III. MAIN RESULTS A. The Case of Idealized Duty Cycle Constraint B. The Case of Realistic Duty Cycle Constraint Let µ denote the distribution of the channel input X . The set of distributions with duty cycle constraint ( q, 0) and power constraint γ is denoted by It should be understood that µ is a probability measure defined on the Borel algebra on the real number set, denoted by B ($_{R}$) . Theorem 1: The capacity of the additive white Gaussian noise channel (1) with its idealized duty cycle no greater than 1 − q and the average power no greater than γ is In particular, the following properties hold: The proof of Theorem 1 is relegated to Section IV. Property (b) suggests that the capacity-achieving input always exhausts the power budget. Property (c) indicates that the capacity-achieving input can be well approximated by some discrete inputs with finite alphabet, which can be computed using numerical methods. The achievable rate of numerically optimized input distribution is studied in Section VI. b) µ$_{0}$ is symmetric about 0 and its second moment is exactly equal to γ ; and a) the maximum of (5) is achieved by a unique (capacity-achieving) distribution µ$_{0}$ ∈ Λ( γ, q ) ; c) µ$_{0}$ is discrete with an infinite number of probability mass points, whereas the number of probability mass points in any bounded interval is finite. C ( γ, q ) = max µ ∈ Λ( γ,q ) I ( µ ) . (5) Λ( γ, q ) = $^{{}$µ : µ ( { 0 } ) ≥ q, E$_{µ}$ $^{{}$X 2 $^{}}$≤ γ } . (4) In this paper, let X n k denote the subsequence ( X$_{k}$, X$_{k}$$_{+1}$, · · · , X$_{n}$ ) , where X ∞ k = ( X$_{k}$, X$_{k}$$_{+1}$, · · · ) . We also use shorthand X n = X n $_{1}$. Let µ denote the probability distribution of the process X$_{1}$, X$_{2}$, · · · . We use µ$_{X}$$_{i}$ to denote the marginal distribution of X$_{i}$ , and µ$_{X}$$_{i}$$_{,X}$$_{j}$ to denote the joint probability distribution of ( X$_{i}$, X$_{j}$ ) . Denote the set of n -dimension distribution which satisfy duty cycle constraint ( q, c ) and 6