Optimal quantizer structure for binary discrete input continuous output channels under an arbitrary quantized-output constraint
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Given a channel having binary input X = (x_1, x_2) having the probability distribution p_X = (p_x_1, p_x_2) that is corrupted by a continuous noise to produce a continuous output y ∈ Y = R. For a given conditional distribution p(y|x_1) = ϕ_1(y) and p(y|x_2) = ϕ_2(y), one wants to quantize the continuous output y back to the final discrete output Z = (z_1, z_2, ..., z_N) with N ≤ 2 such that the mutual information between input and quantized-output I(X; Z) is maximized while the probability of the quantized-output p_Z = (p_z_1, p_z_2, ..., p_z_N) has to satisfy a certain constraint. Consider a new variable r_y=p_x_1ϕ_1(y)/ (p_x_1ϕ_1(y)+p_x_2ϕ_2(y)), we show that the optimal quantizer has a structure of convex cells in the new variable r_y. Based on the convex cells property of the optimal quantizers, a fast algorithm is proposed to find the global optimal quantizer in a polynomial time complexity.
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