The circular nature of discrete-time frequency estimates

It is shown that the conventional (linear) definitions of mean and variance can yield absurd results when applied to data defined on a circular domain. Appropriate definitions for the mean, variance, and moments of circular data are given, and these concepts are applied to reformulate a recently proposed frequency estimator to avoid bias and elevated threshold effects. The maximum likelihood frequency estimator is shown to be equivalent to least-squares regression on phase estimates. These results demonstrate the importance of appreciating the true circular nature of many quantities commonly encountered in digital signal processing