Closed-form MSE performance for phase estimation from Gaussian reference signals

In many communications and signal processing applications, phase information carried on Gaussian distributed reference signals is often required for various purposes, such as the carrier frequency offset estimation in orthogonal frequency division multiplexing (OFDM) systems. The performance of phase estimation is usually measured by the mean square error (MSE) which is often infeasible to obtain. Instead, the Cramér-Rao Bound (CRB) and modified Cramér-Rao Bound (MCRB) are used to give lower MSE bounds for the phase estimation. This paper presents closed-form MSE approximations for estimating phase information from Gaussian reference signals, which provide better indications of the MSE performance than the MCRB. It is also shown that the MCRB is only attainable at high signal-to-noise ratios and with large number of observed signal samples. Simulated and analytical results are compared to demonstrate the accuracy and efficiency of the derived MSE formulas.