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 fading, including SISO
systems with circularly symmetric stationary and ergodic Gaussian
fading. The fading number of the latter is determined by the fading
mean,fading variance, and the mean squared-error in predicting the
present fading from its past; it is not directly related to the
Doppler spread.

A key ingredient in the analysis of the fading number is played by
the notion of "capacity achieving input distributions that escape
to infinity." This is a general property that many cost-constrained
channel possess, and it is hoped that it will find use in the
analysis of the high SNR behavior of other channels too.

For some specific channels, e.g., the Rayleigh, Ricean, and
Multi-Antenna Rayleigh fading channels we also present firm upper
and lower bounds on channel capacity. These bounds are
asymptotically tight in the sense that their difference from
capacity approaches zero at high SNR, and their ratio to capacity
approaches one at low SNR.


Channel capacity, upper bounds, duality, fading channels, flat fading,
multi-antenna fading number, high SNR, Rayleigh fading, Ricean fading,
non-central chi-square.

-||-   _|_ _|_     /    __|__   Stefan M. Moser
[-]     --__|__   /__\    /__   Senior Researcher & Lecturer, ETH Zurich, Switzerland
_|_     -- --|-    _     /  /   Adj. Professor, National Chiao Tung University (NCTU), Taiwan
/ \     []  \|    |_|   / \/    Web: http://moser-isi.ethz.ch/

Last modified: Wed May 10 13:01:03 2006