1. Design of Multiple Antenna Coding Schemes with Channel Feedback
Krishna Kiran Mukkavilli, Ashutosh Sabharwal,
and Behnaam Aazhang
Department of Electrical and Computer Engineering
Rice University, Houston, Texas

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]
Capacity achieved via space-time codes

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
Role of phase information
Role of amplitude 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
Transmission schemes with phase and amplitude feedback

6. System and Channel Model
h1,i L Y X
Block Fading
m tx
antennas
n rx
antennas
hm,i
Channel realization known at the receiver
Error free feedback channel

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

8. Feedback Cases Dominant eigenvector solution
Captures relevant channel phase and amplitude information
(2m – 2) real numbers
Analyze the cases of
Phase information only
Amplitude information only
Outage probability performance

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

10. Special Cases
m =2 ,n = 1
m=2, n = n
m=m, n = 1

11. 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
Loss is about 1.05 dB for 1 rx antenna and large m

12. Beamforming with Amplitude Information
Problem: find such that minimizes error probability where x is the information vector;
Solution (selection diversity) : Set hi= 1 for the antenna with the “best channel”, hi =0 for others
Additional amplitude information does not help

13. Features of Selection Diversity
Finite feedback of log(m) bits per frame
Maximum diversity achieved
Performance loss compared to dominant eigenvector, for n=1, given by
For large m, with n=1, loss approximated by

14. Simulation Results (1) 2 transmit antennas and 2 receive antennas
Rayleigh block fading
Antipodal signaling
No channel code

15. 3 transmit antennas and 2 receive antennas
Simulation Results (2)
3 transmit antennas and 2 receive antennas

16. Outage Probability Analysis
Useful concept for non-ergodic (delay constrained) channels
Lower bound on frame error rate
Same slope for outage for all the schemes
Given by the maximum spatial diversity
Sufficient condition based on channel norm for attaining maximum slope for outage probability [ISIT 2002]

17. Conclusion and Future Work
Summary
Designed and analyzed various transmission and feedback schemes
Future work
Address tradeoff between phase and amplitude information with finite feedback
Framework for performance analysis normalized by the extent of feedback