1. 



Ogilvie 



A PROCESSING SCHEME FOR THE ESTIMATION 

 OF FREQUENCY RESPONSE FUNCTIONS 



Beginning with the work of Blackman and Tukey [l], many contributions 

 have been made to the problem of the estimation of the statistical characteris- 

 tics of time series. Most of them, however, treat mainly the estimation of the 

 spectrum, and very few have been concerned with the frequency response itself. 

 For the sake of establishing a standard procedure for obtaining the frequency 

 response, our group did some work [2] that followed the results of Dr. Akaike 

 and myself [3]. 



Skipping over the items that are already commonly clear in text books, 

 several items closely related to the selection of parameters in the estimation 



of the frequency response fvinction will be 

 mentioned and examples for the case of ship 

 oscillations will be given. 



Choice of m 



If m is the maximum number of lags in 

 the correlation fvinction used for the compu- 

 tation of the estimates, then m should be 

 chosen so as to satisfy 



mAt > 27T/B , 



where B is the bandwidth of the peak of |H(a)) 

 of main concern as defined by Fig. 1. 



An excessively large m implies an in- 

 crease of sample variance, and an exces- 

 sively small ra implies that bias occurs and 

 usually gives an underestimate of |H(aj) | . 



Also, the amount of shift, mentioned later, should be taken into account in 



choosing m. 



For the case of ship rolling where 



4> + 2ku>4> + '^J'^ = ^o ^^ ' 



under the condition where k « l, the bandwidth of the peak of |H(w) | is calcu- 

 lated as B = 2kco. 



Figure 1 



The Effect of Windows 



Before starting to discuss the choice of the amount of shift, the effect of 

 various windows is examined approximately. 



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