Mean Squared Error Analysis of the One-Bit Signal Power Estimator

One-bit quantization is a promising technique in signal processing which can be applied to various resource-limited contexts. In order to recover the complete covariance matrix of the received signal, non-zero thresholds are needed to enable the estimation of both variance and correlation coefficients. Due to the absence of a closed-form for the Q function, it is extremely difficult to analyze the mean squared error (MSE) of the variance estimator, preventing us from finding the optimal clipping threshold. In this paper, we propose a novel exponential approximation of the Q function that ensures accuracy around a certain point. Based on this approximation, a closed-form formulation of the variance estimator's MSE is derived, which allows us to identify the optimal threshold through a straightforward numerical searching. Simulation results demonstrate that the MSE approximation is reasonably accurate. A comparison with the Cramer-Rao lower bound verify that the estimator is asymptotically efficient.