Upper Bounds on MIMO Channel Capacity with Channel Frobenius Norm Constraints

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

8
Point-to-Point Channel

• Iterative water-filling between
• Optimal solution
• Equal channel eigenmodes
• Equal power allocation
• Number of non-zero eigenmodes optimized

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, M10
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