Digital Signal Processing Reference
In-Depth Information
spectrum for the horizontally polarized interferers shows random peaks. This demon-
strates that it fails to perform direction finding of the interferers that are suppressed by
the array orthogonal dual-polarization property.
14.6
Constrained Adaptive Algorithm
In this section, we recall a simple adaptive algorithm (constrained LMS algorithm) and
use it to provide the cancellation weight matrix without the need to estimate the data
covariance matrix. The constraint LMS algorithm is based on the gradient-descent algo-
rithm. It iteratively adapts the weights of an antenna array to minimize the noise power
at the output while retaining a unit response at the look directions. The multiple MVDR
beamforming method solves for
w
according to
H
H
min
wrw
subjectto =1
c
w
.
(14 .43)
w
Denote
F
=
c
(
c
H
c
)
-1
and
P
=
I
-
c
(
c
H
c
)
-1
c
H
. The weight vector can be recursively updated
as [40]
w
(
k
+= −
1
)
PwPrw
()
k
µ
()
k
+ =
FPw
(()
k
−
µ
wF
(
k
))
+
(14 .4 4)
=
Pw
(
(()
kykk
−
µ
x
()()).
+
F
The constraint LMS algorithm at the
k
t h
time sample can be expressed as
wF
()
0
=
(14 .45)
w
(
k
+= −
1
)
P w
(()
k
µ
yk
(
)( ))
x
k
+
F
,
where μ is the adaptation step size. The output power of the antenna array is given by
yy
H
H
H
H
P ut
_ == =
wxxw wrw
H
−
1
−
1
rc
cr
rc
crc
=
r
(14 .4 6)
H
−
1
c
H
−
1
H
−
1
−
1
H
−
1
cr
cr
rc
crc
crc
crccrc
1
=
c
r
=
=
−1
c
.
H
−
1
H
−
1
H
−
1
H
−
1
H
(
)(
)
cr
Details of the derivations of equations (14.44)-(14.46) can be found in [40]. The opti-
mum nulling weight vector in equation (14.25) then becomes
ww
ai
=
P
.
(14 .47)
i
ut i
_
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