Coherence Function in Noisy Linear System

The coherence function provides a measure of spectral similarity of two signals, but measurement noise decreases the values of measured coherence. When the two signals are the input and output of a linear system, any system noise also decreases the measured coherence values. In digital computations, useful coherence values require some degree of averaging to increase the degrees of freedom to more than two. These fundamental issues are presented with application to system input-output coherence and two random signals with a common component. Finally, estimated coherence of the two random signals, with varying degrees of freedom, are shown with empirical adjustments that can improve the estimate of coherence. Coherence has a wide range of biomedical applications, but this article focuses on the fundamental properties of the coherence function.