Digital Signal Processing Reference
In-Depth Information
3.7.1 Constrained Optimal Filters
A typical problem in which a reference signal is replaced by a set of suitable
constraints is the so-called linearly constrained minimum variance (LCMV)
filter, in which the minimization process is not carried out with respect to
an error signal, but directly over the output signal. In order to present the
problem, let us consider again the output of a linear combiner, given by
w
H
x
y
(
n
)
=
(
n
)
(3.86)
Notice that, in this case, we are assuming that all signals are complex-valued.
Clearly, the direct minimization of
E
y
2
leads to the trivial solu-
(
n
)
tion
w
0. However, a set of nontrivial coefficients can be obtained by the
following procedure:
Minimize
E
y
=
2
E
w
H
x
w
)
x
H
w
H
Rw
(
n
=
(
n
)
(
n
)
=
(3.87)
subject to
C
H
w
=
g
(3.88)
The resulting optimization problem corresponds to the minimization of
a quadratic form given a set of linear constraints, which can be performed
in accordance with the method of
Lagrange multipliers
. The use of Lagrange
multipliers requires the minimization of the following expression [139]:
λ
H
C
H
w
g
1
2
w
H
Rw
+
−
(3.89)
In order to find all the relevant parameters, we must set to zero the gradient
of (3.89), which leads to
Rw
+
Cλ
=
0
(3.90)
Then, from (3.89), the Lagrange multipliers are
C
H
R
−
1
C
−
1
g
λ
=−
(3.91)
and the optimal coefficients are given by
R
−
1
C
C
H
R
−
1
C
)
−
1
g
=
(
w
opt
(3.92)
Hence, we reach a closed-form solution, which depends on the autocor-
relation matrix, as well as in (3.14), but also on the parameters defining the
