Biomedical Engineering Reference
In-Depth Information
Substituting this equation into Eq. (
3.22
), we can derive the beamformer expression,
T
s
(
r
j
,
t
)
=
w
(
r
j
)
y
(
t
),
(3.27)
where the weight vector is expressed as
ʣ
−
y
l
j
l
j
ʣ
−
1
w
(
r
j
)
=
l
j
.
(3.28)
y
If the model data covariance matrix
ʣ
y
is replaced with the sample data covariance
matrix
R
, the weight equation (
3.28
) becomes
R
−
1
l
T
(
r
j
)
w
(
r
j
)
=
r
j
)
,
(3.29)
l
T
R
−
1
l
(
r
j
)
(
which is exactly equal to the weight expression of the minimum-variance beam-
former. To derive the array-gain constraint beamformer, we use the relationship,
2
=
ʱ
−
1
j
s
(
r
j
,
t
)
l
(
r
j
)
,
(3.30)
The weight vector in this case is obtained as
l
R
−
1
(
r
j
)
w
(
r
j
)
=
,
(3.31)
[
l
T
l
R
−
1
(
r
j
)
(
r
j
)
]
which is equal to the weight expression of the array-gain constraint minimum-
variance beamformer.
3.4 Diagonal-Loading and Bayesian Beamformers
The sample covariance matrix sometimes has a large condition number. This situation
happens, for example, when the number of time samples is considerably fewer than
the size of the matrix or when the SNR of the sensor data is very high. In such
cases, since the direct inversion of the sample covariance
R
−
1
may cause numerical
instability, the regularized inverse,
)
−
1
, may be a better approximation of the
(
R
+
ʺ
I
ʣ
−
1
inverse of the model data covariance
is a small positive real number
called the regularization constant. This gives the weight expression,
, where
ʺ
y
)
−
1
l
(
R
+
ʺ
I
(
r
)
w
(
r
)
=
.
(3.32)
l
T
[
(
)(
+
ʺ
)
−
1
l
(
)
]
r
R
I
r
