Biomedical Engineering Reference
In-Depth Information
L
T
T
ʷ
(
r
)
[
(
r
)
L
(
r
)
]
ʷ
(
r
)
ʷ
opt
(
r
)
=
argmax
ʷ
(
,
(3.39)
L
T
R
−
1
L
T
ʷ
(
r
)
[
(
r
)
(
r
)
]
ʷ
(
r
)
r
)
which can be computed using
L
T
R
−
1
L
L
T
ʷ
opt
(
r
)
=
ˑ
min
{
(
r
)
(
r
),
(
r
)
L
(
r
)
}
.
(3.40)
Once
ʷ
opt
(
r
)
is obtained, the weight vector is obtained using Eq. (
3.10
) with
l
(
r
)
=
and
l
L
(
r
)
ʷ
opt
(
r
)
(
r
)
=
l
(
r
)/
l
(
r
)
. The output power of this scalar beamformer is
given by
1
2
s
(
r
,
t
)
=
)
}
.
(3.41)
L
T
R
−
1
L
L
T
S
min
{
(
r
)
(
r
),
(
r
)
L
(
r
For the scalar-type weight-normalized minimum-variance beamformer, the opti-
mum orientation is derived using
T
L
T
R
−
1
L
ʷ
(
r
)
[
(
r
)
(
r
)
]
ʷ
(
r
)
ʷ
opt
(
r
)
=
argmax
ʷ
(
,
(3.42)
L
T
R
−
2
L
ʷ
T
(
r
)
[
(
r
)
(
r
)
]
ʷ
(
r
)
r
)
which can be computed using
L
T
R
−
2
L
L
T
R
−
1
L
ʷ
opt
(
r
)
=
ˑ
min
{
(
r
)
(
r
),
(
r
)
(
r
)
}
.
(3.43)
Once
ʷ
opt
(
r
)
is obtained, the weight vector is obtained using Eq. (
3.17
) with
l
(
r
)
=
L
(
r
)
ʷ
opt
(
r
)
. The output power of this scalar beamformer is given by
1
2
(
,
)
=
)
}
.
s
r
t
(3.44)
L
T
R
−
2
L
L
T
R
−
1
L
S
min
{
(
)
(
),
(
)
(
r
r
r
r
3.6 Vector-Type Adaptive Beamformer
3.6.1 Vector Beamformer Formulation
The vector adaptive beamformer is another type of adaptive beamformers that can
reconstruct the source orientation as well as the source magnitude. It uses a set of
three weight vectors
w
x
(
r
)
,
w
y
(
r
)
, and
w
z
(
r
)
, which, respectively, detect the
x
,
y
,
and
z
components of the source vector
s
(
r
,
t
)
. That is, the weight matrix
W
(
r
)
is
defined as
W
(
r
)
=[
w
x
(
r
),
w
y
(
r
),
w
z
(
r
)
]
,
(3.45)
