where h(nAt) is the impulse response function and w(nAt) is the hanning 

 window function. The final filter response function is defined as: 



F(f^) = L B(nAt)e-J2™Vt, 

 n 



where j = /^ and At is the constant time interval between samples. 



The transition band or the frequency increment traversed by the 

 cutoff of the filter function can be approximated by: 



A = ~- = 0.078 



and the maximum stopband attenuation for the hanning window is 55 de- 

 cibels. These values can only be achieved through proper filter design. 

 The actual values for the filters used are A =0.08 and a maximum atten- 

 uation of greater than 55 decibels. The ripples in the passband for each 

 filter used were below 0.01 percent. These values could be improved on ~ 

 by increasing the number of points used for the filter response function 

 estimate. Also the stopband attenuation could be improved, at the ex- 

 pense of a wider transition band for a given size filter function by 

 using the Blackman window function. However, the accuracy of the filter 

 response functions used exceeds that of the measurements and is suffi- 

 cient for this application. 



After initial processing and prior to all spectral calculations a 

 tapered cosine data window was applied to the first and last 10 percent 

 of the data to reduce spillover of spectral energy to adjacent frequency 

 points. For data stretching from n = 1 to n = N, the formulas for the 

 data window are: 



4 (1 - cos IT ^^) for 1 < n < O.IN 



O.IN-" 



for O.IN < n < 0.9N (14) 



cos IT 5^) for 0.9N < n < N. 

 O.IN — — 



The data were then transformed directly using fast Fourier transformation 

 procedures and smoothed by averaging adjacent raw spectral components. 

 Initial sampling was performed at 0.5-second intervals with 2,048 samples 

 per record, and 20 adjacent points averaged together in the autospectral 

 calculations to get the final smoothed spectral estimates. This gives a 

 frequency resolution of 0.0195 hertz with 40 degrees of freedom per spec- 

 tral estimate. 



All of the wave data was high-pass filtered, using the filtering 

 techniques previously outlined with a cutoff frequency of 0.05 hertz. 

 This was done to remove the tidal influence on the transmit5ed wave 

 staffs and to eliminate any possilbe buoy motion in the incident wave re- 

 cords. Also the anchor cable force data were separated into a low-and 



57 



