Understanding and Prediction of Ship Motions 



Assume that the output y(t) and input x(t), as well as the noise n(t) that 

 contaminates the output y(t), have the Fourier transform. Then 



V (f m) = H (f m) ^ X (f m) * N (f .) . 



Accordingly, the estimate of R(co) at w = (27T/2T)/a evaluated from the cross 

 spectrum is 



2W Y • X 



V IX- V IX- V 



l2T / 2W X 



V 



IX- V fl- V 



^W H & i^i- V)} X„ ^ • X SW H • X 



2W X -X SW X ^^ 



V ix-v IX-V V ix-v ix-v 



where W^ are the weight factors that describe various windows. The noise and 

 the input can be considered as mutually uncorrelated. Therefore 



[H(|f.)] 



ZW h(-^ (/x- v)| -x -x 



^ ^ sw x ^"x 



V IX-V IX-V 



2T ^xx V2T ^j 



^ /27T 



'xx \2T 



Here the variation of s^^(a;) around w = (277/2T)^i is assumed to be smaller than 

 that of H(w). Accordingly 



^["(f -)] ^ f »."{#"'-■'>} 



Now let us consider the case where H(c^) in the form \y{(co) |e^'^^'") has the 

 local approximation 



«{!?'-■'>} = "(#''){-«{f*')^Mf-)'} 



^W^'^M 



where k^ = - dc7(aj)/daj as in Fig. 2. We have assumed that |H(a)) | varies locally 

 as a 2nd order curve, and that the phase cr(co) increases linearly with frequency. 

 Then, 



