constrained capacity. Furthermore, Smith [3] investigated the capacity of a scalar AWGN channel under both peak power constraint and average power constraint. The input distribution that achieves the capacity is shown to be discrete with a finite number of probability mass points. The discreteness of capacity- achieving distributions for various channels, including quadrature Gaussian channels, and Rayleigh-fading channels is also established in [4]–[9]. Chan [10] studied the capacity-achieving input distribution for conditional Gaussian channels which form a general channel model for many practical communication systems. Until now, the impact of duty cycle constraint on capacity-achieving signaling is underexplored in the literature. The main results of this paper are summarized in Section III. In the case of the idealized duty cycle constraint, because all costs associated with the constraints can be decomposed into per-letter costs, the optimal input distribution is independent and identically distributed (i.i.d.). We use a similar approach as in [3] and [10] to show that the capacity-achieving input distribution for an AWGN channel with duty cycle constraint and average power constraints is discrete. Unlike in [3] and [10], the optimal distribution has an infinite number of probability mass points, whereas only a finite number of the points are found in every bounded interval. This allows efficient numerical optimization of the input distribution. The case of realistic duty cycle constraint is more challenging. Because the constraint concerns symbol transitions, the capacity-achieving input distribution is no longer independent over time, and becomes hard to compute. We develop a good lower bound of the input-output mutual information as a function of the input distribution. It is proved that, under the realistic duty cycle constraint, a first-order Markov process maximizes the lower bound, the distribution of which is also discrete and can be computed efficiently. The main theorems for the cases of idealized and realistic duty cycle constraints are proved in Section IV and V, respectively. We devote Section VI to the numerical methods and results. In order to compute the achievable rate when the input is a Markov Chain, a Monte Carlo method is introduced in Section VI-A to numerically compute the differential entropy rate of hidden Markov processes. Numerical results in Section VI-B demonstrate that in the case of idealize duty cycle constraint using a numerically optimized discrete signaling achieves higher rates than using Gaussian signaling over a deterministic transmission schedule. For example, if the radio is allowed to transmit no more than half the time, i.e., the duty cycle is no greater than 50%, a near-optimal discrete input achieves 50% higher rate at 10 dB signal-to-noise ratio 3