  The capacity of a discrete-time memoryless channel, in which successive
symbols fade independently, and where the channel state information (CSI) is
neither available at the transmitter nor at the receiver, is considered at low
SNR. We derive a closed form expression of the optimal capacity-achieving input
distribution at low signal-to-noise ratio (SNR) and give the exact capacity of
a non-coherent channel at low SNR. The derived relations allow to better
understanding the capacity of non-coherent channels at low SNR and bring an
analytical answer to the peculiar behavior of the optimal input distribution
observed in a previous work by Abou Faycal, Trott and Shamai. Then, we compute
the non-coherence penalty and give a more precise characterization of the
sub-linear term in SNR. Finally, in order to better understand how the optimal
input varies with SNR, upper and lower bounds on the capacity-achieving input
are given.
