njit-etd2003-081 - New Jersey Institute of Technology

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when there is significant correlation between the two signals. In practice
used to indicate the regions where Фx1(f) has a valid interpretation. Phase estimates
can be interpreted as the phase difference between harmonics of N 1 and x at frequency f.
The arctan function can be used to obtain the argument of the cross-spectrum, resulting
in a phase estimate over the range [—Tr / 2,71- /2] radians. However, the signs of the real
and imaginary parts of Px1(f ) can be used to determine in which quadrant the arctangent
falls, so extending the range to [
-π, π] radians.
Phase estimates can often be interpreted according to different theoretical
models. A useful model is the phase curve for two signals, which are correlated with a
fixed time delay, where the theoretical phase curve is a straight line, passing through the
origin (0 radians at 0 frequency) with slope equal to the delay, and a positive slope for a
phase lead, and negative slope for a phase lag (see [67]). In situations where there is
significant correlation over a wide range of frequencies and a delay between two signals,
it is reasonable to extend the phase estimate outside the range [-π,π] radians, which
avoids discontinuities in phase estimates. Such a phase estimate is often referred to as
an unconstrained phase estimate. The representation of phase estimates is discussed in
[66]. Different theoretical phase curves for other forms of correlation structure are
discussed in [67]. Details for the construction of confidence limits about estimated
phase values can be found in [64]. In situations where the correlation structure between
two signals is dominated by a delay it is possible to estimate this delay from the phase
curve. Such an approach, based on weighted least squares regression, is described in the
Appendix in [62]. This method has the advantage of providing an estimate of the