in which 



2tt 2tt 2tt 



1 record Length T NAt 

 t- = jAt = a discrete time where j is the integer time step 

 F(n) = a(n) - ib(n) 



•(T^)--(T^r)..-*o,l 



a(0) = mean of sampled record 



b(0) = b(|)=0 



Because negative indexes are not readily handled by most FORTRAN compilers, 

 the summation extends over the interval < n < N - 1, rather than over the 

 symmetric interval - N/2 < n < N/2. From the definition of the coefficients 

 above, it is clear that the coefficients a(n) and b(n) for n > N/2 contain 

 no additional information. 



The inverse relationship completing the FFT pair is 



N 

 N 



1 

 F(n) = — I f(j) exp(-ina) jAt) 



XT . 1 



J=l 



As an enumeration of the complex FFT coefficients, suppose the series of 8 

 values is considered, N = 8. The coefficients would be 



F(0) = a(0) 



F(l) = a(l) - ib(l), F(7) = a(7) - ib(7) = a(l) + ib(l) 



F(2) = a(2) - ib(2), F(6) = a(6) - ib(6) = a(2) + ib(2) 



F(3) = a(3) = ib(3), F(5) = a(5) - ib(5) = a(3) + ib(3) 



F(4) = a(4) 



This pattern prevails for all sets of FFT coefficients, regardless of the 

 value of N. Both F(0) and F(N/2) are real and, as noted previously, the 

 coefficients F(n) for n > N/2 really contain no additional information. 

 The FFT subroutine used here requires that the number of data points, N, 

 provided be an integral power of 2, i.e., 



N = 2 K 



where K is an integer. Thus analyses of 512, 1,024, or 2,048 data points 

 (K = 9, 10, 11) would be suitable with this subroutine. 



23 



