which is prohibitive. Assuming any upper bound on the number of probability mass points, however, a numerical optimization over the mutual information can yield a suboptimal input distribution and a lower bound on the channel capacity. As we increase the number of mass points, the lower bound can be further refined. We take this approach to numerically compute a good approximation of the channel capacity by optimizing over a sufficient number of probability mass points. Given the duty cycle and power constraints, we first numerically optimize the mutual information by a 3-point input distribution (including a mass at 0), then increase the number of probability mass points by 2 at a time to improve the mutual information, until the improvement is less than 10 − $^{3}$. First consider the case that the duty cycle is no greater than 70% , i.e., P ( X = 0) ≥ q = 0 . 3 . For different SNRs, the mass points of the near-optimal input distribution with finite support along with the corresponding probability masses are shown in Fig. 1. Due to symmetry, only the positive half of the input distribution is plotted. We can see that as the SNR increases, more masses are put on higher-amplitude points, whereas the probability mass at zero achieves its lower bound 0 . 3 eventually. In Fig 2, we compare the rate achieved by the near-optimal input distribution and the rate achieved by a conventional scheme using Gaussian signaling over a deterministic schedule, which is (1 − q ) times the Gaussian channel capacity without duty cycle constraint. It is shown in the figure that there is substantial gain for both 0 dB and 10 dB SNRs by using discrete input over Gaussian signaling with a deterministic schedule. For example, when the SNR is 10 dB, given the duty cycle is no more than 50%, the discrete input distribution achieves 50% higher rate. Hence departing from the usual paradigm of intermittent packet transmissions may yield significant gains. We also plot in Fig 2 the achievable rate by a superposition coding, where the input distribution is a mixture of Gaussian and a point mass at 0. We first decode the support of the input to find out the positions of nonzero symbols, and then the Gaussian codeword conditioned on the support. It is shown in the figure that the near-optimal discrete input achieves higher rate compared with the mixture input. 2) Realistic duty cycle constraint ( q, c ) : In this subsection the numerical results of lower bound of capacity and suboptimal distribution are provided based on the results in Section V and VI-A. We first seek a discrete Markov chain with finite alphabet that maximizes the objective L ( µ ) defined in (11). Once the optimal Markov distribution µ ∗ is determined, we compute the achievable rate I ( µ $^{∗}$) according to (85). 29