Subspace-based Estimators - Spectral Analysis: Parametric and Non-Parametric Digital Methods - page 216

Let us note that the bigger M - P compared to P , the less constraining this
additional hypothesis on A 2 , i.e. when the number of observations is bigger than the
number of frequencies to estimate.

The estimation of the noise variance is obtained using the following comments.
By partitioning Γ xx of [8.4] in accordance with:

GH
GH

}
}

P
M

⎡ ⎤

1 1

Γ = ⎢ [8.52]

⎥

xx

− P

⎣ ⎦

2 2

where G 1 , G 2 , H 1 and H 2 are matrices of dimensions P × P , ( M - P ) × P , P × ( M -
P ) and ( M - P ) × ( M - P ) respectively, we have:

H 2

⎧ =

GA A Γ + σ I

1 1 ss 1 P

⎪
⎪ =

H

GA A Γ

⎪ ⎨

2 2 ss 1

[8.53]

H

HA A
HA A

= Γ

Γ

⎪ ⎪

1 1 ss 2

H 2

= + σ I

⎪ ⎩

2 2 ss 2 M − P

It is then easy to see that:

{ }

{}

tr H Π

Π

2 2

σ = [8.54]

tr

† †

where I GGI A Π and where tr {.} indicates the trace

operator of a matrix and † the pseudo inverse.

= − = −

MP − 2 MP − 2

2 2

A possible estimation of σ 2 is thus the following [MAR 90b, MAR 97]:

{ }
{}

⎧ ˆ ˆ ⎫

tr H Π

⎪ ⎪

2

2

ˆ

σ = ⎨ Ré [8.55]

⎬

ˆ

tr Π

⎪ ⎪

⎩ ⎭

ˆ = IGG ˆˆ

†

ˆ HG enter the partition of ˆ ˆ x Γ :

where Π and where

2 2 22

ˆ ˆ

⎡ ⎤

GH

ˆ

1 1

Γ = ⎢ ⎥ [8.56]

xx

ˆ ˆ

⎢ GH ⎥

⎣ ⎦

2 2