Biomedical Engineering Reference
In-Depth Information
R
−
1
L
(
r
)
ʷ
w
(
r
,
ʷ
)
=
)
ʷ
]
,
(3.33)
T
L
T
R
−
1
L
[
ʷ
(
r
)
(
r
and the output power obtained using this weight is given by
1
2
s
(
r
,
ʷ
)
=
)
]
ʷ
.
(3.34)
L
T
R
−
1
L
ʷ
T
[
(
r
)
(
r
The source orientation is then determined by maximizing this output power. That is,
the optimum orientation
ʷ
opt
(
r
)
is derived as [
5
]
1
ʷ
opt
(
r
)
=
argmax
ʷ
(
r
)
.
(3.35)
R
−
1
L
T
L
T
ʷ
(
r
)
(
r
)
(
r
)
ʷ
(
r
)
According to the Rayleigh-Ritz formula in Sect. C.9, the orientation
ʷ
opt
(
r
)
is
obtained as
ʷ
T
L
T
R
−
1
L
L
T
R
−
1
L
ʷ
opt
(
)
=
(
)
[
(
)
(
)
]
ʷ
(
)
=
ˑ
min
{
(
)
(
)
}
,
r
argmin
ʷ
(
r
r
r
r
r
r
r
)
(3.36)
where
indicates the eigenvector corresponding to the minimum eigenvalue
of the matrix in the curly braces.
3
Namely, the optimum orientation
ˑ
min
{·}
ʷ
opt
(
r
)
is given
by the eigenvector corresponding to the minimum eigenvalue of
L
T
R
−
1
L
(
)
(
)
r
r
.
is obtained, the explicit form of the weight vector for the scalar
minimum-variance beamformer is expressed as
Once
ʷ
opt
(
r
)
R
−
1
L
(
r
)
ʷ
opt
(
r
)
w
(
r
)
=
)
]
.
(3.37)
opt
(
L
T
R
−
1
L
[
ʷ
r
)
(
r
)
(
r
)
ʷ
opt
(
r
The output power is given by
1
1
2
s
(
r
,
t
)
=
)
]
=
)
}
,
(3.38)
T
L
T
R
−
1
L
L
T
R
−
1
L
[
ʷ
opt
(
r
)
(
r
)
(
r
)
ʷ
opt
(
r
S
min
{
(
r
)
(
r
where
S
min
{·}
is the minimum eigenvalue of the matrix in the curly braces.
3.5.2 Expressions for the Array-Gain and Weight-Normalized
Beamformers
For the scalar-type array-gain minimum-variance beamformer, the optimum orien-
tation is derived using
3
The notations such as
ˑ
{·}
and
S
min
{·}
are defined in Sect. C.9 in the Appendix.
