Digital Signal Processing Reference
In-Depth Information
Fig. 2.2
Block diagram of
the RJIO scheme
ensures that the beamforming algorithm is robust against steering vector mismatches
and short data records.
In order to describe the RJIO method, let us first consider the structure of the
M
×
D
rank-reduction matrix
=
s
1
(i)
s
D
(i)
,
S
D
(i)
|
s
2
(i)
|
...
|
(2.14)
where the columns
s
d
(i)
for
d
=
1
,...,D
constitute a bank of
D
robust beamform-
ers with dimensions
M
×
1 as given by
s
d
(i)
=
s
1
,d
(i)s
2
,d
(i) ... s
M,d
(i)
T
.
The output
x(i)
of the RJIO scheme can be expressed as a function of the input
vector
r
(i)
, the matrix
S
D
(i)
and the reduced-rank beamformer
¯
w
(i)
:
¯
w
H
(i)
S
D
(i)
r
(i)
w
H
(i)
x(i)
¯
= ¯
= ¯
r
(i).
¯
(2.15)
It is interesting to note that for
D
=
1, the RJIO scheme becomes a robust full-rank
LCMV beamforming scheme with an additional weight parameter
w
D
that provides
an amplitude gain. For
D>
1, the signal processing tasks are changed and the robust
full-rank LCMV beamformers compute a subspace projection and the reduced-rank
beamformer provides a unity gain in the direction of the SoI. This rationale is funda-
mental to the exploitation of the low-rank nature of signals in typical beamforming
scenarios.
The robust LCMV expressions for
S
D
(i)
and
w
(i)
can be computed via the fol-
¯
lowing optimization problem
minimize
E
w
H
(i)
S
D
(i)
r
(i)
2
ε
2
S
D
(i)
w
(i)
2
+
w
H
(i)
S
D
(i)
RS
D
(i)
w
H
(i)
S
D
(i)
S
D
(i)
ε
2
(2.16)
= ¯
w
(i)
¯
+
¯
w
(i)
¯
w
H
(i)
S
D
(i)
a
p
(θ
k
)
¯
=
subject to
1
.
In order to solve the above problem, we resort to the method of Lagrange mul-
tipliers [
3
] and transform the constrained optimization into an unconstrained one
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