Denote the relative entropy D $^{(}$p$_{Y}$$_{|}$$_{X}$ ( ·| x ) ‖ p$_{Y}$ ( · ; µ ) by d ( x ; µ ) , which is expressed as The mutual information I ( µ ) = I ( X ; Y ) is then The capacity of the AWGN channel under per-letter duty cycle constraint and power constraint is evidently given by the supremum of the mutual information I ( µ ) where µ ∈ Λ( γ, q ) . The achievability and converse of this result can be established using standard techniques in information theory. The proof of property (a) is presented in Section IV-A. Now suppose µ$_{0}$ is the unique capacity- achieving distribution, property (b) is established as follows. Since the mirror reflection of µ$_{0}$ about 0 is evidently also a maximizer of (5), the uniqueness requires that µ$_{0}$ be symmetric. Note that linear scaling of the input to increase its power maintains its duty cycle and cannot reduce the mutual information, as the receiver can add noise to maintain the same SNR. By the uniqueness of the maximizer µ$_{0}$ , the power constraint must be binding, i.e., the second moment of µ$_{0}$ must be equal to γ . In order to prove property (c), we first establish a sufficient and necessary condition for µ$_{0}$ in Section IV-B and then apply it to show the discreteness of µ$_{0}$ in Section IV-C. Let P denote the collection of all Borel probability measures defined on ($_{R}$ , B ($_{R}$)) , which is a topological space with the topology of weak convergence [15]. We first establish the following lemma. Lemma 1: Λ( γ, q ) is compact in the topological space P . Proof: According to [15], the topology of weak convergence on P is metrizable. Therefore, by Prokhorov’s theorem [16], in order to prove that Λ( γ, q ) is compact in P , it suffices to show that it is both tight and closed. For any ϵ > 0 , there exits an a$_{ϵ}$ > 0 , such that for all µ ∈ Λ$_{γ}$ , A. Existence and Uniqueness of µ$_{0}$ µ ( | X | > a$_{ϵ}$ ) ≤ E$_{µ}$ $^{{}$X 2 } a 2 ϵ ≤ γ a 2 ϵ < ϵ (18) 9 d ( x ; µ ) = ∫ ∞ −∞ p$_{Y}$$_{|}$$_{X}$ ( y | x ) log p$_{Y}$$_{|}$$_{X}$ ( y | x ) p$_{Y}$ ( y ; µ ) d y . (16) I ( µ ) = ∫ d ( x ; µ ) µ (d x ) = E$_{µ}$ { d ( X ; µ ) } . (17)