Digital Signal Processing Reference
In-Depth Information
THEOREM 8.1.
Let the following partition be
[
]
[
]
[
]
=
,
Λ
=
diag
λ
…
λ
Λ
=
diag
λ
…
λ
VVV
,
,
,
,
,
,
s
b
s
1
P
b
P
+
1
M
where
V
s
and
V
b
are of respective dimensions M
×
P and M
×
(
M - P), with
P <M. Using the hypothesis that the covariance matrix
s
Γ
of the amplitudes of the
complex sine waves is of full rank P, the minimum eigenvalue of
x
Γ
is
σ
2
and is of
multiplicity M - P and the columns of
V
s
span the same subspace as
A
, i.e.:
σ
Λ
=
=
I
b
{} {}
{}
esp
V
esp
A
[8.13]
s
b
{}
esp
V
⊥
esp
A
where
⊥
indicates the orthogonality.
DEMONSTRATION. In the noise absence, the covariance matrix
x
Γ
is written:
H
Γ
=
AA
Γ
[8.14]
xx
ss
The matrices
A
and
s
Γ
being of the respective dimensions (
M
,
P
)
and (
P
,
P
),
with
P
<
M
,
the matrix
x
Γ
of dimension (
M
,
M
),
is of rank
at most equal to
P.
In
Theorem 8.1 we suppose that
A
and
x
Γ
are of full rank, therefore
x
Γ
is of rank
P.
On the other hand,
x
Γ covariance matrix, being a non-negative defined hermitian
matrix, has eigenvalues which are real and positive or null. The hypothesis
P
<
M
implies that
x
Γ
has
M - P
null eigenvalues and
P
strictly positive eigenvalues.
Let
v
be an eigenvector of
x
Γ
associated with the null eigenvalue; then:
H
Γ
vA Av
=
Γ
=
0
[8.15]
xx
ss
The matrices
A
and
x
Γ
being of full rank, it comes:
H
Av
=
0
implying that
v
belongs to the kernel of the application associated to the matrix
A
H
,
given
esp
{
A
}
⊥
, the subspace orthogonal to the subspace spanned by the columns of
A
. If we call
esp
{
V
b
}
the set of the eigenvectors associated with the null
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