Biomedical Engineering Reference
In-Depth Information
2
2
|
d
1
|
˃
T
|
+
d
2
|
˃
S
|
|
|
ˆ
|=
|
˃
S
|
.
(7.21)
|
˃
T
|
d
1
|
˃
T
|
d
2
|
˃
S
|
2
2
2
2
+
+
7.4.2 Leakage Effects on the Imaginary Coherence
We next analyze the effects of the algorithm leakage on the imaginary coherence.
Using Eq. (
7.18
) and the relationship
˃
S
˃
T
,
˃
T
˃
S
+
˃
S
˃
T
=
2
˃
T
˃
S
the cross spectrum
can be expressed as
2
d
1
d
2
˃
T
˃
S
.
(7.22)
˃
T
˃
S
=
(
d
1
d
2
)
˃
T
˃
S
+
2
2
1
−
d
1
|
˃
T
|
+
d
2
|
˃
S
|
+
By taking the imaginary part of Eq. (
7.22
), we can derive
˃
T
˃
S
=
(
d
1
d
2
)
˃
T
˃
S
.
1
−
(7.23)
(ˆ)
We can then obtain the imaginary part of the estimated coherence
as
˃
T
˃
S
d
1
d
2
)
˃
T
˃
S
=
(
1
−
(ˆ)
=
=
ʛ
(ˆ) ,
(7.24)
2
2
2
2
|
˃
T
|
|
˃
S
|
|
˃
T
|
|
˃
S
|
where
(ˆ)
indicates the true value of the imaginary coherence. Using Eqs. (
7.19
)
and (
7.20
),
ʛ
is obtained as
ʛ
=
(
1
d
1
d
2
)
√
˄
1
˄
2
−
(7.25)
where
2
d
1
˃
T
˃
S
|
˃
S
|
2
d
1
|
˃
T
|
˄
1
=
1
+
+
,
(7.26)
2
2
|
˃
S
|
and
2
d
2
˃
T
˃
S
|
˃
T
|
2
d
2
|
˃
S
|
˄
2
=
1
+
+
.
(7.27)
|
˃
T
|
2
2
(ˆ)
=
0, indicating that no
spurious imaginary coherence has been caused. However, Eq. (
7.24
) also indicates
that the value of
Equation (
7.24
) shows that when
(ˆ)
=
0, we have
(ˆ)
, i.e., the intensity of the
estimated imaginary coherence is biased and the bias is represented by
differs from the true value
(ˆ)
ʛ
inEq. (
7.25
).
