A graphbased framework for transmission of correlated information
A graph-based framework for transmission of correlated information sources over multiuser channels S. Sandeep Pradhan University of Michigan, Ann Arbor, MI
Acknowledgements • Suhan Choi • Kannan Ramchandran • David Neuhoff
Outline • • • Introduction Motivation Problem Formulation Main Result Conclusions
Multiuser Communication
Multiuser Communication Many-To-One Communications One-To-Many Communications • Practical Applications – Sensor Networks – Wireless Cellular Systems, Wireless LAN – Broadcasting Systems
Motivation (1) Near lossless transmission of correlated sources over multiuser channels: Encoder/ Encoders Channel Decoder/ Decoders Source: Discrete Memoryless Vector Channel: Discrete Memoryless (without feedback)
Motivation (2): Example Encoder Channel Decoder Encoder S: temparature readings in Ann Arbor T: temparature readings in Detroit Channel: wireless channel to Lansing.
Motivation (3): Point-to-point Communication Near lossless transmission of a source over a channel: Channel Decoder Encoder Separation Approach: [Shannon 1959] Source Encoder Channel Decoder Source Decoder Reliable transmission Entropy of source < Capacity of channel
Motivation (4) • Separation Approach: source coding+channel 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.
Motivation (5): Example Source Encoder Channel Encoder C H A N N E L Channel Decoder Source Decoder
Motivation (5): Example Indexes Source Encoder Channel Encoder Indexes C H A N N E L Channel Decoder Source Decoder Indexes
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
Motivation (7): Example [Cover, El Gamal, Salehi, 1980] S 0 1/3 1 0 1/3 T 0
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?
Motivation (9) Q: What makes separation work in point-to-point setting? A: Typicality. Non-typical set
Motivation (10): Example • Bernoulli source with Pr(S=1)=0. 2. • Typical sequences are binary sequences with fraction of “heads” nearly equal to 0. 2. • If you toss a biased coin (bias=0. 2) many times, you will most likely see a sequence which is typical.
Motivation (11)
Motivation (12):
Motivation (13) • Not all pairs of S-typical and T-typical sequences are jointly typical. • Because H(S, T)<H(S)+H(T).
Motivation (14): Joint typicality can be captured by a graph n Typicality Graph n Graph Nearly Semi-regular Bipartite Graph
Motivation (15) • Could nearly semi-regular bipartite graphs be used as discrete interface for multiterminal communication?
Graph-based separation Approach ? Source Encoder Channel Encoder C H A N N E L Channel Decoder Source Decoder
Graph-based separation Approach ? Edges of A graph Source Encoder Channel Encoder C H A N N E L Channel Decoder Source Decoder Related Work: [Slepian, Wolf, 73, BSTJ], [Ahlswede, Han, 83, IT]
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.
Definitions: Bipartite Graphs 1 A B 2 C
Definition: Nearly Semi-Regular Bipartite Graphs 1 1 2 2 3 3 4 5 4 6
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.
Examples Two graphs that belong to the same equivalence class Two graphs that belong to different equivalence classes 1 1 2 2 3 3 4 4
Today: • A characterization of the set of nearly semi-regular graphs whose edges can be transmitted over a multiple-access channel.
Multiple-Access Channel Encoder C H A N N E L Channel Decoder • Input Alphabets : • Output Alphabet : • Stationary Discrete Memoryless Channel without feedback • An ordered tuple:
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.
Multiple-Access Channel Capacity [Ahlswede, Liao, 1971]
Problem Formulation: Transmission System
Example 100100000 010100010 10000010100010 100010100 (2, 2) (1, 1) (2, 3) (1, 2) (3, 3) (3, 1)
In other words The messages have the distribution:
Definition of Achievable Rates
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.
Main Result
Remark on 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)
Gaussian Example
Gaussian Example Contd.
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 Source Encoder Channel Encoder Edges of a graph Channel Decoder Source Decoder
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.
Conclusions • A graph-based framework for transmission of correlated sources over multiple-access channels. • A characterization of a set of nearly semiregular bipartite graphs whose edges can be transmitted over a multiple-access channel.
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