Digital Signal Processing Reference
In-Depth Information
and the components of
C
are the
central second-order moments
.If
n
=1,
then
m
X
)
2
)=
C
X
var
X
:=
σ
X
:=
E
((
X
−
is called the
variance
of
X
. Its square root
σ
X
is called the
standard
deviation
of
X
.
The central moments and the general second-order ones are related
as follows:
R
X
=
C
X
+
m
X
m
X
.
Decorrelation and Independence
We are interested in analyzing the structure of random vectors. A simple
question to ask is how strongly they depend on each other. This we can
measure in first approximation using correlations. By taking into account
higher-order correlations, we later arrive at the notion of dependent and
independent random vectors.
n
be an arbitrary random vector.
If Cov(
X
) is diagonal, then
X
is called (mutually)
decorrelated
.
X
is
said to be white or
whitened
if
E
(
X
)=0andCov(
X
)=
I
(i.e. if
X
is
centered and decorrelated with unit variance components). A
whitening
transformation
of
X
is a matrix
W
Definition 3.7:
Let
X
:Ω
→ R
∈
Gl(
n
) such that
WX
is whitened.
Note that
X
is white if and only if
AX
is white for an orthogonal
matrix
A
AA
=
I
∈
O
(
n
)=
{
A
∈
Gl(
n
)
|
}
, which follows directly from
Cov(
AX
)=
A
Cov(
X
)
A
.
Lemma 3.2: Given a centered random vector
X
with nondeterministic
components, there exists a whitening transformation of
X
, and it is
unique modulo
O
(
n
).
Proof
Let
C
:= Cov(
X
) be the covariance matrix of
X
.
C
is symmetric,
so there exists
V
O
(
n
) such that
VCV
=
D
with
D
Gl(
n
) diagonal
and positive. Set
W
:=
D
−1/2
V
,where
D
−1/2
denotes a diagonal matrix
(square root) with
D
−1/2
D
−1/2
=
D
−1
. Then, using the fact that
X
is
∈
∈
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