Biomedical Engineering Reference
In-Depth Information
The beamformer using the above weight is called the unit-gain (constraint) minimum-
norm filter. Using the optimization
2
T
r
)
−
ʴ(
r
)
d
r
w
(
r
)
=
argmin
w
(
r
)
w
(
r
)
l
(
r
−
Ω
T
subject to
w
(
r
)
l
(
r
)
=
l
(
r
)
,
(3.91)
we obtain the array-gain (constraint) minimum-norm filter, such that
)
=[
l
T
l
l
G
−
1
)
]
−
1
G
−
1
w
(
r
(
r
)
(
r
(
r
),
(3.92)
where
l
(
r
)
=
l
(
r
)/
l
(
r
)
.
3.9.2 Derivation of RENS Beamformer
The recursive null-steering (RENS) beamformer is derived from the following opti-
mization [
18
]:
2
T
r
)
−
ʴ(
r
)
2
d
r
w
(
)
=
w
(
)
(
−
(
,
)
r
argmin
w
(
r
l
r
s
r
t
r
)
Ω
T
subject to
w
(
r
)
l
(
r
)
=
l
(
r
)
,
(3.93)
2
is the instantaneous source power. The idea behind the above optimiza-
tion is that we wish to impose a delta-function-like property on the beam response
only over a region where sources exist, instead of imposing that property over the
entire source space.
The filter weight is obtained as
where
s
(
r
,
t
)
)
=[
l
T
)
G
−
1
G
−
1
l
)
]
−
1
l
w
(
r
(
r
(
r
(
r
),
(3.94)
where
G
2
l
l
T
=
s
(
r
,
t
)
(
r
)
(
r
)
d
r
.
(3.95)
Ω
However, to compute the weight in Eq. (
3.94
), we need to know the source magni-
tude distribution
s
2
, which is unknown. Therefore, we use an estimated source
(
r
,
t
)
G
, i.e.,
2
magnitude
s
(
r
,
t
)
when computing
G
2
l
l
T
=
s
(
r
,
t
)
(
r
)
(
r
)
d
r
.
(3.96)
Ω
