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

Outline  Background and Motivation 2  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

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 … TransmittersReceivers … 4

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) SourceDestination Relay 5

Single-Relay Channel 6 Compress-and-Forward (C-F) (Cover & El Gamal 1979): Decode-and-Forward (D-F)

Decode-and-Forward 012 for node 1for node 2 7 (Cover and El Gamal `79) Achievable Rate:

Compress-and-Forward 8 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:

9 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)

10 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

11 (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

Multi-level D-F 12 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

In generalizing C-F to multiple-relay case… 13 Compression-Message successive decoding can be firstly decoded Based on and, can be decoded … (Kramer, Gastpar, Gupta `05) (Wu and Xie `10) Achievable Rate:

In generalizing C-F to multiple-relay case… 14 … Unaddressed issues: Joint decoding ? Joint decoding V.S. Successive decoding ? Any better C-F scheme ? (Kramer, Gastpar, Gupta `05) (Wu and Xie `10)

15 Generalizing C-F to multiple-relay case… Joint decoding ? Joint decoding V.S. Successive decoding ? Any better C-F scheme ?

Joint Decoding in Multiple-Relay Case 16 … Achievable Rate Theorem: (Wu & Xie `10) Compression-Message joint decoding -- No constraint for  more freedom in choosing compressions

17 Generalizing C-F to multiple-relay case Joint decoding ? Joint decoding V.S. Successive decoding ? Any better C-F scheme ?

18 Successive Decoding vs. Joint Decoding Successive DecodingJoint Decoding

19 Successive Decoding vs. Joint Decoding Successive DecodingJoint Decoding ?

20 Successive Decoding vs. Joint Decoding Successive DecodingJoint 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!

21 Generalizing C-F to multiple-relay case Joint decoding ? Successive decoding V.S. joint decoding ? Any better C-F scheme ?

Recent Advances on C-F 22 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

Recent Advances on C-F 23 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 ?

Recent Advances on C-F 24  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 25 NNC

A Unified Relay Framework 26

A unified relay framework is needed! 27 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 28 (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 29 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 30 … Relay node 1: D-F Relay 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: 31 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

A two-relay channel: 32 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)

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

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

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

36 Our unified scheme v.s. Unified scheme in (Kramer, Gastpar, Gupta 2005) D-F C-F

37 Our unified scheme v.s. Unified scheme in (Behboodi, Piantanida 2012)

Conclusion 38  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 39  To further the research in the thesis Cover’s open problem on capacity of relay channel  Converse part of the relay problem

Future Work 40  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 41 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 42  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 43 (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

Thank you! 44

Backup Slides 45

Hybrid Schemes? 46 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

47 Tradeoff and Joint decoding! When to Use Joint Decoding Relay node Multiple-destination case