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 ∈ {x<inf>1</inf>,x<inf>2</inf>} with the probability mass function (pmf) p<inf>X</inf> = [p<inf>x1</inf>,p<inf>x2</inf>] and the output is a continuous random variable Y ∈ R as a result of a continuous noise, characterized by the channel conditional densities p<inf>y|x1</inf> = ϕ<inf>1</inf>(y) and p<inf>y|x2</inf> = ϕ<inf>2</inf>(y). A quantizer Q is used to map Y back to a discrete set Z ∈ {z<inf>1</inf>,z<inf>2</inf>,...,z<inf>N</inf>}. 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 p<inf>Z</inf> = [p<inf>z1</inf>,p<inf>z2</inf>,...,p<inf>zN</inf>] satisfies a certain constraint. In particular, we are interested in designing a quantizer that maximizes βI(X;Z)−C(p<inf>Z</inf>) where β is a tradeoff parameter and C(p<inf>Z</inf>) is an arbitrary cost function of p<inf>Z</inf>. 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 r<inf>y</inf> 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.