Optimum one-bit quantization

Abstract

This paper deals with discrete input one-bit output quantization. A discrete input signal is subject to additive noise and is then quantized to zero or one by comparison with a threshold q. For finitely many fixed support points and fixed threshold q we first determine the mutual information of this channel. The capacity-achieving input distribution is shown to be concentrated on merely two extreme support points. Furthermore, an elegant representations of the corresponding probabilities is found. Finally, we set out to determine the optimum threshold q, which is an extremely hard problem. By means of graphical representations a completely different behavior of the objective function is revealed, depending on the choice of parameters and the noise distribution.