Biomedical Engineering Reference
In-Depth Information
The source power estimate is obtained by
tr
)
]
−
1
[
L
T
2
2
R
−
1
L
s
(
r
,
t
)
=
s
(
r
,
t
)
=
(
r
)
(
r
.
(3.54)
The vector-type beamformer can be formulated with the unit-noise-gain constraint,
and the detail of the formulation is found in [
6
,
12
].
3.6.2 Semi-Bayesian Formulation
The weight matrix of a vector-type adaptive beamformer can be derived using the
same Bayesian formulation in Sect.
3.3
. The prior distribution in Eq. (
3.20
) and the
noise assumption in Eq. (
3.19
) lead to the Bayesian estimate of
x
in Eq. (
3.21
), which
is written as
)
=
ʦ
−
1
F
T
ʣ
−
1
x
¯
(
t
y
(
t
),
(3.55)
y
¯
(
)
where,
x
t
is expressed as
⊡
⊤
s
(
r
1
,
t
)
⊣
⊦
.
s
(
r
2
,
t
)
x
¯
(
t
)
=
(3.56)
.
s
(
r
N
,
t
)
ʦ
−
1
which is a 3
N
To derive the vector beamformer, we use the matrix
×
3
N
Block
diagonal matrix such that
⊡
⊣
⊤
⊦
ʥ
1
0
···
0
.
0
ʥ
2
·
ʦ
−
1
=
,
(3.57)
.
.
.
.
·
0
0
···
0
ʥ
N
where the 3
ʥ
j
is the prior covariance matrix of the source vector at the
j
th voxel. Thus, the estimated source vector for the
j
th voxel is obtained as
×
3matrix
)
=
ʥ
j
L
j
ʣ
−
1
s
(
r
j
,
t
y
(
t
).
(3.58)
y
s
T
Computing the estimated source-vector covariance matrix,
s
(
r
j
,
t
)
(
r
j
,
t
)
,
leads to
