Thresholds in frequency estimation

It is well known that frequency estimation exhibits a threshold effect: as the SNR dips below a certain value (while the number of data points remains fixed) the estimate's variance increases rapidly, and the Cramer-Rao bound is no longer attainable. In this paper, a simple, explicit formula for this threshold will be derived, for the case of a single complex sinusoid in white Gaussian noise. Since the presence of multiple frequencies raises this threshold, the formula serves as a lower bound for more complex problems. Because of the duality between time series and sensor arrays, the threshold is also applicable to angle of arrival estimation of a plane wave. Frequency estimation thresholds have been studied for some time. For instance, in Van Trees classic textbook, "Detection, Estimation, and Modulation Theory", plots of SNR versus estimation variance are shown which exhibit obvious thresholding behavior. Previous studies, however, lack formulas for the threshold locations. The case to which the formula applies (one complex sinusoid) is clearly the simplest to analyze. Unfortunately, more complex signal scenarios (such as one or more real sinusoids) are more commonplace. The methodology employed in deriving our formula is sufficiently flexible so as to suggest that similar thresholds are attainable for more complex signals.