definitions made, the complex DFT Z^ can be inverse transformed to recover a 

 complex array for which the real part is the time series x^ and the imagi- 

 nary part is the second time series y^ . 



36. In summary, two time series can be synthesized from two distinct 

 spectral density definitions using a single algorithm. Complex numbers are 

 formed from the Fourier coefficients of the time series. For one time series, 

 the complex Fourier coefficients are formed from 



'^-"--(^]'"^-"""'''°'" ^-^-^ I <"' 



and for the other time series, the Fourier coefficients are 



Equations 59 and 60 are then combined to define the DFT of the complex sum of 

 the two time series in the form 



and 



= I [(A.k - iBxk) + i(V - iV)] ^ = 2, 3, 



ZN-k+2 = I [(Axk - iBxk) - i(V - iByk)] k = 2, 3, 



N 

 ' 2 



(61) 



(62) 



with Z^ = Z[j/2+i = arbitrarily assigned without loss of generality in the 

 present application for the mean values and Nyquist frequency values, respec- 

 tively. Finally, the complex array Z|^ (k = 1, 2 N) is inverse 



Fourier transformed to recover the time series through the sum 



Xn + iyn = \ Z^ ^^i^■^v^-rn^-^)m n = 1, 2 N (63) 



k=l 



In Equation 63, the real part is the time series v^ and the imaginary part 

 is the time series y^ . Where a collection of such time series (or their 

 properties) is useful for statistical purposes, the above algorithm can be 

 applied repeatedly, with the procedure providing two new time series in each 

 application. Computer techniques known as Fast Fourier Transforms enable sums 

 like those on the right sides of Equations 19, 20, and 63 to be computed very 



21 



