Feedback increases neither the fading number nor the pre-log

The feedback capacity of a single-input single-output fading channel with memory is investigated. It is shown that, asymptotically, as the signal-to-noise ratio (SNR) tends to infinity, noiseless feedback does not increase capacity. More specifically, for finite-energy stationary and ergodic fading processes that are regular (i.e., that have a finite differential entropy rate), feedback does not increase the fading number; and, for Gaussian fading processes that are non-regular, feedback does not increase the pre-log.