Zk = Xk + iYk 



= I (^ + iyj e-^[2.(k-l)(n-l)/Nl k = 1, 2, ..., ^ 

 n=l ^ 



= I (A,, - iB,k) + i| (V - iByk) k = 1, 2, ..., I (54) 



where Equations 32 and 34 were used to obtain the last line. The complex 

 conjugate of element N - k + 2 of Equation 54 has the form 



^N-k+Z ^ •XN-k+2 ■ ^^N-k+Z 



= y (x„ - iy ) e^'^"^'*''^'^^"^*^'''^^'"^ 



n=l 



= f: (x„ - iyj e-^[2''<'^-^)<"-^>"'l k = 1, 2, ..., I (55) 



n=l ^ 



where use is made of the identity e^'^""^^""-"-'"^' = 1 . Again using Equations 32 

 and 34, Equation 55 can be written 



z;-k.2 = I (A.k - iBxk) - i f (\k - iByk) k = 1, 2 \ (56) 



35. Adding Equations 54 and 56 and multiplying by 1/N yields an 

 expression for just the Fourier coefficients of time series x„ , i.e., 



Axk - iBxk = I [Zk + 'Ll-vJ^ (57) 



Subtracting Equation 56 from Equation 54 and multiplying by - i/N yields an 

 expression for just the Fourier coefficients of time series y„ in the form 



Ayk ■ i^yk ~ N l^*^ ' ^N-k+zl 



(58) 



Equations 57 and 58 can be inverted to recover Equation 54 representing Z^ 

 for k = 1, 2, ..., N/2 and the complex conjugate of Equation 56 representing 

 Zk for k = N/2 + 2, N/2 + 3, ..., N . The k^^ and B^,, for time series 

 Xn are assigned values from user-defined spectral densities Sj^^ and a set 

 of random phases l-K\:i^^\Q ,\\ following Equations 49 and 50. A similar set of 

 relations defines the Ay^ and By^ for time series y„ . With these 



20 



