1. Transmit Diversity with Channel Feedback Krishna K. Mukkavilli, Ashutosh Sabharwal, Michael Orchard and Behnaam Aazhang Department of Electrical and Computer Engineering Rice University, Houston, Texas, USA

2. Introduction  Demand for high data rates in wireless communications  High spectral efficiency schemes  Fading phenomenon  Diversity schemes  Multiple antenna  Overcomes fading  Capacity grows linearly with min(t,r) [Telatar95]  Achieved via space-time codes [Tarokh98]

3. Background  Slow (block) fading channel  Feedback  Increase spectral efficiency  Decrease frame error rate  Spatial water filling [Telatar95]  Maximizes mutual information  Requires substantial channel information  No guarantees for practical low dimensional codebooks

4. Our Focus  Practical management of unknown channel condition  Limited feedback  Practical codebook design issues  Our approach  Role of feedback in codebook design  Phase information  Quantized phase information

5. Codebook Design with Feedback  Objective: Error minimizing codebooks  Unknown channel condition with reduced dimension feedback  Issues: what feedback? what codebook?  Dominant spatial direction is the key parameter  Chernoff bound analysis  Comparison with mutual information solution

6. System and Channel Model Block Fading n rx antennas X Y L h 1,i h m,i  Channel realization known at the receiver  Error free feedback channel m tx antennas

7. Chernoff Bound  Minimize pairwise error probability given H Use X = W d x where W d is the eigenvector corresponding to the maximum singular value of H, max  Observations:  Reduce dimension feedback required  Gaussian channel codebooks

8. Comparison with Mutual Information Solution Achievable rates with two transmit and two receive antennas 2468101214161820 2 3 4 5 6 7 8 9 10 11 12 SNR bits/sec/Hz Beamforming Spatial water-filling

9. Phase Feedback  Dominant eigenvector solution  Requires phase and amplitude information  (2m – 2) real numbers  Proposed: using only phase information  Chernoff bound  Simulations

10. Beamforming with Phase Information  Problem : find such that minimizes error probability where x is the information vector  Solution: Choose whose components satisfy

11. Beamforming with Phase Information X X X x Y H  Beamforming vector

12. Special Cases  m =2,n = 1  m=2, n = n  m=m, n = 1  Solved for the case of m = 3 and any n

13. Features of Beamforming with Phase Feedback  Less feedback information (m-1 real numbers)  No need for singular value decomposition  Performance loss compared to dominant eigenvector solution, for n=1  Loss is 0.49 dB for 2 tx and 1 rx antenna

14. Quantized Phase Feedback  Vector quantization design in phase space  Reduces required feedback to Log 2 K bits  Case 2 tx antenna  Scalar quantization  1 or 2 bits suffice 0.5 dB loss for 1 bit

15. Simulation Results (1) 2 transmit antennas and 2 receive antennas  Rayleigh block fading  L=130 symbols  Antipodal signaling  No channel code

16. Simulation Results (2) 3 transmit antennas and 2 receive antennas

17. Conclusions Circle the point that does not fit the talk: a.Simple design b. Small feedback c. Low down payment d. Loss analysis e. Colorful graphs