Capacity Bounds via Duality with Applications to Multi-Antenna Systems on Flat Fading Channels

A general technique is proposed for the derivation of upper bounds on channel capacity. The technique is based on a dual expression for channel capacity where the maximization (of mutual information) over distributions on the channel input alphabet is replaced with a minimization (of average relative entropy) over distributions on the channel output alphabet. Every choice of an output distribution — even if not the channel image of some input distribution — leads to an upper bound on mutual information. The proposed approach is used in order to study multi-antenna flat fading channels with memory where the realization of the fading process is unknown at the transmitter and unknown (or only partially known) at the receiver. It is demonstrated that, for high signal-to-noise ratio (SNR), the capacity of such channels typically grows only double-logarithmically in the SNR. This is in stark contrast to the case with perfect receiver side information where capacity grows logarithmically in the SNR. To better understand this phenomenon and the rates at which it occurs, we introduce the fading number as the second order term in the high SNR asymptotic expansion of capacity, and derive estimates on its value for various systems. It is suggested that at rates that are significantly higher than the fading number, communication becomes extremely power inefficient, thus posing a practical limit on the achievable rates. In an attempt to better understand the dependence of channel capacity on the fading law and on the number of antennae, we derive upper and lower bounds on the system’s fading number. For Single-Input Single-Output (SISO) systems we present a complete characterization of the fading number for general stationary and ergodic fading processes. We also demonstrate that for memoryless Multi-Input Single-Output (MISO) channels, the fading number is achievable using beam-forming, and we derive an expression for the optimal beam direction. This direction depends on the fading law and is, in general, not the direction that maximizes the SNR on the induced SISO channel. Using a closed-form expression for the expectation of the logarithm of a non-central chi-square distributed random variable we provide some closed-form expressions for the fading number of some systems with Gaussian ∗Both authors are with the Department of Electrical Engineering at the Swiss Federal Institute of Technology (ETH) in Zurich, CH-8092, Switzerland. This research was conducted in part while A. Lapidoth was a resident at the Rockefeller Foundation Bellagio Study and Conference Center. These results were presented in part at the 2001 IEEE International Symposium on Information Theory, Washington, D.C.; the 2001 Workshop on Information Theory, Cairns, Australia; and the 2002 IEEE International Symposium on Information Theory, Lausanne, Switzerland.