Many wireless communication systems are subject to duty cycle constraint, that is, a radio only actively transmits signals over a fraction of the time. For example, it is desirable to have a small duty cycle in some low-power systems; a half-duplex radio cannot keep transmitting if it wishes to receive useful signals; and a cognitive radio needs to listen to the channel frequently to detect primary users. Zhang and Guo have shown that the capacity of a Gaussian channel subject to an idealized duty cycle constraint as well as average transmission power constraint is achieved by discrete independent and identically distributed (i.i.d.) on-off signaling in lieu of Gaussian signaling. This paper extends the previous results by considering a more realistic duty cycle constraint where the extra cost of transitions between transmissions and nontransmissions due to pulse shaping is accounted for. The capacity-achieving input is no longer independent over time and is hard to compute. A lower bound of the input-output mutual information as a function of the input distribution is developed, which is shown to be maximized by a first-order Markov process, the distribution of which is also discrete and can be computed efficiently. Simulation results show that the Markov input is superior to i.i.d. inputs for the Gaussian channel subject to the realistic duty cycle and average power constraints.