where 



exp (-jVl8) 



(32) 



These weights are, thus, Gaussian smoothers with a standard deviation of 

 a = 3Af = 0.00366 sec." . The numerical values of the w-: are given in 

 Table 2. In the vicinity of zero frequency, the subscript j in equation 

 (31) was summed over the possible values and the divisor normed the weights 

 appropriately. At m = 6, for example, j was summed over -6_<j<_13 (the 

 left tail of the moving average was truncated off). 



Table 2. Weights used in the moving average 

 of the spectral lines to produce 

 the estimates of the spectral density. 



j 



Wj 









1.0000 







1 



0.9460 







2 



0.8007 







3 



0.6065 







4 



0.4111 







5 



0.2494 







6 



0.1353 







7 



0.0657 







8 



0.0286 







9 



0.0111 







10 



0.0039 







11 



0.0012 







12 



0.0003 







13 



0.0001 





Note : w_ ■ = wj . 



The effective width of the Gaussian smoother in equation (32) is /2tt a 

 where o = 3Af, if lags are measured on the frequency scale and a =3, 

 if lags are measured relative to the number of spectral lines. The effec- 

 tive width in terms of number of spectral lines encompassed is thus 

 3 /2tt = 7.52, or rounding to the nearest integer, 8 spectral lines. By 

 equation (28) , the FFT spectral density estimates so produced would be 

 comparable to covariance spectral density estimates with maximum covariance 

 lag derived from: 



N 



(33) 



4,096 



512 



(34) 



17 



