On the discreteness of capacity-achieving distributions for fading and signal-dependent noise channels with amplitude-limited inputs

We address the problem of finding the capacity of two classes of channels with amplitude-limited inputs. The first class is frequency flat fading channels with an arbitrary (but finite support) channel gain with the channel state information available only at the receiver side; while the second one we consider is the class of additive noise channels with signal-dependent Gaussian noise. We show that for both channel models and under some regularity conditions, the capacity-achieving distribution is discrete with a finite number of mass points. Furthermore, finding the capacity-achieving distribution turns out to be a finite-dimensional optimization problem, and efficient numerical algorithms can be developed using standard optimization techniques to compute the channel capacity. We demonstrate our findings via several examples. In particular, we present an example for a block fading channel where the channel gain follows a truncated Rayleigh distribution, and two instances of signal-dependent noise that are used in the literature of magnetic recording and optical communication channels.