1. Upper Bounds on MIMO Channel Capacity with Channel Frobenius Norm Constraints
Zukang Shen, Jeffrey Andrews, and Brian Evans
The University of Texas at Austin
Nov. 30, 2005
IEEE Globecom 2005

2. Multi-Antenna Systems
Exploit spatial dimension with multiple antennas
Improve transmission reliability – diversity
Combat channel fading [Jakes, 1974]
Combat co-channel interference [Winters, 1984]
Increase spectral efficiency – multiplexing
Multiple parallel spatial channels created with multiple antennas at transmitter and receiver [Winters, 1987] [Foschini et al., 1998]
Theoretical results on point-to-point MIMO channel capacity
[Telatar, 1999]
Tradeoff between diversity and multiplexing
Theoretical treatment [Zheng et al., 2003]
Switching between diversity and multiplexing [Heath et al. 2005]

3. Point-to-Point MIMO Systems
Narrowband system model
Rayleigh model
Each element in is i.i.d. complex Gaussian
Channel energy scales sub-linearly in the number of antennas [Sayeed et al., 2004]
Ray-tracing models [Yu et al., 2002]

4. MIMO Gaussian Broadcast Channels
Duality between MIMO Gaussian broadcast and multiple access channels [Vishwanath et al., 2003]
Dirty paper coding [Costa 1983]
Sum capacity achieved with DPC [Vishwanath et al., 2003]
Iterative water-filling [Yu et al., 2004] [Jindal et al., 2005]
Capacity region of MIMO Gaussian broadcast channels [Weingarten et al., 2004]

5. Joint Transmitter-Channel Optimization
Joint transmit-channel optimization
Point-to-point
Broadcast channel
Transmit signal covariance only
Point-to-point
Broadcast channel

6. Motivations and Related Work
Joint transmit signal and channel optimization
Obtain upper bounds on MIMO channel capacity
Reveal best channel characteristics
Direct antenna configurations
Related work
Point-to-point case [Chiurtu, et al., 2000]
Convex optimization
Equal energy in every MIMO channel eigenmode
Equal power allocated for each channel eigenmode
Game theoretic approach [Palomar et al., 2003]
No transmit channel state information
Equal power distribution

7. Point-to-Point Channel
Denote
Notice
Reformulated problem
TX power allocated for the ith eigenmode
The ith eigenvalue of
Number of transmit antennas
Number of receive antennas

8. Point-to-Point Channel
Iterative water-filling between
Optimal solution
Equal channel eigenmodes
Equal power allocation
Number of non-zero eigenmodes optimized
Water-level for TX power
Water-level for channel
Number of non-zero channel
eigenmodes

9. Broadcast Channel Cooperative channel
User cooperation
Upper bound on BC sum capacity
Effective point-to-point channel
Upper bound for Joint TX-H optimization

10. Broadcast Channel When for some integer and , the bound is tight
Construct a set of
Each has non-zero singular values of
Equal TX power for non-zero eigenmodes
Bound is asymptotically tight for high SNR when and

11. Numerical Results Maximum capacity vs. SNR
Optimal # of eigenmodes vs. SNR, M=10

12. Summary
Jointly optimize transmit signal covariance and MIMO channel matrix
Obtain upper bounds on MIMO channel capacity
Reveal best channel characteristics
Direct antenna configurations
Re-derive the optimal solution for point-to-point MIMO channels with iterative water-filling
Equal MIMO eigenmode gains
Equal transmit power in every MIMO eigenmode
Number of eigenmodes optimized to SNR
Upper bound sum capacity of MIMO broadcast channels with cooperative point-to-point channels
Orthogonalize user channels
Optimize number of user channel eigenmodes