# 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 $$\pi - \beta$$.

For example, the orange plot in the figure below represents the variation of the estimates for a true value around zero. This variation is captured by the first term.

The blue plot in the figure below represents the variation of the estimates for the true value around the end of the period. The trouble is that some estimates end up all the way at the other end of the periodic region. This causes a bias to appear. The second term in (36) is aimed at capturing that bias.

• Say that you are using the CMSE formula to compare the relative performance of several frequency estimation algorithms (this is actually the reason it was introduce in [2]). The bias correction term will then make it possible to compare the performance of these estimators in regions with lower SNRs. Is this a correct statement? – Arrigo May 26 '20 at 18:06