Single-bit quantization of binary-input, continuous-output channels

Abstract

A binary-input, memoryless channel with a continuous-valued output quantized to one bit is considered. For arbitrary noise models, conditions on an optimal quantizer, in the sense of maximizing mutual information between the channel input and the quantizer output, are given. This result is obtained by considering the “backward” channel and applying Burshtein et al.'s theorem on optimal classification. In this backward channel, there exists an optimal quantizer for which the quantizer preimage is convex. It is possible no optimal forward quantizer is convex, but by working with the backward channel, the optimal quantizer may be found. However, if the channel satisfies a certain condition, then a convex optimal forward quantizer exists.