Digital Signal Processing Reference
In-Depth Information
Since the mean-square value of
y
(
n
)
is necessarily nonnegative, we have
w
H
Rw
≥
0
(A.9)
From (A.9), we conclude that the matrix
R
is positive semidefinite [139],
and, as a consequence, all of its eigenvalues are nonnegative.
A.3 The Correlation Matrix in the Context of Temporal Filtering
Some additional features may be commented about the structure of
R
when
the input vector is formed by delayed versions of a same signal, i.e., in the
case of temporal filtering.
As already presented, we have in such case
]
T
x
(
n
)
=
[
x
(
n
)
,
x
(
n
−
1
)
,
...
,
x
(
n
−
K
+
1
)
(A.10)
so that the elements of the correlation matrix are given by
E
x
)
x
∗
(
r
(
k
)
=
(
n
)
n
−
k
(A.11)
are a function
exclusively of the lag between samples, and do not depend on the time
instant. Consequently, the columns and rows of matrix
R
are formed from
a same vector
r
Since
x
(
n
)
is assumed to be stationary, the elements
r
(
k
)
]
T
, the elements of which are
r
=
[
r
(
0
)
,
r
(
1
)
,
...
,
r
(
K
−
1
)
(
k
)
,
for
k
,
N
, by performing subsequent circular shifts and appropriate
complex conjugation:
=
1,
...
⎡
⎤
)
∗
)
∗
r
(
0
)
r
(
1
···
r
(
K
⎣
⎦
)
∗
r
(
1
)
r
(
0
)
···
r
(
K
−
1
R
=
(A.12)
.
.
.
.
.
.
.
.
.
.
.
.
r
(
K
)
r
(
K
−
1
)
···
r
(
0
)
This fact implies the so-called Toeplitz structure of
R
in this case. This
property is particularly interesting in recursive methods for matrix inversion
and for solving linear equation systems. For instance, the Levinson-Durbin
algorithm, which recursively yields the coefficients of an optimal linear
predictor, is based on the Toeplitz structure of the correlation matrix.
