Digital Signal Processing Reference
In-Depth Information
Appendix A: Some Properties of the
Correlation
Matrix
Let us first consider the context of the linear combiner as described in
Figure A.1 and the general case of complex-valued signals.
Let
x
denote the input vector of the linear combiner. The correlation
matrix associated therewith is defined as
(
n
)
E
x
x
H
R
=
(
n
)
(
n
)
(A.1)
where
E
[
] is the statistical expectation operator. A first remark is that the
correlation matrix is a
K
·
×
K
square matrix,
K
being the number of input
signals that compose
x
(
n
)
.
A.1 Hermitian Property
In accordance with (A.1), it is straightforward to verify that the matrix
R
must be Hermitian, as the correlation between, for instance, inputs
x
1
(
n
)
and
x
2
(
n
)
, is equal to the complex conjugate of the correlation between
x
2
(
n
)
and
x
1
(
n
)
. In other words,
E
x
1
(
)
=
E
x
2
(
)
∗
x
2
(
x
1
(
n
)
n
n
)
n
(A.2)
A.2 Eigenstructure
First, let us present the classical definition of eigenvalues and eigenvectors
of a matrix
R
:
Rq
i
=
λ
i
q
i
for i
=
1,
...
,
K
(A.3)
where λ
i
and
q
i
represent, respectively, the eigenvalues and eigenvectors
of
R
.
The
K
eigenvectors
q
1
,
,
q
K
are orthogonal to each other, thus form-
ing a suitable basis for signal representation, which establishes the so-called
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