On Kay's Frequency Estimator

Kay has proposed a technique for estimating the frequency of a complex sinusoid in additive noise, the real and imaginary parts of which are independent and normally distributed with means zero and the same variance. For fixed sample size the estimator achieves the Cramer–Rao lower bound for unbiased estimators of the frequency in the limit as the signal to noise ratio approaches infinity. It has been noted by Lovell and Williamson, however, that the estimator is biased. It is shown in this paper that under Kay's assumptions the estimator is not consistent (for fixed signal to noise ratio the estimator converges almost surely, as the sample size N increases, to a number that is not the true frequency, no matter how large the signal to noise ratio). A class of distributions for the additive noise is proposed under which the technique is strongly consistent and has the correct order of asymptotic variance, namely N−3, for the case where there is some a priori knowledge concerning the range of the frequency. For this class a normal central limit theorem is developed.