Capacity achieving quantizer design for multiple-input multiple-output thresholding channels

We consider a communication channel whose input is modeled as a discrete random variable X with distribution p
X
. X is transmitted over a noisy channel and distorted by a continuous-valued noise to result in a continuous-valued output signal U at the receiver. A thresholding quantizer Q is applied to reconstruct a discrete signal V = Q(U) from the continuous-valued U. Our goal is to jointly design both the input distribution p
X
and the thresholding quantizer Q to maximize the mutual information I(X; V) between the input X and V since the accuracy of any decoding algorithm that estimates X from V fundamentally depends on I(X; V). In this paper, an alternating maximization algorithm is proposed that guarantees to achieve a locally optimal solution. In addition, we numerically show that by randomly selecting a set of initial starting points, the proposed algorithm is capable of achieving the globally optimal solution. Both the theoretical and numerical results are provided to justify our approach.