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We address the capacity of a discrete-time memoryless Gaussian channel, wherethe channel state information (CSI) is neither available at the transmitter norat the receiver. The optimal capacity-achieving input distribution at lowsignal-to-noise ratio (SNR) is precisely characterized, and the exact capacityof a non-coherent channel is derived. The derived relations allow to betterunderstanding the capacity of non-coherent channels at low SNR. Then, wecompute the non-coherence penalty and give a more precise characterization ofthe sub-linear term in SNR. Finally, in order to get more insight on how theoptimal input varies with SNR, upper and lower bounds on the non-zero masspoint location of the capacity-achieving input are given.

Capacity of The Discrete-Time Non-Coherent Memoryless Gaussian Channels at Low SNR