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Given a channel having binary input X = (x_1, x_2) having the probabilitydistribution p_X = (p_{x_1}, p_{x_2}) that is corrupted by a continuous noiseto produce a continuous output y \in Y = R. For a given conditionaldistribution p(y|x_1) = \phi_1(y) and p(y|x_2) = \phi_2(y), one wants toquantize the continuous output y back to the final discrete output Z = (z_1,z_2, ..., z_N) with N \leq 2 such that the mutual information between input andquantized-output I(X; Z) is maximized while the probability of thequantized-output p_Z = (p_{z_1}, p_{z_2}, ..., p_{z_N}) has to satisfy acertain constraint. Consider a new variable r_y=p_{x_1}\phi_1(y)/(p_{x_1}\phi_1(y)+p_{x_2}\phi_2(y)), we show that the optimal quantizer has astructure of convex cells in the new variable r_y. Based on the convex cellsproperty of the optimal quantizers, a fast algorithm is proposed to find theglobal optimal quantizer in a polynomial time complexity.

Optimal quantizer structure for binary discrete input continuous output channels under an arbitrary quantized-output constraint