Biomedical Engineering Reference
In-Depth Information
According to Eq. (
7.6
), we have
∞
∞
∞
ʨ(
)
d
f
˃
T
(
)
d
f
)˃
S
(
R
(
0
)
=
ʨ(
f
)
d
f
=
f
=
f
f
,
(7.8)
−∞
−∞
−∞
where
] indicates the real part of the complex number in the square brackets.
The right-hand side of the above equation holds because the real part of
[
·
ʨ(
f
)
is an
even function and the imaginary part is an odd function. (This is because
R
(˄ )
is a
real-valued function.) Therefore, for a narrow-band signal,
∞
˃
T
(
)
d
f
≈
˃
T
(
)
.
)˃
S
(
)˃
S
(
R
(
0
)
=
f
f
f
f
(7.9)
−∞
Here,
R
, which represents the zero-time-lag correlation, is related to the real part
of the cross spectrum, and thus, the real part of coherence represents the instantaneous
interaction.
Applying Parseval's theorem to the Fourier transform relationship in Eq. (
7.6
),
we get
(
0
)
∞
∞
∞
2
d
2
d
f
)˃
S
(
2
d
f
R
(˄ )
˄
=
|
ʨ(
f
)
|
=
|
˃
T
(
f
f
)
|
.
(7.10)
−∞
−∞
−∞
Thus, for a narrow-band signal, the following relationship holds:
∞
=
˃
T
(
)
+
˃
T
(
)
,
2
d
)˃
S
(
2
)˃
S
(
)˃
S
(
R
(˄ )
˄
≈|
˃
T
(
f
f
)
|
f
f
f
f
−∞
(7.11)
where
] indicate the imaginary part of the complex number in the square brackets.
Using Eq. (
7.9
), we have
[
·
∞
˃
T
(
)
≈
)˃
S
(
2
d
2
2
f
f
R
(˄ )
˄
−
R
(
0
)
=
R
(˄ )
.
(7.12)
˄
=
0
−∞
The imaginary part of the cross spectrum is equal to the sum of nonzero-lag corre-
lations. Therefore, we can see that the imaginary part of coherence represents the
non-instantaneous interaction.
