This article introduces a novel estimation framework for the parameters in a linear-in-the-parameter estimation problem when one-bit measurements are processed. We consider a periodic signal, whose components have unknown amplitudes and phases. This signal is assumed to be quantized by a single comparator under various problem settings. To provide enough information for the estimation of the signal parameters based on one-bit quantized signal measurements, the threshold in the one-bit comparator is assumed known. Several problem settings are considered. They include synchronous/asynchronous sampling, presence or absence of deterministic or stochastic dither, and presence or absence of additive noise. The results obtained by applying three alternative methods are compared and analyzed. Experimental results on a two-component 1.2-GHz signal validate the theoretical analysis. It is shown that several estimation approaches are available, which provides different performance levels, in terms of final estimation accuracy and computational complexity.