1. Amplify-and-Forward in Wireless Relay Networks Samar Agnihotri, Sidharth Jaggi, Minghua Chen Institute of Network Coding The Chinese University of Hong Kong

2. In the Beginning … … there was ITW 2011 Analog network coding in the high-SNR regime - Marić, Goldsmith, Médard. WiNC 2010 - Layered networks - High SNR 2

3. ITW 2011 Relay Channel s t Capacity is known only for some special cases Capacity of the general relay channel is not known st 3

4. Achievability Schemes Decode-and-Forward (Cover/El Gamal 1979) Compress-and-Forward (Cover/El Gamal 1979) Amplify-and-Forward (Laneman/Tse 2002) Compute-and-Forward (Nazer/Gastpar 2006) Quantize-map-and-Forward (ADT 2010) ITW 20114

5. Achievability Schemes Decode-and-Forward (Cover/El Gamal 1979) Compress-and-Forward (Cover/El Gamal 1979) Amplify-and-Forward (Laneman/Tse 2002) Compute-and-Forward (Nazer/Gastpar 2006) Quantize-map-and-Forward (ADT 2010) ITW 20115

6. Amplify-and-Forward Relaying ITW 20116

7. Amplify-and-Forward Relaying Cooperative communication Capacity estimation ANC ITW 20117

8. General Wireless Networks Any Size Topology Received SNR s t ITW 20118

9. Network Model ITW 2011 -Bidirectional links -Single antenna -Full-duplex 9 -Fixed channel gains, known throughout

10. Amplify-and-Forward in Wireless Networks st “Intersymbol Interference Channel with Colored Gaussian Noise” ITW 201110

11. Achievable Rate for AF Relay Network ITW 2011 Lemma (Achievable rate for AF relay network): For an AF-relay network with M nodes, the rate achievable with a given amplification vector β is Maximum Achievable rate: W. Hirt, J. L. Massey, Capacity of the discrete-time Gaussian channel with intersymbol interference, Trans. IT, vol IT-34, 1988. 11 Proof technique: circular convolution, DFT

12. “Shout Only If You Make Sense” ITW 2011 s t R2 R1 1 1 1 0.1 Ps = P1 = P2 = 10 Scale-and-Forward Amplify-and-Forward 12

13. Approximating I AF (P s ) ITW 2011 Computing I AF (P s ) is ``hard’’ Relay without Delay Approximation 13 S. Katti, I. Marić, A. Goldsmith, D. Katabi, M. Médard, Joint relaying and network coding in wireless networks, Proc. IEEE ISIT 2007, Nice, France, June 2007. *

14. Lower Bound Computation-I ITW 2011 β i = β, 1≤ i ≤ M No Attenuation Constant Total Relay Power Type-A Network 14

15. Lower Bound Computation-II ITW 2011 β i = β, 1≤ i ≤ M No Attenuation Constant Total Relay Power Type-B Network 15

16. Cut-set Upper Bounds ITW 2011 C ≤ min(C BC, C MAC ) 16 M Relays BC Cut MAC Cut s t

17. Asymptotic Capacity ITW 2011 No Attenuation, Constant Total Relay Power (Type-A Network) (Type-B Network) 17

18. Conclusions A unified framework for AF relay networks ITW 2011 Tighter lower bounds for AF relay networks AF relaying can be capacity achieving for a broader class of networks ANC in a class of general networks 18

19. Current and Future Work Half-duplex networks, multiple-antennas/node, … ITW 2011 Distributed schemes Resource-Performance trade-off – rates beyond AF 19

20. ITW 2011 Thank You! 20 Samar Agnihotri Email: samar@ie.cuhk.edu.hk Web: http://personal.ie.cuhk.edu.hk/~samar/ https://sites.google.com/site/samaragnihotri/