Biomedical Engineering Reference
In-Depth Information
7.7.2 Residual Envelope Correlation
Residual envelope correlation, which is free from the problem of seed blur, is pro-
posed in [
15
]. To compute the residual envelope correlation, we first compute the
residual time course
u
R
(
t
)
using
∞
e
i
2
ˀ
ft
d
f
u
R
(
t
)
=
v(
f
)
.
(7.103)
−∞
Here
v(
f
)
is the residual spectrum obtained using Eq. (
7.48
). Since
v(
f
)
is defined
only for
f
≥
0, we create the residual spectrum for
f
<
0, such that
)
∗
,
v(
−
f
)
=
v(
f
where the superscript
∗
indicates the complex conjugation. Since the Hermitian
symmetry holds for
v(
f
)
,
u
R
(
t
)
is guaranteed to be a real-valued time course. The
envelope of
u
R
(
t
)
is computed using
e
i
ʸ
R
(
t
)
,
A
[
u
R
(
t
)
]=
A
R
(
t
)
(7.104)
where
A
R
(
t
)
is the envelope of
u
R
(
t
)
called the residual envelope. Once
A
R
(
t
)
is
ʘ
R
is computed using
computed, the residual envelope correlation
j
=
1
A
R
(
t
j
)
A
S
(
t
j
)
ʘ
R
=
j
=
1
A
R
(
2
.
(7.105)
2
j
=
1
A
S
(
t
j
)
t
j
)
Since in the residual time course the seed signal is regressed out, the residual envelope
correlation is free from the spurious correlation caused by the algorithm leakage. Note
that a method similar to the one mentioned here was reported in [
16
].
7.7.3 Envelope Coherence
The coherence can be computed between the envelope time courses
A
T
(
t
)
and
A
S
(
t
)
.
Let us define the spectra obtained from the envelope
A
S
(
t
)
and
A
T
(
t
)
as
ʣ
S
(
f
)
and
ʣ
T
(
f
)
, respectively. The complex envelope coherence is computed using
)ʣ
S
(
ʣ
T
(
f
f
)
ˆ
E
(
f
)
=
.
(7.106)
2
2
|
ʣ
T
(
f
)
|
|
ʣ
S
(
f
)
|
