Estimation in Spectral Analysis - Spectral Analysis: Parametric and Non-Parametric Digital Methods - page 94

()

' J ω can be done using analysis of the perturbation on
projection operators presented in [KRI 96]:

The calculation of

0

b

ΠΠΠ+ =+ δ

b b

By limiting ourselves to the first order, we thus obtain:

{ }

H

() () ()

J ' Re a Π d

[3.31]

ω ω δ ω

N 0 b

with:

( ) ( )

2 H

ˆ

# #

ˆ

()()

δ

ΠΠ =−

RRS

− − ∆ −

S RR

Π

, S=Aa ω ω a

b s s

where S # designates the pseudo-inverse of S . By substituting [3.31] and [3.30] in
[3.29] we thus obtain an expression of the error on ˆ ω according to ˆ RR which
helps link the variance of ˆ ω to the variance of R . Then, the asymptotic arguments
(for example. results 3.6, 3.7 and 3.8) help complete the calculation of the
asymptotic variance of ˆ ω (see [STO 91 ] for a detailed analysis).

3.5. Conclusion

In this chapter, we studied the methods of analysis of estimators commonly used
in spectral analysis. The approach consists most often of linking the estimation
errors on the parameters θ to the estimation errors on the moments of the signal and
use the asymptotic arguments based on Taylor expansions. If this approach does not
cover all the cases envisaged, it provides answers to a large number of problems of
spectral analysis. See the topics mentioned in the bibliography for information on
how to tackle other possible cases.

3.6. Bibliography

[AND 71] ANDERSON T., The Statistical Analysis of Time Series, John Wiley, New York,
1971.

[BRI 81] BRILLINGER D., Time Series: Data Analysis and Theory, McGraw-Hill, 1981.

[BRO 91] BROCKWELL P. J., DAVIS R. Α., Time Series: Theory and Methods, Springer
Series in Statistics, Springer Verlag, Berlin, 2nd edition, 1991.