Biomedical Engineering Reference
In-Depth Information
l
T
R
−
1
y
(
r
)
(
t
)
s
(
r
,
t
)
=
,
(3.11)
[
l
T
l
R
−
1
(
r
)
(
r
)
]
and the output power is expressed as
l
T
(
r
)
l
(
r
)
2
s
(
r
,
t
)
=
.
(3.12)
l
T
R
−
1
l
[
(
)
(
)
]
r
r
3.2.3 Minimum-Variance Beamformer
with Unit-Noise-Gain Constraint
Another possible constraint is the unit-noise-gain constraint, which is expressed as
w
T
(
r
)
w
(
r
)
=
1. That is, the filter weight is obtained using
T
T
w
(
r
)
=
argmin
w
(
r
)
w
(
r
)
R
w
(
r
),
subject to
w
(
r
)
l
(
r
)
=
˄ ,
(3.13)
T
and
w
(
r
)
w
(
r
)
=
1
,
T
where the minimization problem is solved with the first constraint,
w
(
r
)
l
(
r
)
=
˄
.
T
˄
w
(
)
w
(
)
=
The scalar constant
1. To
obtain the weight vector derived from the above minimization, we first calculate the
weight using
is determined by the second constraint,
r
r
T
T
w
(
r
)
=
argmin
w
(
w
(
r
)
R
w
(
r
)
subject to
w
(
r
)
l
(
r
)
=
˄ .
(3.14)
r
)
Following the same steps from Eqs. (
3.3
)to(
3.6
), the weight satisfying Eq. (
3.14
)is
obtained as
R
−
1
l
(
r
)
w
(
r
)
=
˄
.
(3.15)
l
T
R
−
1
l
[
(
r
)
(
r
)
]
T
Substituting this expression to
w
(
r
)
w
(
r
)
=
1 leads to
l
T
R
−
1
l
(
r
)
(
r
)
˄
=
l
T
,
(3.16)
R
−
2
l
(
r
)
(
r
)
and the weight is given by:
R
−
1
l
(
r
)
w
(
r
)
=
l
T
.
(3.17)
R
−
2
l
(
r
)
(
r
)
