njit-etd2003-081 - New Jersey Institute of Technology

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after removal of the effects of process y from each, may be written (suppressing the
dependencies on ω) as [61, 62, 64, 66].
Expansion of this expression in a similar manner to (3.84a) leads to equation
(3.89). The two terms (Pxy/Pyy ) and (P0y/Pyy ) represent the regression coefficients
which give the optimum linear prediction of Fx (ω) and FNo(ω)), respectively, in terms
of Fy (ω). Estimates of |γx0/y(ω)|^2 test the hypothesis that the coupling between N 0 and
x can be predicted by process y, in which case the parameter will have the value zero.
The partial coherence defined in (3.89) and (3.91) is a first order partial coherence,
which examines the correlation between two signals after removing the common effects
of a single predictor. This framework can be extended to define and estimate partial
coherence functions of any order. Full details, including estimation procedures and the
setting of confidence limits can be found in [64].
For further in-depth details about these calculations, see Bendat and Piersol [50].
For all the partial coherence equations used in the LabVIEW program, refer to Appendix
C.