Estimation in Spectral Analysis - Spectral Analysis: Parametric and Non-Parametric Digital Methods - page 69

that is to say:

1 ⎧ 1 ⎫

T

− 1

( ) () () ()

p x ; θ = exp − ⎡ x − s θ ⎤ R θ ⎡ x − s θ ⎤

⎨ ⎬

⎣ ⎦ ⎣ ⎦

N N N N N N

1/ 2

N /2

2

() ()

⎩ ⎭

2π R θ

N

() () ()

with R N θ as the determinant of the matrix

R θ . s θ denotes the mean

N N

()

(which can be a useful deterministic signal) and

R N θ the covariance matrix. Thus,
by applying theorem 3.1, we can obtain the following simplified expression:

T

() ()

⎡ ⎤

⎡ ∂ s θ ⎤ ∂ s θ

N − 1 N

() ()

F θ = R θ

⎢ ⎥

⎢ ⎥

N N

ij

∂θ ∂θ

⎢ ⎥

⎣ ⎦

i

⎣

j

⎦

() ()

⎧ ⎫

∂ R θ ∂ R θ

1 Tr
2

⎪ ⎪

− 1 N − 1 N

() ()

+ R θ R θ

⎨ ⎬

N N

∂θ ∂θ

⎪ ⎪

⎩

i j

⎭

N
k

where {}

( )

= ∑ Tr A A kk , stands for the trace of the matrix A .

= 1

We now consider the two following cases.

Case 1
The signal is the sum of a useful deterministic signal ()

s N θ and a noise whose

()

covariance matrix

R θ = R

does not depend on θ.

N N

Case 2
The signal is a random process with zero mean [ () ≡

s N θ 0

] and covariance

()

matrix

R N θ .

Let us consider the first case. Thus, from the previous formula, we obtain:

T
N () ()

∂ s θ ∂ s θ

− 1 N

()

F θ = R

N N

T

∂ θ

∂ θ

as the covariance matrix does not depend on θ. Now let us concentrate on the
existence of an effective estimator. We can write:

( )

T

()

T

()

∂ ln p

x

; θ

∂

s

θ ∂

s

θ

N N − 1 N − 1

()

= R x

−

R s

θ

NN NN

∂ θ ∂ θ ∂ θ