Biomedical Engineering Reference
In-Depth Information
3.2 Classical Derivation of Adaptive Beamformers
3.2.1 Minimum-Variance Beamformers
with Unit-Gain Constraint
The weight vector of the adaptive beamformer is derived using the optimization:
T
T
w
(
r
)
=
argmin
w
(
r
)
w
(
r
)
R
w
(
r
),
subject to
w
(
r
)
l
(
r
)
=
1
.
(3.2)
y
T
Here,
R
is the data covariance matrix obtained using
y
(
t
)
(
t
)
where
·
indi-
T
cates the ensemble average. In Eq. (
3.2
), the inner product
represents the
beamformer output from a unit-magnitude source located at
r
. Therefore, setting
w
w
(
r
)
l
(
r
)
T
1 guarantees that the beamformer passes the signal from
r
with the
gain equal to one. The constraint
(
r
)
l
(
r
)
=
T
w
(
r
)
l
(
r
)
=
1 is called the unit-gain constraint.
T
generally contains not only the
noise contributions but also unwanted contributions such as the influence of sources
at locations other than
r
. Accordingly, by minimizing the output power with this
unit-gain constraint, we can derive a weight that minimizes such unwanted influence
without affecting the signal coming from
r
, the pointing location of the beamformer.
This constrained minimization problem can be solved using a method of the
Lagrange multiplier. We define the Lagrange multiplier as a scalar
The output power of the beamformer
w
(
r
)
R
w
(
r
)
ʶ
, and the
Lagrangian as
L
(
w
, ʶ)
, such that
T
R
T
l
L
(
w
, ʶ)
=
w
w
+
ʶ(
w
(
r
)
−
1
),
(3.3)
where the explicit notation of
for simplicity. The weight
vector satisfying Eq. (
3.2
) can be obtained by minimizing the Lagrangian
(
r
)
is omitted from
w
(
r
)
L
(
w
, ʶ)
in Eq. (
3.3
) with no constraints.
The derivative of
L
(
w
, ʶ)
w
with respect to
is given by:
∂
L
(
w
, ʶ)
∂
w
=
w
+
ʶ
(
).
2
R
l
r
(3.4)
By setting the right-hand side of the above equation to zero, we obtain
R
−
1
l
w
=−
ʶ
(
r
)/
2
.
(3.5)
T
l
Substituting this relationship back into the constraint equation
w
(
r
)
=
1, we
l
T
R
−
1
l
get
ʶ
=−
2
/
[
(
r
)
(
r
)
]
. Substituting this
ʶ
into Eq. (
3.5
), the weight vector
satisfying Eq. (
3.2
) is obtained as
R
−
1
l
(
r
)
w
(
r
)
=
)
]
.
(3.6)
l
T
R
−
1
l
[
(
r
)
(
r
