available, but sometimes they do not say anything about “windows.” We have to be care- 
ful in our choice of the window to be used in the analysis to obtain a reliable spectrum. 
We can cut the number of computations from the order of N° for autocorrelation 
methods, N being the number of data, to N(r; +r2+... rg) for the F.F.T. method when 
N is factored toN =7r,-r2-... . 7g. Very commonly, N = 2? is used and then the num- 
g: Very y 
ber of computations is on the order of 2pN for F.E.T. For example, when N = 1024 = 2!°, 
the number of computations is reduced from the order of 1,000,000 to 20,000, or about 
1/50. 
The spectrum function is a powerful expression for showing the characteristics of a 
time series. However, we sometimes need the autocorrelation function to find the proper 
pair of lag and spectral windows, to decide the size of M compared with N, or to find an 
adequate statistical model from which we can go to the parametric analysis of the pro- 
cess, as will be mentioned in detail in Part II, Chapter 5. 
The autocorrelation function R(s) can be obtained as the Fourier transform of 
s(@) or as the periodogram Py(w) where, from Eq. 2.90, 
S(@) = | Pn) Wn(w —-¢) do. 
I- 
From Egs. 2.64, 2.63, and 2.66, 
Py(wp) = |Fx(wp)l’, (2.64’) 
ih pee N 
F,(p) Tai X, eM, p=0,1, ... > (2.63”) 
t=1 
1 N-1 . 
Py@,)=— > Rr) er’, : (2.66’) 
r=—(N-1) 
N 
Analogously, from Eq. 2.64’ or from Eq. 2.66’, we see that He ordinates of Py(w,) it 
is impossible to get N values of R(r),r =1, ...N. We get only R(r) values at r < ~. 
In order to get R(r) values at r = 0 toN, we need N ordinates of the spectrum or the 
47 
