For the entire discrete process, 
N-1 
1 A 
E [Py@)]}=— > E [R(r)] cos ro, (2.67) 
27 
r=—(N-1) 
which, from Eq. 2.24 
N-1 
1 Ir| 
=— y (ese R(r) cos ro. (2.68) 
2m, (W-1) 
From these relationships, the spectrum computation from the periodogram is the 
same as the spectrum estimation from auto correlation. The Fast Fourier Transform gives 
the periodogram directly through direct Fourier transfer and is statistically the same as 
this periodogram analysis. Accordingly, care should be taken in selecting the spectrum 
windows to be used in the F-F.T. as will be discussed later. 
From Eq. 2.68 if N becomes ©, surely 
E [Pn(@)] > s(@) . (2.69) 
Then, from the periodogram, we can estimate the spectrum. Py(w) is an unbiased 
estimate of s(w). Here, however, R(r) is the theoretical auto correlation, and we think the 
spectrum is continuous and 
‘4 
Rr) = | so) cosrwdw , r=0,+1,+2,... (2.70) 
10 
Thus 
ge ae nlf 
E [Pa(w)] = — 1-— | suo cos ru du cos rw 
21 N 
r=—(N-1) 
—I 
rs 
tie ani 1 
= fis(i) = oe [N—tri}—{cos ru +) +cos r(u—w)} du. (2.71) 
2 N 2x SEI) 2 
Here we use the following relations on summations of digital quantities, which can 
easily be proved, 
24 
