Modified Arcsine Law for One-Bit Sampled Stationary Signals with Time-Varying Thresholds

Abstract

One-bit quantization has attracted considerable attention in signal processing for communications and sensing. The arcsine law is a useful relation often used to estimate the normalized covariance matrix of zero-mean stationary input signals when they are sampled by one-bit analog-to-digital converters (ADCs)-practically comparing the signals with a given threshold level. This relation, however, only considers a zero threshold which can cause a remarkable information loss. For the first time in the literature, this paper introduces an approach to extending the arcsine law to the case where one-bit ADCs apply time-varying thresholds. In particular, the proposed method is shown to accurately recover the variance and autocorrelation of the stationary signals of interest.

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

“…By substituting (9) in (8), the output autocorrelation function R y (i, j) can be evaluated as [1], d) dw i dw j (10) where λ(d) is defined as follows:…”

“…Clearly, the determinant of the Hankle matrix must be non-zero to permit a unique solution to the linear system. The coefficients {a n } are obtained by backsubstitution [1], [19], [20]:…”

“…In this subsection, p 0 and {p l } are estimated by formulating a minimization problem. For this purpose, one may consider the following criterion [1]:…”

“…In this paper, we present a new approach to extending the arcsine law in the context of time-varying sampling thresholds, which can recover the covariance values of the input unquantized signal with accuracy. In particular, we further expand on the ideas we presented in [1] by employing several competing recovery approaches. Moreover, we propose a new formalism for the Bussgang law [4], [17] in the context of time-varying thresholds, which is referred to as the modified Bussgang law.…”

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

“…The zero threshold values can give rise to some difficulties in signal amplitude recovery and a considerable portion of signal information may be lost. As a natural alternative, time-varying thresholds are utilized in recent works which can lead to enhancements in recovery performance [1], [21]- [26].…”