Coding Schemes for Multiple-Relay Channels 1 Ph.D. Defense Department of Electrical and Computer Engineering University of Waterloo Xiugang Wu December 4, 2013
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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
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
•
•
•
•
• 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 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)
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
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