Digital Signal Processing Reference
In-Depth Information
T
−
ωτ
0
α
N,n
···e
−
(
M −
1)
ωτ
0
α
N,n
d
(
ω,α
N,n
)=
,n
=1
,
2
,...,N
1
e
(6.3)
is a steering vector of length
M
,
T
h
(
ω
)=
H
1
(
ω
)
H
2
(
ω
)
···H
M
(
ω
)
(6.4)
is a filter of length
M
, and
T
Α
=
1
α
N,1
···α
N,N
(6.5)
T
Β
=
1
β
N,1
···β
N,N
(6.6)
are vectors of length
N
+1 containing the design coecients of the directional
pattern. In all previous chapters, only the case
M
=
N
+ 1 was considered.
This is also the case in all known approaches in the literature [2].
In adaptive beamforming, we minimize the residual noise at the beam-
former output subject to some constraints. Here, the constraints are summa-
rized in (6.1). Mathematically, this procedure is equivalent to
h(ω)
h
H
(
ω
)
Φ
v
(
ω
)
h
(
ω
) subject to
D
(
ω,Α
)
h
(
ω
)=
Β,
min
(6.7)
for which the solution is
−1
−1
v
(
ω
)
D
H
(
ω,Α
)
−1
v
(
ω
)
D
H
(
ω,Α
)
h
LCMV
(
ω
)=
Φ
D
(
ω,Α
)
Φ
Β,
(6.8)
where we recognize the well-known linearly constrained minimum vari-
ance (LCMV) filter [3], [4], [5]. We observe that for the matrix
D
(
ω,Α
)
Φ
v
(
ω
)
D
H
(
ω,Α
) in (6.8) to be full rank, we must have
N
+1
≤ M
,
which is the same condition to design a differential array of order
N
. It can
also be shown that (6.8) can be expressed as
−1
−1
−1
y
(
ω
)
D
H
(
ω,Α
)
−1
y
(
ω
)
D
H
(
ω,Α
)
h
LCMV
(
ω
)=
Φ
D
(
ω,Α
)
Φ
Β,
(6.9)
y
(
ω
)
y
H
(
ω
)
where
Φ
y
(
ω
)=
E
is the correlation matrix of the observation
signal vector,
y
(
ω
). However, from an implementation point of view, it is
preferable to use (6.8) than (6.9) since
Φ
v
(
ω
) is usually better conditioned
than
Φ
y
(
ω
).
For
M
=
N
+ 1, we easily deduce from (6.8) [or (6.9)] that
−1
(
ω,Α
)
Β,
h
LCMV
(
ω
)=
D
(6.10)
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