Task-Based Quantization for Recovering Quadratic Functions Using Principal Inertia Components

Quantization allows physical signals to be processed using digital devices. Quantizers are commonly implemented using analog-to-digital converters (ADCs), which operate in a serial and scalar manner and are designed to yield an accurate digital representation of the observed signal. However, in many practical scenarios quantization is part of a system whose task is not to recover the observed signal, but some function of it. Recent works have shown that properly designed task-based quantizers, which include pre-quantization analog combining as well as digital processing, can achieve notable gains in recovering linear functions of the observations. In this work we focus on quantization for the task of recovering quadratic functions. Our analysis is based on principal inertia components (PICs), which form a basis for decomposing the statistical dependence between random quantities. Using PICs, we identify a practical structure of the pre-quantization mapping for recovering quadratic functions, which allows us to design a task-based quantization system capable of accurately estimating these functions. Our numerical study demonstrates that, when using scalar ADCs, notable performance gains that can be achieved using the proposed design over intuitive approaches such as quantizing the quadratic function directly as well as task-ignorant quantization.