,^ tr I \ ^ r27r(k - l)(n - 1) ,1 ,,„ . 



Cxk cos(27r4t„ + 4>^^) = C,^ cos|^-^ ^ '- + ^^^J (^8a) 



Equation 48a is recognized as the k'^'^ contribution of the Fourier series of 

 Xjj given by Equation 35. The Fourier coefficients A^^j^ and B^^ of Equa- 

 tions 35 and 37, respectively, then become 



^-•^ = (l^)' ' sin{2;rU,k[0,l]) (50) 



Using these Fourier coefficients in Equations 32 and solving for X^ yields 



N [(2 S V^ (2 S V/2 "1 



^'^ = 2 [[l^J ^°^f27rU,,[0,l]) -i l^-^j^J sin{27rU,,[0,l])J (51) 



as an expression for the k*^^ element of the DFT of time series x^^ . Equa- 

 tion 51 can be written in more abbreviated form by combining terms and using 

 Euler notation, which results in 



A similar set of steps can be created for time series y^^ . If the k"^"^ 

 component of the frequency spectrum of y^^ is Sy,^ and its k*^^ initial phase 

 is 2irUyi^[0,l] , the corresponding DFT element Y^ is given by 



34. By taking advantage of complex notation and certain properties of 

 DFT's, an efficient means of computing multiple time series is achieved. If 

 the DFT for time series y^^ (Equation 20) is multiplied by i and added to 

 the DFT for time series x^^ (Equation 19) and the k'^^ element of the result 

 is denoted as Z^ , then 



19 



