Biomedical Engineering Reference
In-Depth Information
⊡
⊤
⊡
⊤
x
1
(
f
)
y
1
(
f
)
⊣
⊦
⊣
⊦
.
x
2
(
f
)
y
2
(
f
)
x
(
f
)
=
and
y
(
f
)
=
(7.57)
.
x
p
(
.
y
q
(
f
)
f
)
In the following arguments, the notation
is omitted for simplicity. Using the same
arguments as those for the canonical correlation in Sect. C.3, the random vectors
x
and
y
are, respectively, projected in the directions of
a
and
b
where
a
and
b
are
complex-valued
p
(
f
)
a
H
x
and
×
1 and
q
×
1 column vectors. That is, defining
x
=
b
H
y
, the coherence between
y
=
x
and
y
,
ˆ
, is given by
y
∗
a
H
xy
H
x
b
ˆ
=
=
x
∗
y
∗
x
y
b
H
a
H
xx
H
yy
H
[
a
][
b
]
a
H
ʣ
xy
b
=
,
(7.58)
b
H
[
a
H
ʣ
xx
a
][
ʣ
yy
b
]
where the superscript
H
indicates the Hermitian transpose. Here the cross-spectral
matrices are defined as
xy
H
xx
H
yy
H
ʣ
xy
=
ʣ
xx
=
ʣ
yy
=
.We
derive a formula to compute the canonical magnitude coherence. Using Eq. (
7.58
),
the magnitude coherence between
,
, and
x
and
y
is expressed as
a
H
ʣ
xy
b
2
2
|
ˆ
|
=
]
.
(7.59)
b
H
a
H
[
ʣ
xx
a
][
ʣ
yy
b
2
is defined as the maximum of
2
The canonical squared magnitude coherence
|
ˈ
|
|
ˆ
|
(with respect to
a
and
b
), which is obtained by solving the optimization problem
ʣ
xy
b
2
2
a
H
subject to
a
H
1 and
b
H
|
ˈ
|
=
max
a
ʣ
xx
a
=
ʣ
yy
b
=
1
.
(7.60)
,
b
This constrained optimization problem can be solved in the following manner.
Since
a
H
ʣ
xy
b
2
is a scalar, we have the relationship,
ʣ
xy
b
a
H
ʣ
xy
b
H
a
H
ʣ
xy
b
2
a
H
b
H
H
xy
aa
H
=
=
ʣ
ʣ
xy
b
.
(7.61)
We first fix
a
, and solve the maximization problem with respect to
b
. The maximiza-
tion problem is
b
H
xy
aa
H
subject to
b
H
max
b
ʣ
ʣ
xy
b
ʣ
yy
b
=
1
.
(7.62)
