Digital Signal Processing Reference
In-Depth Information
information. One may therefore proceed to estimate the interferer DOAs by applying
subspace methods, such as the MUSIC technique, to a matrix made of the weight vectors
corresponding to different satellites. These weights are already generated for interfer-
ence nulling purposes and can be obtained recursively through digital, or even analog,
implementations. It is noted that this approach is different from beamspace MUSIC,
where the covariance matrix of the beam data outputs is considered for eigendecomposi-
tion [45].
To elaborate on the combined cascade implementation of interference nulling and
DOA estimation, we consider the optimum nulling weight matrix solution, given by
WW
a
op
=α
(14.25)
where
=
−1
Wra
a
()θ
(14.26)
D
he 2
N
-by-
D
matrix
W
opt
= [
w
1_
opt
w
2_
opt
…
w
D
_
opt
] consists of the weight vectors in the
multiple MVDR beamforming approach, and
11
1
α=
diag
…
αα α
1
2
D
is a diagonal
D
-by-
D
matrix, where α
i
(
i
= 1, 2, …,
D
) is the output power at each beam-
former, α
i
=
y
i
y
i
*
.
The spatial covariance matrix of the received data can be written as
H
H
H
σ
2
rxxaPa
== +
aPaI
+ =++
rr r
,
(14.27)
Ds D
II I
s
I
n
where
P
s
=
diag
(
P
si
) (
i
= 1, 2, …,
D
) and
P
I
=
diag
(
P
Ii
) (
i
= 1, 2, …,
M
) represent the diago-
nal power matrix of the GPS signals and the interferers, respectively. Due to the low
power of the GPS signals, matrix
r
s
is negligible compared to the interference and noise
covariance matrix. Therefore, equation (14.27) can be simplified to
H
σ
2
raPa
=
+
I
.
(14.28)
II I
The eigendecomposition of (14.28) is
=Λ
H
,
ree
(14.29)
where
Λ
is the diagonal eigenvalue matrix,
Λ
=
diag
{σ
1
2
… σ
2
σ
n
2
… σ
n
2
}, and
e
is the
corresponding eigenvector matrix. In the above equation, σ
i
2
(
i
= 1, 2, …,
M
) represents
the
i
t h
significant eigenvalue, and σ
n
2
represents the noise eigenvalue. The first
M
col-
umns of
e
span the interference subspace, i.e.,
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