Estimation in Spectral Analysis - Spectral Analysis: Parametric and Non-Parametric Digital Methods - page 89

as

that (

)

− 1

θθ ( )

ˆ

a − ∼ 0 Σ , dlim a = 0

then the result 3.8 helps us

NN N →∞ N

immediately conclude that:

as

( )

( )

− 1

ˆ

T

( ) ( ) ( ) ( )

aSf , θ − Sf ∼ , θ 0 , d f , θ Σ d f , θ

Nx x

N

( ) ( )

where

d θ θ θ . Using similar reasoning and by basing ourselves
on the corollary 3.1, Friedlander and Porat suggest in [FRI 84b] the use of:

f , =∂ S f , / ∂

x

T − 1

( ) ( ) ( )

d

f

,

θ

F

θ

d

f

,

θ

N

where the Fisher matrix is obtained from Whittle's formula [3.7], as the lower bound
to the variance of (

ˆ

)

Sf θ :

,

x

N

− 1

⎧

T

⎫

( ) ( )

∫ d

f , θ d f , θ

2

1/ 2

( )

⎪ ⎪

ˆ

( )

T

( ) ( )

, θ , θ , θ

var

Sf ≥ d f f d f

⎨ ⎬

x

N 2

N

( )

− 1/ 2

Sf , θ

⎪ ⎪

⎩ ⎭

x

Results 3.6, 3.7 and 3.8 also help analyze the performance of a large class of
estimators based on ˆ c The aim of this section is not to be exhaustive but rather to
focus on an analytical approach; we will illustrate the latter on two specific cases:
1) estimation of parameters of an AR model;
2) estimation of the frequency of a noisy cisoid by subspace methods.

3.4.1. Estimation of parameters of an AR model

We thus analyze the estimation of parameters of an autoregressive model and the
estimation of the frequencies of the poles of this model. Let us suppose that x ( k ) is
an autoregressive process of order ρ whose power spectral density may be
written as:

2

σ

()

Sf =

x

() ()

Af A f *

p

=+ ∑

()

− j 2 π f

Af 1 ae

k

k = 1