Subspace-based Estimators - Spectral Analysis: Parametric and Non-Parametric Digital Methods - page 219

We then introduce:

()

⎡ x k ⎤

∼
=Α

() () ()

z = s + n [8.64]

k k k

⎢ ⎥

( )

x k + 1

⎣ ⎦

where:

()

⎡ ⎤ ⎡ b k ⎤

A

()

A = and n k = [8.65]

⎢ ⎥

⎢ ⎥

( )

b k + 1

A Φ

⎣ ⎦ ⎣ ⎦

It is thus the structure of the matrix A of di mension 2 MP × P , which will be
exploited to estimate Φ without having to know A .

It is easy to see that the covariance matrix Γ zz of all the observations contained in
z ( k ) is of dimension 2 M × 2 M and is written:

⎡

()() H

⎤

H2

Γ = Ekk zz =AA Γ + σ I [8.66]

zz

⎢ ⎥

ss

⎣ ⎦

where Γ ss is the covariance matrix of dimension P × P of the amplitudes of the
complex sine waves and I is the identity matrix of dimension 2 M × 2 M. Thus, the
structure of the covariance matrix is identical to that of the observation [8.4] and we
can then apply the theorem of the eigendecomposition and the definition of the
signal and noise subspaces. In particular, let V s be the matrix of dimension 2 M × P
of the eigenvectors of the covariance matrix Γ xx , associated with the eigenvalues
which are strictly superior to the variance of the noise σ 2 , it results that the columns
of V s and the columns of A define the same signal subspace. There is thus a unique
operator T as:

s V T

[8.67]

By decomposing V s as:

V

⎡⎤⎡ ⎤ AT

x

V = = [8.68]

⎢⎥⎢ ⎣ ⎦

s

V

A Φ

⎣⎦

y

where the matrices V x and V y are of dimension M × P , we can see that the subspaces
esp V x = esp V y = esp A are the same. Let Ψ be the unique matrix P × P of
transition from the basis of the columns of V x to that of the columns of V y , we have:

VV Ψ [8.69]

y x