CHAPTER 3 
CONSIDERATIONS ON THE IMPROVEMENT OF COHERENCY FUNCTIONS 
3.1 INTRODUCTION 
Coherency functions y(w) can be considered good clues to finding the extent to 
which the system can be approximated as linear to the input X, which can also be consid- 
ered linear. If there is noise in the output Y,, as was discussed in Section 2.6.3, the 
coherency function will be reduced, usually to coherencies of less than 1. It is difficult to 
find the real reason for this reduction, although Eqs. 2.175 and 2.176 tell us it is the effect 
of noise. Here not only noise in the output but also computational error, statistical bias, 
nonlinearity of the response characteristics, and the effect of feedback are all counted as 
noise in the result. All the effects that cannot be accounted for by linear, open relations 
are counted as noise. The author has made a few suggestions for improving the nonpara- 
metric spectrum analysis and the resulting computation of coherencies. 
3.2 SHIFT OF THE OUTPUT IN CALCULATING THE CROSS SPECTRUM 
R;(r), the auto correlation function of a real process i(f), is an even function and 
shows a maximum value at r = 0, the origin of the lag r. The lag window applied to this 
auto correlation to get consistent estimates of the spectrum ordinates is also usually an 
even function and has its peak at r = 0, and w(0) = 1, as in Fig 2.11. Accordingly, R(0) 
remains unchanged and then gradually decreases toward r=+ M. This procedure keeps 
the most reliable and important part of the correlation near r = 0 almost unchanged and 
reduces the contribution of the correlation at larger values of r near + M, where the 
correlation is less reliable, to nearly zero, preventing the formation of large negative lobes 
in the spectral window. This way is reasonable to get consistent estimates of the spectral 
ordinates and make the spectral window fulfill conditions 1—5 in Section 2.5.4. 
However, in getting the cross correlation, if we use lag windows in the same way, 
sometimes important information is lost, and the result is an apparent reduction in coher- 
encies. This was studied theoretically by N. Akaike and Y. Yamanouchi”? (1962), but a 
more intuitive explanation by this author [Yamanouchi*° (1961)] is given here. 
In cross correlation R;,(r), if the phase relation of the output j(r) lags or leads the 
input i(r) considerably, the maximum value of the cross correlation, which is not the even 
function, will no longer lie at the origin r = 0 but will lie at a larger or minus value of r 
(distant from the origin). In applying lag windows, if we set the maximum of the lag win- 
dow at r = 0, sometimes important information is lost at the peak of cross correlation. A 
small value of the lag window will be multiplied to the peak value of the cross correlation 
at a larger or minus value of r, and will keep the less important part of the cross correla- 
tion almost unchanged by the windows, as shown in the example in Fig. 3.1. This is not 
reasonable, so the author suggested shifting the origin of the cross correlation to its peak 
point and then applying the lag windows. The amount of the change of phase from this 
shift ro should, of course, be used later to modify the phase relation by row. 
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