Biomedical Engineering Reference
In-Depth Information
and the source vector can be estimated using
T
W
T
s
(
r
,
t
)
=[
s
x
(
r
,
t
),
s
y
(
r
,
t
),
s
z
(
r
,
t
)
]
=
(
r
)
y
(
t
).
(3.46)
To derive the weight matrix of a vector-type minimum-variance beamformer, we
use the optimization
W
T
W
T
W
(
r
)
=
argmin
W
(
r
)
tr
[
(
r
)
RW
(
r
)
]
,
subject to
(
r
)
L
(
r
)
=
I
.
(3.47)
A derivation similar to that for Eq. (
3.6
) leads to the solution for the weight matrix
W
(
r
)
, which is expressed as [
4
]
R
−
1
L
L
T
R
−
1
L
)
]
−
1
W
(
r
)
=
(
r
)
[
(
r
)
(
r
.
(3.48)
The beamformer that uses the above weight is called the vector-type adaptive beam-
former. Using the weight matrix above, the source-vector covariance matrix is esti-
mated as
s
T
L
T
R
−
1
L
)
]
−
1
s
(
r
,
t
)
(
r
,
t
)
=[
(
r
)
(
r
,
(3.49)
and the source power estimate is obtained using
tr
)
]
−
1
R
−
1
L
2
2
L
T
s
(
r
,
t
)
=
s
(
r
,
t
)
=
[
(
r
)
(
r
.
(3.50)
This vector-type beamformer is sometimes referred to as the linearly constrained
minimum-variance (LCMV) beamformer [
4
].
The weight matrix of the array-gain vector beamformer is derived, such that
tr
W
T
W
(
r
)
=
argmin
W
(
r
)
RW
(
r
)
,
(
r
)
W
T
subject to
(
r
)
L
(
r
)
=
L
(
r
)
.
(3.51)
The vector version of the array-gain minimum-variance beamformer is expressed as
)
[
L
T
L
L
R
−
1
R
−
1
)
]
−
1
(
)
=
(
(
)
(
,
W
r
r
r
r
(3.52)
where
L
is the normalized lead field matrix defined as
L
(
r
)
(
r
)
=
L
(
r
)/
L
(
r
)
.
The source-vector covariance matrix is estimated as
)
=[
L
T
s
T
R
−
1
L
)
]
−
1
s
(
r
,
t
)
(
r
,
t
(
r
)
(
r
.
(3.53)
