Outage Probability Under Channel Distribution Uncertainty

Outage probability and capacity of a class of block fading MIMO channels are considered under partial channel distribution information. Specifically, the channel or its distribution is not known but the latter is known to belong to a class of distributions where each member is within a certain distance (uncertainty) from a nominal distribution. Relative entropy is used as a measure of distance between distributions. Compound outage probability defined as min (over the transmitted signal distribution) -max (over the channel distribution class) outage probability is introduced and investigated. This generalizes the standard outage probability to the case of partial channel distribution information. Compound outage probability characterization (via 1-D convex optimization and in a closed form), its properties, and approximations are given. It is shown to have two-regime behavior: when the nominal outage probability decreases (e.g., by increasing the SNR), the compound outage first decreases linearly down to a certain threshold (related to the relative entropy distance; this is the nominal outage-dominated regime) and then only logarithmically (i.e., very slowly; this is the uncertainty-dominated regime) so that no significant further decrease is possible. This suggests the following design guideline: the outage probability is decreased by increasing the SNR or optimizing the transmitted signal distribution (both decrease nominal outage) in the first regime and by reducing the channel distribution uncertainty (e.g., via better estimation) in the second one. The compound outage depends on the relative entropy distance and the nominal outage only, all other details (nominal fading and noise distributions) being irrelevant. The transmit signal distribution optimized for the nominal channel distribution is shown to be also optimal for the whole class of distributions. The effect of swapping the distributions in relative entropy is investigated and an error floor effect is established. The compound outage probability under L
_p
distance constraint is also investigated. The obtained results hold in full generality, i.e., for the general channel model with arbitrary nominal fading and noise distributions.