A Graph-based Framework for
Transmission of Correlated Sources
over Multiuser Channels
Suhan Choi
May 2006
Multiuser Communication Scenarios
Multiuser Communication Scenarios
Many-To-One
Communications

One-To-Many
Communications
Practical Applications



Sensor Networks
Wireless Cellular Systems, Wireless LAN
Broadcasting Systems
Contents 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 Presentation


Many-to-One Communications

Preliminaries

Channel Coding Problem

Source Coding Problem

Motivation & Remarks

Example
Conclusion & Future Research Issues
Outline


Many-to-One Communications

Preliminaries

Channel Coding Problem

Source Coding Problem

Motivation & Remarks

Example
Conclusion & Future Research Issues
Definition of Bipartite Graphs
1
2
A
B
C
Semi-Regular Bipartite Graphs
1
2
3
4
1
2
3
4
5
6
1
2
3
4
1
2
3
4
5
6
Nearly Semi-Regular Bipartite Graphs
1
1
2
2
3
3
4
5
4
6
Strongly Typical Sequences

Non-typical set
Strongly Jointly Typical Sequences
Outline


Many-to-One Communications

Preliminaries

Channel Coding Problem

Source Coding Problem

Motivation & Remarks

Example
Conclusion & Future Research Issues
Problem Formulation:
MAC with Correlated Messages
Channel
Encoder 1
Channel
Encoder 2
MAC
Channel
Decoder
Correlated
1
1
2
2
3
3
1
1
2
2
3
3
Independent
Problem Formulation:
Transmission System
Channel
Encoder 1
Channel
Encoder 2
MAC
Channel
Decoder
Definition of Achievable Rates
Remark on Achievable Rates
& 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
MAC with Correlated Messages
Remark on the Theorem 1
Sketch of the Proof of Theorem 1 (1)
Sketch of the Proof of Theorem 1 (2)
Sketch of the Proof of Theorem 1 (3)
n
Sender 1
Codewords
n
Sender 2
Codewords
Sketch of the Proof of Theorem 1 (4)
Sketch of the Proof of Theorem 1 (5)
n
n
graph
generation
Sender 1
Codewords
Sender 2
Codewords
1
1
2
2
3
3
4
4
Sketch of the Proof of Theorem 1 (6)
Converse Theorem for the Sum-Rate
of the MAC with Correlated Messages
Outline


Many-to-One Communications

Preliminaries

Channel Coding Problem

Source Coding Problem

Motivation & Remarks

Example
Conclusion & Future Research Issues
Source Coding Problem
(Representation of Correlated Sources using
nearly semi-regular bipartite graphs)
Source
Encoder 1
Source
Encoder 2
Source
Decoder
Problem Formulation:
Transmission System
Definition of Achievable Rates
Remark on Achievable Rates
& 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 Region
Sketch of the Proof of Theorem 3 (1)
(Direct Part)
Sketch of the Proof of Theorem 3 (2)
Sketch of the Proof of Theorem 3 (3)
graph
generation
1
1
2
2
3
3
4
4
Sketch of the Proof of Theorem 3 (4)
Sketch of the Proof of Theorem 3 (5)
Outline


Many-to-One Communications

Preliminaries

Channel Coding Problem

Source Coding Problem

Motivation & Remarks

Example
Conclusion & Future Research Issues
Motivation: Why we choose graphs?

Jointly Typicality can be captured by the graph
n
Typicality Graph
Graph
n
1
1
1
1
2
2
2
2
Nearly Semi-regular
Bipartite Graph
Transmission of Correlated Sources over a
Multiple Access Channel (MAC)
Encoder 1
MAC
Decoder
Encoder 2

Source
Encoder 1
Channel
Encoder 1
Source
Encoder 2
Channel
Encoder 2
MAC
Channel
Decoder
Source
Decoder
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
Outline


Many-to-One Communications

Preliminaries

Channel Coding Problem

Source Coding Problem

Motivation & Remarks

Example
Conclusion & Future Research Issues
Gaussian MAC with Jointly Gaussian
Channel Input

Gaussian MAC
Z
X1
X2

Y
A Special Case in the Gaussian MAC
A Special Case in the Gaussian MAC
Gaussian MAC with Correlated
Messages

Independent vs. Correlated Codewords
Outline


Many-to-One Communications

Preliminaries

Channel Coding Problem

Source Coding Problem

Motivation & Remarks

Example
Conclusion & Future Research Issues
Conclusion

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 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!