Capacity-approaching signals for the scalar Gaussian channel

We consider the scalar Gaussian channel with power constraint P. A gap exists between the channel capacity and the highest achievable rate of equiprobable uniformly spaced signal. Several approaches enable to overcome this limitation such as constellations with non-uniform probability or constellation shaping. In this short paper, we focus on constellation shaping. We present a method to construct one-dimensional constellations with equiprobable signaling that achieve the Gaussian capacity as the number of signal points goes to infinity.