Covariance Matrix Recovery From One-Bit Data With Non-Zero Quantization Thresholds: Algorithm and Performance Analysis

Abstract

Covariance matrix recovery is a topic of great significance in the field of one-bit signal processing and has numerous practical applications. Despite its importance, the conventional arcsine law with zero threshold is incapable of recovering the diagonal elements of the covariance matrix. To address this limitation, recent studies have proposed the use of non-zero clipping thresholds. However, the relationship between the estimation error and the sampling threshold is not yet known. In this paper, we undertake an analysis of the mean squared error by computing the Fisher information matrix for a given threshold. Our results reveal that the optimal threshold can vary considerably, depending on the variances and correlation coefficients. As a result, it is inappropriate to adopt a constant threshold to encompass parameters that vary widely. To mitigate this issue, we present a recovery scheme that incorporates timevarying thresholds. Our approach differs from existing methods in that it utilizes the exact values of the threshold, rather than its statistical properties, to increase the estimation accuracy. Simulation results, including those of the direction-of-arrival estimation problem, demonstrate the efficacy of the developed scheme, especially in complex scenarios where the covariance elements are widely separated.

UNO: Unlimited Sampling Meets One-Bit Quantization

“…For example, the covariance matrix formulation of [30] employs the cyclic optimization method to recover the parameters. A convex program based on the Gauss-Legendre integration to recover the input covariance matrix from onebit sampled data was suggested in [13,27,28]. Other recent works exploit sparsity of the signal and apply techniques such as ℓ 1 -norm minimization [66,67], ℓ 1 -regularized MLE formulation [51,52], and log-relaxation [68] to lay the ground for signal reconstruction.…”

“…This yields the quantized signal as r k = Q(x k ). In one-bit quantization, compared to zero or constant thresholds, time-varying sampling thresholds yield a better reconstruction performance [13,27]. These thresholds may be chosen from any distribution.…”

“…Conventional multi-bit ADCs require a very large number of quantization levels to represent the original continuous signal in high resolution settings. Sampling at high data rates with high resolution ADCs, however, would dramatically increase the overall power consumption and the manufacturing cost of such ADCs [13]. This problem is exacerbated in systems that require multiple ADCs such as large array receivers [14].…”