PART III: TEST DATA GENERATION 



26. In this study, wave height distributions are extracted from 

 synthetic time series that have specific spectral definitions in the frequency 

 domain. A very efficient computational scheme for synthesizing a time series 

 from spectra is by way of inverse Fourier transforms. In this chapter, this 

 technique and the nature of the spectra used in this study are described. 



Inverse Fourier Transform Technique 



27. Following the description by Bendat and Piersol (1971), a time 

 series of N water surface elevations or an artificial representation thereof 

 can be represented by the notation x^^ = ^C^n) = x[(n - l)At] , for n = 1, 2, 

 3, . . . , N - 1, N where ^ is the n*^"^ discrete sample of a continuous 

 function x(t) and t^ is the n"^^ discrete increment of time At after an 

 initial time of zero for the first sample (n = 1) . A second, independent, 

 N-point time series can be represented by the notation y^ = y(t^) 



= y[(n - l)At] , for n = 1, 2, ..., N . Each of these time series can be 

 subjected to a discrete Fourier transformation (DFT) . For time series x^^ , 

 the k*^^ element of the DFT is denoted by X^^ , and the whole set is defined by 



Xk = I ^ ^-2nlin-l)i^-l)IU k=l, 2, .... N (19) 



n=l 



For the second time series, DFT elements are denoted by Y;^ and are defined 

 by 



\= I y^ e-2^i(n-i)ck-i)/N k = 1, 2, ..., N (20) 



n=l 



28. Because each time series represents a piecewise continuous func- 

 tion, each can also be represented as a Fourier series of points. The first 

 time series can thus be written 



. - S {a. cos P"^ ■ ;;><" - ^>] 



. B,, sl„ p'<'= - »<" - ^> ]} „-l, 2 I (21) 



14 



