Biomedical Engineering Reference
In-Depth Information
s
T
)
=
ʥ
j
L
j
ʣ
−
1
y
T
)
ʣ
−
1
y
(
r
j
,
)
(
r
j
,
(
)
(
L
j
ʥ
j
s
t
t
y
t
t
y
=
ʥ
j
L
j
ʣ
−
1
L
j
ʥ
j
,
(3.59)
y
y
T
where we assume
)
=
ʣ
y
.
In this vector case, the unit-gain constraint is expressed as
y
(
t
)
(
t
s
T
s
(
r
j
,
t
)
(
r
j
,
t
)
=
ʥ
j
.
(3.60)
Imposing this relationship, we get,
=
ʥ
j
L
j
ʣ
−
1
ʥ
j
L
j
ʥ
j
,
y
and
L
j
ʣ
−
1
L
j
−
1
ʥ
j
=
.
(3.61)
y
Substituting the above equation into Eq. (
3.58
), we obtain
L
j
ʣ
−
1
L
j
−
1
L
j
ʣ
−
1
s
(
r
j
,
t
)
=
y
(
t
).
(3.62)
y
y
Assuming that the model data covariance can be replaced by the sample data covari-
ance
R
,Eq.(
3.62
) is rewritten as
W
T
s
(
r
,
t
)
=
(
r
j
)
y
(
t
),
(3.63)
where the weight matrix is given by
L
T
−
1
R
−
1
L
R
−
1
L
W
(
r
j
)
=
(
r
j
)
(
r
j
)
(
r
j
)
.
(3.64)
The weight matrix of the vector array-gain minimum-variance beamformer can be
derived using the array-gain constraint,
s
T
s
(
r
j
,
t
)
(
r
j
,
t
)
=
ʥ
j
L
(
r
j
)
.
(3.65)
Substituting this relationship into Eq. (
3.59
) and using Eq. (
3.58
), the resultant weight
matrix is expressed as
L
T
−
1
R
−
1
L
R
−
1
L
W
(
r
j
)
=
(
r
j
)
(
r
j
)
(
r
j
)
.
(3.66)
