One-bit compressive sampling with time-varying thresholds: Maximum likelihood and the Cramér-Rao bound

This paper considers the problem of estimating the parameters of a noisy signal which has been quantized to one-bit via a time-varying thresholding operation. An expression for the Fisher information matrix (FIM) is derived for a generic deterministic signal parameterized by a vector β when the noise is independent and identically distributed (i.i.d.) Gaussian with either known or unknown variance. The case of single sinusoidal parameter estimation is considered as a particular example, and the Cramér-Rao bounds (CRB) for amplitude, frequency, and phase estimators are computed for a variety of parameter values. A maximum likelihood (ML) estimator for the sinusoidal signal parameters is proposed, and its performance is compared with the CRB as a function of the number of observations.