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As is well known, coherence does not distinguish the relative phase of a pair of real, sinusoidal time series; the coherence between them is always unity. This behavior can limit the applicability of coherence analysis in the special case where the time series are band-limited (nearly-monoch- romatic) and where sensitivity to phase differences is advantageous. We propose a simple mod-ification to the usual formula for coherence in which the cross-spectrum is replaced by its real part. The resulting quantity behaves similarly to coherence, except that it is sensitive to relative phase when the signals being compared are strongly band-limited. Furthermore, it has a useful interpretation in terms of the zero-lag cross-correlation of real band-passed versions of the time series.

In this paper, we examine the well-known formula for the frequency-dependent coherence

The issue considered here is how best to quantify the similarity between time series that are 1) real (as contrasted to complex) and 2) band-limited (in the sense of being nearly monochromatic). Such time series constitute important special cases because most natural phenomena are described using real numbers and many are dominated by a single period of oscillation. For example, the daily period often contributes strongly to physiological and meteorological signals, the annual period to environmental and climatic signals, the precessional period (25.7 ka) [

An important property of nearly-monochromatic signals is their relative phase. Whether two time series that are in-phase (as in

Traditional coherence analysis [

When asking why any quantity encountered in time series analysis, such as coherence, behaves in a certain way, one must contend with the fact that most, if not all, such quantities can be derived from several different perspectives. Any answer will probably make sense only from one of these points of view. Consider, for example, the estimated mean of a time series. This deceptively simple quantity can be understood, alternately, as arising through the minimizing of error (a deterministic derivation) [

The coherence between two time series, at frequency,

This is but one perspective among many, but one we find helpful because it brings out a relationship to the cross-correlation, another quantity useful in assessing the similarity between two time series. Cross-correlation is defined in the time-domain, as contrasted to coherence, which is defined in the frequency-domain, so the link provides complimentary information.

The appearance of a one-sided filter (

frequency components of the time series behave completely differently from one another. This is contrast to real-time series, where they are complex conjugate pairs. However, in being general, it cannot exploit an important property of real signals: that sines and cosines are distinguishable from one another. As we show below, substituting a two-sided filter produces a version of coherence that distinguishes sines from cosines; that is, one that is sensitive to the relative phase of band-limited signals.

The problem we consider is how to quantify the similarity of two real, transient time series,

The convolution,

Note that at zero-lag, cross-correlation is just the area beneath the product of the two time series:

Note also that definition of the convolution and cross-correlation in (1a), (1b) differ only by a sign of

At zero lag, the cross-correlation is proportional the integral of its Fourier transform,

Inserting (3) into (4) and using the rule that the Fourier transform of a convolution is the product of the transforms ([

Here

is defined for

The normalized measure of similarity, say

Note that the quantity,

As asserted in the Introduction, the usual formula for coherence can be obtained simply by switching to a one-sided filter, a single unit step function of width

As an aside, we note that our derivations of

Suppose that time series

The similarity,

Thus,

The coherence,

We then find:

This is the well-known result that the coherence,

We consider the example of a sequence of nearly-monochromatic wavelets, formed by taking the product of a phase-shifted sinusoid of frequency,

and then by adding a small amount of uncorrelated random noise. Figures 3(a)-(c) illustrate pairs of these wavelets with different phase relationships. Note that the wavelets are not merely time-shifted versions of one another, since the position of the zeros crossings of the sinusoid (parameterized by

We have not performed an exhaustive analysis of the differences between

where the Fourier transforms are written in terms of their real and imaginary parts,

We might expect in the case that

In summary, we recommend this simple modification of coherence in cases where the time series that are being compared are narrow-band and where phase relationships between them are considered important. For pure sinusoids differing by phase,

The rule that the transform of

Here we have utilized the substitution