Biomedical Engineering Reference
In-Depth Information
−
ʳ
∗
The cross correlation of the two innovation components,
e
x
(
f
)
and
Σ
x
e
x
(
f
)
+
e
y
(
f
)
is zero, because
e
x
(
)
∗
−
ʳ
∗
Σ
x
e
x
(
f
)
+
e
y
(
f
)
f
=−
ʳ
∗
)
∗
=−
ʳ
∗
2
Σ
x
Σ
x
+
ʳ
∗
=
Σ
x
|
e
x
(
f
)
|
+
e
y
(
f
)
e
x
(
f
0
(8.57)
We can thus derive
2
)
∗
|
(
)
|
(
)
(
x
f
x
f
y
f
)
∗
|
2
y
(
f
)
x
(
f
y
(
f
)
|
⊡
⊤
H
xx
+
ʳ
∗
Σ
x
ʳ
ʳ
Σ
x
Σ
x
H
xy
H
yx
+
H
yy
H
xx
+
H
xy
H
xy
Σ
x
0
⊣
⊦
=
.
2
Σ
x
Σ
y
−
|
ʳ
|
ʳ
∗
Σ
x
H
xy
H
yy
0
H
yx
+
H
yy
H
yy
(8.58)
2
Therefore, the signal power
|
x
(
f
)
|
is factorized to
H
xy
.
(8.59)
On the right-hand side of this equation, the first term is interpreted as the intrin-
sic term, which represents the influence of the past of
x
H
xx
+
H
xy
Σ
x
H
xx
+
H
xy
H
xy
ʳ
∗
Σ
x
2
Σ
x
ʳ
Σ
x
Σ
y
−
|
ʳ
|
2
|
x
(
f
)
|
=
+
(
t
)
on the power spectrum
2
|
. The second term is interpreted as the causal influence term, which rep-
resents the influence of the past of
y
x
(
f
)
|
2
(
)
|
(
)
|
t
on
x
f
. Thus, the spectral Granger
causality,
f
y
ₒ
x
(
)
f
, is defined as
2
|
x
(
f
)
|
f
y
ₒ
x
(
f
)
=
log
ʳ
∗
Σ
x
H
xy
)Σ
x
(
ʳ
H
xx
+
Σ
x
H
xy
)
(
H
xx
+
2
H
xy
(Σ
y
−
|
ʳ
|
2
H
xy
log
|
x
(
f
)
|
−
x
)
Σ
=−
2
|
x
(
f
)
|
⊡
⊤
2
Σ
x
)
|
(Σ
y
−
|
ʳ
|
2
H
xy
(
f
)
|
⊣
1
⊦
.
=−
log
−
(8.60)
|
x
(
f
)
|
2
f
x
ₒ
y
(
)
The spectral Granger causality,
f
, can be derived in exactly the same
manner, using
1
−
ʳ
∗
/Σ
y
01
ʠ
=
,
