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Coding Schemes for Multiple-Relay Channels

Ph.D. Defense

Department of Electrical and Computer Engineering

University of Waterloo

Xiugang Wu

December 4, 2013

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Outline

Background and Motivation

Conclusion and Future work

Main Results:

• Generalizing C-F from single- to multiple-relay case

• Unifying D-F and C-F

• Decode-and-Forward (D-F) and Compress-and-Forward (C-F)

Shannon’s Information Theory

• Discrete Memoryless Channel (DMC):

Channel Coding Theorem

• Channel Capacity:

A Mathematical Theory of Communication, Shannon 1948

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Network Information Theory

• Fundamental questions:

-- The capacity region of the network ?

-- The coding schemes to achieve it ?

• New elements: cooperation, competition, feedback…

A complete theory is yet to be developed!

Network

…

Transmitters Receivers

…

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State of The Art of Network Information Theory

Some successes:

However, little else is known…

Multiple access channel Degraded broadcast channel

Single-relay channel Multiple-relay channel

Source Destination

Relays

(capacity open after decades’ effort)

(Alshwede `71; Liao `72)

(Cover `72; Bergmans `73; Gallager `74)

Source Destination

Relay

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Single-Relay Channel

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• Compress-and-Forward (C-F)

(Cover & El Gamal 1979):

• Decode-and-Forward (D-F)

0

1

2 0

1

2

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Decode-and-Forward

0 1 2

for node 1 for node 2

(Cover and El Gamal `79)

Achievable Rate:

Compress-and-Forward

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• Compression-Message successive decoding

-- Step 2: then decode based on and

No need to decode!

can be firstly decoded

Based on and , can be decoded

(Cover and El Gamal `79)

(Compression)

(Message)

-- Step 1: decode

• Achievable Rate:

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• Compression-Message joint decoding

-- Jointly decode and w/o completely determining

Compress-and-Forward

• Achievable Rate:

No need to decode!

(Compression)

Theorem:For single-relay channels, two schemes achieve the same rate.

-- No constraint for more freedom in choosing compression

-- Q): Will this freedom improve the achievable rate ?

(El Gamal, Kim`10)

(Message)

(Xie `09)(El Gamal, Kim `10)(Wu, Xie `10)

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Extension to Multiple Relays

(Aref `80) (Gupta and Kumar `03)(Reznik, Kulkarni, Verdu `04)(Xie and Kumar `04, `05)(Kramer, Gastpar, Gupta `05)(Razaghi, Yu `09)

Relay nodes set

Generalization of D-F

(Kramer, Gastpar, Gupta `05)(Wu, Xie `10)

Generalization of C-F

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(Aref `80) (Gupta and Kumar `03)(Reznik, Kulkarni, Verdu `04)(Xie and Kumar `04, `05)(Kramer, Gastpar, Gupta `05)(Razaghi, Yu `09)

Relay nodes set

Resolved

Some fundamental issues unaddressed!

Extension to Multiple Relays

Generalization of D-F

(Kramer, Gastpar, Gupta `05)(Wu, Xie `10)

Generalization of C-F

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Multi-level D-F

Upstream Downstream

(Xie and Kumar `04, `05) (Kramer, Gastpar, Gupta `05)(Razaghi, Yu `09)

…

• Achievable Rate:

• Upstream nodes decode before downstream nodes

• Information passed along some route

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In generalizing C-F to multiple-relay case…

Compression-Message successive decoding

can be firstly decoded

Based on and , can be decoded

…

(Kramer, Gastpar, Gupta `05)(Wu and Xie `10)

Achievable Rate:

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In generalizing C-F to multiple-relay case…

…Unaddressed issues:

• Joint decoding ?

• Joint decoding V.S. Successive decoding ?

• Any better C-F scheme ?

(Kramer, Gastpar, Gupta `05)(Wu and Xie `10)

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Generalizing C-F to multiple-relay case…

• Joint decoding ?

• Joint decoding V.S. Successive decoding ?

• Any better C-F scheme ?

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Joint Decoding in Multiple-Relay Case

…Achievable Rate Theorem: (Wu & Xie `10)

Compression-Message joint decoding

-- No constraint for more freedom in choosing compressions

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Generalizing C-F to multiple-relay case

• Joint decoding ?

• Joint decoding V.S. Successive decoding ?

• Any better C-F scheme ?

•

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Successive Decoding vs. Joint Decoding

Successive Decoding Joint Decoding

•

•

•

•

•

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Successive Decoding vs. Joint Decoding

Successive Decoding Joint Decoding

•

•

•

• ?

•

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Successive Decoding vs. Joint Decoding

Successive Decoding Joint Decoding

•

•

•

•

• Optimal rate with joint decoding can be achieved only when

Theorem:

• Two schemes achieve the same rate even in multiple-relay case

(Wu & Xie `10)

-- Optimal compressions should support successive decoding!

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Generalizing C-F to multiple-relay case

• Joint decoding ?

• Successive decoding V.S. joint decoding ?

• Any better C-F scheme ?

