Biomedical Engineering Reference
In-Depth Information
A
1
,
2
(
f
)
H
1
,
2
(
f
)
=
|
¯
A
(
f
)
|
A
1
,
2
(
indicates that the directed transfer function is equal to
f
)
aside from the scaling
|
¯
A
constant
.
One advantage of this definition is that the extension to a general multivariate case
is straightforward. In a general multivariate case, the unnormalized transfer function
is equal to
(
f
)
|
)
|=
|
M
2
,
1
(
f
)
|
|
H
1
,
2
(
f
,
(8.64)
|
¯
A
(
f
)
|
¯
A
where
M
2
,
1
(
. In a trivariate
case, using Eq. (
8.64
), the unnormalized transfer function is derived as
f
)
is a minor of the
(
2
,
1
)
element of the matrix
(
f
)
)
|=
|
A
1
,
2
A
3
,
3
−
A
3
,
1
A
2
,
3
|
|
H
1
,
2
(
f
,
(8.65)
|
¯
A
(
f
)
|
where the explicit notation
is omitted in the numerator. On the right hand side,
A
1
,
2
represents the direct influence from channel #1 to channel #2. The equation
above shows that even when
(
f
)
A
1
,
2
is equal to zero,
cannot equal zero because
the term
A
3
,
1
A
2
,
3
can also not be zero. This term represents the indirect influence of
channel #1 on channel #2 via channel #3. That is, DTF contains the indirect causal
influence, as well as the direct causal influence.
The normalized DTF can be defined by normalizing the causal influence on the
j
th
channel from all other channels. That is, when total
q
channels exist, the normalized
DTF is defined as
|
H
1
,
2
|
H
j
,
k
(
f
)
H
j
,
k
(
f
)
μ
k
ₒ
j
=
=
q
m
2
.
(8.66)
h
j
[
h
j
]
1
|
H
j
,
m
(
f
)
|
=
8.5.2 Relationship Between DTF and Coherence
The coherence can be expressed as a result of factorization of the normalized DTF.
To show this, we define the directed coherence
ʳ
j
,
k
such that
H
j
,
k
(
f
)
ʳ
j
,
k
=
h
j
h
j
.
(8.67)
ʣ
We then define the column vector
)
=[
ʳ
j
,
1
,...,ʳ
j
,
q
]
T
ʳ
j
(
f
,
(8.68)
