated Sources into graphs

Graph-based framework for transmission of correlated

sources over multiuser channels Modular architecture

Interface between source and channel coding

→ Nearly semi-regular graphs

Future Research IssuesFuture Research Issues

More detailed characterization of the structure of

bipartite graphs

Number of different equivalence class with particular

parameters

Relation between probability distributions and

equivalence classes

Construction of practical codes for MAC and BC

with correlated sources

Thank you!Thank you!