N—o 
E [Pn(@)] = [so O0@-w) dp = sa). (2.80) 
When N is finite, E[Py(w)] is a kind of filtered spectrum, filtered by the Fejer kernel 
function F,,. 
Though the details of manipulation by statistical mathematics are not shown here, 
we know that the variance of the periodogram can be calculated as 
var. [Py(w)] is on the order of Sp), (2.81) 
27, N 
Oh Sa aL De Ue 
and also that Py(w) follow 1G , X° distribution with degree of freedom v = 2. As the 
mean and the variance of x arev and 2y (here 2 and 4, respectively), then the signal to 
noise ratio S/N is 
mean 2 
— ratio = = il. 2.82 
N ( ) 
standard variation _ (2 X 2)i/ 2 
Equations 2.80 and 2.81 can also be used to check Eq. 2.82. This S/N ratio of 1 means 
that this value of Py(@) is a very poor estimate of the spectrum. Further, the fact that 
var. [Py(@)] does not tend to 0, even when N tends to ©, means this estimate of 
E[Pn(@)] is not a consistent estimate of s(w). 
Furthermore, at two fixed neighboring frequencies w, and@ 2, cov [Py(w), 
Py(W2)] can be calculated to decrease by the increase in N. 
These facts are reflected in the erratic and widely fluctuating form of Py(w). 
Py(@) may produce a spurious peak in the region in which Py(w) is large, as shown in 
Fig. 2.10. 
25.2 Consistent Estimation of the Spectrum 
In the expression of the periodogram, 
27 
