The magnitude squared coherence estimate: A geometric view

The magnitude squared coherence estimate is often used as a means for detecting the presence of a common signal on two different channels. The value of this detection statistic is enhanced by understanding its behavior when a common signal is not present on both channels and the channel sequences are statistically independent. In this paper the distribution of the magnitude-squared-coherence (MSC) estimate is examined from a geometric point of view. A geometric model is used to show that statistical independence of the channel sequences and spherical symmetry of the distribution function of one of the two sequences used in the coherence estimate are sufficient conditions for the distribution of the coherence estimate to be invariant to second channel statistics. This is a generalization of results obtained by Nuttall [1]. In addition a connection between spherical symmetry and the Gaussian distribution is discussed. A geometric method for the derivation for the MSC estimate is also presented.