Coherence estimate between a random and a periodic signal: Bias, variance, analytical critical values, and normalizing transforms

The present work deals with recent results on the sampling distribution of the magnitude-squared coherence (also called just coherence) estimate between a random (Gaussian) and a periodic signal, in order to obtain analytical critical values, alternative expressions for the probability density function (PDF) as well as the variance and bias of the estimate. A comparison with the more general case of coherence estimation when both signals are Gaussian was also provided. The results indicate that the smaller the true coherence (TC) values the closer both distributions become. The behaviour of variance and bias as a function of the number of data segments and the TC is similar for both coherence estimates. Additionally, the effect of a normalizing function (Fisher's z transform) in the coherence estimated between a random and a periodic signal was also evaluated and normality has been nearly achieved. However, the variance was less equalized in comparison with coherence estimate between two Gaussian signals.