Digital Signal Processing Reference
In-Depth Information
for a signal impinging at angle
θ
l
,
l
=
1
,
2
,...,K
, where
d
s
=
λ
c
/
2 is the inter-
)
T
denotes the transpose operation. The
vector
n
(i)
denotes the complex vector of sensor noise, which is assumed to be
zero-mean and Gaussian with covariance matrix
σ
2
I
.
·
element spacing,
λ
c
is the wavelength and
(
2.3 Problem Statement and Design of Adaptive Beamformers
In this section, the problem of designing robust beamforming algorithms against
steering vector mismatches is stated. The design of robust full-rank and reduced-
rank LCMV beamformers is introduced along with the modeling of steering vector
mismatches. The presumed array steering vector for the
k
th desired signal is given
by
a
p
(θ
k
)
=
1 mismatch vector and
a
(θ
k
)
is the actual
array steering vector which is unknown for the system. By using the presumed array
steering vector
a
p
(θ
k
)
, the performance of a conventional LCMV beamformer can
be degraded significantly. The problem of interest is how to design a beamformer
that can deal with the mismatch and minimize the performance loss due to the un-
certainty.
a
(θ
k
)
+
e
, where
e
is the
M
×
2.3.1 Adaptive LCMV Beamformers
In order to perform beamforming with a full-rank LCMV beamformer, we linearly
combine the data vector
r
(i)
=[
r
(i
1
r
(i)
... r
(i
M
]
T
with the full-rank beamformer
2
T
to yield
w
=[
w
1
w
2
...w
M
]
w
H
r
(i).
x(i)
=
(2.3)
The optimal LCMV beamformer is described by the
M
×
1 vector
w
, which is
designed to solve the following optimization problem
minimize
E
w
H
r
(i)
=
2
w
H
Rw
(2.4)
w
H
a
(θ
k
)
subject to
=
1
.
The solution to the problem in (
2.4
) is given by [
3
,
4
]
R
−
1
a
(θ
k
)
a
H
(θ
k
)
R
−
1
a
(θ
k
)
,
w
opt
=
(2.5)
where
a
(θ
k
)
is the steering vector of the SoI,
r
(i)
is the received data, the covariance
matrix of
r
(i)
is described by
R
r
(i)
r
H
(i)
)
H
denotes Hermitian transpose
=
E
[
]
,
(
·
and
E
[·]
stands for the expected value. The beamformer
w
(i)
can be estimated via
SG or RLS algorithms [
3
]. However, the laws that govern their convergence and
tracking behaviors imply that they depend on
M
and on the eigenvalue spread of
R
.
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