A Biased Look at Phase Locking: Brief Critical Review and Proposed Remedy

A number of popular measures of dependence between pairs of bandlimited signals rely on analytic phase. A common misconception is that the dependence revealed by these measures must be specific to the spectral range of the filtered input signals. Implicitly or explicitly, obtaining analytic phase involves normalizing the signal by its own envelope, which is a nonlinear operation that introduces broad spectral leakage. We review how this generates bias and complicates the interpretation of commonly used measures of phase locking. A specific example of this effect may create spurious phase locking as a consequence of nonzero circular mean in the phase of input signals, which can be viewed as spectral leakage to 0 Hz. Corrections for this problem which recenter or uniformize the distribution of phase may fail when the amplitudes of the compared signals are correlated. To address the more general problem of spectral bias, a novel measure of phase locking is proposed, the amplitude-weighted phase-locking value. This measure is closely related to coherence, but it removes ambiguities of interpretation that detract from the latter.