Table 2.4. Coefficients o; for various windows. 
ee 
nies 683 
0.25 ae 0.25 
| o.008 | 0.657 | 0.164 | 0.008 | 0.008 
TUKEY 
GENERAL 
HAMMING 
HANNING 
PARZEN 
AKAIKE’S 
—W, 
—W, 
—W, 
— @ 
WIDE 
4 
NARROW 
25.7 Use of the Fast Fourier Transformation (F F.T.) Method 
There are misunderstandings sometimes that spectrum analysis by the Fast Fourier 
Transformation (F.F.T.) technique is completely different from getting the spectrum from 
analysis of auto correlation functions. However, Eq. 2.64 shows that this method is mere- 
ly the one that gives the spectrum through a periodogram, and Eq. 2.66 shows that it is 
the same as calculating the spectrum from the Fourier transform of the auto correlation 
function. 
Through the F.F.T. we get the spectrum ordinates at frequency = , or in the range 
. 2p Nig : 
-% = @ <1, at the frequency points Gee jo) 6s hes ys 65 ES Ao the dis- 
N 
crete case. This procedure corresponds to getting ms +1 ordinates in the frequency range 
0 toz, or N+1 ordinates in the frequency range —z to z. It is the same as using 
N 
ao for a rectangular spectrum window and as a result, the individual spectrum ordi- 
N 
nates of a +1 are so unreliable, statistically, and the variance so large that SNR = 100%, 
equivalent to a degree of freedom v = 2. Accordingly, the same considerations on the 
use of the spectral window as were necessary for auto correlation methods are necessary 
for the F.F.T. method after getting the “raw” ordinates at N+ 1 frequency points. Com- 
mercial programs or even a specialized spectrum analyzer through F.F.T. are now 
46 
