EjPN(w)] 
S(w) 
S(w) 
ve E[PN(w)] 
—— WwW 
Fig. 2.10. Behavior of E[Py(a)] . 
, Re, 1 N-1 
Pw) =— >  R(r) cos rw =—4RO) +2) Rr) cos rw}, (2.83) 
Dye 2m re 
r=—(N-1) r=1 
A = 1 
var. [R(r)] is on the order of O a 
and var. [Pa({q@)] is on the order of O(1) (2.84) 
are known. The reason for the large var. [Py(@)] is, as shown in Eq. 2.83, that Py(w) 
adds too many terms in TR. These terms are slightly correlated, but the basic effect of 
too many terms of YR) in Eq. 2.83 remains the same. 
Accordingly, the way to reduce this large variance in Py(w) is to reduce the number 
of additions from N to M and omit the term N > M in Eq. 2.83, 
M 
; 1 R 
E [So(@)] =— >. E [R()] cosra. (2.85) 
2 
r=-M 
This reduction decreases the variance, and from Eq. 2.84 intuitively we find 
var. [s,(@)] is on the order of 0(M/N). This can be proved by statistical mathematics, 
although the manipulation is not shown here. Substituting Eq. 2.24 into Eq. 2.85 gives 
28 
