Non-Linear Analog Processing Gains in Task-Based Quantization

In task-based quantization, a multivariate analog signal is transformed into a digital signal using a limited number of low-resolution analog-to-digital converters (ADCs). This process aims to minimize a fidelity criterion, which is assessed against an unobserved task variable that is correlated with the analog signal. The scenario models various applications of interest such as channel estimation, medical imaging applications, and object localization. This work explores the integration of analog processing components -- such as analog delay elements, polynomial operators, and envelope detectors -- prior to ADC quantization. Specifically, four scenarios, involving different collections of analog processing operators are considered: (i) arbitrary polynomial operators with analog delay elements, (ii) limited-degree polynomial operators, excluding delay elements, (iii) sequences of envelope detectors, and (iv) a combination of analog delay elements and linear combiners. For each scenario, the minimum achievable distortion is quantified through derivation of computable expressions in various statistical settings. It is shown that analog processing can significantly reduce the distortion in task reconstruction. Numerical simulations in a Gaussian example are provided to give further insights into the aforementioned analog processing gains.