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

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

8. Water-level for TX power Water-level for channel Number of non-zero channel eigenmodes 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, 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