Biomedical Engineering Reference
In-Depth Information
(
)
(
×
)
where
A
p
is a
2
2
coefficient matrix. We can derive Eq. (
8.7
), which is explicitly
written as
x
H
xx
H
xy
H
yx
H
yy
e
x
(
(
f
)
f
)
=
.
(8.46)
y
(
f
)
e
y
(
f
)
In the expressions above, the explicit notation of the frequency
f
is omitted from
the components of
H
(
f
)
. Denoting the residual covariance matrix such that
Σ
x
2
)
∗
|
e
x
(
f
)
|
e
x
(
f
)
e
y
(
f
ʳ
=
,
(8.47)
)
∗
|
ʳ
∗
2
e
y
(
f
)
e
x
(
f
e
y
(
f
)
|
Σ
y
where the superscript
∗
indicates the complex conjugation. Equation (
8.8
) is explicitly
written as
H
Σ
x
H
H
2
)
∗
|
x
(
f
)
|
x
(
f
)
y
(
f
ʳ
=
.
(8.48)
)
∗
|
2
ʳ
∗
y
(
f
)
x
(
f
y
(
f
)
|
Σ
y
8.4.2 Total Interdependence and Coherence
We first define the total interdependence between
x
and
y
, which corresponds to
I
{
x
,
y
}
in Eq. (
8.40
). Using the arguments in Sect.
8.3.3
, the total interdependence
between
x
and
y
in the spectral domain is defined as
2
|
x
(
f
)
|
0
2
0
|
y
(
f
)
|
f
}
=
log
{
x
,
y
|
S
(
f
)
|
2
2
|
x
(
f
)
|
|
y
(
f
)
|
=
log
2
.
(8.49)
2
2
)
∗
|
|
x
(
f
)
|
|
y
(
f
)
|
−|
x
(
f
)
y
(
f
that under the Gaussianity assumption, the spectral domain total interdependence is
arguments, the total interdependence
f
is related to the magnitude coherence
{
x
,
y
}
|
ˆ(
f
)
|
through
2
f
}
(
f
)
=−
log
(
1
−|
ˆ(
f
)
|
),
(8.50)
{
x
,
y
where
)
∗
|
2
|
x
(
f
)
y
(
f
2
|
ˆ(
f
)
|
=
.
(8.51)
2
2
|
x
(
f
)
|
|
y
(
f
)
|