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Recent Advances on C-F

• Repetitive encoding/all blocks united decoding

Noisy network coding (NNC) (Lim, Kim, El Gamal, Chung `11)

V.S. the classical: Cumulative encoding/block-by-block forward decoding

(Cover and El Gamal `79)

Repetitive encoding/all blocks united decoding

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Recent Advances on C-F

• Repetitive encoding/all blocks united decoding

Noisy network coding (NNC) (Lim, Kim, El Gamal, Chung `11)

• Compression-Message joint decoding

• Achievable Rate:

-- same as classical C-F with forward decoding in single-relay case

-- in general better than classical C-F in multiple-relay case

Not necessary! (Wu, Xie `11)

-- improvement due to repetitive encoding and joint decoding ?

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Recent Advances on C-F

Cumulative encoding/block-by-block backward decoding (Wu, Xie `11)

• Two decoding modes: Successive decoding; Joint decoding

• Both modes achieve the same rate as Noisy Network Coding

Theorem:

• Successive decoding achieves same rate as joint decoding

(Wu & Xie `11)

-- Reveals essential reason for improvement: not repetitive encoding, not joint decoding, but delayed decoding until all blocks finished

-- Backward decoding + successive decoding is the simplest choice in achieving the highest C-F rate

Implications

Single Relay Multiple Relays

Existing Work

D-F Multi-level D-F

Our Work

C-F

Successive Decoding

Joint Decoding

Summary

Forward Decoding

Successive Decoding

Joint Decoding

Backward Decoding

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NNC

A Unified Relay Framework

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A unified relay framework is needed!

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So far,

However…

D-F Relays C-F Relays

all the relays perform the same relay strategy, either D-F or C-F

to obtain higher rates, freedom of choosing D-F or C-F may be necessary

Source Destination

Challenge: Can we fully incorporate the best known D-F and C-F ?

Existing Works on Unifying D-F and C-F

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(Kramer, Gastpar, Gupta, `05), (Behboodi, Piantanida, `12)

• In (Kramer, Gastpar, Gupta, `05)

-- the recent advances on C-F not reflected

• In (Behboodi, Piantanida, `12)

-- multi-level D-F not utilized

-- D-F nodes didn’t utilize help of C-F nodes

Major Difficulty

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• Upstream D-F node has to decode before downstream node

A seeming contradiction:

• Decoding at D-F nodes has to wait until all blocks finished

Our solution:Nested blocks + Backward decoding

(Kramer, Gastpar, Gupta, `05)(Xie, Kumar, `07)

Nested blocks + Backward decoding

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… Relay node 1: D-FRelay nodes 2 - : C-F

Decoding at D-F node 1:

A total of blocks will be used (instead of blocks)

Decoding at node :

Achievable Rate Theorem:

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where is the largest subset of s.t.

• Combines both best known D-F and C-F rates

Our Achievable Rate

(Wu, Xie `12)

• Includes them as special cases

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A two-relay channel:

An Example of Gaussian Networks

• Pass-loss exponent ; Uniform power constraint

• Compare:

-- Our unified scheme vs. D-F or C-F alone

-- Our unified scheme vs. Unified scheme in (Kramer, Gastpar, Gupta, 2005)

-- Our unified scheme vs. Unified scheme in (Behboodi, Piantanida, 2012)

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Our unified scheme v.s. D-F or C-F alone

D-F C-F

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Our unified scheme v.s. D-F or C-F alone

D-F C-F

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Our unified scheme v.s. D-F or C-F alone

D-F C-F

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Our unified scheme v.s.

Unified scheme in (Kramer, Gastpar, Gupta 2005)

D-F C-F

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Our unified scheme v.s.

Unified scheme in (Behboodi, Piantanida 2012)

Conclusion

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On the optimal compressions in C-F schemes

• Successive decoding achieves same rate as joint decoding

• Optimal compressions should support successive decoding

A unified relay framework

A new C-F scheme with backward decoding

• Simplest choice in achieving the highest C-F rate

• Reveals the essential reason for the improvement

• Fully incorporate the best D-F and C-F schemes

• Better than existing unified schemes, and D-F or C-F alone

Future Work

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To further the research in the thesis

• Cover’s open problem on capacity of relay channel

Converse part of the relay problem

Future Work

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To further the research in the thesis

• Cover’s open problem on capacity of relay channel

Converse part of the relay problem

Future Work: Part I

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To further the research in the thesis:

Extend the unified relay framework to multiple-source case

On achieving capacity of relay networks to within constant gap

• Current best gap: (based on NNC)

• Can our unified scheme achieve better or universal gap that is independent of node number ?

-- Limitation: gap grows with # of nodes

-- Reason: compression based scheme noise accumulated

-- independent of channel gain, SNR, network topology

Future Work

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To further the research in the thesis

• Cover’s open problem on capacity of relay channel

Converse part of the relay problem

Open Problem on Capacity of Relay Channel

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(Cover, Open problems in communication and computation, 1987)

Q) : The minimum needed s.t. ?

• Non-trivial even in binary symmetric case…

• By C-F with Slepian-Wolf coding,

• Is C-F optimal such that ?

BSC

BSC

Thank you!

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Backup Slides

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Hybrid Schemes?

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• Involves superposition coding which induces auxiliary RV

Focus on ``pure’’ D-F or C-F strategies

• Partially decodes and compresses the rest, e.g., Thm 7 in (Cover, El Gamal `79)

• Complicated expression and evaluation of achievable rates, especially in multiple-relay case

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Tradeoff and Joint decoding!

When to Use Joint Decoding

Relay node

Multiple-destination case