biased by (I7l|/N) R(r). WhenN — ©, E[R(r)] — R(r), and is therefore asymptotically 
unbiased. 
By statistical mathematics, we can get 
var. [R°(r)] > O(1/(N=Irl)) (2.25) 
var. [R(r)] > O(1/N). (2.26) 
1. When r is small relative to N, the difference between R%(r) and R(r) is 
small, and the bias of [R(r)] is also small. 
2. When r becomes large relative to N and approaches N-1, 
bias [R(n)] — R(r). However, whenr > ~, R(r) — 0. Therefore, when 
N is very large, the bias remains small at all r. 
3. When r > (N-1), 
var. [R(r)] — 0(1) (2.27) 
var. [R(r)] — O(1/N). (2.28) 
Therefore, the tail of correlation Rr) shows a wild and erratic behavior. Also, from 
Statistical mathematics, cov. [R(r) R(r + s)] was computed and fairly high correlations 
between neighboring points were found when s is small. When 7 tends toward infinity, 
R(r) — O. It can be concluded here that Rr) will be less damped than R(r) and will not 
decay as quickly as R(r). 
2.4 SPECTRUM ANALYSIS 
The spectrum function decomposes a time varying quantity into a sum (or integral) 
of sine and cosine functions. 
2.4.1 Spectrum for Various Processes 
a. For Deterministic Periodic Functions—Fourier Series 
For periodic functions with period T, 
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