Covariance Recovery for One-Bit Sampled Data With Time-Varying Sampling Thresholds-Part I: Stationary Signals

TL;DR

This paper extends covariance recovery methods for one-bit sampled data by incorporating time-varying thresholds, improving information retention over traditional zero-threshold approaches in stationary signal processing.

Contribution

It introduces three approaches for autocorrelation recovery and modifies the Bussgang law to handle time-varying thresholds in one-bit ADCs for stationary signals.

Findings

Proposed three new methods for autocorrelation recovery with time-varying thresholds.

Extended the Bussgang law to accommodate non-zero, time-varying thresholds.

Compared and analyzed the effectiveness of the approaches in various scenarios.

Abstract

One-bit quantization, which relies on comparing the signals of interest with given threshold levels, has attracted considerable attention in signal processing for communications and sensing. A useful tool for covariance recovery in such settings is the arcsine law, that estimates the normalized covariance matrix of zero-mean stationary input signals. This relation, however, only considers a zero sampling threshold, which can cause a remarkable information loss. In this paper, the idea of the arcsine law is extended to the case where one-bit analog-to-digital converters (ADCs) apply time-varying thresholds. Specifically, three distinct approaches are proposed, investigated, and compared, to recover the autocorrelation sequence of the stationary signals of interest. Additionally, we will study a modification of the Bussgang law, a famous relation facilitating the recovery of the cross-correlation between the one-bit sampled data and the zero-mean stationary input signal. Similar to the case of the arcsine law, the Bussgang law only considers a zero sampling threshold. This relation is also extended to accommodate the more general case of time-varying thresholds for the stationary input signals.