# Derivation of Circular Mean Square Error

I would like to understand how Eq. (36) in [2] was derived:

The rationale behind the definition of circular sample mean in Eq. (37) is clear, but there is no motiviation for the CMSE definition in Eq. (36).

[2] Lovell, Brian C., and Robert C. Williamson. "The statistical performance of some instantaneous frequency estimators." IEEE Transactions on Signal Processing 40.7 (1992): 1708-1723.

• The first term in Eq. (36) is the Mean Resultant Length (MRL) [Kutil, Biased and unbiased estimation of the circular mean resultant length and its variance] mapped to [0,+inf]. I still don't know where the second term (bias correction) is coming from. – Arrigo May 22 '20 at 17:38

The first term in $$e^2_p$$ is just the variation of the estimates around the true value.
The second term is due to the problem that happens at the end of the periodic region. Sometimes, the noise is enough to move the estimate from $$-\pi+\alpha$$ to $$\pi - \beta$$.