  This paper studies the structure of downlink sum-rate maximizing selective
decentralized feedback policies for opportunistic beamforming under finite
feedback constraints on the average number of mobile users feeding back.
Firstly, it is shown that any sum-rate maximizing selective decentralized
feedback policy must be a threshold feedback policy. This result holds for all
fading channel models with continuous distribution functions. Secondly, the
resulting optimum threshold selection problem is analyzed in detail. This is a
non-convex optimization problem over finite dimensional Euclidean spaces. By
utilizing the theory of majorization, an underlying Schur-concave structure in
the sum-rate function is identified, and the sufficient conditions for the
optimality of homogenous threshold feedback policies are obtained. Applications
of these results are illustrated for well known fading channel models such as
Rayleigh, Nakagami and Rician fading channels, along with various engineering
and design insights. Rather surprisingly, it is shown that using the same
threshold value at all mobile users is not always a rate-wise optimal feedback
strategy, even for a network with identical mobile users experiencing
statistically the same channel conditions. For the Rayleigh fading channel
model, on the other hand, homogenous threshold feedback policies are proven to
be rate-wise optimal if multiple orthonormal data carrying beams are used to
communicate with multiple mobile users simultaneously.
