# Technical Program

## Paper Detail

 Paper ID G.5.3 Paper Title Structure of Optimal Quantizer for Binary-Input Continuous-Output Channels with Output Constraints Authors Thuan Nguyen, Thinh Nguyen, Oregon State University, United States Session G.5: Signal Processing Presentation Lecture Track Graphs, Games, Sparsity, and Signal Processing Manuscript Click here to download the manuscript Virtual Presentation Click here to watch in the Virtual Symposium Abstract In this paper, we consider a channel whose the input is a binary random source $X \in \{x_1,x_2\}$ with the probability mass function (pmf) $\textbf{p}_X = [p_{x_1}, p_{x_2}]$ and the output is a continuous random variable $Y \in \mathbb{R}$ as a result of a continuous noise, characterized by the channel conditional densities $p_{y|x_1}=\phi_1(y)$ and $p_{y|x_2}=\phi_2(y)$. A quantizer $Q$ is used to map $Y$ back to a discrete set $Z \in \{z_1, z_2, \dots, z_N\}$. 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 $\textbf{p}_Z = [p_{z_1}, p_{z_2}, \dots, p_{z_N}]$ satisfies a certain constraint. In particular, we are interested in designing a quantizer that maximizes $\beta I(X;Z) - C(\textbf{p}_Z)$ where $\beta$ is a trade-off parameter and $C(\textbf{p}_Z)$ is an arbitrary cost function of $\textbf{p}_Z$. Let the posterior probability $p_{x_1|y} = r_y=\dfrac{p_{x_1} \phi_1(y)}{p_{x_1} \phi_1(y) + p_{x_2} \phi_2(y)}$, our result shows that the structure of the optimal quantizer separates $r_y$ into convex cells. In other words, the optimal quantizer has the form: $Q^*(r_y)= z_i, \text{ if } a^*_{i-1} \leq r_y < a^*_i,$ for some optimal thresholds $a^*_0=0 < a^*_1 < a^*_2< \dots < a^*_{N-1} < a^*_N = 1.$ Based on this optimal structure, we describe some fast algorithms for determining the optimal quantizers.