Understanding and Prediction of Ship Motions 



Figure 3 



iw^h{-^(m-^)} = 



This window also causes bias in phase estimation. Generally, however, /3 is 

 much larger than a, when \}i(co)\ has a peak, and then da(oj)/dco usually has a 

 large value. Accordingly the effect of (a/27T)w(k^) is rather small compared to 

 that of w(k ) and [/3/(277)2]w(k ) . This shows that the effect of a window on the 

 phase shift, cr(aj), is rather small compared to the effect on the amplitude gain, 



|H(a))|. 



If we shift the data window by K and bring the origin of the window to k , 

 then w(t) = and w(r) = o. Moreover if we can choose the shape of the window 

 so as to have w(t) = 0, then, as the result, the bias due to phase shift can be 

 eliminated completely. 



As the natural results of the above-mentioned considerations, the following 

 results are obtained. In these results sAt is the amount of shift of the data 

 window for the computation of the sample cross spectra. 



Choice of sAt 



To compensate for the bias due to the phase shift, sAt should be chosen so 

 that we have 



sAt 



I dco ) 



for W, 



namely if 



dcr( O)) 

 dw 



101 



