Capacity of power constrained memoryless AWGN channels with fixed input constellations

We propose a numerical method to compute the capacity of a power constrained memoryless additive white Gaussian noise (AWGN) channel with finite and fixed input alphabets. The method is based on a two-part algorithm. The first part is a modified version of the Blahut-Arimoto algorithm and the second part is a simple maximization algorithm over a single parameter. The optimal input distribution we obtain can be utilized to construct probabilistic codes for this channel. These codes promise to bridge the shaping gap between the uniform-input information rate and the capacity of the channel.