Threshold optimization for capacity-achieving discrete input one-bit output quantization

In this paper, we consider one-bit output quantization of a discrete signal with m real signaling points subject to arbitrary additive noise. First, the capacity-achieving distribution is determined for the corresponding channel. For any fixed quantization threshold q it concentrates on the two most distant signaling points, hence leading to an interpretation as binary asymmetric channel. The direct proof of this result allows for an explicit form of the capacity as a function of threshold q. We characterize stationary points as candidates for optimal thresholds by a condition on the differential quotient of the derivative of the binary entropy function. In contrast to intuition, symmetry of the noise distribution does not ensure a unique optimum antipodal threshold.