Understanding and Prediction of Ship Motions 



As an example, these windows can be applied to the estimation of the frequency 

 response of a simple oscillation such as linear rolling. 



H(a;) = 



Wj, W2, W3 have been checked to be adequate for use if 



p 



Koj mAt 



is kept less than 1. (Usually this is satisfied if mAt > 2VB, where 



2km , 



p = 



1 ^ _77 B_ 



mAt - Kco 277 



-- 1) 



The window Q, which is a modification of Wj, is generally recommended for the 

 estimation of the power spectra, cross spectra and of the frequency response 

 function of a linear time -invariant system. 



If the very careful estimation is necessary, the following is recommended. 

 Apply the windows Wj, Wj and W3 successively, and if there is a significant dif- 

 ference, say, of order greater than 10% between the results, it is advisable to 

 repeat the whole computing process using 2m in place of m, and, at the same 

 time, use a correspondingly increased N, if necessary. W3 tends to produce the 

 deepest troughs and highest peaks, Wj the next deepest and highest, and Wj the 

 shallowest. 



R(a^): RELATIVE ERROR OF H(co); CONFIDENCE BAND 



The relative error of the estimation of H(a;), obtained by the procedure 

 mentioned above was evaluated relative to the estimated value of coherency. 

 The results are as follows: Assuming that 



Y(aj) = X(aj) H(w) + N(«) 



105 



