On the fading number of multi-antenna systems

It has recently been shown that at high signal-to-noise ratios (SNR) the capacity of multi-antenna systems over flat fading channels (without receiver or transmitter side-information) typically grows only double-logarithmically in the SNR. Here we further refine the analysis and study the "fading number" χ, which we define as the limit of the difference between channel capacity and log(1+log(1+SNR)). It is suggested that at high SNR, i.e., at rates that significantly exceed the fading number, a capacity increase of one bit per channel use requires the squaring of the SNR, or equivalently, the doubling of the SNR as expressed in decibels. In this loose sense, the fading number can be viewed as the channel limiting rate for power-efficient communication. Note, however, that the fading number may be negative. While the use of multiple antennas does not typically change the double-logarithmic asymptotic dependence of channel capacity on the SNR, multiple antennas do typically increase the fading number, albeit at times (e.g., in the Rayleigh fading case) only in an additive way that grows only logarithmically with the number of antennas