njit-etd2003-081 - New Jersey Institute of Technology

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3.10 Coherence Analysis
Within the present framework, the dependence between two signals can be characterized
by parameters, which assess the correlation between the signals. In the frequency
domain it is customary to consider the magnitude squared of the correlation between the
Fourier transforms of the two signals under consideration. For the bivariate point
processes (N0 , N 1 ) this leads to [61][62]
as a measure of the correlation between processes N 0 and N 1 . This quantity is called
the coherence function [63], denoted by 1710 (f )1 2
, estimates of which provide a measure
of the strength of correlation between N 0 and N1 as a function of frequency. The
definition of the correlation corr{FN1(ω),FN0 (ω)} between the Fourier transforms of the
two point processes N 0 and N 1 in terms of variance and covariance, given by the
leads to the alternative definition for the coherence function between point processes N 0
and N 1 as
Coherence functions provide a normative measure of linear association between
two processes on a scale from 0 to 1, with 0 occurring in the case of independent
processes [61][62]. Equation (3.84b) leads to an estimation procedure by substitution of
the appropriate spectral estimates to give