Structure of Optimal Quantizer for Binary-Input Continuous-Output Channels with Output Constraints

In this paper, we consider a channel whose the input is a binary random source X ∈ {x1,x2} with the probability mass function (pmf) pX = [px1,px2] and the output is a continuous random variable Y ∈ R as a result of a continuous noise, characterized by the channel conditional densities py|x1 = ϕ1(y) and py|x2 = ϕ2(y). A quantizer Q is used to map Y back to a discrete set Z ∈ {z1,z2,...,zN}. To retain most amount of information about X, an optimal Q is one that maximizes I(X;Z). On the other hand, our goal is not only to recover X but also ensure that pZ = [pz1,pz2,...,pzN] satisfies a certain constraint. In particular, we are interested in designing a quantizer that maximizes βI(X;Z)−C(pZ) where β is a tradeoff parameter and C(pZ) is an arbitrary cost function of pZ. Let the posterior probability ${p_{{x_1}\mid y}} = {r_y} = \frac{{{p_{{x_1}}}{\phi _1}(y)}}{{{p_{{x_1}}}{\phi _1}(y) + {p_{{x_2}}}{\phi _2}(y)}}$, our result shows that the structure of the optimal quantizer separates ry into convex cells. In other words, the optimal quantizer has the form: ${Q^{\ast}}\left( {{r_y}} \right) = {z_i}$, if $a_{i - 1}^{\ast} \leq {r_y} < a_i^{\ast}$ for some optimal thresholds $a_0^{\ast} = 0 < a_1^{\ast} < a_2^{\ast} < \cdots < a_{N - 1}^{\ast} < a_N^{\ast} = 1$. Based on this optimal structure, we describe some fast algorithms for determining the optimal quantizers.