FT. DEG. SEC. CROSS SPECTRUM 
WITHOUT B.K. 
V=0 
SPECTRUM II 
m= 40---e-- 
ao (FEET)*-SEC. 
SPECTRUM AUN a CeO 
VANES WITHOUT BK. 
vV=0 
SPECTRUM II 
ONLY m = 60 WITH SHIFT 
m = 40, 120 WITH— 
OUT SHIFT 
te) 1 2 3 4 5 6 7 8 9 10 tls) 
—_— SEC 
Fig. 3.6. Auto and cross spectra of wave, roll, rol-ewave (run 832) 
(m=60 with shift, and m=40 and 120 without shift). 
(From Yamanouchi.3") 
diagonal and is symmetrical around the diagonal. The solution of Eq. 3.9 gives g,, over 
the range u =—n to n. This author [Yamanouchi*?] believes this method provides the lag 
window free estimate of the impulse response function, or the spectral window free esti- 
mate of the frequency response function from Eq. 3.6. The choice of the lag window or 
the spectrum window is a serious problem in getting a good estimation of the spectra and 
the frequency response relations of the output to the input, as was discussed in Sections 
2.5.2—2.5.6. This method eliminates the problem of windows and frees the calculation 
from their blurring effect. The use of Eq. 3.9 might eliminate also some of the uncertain- 
ties and errors in the algorithm for the correlation. 
Figures 3.8—3.11 show the analysis of the rolling of model ships afloat in irregular 
beam seas. g, for—90 to 90 was analyzed using Ryx(w) of — 90 to 90 and Ryyx of 0 to 
180. The correlations in Fig. 3.8, where the cross correlations are already shifted by 9, 
are shown in normalized form and the calculated g,, are shown at the top of Fig. 3.9 for 
u of — 30 to + 30 only, though the g,, were computed for u of — 90 to + 90. For compari- 
son, the impulse response function obtained as the Fourier inverse transform of the 
frequency response function calculated by cross and auto spectrum analysis is shown at 
the bottom of Fig. 3.9. They look very similar. 
The frequency response function obtained as the Fourier transform of the impulse 
response function g,, calculated by this method is shown in Fig. 3.10 together with the 
results of cross and auto spectral analysis. Again, the frequency response function 
73 
