Digital Signal Processing Reference
In-Depth Information
followed by peak finding and non-LHCP interference DOA estimation. This technique
is known as concurrent adaptive nulling and localization (CANAL) [43] (although it
performs nulling and localization in a sequential manner). In practice, however, only a
short observation interval of the data is available, and as such, the noise perturbation
and estimation inaccuracies will violate the interference subspace in equation (14.33),
rendering equation (14.34) nonapplicable. For illustration, define
=
11
2
λ
i
2
σσ
i
for
i
= 1, 2, …,
M
, and
=
11
2
λ
j
σσ
2
nj
for
j
=
M
+ 1, …, 2
N
, where σ
n
2
represents the
j
t h
noise eigenvalue obtained from the
estimated data covariance matrix
ˆ
, and it is a perturbed version of σ
2
. If σ
2
≠ σ
n
2
, then
equation (14.32) can be rewritten as
M
2N
1
∑∑
1
1
H
H
H
w
=
EE
−
e e
−
e e
a
()
θ
ck
i
i
j
j
k
2
2
2
σ
σ
σ
i
nj
i
=
1
jM
= +
1
(14 . 35)
M
2N
=
( )
+
=
∑
∑
( )
H
H
λ
ea
()
θ
e
λ
e
a
(()
e
j
.
i
i
k
i
j
j
k
i
1
jM
1
=+
It is clear that the cancellation weight vector is a linear combination of all interference
and the noise subspace vectors
e
i
(
i
= 1, 2, …, 2
N
) pertaining to the spatial covariance
matrix. Proceeding with the singular value decomposition (SVD) for
W
c
yields
WuV
c
=Σ ,
(14 . 36)
where
u
is a 2
N
-by-2
N
unitary matrix,
V
is a
D
-by-
D
unitary matrix, and
Σ
is a 2
N
-by-
D
diagonal matrix with singular values, γ
i
,
i
= 1, …,
D
,
γ
0
0
1
0
…
0
00
0
γ
D
Σ=
.
(14 . 37)
…
…
…
0
0
0
0
0
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