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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]
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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]
Space-Time
Transmitter
Space-Time
Receiver
User Data User Data
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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]
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Joint Transmitter-Channel Optimization
Transmit signal covariance onlyPoint-to-point
Broadcast channel
Joint transmit-channel optimizationPoint-to-point
Broadcast channel
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Motivations and Related WorkJoint transmit signal and channel optimization
Obtain upper bounds on MIMO channel capacityReveal best channel characteristicsDirect antenna configurations
Related workPoint-to-point case [Chiurtu, et al., 2000]
Convex optimizationEqual energy in every MIMO channel eigenmodeEqual power allocated for each channel eigenmode
Game theoretic approach [Palomar et al., 2003]
No transmit channel state informationEqual power distribution
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Point-to-Point Channel
DenoteNoticeReformulated problem
TX power allocated for the ith eigenmodeThe ith eigenvalue of
Number of transmit antennasNumber of receive antennas
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Point-to-Point ChannelIterative water-filling between
Optimal solutionEqual channel eigenmodesEqual power allocation Number of non-zero eigenmodes optimized
Water-level for TX power
Water-level for channel
Number of non-zero channeleigenmodes
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Broadcast ChannelCooperative channel
User cooperationUpper bound on BC sum capacityEffective point-to-point channel
Upper bound for Joint TX-H optimization
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Broadcast Channel
When for some integer and , the bound is tightConstruct a set ofEach has non-zero singular values ofEqual TX power for non-zero eigenmodes
Bound is asymptotically tight for high SNR when and
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Numerical Results
Maximum capacity vs. SNR Optimal # of eigenmodes vs. SNR, M=10
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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