1. Ergodic Capacity of MIMO Relay Channel Bo Wang and Junshan Zhang Dept. of Electrical Engineering Arizona State University Anders Host-Madsen Dept. of Electrical Engineering University of Hawaii CISS’ 2004

2. Outline Introduction Review of capacity bounds for fixed channel case Bounds on ergodic capacity over Rayleigh fading Discussions on achievability on ergodic capacity:  High SNR case  Scalar channel case Numerical results Conclusion

3. MIMO Relay Channel Model Vector relay channel -- source, destination and/or relay are equipped with multiple antennas

4. Cont’d Signal model , and :, and independent matrices  SNR parameters:  Power constraints:  Noise vectors:

5. Capacity Bounds of Relay Channel Upper bound (max-flow min-cut)  General channel [Cover & El Gamal 79]  Degraded channel: achieve upper bound Lower bound – achievable rates

6. MIMO Relay Channel Capacity Challenges  Non-degraded  Vector channel: maximization over matrices We study capacity bounds [Wang-Zhang03]  Fixed Channel case  Rayleigh fading channel case

7. Upper Bound: Fixed Channel Case Theorem 1: An upper bound on capacity of MIMO relay channel is given by where and

8. Capacity Bounds: Rayleigh Fading Case Upper bound on ergodic capacity over Rayleigh fading (receiver CSI only) Theorem 2: a). An upper bound on ergodic capacity is given by

9. Cont’d b). A lower bound on ergodic capacity is given by

10. Some Intuition Upper bound and lower bound can “meet” under certain conditions Ergodic capacity can be characterized exactly; previously, this was shown only for degraded relay channel (fixed channel case) Independent codebooks at source and relay Channel uncertainty (randomness) at transmitters make and independent Relaying improves capacity by achieving MAC gain and BC gain

11. Some Intuition ( cont’d) Question: sufficient conditions for achieving ergodic capacity?  Recall upper bound and lower bound: common termupper bound lower bound  Ergodic Capacity can be  Observation: If and, upper bound meets lower bound at

12. Outline of Proof: Upper Bound on Ergodic Capacity Apply Gaussian codebooks,

13. Cont’d Choosing maximizes : The same distributions maximize Thanks Dr. Kramer for his comments on this proof.

14. Outline of Proof: Lower Bound on Ergodic Capacity Without relay: single-user MIMO channel With relay, following rate can be achieved  Consider fading:  Independent input signals maximize above rates

15. Conditions on Capacity Achievability Numbers of antennas = 2 in all cases Case I: Case II: Case III:

16. Case I:

17. Case II: and

18. Case III : and, upper bound meets lower bound

19. Discussions on Capacity Achievability Assume numbers of antennas Upper bound is given by, iff Remains to find conditions for An upper bound on Study two cases:  High SNR Regime  Scalar Channel case

20. High SNR Regime Approximate by Approximate upper bound on by

21. High SNR Regime (cont’d) Example:

23.  Sufficient conditions for achieving capacity can be viewed as a generalization of “degradedness” to fading channels

24. Scalar Channel Compute and Compare them to find sufficient conditions

25. Conclusion and Future Work Study upper bounds and lower bounds on capacity of MIMO relay channel over Rayleigh fading (full version at www.eas.asu.edu/~junshan/) For equal numbers of antenna cases:  Find sufficient conditions for achieving ergodic capacity  Sufficient conditions can be viewed as a generalization of “degradedness” to fading channels Future work on correlated fading channel and partial CSI cases, and study sufficient conditions for achieving ergodic capacity Other results: ergodic capacity and power allocation for relay channel over Rayleigh fading, for both full-duplex and time-division cases. (Host-Madsen & Zhang03)

26. Questions? & Thank You!

27. Single-user MIMO Channel Capacity Channel Model: Capacity of fixed channel: Ergodic capacity: time varying, receiver CSI only [Telatar 99, Foschini & Gans 98]