  We address the capacity of a discrete-time memoryless Gaussian channel, where
the channel state information (CSI) is neither available at the transmitter nor
at the receiver. The optimal capacity-achieving input distribution at low
signal-to-noise ratio (SNR) is precisely characterized, and the exact capacity
of a non-coherent channel is derived. The derived relations allow to better
understanding the capacity of non-coherent channels at low SNR. Then, we
compute the non-coherence penalty and give a more precise characterization of
the sub-linear term in SNR. Finally, in order to get more insight on how the
optimal input varies with SNR, upper and lower bounds on the non-zero mass
point location of the capacity-achieving input are given.
