Assume a sinusoid with frequency given as: 



2t7(tii + 6) 



^ = — NKi ' 



(6) 



where At is the interval of time between samples, |6| is less than 

 or equal to 1/2, and NAt = T, the record duration. 



Equation (6) provides for assigning frequencies which differ from the 

 spectral frequencies. The contribution to the variance at the spectral 

 frequencies of the sampled record is given by S^ as : 



where a^ and b^ are the Fourier coefficients. 



Harris (1974) showed that for values of m near in (i.e., for spec- 

 tral frequencies near the frequency of the sinusoid) , and for in far 

 removed from one and N/2, the approximations below are good estimates 

 to the coefficients. 



• A sin t t6 cos ((j) - tt6) 



"^m "^ p: rr > 



TT (m - m + o) 



. A sin 7t6 sin(i^ - it6) 



bffl 77 • (83 



T7(.m - m + o) 



Slow convergence of the energy toward the spectral period closest to 

 the assigned period is clearly indicated. Thus, the energy is spread 

 over adjacent spectral periods. This spreading, due to the finiteness 

 of the record, is usually referred to as spillover. 



The technique routinely used at CERC to decrease spillover is to 

 apply the cosine bell data window as defined by: 



h-k 



1 - COS ^ 



y^ , i = 1, . . . N , (9) 



where y^ are the values in the original record. The Fourier coefficient 

 for the resulting function yi, is given by: 



'^ ^ A sin tt6 cos(({) - ttS) 



27r(m - m + 6) [(m - m + 6)2 - i] 



A sin T76 sin((|) - 7t6) 

 2iT(m - m + 6)I(in - m + 6)^ - 1] 



Thus, convergence is greatly increased and spillover is effectively 

 reduced to three adjacent spectral periods. 



22 



