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]

Space-Time

Transmitter

Space-Time

Receiver

User Data User Data

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

Transmit signal covariance onlyPoint-to-point

Broadcast channel

Joint transmit-channel optimizationPoint-to-point

Broadcast channel

6

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

7

Point-to-Point Channel

DenoteNoticeReformulated problem

TX power allocated for the ith eigenmodeThe ith eigenvalue of

Number of transmit antennasNumber of receive antennas

8

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

9

Broadcast ChannelCooperative channel

User cooperationUpper bound on BC sum capacityEffective point-to-point channel

Upper bound for Joint TX-H optimization

10

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

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