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We consider a channel with discrete binary input X that is corrupted by agiven continuous noise to produce a continuous-valued output Y. A quantizer isthen used to quantize the continuous-valued output Y to the final binary outputZ. The goal is to design an optimal quantizer Q* and also find the optimalinput distribution p*(X) that maximizes the mutual information I(X; Z) betweenthe binary input and the binary quantized output. A linear time complexitysearching procedure is proposed. Based on the properties of the optimalquantizer and the optimal input distribution, we reduced the searching rangethat results in a faster implementation algorithm. Both theoretical andnumerical results are provided to illustrate our method.

Single-bit Quantization Capacity of Binary-input Continuous-output Channels